Contents

Cover

Title Page

Copyright

Dedication

About the Author

Preface

Chapter 1: Basics1.1 Introduction1.2 Lift and Drag1.3 Monoplane Aircraft1.4 Biplane1.5 Triplane1.6 Aspect Ratio1.7 Camber1.8 Incidence1.9 Aerodynamic Force1.10 Scale Effect1.11 Force and Moment Coefficients1.12 The Boundary Layer1.13 SummaryReference

Chapter 2: Essence of Fluid Mechanics2.1 Introduction2.2 Properties of Fluids2.3 Thermodynamic Properties2.4 Surface Tension2.5 Analysis of Fluid Flow

2.6 Basic and Subsidiary Laws2.7 Kinematics of Fluid Flow2.8 Streamlines2.9 Potential Flow2.10 Combination of Simple Flows2.11 Flow Past a Circular Cylinder without Circulation2.12 Viscous Flows2.13 Compressible Flows2.14 SummaryReferences

Chapter 3: Conformal Transformation3.1 Introduction3.2 Basic Principles3.3 Complex Numbers3.4 SummaryExercise Problems

Chapter 4: Transformation of Flow Pattern4.1 Introduction4.2 Methods for Performing Transformation4.3 Examples of Simple Transformation4.4 Kutta−Joukowski Transformation4.5 Transformation of Circle to Straight Line4.6 Transformation of Circle to Ellipse4.7 Transformation of Circle to Symmetrical Aerofoil4.8 Transformation of a Circle to a Cambered Aerofoil4.9 Transformation of Circle to Circular Arc4.10 Joukowski Hypothesis4.11 Lift of Joukowski Aerofoil Section4.12 The Velocity and Pressure Distributions on theJoukowski Aerofoil4.13 The Exact Joukowski Transformation Process and ItsNumerical Solution4.14 The Velocity and Pressure Distribution

4.15 Aerofoil Characteristics4.16 Aerofoil Geometry4.17 Wing Geometrical Parameters4.18 Aerodynamic Force and Moment Coefficients4.19 SummaryExercise ProblemsReferences

Chapter 5: Vortex Theory5.1 Introduction5.2 Vorticity Equation in Rectangular Coordinates5.3 Circulation5.4 Line (point) Vortex5.5 Laws of Vortex Motion5.6 Helmholtz's Theorems5.7 Vortex Theorems5.8 Calculation of uR, the Velocity due to Rotational Flow5.9 Biot-Savart Law5.10 Vortex Motion5.11 Forced Vortex5.12 Free Vortex5.13 Compound Vortex5.14 Physical Meaning of Circulation5.15 Rectilinear Vortices5.16 Velocity Distribution5.17 Size of a Circular Vortex5.18 Point Rectilinear Vortex5.19 Vortex Pair5.20 Image of a Vortex in a Plane5.21 Vortex between Parallel Plates5.22 Force on a Vortex5.23 Mutual action of Two Vortices5.24 Energy due to a Pair of Vortices5.25 Line Vortex5.26 Summary

Exercise ProblemsReferences

Chapter 6: Thin Aerofoil Theory6.1 Introduction6.2 General Thin Aerofoil Theory6.3 Solution of the General Equation6.4 The Circular Arc Aerofoil6.5 The General Thin Aerofoil Section6.6 Lift, Pitching Moment and Center of PressureCoefficients for a Thin Aerofoil6.7 Flapped Aerofoil6.8 SummaryExercise ProblemsReferences

Chapter 7: Panel Method7.1 Introduction7.2 Source Panel Method7.3 The Vortex Panel Method7.4 Pressure Distribution around a Circular Cylinder bySource Panel Method7.5 Using Panel Methods7.6 SummaryReferences

Chapter 8: Finite Aerofoil Theory8.1 Introduction8.2 Relationship between Spanwise Loading and TrailingVorticity8.3 Downwash8.4 Characteristics of a Simple Symmetrical Loading –Elliptic Distribution8.5 Aerofoil Characteristic with a More General Distribution8.6 The Vortex Drag for Modified Loading8.7 Lancaster –Prandtl Lifting Line Theory

8.7 Lancaster –Prandtl Lifting Line Theory8.8 Effect of Downwash on Incidence8.9 The Integral Equation for the Circulation8.10 Elliptic Loading8.11 Aerodynamic Characteristics of Asymmetric Loading8.12 Lifting Surface Theory8.13 Aerofoils of Small Aspect Ratio8.14 Lifting Surface8.15 SummaryExercise Problems

Chapter 9: Compressible Flows9.1 Introduction9.2 Thermodynamics of Compressible Flows9.3 Isentropic Flow9.4 Discharge from a Reservoir9.5 Compressible Flow Equations9.6 Crocco's Theorem9.7 The General Potential Equation for Three-DimensionalFlow9.8 Linearization of the Potential Equation9.9 Potential Equation for Bodies of Revolution9.10 Boundary Conditions9.11 Pressure Coefficient9.12 Similarity Rule9.13 Two-Dimensional Flow: Prandtl-Glauert Rule forSubsonic Flow9.14 Prandtl-Glauert Rule for Supersonic Flow: Versions Iand II9.15 The von Karman Rule for Transonic Flow9.16 Hypersonic Similarity9.17 Three-Dimensional Flow: The Gothert Rule9.18 Moving Disturbance9.19 Normal Shock Waves9.20 Change of Total Pressure across a Shock9.21 Oblique Shock and Expansion Waves

9.22 Thin Aerofoil Theory9.23 Two-Dimensional Compressible Flows9.24 General Linear Solution for Supersonic Flow9.25 Flow over a Wave-Shaped Wall9.26 SummaryExercise ProblemsReferences

Chapter 10: Simple Flights10.1 Introduction10.2 Linear Flight10.3 Stalling10.4 Gliding10.5 Straight Horizontal Flight10.6 Sudden Increase of Incidence10.7 Straight Side-Slip10.8 Banked Turn10.9 Phugoid Motion10.10 The Phugoid Oscillation10.11 SummaryExercise Problems

Further Readings

Index

This edit ion first published 2013

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Library of Congress Cataloging-in-Publication Data

Rathakrishnan, E.

Theoret ical aerodynamics / Ethirajan Rathakrishnan.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-47934-6 (cloth)

1. Aerodynamics. I. Tit le.

TL570.R33 2013

629.132′3–dc23

2012049232

This book is dedicated to my parents,Mr Thammanur Shunmugam Ethirajan

andMrs Aandaal Ethirajan

Ethirajan Rathakrishnan

About the AuthorEthirajan Rathakrishnan is Professor of Aerospace Engineering at the Indian Inst itute ofTechnology Kanpur, India. He is well-known internat ionally for his research in the area of high-speed jets. The limit for the passive control of jets, called Rathakrishnan Limit, is hiscontribut ion to the field of jet research, and the concept of breathing blunt nose (BBN), whichreduces the posit ive pressure at the nose and increases the low-pressure at the basesimultaneously, is his contribut ion to drag reduct ion at hypersonic speeds. He has published alarge number of research art icles in many reputed internat ional journals. He is a fellow of manyprofessional societ ies, including the Royal Aeronaut ical Society. Professor Rathakrishnanserves as editor-in-chief of the International Review of Aerospace Engineering (IREASE)Journal. He has authored nine other books: Gas Dynamics, 4th ed. (PHI Learning, New Delhi,2012); Fundamentals of Engineering Thermodynamics, 2nd ed. (PHI Learning, New Delhi, 2005);Fluid Mechanics: An Introduction, 3rd ed. (PHI Learning, New Delhi, 2012); Gas Tables, 3rd ed.(Universit ies Press, Hyderabad, India, 2012); Instrumentation, Measurements, and Experimentsin Fluids (CRC Press, Taylor & Francis Group, Boca Raton, USA, 2007); Theory ofCompressible Flows (Maruzen Co., Ltd., Tokyo, Japan, 2008); Gas Dynamics Work Book(Praise Worthy Prize, Napoli, Italy, 2010); Applied Gas Dynamics (John Wiley, New Jersey, USA,2010); and Elements of Heat Transfer, (CRC Press, Taylor & Francis Group, Boca Raton, USA,2012).

Preface

This book has been developed to serve as a text for theoret ical aerodynamics at theintroductory level for both undergraduate courses and for an advanced course at graduatelevel. The basic aim of this book is to provide a complete text covering both the basic andapplied aspects of aerodynamic theory for students, engineers, and applied physicists. Thephilosophy followed in this book is that the subject of aerodynamic theory is covered bycombining the theoret ical analysis, physical features and applicat ion aspects.

The fundamentals of fluid dynamics and gas dynamics are covered as it is t reated at the

undergraduate level. The essence of fluid mechanics, conformal t ransformat ion and vortextheory, being the basics for the subject of theoret ical aerodynamics, are given in separatechapters. A considerable number of solved examples are given in these chapters to fix theconcepts introduced and a large number of exercise problems along with answers are listed atthe end of these chapters to test the understanding of the material studied.

To make readers comfortable with the basic features of aircraft geometry and its flight , vitalparts of aircraft and the preliminary aspects of its flight are discussed in the first and finalchapters. The ent ire spectrum of theoret ical aerodynamics is presented in this book, withnecessary explanat ions on every aspect. The material covered in this book is so designed thatany beginner can follow it comfortably. The topics covered are broad based, start ing from thebasic principles and progressing towards the physics of the flow which governs the flowprocess.

The book is organized in a logical manner and the topics are discussed in a systemat ic way.First , the basic aspects of the fluid flow and vort ices are reviewed in order to establish a firmbasis for the subject of aerodynamic theory. Following this, conformal t ransformat ion of flows isintroduced with the elementary aspects and then gradually proceeding to the vital aspectsand applicat ion of Joukowski t ransformat ion which transforms a circle in the physical plane tolift generat ing profiles such as symmetrical aerofoil, circular arc and cambered aerofoil in thetranformed plane. Following the transformat ion, vortex generat ion and its effect on lift anddrag are discussed in depth. The chapter on thin aerofoil theory discusses the performance ofaerofoils, highlight ing the applicat ion and limitat ions of the thin aerofoils. The chapter on panelmethods presents the source and vortex panel techniques meant for solving the flow aroundnonlift ing and lift ing bodies, respect ively.

The chapter on finite wing theory presents the performance of wings of finite aspect rat io,where the horseshoe vortex, made up of the bound vortex and t ip vort ices, plays a dominantrole. The procedure for calculat ing the lift , drag and pitching moment for symmetrical andcambered profiles is discussed in detail. The consequence of the velocity induced by thevortex system is presented in detail, along with solved examples at appropriate places.

The chapter on compressible flows covers the basics and applicat ion aspects in detail forboth subsonic and supersonic regimes of the flow. The similarity considerat ion covering theParandt l-Glauert I and II rules and Gothert rule are presented in detail. The basic governingequat ion and its simplificat ion with small perturbat ion assumption is covered systemat ically.Shocks and expansion waves and their influence on the flow field are discussed in depth.Following this the shock-expansion theory and thin aerofoil theory and their applicat ion tocalculate the lift and drag are presented.

In the final chapter, some basic flights are introduced briefly, covering the level flight , glidingand climbing modes of flight . A brief coverage of phugoid mot ion is also presented.

The selected references given at the end are, it is hoped, a useful guide for further study ofthe voluminous subject .

This book is the outgrowth of lectures presented over a number of years, both atundergraduate and graduate level. The student, or reader, is assumed to have a background inthe basic courses of fluid mechanics. Advanced undergraduate students should be able tohandle the subject material comfortably. Sufficient details have been included so that the textcan be used for self study. Thus, the book can be useful for scient ists and engineers working inthe field of aerodynamics in industries and research laboratories.

My sincere thanks to my undergraduate and graduate students in India and abroad, who aredirect ly and indirect ly responsible for the development of this book.

I would like to express my sincere thanks to Yasumasa Watanabe, doctoral student ofAerospace Engineering, the University of Tokyo, Japan, for his help in making some solvedexamples along with computer codes. I thank Shashank Khurana, doctoral student ofAerospace Engineering, the University of Tokyo, Japan, for crit ically checking the manuscript ofthis book. Indeed, incorporat ion of the suggest ions given by Shashank great ly enhanced the

clarity of manuscript of this book. I thank my doctoral students Mrinal Kaushik and Arun Kumar,for checking the manuscript and the solut ions manual, and for giving some useful suggest ions.

For instructors only, a companion Solut ions Manual is available from John Wiley and containstyped solut ions to all the end-of-chapter problems can be found atwww.wiley.com/go/rathakrishnan. The financial support extended by the Cont inuing Educat ionCentre of the Indian Inst itute of Technology Kanpur, for the preparat ion of the manuscript isgratefully acknowledged.

Ethirajan Rathakrishnan

1

Basics

1.1 IntroductionAerodynamics is the science concerned with the mot ion of air and bodies moving through air. Inother words, aerodynamics is a branch of dynamics concerned with the study of mot ion of air,part icularly when it interacts with a moving object . The forces act ing on bodies moving throughthe air are termed aerodynamic forces. Air is a fluid, and in accordance with Archimedesprinciple, an aircraft will be buoyed up by a force equal to the weight of air displaced by it . Thebuoyancy force Fb will act vert ically upwards. The weight W of the aircraft is a force which actsvert ically downwards; thus the magnitude of the net force act ing on an aircraft , even when it isnot moving, is . The force will act irrespect ive of whether the aircraft is atrest or in mot ion.

Now, let us consider an aircraft flying with constant speed V through st ill air, as shown inFigure 1.1, that is, any mot ion of air is solely due to the mot ion of the aircraft . Let this mot ion ofthe aircraft is maintained by a t ract ive force T exerted by the engines.

Figure 1.1 Forces act ing on an aircraft in horizontal flight .

Newton's first law of mot ion asserts that the resultant force act ing on the aircraft must bezero, when it is at a steady flight (unaccelerated mot ion). Therefore, there must be anaddit ional force F ad, say, such that the vectorial sum of the forces act ing on the aircraft is:

Force F ad is called the aerodynamic force exerted on the aircraft . In this definit ion ofaerodynamic force, the aircraft is considered to be moving with constant velocity V in stagnantair. Instead, we may imagine that the aircraft is at rest with the air streaming past it . In thiscase, the air velocity over the aircraft will be −V. It is important to note that the aerodynamicforce is theoret ically the same in both cases; therefore we may adopt whichever point of viewis convenient for us. In the measurement of forces on an aircraft using wind tunnels, this

principle is adopted, that is, the aircraft model is fixed in the wind tunnel test-sect ion and theair is made to flow over the model. In our discussions we shall always refer to the direct ion of Vas the direction of aircraft motion, and the direct ion of −V as the direction of airstream orrelative wind.

1.2 Lift and DragThe aerodynamic force F ad can be resolved into two component forces, one at right angles toV and the other opposite to V, as shown in Figure 1.1. The force component normal to V iscalled lift L and the component opposite to V is called drag D. If θ is the angle between L and Fad, we have:

The angle θ is called the glide angle. For keeping the drag at low value, the gliding angle has tobe small. An aircraft with a small gliding angle is said to be streamlined.

At this stage, it is essent ial to realize that the lift and drag are related to vert ical andhorizontal direct ions. To fix this idea, the lift and drag are formally defined as follows:

“Lift is the component of the aerodynamic force perpendicular to the direction of motion.”“Drag is the component of the aerodynamic force opposite to the direction of motion.”

Note: It is important to understand the physical meaning of the statement, “an aircraft with asmall gliding angle θ is said to be streamlined.” This explicit ly implies that when θ is large theaircraft can not be regarded as a streamlined body. This may make us wonder about thenature of the aircraft geometry, whether it is streamlined or bluff. In our basic courses, welearned that all high-speed vehicles are streamlined bodies. According to this concept, anaircraft should be a streamlined body. But at large θ it can not be declared as a streamlinedbody. What is the genesis for this drast ic conflict? These doubts will be cleared if we get thecorrect meaning of the bluff and streamlined geometries. In fluid dynamics, we learn that:

“a streamlined body is that for which the skin friction drag accounts for the major portion of thetotal drag, and the wake drag is very small.”“A bluff body is that for which the wake drag accounts for the major portion of the total drag,and the skin friction drag is insignificant.”

Therefore, the basis for declaring a body as streamlined or bluff is the relat ive magnitudes ofskin frict ion and wake drag components and not just the geometry of the body shape alone.Indeed, somet imes the shape of the body can be misleading in this issue. For instance, a thinflat plate kept parallel to the flow, as shown is Figure 1.2(a), is a perfect ly streamlined body, butthe same plate kept normal to the flow, as shown is Figure 1.2(b), is a typical bluff body. Thisclearly demonstrates that the streamlined and bluff nature of a body is dictated by thecombined effect of the body geometry and its orientat ion to the flow direct ion. Therefore, eventhough an aircraft is usually regarded as a streamlined body, it can behave as a bluff bodywhen the gliding angle θ is large, causing the format ion of large wake, leading to a large valueof wake drag. That is why it is stated that, “for small values of gliding angle θ an aircraft is saidto be streamlined.” Also, it is essent ial to realize that all commercial aircraft are usuallyoperated with small gliding angle in most port ion of their mission and hence are referred to asstreamlined bodies. All fighter aircraft , on the other hand, are designed for maneuvers such asfree fall, pull out and pull up, during which they behave as bluff bodies.

Figure 1.2 A flat plate (a) parallel to the flow, (b) normal to the flow.

Example 1.1An aircraft of mass 1500 kg is in steady level flight . If the wing incidence with respect to thefreestream flow is 3 , determine the lift to drag rat io of the aircraft .

SolutionGiven, m = 1500 kg and θ = 3 .In level flight the weight of the aircraft is supported by the lift . Therefore, the lift is:

The relat ion between the aerodynamic force, F ad, and lift , L, is:

The aerodynamic force becomes:

The relat ion between the aerodynamic force, F ad, and drag, D, is:

Therefore, the drag becomes:

The lift to drag rat io of the aircraft is:

Note: The lift to drag rat io L/D is termed aerodynamic efficiency.

1.3 Monoplane AircraftA monoplane is a fixed-wing aircraft with one main set of wing surfaces, in contrast to abiplane or t riplane. Since the late 1930s it has been the most common form for a fixed wingaircraft .

The main features of a monoplane aircraft are shown in Figure 1.3. The main lift ing systemconsists of two wings; the port (left ) and starboard (right) wings, which together const itute theaerofoil. The tail plane also exerts lift . According to the design, the aerofoil may or may not beinterrupted by the fuselage. The designer subsequent ly allow for the effect of the fuselage asa perturbat ion (a French word which means disturbance) of the propert ies of the aerofoil. Forthe present discussion, let us ignore the fuselage, and treat the wing (aerofoil) as one

cont inuous surface.

Figure 1.3 Main features of a monoplane aircraft .

The ailerons on the right and left wings, the elevators on the horizontal tail, and the rudderon the vert ical tail, shown in Figure 1.3, are control surfaces. When the ailerons and rudder arein their neutral posit ions, the aircraft has a median plane of symmetry which divides the wholeaircraft into two parts, each of which is the opt ical image of the other in this plane, consideredas a mirror. The wings are then the port ions of the aerofoil on either side of the plane ofsymmetry, as shown in Figure 1.4.

Figure 1.4 Typical geometry of an aircraft wing.

The wing tips consist of those points of the wings, which are at the farthest distance fromthe plane of symmetry, as illustrated in Figure 1.4. Thus, the wing t ips can be a point or a line oran area, according to the design of the aerofoil. The distance between the wing t ips is calledthe span. The sect ion of a wing by a plane parallel to the plane of symmetry is called a profile.The shape and general orientat ion of the profile will usually depend on its distance from theplane of symmetry. In the case of a cylindrical wing, shown in Figure 1.5, the profiles are thesame at every locat ion along the span.

Figure 1.5 A cylindrical wing.

1.3.1 Types of MonoplaneThe main dist inct ion between types of monoplane is where the wings at tach to the fuselage:

Low-wing: the wing lower surface is level with (or below) the bottom of the fuselage.Mid-wing: the wing is mounted mid-way up the fuselage.High-wing: the wing upper surface is level with or above the top of the fuselage.Shoulder wing: the wing is mounted above the fuselage middle.Parasol-wing: the wing is located above the fuselage and is not direct ly connected to it ,structural support being typically provided by a system of struts, and, especially in thecase of older aircraft , wire bracing.

1.4 BiplaneA biplane is a fixed-wing aircraft with two superimposed main wings. The Wright brothers'Wright Flyer used a biplane design, as did most aircraft in the early years of aviat ion. While abiplane wing structure has a structural advantage, it generates more drag than a similarmonoplane wing. Improved structural techniques and materials and the quest for greaterspeed made the biplane configurat ion obsolete for most purposes by the late 1930s.

In a biplane aircraft , two wings are placed one above the other, as in the Boeing StearmanE75 (PT-13D) biplane of 1944 shown in Figure 1.6. Both wings provide part of the lift , althoughthey are not able to produce twice as much lift as a single wing of similar size and shapebecause both the upper and lower wings are working on nearly the same port ion of theatmosphere. For example, in a wing of aspect rat io 6, and a wing separat ion distance of onechord length, the biplane configurat ion can produce about 20% more lift than a single wing ofthe same planform.

Figure 1.6 Boeing Stearman E75 (PT-13D) biplane of 1944.

In the biplane configurat ion, the lower wing is usually at tached to the fuselage, while theupper wing is raised above the fuselage with an arrangement of cabane struts, although otherarrangements have been used. Almost all biplanes also have a third horizontal surface, thetailplane, to control the pitch, or angle of at tack of the aircraft (although there have been a fewexcept ions). Either or both of the main wings can support flaps or ailerons to assist lateralrotat ion and speed control; usually the ailerons are mounted on the upper wing, and flaps (ifused) on the lower wing. Often there is bracing between the upper and lower wings, in the formof wires (tension members) and slender inter-plane struts (compression members) posit ionedsymmetrically on either side of the fuselage.

1.4.1 Advantages and DisadvantagesAircraft built with two main wings (or three in a t riplane) can usually lift up to 20% more than

can a similarly sized monoplane of similar wingspan. Biplanes will therefore typically have ashorter wingspan than a similar monoplane, which tends to afford greater maneuverability. Thestruts and wire bracing of a typical biplane form a box girder that permits a light but very strongwing structure.

On the other hand, there are many disadvantages to the configurat ion. Each wingnegat ively interferes with the aerodynamics of the other. For a given wing area the biplanegenerates more drag and produces less lift than a monoplane.

Now, one may ask what is the specific difference between a biplane and monoplane? Theanswer is as follows.

A biplane has two (bi) sets of wings, and a monoplane has one (mono) set of wings. The twosets of wings on a biplane add lift , and also drag, allowing it to fly slower. The one set of wingson a monoplane do not add as much lift or drag, making it fly faster, and as a result , all fastplanes are monoplanes, and most planes these days are monoplanes.

1.5 TriplaneA triplane is a fixed-wing aircraft equipped with three vert ically-stacked wing planes. Tailplanesand canard fore-planes are not normally included in this count, although they may occasionallybe. A typical example for t riplane is the Fokker Dr. I of World War I, shown in Figure 1.7.

Figure 1.7 Fokker Dr. I of World War I.

The triplane arrangement may be compared with the biplane in a number of ways. A triplanearrangement has a narrower wing chord than a biplane of similar span and area. This giveseach wing plane a slender appearance with a higher aspect rat io, making it more efficient andgiving increased lift . This potent ially offers a faster rate of climb and t ighter turning radius, bothof which are important in a fighter plane. The Sopwith Triplane was a successful example,having the same wing span as the equivalent biplane, the Sopwith Pup.

Alternat ively, a t riplane has a reduced span compared with a biplane of given wing area andaspect rat io, leading to a more compact and lightweight structure. This potent ially offers bettermaneuverability for a fighter plane, and higher load capacity with more pract ical groundhandling for a large aircraft type.

The famous Fokker Dr.I t riplane was a balance between the two approaches, havingmoderately shorter span and moderately higher aspect rat io than the equivalent biplane, theFokker D.VI.

Yet a third comparison may be made between a biplane and triplane having the same wingplanform—the triplane's third wing provides increased wing area, giving much increased lift .The extra weight is part ially offset by the increased depth of the overall structure, allowing amore efficient construct ion. The Caproni Ca.4 series had some success with this approach.

These advantages are offset , to a greater or lesser extent in any given design, by the extraweight and drag of the structural bracing, and the aerodynamic inefficiency inherent in thestacked wing layout. As biplane design advanced, it became clear that the disadvantages ofthe triplane outweighed the advantages.

Typically the lower set of wings are approximately level with the underside of the aircraft 'sfuselage, the middle set level with the top of the fuselage, and the top set supported abovethe fuselage on cabane struts.

1.5.1 Chord of a ProfileA chord of any profile is generally defined as an arbit rarily fixed line drawn in the plane of theprofile, as illustrated in Figure 1.8. The chord has direct ion, posit ion, and length. The mainrequisite is that in each case the chord should be precisely defined, because the chord entersinto the constants such as the lift and drag coefficients, which describe the aerodynamicpropert ies of the profile. For the profile shown in Figure 1.8(a), the chord is the line joining thecenter of the circle at the leading and trailing edges.

For the profile in Figure 1.8(b), the line joining the center of the circle at the nose and the t ipof the tail is the chord. For the profile in Figure 1.8(c), the line joining the t ips of leading andtrailing edges is the chord.

Figure 1.8 Illustrat ion of chord for different shapes of leading and trailing edges.

A definit ion which is convenient is: the chord is the project ion of the profile on the doubletangent to its lower surface (that is, the tangent which touches the profile at two dist inctpoints), as shown in Figure 1.9. But this definit ion fails if there is no such double tangent.

Figure 1.9 Chord of a profile.

1.5.2 Chord of an AerofoilFor a cylindrical aerofoil (that is, a wing for which the profiles are the same at every locat ionalong the span, as shown in Figure 1.5), the chord of the aerofoil is taken to be the chord of theprofile in which the plane of symmetry cuts the aerofoil. In all other cases, the chord of theaerofoil is defined as the mean or average chord located in the plane of symmetry.

Let us consider a wing with rectangular Cartesian coordinate axes, as shown in Figure 1.10.The x-axis, or longitudinal axis, is in the direct ion of mot ion, and is in the plane of symmetry; they-axis, or lateral axis, is normal to the plane of symmetry and along the (straight) t railing edge.The z-axis, or normal axis, is perpendicular to the other two axes in the sense that the three

axes form a right-handed system. This means, in part icular, that in a straight horizontal flightt he z-axis will be directed vert ically downwards. Consider a profile whose distance from theplane of symmetry is |y|. Let c be the chord length of this profile, θ be the inclinat ion of thechord to the xy plane, and (x, y, z) be the coordinates of the quarter point of the chord, that is,the point of the chord at a distance c/4 from the leading edge of the profile. This point isusually referred to as the quarter chord point. Since the profile is completely defined when y isgiven, all quant it ies characterizing the profile, namely, the mean chord, its posit ion andinclinat ion to the flow, are funct ions of y.

Figure 1.10 A wing with Cartesian coordinates.

The chord of an aerofoil is defined by averaging the distance between the leading andtrailing edges of the profiles at different locat ions along the span. Thus, if cm is the length ofthe mean chord, (xm, 0, zm) its quarter point , and θm its inclinat ion, we take the average ormean chord as:

These mean values completely define the chord of the aerofoil in length (cm), direct ion (θm),and posit ion (xm, zm).

1.6 Aspect RatioAspect rat io of a wing is the rat io of its span 2b to chord c. Consider a cylindrical wing shown inFigure 1.10. Imagine this to be projected on to the plane (xy-plane), which contains the chordsof all the sect ions (this plane is perpendicular to the plane of symmetry (xz-plane) and containsthe chord of the wing). The project ion in this case is a rectangular area S, say, which is calledthe plan area of the wing. The plan area is different from the total surface area of the wing.The simplest cylindrical wing would be a rectangular plate, and the plan area would then behalf of the total surface area.

The aspect ratio of the cylindrical wing is then defined by:

where S = span × chord = 2b × c.In the case of a wing which is not cylindrical, the plan area is defined as the area of the

project ion on the plane through the chord of the wing (mean chord) perpendicular to the planeof symmetry, and the aspect rat io is defined as:

A representat ive value of aspect rat io is 6.

Example 1.2The semi-span of a rectangular wing of planform area 8.4 m2 is 3.5 m. Determine the aspectrat io of the wing.

SolutionGiven, S = 8.4 m2 and b = 3.5 m.

The planform area of a wing is S = span × chord. Therefore, the wing chord becomes:

The aspect rat io of the wing is:

1.7 CamberCamber is the maximum deviat ion of the camber line (which is the bisector of the profilethickness) from the chord of the profile, as illustrated in Figure 1.11.

Figure 1.11 Illustrat ion of camber, camberline and chord of aerofoil profile.

Let zu and zl be the ordinates on the upper and lower parts of the profile, respect ively, forthe same value of x. Let c be the chord, and the x-axis coincide with the chord. Now, the upperand lower camber are defined as:

where the subscript “max” refers to that ordinate which is numerically the greatest . Camber is

taken as posit ive or negat ive according to the sign of and . Also, at a given x,the magnitudes of and may be different for unsymmetrical profiles.

T h e camber line is defined as the locus of the point . In the case ofsymmetrical profile zu + zl = 0, and the camber line is straight and coincides with the chord.Denot ing the numerically greatest ordinate of the camber line by z max, we define:

Note that the mean camber, in general, is not the same as the mean of upper and lowercamber, and the mean camber of a symmetrical profile is zero. Usually the word camber refersto the mean camber.

The thickness rat io of an aerofoil is the rat io of the maximum thickness (measuredperpendicular to the chord) to the chord. The thickness rat io is essent ially t max/c.

From the above discussions, it is evident that :Camberline of an aerofoil is essent ially the bisector of its thickness.Camber is the deviat ion of the camberline from the chord, namely the shortest line joiningthe leading and trailing edges of the aerofoil profile.The local camber can vary cont inuously from the leading edge to the trailing edge.Therefore, the maximum camber is taken as the representat ive camber. That is, themaximum ordinate of the camberline from the chord is taken as the camber of an aerofoil.The thickness of an aerofoil profile also varies cont inuously from the leading edge to thetrailing edge. Therefore, the rat io of the maximum thickness:

to chord c is used to represent the thickness-to-chord rat io of an aerofoil.

1.8 IncidenceWhen an aircraft t ravels in the plane of symmetry (that is, the direct ion of flight is parallel to theplane of symmetry), the angle between the direct ion of mot ion and the direct ion of the chordof a profile, as shown in Figure 1.12, is called the geometrical incidence of the profile, denotedby the Greek let ter α. The angle α is also called angle of attack.

Figure 1.12 Illustrat ion of geometrical incidence.

For an airplane as a whole the geometrical incidence will be defined as the angle betweenthe direct ion of mot ion and the chord of the aerofoil. When the chords of various profiles of awing are parallel the incidence is the same at each sect ion. When the chords are not parallelthe incidence varies from sect ion to sect ion and the wing has twist. The value of thegeometrical incidence would be altered if a different line were chosen as chord.

In this situat ion, it will be beneficial to understand the difference between the wing with thechords of its profiles at different locat ions along the span parallel to each other and the wingwith the chords of its profiles at different locat ions along the span not parallel. We know thatthe profiles are the cross-sect ions of the wing geometry, at different locat ions of the span, inplanes parallel to the mid-plane (xz-plane in Figure 1.3) passing through the nose and tail t ipsof the airplane. Therefore, only for a wing which has its left and right wings parallel to the y-axisin Figure 1.3 the chords of its profile will be parallel, and the wing will be termed cylindrical wing.For a wing with its left and right parts not parallel to the y-axis, the chords will not be parallel,and the wing will be termed a twisted wing.

1.9 Aerodynamic ForceAerodynamic force act ing on an aircraft is the force due to the pressure distribut ion around it ,caused by the mot ion of the aircraft . Thus, the gravity does not enter into the specificat ion ofaerodynamic force. Assuming the mot ion of the aircraft to be steady without rotat ion, theaerodynamic force on the wing or on the complete aircraft may be expected to depend on theforward speed V, air density ρ, speed of sound a and kinematic viscosity ν, of the environmentin which it is flying, and the total length l of the aircraft .

If the air is assumed to be incompressible and inviscid, we have the density ρ = constant andthe viscosity coefficient μ = 0. Therefore, the speed of sound becomes:

Assuming the flow over the aircraft to be isentropic, we have:

Different iat ing with respect to ρ, we have:

Now, replacing the “constant” with , we get:

Subst itut ing this, we get the speed of sound as:

For incompressible flow with dρ = 0, we have the speed of sound as:

For inviscid fluid, the kinematic viscosity becomes:

Therefore, for incompressible flows, the aerodynamic force F ad does not depend on the speedof sound a and kinematic viscosity ν. Thus, F ad can be assumed to depend only on ρ, V and l.The F ad would be given by a formula such as:

(1.1) where is a dimensionless number and the indices a, b, c on the right-hand-side can bedetermined by dimensional theory as follows.

In terms of the fundamental dimensions of mass (M), length (L) and t ime (T), we can expressEquat ion (1.1) as:

Equat ing the dimensions M, L, T on the left -hand-side and right-hand-side, we get:

Solving for a, b and c, we get:

Subst itut ing for a, b, c into Equat ion (1.1), we get:

(1.2) This is valid only for steady incompressible and inviscid flows. If we wish to account forcompressibility and viscosity, a and ν should be included in Equat ion (1.1) and expressed as:

(1.3) where is a dimensionless number, and each side must have the dimension of force. Here∑ denotes the sum of all allowable terms. In terms of basic dimensions M, L and T, Equat ion(1.3) becomes:

Equat ing the dimensions M, L, T on the left -and right-hand-sides, we have:

Solving for a, b and c, in terms of d and e, we get:

Thus, Equat ion (1.3) becomes:

or

(1.4) The rat io (V/a) is called the Mach number M, which is essent ially a dimensionless speed. Machnumber is the ratio of local flow speed to the local speed of sound or the ratio of inertial force toelastic force. It is a measure of compressibility. For an incompressible fluid, M = 0.

The dimensionless group (Vl/ν) is called the Reynolds number Re. Reynolds number is theratio of inertial force to viscous force. For an inviscid fluid Re = ∞. For air, the kinematic viscosityν is small and Re is large unless Vl is small.

Thus, Equat ion (1.4) becomes:

where l2 has been replaced by the plan area S, a proport ional number of the same dimensions,and f(M, Re) is a funct ion, whose form is not determined by the present method, with valueswhich are independent of physical units.

The dimensionless number:

(1.5) is called the (dimensionless) coefficient of the aerodynamic force F ad. The effect ofcompressibility can usually be neglected if M < 0.3, and the flow is termed incompressible.Thus, for an incompressible flow, the aerodynamic force coefficient is a funct ion of Reynoldsnumber only. That is:

At this stage, we may wonder about the definit ion of incompressible flow. The mathematicaldefinit ion of incompressible flow is that “it is a flow with Mach number zero.” But it is obviousthat, for M = 0, the flow velocity is zero, and hence there is no flow. But mathematics, as anabstract science, st ipulates the limit of M = 0, with the sole idea of rendering the density tobecome invariant. But when V = 0, engineering science will declare it as a stagnant field andnot as a flow field. Therefore, the engineering definit ion of incompressible flow is drast icallydifferent from the mathematical definit ion. From an engineering point of view, when the densitychange associated with V is insignificant the flow can be termed incompressible. Also, forengineering applicat ions, any change less than 5% is usually regarded as insignificant. Withthis considerat ion, any flow with density change less than 5% can be called incompressible. Forair flow at standard sea level condit ions (p = 101325 Pa and T = 288 K), 5% density changecorresponds to M = 0.3 [1]. Therefore, flows with Mach number less than 0.3 are regarded asincompressible flows and the density ρ0 corresponding to the stagnat ion state is taken as thedensity of an incompressible flow.

1.10 Scale EffectFrom our studies on similarity analysis in fluid mechanics, we know that, for dynamic similaritybetween the forces act ing on an actual (or full-scale) machine and a scaled-down model usedfor test ing (usually wind tunnel tests), the actual machine and the scale model must sat isfygeometric and kinematic similarit ies. Thus, the test model and the actual machine should begeometrically similar, and if the model tests give an aerodynamic coefficient C ad,m for a testconducted at a Reynolds number Rem, the scale effect on the aerodynamic force coefficient Cad of the actual machine is given by:

where Re is the Reynolds number of the flow around the actual machine and Rem is theReynolds number of the flow around the model. The model tests will give aerodynamiccoefficient (C ad = C ad,m ) direct ly, if Re = Rem. If the viscosity μ and density ρ are kept thesame in the flow fields of the actual machine and its scale model, then both the flow velocity Vand the characterist ic length (for example, chord for an aerofoil) should be adjusted in such away to keep Re = Rem. But the characterist ic length lm for the model will be, usually, smallerthan the l for the actual machine. Therefore, the test speed for the model has to be greaterthan the speed of the actual machine.

If there is provision to use compressed air wind tunnel, then the density ρ also can beincreased to adjust the model Reynolds number to match the Reynolds number of the actualmachine. In this kind of studies, it is essent ial to make a statement about the length scale usedfor calculat ing the Reynolds number.

Example 1.3An aircraft wing profile has to be tested in a wind tunnel. If the actual wing of mean chord 1.2 mhas to fly at an alt itude, where the pressure and temperature are 50 kPa and 2 C,respect ively, with a speed of 250 km/h. Determine the chord of the wing model to be tested inthe wind tunnel, ensuring dynamic similarity, if the test-sect ion condit ions are 90 m/s, p = 100kPa, T = 22 C.SolutionLet the subscripts p and m refer to the prototype (actual) wing and the wing model to betested in the wind tunnel, respect ively.

Given, m, pp = 50 kPa, Tp = 2 + 273.15 = 275.15 K, Vp = 250/3.6 = 69.44 m/s.Vm = 90 m/s, pm = 100 kPa, Tm = 22 + 273.15 = 295.15 K.

The density and viscosity of the actual and test-sect ion flows are:

The aerodynamic forces, and hence the coefficients of these forces, act ing on the actual wingand model wing will the same if the Reynolds number of the flow field around the actual wingand model wing are the same.The Reynolds number for the prototype is:

This Reynolds number should be equal to Rem. Therefore:

This gives the chord of the wing model as:

1.11 Force and Moment CoefficientsThe important aerodynamic forces and moment associated with a flying machine, such as anaircraft , are the lift L, the drag D, and the pitching moment M. The lift and drag forces can beexpressed as dimensionless numbers, popularly known as lift coefficient CL and drag coefficientCD, by dividing L and D with . Thus:

(1.4a)

(1.4b) The variat ion of CL and CD with the geometrical incidence α is shown in Figure 1.13. The

pitching moment, which is the moment of the aerodynamic force about an axis perpendicularto the plane of symmetry (about y-axis in Figure 1.3), will depend on the part icular axis chosen.Denot ing the pitching moment about the chosen axis by M (note that M is also used fordenot ing Mach number, which is the rat io of local flow speed and local speed of sound), wedefine the pitching moment coefficient as:

(1.4c) where c is the chord of the wing. A typical variat ions of CL, CD and CM with angle of at tack αare shown in Figure 1.13.

Figure 1.13 Variat ion of lift , drag and pitching moment coefficients with geometrical incidence.

Note that the aerodynamic coefficients CL, the drag CD and the moment CM aredimensionless parameters.

Example 1.4An aircraft weighing 20 kN is in level flight at an alt itude where the pressure and temperatureare 45 kPa and 0 C, respect ively. If the flight speed is 400 km/h and the span and mean chordof the wings are 10 m and 1.5 m, determine the lift coefficient .

SolutionGiven, W = 20, 000 N, 2b = 10 m, c = 1.5 m, V = 400/3.6 = 111.11 m/s, p = 45 kPa, T = 0 +273.15 = 273.15 K.The density of air is:

The planform area of the wing is:

In level flight , the weight of the aircraft is equal to the lift . Thus:

Therefore, by Equat ion (1.4)a, the lift coefficient becomes:

1.12 The Boundary LayerBoundary layer is a thin layer, adjacent to a solid surface, in which the flow velocity increasesfrom zero to about 99% of the freestream velocity, as shown in Figure 1.14.

Figure 1.14 Boundary layer on a flat plate.

The boundary layer may also be defined as a thin layer adjacent to a solid surface where theviscous effects are predominant. Thus, inside the boundary layer the effect of viscosity ispredominant. Outside the boundary layer the effect of viscosity is negligible. Also, greater theReynolds number the thinner will be the boundary layer, and we have pract ically the case of aninviscid flow past an object . But, however small the viscosity may be, the plate is subjected to atangent ial t ract ion or drag force act ing in the direct ion of flow velocity. This force is known asthe skin friction or the frictional drag, and this force can never be completely eliminated. On theother hand, the flow outside the boundary layer behaves like an inviscid flow.

For flow past a bluff body, such as a circular cylinder, an eddying wake forms behind thecylinder, great ly increasing the drag. The problem of flow separat ion or break away of theboundary layer from a bluff body can be minimized by streamlining the body. For properlystreamlined bodies the boundary layer will not break away and the wake will remain almostinsignificant. This has been achieved in the profiles like that shown in Figure 1.15 which aregenerally referred to as aerofoils.

Figure 1.15 An aerofoil in an uniform flow.

For aerofoils there is a narrow wake but, to a first approximat ion, the problem of the flowpast such a streamlined shape can be assumed as an inviscid flow past the body. In otherwords, the flow past an aerofoil can be regarded as flow without wake. The above

considerat ions give rise to the following general observat ions:1. It is found that to delay the breaking away of the boundary layer from the region wherethe fluid is moving against increasing pressure (that is, adverse pressure gradient, as inthe case of the rear of a circular cylinder) the flow should turn as gradually as possible. Toenable this gradual turning of flow, the body should have a large radius of curvature.2. It is essent ial to keep the surface of the object smooth, because even small project ionsabove the surface (in general) may disturb the boundary layer considerably, causing abreaking away of the flow. Furthermore, a project ion such as a rivet , whose head projectsabove the boundary layer, may ent irely alter the character of the flow. An exaggeratedflow over an aerofoil with such a rivet head is schematically shown in Figure 1.16.3. Good streamlined shapes will have the breaking away of the flow just close to thetrailing edge.

Figure 1.16 Flow separat ion caused by a rivet head project ion.

1.13 SummaryAerodynamics is the science concerned with the mot ion of air and bodies moving through air. Inother words, aerodynamics is a branch of dynamics concerned with the steady mot ion of air,part icularly when it interacts with a moving object . The forces act ing on the bodies movingthrough the air are termed aerodynamic forces.

The aerodynamic force F ad can be resolved into two component forces, one at right anglest o V and the other opposite to V. The force component normal to V is called lift L and thecomponent opposite to V is called drag D.

A streamlined body is that for which the skin frict ion drag accounts for the major port ion ofthe total drag, and the wake drag is very small.

A bluff body is that for which the wake drag accounts for the major port ion of the total drag,and the skin frict ion drag is insignificant.

The main lift ing system of an aircraft consists of two wings which together const itute theaerofoil. The tail plane also exerts lift . The ailerons on the right and left wings, the elevators onthe horizontal tail, and the rudder on the vert ical tail are control surfaces.

The distance between the wing t ips is called the span. The sect ion of a wing by a planeparallel to the plane of symmetry is called a profile.

Chord of any profile is generally defined as an arbit rarily fixed line drawn in the plane of theprofile. The chord has direct ion, posit ion, and length.

For a cylindrical aerofoil (that is, a wing for which the profiles are the same at every locat ionalong the span), the chord of the aerofoil is taken to be the chord of the profile in which theplane of symmetry cuts the aerofoil. In all other cases, the chord of the aerofoil is defined asthe mean or average chord located in the plane of symmetry.

The aspect rat io of a wing is the rat io of its span 2b to chord c.Camberline of an aerofoil is essent ially the bisector of its thickness.Camber is the deviat ion of the camberline from the chord, namely the shortest line joiningthe leading and trailing edges of the aerofoil profile.The local camber can vary cont inuously from the leading edge to the trailing edge.Therefore, the maximum camber is taken as the representat ive camber. That is, themaximum ordinate of the camberline from the chord is taken as the camber of an aerofoil.

When an aircraft t ravels in the plane of symmetry (that is, the direct ion of flight is parallel to

the plane of symmetry), the angle between the direct ion of mot ion and the direct ion of thechord of a profile, is called the geometrical incidence of the profile, denoted by the Greek let terα. The angle α is also called angle of attack.

Aerodynamic force on an aircraft is the force due to the pressure distribut ion around it ,caused by the mot ion of the aircraft . Thus, the gravity does not enter into the specificat ion ofaerodynamic force.

Mach number is the ratio of local flow speed to the local speed of sound or the ratio of inertialforce to elastic force. It is a measure of compressibility. For an incompressible fluid the M = 0.

The dimensionless group (Vl/ν) is called the Reynolds number Re. Reynolds number is theratio of inertial force to viscous force. For an inviscid fluid Re = ∞. For air, ν is small and Re islarge unless Vl is small.

The dimensionless number:

is called the (dimensionless) coefficient of the aerodynamic force F ad.The important aerodynamic forces and moment associated with a flying machine, such as

an aircraft , are the lift L, the drag D, and the pitching moment M. The lift and drag forces canbe expressed as dimensionless numbers, popularly known as lift coefficient CL and dragcoefficient CD, by dividing L and D with . Thus:

The pitching moment, which is the moment of the aerodynamic force about an axisperpendicular to the plane of symmetry (about y-axis in Figure 1.3), will depend on thepart icular axis chosen. Denot ing the pitching moment about the chosen axis by M (note that Mis also used for denot ing Mach number, which is the rat io of local flow speed and local speed ofsound), we define the pitching moment coefficient as:

Boundary layer is a thin layer, adjacent to a solid surface, in which the flow velocity increasesfrom zero to about 99% of the freestream velocity. The boundary layer may also be defined asa thin layer adjacent to a solid surface where the viscous effects are predominant. Thus, insidethe boundary layer the effect of viscosity is predominant. Outside the boundary layer theeffect of viscosity is negligible.

For flow past a bluff body, such as a circular cylinder, an eddying wake forms behind thecylinder, great ly increasing the drag. The problem of flow separat ion or break away of theboundary layer from a bluff body can be minimized by streamlining the body. For properlystreamlined bodies the boundary layer will not break away and the wake will remain almostinsignificant.Exercise Problems

1. An aircraft of total mass 10000 kg cruises steadily at an alt itude. If the aerodynamicefficiency is 4, find the thrust required to propel the aircraft .

[Answer: 24.525 kN]

2. An aircraft of mass 3000 kg in a steady level flight is at an angle of incidence of 5 to thefreestream. Determine the thrust generated by the engine.

[Answer: 2574.8 N]

3. An aircraft weighing 200 kN is in level flight at sea level with a speed of 600 km/h. The wingspan and chord are 8 m and 1.8 m, respect ively. Determine the lift coefficient of the wing.

[Answer: 0.816]

4. Determine the speed of sound in air at sea level condit ions.[Answer: 340.3 m/s]

5. If the aerodynamic efficiency of an aircraft in a steady flight is 10, determine the incidenceof the wing to the freestream direct ion.

[Answer: 5.71 ]

6. A sail plane of mass 270 kg flies straight and level with an incidence of 4 . Determine theaerodynamic force act ing on the wings and the aerodynamic efficiency.

[Answer: 2655.17 N, 14.30]

7. A wing of rectangular planform has 10 m span and 1.2 m chord. In straight and level flightat 240 km/h the total aerodynamic force act ing on the wing is 20 kN. If the aerodynamicefficiency of the wing is 10, calculate the lift coefficient . Assume air density to be 1.2 kg/m3.

[Answer: CL = 0.622]

Reference1. Rathakrishnan, E., Applied Gas Dynamics, John Wiley & Sons Inc., New Jersey, 2010.

2

Essence of Fluid Mechanics

2.1 IntroductionGases and liquids are generally termed fluids. Though the physical propert ies of gases andliquids are different, they are grouped under the same heading since both can be made to flowunlike a solid. Under dynamic condit ions, the nature of the governing equat ions are the samefor both gases and liquids. Hence, it is possible to t reat them under the same heading, namely,fluid dynamics or fluid mechanics. However, certain substances known as viscoelast ic materialsbehave like a liquid as well as a solid, depending on the rate of applicat ion of the force. Pitchand silicone putty are typical examples of viscoelast ic material. If the force is applied suddenly,the viscoelast ic material will behave like a solid, but with gradually applied pressure the materialwill flow like a liquid. The flow of such materials is not considered in this book. Similarly, non-Newtonian fluids, low-density flows, and two-phase flows such as gas liquid mixtures are alsonot considered in this book. The theory presented in this book is for well-behaved simple fluidssuch as air.

2.2 Properties of FluidsFluid may be defined as a substance which will continue to change shape as long as there is ashear stress present, however small it may be. That is, the basic feature of a fluid is that it canflow, and this is the essence of any definit ion of it . Examine the effect of shear stress on a solidelement and a fluid element, shown in Figure 2.1.

Figure 2.1 Solid and fluid elements under shear stress.

It is seen from this figure that the change in shape of the solid element is characterized byan angle Δα, when subjected to a shear stress, whereas for the fluid element there is no suchfixed Δα, even for an infinitesimal shear stress. A cont inuous deformat ion persists as long asshearing stress is applied. The rate of deformat ion, however, is finite and is determined by theapplied shear force and the fluid propert ies.

2.2.1 PressurePressure may be defined as the force per unit area which acts normal to the surface of anyobject which is immersed in a fluid. For a fluid at rest , at any point the pressure is the same inall direct ions. The pressure in a stat ionary fluid varies only in the vert ical direct ion, and isconstant in any horizontal plane. That is, in stat ionary fluids the pressure increases linearlywith depth. This linear pressure distribut ion is called hydrostatic pressure distribution. Thehydrostat ic pressure distribut ion is valid for moving fluids, provided there is no accelerat ion inthe vert ical direct ion. This distribut ion finds extensive applicat ion in manometry.

When a fluid is in mot ion, the actual pressure exerted by the fluid in the direct ion normal tothe flow is known as the static pressure. If there is an infinitely thin pressure t ransducer whichcan be placed in a flow field without disturbing the flow, and made to t ravel with the samespeed as that of the flow then it will record the exact stat ic pressure of the flow. From thisstringent requirement of the probe for stat ic pressure measurement, it can be inferred thatexact measurement of stat ic pressure is impossible. However, there are certain phenomena,such as “the static pressure at the edge of a boundary layer is impressed through the layer,”which are made use of for the proper measurement of stat ic pressure. The pressure which afluid flow will experience if it is brought to rest , isentropically, is termed total pressure. The totalpressure is also called impact pressure. The total and stat ic pressures are used for comput ingflow velocity.

Since pressure is intensity of force, it has the dimensions:

and is expressed in the units of newton per square meter (N/m2) or simply pascal (Pa). Atstandard sea level condit ion, the atmospheric pressure is 101325 Pa, which corresponds to760 mm of mercury column height.

2.2.2 TemperatureIn any form of matter the molecules are cont inuously moving relat ive to each other. In gases

the molecular mot ion is a random movement of appreciable amplitude ranging from about 76 ×10−9 m, under normal condit ions (that is, at standard sea level pressure and temperature), tosome tens of millimeters, at very low pressures. The distance of free movement of a moleculeof a gas is the distance it can travel before colliding with another molecule or the walls of thecontainer. The mean value of this distance for all molecules in a gas is called the molecularmean free path length. By virtue of this mot ion the molecules possess kinet ic energy, and thisenergy is sensed as temperature of the solid, liquid or gas. In the case of a gas in mot ion, it iscalled the static temperature. Temperature has units kelvin (K) or degrees celsius ( C), in SIunits. For all calculat ions in this book, temperatures will be expressed in kelvin, that is, fromabsolute zero. At standard sea level condit ion, the atmospheric temperature is 288.15 K.

2.2.3 DensityThe total number of molecules in a unit volume is a measure of the density ρ of a substance. Itis expressed as mass per unit volume, say kg/m3. Mass is defined as weight divided byaccelerat ion due to gravity. At standard atmospheric temperature and pressure (288.15 K and101325 Pa, respect ively), the density of dry air is 1.225 kg/m3.

Density of a material is a measure of the amount of material contained in a given volume. Ina fluid system, the density may vary from point to point . Consider the fluid contained within asmall spherical region of volume , centered at some point in the fluid, and let the mass offluid within this spherical region be δm. Then the density of the fluid at the point on which thesphere is centered can be defined by:

(2.1) There are pract ical difficult ies in applying the above definit ion of density to real fluidscomposed of discrete molecules, since under the limit ing condit ion the sphere may or may notcontain any molecule. If it contains, say, just a single molecule, the value obtained for thedensity will be fict it iously high. If it does not contain any molecule the resultant value of densitywill be zero. This difficulty can be avoided over the range of temperatures and pressuresnormally encountered in pract ice, in the following two ways:

1. The molecular nature of a gas may be ignored, and the gas is t reated as a cont inuousmedium or cont inuous expanse of matter, termed continuum (that is, does not consist ofdiscrete part icles).2. The decrease in size of the imaginary sphere may be assumed to reach a limit ing size,such that, although it is small compared to the dimensions of any physical object presentin a flow field, for example an aircraft , it is large enough compared to the fluid moleculesand, therefore, contains a reasonably large number of molecules.

2.2.4 ViscosityThe property which characterizes the resistance that a fluid offers to applied shear force istermed viscosity. This resistance, unlike for solids, does not depend upon the deformat ion itselfbut on the rate of deformation. Viscosity is often regarded as the st ickiness of a fluid and itstendency is to resist sliding between layers. There is very lit t le resistance to the movement ofthe knife-blade edge-on through air, but to produce the same mot ion through a thick oil needsmuch more effort . This is because the viscosity of the oil is higher compared to that of air.

2.2.5 Absolute Coefficient of ViscosityThe absolute coefficient of viscosity is a direct measure of the viscosity of a fluid. Consider thetwo parallel plates placed at a distance h apart , as shown in Figure 2.2(a).

Figure 2.2 Fluid shear between a stat ionary and a moving parallel plates.

The space between them is filled with a fluid. The bottom plate is fixed and the other ismoved in its own plane at a speed u. The fluid in contact with the lower plate will be at rest ,while that in contact with the upper plate will be moving with speed u, because of no-slipcondit ion. In the absence of any other influence, the speed of the fluid between the plates willvary linearly, as shown in Figure 2.2(b). As a direct result of viscosity, a force F has to be appliedto each plate to maintain the mot ion, since the fluid will tend to retard the mot ion of themoving plate and will tend to drag the fixed plate in the direct ion of the moving plate. If thearea of each plate in contact with fluid is A, then the shear stress act ing on each plate is F/A.The rate of sliding of the upper plate over the lower is u/h.

These quant it ies are connected by Maxwell's equat ion, which serves to define the absolutecoefficient of viscosity μ. Maxwell's definit ion of viscosity states that:

“the coefficient of viscosity is the tangential force per unit area on either of two parallel platesat unit distance apart, one fixed and the other moving with unit velocity”.

Maxwell's equat ion for viscosity is:

(2.2) Hence,

that is,

Therefore, the unit of μ is kg/(m s). At 0 C the absolute coefficient of viscosity of dry air is1.716 ×10−5 kg/(m s). The absolute coefficient of viscosity μ is also called the dynamic viscositycoefficient.

The Equat ion (2.2), with μ as constant, does not apply to all fluids. For a class of fluids, whichincludes blood, some oils, some paints and so called “thixotropic fluids,” μ is not constant but isa funct ion of du/dh. The derivat ive du/dh is a measure of the rate at which the fluid is shearing.Usually μ is expressed as (N.s)/m2 or gm/(cm s). One gm/(cm s) is known as a poise.

Newton's law of viscosity states that “the stresses which oppose the shearing of a fluid areproportional to the rate of shear strain,” that is, the shear stress τ is given by:

(2.3) where μ is the absolute coefficient of viscosity and ∂u/∂ y is the velocity gradient. The viscosityμ is a property of the fluid. Fluids which obey the above law of viscosity are termed Newtonianfluids. Some fluids such as silicone oil, viscoelast ic fluids, sugar syrup, tar, etc. do not obey theviscosity law given by Equat ion (2.3) and they are called non-Newtonian fluids.

We know that, for incompressible flows, it is possible to separate the calculat ion of velocityboundary layer from that of thermal boundary layer. But for compressible flows it is notpossible, since the velocity and thermal boundary layers interact int imately and hence, theymust be considered simultaneously. This is because, for high-speed flows (compressible flows)

heat ing due to frict ion as well as temperature changes due to compressibility must be takeninto account. Further, it is essent ial to include the effects of viscosity variat ion withtemperature. Usually large variat ions of temperature are encountered in high-speed flows.

The relat ion μ(T) must be found by experiment. The voluminous data available in literatureleads to the conclusion that the fundamental relat ionship is a complex one, and that no singlecorrelat ion funct ion can be found to apply to all gases. Alternat ively, the dependence ofviscosity on temperature can be calculated with the aid of the method of stat ist ical mechanics,but as of yet no completely sat isfactory theory has been evolved. Also, these calculat ions leadto complex expressions for the funct ion μ(T). Therefore, only semi-empirical relat ions appear tobe the means to calculate the viscosity associated with compressible boundary layers. It isimportant to realize that, even though semi-empirical relat ions are not extremely precise, theyare reasonably simple relat ions giving results of acceptable accuracy. For air, it is possible touse an interpolat ion formula based on D. M. Sutherland's theory of viscosity and express theviscosity coefficient , at temperature T, as:

where μ0 denotes the viscosity at the reference temperature T0, and S is a constant, whichassumes the value 110 K for air.

For air the Sutherland's relat ion can also be expressed [1] as:

(2.4) where T is in kelvin. This equat ion is valid for the stat ic pressure range of 0.01 to 100 atm,which is commonly encountered in atmospheric flight . The temperature range in which thisequat ion is valid is from 0 to 3000 K. The absolute viscosity is a funct ion of temperature onlybecause, in the above pressure and temperature ranges, the air behaves as a perfect gas, inthe sense that intermolecular forces are negligible, and that viscosity itself is a momentumtransport phenomenon caused by the random molecular mot ion associated with thermalenergy or temperature.

2.2.6 Kinematic Viscosity CoefficientThe kinematic viscosity coefficient is a convenient form of expressing the viscosity of a fluid. Itis formed by combining the density ρ and the absolute coefficient of viscosity μ, according tothe equat ion:

(2.5) The kinematic viscosity coefficient ν is expressed as m2/s, and 1 cm2/s is known as stoke.

The kinematic viscosity coefficient is a measure of the relat ive magnitudes of viscosity andinert ia of the fluid. Both dynamic viscosity coefficient μ and kinematic viscosity coefficient ν arefunct ions of temperature. For liquids, μ decreases with increase of temperature, whereas forgases μ increases with increase of temperature. This is one of the fundamental differencesbetween the behavior of gases and liquids. The viscosity is pract ically unaffected by thepressure.

2.2.7 Thermal Conductivity of AirAt high-speeds, heat t ransfer from vehicles becomes significant. For example, re-entry vehiclesencounter an extreme situat ion where ablat ive shields are necessary to ensure protect ion ofthe vehicle during its passage through the atmosphere. The heat t ransfer from a vehicledepends on the thermal conduct ivity k of air. Therefore, a method to evaluate k is also

essent ial. For this case, a relat ion similar to Sutherland's law for viscosity is found to be useful,and it is:

where T is temperature in kelvin. The pressure and temperature ranges in which this equat ionis applicable are 0.01 to 100 atm and 0 to 2000 K, respect ively. For the same reason given forviscosity relat ion, the thermal conduct ivity also depends only on temperature.

2.2.8 CompressibilityThe change in volume of a fluid associated with change in pressure is called compressibility.When a fluid is subjected to pressure it gets compressed and its volume changes. The bulkmodulus of elast icity is a measure of how easily the fluid may be compressed, and is defined asthe rat io of pressure change to volumetric strain associated with it . The bulk modulus ofelast icity, K, is given by:

It may also be expressed as:

(2.6) where is specific volume. Since dρ/ρ represents the relat ive change in density brought aboutby the pressure change dp, it is apparent that the bulk modulus of elast icity is the inverse ofthe compressibility of the substance at a given temperature. For instance, K for water and airare approximately 2 GN/m2 and 100 kN/m2, respect ively. This implies that air is about 20,000t imes more compressible than water. It can be shown that, K = a2/ρ, where a is the speed ofsound. The compressibility plays a dominant role at high-speeds. Mach number M (defined asthe rat io of local flow velocity to local speed of sound) is a convenient nondimensionalparameter used in the study of compressible flows. Based on M the flow is divided into thefollowing regimes. When M < 1 the flow is called subsonic, when M ≈ 1 the flow is termedtransonic flow, M from 1.2 to 5 is called supersonic regime, and M > 5 is referred to ashypersonic regime. When flow Mach number is less than 0.3, the compressibility effects arenegligibly small, and hence the flow is called incompressible. For incompressible flows, densitychange associated with velocity is neglected and the density is t reated as invariant.

2.3 Thermodynamic PropertiesWe know from thermodynamics that heat is energy in t ransit ion. Therefore, heat has the samedimensions as energy and is measured in units of joule (J).

2.3.1 Specific HeatThe inherent thermal propert ies of a flowing gas become important when the Mach number isgreater than 0.5. This is because Mach 0.5 corresponds to a speed of 650 km/h for air at sealevel state, therefore for flow above Mach 0.5, the temperature change associated withvelocity becomes considerable. Hence, the energy equat ion needs to be considered in thestudy and owing to this both thermal and calorical propert ies need to be accounted for in theanalysis. The specific heat is one such quant ity. The specific heat is defined as the amount ofheat required to raise the temperature of a unit mass of a medium by one degree. The value ofthe specific heat depends on the type of process involved in raising the temperature of theunit mass. Usually constant volume process and constant pressure process are used forevaluat ing specific heat. The specific heats at constant volume and constant pressure

processes, respect ively, are designated by and cp. The definit ions of these quant it ies are thefollowing:

(2.7) where u is internal energy per unit mass of the fluid, which is a measure of the potent ial andmore part icularly the kinet ic energy of the molecules comprising the gas. The specific heat isa measure of the energy-carrying capacity of the gas molecules. For dry air at normaltemperature, = 717.5 J/(kg K).

The specific heat at constant pressure is defined as:

(2.8) where , the sum of internal energy and flow energy is known as the enthalpy or totalheat constant per unit mass of fluid. The specific heat at constant pressure cp is a measure ofthe ability of the gas to do external work in addit ion to possessing internal energy. Therefore,cp is always greater than . For dry air at normal temperature, cp = 1004.5 J/(kg K).Note: It is essent ial to understand what is meant by normal temperature. For gases, up tocertain temperature, the specific heats will be constant and independent of temperature. Up tothis temperature the gas is termed perfect , implying that cp, and their rat io γ are constants,and independent of temperature. But for temperatures above this limit ing value, cp, willbecome funct ions of T, and the gas will cease to be perfect . For instance, air will behave asperfect gas up to 500 K. The temperature below this liming level is referred to as normaltemperature.

2.3.2 The Ratio of Specific HeatsThe rat io of specific heats:

(2.9) is an important parameter in the study of high-speed flows. This is a measure of the relativeinternal complexity of the molecules of the gas. It has been determined from kinet ic theory ofgases that the rat io of specific heats can be related to the number of degrees of freedom, n, ofthe gas molecules by the relat ion:

(2.10) At normal temperatures, there are six degrees of freedom, namely three translat ional andthree rotat ional, for diatomic gas molecules. For nit rogen, which is a diatomic gas, n = 5 sinceone of the rotat ional degrees of freedom is negligibly small in comparison with the other two.Therefore:

Monatomic gases, such as helium, have 3 t ranslat ional degrees of freedom only, and therefore:

This value of 1.67 is the upper limit of the values which the rat io of specific heats γ can take. Ingeneral γ varies from 1 to 1.67, that is:

The specific heats of a gas are related to the gas constant R. For a perfect gas this relat ion is:

2.4 Surface TensionLiquids behave as if their free surfaces were perfect ly flexible membranes having a constanttension σ per unit width. This tension is called the surface tension. It is important to note thatthis is neither a force nor a stress but a force per unit length. The value of surface tensiondepends on:

the nature of the fluid;the nature of the surface of the substance with which it is in contact ;the temperature and pressure.

Consider a plane material membrane, possessing the property of constant tension σ per unitlength. Let the membrane have a straight edge of length l. The force required to hold the edgestat ionary is:

(2.11) Now, suppose that the edge is pulled so that it is displaced normal to itself by a distance x inthe plane of the membrane. The work done, F, in stretching the membrane is given by:

(2.12) where A is the increase in the area of the membrane. It is seen that σ is the free energy of themembrane per unit area. The important point to be noted here is that , if the energy of asurface is proport ional to its area, then it will behave exact ly as if it were a membrane with aconstant tension per unit width, and this is totally independent of the mechanism by which theenergy is stored. Thus, the existence of surface tension, at the boundary between twosubstances, is a manifestat ion of the fact that the stored energy contains a term proport ionalto the area of the surface. This energy is at t ributable to molecular at t ract ions.

An associated effect of surface tension is the capillary deflect ion of liquids in small tubes.Examine the level of water and mercury in capillaries, shown in Figure 2.3.

Figure 2.3 Capillary effect of water and mercury.

When a glass tube is inserted into a beaker of water, the water will rise in the tube anddisplay a concave meniscus, as shown in Figure 2.3(a). The deviat ion of water level h in thetube from that in the beaker can be shown to be:

(2.13) where θ is the angle between the tangent to the water surface and the glass surface. In otherwords, a liquid such as water or alcohol, which wets the glass surface makes an acute anglewith the solid, and the level of free surface inside the tube will be higher than that outside. Thisis termed capillary action. However, when wett ing does not occur, as in the case of mercury inglass, the angle of contact is obtuse, and the level of free surface inside the tube is depressed,as shown in Figure 2.3(b).

Another important effect of surface tension is that a long cylinder of liquid, at rest or inmot ion, with a free surface is unstable and breaks up into parts, which then assume an

approximately spherical shape. This is the mechanism of the breakup of liquid jets intodroplets.

2.5 Analysis of Fluid FlowBasically two treatments are followed for fluid flow analysis. They are the Lagrangian andEulerian descript ions. Lagrangian method describes the mot ion of each part icle of the flowfield in a separate and discrete manner. For example, the velocity of the n th part icle of anaggregate of part icles, moving in space, can be specified by the scalar equat ions:

(2.14a)

(2.14b)

(2.14c) where Vx, Vy, Vz are the velocity components in x- , y- , z-direct ions, respect ively. They areindependent of the space coordinates, and are funct ions of t ime only. Usually, the part icles aredenoted by the space point they occupy at some init ial t ime t0. Thus, T(x0, t) refers to thetemperature at t ime t of a part icle which was at locat ion x0 at t ime t0.

This approach of ident ifying material points, and following them along is also termed theparticle or material description. This approach is usually preferred in the descript ion of low-density flow fields (also called rarefied flows), in describing the mot ion of moving solids, such asa project ile and so on. However, for a deformable system like a cont inuum fluid, there areinfinite number of fluid elements whose mot ion has to be described, the Lagrangian approachbecomes unmanageable. For such cases, we can employ spat ial coordinates to help to ident ifypart icles in a flow. The velocity of all part icles in a flow field, therefore, can be expressed in thefollowing manner:

(2.15a)

(2.15b) (2.15c)

This is called the Eulerian or field approach. If propert ies and flow characterist ics at eachposit ion in space remain invariant with t ime, the flow is called steady flow. A t ime dependentflow is referred to as unsteady flow. The steady flow velocity field would then be given as:

(2.16a)

(2.16b) (2.16c)

2.5.1 Local and Material Rates of ChangeThe rate of change of propert ies measured by probes at fixed locat ions are referred to as localrates of change, and the rate of change of propert ies experienced by a material part icle istermed the material or substantive rates of change.

The local rate of change of a property η is denoted by ∂η(x, t)/∂ t, where it is understood thatx is held constant. The material rate of change of property η shall be denoted by Dη/Dt. If η isthe velocity V, then DV/Dt is the rate of change of velocity for a fluid part icle and thus is theaccelerat ion that the fluid part icle experiences. On the other hand, ∂V/∂ t is just a local rate ofchange of velocity recorded by a stat ionary probe. In other words, DV/Dt is the part icle ormaterial accelerat ion and ∂V/∂ t is the local accelerat ion.

For a fluid flowing with a uniform velocity V∞, it is possible to write the relat ion between thelocal and material rates of change of property η as:

(2.17) Thus, the local rate of change of η is due to the following two effects:

1. Due to the change of property of each part icle with t ime.2. Due to the combined effect of the spat ial gradient of that property and the mot ion ofthe fluid.

When a spat ial gradient exists, the fluid mot ion brings different part icles with different valueso f η to the probe, thereby modifying the rate of change sensed by the probe. This effect istermed convection effect. Therefore, V∞(∂ η/∂ x) is referred to as the convect ive rate of changeof η. Even though Equat ion (2.17) has been obtained with uniform velocity V∞, note that in thelimit δt → 0 it is only the local velocity V which enters into the analysis and Equat ion (2.17)becomes:

(2.18) Equat ion (2.18) can be generalized for a three-dimensional space as:

(2.19) where ∇ is the gradient operator (≡ i ∂/∂ x + j ∂/∂ y + k ∂/∂ z) and (V · ∇) is a scalar product (= Vx∂/∂ x + Vy ∂/∂ y + Vz ∂/∂ z). Equat ion (2.19) is usually writ ten as:

(2.20) when η is the velocity of a fluid part icle, DV/Dt gives accelerat ion of the fluid part icle and theresultant equat ion is:

(2.21) Equat ion (2.21) is known as Euler's acceleration formula.Note that the Euler's accelerat ion formula is essent ially the link between the Lagrangian andEulerian descript ions of fluid flow.

2.5.2 Graphical Description of Fluid MotionThe following are the three important concepts for visualizing or describing flow fields:

The concept of pathline.The concept of streakline.The concept of streamline.

PathlinePathline may be defined as a line in the flow field describing the trajectory of a given fluidpart icle. From the Lagrangian view point , namely, a closed system with a fixed ident ifiablequant ity of mass, the independent variables are the init ial posit ion, with which each part icle isident ified, and the t ime. Hence, the locus of the same part icle over a t ime period from t0 to tn iscalled the pathline.

StreaklineStreakline may be defined as the instantaneous loci of all the fluid elements that have passedthe point of inject ion at some earlier t ime. Consider a cont inuous tracer inject ion at a fixedpoint Q in space. The connect ion of all elements passing through the point Q over a period oft ime is called the streakline.

StreamlinesStreamlines are imaginary lines, in a fluid flow, drawn in such a manner that the flow velocity isalways tangent ial to it . Flows are usually depicted graphically with the aid of streamlines.These are imaginary lines in the flow field such that the velocity at all points on these lines arealways tangent ial. Streamlines proceeding through the periphery of an infinitesimal area atsome instant of t ime t will form a tube called streamtube, which is useful in the study of fluidflow.

From the Eulerian viewpoint , an open system with constant control volume, all flowpropert ies are funct ions of a fixed point in space and t ime, if the process is t ransient. The flowdirect ion of various part icles at t ime ti forms streamline. The pathline, streamline and streaklineare different in general but coincide in a steady flow.

TimelinesIn modern fluid flow analysis, yet another graphical representat ion, namely timeline, is used.When a pulse input is periodically imposed on a line of t racer source placed normal to a flow, achange in the flow profile can be observed. The tracer image is generally termed t imeline.Timelines are often generated in the flow field to aid the understanding of flow behavior suchas the velocity and velocity gradient.

From the above ment ioned graphical descript ions, it can be inferred that:There can be no flow through the lateral surface of the streamtube.An infinite number of adjacent streamtubes arranged to form a finite cross-sect ion isoften called a bundle of streamtubes.Streamtube is a Eulerian (or field) concept.Pathline is a Lagrangian (or part icle) concept.For steady flows, the pathline, streamline and streakline are ident ical.

2.6 Basic and Subsidiary LawsIn the range of engineering interest , four basic laws must be sat isfied by any cont inuousmedium. They are:

Conservat ion of matter (cont inuity equat ion).Newton's second law (momentum equat ion).Conservat ion of energy (first law of thermodynamics).Increase of entropy principle (second law of thermodynamics).

In addit ion to these primary laws, there are numerous subsidiary laws, somet imes calledconst itut ive relat ions, that apply to specific types of media or flow processes (for example,equat ion of state for perfect gas, Newton's viscosity law for certain viscous fluids, isentropicand adiabat ic process relat ions are some of the commonly used subsidiary equat ions in flowphysics).

2.6.1 System and Control VolumeIn employing the basic and subsidiary laws, any one of the following modes of applicat ion maybe adopted:

The act ivit ies of each and every given element of mass must be such that it sat isfies thebasic laws and the pert inent subsidiary laws.The act ivit ies of each and every elemental volume in space must be such that the basiclaws and the pert inent subsidiary laws are sat isfied.

In the first case, the laws are applied to an ident ified quant ity of matter called the control masssystem. A control mass system is an ident ified quant ity of matter, which may change shape,

posit ion, and thermal condit ion, with t ime or space or both, but must always entail the samematter.

For the second case, a definite volume called control volume is designated in space, and theboundary of this volume is known as control surface. The amount and ident ity of the matter inthe control volume may change with t ime, but the shape of the control volume is fixed, that is,the control volume may change its posit ion in t ime or space or both, but its shape is alwayspreserved.

2.6.2 Integral and Differential AnalysisThe analysis in which large control volumes are used to obtain the aggregate forces or t ransferrates is termed integral analysis. When the analysis is applied to individual points in the flowfield, the result ing equat ions are different ial equat ions, and the method is termed differentialanalysis.

2.6.3 State EquationFor air at normal temperature and pressure, the density ρ, pressure p and temperature T areconnected by the relat ion p = ρRT, where R is a constant called gas constant. This is knownas the thermal equation of state. At high pressures and low temperatures, the above stateequat ion breaks down. At normal pressures and temperatures, the mean distance betweenmolecules and the potent ial energy arising from their at t ract ion can be neglected. The gasbehaves like a perfect gas or ideal gas in such a situat ion. At this stage, it is essent ial tounderstand the difference between the ideal and perfect gases. An ideal gas is frictionlessand incompressible. The perfect gas has viscosity and can therefore develop shear stresses,and it is compressible according to state equat ion.

Real gases below crit ical pressure and above the crit ical temperature tend to obey theperfect-gas law. The perfect-gas law encompasses both Charles' law and Boyle's law. Charles'law states that at constant pressure the volume of a given mass of gas varies directly as itsabsolute temperature. Boyle's law (isothermal law) states that for constant temperature thedensity varies directly as the absolute pressure.

2.7 Kinematics of Fluid FlowKinematics is the branch of physics that deals with the characterist ics of mot ion withoutregard for the effects of forces or mass. In other words, kinematics is the branch of mechanicsthat studies the mot ion of a body or a system of bodies without considerat ion given to itsmass or the forces act ing on it . It describes the spat ial posit ion of bodies or systems, theirvelocities, and their acceleration. If the effects of forces on the mot ion of bodies are accountedfor the subject is termed dynamics. Kinematics differs from dynamics in that the lat ter takesthese forces into account. To simplify the discussions, let us assume the flow to beincompressible, that is, the density is t reated as invariant. The basic governing equat ions foran incompressible flow are the cont inuity and momentum equat ions. The cont inuity equat ion isbased on the conservat ion of matter. For steady incompressible flow, the cont inuity equat ionin different ial form is:

(2.22) where Vx, Vy and Vz are the velocity components along x-, y-and z-direct ions, respect ively.

Equat ion (2.22) may also be expressed as:

where:

and V = i Vx + j Vy + k Vz.The momentum equat ion, which is based on Newton's second law, represents the balance

between various forces act ing on a fluid element, namely:1. Force due to rate of change of momentum, generally referred to as inert ia force.2. Body forces such as buoyancy force, magnet ic force and electrostat ic force.3. Pressure force.4. Viscous forces (causing shear stress).

For a fluid element under equilibrium, by Newton's second law, we have the momentumequat ion as:

(2.23) For a gaseous medium, body forces are negligibly small compared to other forces and hencecan be neglected. For steady incompressible flows, the momentum equat ion can be writ ten as:

(2.23a)

(2.23b)

(2.23c) Equat ion (2.23)a , (2.23)b , (2.23)c are the x, y, z components of momentum equat ion,respect ively. These equat ions are generally known as Navier–Stokes equat ions. They arenonlinear part ial different ial equat ions and there exists no known analyt ical method to solvethem. This poses a major problem in fluid flow analysis. However, the problem is tackled bymaking some simplificat ions to the equat ion, depending on the type of flow to which it is to beapplied. For certain flows, the equat ion can be reduced to an ordinary different ial equat ion of asimple linear type. For some other type of flows, it can be reduced to a nonlinear ordinarydifferent ial equat ion. For the above types of Navier–Stokes equat ion governing specialcategory of flows, such as potent ial flow, fully developed flow in a pipe or channel, andboundary layer over flat plates, it is possible to obtain analyt ical solut ions.

It is essent ial to understand the physics of the flow process before reducing the Navier-Stokes equat ions to any useful form, by making appropriate approximat ions with respect tothe flow. For example, let us examine the flow over an aircraft wing, shown in Figure 2.4.

Figure 2.4 Flow past a wing.

This kind of problem is commonly encountered in fluid mechanics. Air flow over the wingcreates higher pressure at the bottom, compared to the top surface. Hence, there is a netresultant force component normal to the freestream flow direct ion, called lift, L, act ing on thewing. The velocity varies along the wing chord as well as in the direct ion normal to its surface.The former variat ion is due to the shape of the aerofoil, and the lat ter is due to the no-slipcondit ion at the wall. In the direct ion normal to wing surface, the velocity gradients are verylarge in a thin layer adjacent to the surface and the flow reaches asymptot ically to the

freestream velocity within a short distance, above the surface. This thin region adjacent to thewall, where the velocity increases from zero to freestream value, is known as the boundarylayer. Inside the boundary layer the viscous forces are predominant. Further, it so happens thatthe stat ic pressure outside the boundary layer, act ing in the direct ion normal to the surface, istransmit ted to the boundary through the boundary layer, without appreciable change. In otherwords, the pressure gradient across the boundary layer is zero. Neglect ing the inter-layerfrict ion between the streamlines, in the region outside the boundary layer, it is possible to t reatthe flow as inviscid. Inviscid flow is also called potential flow, and for this case the Navier-Stokes equat ion can be simplified to become linear. It is possible to obtain the pressures in thefield outside the boundary layer and treat this pressure to be invariant across the boundarylayer, that is, the pressure in the freestream is impressed through the boundary layer. For low-viscous fluids such as air, we can assume, with a high degree of accuracy, that the flow isfrict ionless over the ent ire flow field, except for a thin region near solid surfaces. In the vicinityof solid surface, owing to high velocity gradients, the frict ional effects become significant. Suchregions near solid boundaries, where the viscous effects are predominant, are termedboundary layers.

In general, boundary layer over streamlined bodies are extremely thin. There may be laminarand turbulent flow within the boundary layer, and its thickness and profile may change alongthe direct ion of the flow. Consider the flow over a flat plate shown in Figure 2.5. Different zonesof boundary layer over a flat plate are shown in Figure 2.5. The laminar sublayer is that zoneadjacent to the boundary, where the turbulence is suppressed to such a degree that only thelaminar effects prevail. The various regions shown in Figure 2.5 are not sharp demarcat ions ofdifferent zones. There is actually a gradual t ransit ion from one region, where certain effectpredominates, to another region, where some other effect is predominant.

Figure 2.5 Flow past a flat plate.

Although the boundary layer is thin, it plays a vital role in fluid dynamics. The drag on ships,aircraft and missiles, the efficiency of compressors and turbines of jet engines, theeffect iveness of ram jets and turbojets, and the efficiencies of numerous other engineeringdevices, are all influenced by the boundary layer to a significant extent. The performance of adevice depends on the behavior of boundary layer and its effect on the main flow. Thefollowing are some of the important parameters associated with boundary layers.

2.7.1 Boundary Layer ThicknessBoundary layer thickness δ may be defined as the distance from the wall in the direct ionnormal to the wall surface, where the fluid velocity is within 1% of the local main streamvelocity. It may also be defined as the distance δ, normal to the surface, in which the flowvelocity increases from zero to some specified value (for example, 99%) of its local mainstream flow velocity. The boundary layer thickness δ may be shown schematically as in Figure2.6.

Figure 2.6 Illustrat ion of boundary layer thickness.

2.7.2 Displacement ThicknessDisplacement thickness δ∗ may be defined as the distance by which the boundary would haveto be displaced if the ent ire flow field were imagined to be frict ionless and the same mass flowis maintained at any sect ion.

Consider unit width in the flow over an infinite flat plate at zero angle of incidence, and letthe x-component of velocity to be Vx and the y-component of velocity be Vy. The volume flowrate through this boundary layer segment of unit width is given by:

where Vm is the main stream (frict ionless flow) velocity component and Vx is the actual localvelocity component. To maintain the same volume flow rate for the frict ionless case, as in theactual case, the boundary must be shifted out by a distance δ∗ so as to cut off the amount of volume flow rate. Thus:

(2.24) The displacement thickness is illustrated in Figure 2.7. The main idea of this postulat ion is to

permit the use of a displaced body in place of the actual body such that the frict ionless massflow around the displaced body is the same as the actual mass flow around the real body. Thedisplacement thickness concept is made use of in the design of wind tunnels, air intakes for jetengines, and so on.

Figure 2.7 Illustrat ion of displacement thickness.

T h e momentum thickness θ and energy thickness δe are other (thickness) measurespertaining to boundary layer. They are defined mathematically as follows:

(2.25)

(2.26) where Vm and ρm are the velocity and density at the edge of the boundary layer, respect ively,and Vx and ρ are the velocity and density at any y locat ion normal to the body surface,respect ively. In addit ion to boundary layer thickness, displacement thickness, momentumthickness and energy thickness, we can define the transit ion point and separat ion point alsowith the help of boundary layer.

A closer look at the essence of the displacement, momentum and energy thicknesses of aboundary layer will be of immense value from an applicat ion point of view. First of all, δ*, θ andδe are all length parameters, in the direct ion normal to the surface over which the boundarylayer prevails. Physically, they account for the defect in mass flow rate, momentum and kinet icenergy, caused by the viscous effect . In other words:

The displacement thickness is the distance by which the boundary, over which theboundary layer prevails, has to be hypothet ically shifted, so that the mass flow rate of theactual flow through distance δ and the ideal (inviscid) flow through distance (δ − δ*),illustrated in Figure 2.7, will be the same.The momentum thickness is the distance by which the boundary, over which theboundary layer prevails, has to be hypothet ically shifted, so that the momentumassociated with the mass passing through the actual thickness (distance) δ and thehypothet ical thickness (δ − θ) will be the same.The energy thickness is the distance by which the boundary, over which the boundarylayer prevails, has to be hypothet ically shifted, so that the kinet ic energy of the flowpassing through the actual thickness (distance) δ and the hypothet ical thickness (δ − δe)will be the same.

2.7.3 Transition PointTransit ion point may be defined as the end of the region at which the flow in the boundarylayer on the surface ceases to be laminar and begins to become turbulent. It is essent ial tonote that the t ransit ion from laminar to turbulent nature takes place over a length, and not ata single point . Thus the transit ion point marks the beginning of the t ransit ion process fromlaminar to turbulent nature.

2.7.4 Separation PointSeparat ion point is the posit ion at which the boundary layer leaves the surface of a solid body.If the separat ion takes place while the boundary layer is st ill laminar, the phenomenon istermed laminar separation. If it takes place for a turbulent boundary layer it is called turbulentseparation.

The boundary layer theory makes use of Navier-Stokes Equat ion (2.23) with the viscousterms in it but in a simplified form. On the basis of many assumptions such as, boundary layerthickness is small compared to the body length and similarity between velocity profiles in alaminar flow, the Navier-Stokes equat ion can be reduced to a nonlinear ordinary different ialequat ion, for which special solut ions exist . Some such problems for which Navier-Stokesequat ions can be reduced to boundary layer equat ions and closed form solut ions can beobtained are: flow past a flat plate or Blassius problem; Hagen-Poiseuille flow through pipes;Couette flow between a stat ionary and moving parallel plates; and flow between rotat ingcylinders.

2.7.5 Rotational and Irrotational Motion

When a fluid element is subjected to a shearing force, a velocity gradient is producedperpendicular to the direct ion of shear, that is, a relat ive mot ion occurs between two layers. Toencounter this relat ive mot ion the fluid elements have to undergo rotat ion. A typical exampleof this type of mot ion is the mot ion between two roller chains rubbing each other, but movingat different velocit ies. It is convenient to use an abstract quant ity called circulation Γ, definedas the line integral of velocity vector between any two points (to define rotat ion of the fluidelement) in a flow field. By definit ion:

(2.27) where dl is an elemental length, c is the path of integrat ion.

Circulat ion per unit area is known as vorticity ζ,

(2.28) In vector form, ζ becomes:

(2.29) where V is the flow velocity, given by V = i Vx + j Vy, and:

For a two-dimensional flow in xy-plane, vort icity ζ becomes:

(2.30a) where ζz is the vort icity about the z-direct ion, which is normal to the flow field. Likewise, theother components of vort icity about x-and y-direct ions are:

(2.30b)

(2.30c) If ζ = 0, the flow is known as irrotational flow. Inviscid flows are basically irrotat ional flows.

2.8 StreamlinesStreamlines are imaginary lines in the flow field such that the velocity at all points on theselines are always tangent ial to them. Flows are usually depicted graphically with the aid ofstreamlines. Streamlines proceeding through the periphery of an infinitesimal area at somet ime t forms a tube called streamtube, which is useful for the study of fluid flow phenomena.From the definit ion of streamlines, it can be inferred that:

Flow cannot cross a streamline, and the mass flow between two streamlines isconserved.Based on the streamline concept, a funct ion ψ called stream function can be defined. Thevelocity components of a flow field can be obtained by different iat ing the stream funct ion.

In terms of stream funct ion ψ, the velocity components of a two-dimensional incompressibleflow are given as:

(2.31) If the flow is compressible the velocity components become:

(2.32) It is important to note that the stream funct ion is defined only for two-dimensional flows, and

the definit ion does not exist for three-dimensional flows. Even though some books define ψ for

axisymmetric flow, they again prove to be equivalent to two-dimensional flow. We must realizethat the definit ion of ψ does not exist for three-dimensional flows, because such a definit iondemands a single tangent at any point on a streamline, which is not possible in three-dimensional flows.

2.8.1 Relationship between Stream Function and VelocityPotential

For irrotat ional flows (the fluid elements in the field are free from rotat ion), there exists afunct ion ϕ called velocity potential or potential function. For a steady two-dimensional flow, ϕmust be a funct ion of two space coordinates (say, x and y). The velocity components are givenby:

(2.33) From Equat ions (2.31) and (2.33), we can write:

(2.34) These relat ions between stream funct ion and potent ial funct ion, given by Equat ion (2.34), arethe famous Cauchy-Riemann equations of complex-variable theory. It can be shown that thelines of constant ϕ or potent ial lines form a family of curves which intersect the streamlines insuch a manner as to have the tangents of the respect ive curves always at right angles at thepoint of intersect ion. Hence, the two sets of curves given by ψ = constant and ϕ = constantform an orthogonal grid system or flow-net. That is, the streamlines and potent ial lines in flowfield are orthogonal.

Unlike stream funct ion, potent ial funct ion exists for three-dimensional flows also, becausethere is no condit ion like the local flow velocity must be tangent ial to the potent ial linesimposed in the definit ion of ϕ. The only requirement for the existence of ϕ is that the flow mustbe potent ial.

2.9 Potential FlowPotent ial flow is based on the concept that the flow field can be represented by a potent ialfunct ion ϕ such that:

(2.35) This linear part ial different ial equat ion is popularly known as Laplace equation. Derivat ives of ϕwith respect to the space coordinates x, y and z give the velocity components Vx, Vy and Vz,respect ively, along x- , y-and z-direct ions. Unlike the stream funct ion ψ, the potent ial funct ioncan exist only if the flow is irrotational, that is, when viscous effects are absent. All inviscidflows must sat isfy the irrotat ionality condit ion:

(2.36) For two-dimensional potent ial flows, by Equat ion (2.30), we have the vort icity ζ as:

Using Equat ion (2.33), we get the vort icity as:

This shows that the flow is irrotat ional. For two-dimensional incompressible flows, thecont inuity equat ion is:

In terms of the potent ial funct ion ϕ, this becomes:

that is:

This linear equat ion is the governing equat ion for potent ial flows.For potent ial flows, the Navier-Stokes equat ions (2.23) reduce to:

(2.37a)

(2.37b)

(2.37c) Equat ion (2.37) is known as Euler's equation.

At this stage, it is natural to have the following doubts about the streamline and potent ialfunct ion, because we defined the streamline as an imaginary line in a flow field and potent ialfunct ion as a mathematical funct ion, which exists only for inviscid flows. The answers to thesevital doubts are the following:

Among the graphical representat ion concepts, namely the pathline, streakline andstreamline, only the first two are physical, and the concept of streamline is onlyhypothet ical. But even though imaginary, the streamline is the only useful concept,because it gives a mathematical representat ion for the flow field in terms of streamfunct ion ψ, with its derivat ives giving the velocity components. Once the velocitycomponents are known, the resultant velocity, its orientat ion, the pressure andtemperature associated with the flow can be determined. Thus, streamline plays adominant role in the analysis of fluid flow.Knowing pret ty well that no fluid is inviscid or potent ial, we introduce the concept ofpotent ial flow, because this gives rise to the definit ion of potent ial funct ion. Thederivat ives of potent ial funct ion with the spat ial coordinates give the velocitycomponents in the direct ion of the respect ive coordinates and the subst itut ion of thesevelocity components in the cont inuity equat ion results in Laplace equat ion. Even thoughthis equat ion is the governing equat ion for an impract ical or imaginary flow (inviscid flow),the fundamental solut ions of Laplace equat ion form the basis for both experimental andcomputat ional flow physics. The basic solut ions for the Laplace equat ion are the uniformflow, source, sink and free or potential vortex. These solut ions being potent ial can besuperposed to get the mathematical funct ions represent ing any pract ical geometry ofinterest . For example, superposit ion of a doublet (source and a sink of equal strength inproximity) and uniform flow would represent flow past a circular cylinder. In the samemanner, suitable distribut ion of source and sink along the camberline and superposit ion ofuniform flow over this distribut ion will mathematically represent flow past an aerofoil.Thus, any pract ical geometry can be modeled mathematically, using the basic solut ions ofthe Laplace equat ion.

2.9.1 Two-dimensional Source and SinkSource is a potent ial flow field in which flow emanat ing from a point spreads radially outwards,as shown in Figure 2.8(a). Sink is potent ial flow field in which flow gushes towards a point fromall radial direct ions, as illustrated in Figure 2.8(b).

Figure 2.8 Illustrat ion of two-dimensional (a) source and (b) sink.

Consider a source at origin, shown in Figure 2.8(a). The volume flow rate crossing a circularsurface of radius r and unit depth is given by:

(2.38) where Vr is the radial component of velocity. The volume flow rate is referred to as thestrength of the source. For a source, the radial lines are streamlines. Therefore, the potent iallines must be concentric circles, represented by:

(2.39) where A is a constant. The radial velocity component Vr = ∂ ϕ/∂ r = A/r.

Subst itut ing this into Equat ion (2.39), we get:

or

Thus, the velocity potent ial for a two-dimensional source of strength becomes:

(2.40) In a similar manner as above, the stream funct ion for a source of strength can be obtained

as:

(2.41) where θ is the orientat ion (inclinat ion) of the streamline from the x-direct ion, measured in thecounter-clockwise direct ion, as shown in Figure 2.8(a). Similarly, for a sink, which is a type of flowin which the fluid from infinity flows radially towards the origin, we can show that the potent ialand stream funct ions are given by:

and

where is the strength of the sink. Note that the volume flow rate is termed the strength ofsource and sink. Also, for both source and sink the origin is a singular point .

2.9.2 Simple VortexA simple or free vortex is a flow field in which the fluid elements simply t ranslate alongconcentric circles, without spinning about their own axes. That is, the fluid elements have onlytranslatory mot ion in a free vortex. In addit ion to moving along concentric circular paths, if thefluid elements spin about their own axes, the flow field is termed forced vortex.

A simple or free vortex can be established by select ing the stream funct ion, ψ, of the sourceto be the potent ial funct ion ϕ of the vortex. Thus, for a simple vortex:

(2.42) It can be easily shown from Equat ion (2.42) that the stream funct ion for a simple vortex is:

(2.43) It follows from Equat ions (2.42) and (2.43) that the velocity components of the simple vortex,shown in Figure 2.9, are:

(2.44) Here again the origin is a singular point , where the tangent ial velocity Vθ tends to infinity, asseen from Equat ion (2.44). The flow in a simple or free vortex resembles part of the commonwhirlpool found while paddling a boat or while emptying water from a bathtub. An approximateprofile of a whirlpool is as shown in Figure 2.10. For the whirlpool, shown in Figure 2.10, thecirculat ion along any path about the origin is given by:

Figure 2.9 A simple or potent ial vortex flow.

By Equat ion (2.44), , therefore, the circulat ion becomes:

Since there are no other singularit ies for the whirlpool, shown in Figure 2.10, this must be thecirculat ion for all paths about the origin. Consequent ly, in the case of vortex is the measure ofcirculat ion about the origin and is also referred to as the strength of the vortex.

Figure 2.10 A whirlpool flow field.

2.9.3 Source-Sink Pair

This is a combinat ion of a source and sink of equal strength, situated (located) at a distanceapart . The stream funct ion due to this combinat ion is obtained simply by adding the streamfunct ions of source and sink. When the distance between the source and sink is madenegligibly small, in the limit ing case, the combinat ion results in a doublet.

2.9.4 DoubletA doublet or a dipole is a potent ial flow field due to a source and sink of equal strength,brought together in such a way that the product of their strength and the distance betweenthem remain constant. Consider a point P in the field of a doublet formed by a source and asink of strength and , respect ively, kept at a distance ds, as shown in Figure 2.11, with sinkat the origin.

Figure 2.11 Source and sink.

By Rankine's theorem, the velocity potent ial of the doublet , ϕD, can be expressed as thesum of the velocity potent ials of the source and sink. Thus, we have:

Expanding , we get:

But , therefore, neglect ing the second and higher order terms, we get the potent ialfunct ion for a doublet as:

By the definit ion of doublet , ds → 0, therefore:

Hence,

Also, for a doublet , by definit ion, = constant. Let this constant, known as the strength of thedoublet be denoted by m, then:

and

(2.45) In Cartesian coordinates, the velocity potent ial for the doublet becomes:

From the above equat ions for ϕD, the expression for the stream funct ion ψD can be obtainedas:

In Cartesian coordinates, the stream funct ion becomes:

If the source and sink were placed on the x-axis, the streamlines of the doublet will be asshown in Figure 2.12.

Figure 2.12 Doublet with source and sink on the x-axis (source located on the left and sink onthe right of the origin).

If the source and sink are placed on the y-axis, the result ing expressions for the ϕD and ψDwill become:

The streamlines of the doublet will be as shown in Figure 2.13. The expression for thestream funct ion:

can be arranged in the form:

where c = m/(2π), is a constant. This can be expressed as:

Figure 2.13 Doublet with source and sink on the y-axis (source located below the origin andsink above the origin).

Thus, the streamlines represented by ψD(yy) = constant are circles with their centers lying ont he x-axis and are tangent to the y-axis at the origin (Figure 2.13). Direct ion of flow at theorigin is along the negat ive y-axis, point ing outward from the source of the limit ing source-sinkpair, which is called the axis of the doublet .

The potent ial and stream funct ions for the concentrated source, sink, vortex, and doubletare all singular at the origin. It will be shown in the following sect ion that several interest ing flowpatterns can be obtained by superposing a uniform flow on these concentrated singularit ies.

2.10 Combination of Simple FlowsIn Sect ion 2.9 we saw that flow past pract ical shapes of interest can be represented orsimulated with suitable combinat ion of source, sink, free vortex and uniform flow. In this sect ionlet us discuss some such flow fields.

2.10.1 Flow Past a Half-BodyAn interest ing pattern of flow past a half-body, shown in Figure 2.14, can be obtained bycombining a source and a uniform flow parallel to x-axis. By definit ion, a given streamline (ψ =constant) is associated with one part icular value of the stream funct ion. Therefore, when wejoin the points of intersect ion of the radial streamlines of the source with the rect ilinearstreamlines of the uniform flow, the sum of magnitudes of the two stream funct ions will beequal to the streamline of the result ing combined flow pattern. If this procedure is repeated fora number of values of the combined stream funct ion, the result will be a picture of thecombined flow pattern.

Figure 2.14 Irrotat ional flow past a two-dimensional half-body.

The stream funct ion for the flow due to the combinat ion of a source of strength at theorigin, immersed in a uniform flow of velocity V∞, parallel to x-axis is:

The streamlines of the result ing flow field will be as shown in Figure 2.14.The streamline passing through the stagnat ion point S is termed the stagnat ion streamline.

The stagnat ion streamline resembles a semi-ellipse. This shape is popularly known asRankine's half-body. The streamlines inside the semi-ellipse are due to the source and thoseoutside the semi-ellipse are due to the uniform flow. The boundary or stagnat ion streamline is

outside the semi-ellipse are due to the uniform flow. The boundary or stagnat ion streamline isgiven by:

It is seen that S is the stagnat ion point where the uniform flow velocity V∞ cancels the velocityof the flow from the source. The stagnat ion point is located at (a, π). At the stagnat ion point ,both Vr and Vθ should be zero. Thus:

This gives:

Therefore, the stream funct ion of the stagnat ion point is:

At the stagnat ion point S, r = a and θ = π, therefore:

The equat ion of the streamline passing through the stagnat ion point is obtained by sett ing , result ing in:

(2.46) A plot of the streamlines represented by Equat ion (2.46) is shown in Figure 2.14. It is a semi-

infinite body with a smooth nose, generally called a half-body. The stagnat ion streamlinedivides the field into a region external to the body and a region internal to it . The internal flowconsists ent irely of fluid emanat ing from the source, and the external region contains theoriginally uniform flow. The half-body resembles several shapes of theoret ical interest , such asthe front part of a bridge pier or an aerofoil. The upper half of the flow resembles the flow overa cliff or a side contour of a wide channel.

The half-width of the body is given as:

As θ → 0, the half-width tends to a maximum of , that is, the mass flux fromthe source is contained ent irely within the half-body, and at a largedownstream distance where the local flow velocity u = V∞.

The pressure distribut ion can be found from the incompressible Bernoulli's equat ion:

where p and u are the local stat ic pressure and velocity of the flow, respect ively.The pressure can be expressed through the nondimensional pressure difference called the

pressure coefficient, defined as:

where p and p∞ are the local and freestream stat ic pressures, respect ively, ρ∞ is freestreamdensity and V∞ is freestream velocity.

A plot of Cp distribut ion on the surface of the half-body is shown in Figure 2.15. It is seen

that there is a posit ive pressure or compression zone near the nose of the body and thepressure becomes negat ive or suct ion, downstream of the posit ive pressure zone. Thisposit ive pressure zone is also called pressure-hill. The net pressure force act ing on the bodycan easily be shown to be zero, by integrat ing the pressure p act ing on the surface. The half-body is obtained by the linear combinat ion of the individual stream funct ions of a source and auniform flow, as per the Rankine's theorem which states that:

Figure 2.15 Pressure distribut ion for potent ial flow over a half-body.

“the resulting stream function of n potential flows can be obtained by combining the streamfunctions of the individual flows.”

The half-body shown in Figure 2.15 is also referred to as Rankine's half-body.

Example 2.1A two-dimensional source of strength 4.0 m2/s is placed in a uniform flow of velocity 1 m/sparallel to x-axis. Determine the flow velocity and its direct ion at r = 0.8 m and θ = 140 .

SolutionThe given flow is an ideal flow around a half-body. The stream funct ion for the flow around ahalf-body, by Equat ion (2.46), is:

Given that V∞ = 1 m/s and = 4 m2/s. Thus:

The tangent ial and radial components of velocity, respect ively, are:

and

The resultant velocity, at r = 0.8 and θ = 140 , is:

The velocity field is as shown in the Figure 2.16.

Figure 2.16 Velocity field.

If θ is the angle the velocity makes with the horizontal, as shown in the figure, then:

Also:

Thus:

Example 2.2A two-dimensional flow field is made up of a source at the origin and a flow given by ϕ = r2 cos2θ. Locate any stagnat ion points in the upper half of the coordinate plane (0 ≤ θ ≤ π).

SolutionThe potent ial funct ion of the flow is:

where is the velocity potent ial for the source, with strength m m2/s.The velocity components are given by:

At the stagnat ion point , Vr = 0 and Vθ = 0. Thus, we have:

(i) and

Thus, θs = π/2 at the stagnat ion point . Subst itut ion of this into Equat ion (i.) gives:

Example 2.3A certain body has the shape of Rankine's half-body of maximum thickness of 0.5 m. If thisbody is to be placed in an air stream of velocity 30 m/s, find the source strength required tosimulate flow around the body?

SolutionThe half-body can be represented as a combinat ion of a source and a uniform flow.The result ing stream funct ion is:

(i) where m is source strength. At the stagnat ion point S:

That is:

But at S, θ = π, thus:

If x = b is the stagnat ion point , then at r = b:

or

The value of the stream funct ion of the streamline passing through the stagnat ion point canbe obtained by evaluat ing Equat ion (i.), at r = b and θ = π, which yields:

Thus, from Equat ion (i), we get:

Solving, we get:

(ii) The width of the half-body asymptot ically approaches 2 πb, as shown below. Equat ion (ii) canbe writ ten as:

But y = r sin θ, thus:

(iii) From Equat ion (iii.), it is seen that as θ → 0 or θ → 2 π the half-width approaches ± bπ.For θ = 0, Equat ion (iii) gives:

But:

Therefore:

or

For U = 30 m/s and y = 0.5 m, we have:

The source strength required is 15 m2/s.

Example 2.4Check whether the flow represented by the stream funct ion:

where is the volume flow rate, which is a constant, is irrotat ional.

SolutionThe radial and tangent ial velocity components of the given flow are:

The irrotat ionality condit ion given by Equat ion (2.36) is:

In terms of r and θ, this becomes:

Thus:

The irrotat ional condit ion is sat isfied and hence the flow is irrotat ional.

2.11 Flow Past a Circular Cylinder withoutCirculation

A flow pattern equivalent to an irrotat ional flow over a circular cylinder can be obtained bycombining a uniform stream and a doublet with its axis directed against the stream, as shownin Figure 2.17.

Figure 2.17 Irrotat ional flow past a circular cylinder without circulat ion.

The points S1 and S2 are the stagnat ion points. The combined stream funct ion becomes:

The potent ial funct ion for the flow is:

(2.47) It is seen that ψ = 0 for all values of θ, showing that the streamline ψ = 0 represents a circularcylinder of radius . Let . For a given velocity of theuniform flow and a given strength of the doublet , the radius a is constant. Thus, the streamfunct ion and potent ial funct ion of the flow past a cylinder can also be expressed as:

(2.48)

(2.49) From the flow pattern shown in Figure 2.17, it is evident that the flow inside the circle has noinfluence on the flow outside the circle. The normal and tangent ial components of velocityaround the cylinder, respect ively, are:

The flow speed around the cylinder is given by:

where what is meant by |Vθ| is the posit ive value of sin θ. This shows that there are stagnat ionpoints on the surface at (a, 0) and (a, π). The flow velocity reaches a maximum of 2V∞ at thetop and bottom of the cylinder, where θ = π/2 and 3π/2, respect ively.

The nondimensional pressure distribut ion over the surface of the cylinder is given by:

(2.50) Pressure distribut ion at the surface of the cylinder is shown by the cont inuous line in Figure2.18. The symmetry of the pressure distribut ion in an irrotat ional flow implies that “a steadilymoving body experiences no drag.” This result , which is not t rue for actual (viscous) flowswhere the body experiences drag, is known as d’Alembert's paradox. This discrepancybetween the results of inviscid and viscous flows is because of:

Figure 2.18 Comparison of irrotat ional and actual pressure over a circular cylinder.

the existence of tangent ial stress or skin friction anddrag due to the separat ion of the flow from the sides of the body and the result ingformat ion of wake dominated by eddies, in the case of bluff bodies, in the actual flowwhich is viscous.

The surface pressure in the wake of the cylinder in actual flow is lower than that predicted byirrotat ional or potent ial flow theory, result ing in a pressure drag.Note: For flow past a circular cylinder, there are two limits for the Cp, as shown in Figure 2.18.These two limits are Cp = + 1 and Cp = − 3, at the forward and rear stagnat ion points (at 0and 180 , respect ively), and at the top and bottom locat ions of the cylinder (at 90 and 270 ,respect ively). At this stage, it is natural to quest ion about the validity of these limit ing values ofthe pressure coefficient Cp for flow past geometries other than circular cylinder. Clarifyingthese doubts is essent ial from both theoret ical and applicat ion points of view.

The posit ive limit of +1 for Cp, at the forward stagnat ion point , is valid for all geometriesand for both potent ial and viscous flow, as long as the flow speed is subsonic.When the flow speed becomes supersonic, there will be a shock ahead of or at the noseof a blunt-nosed and sharp-nosed bodies, respect ively. Hence, there are two differentspeeds at the zones upstream and downstream of the shock. Therefore, the freestreamstat ic pressure p∞ and dynamic pressure to be used in the Cp relat ion:

have two opt ions, where p is the local stat ic pressure. This makes the Cp at theforward stagnat ion point sensit ive to the freestream stat ic and dynamic pressures,used to calculate it . Therefore, Cp = + 1 can not be taken as the limit ing maximum ofCp, when the flow speed is supersonic.

The limit ing minimum of −3, for the Cp over the cylinder in potent ial flow, is valid only forcircular cylinder. The negat ive value of Cp can take values lower than −3 for othergeometries. For example, for a cambered aerofoil at an angle of incidence can have Cp as

low as −6.Another important aspect to be noted for viscous flow is that there is no specific locat ionfor rear stagnat ion point on the body. The flow separates from the body and establishesa wake. The separat ion is taking place at two locat ions, above and below the horizontalaxis passing through the center of the body. Also, these upper and lower separat ions arenot taking place at fixed points, but oscillate around the separat ion locat ion, because ofvortex format ion. Therefore, the negat ive pressure at the rear of the body does notassume a specified minimum at any fixed point , as in the case of potent ial flow. For manycombinat ions of the geometries and flow Reynolds numbers, the negat ive Cp at theseparated zone of the body can assume comparable magnitudes over a large port ion ofthe wake.

2.11.1 Flow Past a Circular Cylinder with CirculationWe saw that there is no net force act ing on a circular cylinder in a steady irrotat ional flowwithout circulat ion. It can be shown that a lateral force ident ical to a lift force on an aerofoil,results when circulat ion is introduced around the cylinder. When a clockwise line vortex ofcirculat ion Γ is superposed around the cylinder in an irrotat ional flow, the stream funct ion [sumof Equat ions (2.43) and (2.48)] becomes:

The tangent ial velocity component at any point in the flow is:

(2.51) At the surface of the cylinder of radius a, the tangent ial velocity becomes:

At the stagnat ion point , Vθ = 0, thus:

(2.52) For Γ = 0, the potent ial flow past the cylinder is symmetrical about both x-and y-direct ions,

as shown in Figure 2.19(a). For this case there is no drag act ing on the cylinder.

Figure 2.19 Flow past a circular cylinder (a) without circulat ion and (b), (c) and (d) withcirculat ion.

For Γ < 4 π a V∞, two values of θ sat isfy Equat ion (2.52). This implies that there are twostagnat ion points on the surface, as shown in Figure 2.19(b).

When Γ = 4 π a V∞, the stagnat ion points merge on the negat ive y-axis, as shown in Figure2.19(c).

For Γ > 4 π a V∞ the stagnat ion points merge and stay outside the cylinder, as shown inFigure 2.19(d). The stagnat ion points move away from the cylinder surface, since sin θ cannotbe greater than 1. The radial distance of the stagnat ion points for this case can be found from:

This gives:

One root of this is r > a, and the flow field for this is as shown in Figure 2.19(d), with thestagnat ion points S1 and S2, overlapping and posit ioned outside the cylinder. The second rootcorresponds to a stagnat ion point inside the cylinder. But the stagnat ion point for flow past acylinder cannot be inside the cylinder. Therefore, the second solut ion is an impossible one.

As shown in Figure 2.19, the locat ion of the forward and rear stagnat ion points on thecylinder can be adjusted by controlling the magnitude of the circulat ion Γ. The circulat ion whichposit ions the stagnat ion points in proximity, as shown in Figure 2.19(b), is called subcriticalcirculation, the circulat ion which makes the stagnat ion points coincide at the surface of thecylinder, as shown in Figure 2.19(c), is called critical circulation, and the circulat ion which makesthe stagnat ion points coincide and take a posit ion outside the surface of the cylinder, asshown in Figure 2.19(d), is called supercritical circulation.

To determine the magnitude of the t ransverse force act ing on the cylinder, it is essent ial tofind the pressure distribut ion around the cylinder. Since the flow is irrotat ional, Bernoulli'sequat ion can be applied between a point in the freestream flow and a point on the surface ofthe cylinder. Bernoulli's equat ion for incompressible flow is:

Using Equat ion (2.52), the surface pressure can be found as follows.At the surface r = a and Equat ion (2.51) gives the local velocity at any point on the surface

as:

Subst itut ing this into Bernoulli's equat ion, we get:

that is:

(2.53) The symmetry of flow about the y-axis implies that the pressure force on the cylinder has nocomponent along the x-axis. The pressure force act ing in the direct ion normal to the flow(along y-axis) is called the lift force L in aerodynamics.

Consider a cylinder of radius a in a uniform flow of velocity V∞, shown in Figure 2.20.

Figure 2.20 Circular cylinder in a uniform flow.

The lift act ing on the cylinder is given by:

Subst itut ing Equat ion (2.53), and integrat ing we obtain the lift as:

(2.54) where we have used:

It can be shown that Equat ion (2.54) is valid for irrotat ional flows around any two-dimensionalshape, not just for circular cylinders alone. The expression for lift in Equat ion (2.54) shows thatthe lift force proport ional to circulat ion Γ is of fundamental importance in aerodynamics.Wilhelm Kutta (1902), the German mathematician, and Nikolai Zhukovsky (1906), the Russianaerodynamicist , have proved the relat ion for lift , given by Equat ion (2.54), independent ly; this iscalled the Kutta-Zhukovsky lift theorem (the name Zhukovsky is t ransliterated as Joukowsky inolder Western texts). The circulat ion developed by certain two-dimensional shapes, such asaerofoil, when placed in a stream can be explained with vortex theory. It can be shown that theviscosity of the fluid is responsible for the development of circulat ion. The magnitude ofcirculat ion, however, is independent of viscosity, and depends on the flow speed V∞, the shapeand orientat ion of the body to the freestream direct ion.

For a circular cylinder in a potent ial flow, the only way to develop circulat ion is by rotat ing it ina flow stream. Although viscous effects are important in this case, the observed pattern forhigh rotat ional speeds displays a striking similarity to the ideal flow pattern for Γ > 4πaV∞.When the cylinder rotates at low speeds, the retarded flow in the boundary layer is not able toovercome the adverse pressure gradient over the rear surface (downstream of θ = 90 ) thecylinder. This leads to the separat ion of the real (actual) flow, unlike the irrotat ional flow whichdoes not separate. However, even in the presence of separat ion, observed speeds are higheron the upper surface of the cylinder, implying the existence of a lift force.

A second reason for a rotat ing cylinder generat ing lift is the asymmetry to the flow pattern,caused by the delayed separat ion on the upper surface of the cylinder. The asymmetry resultsin the generat ion of the lift force. The contribut ion of this mechanism is small for two-dimensional objects such as circular cylinder, but it is the only mechanism for the side forceexperienced by spinning three-dimensional objects such as soccer, tennis and golf balls. Thelateral force experienced by rotat ing bodies is called the Magnus effect. The horizontalcomponent of the force on the cylinder, due to the pressure, in general is called drag. For thecylinder, shown in Figure 2.20, the drag is given by:

It is interest ing to note that the drag is equal to zero. It is important to realize that this result isobtained on the assumption that the flow is inviscid. In real (actual or viscous) flows thecylinder will experience a finite drag force act ing on it due to viscous frict ion and flowseparat ion.

2.12 Viscous FlowsIn the previous sect ions of this chapter, we have seen many interest ing concepts of fluid flow.With this background, let us observe some of the important aspects of fluid flow from apract ical or applicat ion point of view.

We are familiar with the fact that the viscosity produces shear force which tends to retardthe fluid mot ion. It works against inert ia force. The rat io of these two forces governs (dictates)many propert ies of the flow, and the rat io expressed in the form of a nondimensionalparameter is known as the famous Reynolds number, ReL:

(2.55) where V, ρ are the velocity and density of the flow, respect ively, μ is the dynamic viscositycoefficient of the fluid and L is a characterist ic dimension. The Reynolds number plays adominant role in fluid flow analysis. This is one of the fundamental dimensionless parameterswhich must be matched for similarity considerat ions in most fluid flow analyses. At highReynolds numbers, the inert ia force is predominant compared to viscous forces. At lowReynolds numbers the viscous effects predominate everywhere, whereas at high Re theviscous effects confine to a thin region, just adjacent to the surface of the object present inthe flow, and this thin layer is termed boundary layer. Since the length and velocity scales arechosen according to a part icular flow, when comparing the flow propert ies at two differentReynolds numbers, only flows with geometric similarity should be considered. In other words,flow over a circular cylinder should be compared only with flow past another circular cylinder,whose dimensions can be different but not the shape. Flow in pipes with different velocit iesand diameters and flow over aerofoils of the same kind are also some geometrically similarflows. From the above-ment ioned similarity considerat ion, we can infer that geometric similarityis a prerequisite for dynamic similarity. That is, dynamically similar flows must be geometricallysimilar, but the converse need not be true. Only similar flows can be compared, that is, whencomparing the effect of viscosity, the changes in flow pattern due to body shape should notinterfere with the problem.

For calculat ing Reynolds number, different velocity and length scales are used. Some popularshapes and their length scales we often encounter in fluid flow studies are given in Table 2.1 .In the descript ion of Reynolds number here, the quant it ies with subscript ∞ are at thefreestream and quant it ies without subscript are the local propert ies. Reynolds number isbasically a similarity parameter. It is used to determine the laminar and turbulent nature of flow.Below a certain Reynolds number the ent ire flow is laminar and any disturbance introducedinto the flow will be dissipated out by viscosity. The limit ing Reynolds number below which the

ent ire flow is laminar is termed the lower critical Reynolds number.T able 2.1 Some popular shapes and their characteristic lengths.

Cylinder:

Re d is cylinder diameter

Aero fo il:

Re c is aero fo il chord

Pipe flo w (fully develo ped):

Re is the average velocity

d is pipe diameter

Channel flo w (t wo -dimensio nal

and fully develo ped):

Re is the average velocity

h is the height o f the channel

Flo w o ver a grid:

Re V is the velocity upstream

or downstream of the grid

m is the mesh size

Bo undary layer:

Re V is the outer velocity

δ is the boundary layer thickness

Re θ is the momentum thickness

Re x is the distance from the leading edge

Some of the well-known crit ical Reynolds number are listed below:Pipe flow - Red = 2300: based on mean velocity and diameter d.

Channel flow - Reh = 1000 (two-dimensional): based on height h and mean velocity.

Boundary layer flow - Reθ = 350: based on freestream velocity and momentum thicknessθ.Circular cylinder - Re (turbulent wake): based on wake width and wake defect .Flat plate - Rex = 5 × 105: based on length x from the leading edge.

Circular cylinder - Red = 1.66 × 105: based on cylinder diameter d.

It is essent ial to note that, the t ransit ion from laminar to turbulent nature does not takeplace at a part icular Reynolds number but over a range of Reynolds number, because anytransit ion is gradual and not sudden. Therefore, incorporat ing this aspect, we can define thelower and upper crit ical Reynolds numbers as follows.

Lower critical Reynolds number is that Reynolds number below which the ent ire flow islaminar.Upper critical Reynolds number is that Reynolds number above which the ent ire flow isturbulent.Critical Reynolds number is that at which the flow field is a mixture of laminar andturbulent flows.

Note: It is important to note that when the Reynolds number is low due to large viscosity μ theflow is termed stratified flow, for example, flow of tar, honey etc. are strat ified flows. When theReynolds number is low because of low density, the flow is termed rarefied flow. For instance,flow in space and very high alt itudes, in the Earth's atmosphere, are rarefied flows.

2.12.1 Drag of BodiesWhen a body moves in a fluid, it experiences forces and moments due to the relat ive mot ion ofthe flow taking place around it . If the body has an arbit rary shape and orientat ion, the flow willexert forces and moments about all the three coordinate axes, as shown in Figure 2.21. Theforce on the body along the flow direct ion is called drag.

Figure 2.21 Forces act ing on an arbit rary body.

The drag is essent ially a force opposing the mot ion of the body. Viscosity is responsible for apart of the drag force, and the body shape generally determines the overall drag. The dragcaused by the viscous effect is termed the frict ional drag or skin friction. In the design oft ransport vehicles, shapes experiencing minimum drag are considered to keep the powerconsumption at a minimum. Low drag shapes are called streamlined bodies and high dragshapes are termed bluff bodies.

Drag arises due to (a) the difference in pressure between the front and rear regions and (b)the frict ion between the body surface and the fluid. Drag force caused by the pressureimbalance is known as pressure drag, and (b) the drag due to frict ion is known as skin frictiondrag or shear drag. A body for which the skin frict ion drag is the major port ion of the total dragis called streamlined body, and that with the pressure drag as the major port ion of the totaldrag is called a bluff body.

2.12.1.1 Pressure DragThe pressure drag arises due to the separat ion of boundary layer, caused by adverse pressuregradient. The phenomenon of separat ion, and how it causes the pressure drag, can beexplained by considering flow around a body, such as a circular cylinder. If the flow is assumedto be potent ial, there is no viscosity and hence no boundary layer. The flow past the cylinderwould be as shown in Figure 2.22, without any separat ion.

Figure 2.22 Potent ial flow past a circular cylinder.

Potent ial flow around a cylinder will be symmetrical about both the horizontal and vert ical

planes, passing through the center of the cylinder. The pressure distribut ion over the front andback surfaces would be ident ical, and the net force along the freestream direct ion would bezero. That is, there would not be any drag act ing on the cylinder. But in real flow, because ofviscosity, a boundary layer is formed over the surface of the cylinder. The flow experiences afavorable pressure gradient from the forward stagnat ion point S1 to the topmost point A onthe cylinder at θ = 90 , shown in Figure 2.22.

Therefore, the flow accelerates from point S1 to A (that is, from θ = 0 to 90 ). However,beyond θ = 90 the flow is subjected to an adverse pressure gradient and hence decelerates.Note that beyond the topmost point A the fluid elements find a larger space to relax.Therefore, in accordance with mass conservat ion (for subsonic flow) [2], as the flow areaincreases the flow speed decreases and the pressure increases. Under this condit ion there is anet pressure force act ing against the fluid flow. This process establishes an adverse pressuregradient, leading to flow separat ion, as illustrated in Figure 2.23. In a boundary layer, thevelocity near the surface is small, and hence the force due to its momentum is unable tocounteract the pressure force. Flow within the boundary layer gets retarded and the velocitynear the wall region reduces to zero at some point downstream of A and then the flow ispushed back in the opposite direct ion, as illustrated in Figure 2.23. This phenomenon is calledflow separation.

Figure 2.23 Illustrat ion of separat ion process.

The locat ion where the flow leaves the body surface is termed separation point. For flowpast a cylinder, there are two separat ion points on either side of the horizontal axis throughthe center of the cylinder. The separated flow is chaot ic and vortex dominated. The separatedflow behind an object is also referred to as wake. Depending on the Reynolds number level, thewake may be laminar or turbulent. An important characterist ics of the separated flow is that itis always unsymmetrical, even for laminar separat ion. This is because of the vort ices prevailingin the separated zone. As we know, for every vortex there is a specific frequency andamplitude. Therefore, when the vort ices formed at the upper and lower separat ion points ofthe cylinder are of the same size and leave the cylinder at the same t ime, the wake must besymmetric. But this kind of format ion of vort ices of ident ical size and leaving the upper andlower separat ion points at the same t ime is possible only when the geometry of the cylinder isperfect ly symmetrical and the freestream flow is absolutely unperturbed and symmetricalabout the horizontal plane bisect ing the cylinder. But in pract ice it is not possible to meetthese stringent requirements of flow and geometrical symmetry to establish symmetricalseparat ion. Owing to this pract ical constraints all separated flows are unsymmetrical. Indeed,the format ion of the vort ices at the upper and lower separat ion points itself is unsymmetrical.When one of them, say the upper one, grows faster, the other one is unable to grow at the

same rate. Therefore, only after the faster growing vortex reaches a limit ing size possible, forthe geometry and Reynolds number combinat ion, and leaves the surface, the growth of thevortex at the opposite side picks up. This retards the growth of the new vortex formed at thelocat ion where the vortex left the surface. Thus, alternat ive shedding of vort ices from theupper and lower separat ion points is established. The alternat ive shedding of vort ices makesthe wake chaot ic.

Across the separated region, the total pressure is nearly a constant and lower than what itwould have been if the flow did not separate. The pressure do not recover completely as in thecase of potent ial flow. Thus, on account of the incomplete recovery of pressure due toseparat ion, a net drag force opposing the body mot ion is generated. We can easily see thatthe pressure drag will be small if the separat ion had taken place later, that is, the area overwhich the pressure unrecovered is small. To minimize pressure drag, the separat ion pointshould be as far as possible from the leading edge or forward stagnat ion point . This is t rue forany shape. Streamlined bodies are designed on this basis and the adverse pressure gradient iskept as small as possible, by keeping the curvature very small. At this stage, it is important torealize that the separated region behind an object is vortex dominated and these vort icescause considerable pressure loss. Thus the total pressure p0,rear behind the object issignificant ly lower than the total pressure p0,face at the face of the object . This difference

, termed pressure loss, is a direct measure of the drag. This drag caused by thepressure loss is called the pressure drag. This is also referred to as form drag, because theform or shape of the moving object dictates the separat ion and the expanse of the separatedzone. The separat ion zone behind an object is also referred to as wake. That is, wake is theseparated region behind an object (usually a bluff body) where the pressure loss is severe. It isessent ial to note that what is meant by pressure loss is total pressure loss, and there isnothing like static pressure loss.

The separat ion of boundary layer depends not only on the strength of the adverse pressuregradient but also on the nature of the boundary layer, namely, laminar or turbulent. A laminarflow has tendency to separate earlier than a turbulent flow. This is because the laminarvelocity profiles in a boundary layer has lesser momentum near the wall. This is conspicuous inthe case of flow over a circular cylinder. Laminar boundary layer separates nearly at θ = 90whereas, for a highly turbulent boundary layer the separat ion is delayed and the at tached flowcont inues up to as far as θ = 150 on the cylinder. The reduct ion of pressure drag when theboundary layer changes from laminar to turbulent is of the order of 5 t imes for bluff bodies. Theflow behind a separated region is called the wake. For low drag, the wake width should besmall.

Although separat ion is shown to take place at well defined locat ions on the body, in theillustrat ion in Figure 2.23, it actually takes places over a zone on the surface which can not beident ified easily. Therefore, theoret ical est imat ion of separat ion especially for a turbulentboundary layer is difficult and hence the pressure drag cannot be easily calculated. Someapproximate methods exist but they can serve only as guidelines for the est imat ion ofpressure drag.

At this stage, we may wonder about the level of stat ic pressure in the separated flow regionor the wake of a body. The total pressure in the wake is found to be lower than that in thefreestream, because of the pressure loss caused by the vort ices in the wake. But the stat icpressure in the wake is almost equal to the freestream level. But it is essent ial to realize thatjust after separat ion, the flow is chaot ic and the streaklines do not exhibit any defined pattern.Therefore, the stat ic pressure does not show any specific mean value in the near-wake regionand keeps fluctuat ing. However, beyond some distance behind the object , the wake stabilizesto an extent to assume almost constant stat ic pressure across its width. This distance isabout 6 t imes the diameter for a circular cylinder. Thus, beyond 6 diameter distance the stat icpressure in the wake is equal to the freestream value.Note: It may be useful to recall what is meant by pressure loss is the total pressure loss andthere is nothing like stat ic pressure loss.

there is nothing like stat ic pressure loss.

2.12.1.2 Skin Friction DragThe frict ion between the surface of a body and the fluid causes viscous shear stress and thisforce is known as skin friction drag. Wall shear stress τ at the surface of a body is given by:

(2.56) where μ is the dynamic viscosity coefficient and ∂Vx/∂ y is the velocity gradient at body surfacey = 0. If the velocity profile in the boundary layer is known, then the shear stress can becalculated.

For streamlined bodies, the separated zone being small, a major port ion of the drag isbecause of skin frict ion. We saw that bodies are classified as streamlined and bluff, based onwhich is dominant among the drag components. A body for which the skin frict ion drag is amajor port ion of the total drag is termed streamlined body. A body for which the pressure(form) drag is the major port ion is termed bluff body. Turbulent boundary layer results in moreskin frict ion than a laminar one. Examine the skin frict ion coefficient cf variat ion with Reynoldsnumber, for a flat plate kept at zero angle of at tack in a uniform stream, plot ted in Figure 2.24.The characterist ic length for Reynolds number is the plate length x, from its leading edge. Itcan be seen from Figure 2.24 that the cf is more for a turbulent flow than laminar flow. Thefrict ion coefficient is defined as:

(2.57) where V∞, ρ are the freestream velocity and density, respect ively, and S is the wetted surfacearea of the flat plate.

Figure 2.24 Skin frict ion coefficient variat ion with Reynolds number.

For bluff bodies, the pressure drag is substant ially greater than the skin frict ion drag, and forstreamlined bodies the condit ion is the reverse. In the case of streamlined bodies, such asaerofoil, the designer aims at keeping the skin frict ion drag as low as possible. Maintaininglaminar boundary layer condit ions all along the surface is the most suitable arrangement tokeep the skin frict ion low. Though such aerofoils, known as laminar aerofoils, have beendesigned, they have many limitat ions. Even a small surface roughness or disturbance canmake the flow turbulent, and spoil the purpose of maintaining the laminar flow over the ent ireaerofoil. In addit ion, for laminar aerofoils there is a tendency for the flow to separate even atsmall angles of at tack, which severely restricts the use of such aerofoils.

2.12.1.3 Comparison of Drag of Various BodiesIn low-speed flow past geometrically similar bodies with ident ical orientat ion and relat iveroughness, the drag coefficient should be a funct ion of the Reynolds number only.

(2.58) The Reynolds number is based upon freestream velocity V∞ and a characterist ic length L ofthe body. The drag coefficient CD could be based upon L2, but it is customary to use acharacterist ic area S of the body instead of L2. Thus, the drag coefficient becomes:

(2.59) The factor , in the denominator of the CD expression, is our t radit ional t ribute to Euler andBernoulli. The area S is usually one of the following three types:

1. Frontal area of the body as seen from the flow stream. This is suitable for thick stubbybodies, such as spheres, cylinders, cars, missiles, project iles, and torpedos.2. Planform area of the body as seen from above. This is suitable for wide flat bodies suchas aircraft wings and hydrofoils.3. Wetted area. This is appropriate for surface ships and barges.

While using drag or other fluid (aerodynamic) force data, it is important to note what lengthand area are being used to scale the measured coefficients.

Table 2.2 gives a few data on drag, based on frontal area, of two-dimensional bodies ofvarious cross-sect ion, at Re ≥104.

T able 2.2 Drag o f two-dimensional bodies at Re ≥104 (Fluid dynamic drag, Hoerner, 1975)

Drag coefficient of sharp-edged bodies, which have a tendency to experience flowseparat ion regardless of the nature of boundary layer, are insensit ive to Reynolds number. Theellipt ic cylinders, being smoothly rounded, have the “laminar-turbulent” t ransit ion effect and are

therefore quite sensit ive to the nature of the boundary layer (that is, laminar or turbulent).Table 2.3 lists drag coefficients of some three-dimensional bodies. For these bodies also we

can conclude that sharp edges always cause flow separat ion and high drag which isinsensit ive to Reynolds number.

T able 2.3 Drag o f three-dimensional bodies at Re ≥104 (Fluid dynamic drag, Hoerner, 1975)

Rounded bodies, such as ellipsoid, have drag which depends upon the point of separat ion,so that both Reynolds number and the nature of boundary layer are important. Increase ofbody length will generally decrease the pressure drag by making the body relat ively moreslender, but sooner or later the skin frict ion drag will catch up. For a flat-faced cylinder, thepressure drag decreases with L/d but the skin frict ion drag increases, so that minimum dragoccurs at about L/d = 2, where L/d is the slenderness rat io of the body.

2.12.2 TurbulenceTurbulent flow is usually described as flow with irregular fluctuat ions. In nature, most of theflows are turbulent. Turbulent flows have characterist ics which are appreciably different fromthose of laminar flows. We have to explain all the characterist ics of turbulent flow tocompletely describe it . Incorporat ing all the important characterist ics, the turbulence may bedescribed as a three-dimensional, random phenomenon, exhibiting multiplicity of scales,possessing vorticity, and showing very high dissipation. Turbulence is described as a three-dimensional phenomenon. This means that even in a one-dimensional flow field the turbulentfluctuat ions are always three-dimensional. In other words, the mean flow may be one-or two-orthree-dimensional, but the turbulence is always three-dimensional. From the above

discussions, it is evident that turbulence can only be described and cannot be defined.A complete theoret ical approach to turbulent flow similar to that of laminar flow is impossible

because of the complexity and apparent ly random nature of the velocity fluctuat ions in aturbulent flow. Nevertheless, semi-theoret ical analysis aided by limited experimental data canbe carried out for turbulent flows, with instruments which have the capacity to detect high-frequency fluctuat ions. For flows at very low-speeds, say around 20 m/s, the frequenciesencountered will be 2 to 500 Hz. Hot-wire anemometer is well suited for measurements in suchflows. A typical hot-wire velocity t race of a turbulent flow is shown in Figure 2.25. Turbulentfluctuat ions are random, in amplitude, phase and frequency. If an instrument such as a pitot-stat ic tube, which has a low frequency response of the order of 30 seconds, is used for themeasurement of velocity, the manometer will read only a steady value, ignoring thefluctuat ions. This means that the turbulent flow consists of a steady velocity componentwhich is independent of t ime, over which the fluctuat ions are superimposed, as shown in Figure2.25(b). That is:

(2.60) where U(t) is the instantaneous velocity, is the t ime averaged velocity, and u ' (t) is theturbulent fluctuat ion around the mean velocity. Since is independent of t ime, the t imeaverage of u ' (t) should be equal to zero. That is:

provided the t ime t is sufficient ly large. In most of the laboratory flows, averaging over a fewseconds is sufficient if the main flow is kept steady.

Figure 2.25 Hot-wire t race of a turbulent flow.

In the beginning of this sect ion, we saw that the turbulence is always three-dimensional innature even if the main flow is one-dimensional. For example, in a fully developed pipe orchannel flow, as far as the mean velocity is concerned only the x-component of velocity alone exists, whereas all the three components of turbulent fluctuat ions arealways present. The intensity of the turbulent velocity fluctuat ions is expressed in the form ofits root mean square value. That is, the velocity fluctuat ions are instantaneously squared, thenaveraged over certain period and finally square root is taken. The root mean square (RMS)value is useful in est imat ing the kinet ic energy of fluctuat ions. The turbulence level for anygiven flow with a mean velocity is expressed as a turbulence number n, defined as:

(2.61)

In the laboratory, turbulence can be generated in many ways. A wire-mesh placed across anair stream produces turbulence. This turbulence is known as grid turbulence. If the incoming airstream as well as the mesh size are uniform then the turbulent fluctuat ions behind the grid areisotropic in nature, that is, u ', , are equal in magnitude. In addit ion to this, the mean velocityis the same across any cross-sect ion perpendicular to the flow direct ion, that is, no shearstress exists. As the flow moves downstream the fluctuat ions die down due to viscous effects.Turbulence is produced in jets and wakes also. The mean velocity in these flows varies andthey are known as free shear flows. Fluctuat ions exist up to some distance and then slowlydecay. Another type of turbulent flow often encountered in pract ice is the turbulent boundarylayer. It is a shear flow with zero velocity at the wall. These flows maintain the turbulence leveleven at large distance, unlike the grid or free shear flows. In wall shear flows or boundary layertype flows, turbulence is produced periodically to counteract the decay.

A turbulent flow may be visualized as a flow made up of eddies of various sizes. Largeeddies are first formed, taking energy from the mean flow. They then break up into smallerones in a sequent ial manner t ill they become very small. At this stage the kinet ic energy getsdissipated into heat due to viscosity. Mathematically it is difficult to define an eddy in a precisemanner. It represents, in a way, the frequencies involved in the fluctuat ions. Large eddy meanslow-frequency fluctuat ions and small eddy means high-frequency fluctuat ions encountered inthe flow. The kinet ic energy distribut ion at various frequencies can be represented by anenergy spectrum, as shown in Figure 2.25(c).

The problem of turbulence is yet to be solved completely. Different kinds of approach areemployed to solve these problems. The well-known method is to write the Navier-Stokesequat ions for the fluctuat ing quant it ies and then average them over a period of t ime,subst itut ing the following [in Navier-Stokes equat ions, Equat ion (2.23)]:

(2.62) (2.62)

where Vx, u ', Vy, , Vz, are the mean and fluctuat ional velocity components along x- , y-andz-direct ions, respect ively, and , p ', respect ively, are the mean and fluctuat ional componentsof pressure p. Bar denotes the mean values, that is, t ime averaged quant it ies.

Let us now consider the x-momentum equat ion [Equat ion (2.23)] for a two-dimensional flow:

(2.63) In Equat ion (2.62), ν is the kinetic viscosity, given by:

Subst itut ing Equat ion (2.62) into Equat ion (2.63), we get:

(2.64) Expanding Equat ion (2.64), we obtain:

(2.65) In this equat ion, t ime average of the individual fluctuat ions is zero. But the product or squareterms of the fluctuat ing velocity components are not zero. Taking t ime average of Equat ion(2.65), we get:

(2.66)

Equat ion (2.66) is slight ly different from the laminar Navier-Stokes equat ion (Equat ion (2.63)).The cont inuity equat ion for the two-dimensional flow under considerat ion is:

This can be expanded to result in:

(2.67a) and

(2.67b) The terms involving turbulent fluctuat ional velocit ies u ' and on the left -hand side ofEquat ion (2.66) can be writ ten as:

Using Equat ion (2.67) the above equat ion can be expressed as:

(2.68) Combinat ion of Equat ions (2.66) and (2.68) results in:

(2.69) The terms and in Equat ion (2.69) are due to turbulence. They are popularlyknown as Reynolds or turbulent stresses. For a three-dimensional flow, the turbulent stressterms are and . Solut ions of Equat ion (2.69) is rathercumbersome. Assumptions like eddy viscosity, mixing length are made to find a solut ion for thisequat ion.

At this stage, it is important to have proper clarity about the laminar and turbulent flows. Thelaminar flow may be described as “a well orderly pattern where fluid layers are assumed to slideover one another,” that is, in laminar flow the fluid moves in layers, or laminas, one layer glidingover an adjacent layer with interchange of momentum only at molecular level. Any tendenciestoward instability and turbulence are damped out by viscous shear forces that resist therelat ive mot ion of adjacent fluid layers. In other words:

“laminar flow is an orderly flow in which the fluid elements move in an orderly manner suchthat the transverse exchange of momentum is insignificant”

and

“turbulent flow is a three–dimensional random phenomenon, exhibiting multiplicity of scales,possessing vorticity, and showing very high dissipation.”

Turbulent flow is basically an irregular flow. Turbulent flow has very errat ic mot ion of fluidpart icles, with a violent t ransverse exchange of momentum.

The laminar flow, though possesses irregular molecular mot ions, is macroscopically a well-ordered flow. But in the case of turbulent flow, there is the effect of a small but macroscopicfluctuat ing velocity superimposed on a well-ordered flow. A graph of velocity versus t ime at agiven posit ion in a pipe flow would appear as shown in Figure 2.26(a), for laminar flow, and asshown in Figure 2.26(b), for turbulent flow. In Figure 2.26(b) for turbulent flow, an averagevelocity denoted as has been indicated. Because this average is constant with t ime, theflow has been designated as steady. An unsteady turbulent flow may prevail when theaverage velocity field changes with t ime, as shown in Figure 2.26(c).

Figure 2.26 Variat ion of flow velocity with t ime.

2.12.3 Flow through PipesFluid flow through pipes with circular and noncircular cross-sect ions is one of the commonlyencountered problems in many pract ical systems. Flow through pipes is driven most ly bypressure or gravity or both.

Consider the flow in a long duct, shown in Figure 2.27. This flow is constrained by the ductwalls. At the inlet , the freestream flow (assumed to be inviscid) converges and enters the tube.

Figure 2.27 Flow development in a long duct.

Because of the viscous frict ion between the fluid and pipe wall, viscous boundary layergrows downstream of the entrance. The boundary layer growth makes the effect ive area ofthe pipe to decrease progressively downstream, thereby making the flow along the pipe toaccelerate. This process cont inues up to the point where the boundary layer from the wallgrows and meets at the pipe centerline, that is, fills the pipe, as illustrated in Figure 2.27.

The zone upstream of the boundary layer merging point is called the entrance or flowdevelopment length Le and the zone downstream of the merging point is termed fully

developed region. In the fully developed region, the velocity profile remains unchanged.Dimensional analysis shows that Reynolds number is the only parameter influencing theentrance length. In the funct ional form, the entrance length can be expressed as:

where ρ, V and μ are the flow density, velocity and viscosity, respect ively, and d is the pipediameter.

For laminar flow, the accepted correlat ion is:

At the crit ical Reynolds number Rec = 2300, for pipe flow, Le = 138d, which is the maximumdevelopment length possible.

For turbulent flow the boundary layer grows faster, and Le is given by the approximaterelat ion:

(2.70) Now, examine the flow through an inclined pipe, shown in Figure 2.28, considering the control

volume between sect ions 1 and 2.

Figure 2.28 Fully developed flow in an inclined pipe.

Treat ing the flow to be incompressible, by volume conservat ion, we have:

where and , respect ively, are the volume flow rates and A1, A2, V1 and V2 are the localareas and velocit ies, at states 1 and 2. The velocit ies V1 and V2 are equal, since the flow isfully developed and also A1 = A2.

By incompressible Bernoulli's equat ion, we have:

(2.71) Since V1 = V2, we can write from Equat ion (2.71) the head loss due to frict ion as:

(2.72) where Δz = (z1 − z2) and Δp = (p1 − p2). That is, the head loss (in a pipe), due to frict ion is equal

to the sum of the change in gravity head and pressure head.By momentum balance, we have:

(2.73) Dividing throughout by (πR2)ρg, we get:

But ΔL sin θ = Δz. Thus:

Using Equat ion (2.72), we obtain:

(2.74) In the funct ional form, the wall shear may be expressed as:

(2.75) where μ is viscosity of the fluid, d is the pipe diameter, and is the wall roughness height. Bydimensional analysis, Equat ion (2.75) may be expressed as:

(2.76) where f is called the Darcy friction factor, which is a dimensionless parameter.

Combining Equat ions (2.74) and (2.76), we obtain the pipe head loss as:

(2.77) This is called the Darcy-Weisbach equat ion, valid for flow through ducts of any cross-sect ion.Further, in the derivat ion of the above relat ion, there was no ment ion about whether the flowwas laminar or turbulent and hence Equat ion (2.77) is valid for both laminar and turbulent flows.The value of frict ion factor f for any given pipe (that is, for any surface roughness and d) at agiven Reynolds number can be read from the Moody chart (which is a plot of f as a funct ion ofRed and /d).

It is essent ial to note that in our discussions here it is ment ioned that, decrease of pipe areadue to boundary layer, results in increase of flow velocity. This is possible only in subsonicflows. When the flow is supersonic, decrease in area will decelerate the flow [2].

2.13 Compressible FlowsIn the preceding sect ions of this chapter, the discussions were for incompressible flows, wherethe density can be regarded as constant. But in many engineering applicat ions, such asdesigning buildings to withstand winds, the design of engines and of vehicles of all kinds –cars,yachts, t rains, aeroplane, missiles and launch vehicles require a study of the flow with velocit iesat which the gas cannot be treated as incompressible. Indeed, the flow becomes compressible.Study of such flows where the changes in both density and temperature associated withpressure change become appreciable is called gas dynamics. In other words, gas dynamics isthe science of fluid flows where the density and temperature changes become important. Theessence of the subject of gas dynamics is that the ent ire flow field is dominated by Machwaves, expansion waves and shock waves, when the flow speed is supersonic. It is throughthese waves that the change of flow propert ies from one state to another takes place. In thetheory of gas dynamics, change of state in flow propert ies is achieved by three means: (a) witharea change, t reat ing the fluid to be inviscid and passage to be frict ionless, (b) with frict ion,t reat ing the heat t ransfer between the surrounding and system to be negligible and (c) withheat t ransfer, assuming the fluid to be inviscid. These three types of flows are called isentropic

flow, frictional or Fanno type flow and Rayleigh type flow, respect ively.All problems in gas dynamics can be classified under the three flow processes described

above, of course with the assumptions ment ioned. Although it is impossible to have a flowprocess which is purely isentropic or Fanno type or Rayleigh type, in pract ice it is just ified inassuming so, since the results obtained with these treatments prove to be accurate enoughfor most pract ical problems in gas dynamics. Even though it is possible to solve problems withmathematical equat ions and working formulae associated with these processes, it is found tobe extremely useful and t ime saving if the working formulae are available in the form of tableswith a Mach number which is the dominant parameter in compressible flow analysis.

2.13.1 Perfect GasIn principle, it is possible to do gas dynamic calculat ions with the general equat ion of staterelat ions, for fluids. But in pract ice most elementary t reatments are confined to perfect gaseswith constant specific heats. For most problems in gas dynamics, the assumption of theperfect gas law is sufficient ly in accord with the propert ies of actual gases, hence it isacceptable.

For perfect gases, the pressure-density-temperature relat ion or the thermal equation ofstate, is given by:

(2.78) where R is the gas constant and T is absolute temperature. All gases obeying the thermalstate equat ion are called thermally perfect gases. A perfect gas must obey at least twocalorical state equat ions, in addit ion to the thermal state equat ion. The cp, relat ions givenbelow are two well-known calorical state equations:

where h is specific enthalpy and u is specific internal energy, respect ively. Further, for perfectgases with constant specific heats, we have:

(2.79) where cp and are the specific heats at constant pressure and constant volume, respect ively,and γ is the isentropic index. For all real gases and γ vary with temperature, but onlymoderately. For example, cp of air increases about 30 percent as temperature increases from 0to 3000 C. Since we rarely deal with such large temperature changes, it is reasonable toassume specific heats to be constants in our studies.

2.13.2 Velocity of SoundIn the beginning of this sect ion, it was stated that gas dynamics deals with flows in which bothcompressibility and temperature changes are important. The term compressibility impliesvariat ion in density. In many cases, the variat ion in density is mainly due to pressure change.The rate of change of density with respect to pressure is closely connected with the velocityof propagat ion of small pressure disturbances, that is, with the velocity of sound “a.”

The velocity of sound may be expressed as:

(2.80) In Equat ion (2.80), the rat io dp/dρ is writ ten as part ial derivat ive at constant entropy because

the variat ions in pressure and temperature are negligibly small, and consequent ly, the processis nearly reversible. Moreover, the rapidity with which the process takes place, together withthe negligibly small magnitude of the total temperature variat ion, makes the process nearlyadiabat ic. In the limit , for waves with infinitesimally small thickness, the process may beconsidered both reversible and adiabat ic, and thus, isentropic.

It can be shown that, for an isentropic process of a perfect gas, the velocity of sound can beexpressed as:

(2.81) where T is absolute stat ic temperature.

2.13.3 Mach NumberMach number M is a dimensionless parameter, expressed as the rat io between themagnitudes of local flow velocity and local velocity of sound, that is:

(2.82) Mach number plays a dominant role in the field of gas dynamics.

2.13.4 Flow with Area ChangeIf the flow is assumed to be isentropic for a channel flow, all states along the channel or streamtube lie on a line of constant entropy and have the same stagnat ion temperature. The state ofzero velocity is called the isentropic stagnation state, and the state with M = 1 is called thecritical state.

2.13.4.1 Isentropic RelationsThe relat ions between pressure, temperature, and density for an isentropic process of aperfect gas are:

(2.83a)

(2.83b) Also, the pressure-temperature density relat ion of a perfect gas is:

(2.84) The temperature, pressure, and density rat ios as funct ions of Mach number are:

(2.85)

(2.86)

(2.87) where T0, p0 and ρ0 are the temperature, pressure and density, respect ively, at the stagnat ionstate. The part icular value of temperature, pressure, and density rat ios at the crit ical state(that is, at the choked locat ion in a flow passage) are found by sett ing M = 1 in Equat ions

(2.85)–(2.87). For γ = 1.4, the following are the temperature, pressure and density rat io at thecrit ical state:

(2.88)

(2.89)

(2.90) where T∗, p∗ and ρ∗ are the temperature, pressure and density, respect ively at the crit icalstate.

The crit ical pressure rat io p∗/p0 is of the same order of magnitude for all gases. It variesalmost linearly with γ from 0.6065, for γ = 1, to 0.4867, for γ = 1.67.

T he dimensionless velocity M∗ is one of the most useful parameter in gas dynamics.Generally it is defined as:

(2.91) where a* is the crit ical speed of sound. This dimensionless velocity can also be expressed interms of Mach number as:

(2.92)

2.13.4.2 Area-Mach Number RelationFor an isentropic flow of a perfect gas through a duct, the area-Mach number relat ion may beexpressed, assuming one-dimensional flow, as:

(2.93) where A* is called the sonic or crit ical throat area.

2.13.4.3 Prandtl-Meyer FunctionThe Prandt l-Meyer funct ion ν is an important parameter to solve supersonic flow problemsinvolving isentropic expansion or isentropic compression. Basically the Prandt l-Meyer funct ionis a similarity parameter. The Prandt l-Meyer funct ion can be expressed in terms of M as:

(2.94) From Equat ion (2.94) it is seen that, for a given M, ν is fixed.

2.13.5 Normal Shock RelationsThe shock may be described as a compression front, in a supersonic flow field, across whichthe flow propert ies jump. The thickness of the shock is comparable to the mean free path ofthe gas molecules in the flow field. When the shock is normal to the flow direct ion it is callednormal shock, and when it is inclined at an angle to the flow it is termed oblique shock. For aperfect gas, it is known that all the flow property rat ios across a normal shock are uniquefunct ions of specific heats rat io, γ, and the upstream Mach number, M1.

Considering the normal shock shown in Figure 2.29, the following normal shock relat ions,assuming the flow to be one-dimensional, can be obtained:

assuming the flow to be one-dimensional, can be obtained:

(2.95)

(2.96)

(2.97)

(2.98) In Equat ion (2.98), h1 and h2 are the stat ic enthalpies upstream and downstream of the shock,respect ively.

Figure 2.29 Flow through a normal shock.

The stagnat ion pressure rat io across a normal shock, in terms of the upstream Machnumber is:

(2.99) The change in entropy across the normal shock is given by:

(2.100)

2.13.6 Oblique Shock RelationsConsider the flow through an oblique shock wave, as shown in Figure 2.30.

Figure 2.30 Flow through an oblique shock.

The component of M1 normal to the shock wave is:

(2.101) where β is the shock angle. The shock in Figure 2.29 can be visualized as a normal shock withupstream Mach number M1 sin β. Thus, replacement of M1 in the normal shock relat ions,Equat ions (2.95) to (2.99), by M1 sin β, results in the corresponding relat ions for the obliqueshock.

(2.102)

(2.103)

(2.104)

(2.105)

(2.106) The entropy change across the oblique shock is given by:

Equat ion (2.102) gives only the normal component of Mach number Mn2 behind the shock. Butthe Mach number of interest is M2. It can be obtained from Equat ion (2.102) as follows:

From the geometry of the oblique shock flow field, it is seen that M2 is related to Mn2 by:

(2.107) where θ is the flow turning angle across the shock. Combining Equat ions (2.102) and (2.107),the Mach number M2 after the shock can be obtained.

2.13.7 Flow with FrictionIn the Sect ion 2.13.4, on flow with area change, it was assumed that the changes in flowpropert ies, for compressible flow of gases in ducts, were brought about solely by area change.That is, the effects of viscosity and heat t ransfer have been neglected. But, in pract ical flowsituat ions like, stat ionary power plants, aircraft engines, high vacuum technology, t ransport ofnatural gas in long pipe lines, t ransport of fluids in chemical process plants, and various typesof flow systems, the high-speed flow travels through passages of considerable length andhence the effects of viscosity (frict ion) cannot be neglected for such flows. In many pract icalflow situat ions, frict ion can even have a decisive effect on the resultant flow characterist ics.

Consider one-dimensional steady flow of a perfect gas with constant specific heats througha constant area duct. Assume that there is neither external heat exchange nor external shaftwork and the difference in elevat ion produces negligible changes in flow propert ies comparedto frict ional effects. The flow with the above said condit ions, namely adiabat ic flow with noexternal work, is called Fanno line flow. For Fanno line flow, the wall frict ion (due to viscosity) isthe chief factor bringing about changes in flow propert ies.

Working Formulae for Fanno Type FlowConsider the flow of a perfect gas through a constant area duct shown in Figure 2.31.Choosing an infinitesimal control volume as shown in the figure, the relat ion between Machnumber M and frict ion factor f can be writ ten as:

(2.108) In this relat ion, the integrat ion limits are taken as (1) the sect ion where the Mach number is M,

and the length x is arbit rarily set equal to zero, and (2) the sect ion where Mach number is unityand x is the maximum possible length of duct, L max and D is the hydraulic diameter, defined as:

On integrat ion, Equat ion (2.108) yields:

(2.109) where is the mean frict ion coefficient with respect to duct length, defined by:

Figure 2.31 Control volume for Fanno flow.

Likewise, the local flow propert ies can be found in terms of local Mach number. Indicat ing thepropert ies at M = 1 with superscripted with “asterisk,” and integrat ing between the ductsect ions with M = M and M = 1, the following relat ions can be obtained [2]:

(2.110)

(2.111)

(2.112)

(2.113)

(2.114)

(2.115) In Equat ion (2.115), the parameter F is called Impulse Function, defined as:

2.13.8 Flow with Simple T0-ChangeIn the sect ion on flow with area change, the process was considered to be isentropic with theassumption that the frict ional and energy effects were absent. In Fanno line flow, only theeffect of wall frict ion was taken into account in the absence of area change and energyeffects. In the present sect ion, the processes involving change in the stagnat ion temperatureor the stagnat ion enthalpy of a gas stream, which flows in a frict ionless constant area duct areconsidered. From a one-dimensional point of view, this is yet another effect producingcont inuous changes in the state of a flowing stream and this factor is called energy effect ,such as external heat exchange, combust ion, or moisture condensat ion. Though a processinvolving simple stagnat ion temperature (T0) change is difficult to achieve in pract ice, manyuseful conclusions of pract ical significance may be drawn by analyzing the process of simpleT0-change. This kind of flow involving only T0-change is called Rayleigh type flow.

Working Formulae for Rayleigh Type FlowConsider the flow of a perfect gas through a constant-area duct without frict ion, shown inFigure 2.32.

Figure 2.32 Control volume for Rayleigh flow.

Considering a control volume, as in Figure 2.32, the normalized expressions (workingformulae) for the flow process involving only heat t ransfer can be obtained as [2].

(2.116)

(2.117)

(2.118)

(2.119)

(2.120)

2.14 SummaryGases and liquids are generally termed fluids. Under dynamic condit ions, the nature ofgoverning equat ions is the same for both gases and liquids.

Fluid may be defined as a substance which will continue to change shape as long as there isa shear stress present, however small it may be.

Pressure may be defined as the force per unit area which acts normal to the surface of anyobject which is immersed in a fluid. For a fluid at rest , at any point the pressure is the same inall direct ions. In stat ionary fluids the pressure increases linearly with depth. This linear pressuredistribut ion is called hydrostatic pressure distribution.

When a fluid is in mot ion, the actual pressure exerted by the fluid in the direct ion normal tothe flow is known as the static pressure. The pressure which a fluid flow will experience if it isbrought to rest , isentropically, is termed total pressure. The total pressure is also called impactpressure. The total and stat ic pressures are used for comput ing flow velocity.

The total number of molecules in a unit volume is a measure of the density ρ of a substance.It is expressed as mass per unit volume, say kg/m3. Mass is defined as weight divided byaccelerat ion due to gravity. At standard atmospheric temperature and pressure (288.15 K and101325 Pa, respect ively), the density of dry air is 1.225 kg/m3.

The property which characterizes the resistance that a fluid offers to applied shear force istermed viscosity. This resistance, unlike for solids, does not depend upon the deformat ion itselfbut on the rate of deformation.

Maxwell's definit ion of viscosity states that:“the coefficient of viscosity is the tangent ial force per unit area on either of two parallel

plates at unit distance apart , one fixed and the other moving with unit velocity.”Newton's law of viscosity states that “the stresses which oppose the shearing of a fluid are

proportional to the rate of shear strain,” that is, the shear stress τ is given by:

Fluids which obey the above law of viscosity are termed Newtonian fluids. Some fluids such assilicone oil, viscoelast ic fluids, sugar syrup, tar, etc. do not obey the viscosity law given byEquat ion (2.3) and they are called non-Newtonian fluids.

For air the viscosity coefficient is expressed as:

where T is in kelvin.The kinematic viscosity coefficient is a convenient form of expressing the viscosity of a fluid.

It is formed by combining the density ρ and the absolute coefficient of viscosity μ, according tothe equat ion:

The kinematic viscosity coefficient ν is expressed as m2/s, and 1 cm2/s is known as stoke.The kinematic viscosity coefficient is a measure of the relat ive magnitudes of viscosity and

inert ia of the fluid.

The change in volume of a fluid associated with change in pressure is called compressibility.The specific heats at constant volume and constant pressure processes, respect ively, are

designated by and cp. The definit ions of these quant it ies are the following:

where u is internal energy per unit mass of the fluid, which is a measure of the potent ial andmore part icularly the kinet ic energy of the molecules comprising the gas. The specific heat isa measure of the energy-carrying capacity of the gas molecules. For dry air at normaltemperature, = 717.5 J/(kg K).

The specific heat at constant pressure is defined as:

The rat io of specific heats:

is an important parameter in the study of high-speed flows. This is a measure of the relativeinternal complexity of the molecules of the gas.

Liquids behave as if their free surfaces were perfect ly flexible membranes having a constanttension σ per unit width. This tension is called the surface tension. It is important to note thatthis is neither a force nor a stress but a force per unit length.

Basically two treatments are followed for fluid flow analysis. They are the Lagrangian andEulerian descript ions. Lagrangian method describes the mot ion of each part icle of the flowfield in a separate and discrete manner.

If propert ies and flow characterist ics at each posit ion in space remain invariant with t ime, theflow is called steady flow. A t ime-dependent flow is referred to as unsteady flow.

The rate of change of a property measured by probes at fixed locat ions are referred to aslocal rates of change, and the rate of change of propert ies experienced by a material part icle istermed the material or substantive rates of change.

For a fluid flowing with a uniform velocity V∞, it is possible to write the relat ion between thelocal and material rates of change of property η as:

when η is the velocity of a fluid part icle, DV/Dt gives accelerat ion of the fluid part icle and theresultant equat ion is:

This is known as Euler's acceleration formula.Pathline may be defined as a line in the flow field describing the trajectory of a given fluid

part icle.Streakline may be defined as the instantaneous loci of all the fluid elements that have

passed the point of inject ion at some earlier t ime.Streamlines are imaginary lines, in a fluid flow, drawn in such a manner that the flow velocity

is always tangent ial to it .In modern fluid flow analysis, yet another graphical representat ion, namely timeline is used.

When a pulse input is periodically imposed on a line of t racer source placed normal to a flow, achange in the flow profile can be observed. The tracer image is generally termed t imeline.

In the range of engineering interest , four basic laws must be sat isfied for any cont inuousmedium. They are:

Conservat ion of matter (cont inuity equat ion).

Newton's second law (momentum equat ion).Conservat ion of energy (first law of thermodynamics).Increase of entropy principle (second law of thermodynamics).

In addit ion to these primary laws, there are numerous subsidiary laws, somet imes calledconst itut ive relat ions, that apply to specific types of media or flow processes (for example,equat ion of state for perfect gas, Newton's viscosity law for certain viscous fluids, isentropicand adiabat ic process relat ions are some of the commonly used subsidiary equat ions in flowphysics).

A control mass system is an ident ified quant ity of matter, which may change shape, posit ion,and thermal condit ion, with t ime or space or both, but must always entail the same matter.

A control volume is a designated volume in space, and the boundary of this volume is knownas control surface. The amount and ident ity of the matter in the control volume may changewith t ime, but the shape of the control volume is fixed, that is, the control volume may changeits posit ion in t ime or space or both, but its shape is always preserved.

The analysis in which large control volumes are used to obtain the aggregate forces ortransfer rates is termed integral analysis. When the analysis is applied to individual points inthe flow field, the result ing equat ions are different ial equat ions, and the method is termeddifferential analysis.

For air at normal temperature and pressure, the density ρ, pressure p and temperature T areconnected by the relat ion p = ρRT, where R is a constant called gas constant. This is knownas the thermal equation of state. An ideal gas is frictionless and incompressible. The perfectgas has viscosity and can therefore develop shear stresses, and it is compressible according tostate equat ion.

The basic governing equat ions for an incompressible flow are the cont inuity and momentumequat ions. For steady incompressible flow, the cont inuity equat ion in different ial form is:

For steady incompressible flows, the momentum equat ion can be writ ten as:

These equat ions are generally known as Navier–Stokes equat ions.Boundary layer thickness δ may be defined as the distance from the wall in the direct ion

normal to the wall surface, where the fluid velocity is within 1% of the local main streamvelocity. It may also be defined as the distance δ, normal to the surface, in which the flowvelocity increases from zero to some specified value (for example, 99%) of its local mainstream flow velocity.

Displacement thickness δ∗ may be defined as the distance by which the boundary wouldhave to be displaced if the ent ire flow fields were imagined to be frict ionless and the samemass flow is maintained at any sect ion.

T h e momentum thickness θ and energy thickness δe are other (thickness) measurespertaining to boundary layer. They are defined mathematically as follows:

Transit ion point may be defined as the end of the region at which the flow in the boundarylayer on the surface ceases to be laminar and becomes turbulent.

Separat ion point is the posit ion at which the boundary layer leaves the surface of a solidbody. If the separat ion takes place while the boundary layer is st ill laminar, the phenomenon istermed laminar separation. If it takes place for a turbulent boundary layer it is called turbulentseparation.

Circulation Γ, is defined as the line integral of velocity vector between any two points (todefine rotat ion of the fluid element) in a flow field. By definit ion:

Circulat ion per unit area is known as vorticity ζ:

In vector form, ζ becomes:

For a two-dimensional flow in xy-plane, vort icity ζ becomes:

where ζz is the vort icity about the z-direct ion, which is normal to the flow field. Likewise, theother components of vort icity about x-and y-direct ions are:

If ζ = 0, the flow is known as irrotational flow. Inviscid flows are basically irrotat ional flows.In terms of stream funct ion ψ, the velocity components of a two-dimensional incompressible

flow are given as:

If the flow is compressible the velocity components become:

For irrotat ional flows (the fluid elements in the field are free from rotat ion), there exists afunct ion ϕ called velocity potential or potential function. For a steady two-dimensional flows, ϕmust be a funct ion of two space coordinates (say, x and y). The velocity components are givenby:

We can relate ψ and ϕ as:

These relat ions between stream funct ion and potent ial funct ion are the famous Cauchy-Riemann equations of complex-variable theory.

Potent ial flow is based on the concept that the flow field can be represented by a potent ialfunct ion ϕ such that:

This linear part ial different ial equat ion is popularly known as Laplace equation.All inviscid flows must sat isfy the irrotat ionality condit ion:

For two-dimensional incompressible flows, the cont inuity equat ion is:

In terms of the potent ial funct ion ϕ, this becomes:

that is:

This linear equat ion is the governing equat ion for potent ial flows.For potent ial flows, the Navier-Stokes equat ions reduce to:

These are known as Euler's equations.Among the graphical representat ion concepts namely, the pathline, streakline andstreamline, only the first two are physical, and the concept of streamline is onlyhypothet ical. But even though imaginary, the streamline is the only useful concept,because it gives a mathematical representat ion for the flow field in terms of streamfunct ion ψ, with its derivat ives giving the velocity components.The fundamental solut ions of Laplace equat ion forms the basis for both experimental andcomputat ion flow physics. The basic solut ions for the Laplace equat ion are the uniformflow, source, sink and free or potential vortex. These solut ions being potent ial, can besuperposed to get the mathematical funct ions represent ing any pract ical geometry ofinterest .

Source is a potent ial flow field in which flow emanat ing from a point spreads radiallyoutwards. Sink is potent ial flow field in which flow gushes towards a point from all radialdirect ions.

The velocity potent ial for a two-dimensional source of strength becomes:

In a similar manner as above, the stream funct ion for a source of strength can be obtained as:

A simple or free vortex is a flow field in which the fluid elements simply move along concentriccircles, without spinning about their own axes. That is, the fluid elements have only t ranslatorymot ion in a free vortex. In addit ion to moving along concentric paths, if the fluid elements spinabout their own axes, the flow is termed forced vortex: For a simple vortex:

The stream funct ion for a simple vortex is:

A doublet or a dipole is a potent ial flow field due to a source and sink of equal strength,brought together in such a way that the product of their strength and the distance betweenthem remain constant. The velocity potent ial for a doublet is:

The stream funct ion for a doublet is:

The stream funct ion for the flow due to the combinat ion of a source of strength at theorigin, immersed in a uniform flow of velocity V∞, parallel to x-axis is:

The streamline passing through the stagnat ion point S is termed the stagnat ion streamline.The stagnat ion streamline resembles a semi-ellipse. This shape is popularly known asRankine's half-body.

The stream funct ion and potent ial funct ion of the flow past a cylinder can be expressed as:

The nondimensional pressure distribut ion over the surface of the cylinder is given by:

The symmetry of the pressure distribut ion in an irrotat ional flow implies that “a steadily movingbody experiences no drag.” This result , which is not t rue for actual (viscous) flows where thebody experiences drag, is known as d’Alembert's paradox.

The posit ive limit of +1 for Cp, at the forward stagnat ion point , is valid for all geometriesand for both potent ial and viscous flow, as long as the flow speed is subsonic.The limit ing minimum of −3, for the Cp over the cylinder in potent ial flow, is valid only forcircular cylinder. The negat ive value of Cp can take values lower than −3 for othergeometries. For example, for a cambered aerofoil at an angle of incidence can have Cp aslow as −6.

There is no net force act ing on a circular cylinder in a steady irrotat ional flow withoutcirculat ion. It can be shown that a lateral force ident ical to a lift force on an aerofoil, resultswhen circulat ion is introduced around the cylinder.

The locat ion forward and rear stagnat ion points on the cylinder can be adjusted bycontrolling the magnitude of the circulat ion Γ. The circulat ion which posit ions the stagnat ionpoints in proximity, as shown in Figure 2.19(b) is called subcritical circulation, the circulat ionwhich makes the stagnat ion points to coincide at the surface of the cylinder, as shown inFigure 2.19(c), is called critical circulation, and the circulat ion which makes the stagnat ionpoints to coincide and take a posit ion outside the surface of the cylinder, as shown in Figure2.19(d), is called supercritical circulation.

For a circular cylinder in a potent ial flow, the only way to develop circulat ion is by rotat ing it ina flow stream. Although viscous effects are important in this case, the observed pattern forhigh rotat ional speeds displays a striking similarity to the ideal flow pattern for Γ > 4πaV∞.When the cylinder rotates at low speeds, the retarded flow in the boundary layer is not able to

overcome the adverse pressure gradient behind the cylinder. This leads to the separat ion ofthe real (actual) flow, unlike the irrotat ional flow which does not separate. However, even in thepresence of separat ion, observed speeds are higher on the upper surface of the cylinder,implying the existence of a lift force.

A second reason for a rotat ing cylinder generat ing lift is the asymmetry to the flow pattern,caused by the delayed separat ion on the upper surface of the cylinder. The asymmetry resultsin the generat ion of the lift force. The contribut ion of this mechanism is small for two-dimensional objects such as circular cylinder, but it is the only mechanism for the side forceexperienced by spinning three-dimensional objects such as soccer, tennis and golf balls. Thelateral force experienced by rotat ing bodies is called the Magnus effect. The horizontalcomponent of the force on the cylinder, due to the pressure, in general is called drag. For thecylinder, shown in Figure 2.20, the drag given by:

It is interest ing to note that the drag is equal to zero. It is important to realize that this result isobtained on the assumption that the flow is inviscid. In real (actual or viscous) flows thecylinder will experience a finite drag force act ing on it due to viscous frict ion and flowseparat ion.

We are familiar with the fact that the viscosity produces shear force which tends to retardthe fluid mot ion. It works against inert ia force. The rat io of these two forces governs (dictates)many propert ies of the flow, and the rat io expressed in the form of a nondimensionalparameter is known as the famous Reynolds number, ReL:

The Reynolds number plays a dominant role in fluid flow analysis. This is one of thefundamental dimensionless parameters which must be matched for similarity considerat ions inmost of the fluid flow analysis. At high Reynolds numbers, the inert ia force is predominantcompared to viscous forces. At low Reynolds numbers the viscous effects predominateeverywhere. Whereas, at high Re the viscous effects confine to a thin region, just adjacent tothe surface of the object present in the flow, and this thin layer is termed boundary layer.

Reynolds number is basically a similarity parameter. It is used to determine the laminar andturbulent nature of flow.

Lower critical Reynolds number is that Reynolds number below which the ent ire flow islaminar.Upper critical Reynolds number is that Reynolds number above which the ent ire flow isturbulent.Critical Reynolds number is that at which the flow field is a mixture of laminar andturbulent flows.

When a body moves in a fluid, it experiences forces and moments due to the relat ive mot ionof the flow taking place around it . The force on the body along the flow direct ion is called drag.

The drag is essent ially a force opposing the mot ion of the body. Viscosity is responsible for apart of the drag force, and the body shape generally determines the overall drag. The dragcaused by the viscous effect is termed the frict ional drag or skin friction. In the design oft ransport vehicles, shapes experiencing minimum drag are considered to keep the powerconsumption at a minimum. Low drag shapes are called streamlined bodies and high dragshapes are termed bluff bodies.

Drag arises due to (a) the difference in pressure between the front and back regions and (b)the frict ion between the body surface and the fluid. Drag force caused by the pressureimbalance is known as pressure drag, and (b) the drag due to frict ion is known as skin frictiondrag or shear drag.

The locat ion where the flow leaves the body surface is termed separation point. For flow

past a cylinder, there are two separat ion points on either side of the horizontal axis throughthe center of the cylinder. The separated flow is chaot ic and vortex dominated. The separatedflow behind an object is also referred to as wake. Depending on the Reynolds number level, thewake may be laminar or turbulent. An important characterist ics of the separated flow is that itis always unsymmetrical, even for laminar separat ion.

The frict ion between the surface of a body and the fluid causes viscous shear stress andthis force is known as skin friction drag. Wall shear stress τ at the surface of a body is given by:

A body for which the skin frict ion drag is a major port ion of the total drag is termedstreamlined body. A body for which the pressure (form) drag is the major port ion is termed bluffbody. Turbulent boundary layer results in more skin frict ion than a laminar one.

The turbulence level for any given flow with a mean velocity is expressed as a turbulencenumber n, defined as:

“A laminar flow is an orderly flow in which the fluid elements move in an orderly manner suchthat the transverse exchange of momentum is insignificant”

and

“A turbulent flow is a three–dimensional random phenomenon, exhibiting multiplicity ofscales, possessing vorticity, and showing very high dissipation.”

Flow through pipes is driven most ly by pressure or gravity or both. In the funct ional form, theentrance length can be expressed as:

For laminar flow, the accepted correlat ion is:

At the crit ical Reynolds number Rec = 2300, for pipe flow, Le = 138d, which is the maximumdevelopment length possible.

For turbulent flow the boundary layer grows faster, and Le is given by the approximaterelat ion:

The pipe head loss is given by:

This is called the Darcy-Weisbach equat ion, valid for flow through ducts of any cross-sect ion.Exercise Problems

1. The turbulence number of a uniform horizontal flow at 25 m/s is 6. If the turbulence isisotropic, determine the mean square values of the fluctuat ions.

[Answer: 6.75 m2/s2]

2. Flow through the convergent nozzle shown in Figure 2.33 is approximated as one-dimensional. If the flow is steady will there be any fluid accelerat ion? If there is accelerat ion,

obtain an expression for it in terms of volumetric flow rate , if the area of cross-sect ion isgiven by A(x) = e−x.

[Answer: ]

Figure 2.33 Flow through a convergent nozzle.

3. Atmospheric air is cooled by a desert cooler by 18 C and sent into a room. The cooled airthen flows through the room and picks up heat from the room at a rate of 0.15 C/s. The airspeed in the room is 0.72 m/s. After some t ime from switching on, the temperature gradientassumes a value of 0.9 C/m in the room. Determine ∂T/∂ t at a point 3 m away from thecooler.

[Answer: − 0 . 498 C/s]

4. For proper funct ioning, an electronic instrument onboard a balloon should not experiencetemperature change of more than ± 0.006 K/s. The atmospheric temperature is given by:

where z is the height in meters above the ground and t is the t ime in hours after sunrise.Determine the maximum allowable rate of ascent when the balloon is at the ground at t = 2hr.

[Answer: 1.12 m/s]

5. Flow through a tube has a velocity given by:

where R is the tube radius and u max is the maximum velocity, which occurs at the tubecenterline. (a) Find a general expression for volume flow rate and average velocity throughthe tube, (b) compute the volume flow rate if R = 25 mm and u max = 10 m/s, and (c) computethe mass flow rate if ρ = 1000 kg/m3.

[Answer: (a) , , (b) 0.00982 m3/s, (c) 9.82 kg/s].

6. A two-dimensional velocity field is given by:

in arbit rary units. At x = 2 and y = 1, compute (a) the accelerat ion components ax and ay, (b)the velocity component in the direct ion θ = 30 , and (c) the direct ions of maximum velocityand maximum accelerat ion.

[Answer: (a) −7 units, 17 units, (b) 2.87 units, (c) V = 4.123 units at 75.96 from x-axis, a =18.385 units at 292.38 from x-axis]

7. A tank is placed on an elevator which starts moving upwards at t ime t = 0 with a constantaccelerat ion a. A stat ionary hose discharges water into the tank at a constant rate as shownin Figure 2.34. Determine the t ime required to fill the tank if it is empty at t = 0.

[Answer: ]

Figure 2.34 A tank on an elevator moving up.

8. Develop the different ial form of cont inuity equat ion for cylindrical polar coordinates shownby taking an infinitesimal control volume, as shown in Figure 2.35.

[Answer: ]

Figure 2.35 Cylindrical polar coordinates.

9. A flow field is given by:

(a) Find the velocity at posit ion (10,6) at t = 3 s. (b) What is the slope of the streamlines forthis flow at t = 0 s? (c) Determine the equat ion of the streamlines at t = 0 up to an arbit raryconstant. (d) Sketch the streamlines at t = 0.

[Answer: (a) V = (30 i + 24 j − 15 k) m/s, (b) 4y/3x, (c) , where c is an arbit raryconstant, (d) At t = 0, the streamlines are straight lines at an angle of 38.66 to the x-axis]

10. For the fully developed two-dimensional flow of water between two impervious flat plates,shown in Figure 2.36, show that Vy = 0 everywhere.

Figure 2.36 Fully developed two-dimensional flow between two impervious flat plates.

11. Water enters sect ion 1 at 200 N/s and exits at 30 angle at sect ion 2, as shown in Figure2.37. Sect ion 1 has a laminar velocity profile , while sect ion 2 has a turbulent

profile . If the flow is steady and incompressible (water), find the maximumvelocit ies um1 and um2 in m/s. Assume uav = 0.5 um, for laminar flow, and uav = 0.82 um, forturbulent flow.

[Answer: 5.2 m/s, 8.79 m/s]

Figure 2.37 Flow through a passage with bend.

12. Consider a jet of fluid directed at the inclined plate shown in Figure 2.38. Obtain the forcenecessary to hold the plate in equilibrium against the jet pressure. Also, obtain the volumeflow rates and in terms of the incoming flow rate . Assume that V0 = V1 = V2 andthe fluid is inviscid.

[Answer: ]

Figure 2.38 Jet impingement on an inclined plate.

13. Consider a laminar fully developed flow without body forces through a long straight pipeof circular cross–sect ion (Poiseuille flow) shown in Figure 2.39. Apply the momentumequat ion and show that:

Assuming (p1 − p2)/l = constant, obtain the velocity profile using the relat ion:

[Answer: ]

Figure 2.39 Fully developed flow in a pipe.

14. A liquid of density ρ and viscosity μ flows down a stat ionary wall, under the influence ofgravity, forming a thin film of constant thickness h, as shown in Figure 2.40. An up flow of airnext to the film exerts an upward constant shear stress τ on the surface of the liquid layer, asshown in the figure. The pressure in the film is uniform. Derive expressions for (a) the filmvelocity Vy as a funct ion of y, ρ, μ, h and τ, and (b) the shear stress τ that would result in a

zero net volume flow rate in the film.

[Answer: (a) , (b) ]

Figure 2.40 Flow down a stat ionary wall.

15. Show that the head loss for laminar, fully developed flow in a straight circular pipe is givenby:

where Re is the Reynolds number defined as .

16. A horizontal pipe of length L and diameter D conveys air. Assuming the air to expandaccording to the law p/ρ = constant and that the accelerat ion effects are small, prove that:

where is the mass flow rate of air through the pipe, f is the average frict ion coefficient , and1 and 2 are the inlet and discharge ends of the pipe, respect ively.

17. In the boundary layer over the upper surface of an airplane wing, at a point A near theleading edge, the flow velocity just outside the boundary layer is 250 km/hour. At anotherpoint B, which is downstream of A, the velocity outside the boundary layer is 470 km/hour. Ifthe temperature at A is 288 K, calculate the temperature and Mach number at point B.

[Answer: 281.9 K, 0.388]

18. A long right circular cylinder of diameter a meter is set horizontally in a steady stream ofvelocity u m/s and made to rotate at an angular velocity of ω radians/second. Obtain anexpression in terms of ω and u for the rat io of pressure difference between the top andbottom of the cylinder to the dynamic pressure of the stream.

[Answer: ]

19. The velocity and temperature fields of a fluid are given by:

Find the rate of change of temperature recorded by a float ing probe (thermocouple) when itis at 3 i + 5 j + 2 k at t ime t = 2 units.

[Answer: 1808]

20. A parachute of 10 m diameter when carrying a load W descends at a constant velocity of5.5 m/s in atmospheric air at a temperature of 18 and a pressure of 105 Pa. Determine theload W if the drag coefficient for the parachute is 1.4.

[Answer: 1.991 kN]

References1. Heiser, W.H. and Prat t , D.T., Hypersonic Air Breathing Propulsion, AIAA Educat ion Series,1994.2. Rathakrishnan, E., Applied Gas Dynamics, John Wiley & Sons Inc., New Jersey, 2010.

3

Conformal Transformation

3.1 IntroductionThe transformat ion technique which transforms an orthogonal geometric pattern (Figure3.1(a)), composed of elements of certain shape, into an ent irely different pattern (Figure 3.1(b)),whilst the elements retain their form and proport ion is termed conformal transformation.

Figure 3.1 Conformal t ransformat ion.

3.2 Basic PrinciplesAs shown in Figure 3.1, the elements will, in the limit , retain their similar geometrical form. Forthis to be true, the angle between the intersect ing lines in plane 1 must remain the same whenthe two lines are t ransformed to plane 2. Let us examine the point p in the (x, iy)-plane (z-plane), referred to as physical plane and the corresponding point P in the (ξ, iη)-plane (ζ-plane),called transformed plane, shown in Figure 3.2.

Figure 3.2 Transformat ion of a general point .

In the z-plane (physical plane) point p is located by z = x + iy, and in the ζ-plane (t ransformedplane), the corresponding point P is located by ζ = ξ + iη. The relat ion between z and ζ is apart icular specified funct ion of ζ, in terms of z. That is:

This funct ion is known as the transformation function.Consider the specific points, located at z1 and z2, on an arc segment p1p2 in the physical

plane, as shown in Figure 3.3(a). The corresponding points in the t ransformed plane are ζ1 andζ2 and the arc segment p1p2 in the z-plane is t ransformed to curve P1P2, shown in Figure3.3(b).

Figure 3.3 Transformat ion of an arc segment.

For t ransforming the points in the z-plane to ζ-plane, the t ransformat ion funct ion used is:

(3.1) Different iat ing Equat ion (3.1), with respect to z, we get:

(3.2) In the limit of arc length p1p2 → 0, δz → dz and in the limit of arc length P1P2 → 0, δζ → dζ.From Equat ion (3.2), it is seen that the length dζ of the segment, in the t ransformed plane,becomes the vector dz, in the physical plane, mult iplied by the vector f ' (z), that is:

Now, to understand this operat ion of the mult iplicat ion of vectors, consider the funct ion f(z)rewrit ten in its exponent ial form, that is:

where r is the modulus of funct ion f(z). Then:

is in the direct ion of dz, after it has been rotated through θ, and the angular displacement off(z) (of the t ransformed element) is equal to the the length of the original element rotatedthrough angle θ and mult iplied by r. The shape of the t ransformed element is given by P1P2, asshown in Figure 3.3(b), and not by Equat ion (3.2).

Consider the arc segments ab and cd, cut t ing each other at point p in the z-plane, as shownin Figure 3.4(a). At point p the angle subtended by the crossing arc ab and cd is β. In thetransformed plane (ζ-plane), in Figure 3.4(b), the corresponding point is P and the transformedcurves AB and CD are crossing with the same angle β, in accordance with the conformaltransformat ion, which st ipulates that the “angle subtended by two crossing arcs in thephysical plane and the angle subtended by the corresponding transformed curves in thetransformed plane must be the same.”

Figure 3.4 Transformat ion of crossing arc segments.

Let us consider the actual elements of the crossing arc segments. Since the transformedelements are crossing at point P, with the same angle of intersect ion as in the z-plane, theirlengths would be affected by the same value of the t ransformat ion funct ion f(z), in thetransformat ion. Therefore:

where f(z) = r eiθ.In the t ransformat ion, both the elements of the crossing arc segments are rotated through

the same angle. Therefore, the angle of intersect ion must remain unchanged during thetransformat ion, that is:

This method can be used to show that a small element abcd in z-plane is t ransformed to ageometrically similar element ABCD in the ζ-plane, as shown in Figure 3.5.

Figure 3.5 Transformat ion of an element from z-plane to ζ-plane.

This type of t ransformat ion sat isfies the condit ion required for conformal t ransformat ion.The transformat ion funct ion is essent ially of the vector type:

where z = x + iy and ζ = ξ + iη.A general form of the t ransformat ion funct ion is:

where A0, A1, etc. and B1, B2, etc. are constants and vectors or combinat ions of constants andvectors, respect ively.

3.2.1 Length Ratios between the Corresponding Elements inthe Physical and Transformed Planes

The length rat io of corresponding elements in the z-and ζ-planes is given by:

The actual length of an element is the modulus of the vector δz. Thus:

(3.3)

3.2.2 Velocity Ratios between the Corresponding Elementsin the Physical and Transformed Planes

The velocity qz at any point p in the z-plane is given by:

where is the complex potent ial at that point , ψ is the stream funct ion and ϕ is thepotent ial funct ion. But with reference to the new (transformed) coordinate axes, the localvelocity at point P is:

At the corresponding points between the original plane (z-plane) and the transformed plane (ζ-plane), considering only the magnitudes, we can express:

(3.4) It is seen that the velocity rat io between corresponding points in the original and transformedplanes is the inverse of the length rat io.

3.2.3 SingularitiesThe relat ion between the corresponding elements in the physical and transformed planes isadequately defined by:

In most situat ions, the correspondence between the elements is the modulus and argument ofthe vector:

as out lined in the previous sect ions. This arrangement clearly breaks down where f ' (z) = dζ/dzis zero or infinite. In both the cases, the conformability of the t ransformat ion is lost . The pointsat which dζ/dz = 0 or ∞, in any transformat ion, are known as singular points, commonlyabbreviated as singularit ies.

3.3 Complex NumbersTo have an understanding about the complex numbers will be of great value to deal with theJoukowski t ransformat ion, to be taken up in Chapter 4. Let us briefly discuss the essent ialaspects of complex numbers in this sect ion.

A complex number may be defined as a number consist ing of a sum of real and imaginaryparts. Let x and y be the real numbers; posit ive or negat ive. Let i be a symbol which obeys theordinary laws of algebra, and also sat isfies the relat ion:

(3.5) The combinat ion of x and y in the following form:

(3.6) is then called a complex number.

A complex number z can be represented by a point p whose Cartesian coordinates are (x, y),as shown in Figure 3.6.

Figure 3.6 Representat ion of a complex number in xy-plane.

The picture, such as Figure 3.6, in which the complex number is represented by a point iscalled the Argand diagram. In this representat ion the complex number z represents the point por (x, y).

The numbers x and y in Equat ion (3.6) are called the real and imaginary parts of the complexnumber z, that is:

When y = 0 the complex number z is said to be purely real and when x = 0 the complex numberz is said to be purely imaginary. Two complex numbers which differ from the sign of i are said tobe conjugates. Usually a conjugate number is represented with an “overline.” For example:

(3.7a) (3.7b)

The simple fact that :

imply the following two simple but important theorems:Theorem 1: The real part of the difference of two conjugate complex numbers is zero.Theorem 2: The imaginary part of the sum of two conjugate complex numbers is zero.

The point p which represents a complex number in the xy-plane can also be described bypolar coordinates (r, θ), in which r is necessarily posit ive. By Euler's theorem, we have:

Therefore:

or

(3.8)

Note that:

When polar coordinates are used the posit ive number r is called the modulus of z, expressedas:

Then the product of two conjugate complex numbers is the square of the modulus of either.The angle θ is called the argument of z, expressed as:

Therefore, all complex numbers whose moduli are the same and whose arguments differ by aninteger mult iple of 2π are represented by the same part in the Argand diagram. The value ofarg z (that is, angle θ) that lies between −π and +π is the principal value. The principal value ofthe argument of a posit ive real number is zero, and the argument of a negat ive real number isπ.

Let us consider a curve C1 encircling the origin and curve C2 which does not encircle theorigin, as shown in Figure 3.7. If θ is the init ial value of the arg z and if z is represented by thepoint P, it is clear that when a point Q originally coinciding with P is moved round C1 in theclockwise direct ion, the corresponding value of its argument increases, and when we finallyreturn to P after going one round, we have the arg z = θ + 2π. On the other hand, if we goround C2, the argument of Q decreases at first unt il OQ becomes a tangent to C2, thenincreases unt il OQ again becomes a tangent and finally decreases to the init ial value. Thus ifarg z has a given value at one point of a curve such as C2 which does not encircle the origin,the value to the argument of z is one-valued at every point inside and on C2, provided whenthe arg z is assumed to vary cont inuously with z.

Now consider:

(3.9) If we take z once round C1, θ increases by 2π and therefore log z increases by 2πi. Thus log zis a many-valued funct ion if z moves inside or upon a curve which encircles the origin. On theother hand, log z can be regarded as a one-valued funct ion if z is restricted to the interior of acurve such as C2 which does not encircle the origin.

If are two complex numbers then their product is:

Thus the modulus of the product is the product of the moduli, while the argument of theproduct is the sum of the argument, that is:

(3.10a) (3.10b)

In applying Equat ion (3.10)b it is well to bear in mind that each of the arguments may be many-valued and therefore the right-hand member is only one of the possible values of arg (z1z2).

Similarly we can express:

(3.11)

Figure 3.7 Two curves in the xy-plane.

3.3.1 Differentiation of a Complex FunctionLet us consider the complex funct ion zn, where n is a posit ive integer. Then we define:

Example 3.1A part icle moves in the xy-plane such that its posit ion (x, y) as a funct ion of t ime t is given by:

Find the velocity and accelerat ion of the part icle in terms of t.

SolutionThe complex velocity is given by:

Given:

Therefore:

The conjugate of is:

Thus:

The accelerat ion is given by:

We have:

Thus:

3.4 SummaryThe transformat ion technique which transforms an orthogonal geometric pattern, composedof elements of certain shape, into an ent irely different pattern, whilst the elements retain theirform and proport ion is termed conformal transformation.

In the z-plane (physical plane) point p is located by z = x + iy, and in the ζ-plane (t ransformedplane), the corresponding point P is located by ζ = ξ + iη. The relat ion between z and ζ is apart icular specified funct ion of ζ, in terms of z. That is:

This funct ion is known as the transformation function. In the t ransformat ion, both the elementsof the crossing arc segments are rotated through the same angle. Therefore, the angle ofintersect ion must remain unchanged during the transformat ion.

The transformat ion funct ion is essent ially of the vector type:

where z = x + iy and ζ = ξ + iη.A general form of the t ransformat ion funct ion is:

where A0, A1, etc. and B1, B2, etc. are constants and vectors or combinat ions of constants andvectors, respect ively.

The length rat io of corresponding elements in the z-and ζ-planes is given by:

The actual length of an element is the modulus of the vector δz. Thus:

The velocity qz at any point p in the z-plane is given by:

where is the complex potent ial at that point , ψ is the stream funct ion and ϕ is thepotent ial funct ion. But with reference to the new (transformed) coordinate axes, the localvelocity at point P is:

At the corresponding points between the original plane (z-plane) and the transformed plane (ζ-plane), considering only the magnitudes, we can express:

It is seen that the velocity rat io between corresponding points in the original and transformedplanes is the inverse of the length rat io.

Exercise Problems1. Find the absolute value of:

[Answer: 1]

2. Find the polar form of (1 + i)2.[Answer: 2eiπ/2]

3. Express:

in (x + iy) form.[Answer: 0.47 − 0.17i]

4. Find x and y if:

[Answer: x = 1, y = ± 1]

5. What is the curve made up of the points (x, y)-plane sat isfying the equat ion |z| = 3?[Answer: a circle of radius 3 units with center at the origin.]

4

Transformation of Flow Pattern

4.1 IntroductionAny flow pattern can be considered to consist of a set of streamlines and potent ial lines (ψand ϕ lines). Thus, t ransformat ion of a flow pattern essent ially amounts to the transformat ionof a set of streamlines and potent ial lines, whilst the t ransformat ion of individual lines impliesthe transformat ion of a number of points.

4.2 Methods for Performing TransformationChoose a t ransformat ion funct ion ζ = f(z) to t ransform the points specified by the Cartesiancoordinates x and y, in the physical plane, given by z = x + iy, to a t ransformed plane given by ζ= ξ + iη. To carry out this t ransformat ion, we need to expand the transformat ion funct ion ζ =f(z) = ξ + iη, equate the real and imaginary parts and find the funct ional form of ξ and η, interms of x and y, that is, find:

Thus, any point p(x, iy) in the physical plane (z-plane) gets t ransformed to point P(ξ, iη) in thetransformed plane (ζ-plane).

Example 4.1Transform a point p(x, iy) in the physical plane to ζ-plane, with the transformat ion funct ion ζ =1/z.

SolutionGiven, ζ = 1/z. Also, z = x + iy.Therefore, from the transformat ion funct ion ζ = 1/z, we get:

But ζ = ξ + iη, therefore:

Comparing the real and imaginary parts, we get the coordinates of the t ransformed point P(ξ,η), in the ζ-plane, as:

Using these expressions for ξ and η, any point in the physical plane, with coordinates (x, y), canbe transformed to a point , with coordinates (ξ, η), in the t ransformed plane. That is, from anypoint of the given flow pattern in the original plane, values of the coordinates x and y can besubst ituted into the expressions of ξ and η to get the corresponding point (ξ, η), in thetransformed plane.

4.2.1 By Analytical MeansFor a given flow pattern in the physical plane, each streamline of the flow can be representedby a separate stream funct ion. Transferring these stream funct ions, using the transformat ionfunct ion, ζ = f(z), the corresponding streamlines in the transformed plane can be obtained. Forexample, the streamlines in the physical plane given by the stream funct ion:

can be expanded, using the transformat ion funct ion:

where z = x + iy, to obtain the following three equat ions:

But ζ = ξ + iη, therefore:

Comparing the real and imaginary parts, we have:

From these equat ions, ξ and η can be isolated, by eliminat ing x and y. The result ing

expressions for ξ and η will represent the t ransformed line in the ζ-plane.

Example 4.2Transform the straight lines, parallel to the x-axis in the physical plane, with the transformat ionfunct ion ζ = 1/z.

SolutionFrom the transformat ion funct ion ζ = 1/z, we have:

Mult iplying and dividing the numerator and denominator of the right-hand side by (x − iy), weget:

But

Therefore:

Equat ing the real and imaginary parts, we get the coordinates for the points on thetransformed lines, in ζ-plane, as:

Taking the flow pattern in the physical plane to be an uniform flow parallel to x-axis, we havethe stream funct ion as:

or

(a) For given values of ψ and Vx, k is a constant. Also:

(b)

(c) From (Equat ion c), we have:

Subst itut ing this into (Equat ion b), we get:

Subst itut ing y = k, we get:

This gives:

This is a circle of radius 1/2k with center at (0, − 1/2k), in the t ransformed plane. Thus, fordifferent values of k, the horizontal streamlines in the physical plane, shown in Figure 4.1(a),can be transformed to circles of radius 1/2k and center at (0, − 1/2k), in the ζ-plane, as shown inFigure 4.1(b), using the transformat ion funct ion ζ = 1/z.

Figure 4.1 Transformat ion of straight lines to circles.

4.3 Examples of Simple TransformationThe main use of conformal t ransformat ion in aerodynamics is to t ransform a complicated flowfield into a simpler one, which is amenable to simpler mathematical t reatment. The mainproblem associated with this t ransformat ion is finding the best t ransformat ion funct ion(formula) to perform the required operat ion. Even though a large number of mathematicalfunct ions can be envisaged for a specific t ransformat ion, in our discussions here, only the wellestablished transformat ions, which are commonly used in aerodynamics, will be considered.One such transformat ion, which generates a family of aerofoil shaped curves, along with theirassociated flow patterns, by applying a certain t ransformat ion to consolidate the theorypresented in the previous sect ions, is the Kutta−Joukowski transformation.

Example 4.3Transform the uniform flow parallel to x-axis of the physical plane, with the transformat ionfunct ion ζ = z2.

SolutionExpressing the transformat ion funct ion ζ = z2, in terms of x and y, we have the following:

But ζ = ξ + iη. Thus:

Equat ing the real and imaginary parts, we get the coordinates ξ and η, in the t ransformedplane, as:

The stream funct ion for uniform flow parallel to x-axis, in the physical plane, is:

Therefore:

Let .Also:

Therefore, ξ becomes:

Replacing y with k, we get:

or:

For a constant value of k, this gives a parabola. Therefore, horizontal streamlines, shown inFigure 4.2(a), in the z-plane, t ransform to parabolas in the ζ-plane, as shown in Figure 4.2(b).Thus, applying the transformat ion funct ion ζ = z2 to an uniform flow parallel to x-axis in thephysical plane, we get parabolas in the ζ-plane.

Figure 4.2 Transformat ion of horizontal lines to parabolas.

Note that the flow zones above or below x-axis, in the z-plane, t ransform to occupy the wholeof the ζ-plane. These zones of the z-plane must be treated separately. In this case, thestreamlines in the lower part of the z-plane, extending along the negat ive y-direct ion, will betaken with the flow streaming from left to right , in Figure 4.2(b). The streamlines for this flow isgiven by:

where y is always negat ive. Thus, the stream funct ion is negat ive in this zone.

Example 4.4Find the transformat ion of the uniform flow parallel to the y-axis, in the z-plane, using thetransformat ion funct ion ζ = z2.

SolutionThe given flow field is as shown in Figure 4.3(a).

Figure 4.3 Transformat ion of vert ical lines to parabolas.

For the transformat ion funct ion ζ = z2, from Example 4.3, we have:

The stream funct ion for the downward uniform flow, parallel to y-axis, shown in Figure 4.3(a), is:

Thus, for a given ψ and Vy:

Let . The coordinates ξ and η can be arranged as follows:

Replacing x with k, we get:

For different values of k this represents a set of parabolas, as shown in Figure 4.3(b).

4.4 Kutta−Joukowski TransformationKutta−Joukowski t ransformat ion is the simplest of all t ransformat ions developed forgenerat ing aerofoil shaped contours. Kutta used this t ransformat ion to study circular-arc wingsect ions, while Joukowski showed how this t ransformat ion could be extended to produce wingsect ions with thickness t as well as camber. In our discussions here, we make anothersimplificat ion that the t ransformat ion is confined to the study of the actual contour of thecircle, and to show how its shape changes on transformat ion.

In our discussion on Kutta−Joukowski t ransformat ion, it is important to note the following:The circle considered, in the physical plane, is a specific streamline. Essent ially the circle isthe stagnat ion streamline of the flow in the original plane 1 (z-plane).The transformat ion can be applied to the circle and all other streamlines, around thecircle, to generate the aerofoil and the corresponding streamlines in plane 2 (ζ-plane) orthe transformed plane. That is, the t ransformat ion can result in the desired aerofoil shapeand the streamlines of the flow around the aerofoil.

It is convenient to use polar coordinates in the z-plane and Cartesian coordinates in ζ-plane.The Kutta−Joukowski transformation function is:

where b is a constant.Now, expressing z as z = reiθ, where r and θ are the polar coordinates, and on expanding, we

get:

Equat ing the real and imaginary parts, we get:

(4.1a)

(4.1b) These expressions for ξ and η are the general expressions for the t ransformat ion of the basicshape, namely the circle in the z-plane, to any desired shape in the ζ-plane. This can beachieved by assigning suitable values to the constant b in the t ransformat ion funct ion (z +b2/z) and locat ing the center of the circle at the origin, or at a suitable locat ion on the x-axis ory-axis, or (usually) in the first quadrant of the z-plane.

4.5 Transformation of Circle to Straight LineFor t ransforming a circle of radius a to a straight line, the constant b in the Joukowskit ransformat ion funct ion should be set equal to a, and the center of the circle should be at theorigin, as shown in Figure 4.4(a).

Figure 4.4 Transformat ion of circle to straight line.

Subst itut ing r = a = b in the ξ and η expressions in Equat ion (4.1a), we get:

(4.2a)

(4.2b) T hese ξ and η represent a straight line coinciding with the ξ-axis in the ζ-plane. Thetransformed line is thus confined to ξ-axis, as shown in Figure 4.4(b), and as θ varies from 0 toπ, point P moves from +2a to −2a. Thus, the chord of the locus of point P is 4a.Note that the singularit ies at z = ± b produce sharp edges at ζ = ± 2a. That is, the extremit ies

of the straight line are sharp.

4.6 Transformation of Circle to EllipseFor t ransforming a circle to an ellipse using the Kutta−Joukowski t ransformat ion funct ion:

the circle should have its center at the origin in the z-plane, but the radius of the circle shouldbe greater than the constant b, in the above transformat ion funct ion, that is, a > b.

With the radius of the circle r = a, we can express the ξ and η expressions in Equat ion (4.1a)as:

Eliminat ing θ in the ξ and η expressions above, we get:

(4.3) This is an ellipse with its major and minor axes, respect ively, along the ξ and η axes in the ζ-plane, as shown in Figure 4.5(b).

Figure 4.5 Transformat ion of circle to ellipse.

The major and minor axes of the ellipse, given by Equat ion (4.3a), are the following:

The fineness rat io of the ellipse, defined as the rat io of the chord to maximum thickness,becomes:

or

(4.3a) From this relat ion it is evident that for every value of the rat io a/b a new ellipse can beobtained.

4.7 Transformation of Circle to SymmetricalAerofoil

To transform a circle into a symmetrical aerofoil, the center of the circle in the z-plane shouldbe shifted from the origin and located slight ly downstream of the origin, on the x-axis, asshown in Figure 4.6(a). This shift would cause asymmetry to the profile (about the ordinates ofthe transformed plane) of the t ransformed shape obtained with the Kutta−Joukowskit ransformat ion funct ion.

Figure 4.6 Transformat ion of circle to a symmetrical aerofoil.

Let the horizontal shift of the center of the circle c, from the origin o, also called as theeccentricity to be e. The actual distance of the center of the circle from the origin is be, asshown in Figure 4.6(a). Thus, the radius of the circle is (b + be). Let us represent the generalpoint p on the circle, in polar coordinates, as shown in Figure 4.7.

Figure 4.7 A general point p on a circle in polar coordinates.

The distance of point p from the origin, shown in Figure 4.7, is:

where e and γ are small. Therefore, the above distance r simplifies to:

Subst itut ing a = (b + be), we get:

This can be arranged as:

But the eccentricity e is very small. Therefore, the term e (1 + cos θ) is very small compared to1. Thus, expanding the right-hand-side of the above equat ion and retaining up to the firstorder terms, we get:

T hus, r and b can be expressed in terms of the horizontal shift e of the circle beingtransformed.

The transformat ion funct ion can be expressed, in terms of r and θ, by replacing z with reiθ,as:

But:

Therefore:

Equat ing the real and imaginary parts, we get the coordinates of the t ransformed profile as:

(4.4a)

(4.4b) These are the coordinates of a symmetrical aerofoil profile. Plot of ξ and η for θ from 0 to π andπ to 2π gives a symmetrical profile shown in Figure 4.6(b). The chord (the shortest distancefrom the leading edge to the trailing edge) of the aerofoil is 4b.

4.7.1 Thickness to Chord Ratio of Symmetrical AerofoilThe maximum thickness of the aerofoil occurs where dη/dθ = 0. Therefore, different iat ing

with respect to θ, and equat ing to zero, we get:

Thus, either:

or

From the above relat ion, we get the following condit ions corresponding to the maximum and

minimum thickness of the aerofoil:cos θ = 1/2, giving θ = π/3 = 60 , at the maximum thickness locat ion.cos θ = − 1, giving θ = π, at the minimum thickness locat ion.

Therefore, the maximum thickness is at the chord locat ion, given by:

This point (b, 0), from the leading edge of the aerofoil, in Figure 4.6(b), is the quarter chordpoint . Thus, the maximum thickness t max is at the quarter chord point . The maximum thicknesst max is given by 2η, with θ = π/3 in Equat ion (4.4b).

That is:

(4.5) Thus, the thickness to chord rat io of the aerofoil becomes:

or

(4.6) From the above relat ion for maximum thickness and thickness-to-chord rat io, it is seen thatthe thickness is dictated by the shift of the center of the circle or eccentricity e. Theeccentricity serves to fix the fineness rat io (t/c rat io) of the profile. For example, a 20% thickaerofoil sect ion would require an eccentricity of:

4.7.2 Shape of the Trailing EdgeAt the trailing edge of the aerofoil, the slope of its upper and lower surfaces merge. Therefore:

At the t railing edge, θ = π; because of this dη/dξ becomes indeterminate. But, by l’Hospital'srule,1 we can expresses dη/dξ as follows:

At θ = π:

Thus, at the t railing edge, both upper and lower surface are tangent ial to the ξ-axis, and

therefore, to each other. In other words, the t railing edge is cusped. This kind of t railing edgewould ensure that the flow will leave the trailing edge without separat ion. But this is possibleonly when the trailing edge is cusped with zero thickness. Thus, this is only a mathematicalmodel. For actual aerofoils, the t railing edge will have a finite thickness, and hence, there isbound to be some separat ion, even for the thinnest possible t railing edge.Note: Transformat ion of a circle with its center at a distance be on the negat ive side of the x-axis, in the physical plane, will result in a symmetrical aerofoil, with its leading edge on thenegat ive side of the ξ-axis (mirror image of the aerofoil profile about the η-axis, in Figure 4.6), inthe transformed plane. Similarly, posit ioning the center of the circle, with an offset , on the y-axis, will get t ransformed to a symmetrical aerofoil, with its leading and trailing edges on the η-axis, in the t ransformed plane.

4.8 Transformation of a Circle to a CamberedAerofoil

For t ransforming a circle to a cambered aerofoil, using Joukowski t ransformat ion, the center ofthe circle in the physical plane has to be shifted to a point in one of the quadrants. Let usconsider the center in the first coordinate of z-plane, as shown in Figure 4.8(a).

Figure 4.8 Transformat ion of a circle to a cambered aerofoil.

As seen in Figure 4.8(a):the center c of the circle is displaced horizontally as well as vert ically from the origin, in thephysical plane.let the horizontal shift of the center be “on = be,” andthe vert ical shift be “cn = h.”

The point p on the circle and its distance from the origin can be represented as shown inFigure 4.9.

Figure 4.9 Locat ion of point p on a circle in polar coordinates.

Both the vert ical shift h and eccentricity e are small. Therefore, the angle β, subtended atpoint m, by om and cm is small, hence, cos β ≈ 1, also cos γ ≈ 1. Therefore, the radius of thecircle becomes:

The vert ical shift of the center can be expressed as:

But e and β are small, therefore, h becomes:

By dropping perpendiculars on to op, from n and c, it can be shown that:

The angle γ is small, therefore, cos γ ≈ 1, thus:

Subst itut ing for a = (b + be) and h = bβ, r becomes:

This can be expressed as:

(4.7a) or

Expanding and retaining only the first order terms, we have:

(4.7b) since e and β are small, their powers are assumed to be negligibly small.

The Joukowski t ransformat ion funct ion is:

Replacing z with r eiθ, we have:

Subst itut ing for and from Equat ions (4.7a) and (4.7b), we get:

or

Equat ing the real and imaginary parts, we get:

(4.8a)

(4.8b) These coordinates represent a cambered aerofoil, in the ζ-plane, as shown in Figure 4.8(b).Thus, the circle with center c in the first quadrant of z-plane is t ransformed to a camberedaerofoil sect ion in the ζ-plane, as shown in Figure 4.8(b), with coordinates ξ and η, given byEquat ions (4.8a) and (4.8b). It is seen that:

The chord of the cambered aerofoil is also 4b, as in the case of symmetrical aerofoil.When β = 0, that is, when there is no vert ical shift for the center of the circle, in thephysical plane, the t ransformat ion results in a symmetrical aerofoil in the t ransformed or ζ-plane.The second term in the η expression, in Equat ion (4.8b), alters the shape of the aerofoilsect ion, because it is always a posit ive addit ion to the η-coordinate (ordinate).The trailing edge is sharp (η = 0, at θ = π), and the t max is at the quarter chord point (ξ =b).

4.8.1 Thickness-to-Chord Ratio of the Cambered AerofoilThickness t of the aerofoil at any locat ion along the chord is the difference between the localthicknesses above and below the mid-plane. That is:

where the subscripts u and l, respect ively, refer to the upper and lower surfaces of the aerofoilprofile. The upper and lower thicknesses of the aerofoil are given, by Equat ion (4.8b), as:

But ηu and ηl are at the same locat ion on the abscissa (ξ-axis), therefore, θl = − θu. Thethickness becomes:

The thickness-to-chord rat io becomes:

or

(4.9)

For (t/c) max, the condit ion is d(t/c)/dθ = 0, and d2(t/c)/dθ2 < 0. Thus:

This gives:

We can express this as:

and

Thus, θu is either π or π/3. At θu = π, t/c = 0, which is the t railing edge of the aerofoil. Hence, t/cshould be maximum at θu = π/3.

The thickness-to-chord rat io is maximum at θu = 60 . Thus:

This maximum is also at the quarter chord point , as in the case of symmetrical aerofoil.

4.8.2 CamberThe camber of an aerofoil is the maximum displacement of the mean camber line from thechord. The mean camber line is the locus of mid-points of lines drawn perpendicular to thechord. In other words, the camber line is the bisector of the aerofoil profile thickness distribut ionfrom the leading edge to the trailing edge. The camber line is given by:

Making this nondimensional, by dividing with the chord, the camber can be expressed as afract ion of the chord:

By Equat ion (4.8b), we have:

Thus:

The chord of the aerofoil is c = 4b. Therefore:

In this relat ion, sin 2θ is maximum at θ = π/2, that is, at the mid-chord. Therefore:

From the above discussions it is evident that the vert ical shift of the circle center is responsiblefor the camber of the aerofoil, and the horizontal shift determines the thickness-to-chord rat ioof the aerofoil.

4.9 Transformation of Circle to Circular ArcTransformat ion of a circle with its center shifted above (or below) the origin, on the ordinate inthe z-plane, with the transformat ion funct ion ζ = z + b2/z results in a circular arc, as shown inFigure 4.10.

Figure 4.10 Transformat ion of circle to circular arc.

A general point p on the circle with center above the origin, on the y-axis, in Figure 4.10, canbe represented as shown in Figure 4.11.

Figure 4.11 The point p on the circle with its center on the ordinate in the physical plane.

From Figure 4.11, it is seen that:

Therefore,

or

But, (e + β cos θ) < 1, therefore, expanding the above, retaining only the first-order terms, weget:

The Joukowski t ransformat ion funct ion is:

Replacing z = reiθ, we have:

Subst itut ing for and , we get:

Comparing the real and imaginary parts, we get:

(4.10a)

(4.10b) Equat ions (4.10a) and (4.10b), respect ively, are the expressions for the abscissa and

ordinates of the circular arc, in the t ransformed plane.The chord is given by:

The camber becomes:

But,

Therefore:

Note: In this analysis, it is essent ial to note that the angle θ, in Figure 4.11, is marked as theangle subtended at the origin o by the line op and the ordinate. This convent ion is differentfrom that followed in Figure 4.9. This change made is just ified by η max at θ = π/2, which has tobe sat isfied.

4.9.1 Camber of Circular ArcFor the transformed circular arc, the chord is 4b. The camber is the maximum deviat ion of thearc line from the chord. Thus, the camber-to-chord rat io becomes:

But for small β, tan β ≈ β. Therefore, the percentage camber for the circular arc becomes 100β/2.

Thus, on transformat ion with , a circle with its origin on the y-axis, as shown inFigure 4.10(a), t ransforms to a circular arc of camber-to-chord rat io β. This is the extreme caseof Joukowski profile, for which the camber is finite and the thickness is zero.

The transformat ion of a circle of radius a in the z-plane with its (a) center at the origin and b= a, (b) center at the origin and b < a, (c) center above the origin and b = om < a, (d) centershifted horizontally and b = om < a and (e) center in the first quadrant and b = om < a, usingthe Joukowski t ransformat ion, ζ = z + b2/z, results in a flat plate, ellipse, circular arc,symmetrical aerofoil and cambered aerofoil, respect ively, as shown in Figures 4.12(a)–4.12(e).

4.10 Joukowski HypothesisJoukowski postulated that “the aerofoil generates sufficient circulat ion to depress the rearstagnat ion point from its posit ion, in the absence of circulat ion, down to the sharp t railingedge.” There is sufficient evidence of a physical nature to just ify this hypothesis, and thefollowing brief descript ion of accelerat ion of an aerofoil from rest may serve as a reminder.Immediately after the state of rest , the streamline pattern around the aerofoil is as shown inFigure 4.13(a).

As seen in Figure 4.13(a), immediately after start ing, the rear stagnat ion point S2 is wellahead of the t railing edge. During the subsequent accelerat ion to steady mot ion, the rearstagnat ion point moves to the trailing edge, a start ing vortex or init ial eddy is cast off andcirculat ion to an equal, but opposite amount, is induced in the flow around the aerofoil, asshown in Figure 4.13(b).

Figure 4.12 Transformat ion of a circle to (a) flat plate, (b) ellipse, (c) circular arc, (d)symmetrical aerofoil, (e) cambered aerofoil.

Figure 4.13 Streamline pattern around an aerofoil (a) just after start , (b) after the start ingvortex is established.

In the potent ial flow model of a sharp-edged aerofoil the exact t railing edge point can bedefined, because the top and bottom profiles are cusped, at the t railing edge, result ing in zerothickness. It is essent ial to note that this kind of sharp t railing edge, with zero thickness, is justa mathematical shape, and cannot be realized in pract ice. Actual wing profiles are with arounded trailing edge of finite thickness. Because of the rounded trailing edge of the wings, inactual flow where viscous boundary layer and wake exist , the posit ion of the rear stagnat ionpoint may differ from the locat ion predicted by potent ial flow theory, and the full Joukowskicirculat ion, may not be established. This is because for realizing full Joukowski circulat ion, thetrailing edge should be of zero thickness and without any wake. This condit ion of realizing fullJoukowski circulat ion, result ing in flow without wake, is known as Kutta condition.

The Kutta condit ion is a principle in steady flow fluid dynamics, especially aerodynamics, thatis applicable to solid bodies which have sharp corners such as the trailing edges of aerofoils. Itis named after the German mathematician and aerodynamicist Mart in Wilhelm Kutta.

The Kutta condit ion can be stated as follows:

“A body with a sharp trailing edge which is moving through a fluid will ‘create about itself acirculation’ of sufficient strength to hold the rear stagnation point at the trailing edge.”In fluid flow around a body with a sharp corner the Kutta condit ion refers to the flow pattern

in which fluid approaches the corner from both direct ions, meets at the corner and then flowsaway from the body. None of the fluid flows around the corner remaining at tached to the body.

The Kutta condit ion is significant when using the Kutta−Joukowski theorem to calculate thelift generated by an aerofoil. The value of circulat ion of the flow around the aerofoil must bethat value which would cause the Kutta condit ion to exist .

4.10.1 The Kutta Condition Applied to AerofoilsWhen a smooth symmetric body, such as a cylinder with oval cross-sect ion, moves with zeroangle of at tack through a fluid it generates no lift . There are two stagnat ion points on the body-one at the front and the other at the back. If the oval cylinder moves with a nonzero angle ofat tack through the fluid there are st ill two stagnat ion points on the body –one on the

underside of the cylinder, near the front edge; and the other on the topside of the cylinder,near the back edge. The circulat ion around this smooth cylinder is zero and no lift is generated,despite the posit ive angle of at tack.

If an aerofoil with a sharp t railing edge begins to move with a posit ive angle of at tackthrough air, the two stagnat ion points are init ially located on the underside near the leadingedge and on the topside near the trailing edge, just as with the cylinder. As the air passing theunderside of the aerofoil reaches the trailing edge it must flow around the trailing edge andalong the topside of the aerofoil toward the stagnat ion point on the topside of the aerofoil.Vortex flow occurs at the t railing edge and, because the radius of the sharp t railing edge iszero, the speed of the air around the trailing edge should be infinitely fast ! Real fluids cannotmove at infinite speed but they can move very fast . The very fast airspeed around the trailingedge causes strong viscous forces to act on the air adjacent to the trailing edge of the aerofoiland the result is that a strong vortex accumulates on the topside of the aerofoil, near thetrailing edge. As the aerofoil begins to move, it carries this vortex, known as the start ing vortex,along with it . Pioneering aerodynamicists were able to photograph start ing vort ices in liquids toconfirm their existence.

The vort icity in the start ing vortex is matched by the vort icity in the bound vortex in theaerofoil, in accordance with Kelvin's circulat ion theorem. As the vort icity in the start ing vortexprogressively increases, the vort icity in the bound vortex also progressively increases, andcauses the flow over the topside of the aerofoil to increase in speed. The stagnat ion point onthe topside of the aerofoil moves progressively towards the trailing edge. After the aerofoil hasmoved only a short distance through the air, the stagnat ion point on the topside reaches thetrailing edge and the start ing vortex is cast off the aerofoil and is left behind, spinning in the airwhere the aerofoil left it . The start ing vortex quickly dissipates due to viscous forces.

As the aerofoil cont inues on its way, there is a stagnat ion point at the t railing edge. The flowover the topside conforms to the upper surface of the aerofoil. The flow over both the topsideand the underside join up at the t railing edge and leave the aerofoil t raveling parallel to oneanother. This is known as the Kutta condit ion.

When an aerofoil is moving with a posit ive angle of at tack, the start ing vortex will be cast off,and the Kutta condit ion will be established. There will be a finite circulat ion of the air aroundthe aerofoil and the aerofoil will generate lift , with magnitude equal to that given by theKutta−Joukowski theorem.

One of the consequences of the Kutta condit ion is that the airflow over the upper surface ofthe aerofoil t ravels much faster than the airflow over the bottom surface. A port ion of air flowwhich approaches the aerofoil along the stagnat ion streamline is split into two parts at thestagnat ion point , one half t raveling over the upper surface and the other half t raveling alongthe bottom surface. The flow over the topside is so much faster than the flow along thebottom that these two halves never meet again. They do not even re-join in the wake longafter the aerofoil has passed. This is somet imes known as “cleavage.” There is a popularfallacy called the equal transit-time fallacy that claims the two halves rejoin at the t railing edgeof the aerofoil. This fallacy is in conflict with the phenomenon of cleavage that has beenunderstood since Mart in Kutta's discovery.

Whenever the speed or angle of at tack of an aerofoil changes there is a weak start ingvortex which begins to form, either above or below the trailing edge. This weak start ing vortexcauses the Kutta condit ion to be re-established for the new speed or angle of at tack. As aresult , the circulat ion around the aerofoil changes and so too does the lift in response to thechanged speed or angle of at tack.

The Kutta condit ion gives some insight into why aerofoils always have sharp trailing edges,even though this is undesirable from structural and manufacturing viewpoints. An aircraft wingwith a smoothly rounded trailing edge would generate lit t le or no lift .

4.10.2 The Kutta Condition in Aerodynamics

The Kutta condit ion allows an aerodynamicist to incorporate a significant effect of viscositywhile neglect ing viscous effects in the underlying conservat ion of momentum equat ion. It isimportant in the pract ical calculat ion of lift on a wing.

The equat ions of conservat ion of mass and conservat ion of momentum applied to aninviscid fluid flow, such as a potent ial flow, around a solid body result in an infinite number ofvalid solut ions. One way to choose the correct solut ion would be to apply the viscousequat ions, in the form of the Navier–Stokes equat ions. However, these normally do not resultin a closed-form solut ion. The Kutta condit ion is an alternat ive method of incorporat ing someaspects of viscous effects, while neglect ing others, such as skin frict ion and some otherboundary layer effects.

The condit ion can be expressed in a number of ways. One is that there cannot be an infinitechange in velocity at the t railing edge. Although an inviscid fluid (a theoret ical concept thatdoes not normally exist in the everyday world) can have abrupt changes in velocity, in realityviscosity smooths out sharp velocity changes. If the t railing edge has a nonzero angle, the flowvelocity there must be zero. At a cusped trailing edge, however, the velocity can be nonzeroalthough it must st ill be ident ical above and below the aerofoil. Another formulat ion is that thepressure must be cont inuous at the t railing edge.

The Kutta condit ion does not apply to unsteady flow. Experimental observat ions show thatthe stagnat ion point (one of two points on the surface of an aerofoil where the flow speed iszero) begins on the top surface of an aerofoil (assuming posit ive effect ive angle of at tack) asflow accelerates from zero, and moves backwards as the flow accelerates. Once the init ialt ransient effects have died out, the stagnat ion point is at the t railing edge as required by theKutta condit ion.

Mathematically, the Kutta condit ion enforces a specific choice among the infinite allowedvalues of circulat ion.

4.11 Lift of Joukowski Aerofoil SectionJoukowski hypothesis direct ly relates the lift generated by a two-dimensional aerofoil to itsincidence, as well as indicates the significance of the thickness to chord rat io and camber ofthe aerofoil in the lift generat ion.

We know that, applying the Joukowski t ransformat ion, ζ = z + b2/z, to an offset circle, asshown in Figure 4.14(a), a cambered aerofoil shape can be obtained, as illustrated in Figure4.14(b).

Figure 4.14 Streamline pattern around (a) a circle in the z-plane, and (b) an aerofoil in the ζ-plane.

If the remaining streamlines of the flow around the circle, in the physical plane, are alsotransformed, they will be distorted in the transformed plane to the shape of the streamlinesaround the aerofoil, as shown in Figure 4.14(b).

If the freestream flow approaches the circle at some angle of incidence α, as shown in Figure4.15(a), but cont inue to t ransform about the original axes, the corresponding flow around thetransformed aerofoil, in the ζ-plane, will be as shown in Figure 4.15(b).

Figure 4.15 Streamline pattern of a freestream flow at some incidence α around (a) a circle inthe z-plane, and (b) the corresponding flow around the transformed aerofoil in the ζ-plane.

This flow (regime) around the aerofoil is not sustained, although it is init iated every t ime anaerofoil starts moving from rest .

In pract ice, the generat ion of start ing vortex shifts (dips) the rear stagnat ion point S2 to thetrailing edge, that is to point M, shown in Figure 4.16(b). As the flow process develops, whichtakes place very quickly, the circulat ion around the aerofoil sect ion is generated and theaerofoil experiences lift .

Figure 4.16 Streamline pattern of a freestream flow and circulat ion around (a) a circle in the z-plane, and (b) the corresponding flow around the transformed aerofoil in the ζ-plane.

In the z-plane, the rear stagnat ion point S2 on the circle must have now been depressed tothe corresponding point m to the t railing edge, as shown in Figure 4.16(a). In the transformedplane, the corresponding rear stagnat ion point S2 of the aerofoil is dipped to point M, as showni n Figure 4.16(b). This depression of the stagnat ion point enables the evaluat ion of themagnitude of the circulat ion.

At the same t ime, because of the circulat ion Γ, the forward stagnat ion point S1 on the circlewould have been depressed by a corresponding amount to point n. The angular displacementbeing (α + β), as illustrated in Figure 4.17.

Figure 4.17 Depression of stagnat ion points.

In Figure 4.17, by geometry, angle subtended by ∠S2cm = ∠ S1cn = (α + β). The velocity atany point p on the circle, with θc measured from the diameter S2cS1, given by Equat ion (2.51),is:

where a is the radius of the circle, V∞ is the freestream velocity and Γ is the circulat ion. At pointn the velocity is zero and θc = − (α + β). Therefore, at n:

This gives the circulat ion as:

Therefore, the velocity becomes:

This simplifies to:

(4.11) The lift per unit span is:

The lift can also be expressed as:

where CL is the lift coefficient and S is the projected area of the wing planform, normal to thedirect ion of freestream flow. The area S is given by:

The chord of the profile is 4b. Therefore, the area of the profile, per unit span, becomes S = 4b.Thus, the lift is:

Equat ing the above two expressions for lift , we get the lift coefficient as:

But,

Therefore,

(4.12) This is the lift coefficient of a two-dimensional aerofoil, in terms of thickness to chord rat io (t/c= 1.299e), percentage camber (100 β/2), and angle of incidence α. Though this relat ion for CL iscompact, there are severe limitat ions for its use, because of the following reasons:

It has been assumed throughout that the fluid is ideal, that is, incompressible and inviscid,that is, the viscous effects are neglected, although in the real flow process creat ingcirculat ion Γ, and hence lift L on an aerofoil, the viscosity (μ) is important.One of the simplest t ransformat ion funct ions, namely the Joukowski t ransformat ionfunct ion, has been used in the analysis. Therefore, the results are applicable only to apart icular family of aerofoils.

However, this result can be of use as a reference value. Different iat ion of Equat ion (4.12), withrespect to the incidence angle α, gives the ideal lift curve slope, a∞I.

For small values of eccentricity e, angle of incidence α and camber β, this simplifies to:

(4.13) This is the theoret ical value of lift curve slope per radian of angle of at tack, α. It is seen that,the lift curve slope is independent of the angle of at tack.

4.12 The Velocity and Pressure Distributions onthe Joukowski Aerofoil

The velocity anywhere on the circle with circulat ion in the z-plane corresponding to the lift ingcambered aerofoil in the ζ-plane, given by Equat ion (4.11), is:

The velocity Va, at the corresponding point on the aerofoil, is obtained direct ly by applying thevelocity rat io between the transformed planes, given by:

(4.14) We know that the t ransformat ion funct ion is:

where b is a constant. Therefore:

Replacing z with reiθ, we get:

Hence:

We have r/b, (Figure 4.9), as:

Subst itut ing this, further simplificat ion can be made. The result ing equat ion is compact forobtaining the velocity distribut ion around the aerofoil profile. However, the velocity appears assquare in the pressure coefficient , Cp, expression, which implies that, comput ing the pressure

coefficient is a tedious process. The approximat ion that the eccentricity e is very smallcompared to unity (e < 1) progressively becomes unrealist ic when thicker and more camberedaerofoil sect ions are required. Use of this approximat ion is just ified only when they producesignificant simplificat ion at the expense of acceptable small deviat ions from the exactsolut ions of the velocity and pressure distribut ions around the aerofoil profile generated. In thepresent case, the ut ility of the approximate method largely ceases after the expressions for ξand η, for the profile:

have been obtained. For obtaining velocity and pressure distribut ion numerical solut ion may beemployed.

4.13 The Exact Joukowski Transformation Processand Its Numerical Solution

The coordinates of the t ransformed aerofoil profile are obtained as follows. The transformat ionfunct ion can be expressed:

Dividing and mult iplying the numerator and denominator, on the right-hand side, by (x − iy), weget:

Comparing the real and imaginary parts, we get the coordinates of the aerofoil sect ions as:

(4.15a)

(4.15b) Referring to Figures 4.8 and 4.9, we have the distance op of point p from the origin in the z-plane, as:

The x and y coordinates of point p are:

With a = (b + be), the above coordinate expressions become:

(4.16a) (4.16b)

where e and β represent the thickness and camber of a given aerofoil, respect ively. Therefore,the values of x and y can be found for the given values of θ ', round the circle, from 0 to 2π.Subst itut ing these x and y in to Equat ions (4.15a) and (4.15b), the aerofoil coordinates ξ and ηcan be obtained.

This method may be used with both singularit ies within the circle, when:

4.14 The Velocity and Pressure DistributionFor finding the velocity distribut ion around a given aerofoil, it is necessary to relate the angle ofincidence α to the circulat ion Γ around the aerofoil. This is done by applying the Joukowskihypothesis. In reality, the full Joukowski circulat ion required to bring the rear stagnat ion point tothe trailing edge is not realized, because of the following:

Air is a viscous fluid, and the flow near the trailing edge of an aerofoil is modified by thepresence of the boundary layer and wake, caused by the viscosity.The zero thickness for the t railing edge, st ipulated by the Joukowski hypothesis, is notpossible in pract ice. Therefore, the t railing edge must be rounded to some degree ofcurvature. The finite thickness of the t railing edge owing to this rounding-off forces therear stagnat ion point to deviate from the posit ion given in the ideal case.

Therefore, if Γ is the full Joukowski circulat ion (theoret ical circulat ion), it can be assumed thatthe pract ical value of circulat ion is only kΓ, where k is less than unity.

The velocity Vc, anywhere on the circle, given by Equat ion (4.11), is:

But the pract ical value of circulat ion is kΓ, therefore:

Subst itut ing Γ = 4πV∞a sin (α + β), we get:

or

The velocity Va at corresponding points on the transformed aerofoil can be found by using therelat ion, Equat ion (4.14):

The Joukowski t ransformat ion funct ion is:

Different iat ing with respect to z, we get:

But z = x + iy. Therefore:

where

and

Therefore,

Thus, the velocity on the aerofoil becomes:

Dividing both sides by the freestream velocity V∞, we get:

Subst itut ing for Vc/V∞, we have:

Now, it is easy to obtain Va/V∞, for different values of θ ', since θc = (θ ' + α).Subst itut ing this, the velocity on the aerofoil becomes:

(4.17) For incompressible flow, the pressure coefficient is given by:

(4.18) Knowing the distribut ion of Va/V∞, over the aerofoil, from Equat ion (4.17), the pressurecoefficient around the aerofoil can be est imated.Note that, with Equat ions (4.14), (4.15a) and (4.17), the aerofoil shape, the velocity around it ,and the Cp distribut ion around it can be computed, for the given values of b and e.

Example 4.5For an aerofoil with b = 100 mm, e = 1/10 and (a) k = 1.0 and (b) k = 0.95, kept at α = 5 ,determine the velocity and pressure around the transformed aerofoil.

SolutionThe aerofoil shape, the streamline pattern over the profile and the pressure coefficientvariat ion around it , for the values b, e and α, listed in the problem were computed for Γ = 1.0and 0.95, with the rout ine given below. The results are given in Figures 4.18(a)–(d).For Γ = 1, the aerofoil shape, the streamlines and the pressure coefficient distribut ion aroundthe aerofoil are shown in Figure 4.18(a). It is interest ing to note that the flow leaving theaerofoil, at the t railing edge, is smooth and there is no wake, in accordance with Joukowski'shypothesis. The Cp distribut ion around the aerofoil, shown in Figure 4.18(b), clearly illustratesthe higher pressure at the bottom and the lower pressure at the top of the profile, causing lift .

Figure 4.18 (a) Aerofoil shape, the streamline pattern over the profile and the pressurecoefficient variat ion around it , (b) Cp distribut ion over the aerofoil for Γ = 1 and (c) thestreamline pattern over the profile and the pressure coefficient variat ion around it , (d) Cpdistribut ion over the aerofoil for Γ = 0.95.

The streamlines and Cp variat ion around the aerofoil, for Γ = 0.95 are shown in Figures 4.18(c)and 4.18(d), respect ively. It is seen that, when Γ is less than 1, the rear stagnat ion point isupstream of the t railing edge, and there is a narrow wake. With decrease of Γ, the wake widthwill increase, leading to higher drag.

The rout ine for calculat ing the aerofoil shape, the streamlines and the pressure distribut ionaround it is given below:

4.15 Aerofoil CharacteristicsWe saw that aerofoil is a streamlined body that would experience the largest value of lift -to-drag rat io, in a given flow, compared to any other body in the same flow. In other words, in agiven flow the aerodynamic efficiency (L/D) of an aerofoil will be the maximum. When an

given flow the aerodynamic efficiency (L/D) of an aerofoil will be the maximum. When anaerofoil is exposed to a flow, due to the pressure act ing normal to the body surface and theshear force, due to viscosity, act ing tangent ial to the body surface, normal and tangent ialforces, respect ively, would act on the aerofoil, as illustrated in Figure 4.19.

Figure 4.19 Normal and shear forces act ing on an aerofoil in a flow field.

The pressure and shear forces can be integrated over the surface of the aerofoil to obtainthe resultant aerodynamic force, F ad, which acts at the center of pressure (kcp) of the aerofoil.

The forces act ing on an aircraft in level flight are the lift L, drag D, thrust T and weight W, asshown in Figure 4.20.

Figure 4.20 Forces act ing in the pitch (xz) plane on an aircraft .

In the xz-plane or pitch plane shown in Figure 4.20, the body-oriented components are theaxial forces (A), which are the forces parallel to the aircraft axis, and the normal forces (N),which are perpendicular to the vehicle axis. As the aircraft t ravels in air, its mot ion isdetermined by its weight, the thrust produced by the engine, and the aerodynamic forcesact ing on the vehicle.

For a steady, unaccelerated level flight in a horizontal plane:The sum of the forces along the flight path is zero.The sum of the forces perpendicular to the flight path is zero.

When the angle of incidence α is small, the component of thrust parallel to the freestream flowdirect ion is only slight ly less than the thrust itself. Therefore, for equilibrium:

Let us consider the case where the lift generated by the wing-body configurat ion Lwb actsahead of the center of gravity (cg), as shown in Figure 4.21.

Figure 4.21 An aircraft in flight with lift act ing ahead of cg.

The lift act ing ahead of center of gravity will produce a nose-up (posit ive) moment about thecenter of gravity. The aircraft is said to be trimmed, when the sum of the moments about thecg is zero, that is:

Thus, a force from a control surface located aft of the cg, for example, the horizontal tailsurface Lt is required to produce a nose-down (negat ive) pitching moment about the cg, whichcould balance the posit ive moment produced by Lwb. The tail surface producing Lt alsoproduces a drag force which is known as the trim drag. The trim drag may vary from 0.5% to5% of the total drag of the aircraft . It is essent ial to note that in addit ion to t rim drag, the tailalso produces drag due to the pressure and shear act ing on its geometry, known as profiledrag. Thus the trim drag is that associated with Lt generated to t rim the vehicle.

In addit ion to the lift and drag act ing in the pitch or xz-plane, there is a side force act ing onthe aircraft . The side force is the component of force in the direct ion perpendicular to both thelift and the drag. The side force act ing towards the starboard (right) wing is referred to asposit ive.

Usually the aerodynamic force will not act through the cg (which is also taken as the origin ofthe airplane's axis system). The moment due to the resultant force acts at a distance from theorigin may be divided into three components, referred to the airplane's axes. The threemoment components are the pitching moment M, the rolling moment L and the yawingmoment N, as shown in Figure 4.22.

Figure 4.22 Illustrat ion of pitching, rolling and yawing moments act ing on an aircraft .

As illustrated in Figure 4.22:Pitching moment is the moment act ing about the lateral axis (y-axis). It is the moment dueto the lift and drag act ing on the aircraft . Pitching moment causing nose-up is regarded asposit ive.Rolling moment is the moment act ing about the longitudinal axis (x-axis) of the aircraft .Rolling moment is generated by a different ial lift generated by the ailerons, located closerto the wingt ips. Rolling moment causing the right (starboard) wingt ip to move downwardis regarded as posit ive.Yawing moment is the moment act ing about the vert ical (z-axis) of the aircraft . Yawingmoment tends to rotate the aircraft nose to the right is regarded posit ive.

4.15.1 Parameters Governing the Aerodynamic ForcesThe primary parameters governing the magnitude of the aerodynamic forces and moments arethe following:

Geometry of the aerofoil.Angle of at tack, namely the aircraft at t itude in the pitch (xz) plane relat ive to the flightdirect ion.Vehicle size.Freestream velocity.Freestream flow density.Reynolds number (viscous effects).

Mach number (compressibility effects).

4.16 Aerofoil GeometryThe geometrical sect ion of a wing obtained by cutt ing it by a vert ical plane parallel to thecenterline of the aircraft is called aerofoil section. The lift generated and the stallcharacterist ics of a wing strongly depends on the geometry of the aerofoil sect ions that makeup the wing. The geometric parameters that dictate the aerodynamic characterist ics of theaerofoil sect ion are; the leading-edge radius, the mean camber line, the maximum thicknessand the thickness distribut ion of the profile, and the trailing-edge angle. These parameters areshown in Figure 4.23.

Figure 4.23 Geometrical parameters of an aerofoil.

4.16.1 Aerofoil NomenclatureThe tests made at Gott ingen during World War I contributed significant ly to the developmentof modern types of wing sect ions. Up to about World War II, most wing sect ions in common usewere derived most ly from the work at Gott ingen. During this period many families of wingsect ions were tested in the laboratories of various countries, but the work of the Nat ionalAdvisory Commit tee for Aeronaut ics (NACA) was outstanding. The NACA invest igat ions werefurther refined by separat ion of the effects of camber and thickness distribut ion, and theexperimental work was performed at higher Reynolds number than were generally obtainedelsewhere. As a result , the geometry of many aerofoil sect ions is uniquely defined by the NACAdesignat ion for the aerofoil.

Aerofoil geometry are usually characterized by the coordinates of the upper and lowersurface. It is often summarized by a few parameters such as: maximum thickness, maximumcamber, posit ion of max thickness, posit ion of max camber, and nose radius (see Figure 4.23).One can generate a reasonable aerofoil sect ion given these parameters. This was done byEastman Jacobs in the early 1930s to create a family of aerofoils known as the NACASect ions. The NACA aerofoils are aerofoil shapes for aircraft wings developed by the Nat ionalAdvisory Commit tee for Aeronaut ics (NACA). The shape of the NACA aerofoils is describedusing a series of digits following the word “NACA.”

The NACA 4-digit and 5-digit aerofoils were created by superimposing a simple mean-lineshape with a thickness distribut ion that was obtained by fit t ing a couple of popular aerofoils ofthe t ime:

The camber-line of 4-digit sect ions was defined as a parabola from the leading edge to theposit ion of maximum camber, then another parabola back to the trailing edge, as illustrated inFigure 4.24.

Figure 4.24 Illustrat ion of the camber line of a 4-digit NACA aerofoil.

NACA 4-Digit Series:The first digit implies the maximum camber in percentage of chord (c), the second digit givesthe posit ion of maximum camber in 1/10 of chord, the last two digits give the maximumthickness in percentage of chord. For example:

1. NACA 4412 aerofoil has a maximum camber of 4% of chord, with the maximum camberlocated at 0.4c and thickness-to-chord rat io 12%.2. NACA 2412 aerofoil has a maximum camber of 2% located 40% (0.4c) from the leadingedge with a maximum thickness of 12% of the chord. Four-digit series aerofoils by defaulthave maximum thickness at 30% of the chord (0.3c) from the leading edge.3. NACA 0015 aerofoil is symmetrical, the 00 indicat ing that it has no camber. The 15indicates that the aerofoil has a 15% thickness to chord length rat io: it is 15% as thick asit is long.

After the 4-digit sect ions came the 5-digit sect ions such as the famous NACA 23012. Thesesect ions had the same thickness distribut ion, but used a camber-line with more curvature nearthe nose. A cubic was faired into a straight line for the 5-digit sect ions.

NACA 5-Digit Series:In NACA 5-digit series the first digit gives approximate maximum camber in percentage ofchord, the second and third digits give the posit ion of maximum camber in 2/100 of chord andthe last two digits give the maximum thickness in percentage of chord. This NACA 23012 is anaerofoil with maximum camber as 2% of c, posit ion of maximum camber at 60% of chord andt/c = 0.12.

Four-and five-digit series aerofoils can be modified with a two-digit code preceded by ahyphen in the following sequence:

1. One digit describing the roundness of the leading edge with 0 being sharp, 6 being thesame as the original aerofoil, and larger values indicat ing a more rounded leading edge.2. One digit describing the distance of maximum thickness from the leading edge in tensof percentage of the chord.

For example, the NACA 1234-05 is a NACA 1234 aerofoil with a sharp leading edge andmaximum thickness as 50% of the chord (0.5 chords) from the leading edge.

In addit ion, for a more precise descript ion of the aerofoil all numbers can be presented asdecimals.

1-series:A new approach to aerofoil design pioneered in the 1930s in which the aerofoil shape wasmathematically derived from the desired lift characterist ics. Prior to this, aerofoil shapes werefirst created and then had their characterist ics measured in a wind tunnel. The 1-seriesaerofoils are described by five digits in the following sequence:

1. The number “1” indicat ing the series.2. One digit describing the distance of the minimum pressure area in tens of percent ofchord.3. A hyphen.

4. One digit describing the lift coefficient in tenths.5. Two digits describing the maximum thickness in percentage of chord.

For example, the NACA 16-123 aerofoil has minimum pressure 60% of the chord back with alift coefficient of 0.1 and maximum thickness of 23% of the chord.

The 6-series of NACA aerofoils departed from this simply-defined family. These sect ionswere generated from a more or less prescribed pressure distribut ion and were meant toachieve some laminar flow.

NACA 6-Digit Series:In NACA 6-digit series the first digit refers to the series, the second digit gives the locat ion ofminimum Cp in 1/10 chord, the third digit gives the half width of low drag bucket in 1/10 of CL,the fourth digit gives the ideal CL in tenths of CL, the fifth and sixth digits give the maxthickness in percentage of chord.

An improvement over 1-series aerofoils with emphasis on maximizing laminar flow. Theaerofoil is described using six digits in the following sequence:

1. The number “6” indicat ing the series.2. One digit describing the distance of the minimum pressure area in tens of percentageof chord.3. The subscript digit gives the range of lift coefficient in tenths above and below thedesign lift coefficient in which favorable pressure gradients exist on both surfaces.4. A hyphen.5. One digit describing the design lift coefficient in tenths.6. Two digits describing the maximum thickness in tens of percentage of chord.

For example, the NACA 612-315 a = 0.5 has the area of minimum pressure 10% of the chordback, maintains low drag 0.2 above and below the lift coefficient of 0.3, has a maximumthickness of 15% of the chord, and maintains laminar flow over 50% of the chord.

After the six-series sect ions, aerofoil design became much more specialized for thepart icular applicat ion. Aerofoils with good transonic performance, good maximum lift capability,very thick sect ions and very low drag sect ions are now designed for each use. Often a wingdesign begins with the definit ion of several aerofoil sect ions and then the ent ire geometry ismodified based on its 3-dimensional characterist ics.

NACA 7-Digit Series:Further advancement in maximizing laminar flow has been achieved by separately ident ifyingthe low pressure zones on upper and lower surfaces of the aerofoil. The aerofoil is describedby seven digits in the following sequence:

1. The number “7” indicat ing the series.2. One digit describing the distance of the minimum pressure area on the upper surface intens of percentage of chord.3. One digit describing the distance of the minimum pressure area on the lower surface intens of percentage of chord.4. One let ter referring to a standard profile from the earlier NACA series.5. One digit describing the lift coefficient in tenths.6. Two digits describing the maximum thickness in tens of percentage of chord.7. “a =” followed by a decimal number describing the fract ion of chord over which laminarflow is maintained. a = 1 is the default if no value is given.

For example, the NACA 712A315 has the area of minimum pressure 10% of the chord back onthe upper surface and 20% of the chord back on the lower surface, uses the standard “A”profile, has a lift coefficient of 0.3, and has a maximum thickness of 15% of the chord.

NACA 8-Digit Series:Eight digit series profiles are supercrit ical aerofoils designed to independent ly maximize airflowabove and below the wing. The numbering is ident ical to the 7-series aerofoils except that thesequence begins with an “8” to ident ify the series.

However, because of the rapid improvements in computer hardware and computer software,and because of the broad use of sophist icated numerical codes, we often encounter aerofoilsect ions being developed that are not described by the standard NACA geometries.

4.16.2 NASA AerofoilsA concerted effort within the Nat ional Aeronaut ics and Space Administrat ion (NASA) duringthe 1960s and 1970s was directed toward developing pract ical aerofoils with two-dimensionalt ransonic turbulent flow and improved drag divergence Mach numbers while retainingacceptable low-speed maximum lift and stall characterist ics and focused on a concept referredto as the supercrit ical aerofoil. This dist inct ive aerofoil shape, based on the concept of localsupersonic flow with isentropic recompression, was characterized by a large leading-edgeradius, reduced curvature over the middle region of the upper surface, and substant ial aftcamber.

The early phase of this effort was successful in significant ly extending drag-rise Machnumbers beyond those of convent ional aerofoils such as the Nat ional Advisory Commit tee forAeronaut ics (NACA) 6-series aerofoils. These early supercrit ical aerofoils (denoted by the SC(phase 1) prefix), however, experienced a gradual increase in drag at Mach numbers justpreceding drag divergence (referred to as drag creep). This gradual buildup of drag was largelyassociated with an intermediate off-design second velocity peak (an accelerat ion of the flowover the rear upper-surface port ion of the aerofoil just before the final recompression at thetrailing edge) and relat ively weak shock waves above the upper surface.

Improvements to these early, phase 1 aerofoils resulted in aerofoils with significant ly reduceddrag creep characterist ics. These early, phase 1 aerofoils and the improved phase 1 aerofoilswere developed before adequate theoret ical analysis codes were available and resulted fromiterat ive contour modificat ions during wind-tunnel test ing. The process consisted of evaluat ingexperimental pressure distribut ions at design and off-design condit ions and physically alteringthe aerofoil profiles to yield the best drag characterist ics over a range of experimental testcondit ions.

The insight gained and the design guidelines that were recognized during these early phase1 invest igat ions, together with t ransonic, viscous, aerofoil analysis codes developed during thesame t ime period, resulted in the design of a matrix of family-related supercrit ical aerofoils(denoted by the SC (phase 2) prefix). Specific details about these profiles can be found inReference 1.

4.16.3 Leading-Edge Radius and Chord LineT he chord line is defined as the shortest (straight) line connect ing the leading and trailingedges. The leading edge of aerofoils used in subsonic applicat ions is rounded, with a radius ofabout 1% of the chord length. The leading edge of an aerofoil is the radius of a circle with itscenter on a line tangent ial to the leading-edge camber connect ing tangency points of theupper and lower surfaces with the leading edge. The magnitude of the leading-edge radius hassignificant effect on the stall characterist ics of the aerofoil sect ion.

The geometrical angle of attack α is the angle between the chord line and the direct ion ofthe undisturbed freestream.

4.16.4 Mean Camber Line

Mean camber line is the locus of the points midway between the upper and lower surfaces ofthe aerofoil. In other words, mean camber line is the bisector of the aerofoil thickness. Theshape of the mean camber line plays an important role in the determinat ion of theaerodynamic characterist ics of the aerofoil sect ion. One of the primary effects of camber is tochange the zero-lift angle of at tack, α0l. For symmetrical aerofoils, zero lift is at α = 0 and forcambered aerofoils, zero lift is at negat ive α for posit ive camber and vice versa.

The camber has a beneficial effect on the maximum value of the sect ion lift coefficient . If themaximum lift coefficient is high, the stall speed will be low, all other factors being the same.However, it is essent ial to note that the high thickness and camber necessary for highmaximum value of sect ion lift coefficient produce low crit ical Mach numbers at high twist ingmoments at high speeds.

4.16.5 Thickness DistributionThe thickness distribut ion and the maximum thickness strongly influence the aerodynamicscharacterist ics of the aerofoil sect ion. The maximum local velocity to which a fluid elementaccelerates as it flows around an aerofoil increases as the maximum thickness increases, inaccordance with the area–velocity relat ion for subsonic flow. Thus the value of the minimumpressure is the smallest for the thickest aerofoil. As a result , the adverse pressure gradientassociated with the decelerat ion of the flow, from the locat ion of this pressure minimum to thetrailing edge, is greatest for the thick aerofoil. As the adverse pressure gradient become larger,the boundary layer becomes thicker. This thickening of boundary layer is likely to cause flowseparat ion, leading to large increase of form drag. Thus, the beneficial effects of increasing themaximum thickness are limited.

For a thin aerofoil sect ion, with relat ively small leading-edge radius, boundary layerseparat ion occurs early, not far from the leading edge of the upper (leeward) surface. Becauseof this, the maximum sect ion lift coefficient for a thin aerofoil sect ion is relat ively small. Themaximum sect ion lift coefficient increases as the thickness rat io increases.

The thickness distribut ion for an aerofoil affects the pressure distribut ion and the characterof the boundary layer. As the locat ion of the maximum thickness moves aft , the velocitygradient in the mid-chord region decreases. The favorable pressure gradient associated withthis decrease of velocity gradient in the mid-chord region promotes the boundary layer stabilityand increases the possibility of boundary layer remaining laminar. As we know, the skin frict iondrag associated with laminar boundary layer is less than that caused by turbulent boundarylayer. Further thicker aerofoils benefit more from the use of high-lift devices but have a lowercrit ical Mach number.

4.16.6 Trailing-Edge AngleThe trailing-edge angle influences the locat ion of the aerodynamic center; the point aboutwhich the sect ion moment coefficient is independent of angle of angle of at tack, α. Theaerodynamic center of this aerofoil sect ion in a subsonic flow is theoret ically located at thequarter-chord point .

4.17 Wing Geometrical ParametersAircraft wings are made up of aerofoil sect ions, placed along the span. In an aircraft , thegeometry of the horizontal and vert ical tails, high-lift ing devices such as flaps on the wings andtails and control surfaces such as ailerons are also made by placing the aerofoil sect ions inspanwise combinat ions.

The relevant parameters used to define the aerodynamic characterist ics of a wing ofrectangular, unswept t rapezoidal, swept and delta configurat ions are illustrated in Figure 4.25.

Figure 4.25 Geometric parameters of some wing planforms.

Wing Area SThis is the plan surface area of the wing. Thus, the representat ive area of the wing may beregarded as the product of the span (2b) and the average chord ( ). Although a port ion of thearea may be covered by fuselage, the pressure distribut ion over the fuselage surface isaccounted in the representat ive wing area.

Wing Span 2bThis is the distance between the t ips of port and starboard wings.

Average Chord This is the geometric average of the chord distribut ion over the length of the wing span.

Aspect Rat io Aspect rat io is the rat io of the span and the average chord. For a rectangular wing, the aspectrat io is:

For a nonrectangular wing:

T he is a fineness rat io of the wing and it varies from 35 for sailplanes to about 2 forsupersonic fighter planes.

Root Chord cr

Root chord is the chord at the wing centerline, that is, at the middle of the span, as shown inFigure 4.25. The tip chord ct is the chord at the wing t ip.

Taper Rat io λTaper rat io is the rat io of the t ip chord to root chord, for the wing planforms with straightleading and trailing edges.

The taper rat io affects the lift distribut ion of the wing. A rectangular wing has a taper rat io of1.0 while the pointed t ip delta wing has a taper rat io of 0.0.

Sweep Angle

Sweep angle is usually measured as the angle between the line of 25% chord and aperpendicular to the root chord. The sweep of a wing affects the changes in maximum lift , thestall characterist ics, and the effects of compressibility.

Mean Aerodynamic Chord macMean aerodynamic chord is an average chord which, when mult iplied by the product of theaverage sect ion moment coefficient , the dynamic pressure, and the wing area, gives themoment for the ent ire wing. The mean aerodynamic chord is given by:

Dihedral AngleDihedral angle is the angle between a horizontal plane containing the root chord and a planemidway between the upper and lower surfaces of the wing. If the wing lies below the horizontalplane, it is termed as anhedral angle. The dihedral angle affects the lateral stability of theaircraft .

Geometric TwistGeometric twist defines the situat ion where the chord lines for the spanwise distribut ion of allthe aerofoil sect ions do not lie in the same plane. Thus, there is a spanwise variat ion in thegeometric angle of incidence for the sect ions. The chord of the root sect ion of the wing shownin Figure 4.26 is inclined at 4 relat ive to the vehicle axis.

Figure 4.26 Unswept, tapered wing with geometric twist (wash-out).

The chord at the wing t ip, however, is parallel to the longitudinal axis of the vehicle. In thiscase, where the incidence of the aerofoil sect ions relat ive to the vehicle axis decrease towardthe t ip, the wing has a “wash-out.” The wings of most subsonic aircraft have wash-out tocontrol the spanwise lift distribut ion and, hence, the boundary layer characterist ics. If the angleof incidence increases toward wing t ip, the wing has “wash-in.”

Example 4.6For a wing with root chord 18 m, t ip chord 3.5 m and span 25 m, calculate the wing area, aspectrat io, taper rat io and the mean aerodynamic chord.

SolutionGiven, cr = 18 m, ct = 3.5 m, 2b = 25 m.The average chord is:

Therefore, the wing area becomes:

The aspect rat io is:

The taper rat io is:

For calculat ing the mac, the expression for the chord as a funct ion of the distance from theplane of symmetry is required. The required expression for the given wing is:

The mean aerodynamic chord becomes:

4.18 Aerodynamic Force and Moment CoefficientsFor thin aerofoils at low angle of at tack, the lift results mainly from the pressure distribut ion(that is, pressure forces), as shown in Figure 4.19. The shear forces due to viscosity actsprimarily in the chordwise direct ion and contributes mainly to the drag. Therefore, we need toconsider only the pressure contribut ions, illustrated in Figure 4.27, to calculate the force in the

z-direct ion.

Figure 4.27 Pressure distribut ion over an aerofoil.

Let us consider a wing sect ion in a flow field, as shown in Figure 4.28. Let p be the pressureact ing on a different ial area (dxdy).

Figure 4.28 Pressure act ing over an elemental surface area of a rectangular wing.

The pressure force component in the z-direct ion is:

(4.19) The net force in the z-direct ion act ing over the ent ire wing surface (that is, over the upper andlower surfaces) is given by:

(4.20) But the resultant force in any direct ion due to a constant pressure over a closed surface iszero. Therefore:

(4.21) where p∞ is the freestream pressure.

Combining Equat ions (4.20) and (4.21), the resultant force component can be expressed as:

(4.22) This equat ion can be nondimensionalized by a suitable reference force. For this, the product offreestream dynamic pressure q∞, wing chord c and wing span 2b can be taken as the referenceforce. Thus dividing the left -hand and right-hand sides of Equat ion (4.22) by q∞c2b, we have:

But c × 2b = S and (p − p∞)/q∞ = Cp, therefore:

(4.23) When the boundary layer is thin, the pressure distribut ion around the aerofoil can be

regarded equivalent to that due to an inviscid flow. Thus, the pressure distribut ion isindependent of Reynolds number and does not depend on whether the boundary layer islaminar or turbulent. When the boundary layer is thin, the pressure coefficient at a part icularlocat ion on the surface, given by the coordinates (x/c, y/2b), is independent of vehicle scale andof the flow condit ions. Over a range of flow condit ions for which the pressure coefficient is aunique funct ion of the dimensionless coordinates (x/c, y/2b), the value of the integral Equat ion(4.23) depends only on the aerofoil geometry and the angle of at tack. Thus, the result ingdimensionless force coefficient is independent of model scale and flow condit ions.

A similar analysis can be used to calculate the lift coefficient, defined as:

(4.24) where L is the lift and S is the wing planform area.

In the same manner, lift coefficient per unit length of wing span, termed section lift coefficientcan be expressed as:

(4.25) where l is the lift per unit length of wing span and c is the chord length.

Typical variat ion of sect ion lift coefficient with angle of at tack is shown in Figure 4.29.

Figure 4.29 Theoret ical variat ion of sect ion lift coefficient with angle of at tack.

It has been experimentally verified that the sect ion lift coefficient is a linear funct ion of angleof at tack α from −10 to +10 . The slope of the linear port ion of the curve is called the two-dimensional lift curve slope. Theoret ical value of two-dimensional lift curve slope is 2π, that is:

The sect ion lift coefficient for a wing which experiences zero lift at α = 0 can be expressed as:

For a wing which experiences zero lift at an angle of α = α0l, the sect ion lift coefficient can beexpressed as:

(4.26)

Example 4.7Air at 300 km/h, 1 atm and 30 C flows over a two-dimensional rectangular wing of chord 1.2 m.If the zero lift angle of at tack for the wing is −2 and Cl,α = 0.10, calculate the left act ing on thewing per unit span when the angle of at tack is 3 .

SolutionGiven, V = 300/3.6 = 83.33 m/s, p = 101325 Pa, T = 30 + 273.15 = 303.15 K, α0,l = − 2 , α = 3 .The sect ion coefficient , by Equat ion (4.26), is:

The corresponding lift is:

The freestream density is:

Therefore:

4.18.1 Moment CoefficientThe moments due to the aerodynamic forces act ing on the wing are usually determined aboutone of the two reference axes, namely the axes passing through the leading edge and theaerodynamic center. The moments obtained are nondimensionalized following a similarprocedure that was used to nondimensionalize the lift . Let us calculate the pitching momentabout the leading edge due to the pressure act ing on the surface of the wing, shown in Figure4.28. Let us assume that the contribut ion of chordwise component of the pressure force to themoment is negligibly small. Thus, the pitching moment about the leading edge due to thepressure force act ing on the surface element of area (dxdy) located at a distance x from theleading edge is:

(4.27) The net pitching moment due to the pressure force act ing on the wing can be obtained byintegrat ing Equat ion (4.27) over the ent ire wing surface. The net pitching moment is given by:

(4.28) when a uniform pressure acts on any closed surface, the resultant force due to this constantpressure is zero. Thus,

(4.29) Combining Equat ions (4.28) and (4.29), the resultant moment about the leading edge can beexpressed as:

(4.30) To make this moment relat ion nondimensional, let us divide both sides by q∞c22b:

The product of c × 2b = S and (p − p∞)/q∞ = Cp, thus:

(4.31) Thus, the pitching moment coefficient becomes:

(4.32) In Equat ion (4.32) the wing chord c is used as a parameter because this derivat ion is for arectangular wing in Figure 4.28. That is the mean aerodynamic chord is used together withwing area S to nondimensionalize the pitching moment.

The pitching moment coefficient for a wing sect ion of unit span, referred to as sectionpitching moment coefficient, becomes:

(4.33) where m0 is the sect ion pitching moment. The sect ion pitching moment coefficient depends onthe camber and thickness rat io of the wing. Similar to sect ion lift coefficient cl, sect ion pitchingmoment coefficient cm about the aerodynamic center is independent of angle of at tack.Typical variat ion of sect ion pitching moment coefficient about the aerodynamic center withangle of at tack is as shown in Figure 4.30.

Figure 4.30 Variat ion of sect ion pitching moment coefficient with angle of at tack.

Thus, the aerodynamic center is that point along the chord where all changes in lifteffect ively take place. Since the moment about the aerodynamic center is the product of aforce (the lift that acts at the center of pressure) and an arm length (the distance from theaerodynamic center to the center of pressure), the center of pressure must move toward theaerodynamic center as the lift increases.

4.19 SummaryTransformat ion of a flow pattern essent ially amounts to the transformat ion of a set ofstreamlines and potent ial lines, whilst the t ransformat ion of individual lines implies thetransformat ion of a number of points.

To transform the points specified by the Cartesian coordinates x and y, in the physical plane,given by z = x + iy, to a t ransformed plane given by ζ = ξ + iη we need to expand thetransformat ion funct ion ζ = f(z) = ξ + iη, equate the real and imaginary parts and find thefunct ional form of ξ and η, in terms of x and y.

For a given flow pattern in the physical plane, each streamline of the flow can berepresented by a separate stream funct ion. Transferring these stream funct ions, using thetransformat ion funct ion, ζ = f(z), the corresponding streamlines in the transformed plane canbe obtained.

The main use of conformal t ransformat ion in aerodynamics is to t ransform a complicated

flow field into a simpler one, which is amenable to simpler mathematical t reatment.The main problem associated with this t ransformat ion is finding the best t ransformat ion

funct ion (formula) to perform the required operat ion. Even though a large number ofmathematical funct ions can be envisaged for a specific t ransformat ion.

A transformat ion, which generates a family of aerofoil shaped curves, along with theirassociated flow patterns, by applying a certain t ransformat ion to consolidate the theorypresented in the previous sect ions, is the Kutta−Joukowski transformation.

Kutta−Joukowski t ransformat ion is the simplest of all t ransformat ions developed forgenerat ing aerofoil shaped contours. Kutta used this t ransformat ion to study circular-arc wingsect ions, while Joukowski showed how this t ransformat ion could be extended to produce wingsect ions with thickness t as well as camber.

In our discussion on Kutta−Joukowski t ransformat ion, it is important to note the following:The circle considered, in the physical plane, is a specific streamline. Essent ially the circle isthe stagnat ion streamline of the flow in the original plane 1 (z-plane).The transformat ion can be applied to the circle and all other streamlines, around thecircle, to generate the aerofoil and the corresponding streamlines in plane 2 (ζ-plane) orthe transformed plane. That is, the t ransformat ion can result in the desired aerofoil shapeand the streamlines of the flow around the aerofoil.

It is convenient to use polar coordinates in the z-plane and Cartesian coordinates in ζ-plane.The Kutta−Joukowski transformation function is:

where b is a constant.Now, expressing z as z = reiθ, where r and θ are the polar coordinates, and on expanding, we

get:

Equat ing the real and imaginary parts, we get:

These expressions for ξ and η are the general expressions for the t ransformat ion of the basicshape, namely the circle in the z-plane, to any desired shape in the ζ-plane.

For t ransforming a circle of radius a to a straight line, the constant b in the Joukowskit ransformat ion funct ion should be set equal to a, and the center of the circle should be at theorigin.

For t ransforming a circle to an ellipse using the Kutta−Joukowski t ransformat ion funct ion:

the circle should have its center at the origin in the z-plane, but the radius of the circle shouldbe greater than the constant b, in the above transformat ion funct ion, that is, a > b.

To transform a circle into a symmetrical aerofoil, the center of the circle in the z-plane shouldbe shifted from the origin and located slight ly downstream of the origin, on the x-axis. This shiftwould cause asymmetry to the profile (about the ordinates of the t ransformed plane) of thetransformed shape obtained with the Kutta−Joukowski t ransformat ion funct ion.

are the coordinates of a symmetrical aerofoil profile. The chord of the aerofoil is 4b. Themaximum thickness of the aerofoil occurs where dη/dθ = 0.

The maximum thickness is at the chord locat ion, given by:

This point (b, 0), from the leading edge of the aerofoil, is the quarter chord point .The thickness to chord rat io of the aerofoil is:

At the t railing edge of the aerofoil, the slope of its upper and lower surfaces merge. This kindof t railing edge would ensure that the flow will leave the trailing edge without separat ion. Butthis is possible only when the trailing edge is cusped with zero thickness. Thus, this is only amathematical model. For actual aerofoils, the t railing edge will have a finite thickness, andhence, there is bound to be some separat ion, even for the thinnest possible t railing edge.

For t ransforming a circle to a cambered aerofoil, using Joukowski t ransformat ion, the centerof the circle in the physical plane has to be shifted to a point in one of the quadrants.

are the coordinates represent ing a cambered aerofoil.The thickness-to-chord rat io for a cambered aerofoil is:

The thickness-to-chord rat io is maximum at θu = 60 . Thus:

This maximum is also at the quarter chord point , as in the case of symmetrical aerofoil.The camber of an aerofoil is the maximum displacement of the mean camber line from the

chord. The mean camber line is the locus of mid-points of lines drawn perpendicular to thechord. In other words, the camber line is the bisector of the aerofoil profile thickness distribut ionfrom the leading edge to the trailing edge.

Transformat ion of a circle with its center shifted above (or below) the origin, on the ordinatein the z-plane, with the transformat ion funct ion ζ = z + b2/z results in a circular arc.

where ξ and η, respect ively, are the expressions for the abscissa and ordinates of the circulararc, in the t ransformed plane.

For the transformed circular arc, the chord is 4b.

But for small β, tan β ≈ β. Therefore, the percentage camber for the circular arc becomes 100β/2.

The postulat ion that “the aerofoil generates sufficient circulat ion to depress the rearstagnat ion point from its posit ion, in the absence of circulat ion, down to the sharp t railingedge” is known as Joukowski hypothesis. This condit ion of realizing full Joukowski circulat ion,

result ing in flow without wake is known as Kutta condition.The Kutta condit ion can be stated as follows:

“A body with a sharp trailing edge which is moving through a fluid will create about itself acirculation of sufficient strength to hold the rear stagnation point at the trailing edge.”The Kutta condit ion is significant when using the Kutta−Joukowski theorem to calculate the

lift generated by an aerofoil. The value of circulat ion of the flow around the aerofoil must bethat value which would cause the Kutta condit ion to exist .

The Kutta condit ion allows an aerodynamicist to incorporate a significant effect of viscositywhile neglect ing viscous effects in the conservat ion of momentum equat ion. It is important inthe pract ical calculat ion of lift on a wing.

The Kutta condit ion does not apply to unsteady flow. Mathematically, the Kutta condit ionenforces a specific choice among the infinite allowed values of circulat ion.

Joukowski hypothesis direct ly relates the lift generated by a two-dimensional aerofoil to itsincidence, as well as indicates the significance of the thickness to chord rat io and camber ofthe aerofoil in the lift generat ion.

As the flow process develops, which takes place very quickly, the circulat ion around theaerofoil sect ion is generated and the aerofoil experiences lift .

where CL is the lift coefficient and S is the projected area of the wing planform, normal to thedirect ion of freestream flow. The area S is given by:

The chord of the profile is 4b. Therefore, the area of the profile, per unit span, becomes S = 4b.Thus, the lift is:

Equat ing the above two expressions for lift , we get the lift coefficient as:

But,

Therefore,

This is the lift coefficient of a two-dimensional aerofoil, in terms of thickness to chord rat io (t/c= 1.299e), percentage camber (100 β/2), and angle of incidence α.

The ideal lift curve slope, a∞I, for small values of eccentricity e, angle of incidence α andcamber β, is:

This is the theoret ical value of lift curve slope per radian of angle of at tack, α. It is seen that,the lift curve slope is independent of the angle of at tack.

The coordinates of the t ransformed aerofoil profile are:

With a = (b + be), the above coordinate expressions become:

For finding the velocity distribut ion around a given aerofoil, it is necessary to relate the angleof incidence α to the circulat ion Γ around the aerofoil. This is done by applying the Joukowskihypothesis. In reality, the full Joukowski circulat ion required to bring the rear stagnat ion point to

the t railing edge is not realized, because of the following:Air is a viscous fluid, and the flow near the trailing edge of an aerofoil is modified by thepresence of the boundary layer and wake, caused by the viscosity.The zero thickness for the t railing edge, st ipulated by the Joukowski hypothesis, is notpossible in pract ice. Therefore, the t railing edge must be rounded to some degree ofcurvature. The finite thickness of the t railing edge owing to this rounding-off forces therear stagnat ion point to deviate from the posit ion given in the ideal case.

Therefore, if Γ is the full Joukowski circulat ion (theoret ical circulat ion), it can be assumed that,the pract ical value of circulat ion is only kΓ, where k is less than unity.

The velocity on the aerofoil is:

For incompressible flow, the pressure coefficient is given by:

Knowing the distribut ion of Va/V∞, over the aerofoil, the pressure coefficient around the aerofoilcan be est imated.

An aerofoil is a streamlined body that would experience the largest value of lift -to-drag rat io,in a given flow, compared to any other body in the same flow. In other words, in a given flow theaerodynamic efficiency (L/D) of an aerofoil will be the maximum.

The pressure and shear forces can be integrated over the surface of the aerofoil to obtainthe resultant aerodynamic force, F ad, which acts at the center of pressure (kcp) of the aerofoil.

The forces act ing on an aircraft in level flight are the lift L, drag D, thrust T and weight W.For a steady, unaccelerated level flight in a horizontal plane:

The sum of the forces along the flight path is zero.The sum of the forces perpendicular the flight path is zero.

When the angle of incidence α is small, the component of thrust parallel to the freestream flowdirect ion is only slight ly less than the thrust itself. Therefore, for equilibrium:

The lift act ing ahead of center of gravity will produce a nose-up (posit ive) moment about thecenter of gravity. The aircraft is said to be trimmed, when the sum of the moments about thecg is zero, that is:

Thus, a force from a control surface located aft of the cg, for example, the horizontal tailsurface Lt is required to produce a nose-down (negat ive) pitching moment about the cg, whichcould balance the posit ive moment produced by Lwb. The tail surface producing Lt alsoproduces a drag force which is known as the trim drag. The trim drag may vary from 0.5% to5% of the total drag of the aircraft .

In addit ion to the lift and drag act ing in the pitch or xz-plane, there is a side force act ing onthe aircraft . The side force is the component of force in the direct ion perpendicular to both thelift and the drag. The side force act ing towards the starboard (right) wing is referred to asposit ive.

Usually the aerodynamic force will not act through the cg (which is also taken as the origin ofthe airplane's axis system). The moment due to the resultant force acts at a distance from theorigin may be divided into three components, referred to the airplane's axes. The threemoment components are the pitching moment M, the rolling moment L and the yawing

moment N:Pitching moment is the moment act ing about the lateral axis (y-axis). It is the moment dueto the lift and drag act ing on the aircraft . Pitching moment causing nose-up is regardedposit ive.Rolling moment is the moment act ing about the longitudinal axis (x-axis) of the aircraft .Rolling moment is generated by a different ial lift generated by the ailerons, located closerto the wingt ips. Rolling moment causing the right (starboard) wingt ip to move downwardis regarded posit ive.Yawing moment is the moment act ing about the vert ical (z-axis) of the aircraft . Yawingmoment tends to rotate the aircraft nose to the right is regarded posit ive.

The primary parameters governing the magnitude of the aerodynamic forces and momentsare the following:

Geometry of the aerofoil.Angle of at tack, namely the aircraft at t itude in the pitch (xz) plane relat ive to the flightdirect ion.Vehicle size.Freestream velocity.Freestream flow density.Reynolds number (viscous effects).Mach number (compressibility effects).

The geometrical sect ion of a wing obtained by cutt ing it by a vert ical plane parallel to thecenterline of the aircraft is called aerofoil section. The lift generated and the stallcharacterist ics of a wing strongly depends on the geometry of the aerofoil sect ions that makeup the wing. The geometric parameters that dictate the aerodynamic characterist ics of theaerofoil sect ion are; the leading-edge radius, the mean camber line, the maximum thicknessand the thickness distribut ion of the profile, and the trailing-edge angle.

The geometry of many aerofoil sect ions is uniquely defined by the NACA designat ion for theaerofoil.

Aerofoil geometry are usually characterized by the coordinates of the upper and lowersurface. It is often summarized by a few parameters such as: maximum thickness, maximumcamber, posit ion of max thickness, posit ion of max camber, and nose radius.

The shape of the NACA aerofoils is described using a series of digits following the word“NACA.”

In NACA 4-Digit series the first digit implies the maximum camber in percentage of chord (c),the second digit gives the posit ion of maximum camber in 1/10 of chord, the last two digits givethe maximum thickness in percentage of chord. For example,

In NACA 5-digit series the first digit gives approximate maximum camber in percentage ofchord, the second and third digits give the posit ion of maximum camber in 2/100 of chord andthe last two digits give the maximum thickness in percentage of chord. This NACA 23012 is anaerofoil with maximum camber as 2% of c, posit ion of maximum camber at 60% of chord andt/c = 0.12.

Four-and five-digit series aerofoils can be modified with a two-digit code preceded by ahyphen in the following sequence:

1. One digit describing the roundness of the leading edge with 0 being sharp, 6 being thesame as the original aerofoil, and larger values indicat ing a more rounded leading edge.2. One digit describing the distance of maximum thickness from the leading edge in tensof percentage of the chord.

1-series:A new approach to aerofoil design pioneered in the 1930s in which the aerofoil shape wasmathematically derived from the desired lift characterist ics. Prior to this, aerofoil shapes were

first created and then had their characterist ics measured in a wind tunnel. The 1-seriesaerofoils are described by five digits in the following sequence:

1. The number “1” indicat ing the series.2. One digit describing the distance of the minimum pressure area in tens of percent ofchord.3. A hyphen.4. One digit describing the lift coefficient in tenths.5. Two digits describing the maximum thickness in percentage of chord.

In NACA 6-digit series the first digit refers to the series, the second digit gives the locat ion ofminimum Cp in 1/10 chord, the third digit gives the half width of low drag bucket in 1/10 of CL,the fourth digit gives the ideal CL in tenths of CL, the fifth and sixth digits give the maxthickness in percentage of chord.

In NACA 7-Digit Series:1. The number “7” indicat ing the series.2. One digit describing the distance of the minimum pressure area on the upper surface intens of percentage of chord.3. One digit describing the distance of the minimum pressure area on the lower surface intens of percentage of chord.4. One let ter referring to a standard profile from the earlier NACA series.5. One digit describing the lift coefficient in tenths.6. Two digits describing the maximum thickness in tens of percentage of chord.7. “a =” followed by a decimal number describing the fract ion of chord over which laminarflow is maintained. a = 1 is the default if no value is given.

In NACA 8-Digit series profiles are supercrit ical aerofoils designed to independent ly maximizeairflow above and below the wing. The numbering is ident ical to the 7-series aerofoils exceptthat the sequence begins with an “8” to ident ify the series.

NASA AerofoilsA concerted effort within the Nat ional Aeronaut ics and Space Administrat ion (NASA) duringthe 1960s and 1970s was directed toward developing pract ical aerofoils with two-dimensionalt ransonic turbulent flow and improved drag divergence Mach numbers while retainingacceptable low-speed maximum lift and stall characterist ics and focused on a concept referredto as the supercrit ical aerofoil. This dist inct ive aerofoil shape, based on the concept of localsupersonic flow with isentropic recompression, was characterized by a large leading-edgeradius, reduced curvature over the middle region of the upper surface, and substant ial aftcamber.

The early phase of this effort was successful in significant ly extending drag-rise Machnumbers beyond those of convent ional aerofoils such as the Nat ional Advisory Commit tee forAeronaut ics (NACA) 6-series aerofoils. These early supercrit ical aerofoils (denoted by the SC(phase 1) prefix), however, experienced a gradual increase in drag at Mach numbers justpreceding drag divergence (referred to as drag creep). This gradual buildup of drag was largelyassociated with an intermediate off-design second velocity peak (an accelerat ion of the flowover the rear upper-surface port ion of the aerofoil just before the final recompression at thetrailing edge) and relat ively weak shock waves above the upper surface.

The chord line is defined as the shortest (straight) line connect ing the leading and trailingedges.

The geometrical angle of attack α is the angle between the chord line and the direct ion ofthe undisturbed freestream.

Mean camber line is the locus of the points midway between the upper and lower surfacesof the aerofoil. One of the primary effects of camber is to change the zero-lift angle of at tack,

α0l. For symmetrical aerofoils, zero lift is at α = 0 and for cambered aerofoils, zero lift is atnegat ive α for posit ive camber and vice versa.

The thickness distribut ion and the maximum thickness strongly influence the aerodynamicscharacterist ics of the aerofoil sect ion. The thickness distribut ion for an aerofoil affects thepressure distribut ion and the character of the boundary layer.

The trailing-edge angle influences the locat ion of the aerodynamic center; the point aboutwhich the sect ion moment coefficient is independent of angle of angle of at tack, α. Theaerodynamic center of this aerofoil sect ions in a subsonic flow is theoret ically located at thequarter-chord point .

Aircraft wings are made up of aerofoil sect ions discussed in the preceding sect ion. Aerofoilsect ions are placed along the span of a wing.

The relevant parameters used to define the aerodynamic characterist ics of a wing arerectangular, unswept t rapezoidal, swept and delta configurat ions.

Wing area is the plan surface area of the wing.Wing span is the distance between the t ips of port and starboard wings.Average chord is the geometric average of the chord distribut ion over the length of the wing

span.Aspect ratio is the rat io of the span and the average chord.Root chord is the chord at the wing centerline. Tip chord the chord at the wing t ip.Taper ratio is the rat io of the t ip chord to root chord.Sweep angle is usually measured as the angle between the line of 25% chord and a

perpendicular to the root chord.Mean aerodynamic chord is an average chord which, when mult iplied by the product of the

average sect ion moment coefficient , the dynamic pressure, and the wing area, gives themoment for the ent ire wing.

Dihedral angle is the angle between a horizontal plane containing the root chord and aplane midway between the upper and lower surfaces of the wing. If the wing lies below thehorizontal plane, it is termed an anhedral angle.

Geometric twist defines the situat ion where the chord lines for the spanwise distribut ion ofall the aerofoil sect ions do not lie in the same plane.

If the incidence of the aerofoil sect ions relat ive to the vehicle axis decrease toward the t ip,the wing has a “wash-out.” If the angle of incidence increases toward wing t ip, the wing has“wash-in.”

For thin aerofoils at low angle of at tack, the lift results mainly from the pressure distribut ion(that is, pressure forces). The shear forces due to viscosity acts primarily in the chordwisedirect ion and contributes mainly to the drag.

When the boundary layer is thin, the pressure distribut ion around the aerofoil can beregarded equivalent to that due to an inviscid flow. Thus, the pressure distribut ion isindependent of Reynolds number and does not depend on whether the boundary layer islaminar or turbulent.

The lift coefficient, is defined:

where L is the lift and S is the wing planform area.Theoret ical value of two-dimensional lift curve slope is 2π, that is:

The moments due to the aerodynamic forces act ing on the wing are usually determinedabout one of the two reference axes, namely the axes passing through the leading edge andthe aerodynamic center. The moments obtained are nondimensionalized following a similarprocedure that was used to nondimensionalize the lift .

The net pitching moment is given by:

The pitching moment coefficient is:

Exercise Problems1. If the pressure coefficient at a point on an aerofoil in a freestream flow of speed 70 m/s is−3.7, determine the flow velocity at that point .

[Answer: 151.76 m/s]

2. A two-dimensional wing of span 20 m and chord 1.1 m, thickness-to-chord rat io 12% andpercentage camber 10% at an angle of at tack of 3 in a uniform air stream at 1 atm and 22

C and 220 km/h experiences a lift of 15 000 N. Determine the circulat ion around the wing andthe lift coefficient , assuming the flow as incompressible and inviscid.

[Answer: Γ = 10.26 m2/s, CL = 0.373]

3. A tapered wing of span 15 m and root chord 4 m has a planform area of 37.5 m2. Find (a)the t ip chord and (b) the mean aerodynamic chord of the wing.

[Answer: (a) 1 m, (b) 2.8 m]

4. A wing of taper rat io 3 has a planform area of 45 m2. If the span is 16 m, determine the rootand t ip chords of the wing.

[Answer: 4.218 m, 1.406 m]

5. If the sect ion lift act ing on a two-dimensional wing of chord 2 m, flying at 250 km/h in sealevel alt itude is 3000 N/m, when the angle of at tack is 4 and sect ion lift curve slope is 0.11,determine the zero lift angle of at tack of the wing.

[Answer: −0 . 618 ]

6. The sect ion lift coefficient of a two-dimensional wing flying at 3 . 2 angle of at tack is 0.6. Ifthe zero lift angle of at tack is −1 . 8 , determine the lift curve slope of the wing.

[Answer: 0.12]

7. A tapered wing of t ip chord 3 m has wing area 220 m2 and aspect rat io 4. Find (a) the rootchord and (b) the wing span. (c) Also, find the expression for the chord in terms of t ransversecoordinate.

[Answer: (a) 11.83 m, (b) 29.66 m, (c) 11.83 − 0.595 y]

8. A NACA 612-415 aerofoil flies at 400 km/h at an alt itude where the air density is 0.082kg/m3. Determine the wing loading of the aerofoil.

[Answer: 202.5 N/m2]

9. Ident ify the number of the NACA profile with 3% maximum camber located at 30% fromthe leading edge, with a thickness of 10% of the chord.

[Answer: NACA 3310]

10. Ident ify the number of the NACA profile with 0 camber and thickness to chord rat io of12%.

[Answer: NACA 0012]

11. If the fineness rat io of the ellipse obtained by t ransforming a circle of unit radius, with

Kutta−Joukowski t ransformat ion, is 4, determine the eccentricity.[Answer: 0.29]

12. A symmetrical aerofoil is obtained by t ransforming a circle of unit radius, withKutta−Joukowski t ransformat ion funct ion. If the eccentricity is 0.1, find the maximum value ofthickness to chord rat io.

[Answer: 0.13]

13. A cambered aerofoil is obtained by t ransforming a circle of unit radius, with Joukowskit ransformat ion funct ion. If the percentage camber is 3.2, determine the locat ion of the circlecenter in the physical plane.

[Answer: (0.024, 0.064)]

14. If the maximum thickness of a 14% cambered Joukowski aerofoil is at π/2.5, determinethe eccentricity.

[Answer: 0.112]

15. If a two-dimensional Joukowski aerofoil of thickness to chord rat io 12% and camber 3% inan ideal flow experiences a lift coefficient of 0.8, determine (a) the angle of incidence and (b)the lift curve slope.

[Answer: (a) 3 . 25 , (b) 6.82]

Note

1. l’Hospital's rule can be used to circumvent the indeterminate forms such as .This rule solves the indeterminate forms by different iat ing the numerator and denominatorseparately t ill a finite form is arrived with the given condit ion.

References1. Harris, C.D., NASA Supercritical Aerofoils –A Matrix of Family-Related Airfoils , NASATechnical Paper 2969, 1990.

5

Vortex Theory

5.1 IntroductionBefore gett ing into the dynamics of vortex mot ion, it is essent ial to have a thoroughunderstanding of rotat ional and irrotat ional flows. Translat ion and rotat ion are the two types ofbasic mot ion in a fluid flow. These two may exist independent ly or simultaneously. When theycoexist they may be considered as one superimposed on the other. It should be emphasizedthat rotat ion refers to the orientat ion of a fluid element and not the path followed by theelement. Thus, for an irrotat ional flow, if a pair of small st icks were placed on a fluid element itcan be observed that the orientat ion is retained even while the fluid element moves along acircular path, as shown in Figure 5.1(a). In other words, in an irrotat ional flow, the fluid elementsdo not rotate about their own axes, that is, fluid elements do not spin in an irrotat ional flow. Butin a rotat ional flow, fluid elements rotate about their axes, as shown in Figure 5.1(b). Thus, in an

irrotat ional flow, like the one shown in Figure 5.1(a), the fluid elements move along circularpaths but do not rotate about their own axes. Thus, the angular velocity of fluid elements in anirrotat ional flow is zero. If the flow field were rotat ing like a rigid body, then the fluid elements inthe field would experience a rotat ion about their own axes, as shown in Figure 5.1(b). This typeof mot ion is termed rotational and cannot be described with a velocity potent ial.

Figure 5.1 (a) Irrotat ional flow, (b) rotat ional flow.

If the possible distort ion of the fluid elements caused by severe viscous tract ion is ignoredthen there are only three possible ways in which a fluid element can move. They are thefollowing:

1. Pure translation –the fluid elements are free to move anywhere in space but cont inueto keep their axes parallel to the reference axes fixed in space, as shown in Figure 5.2(a).The flow in the potent ial flow zone, outside the boundary layer over an aerofoil, issubstant ially this type of flow.

Figure 5.2 (a) Pure t ranslat ional mot ion, (b) pure rotat ional mot ion.

2. Pure rotation –the fluid elements rotate about their own axes which remain fixed inspace, as shown in Figure 5.2(b).3. The general motion in which translat ion and rotat ion are compounded. Such a mot ion isfound, for example, in the wake of a bluff body.

A flow in which all the fluid elements behave as in item (a) above is called potential orirrotational flow. All other flows exhibit , to a greater or lesser extent, the spinning property ofsome of the const ituent fluid elements, and are said to posses vorticity, which is theaerodynamic term for elemental spin. The flow is then termed rotational flow.

From the above descript ions it is evident that a flow is either rotat ional, possessing vort icity,or irrotat ional, for which vort icity is zero. The rotat ional and irrotat ional nature and thepropert ies of a flow can be examined analyt ically, leading to the development of characterist icequat ions governing the flow. Using these equat ions, the nature of any unknown flow can beanalyzed.

5.2 Vorticity Equation in Rectangular CoordinatesIn a two-dimensional mot ion, the vort icity at a point P, which is located perpendicular to theplane, is equal to the limit of the rat io of the circulat ion in an infinitesimal circuit embracing P tothe area of the circuit .

A flow possesses vort icity if any of its elements are rotat ing (spinning). It is a convenient wayto invest igate the mot ion of a circular element, t reat ing it as a solid, at the instant of t imeconsidered. Let P(x, y) be the center of the circular element and u and are the velocity

components, along x-and y-direct ions, respect ively, as shown in Figure 5.3.

Figure 5.3 A fluid element and appropriate coordinates and velocity components.

Let us assume that the fluid element consists of numerous fluid part icles of mass Δm each,such as one at the point Q(x + δx, y + δy). At point Q, the velocity components, along x-and y-direct ions, respect ively, are:

and

The moment of momentum (or angular momentum) of the fluid element about point P(x, y) isthe sum of the moments of momentum of all the part icles such as Q about point P. Taking theant i-clockwise moment as posit ive, we have:

Moment of momentum of the element

For a circular disc, about its center, we have:

Therefore, the angular momentum of the disc becomes:

If the disc were a solid disc, its angular momentum would be Iω, where I is its polar moment ofinert ia about P and ω its angular velocity about P. Thus, assuming the fluid element as a soliddisc, we have:

and

Thus, we have the angular momentum relat ion as:

This gives the angular velocity as:

The quant ity 2ω is the elemental spin, also referred to as vorticity, which is usually denoted as

ζ. Thus,

(5.1) The units of ζ are radian per second. From Equat ion (5.1) and the angular velocity relat ion, it isseen that:

that is, the vorticity is twice the angular velocity.

5.2.1 Vorticity Equation in Polar CoordinatesIn the polar coordinates, the vort icity equat ion can be expressed as:

(5.2) where r and θ are the polar coordinates and qt and qn are the tangent ial and normalcomponents of velocity, respect ively. The derivat ion of Equat ion (5.2) is given in Sect ion 5.3.

If (r, θ, n) are the radial, azimuthal and normal coordinates of a polar coordinates system, thevort icity expression is given by:

where ir, iθ and in are the unit vectors in the direct ions of r, θ and n, respect ively. The vort icitycomponents can be expressed as:

where ur, uθ and un are the velocity components along r, θ and n direct ions, respect ively.

Example 5.1Find the vort icity of the following flows.(a)

(b)

(c)

Solution(a) Given, u = c(x + y), .This is a two-dimensional flow in the xy-plane. Therefore, the vort icity component (withrotat ional axis in the z-direct ion, which is normal to xy-plane) is:

(b) Given, u = x + y + z + t, , .The vort icity components are:

Therefore, the vort icity becomes:

(c) Given, ur = uθ = 0, .

The vort icity components are:

Therefore, the vort icity becomes:

5.3 CirculationCirculation is the line integral of a vector field around a closed plane curve in a flow field. Bydefinit ion:

(5.3) where Γ is circulat ion, V is flow velocity tangent ial to the streamline c, encompassing the closedcurve under considerat ion, and ds is an elemental length. If a line AB forms a closed loop orcircuit in the flow, as shown in Figure 5.4, then the line integral of Equat ion (5.3) taken roundthe circuit is defined as circulat ion, that is:

Figure 5.4 A loop AB in a flow field.

where u is the component of V is the x-direct ion and is that in the y-direct ion. Note thatthe circuit is imaginary and does not influence the flow, that is, it is not a boundary. In Equat ion(5.3), both V and ds are vectors. Therefore, the dot product of V and ds results in aboveexpression for Γ.1

Circulat ion implies a component of rotation of flow in the system. This is not to say that thereare circular streamlines, or the elements, of the fluid are actually moving around some closedloop although this is a possible flow system. Circulat ion in a flow means that, the flow systemcould be resolved into an uniform irrotat ional port ion and a circulat ing port ion. Figure 5.5illustrates concept of circulat ion.

Figure 5.5 Illustrat ion of circulat ion.

This implies that, if circulat ion is present in a fluid mot ion, then vort icity must be present,even though it may be confined to a restricted space, as in the case of the circular cylinderwith circulat ion, where the vort icity at the center of the cylinder may actually be excluded fromthe region of flow considered, namely that outside the cylinder.

An alternat ive equat ion for circulat ion Γ can be obtained by considering the circuit ofintegrat ion made up of a large number of rectangular elements of sides δx and δy, as shown inFigure 5.6.

Figure 5.6 A circuit of integrat ion, c, in the flow field.

Applying the integral round the element abcd with point P(x, y) at its center, where thevelocity components are u and , as shown in Figure 5.6, we get:

Simplificat ion of this results in:

The sum of the circulat ions of all the elemental areas in the circuit const itutes the circulat ionof the circuit as a whole. As the circulat ion ΔΓ of each element is added to the ΔΓ of theneighboring element, the contribut ions of the common sides disappear. Applying this reasoningfrom an element to the neighboring element throughout the area, the only sides contribut ingto the circulat ion, when the ΔΓs of all areas are summed together, are those sides whichactually form the circuit itself. This means that, for the total circuit c, the circulat ion becomes:

and the vort icity ζ is given by:

If the strength of the circulat ion Γ remains constant whilst the circuit shrinks to encompass anelemental area, that is, unt il it shrinks to an area of the size of a rectangular element, then:

Therefore,

(5.4) This is a result which enables an easy derivat ion of the vort icity relat ion in polar coordinates.

Let us consider a segment of a fluid element of width δr, subtending angle δθ, at the originand width δr, as shown in Figure 5.7. If the segment is located at the point P(r, θ), where thenormal and tangent ial velocity components are qn and qt, respect ively, then the velocit iesalong AB, BC, CD, DA are:

where the direct ion qn is along r-direct ion and qt is along θ-direct ion. The lengths of the sidesof the elements are:

The circulat ion about the element is the line integral of the tangent ial component of flowvelocity, that is:

This simplifies to:

We know that:

Figure 5.7 A fluid element.

Also, the area of the element under considerat ion is (r δrδθ). Thus, the vort icity is:

This is the vort icity expression in polar coordinates.

5.4 Line (point) VortexA line vortex is a string of rotat ing part icles. In a line vortex, a chain of fluid part icles arespinning about their common axis and carrying around with them a swirl of fluid part icles whichflow around in circles. A cross-sect ion of such a string of part icles and the associated flowshow a spinning point, outside of which the flow streamlines are concentric circles, as shown inFigure 5.8.

Figure 5.8 (a) Straight line vortex, (b) cross-sect ion showing the associated streamlines.

Vort ices can commonly be encountered in nature. The difference between a real (actual)vortex and theoret ical vortex is that , the real vortex has a core of fluid which rotates like asolid, although the associated swirl outside is the same as the flow outside the point vortex.The streamlines associated with a line vortex are circular, and therefore, the part icle velocity atany point must be only tangent ial.

Stream funct ion of a vortex can easily be obtained as follows. Consider a vortex of strengthΓ, at the origin of a polar coordinate system, as shown in Figure 5.9.

Figure 5.9 A vortex at origin.

Let P(r, θ) be a general point and velocity at P is always normal to OP (tangent ial). The radialvelocity at any point P is zero, that is:

since in polar coordinates, the radial velocity qr and tangent ial velocity qθ, in terms of streamfunct ion ψ are:

For qr = 0, the stream funct ion ψ should be a funct ion of r only. The tangent ial velocity at anypoint P [1] is:

Therefore,

Integrat ing along a convenient boundary, such as from A to P in Figure 5.9, from radius r0(radius of streamline, ψ = 0) to P(r, θ), we get the stream funct ion as:

that is:

(5.5) This is the stream funct ion for a vortex, and the circulat ion Γ of a flow is posit ive when it iscounter-clockwise.2

We know that the streamlines of a line vortex are concentric circles. Therefore, theequipotent ial lines (which are always orthogonal to the streamlines) must be radial linesemanat ing from the center of the vortex. Also, for a vortex, the normal component of velocityqn = 0. Therefore, the potent ial funct ion ϕ must be a funct ion of θ only. Thus:

Therefore:

Integrat ing this, we get:

By assigning ϕ = 0 at θ = 0, we obtain:

(5.6) This is the potent ial funct ion for a vortex.Also, we know that the stream funct ion for a source [1] is:

where m is the strength of the source.Comparing the stream funct ions of a vortex and a source, we see that the streamlines of

the source (the radial lines emanat ing from a point) and the streamlines of the vortex (theconcentric circles) are orthogonal.

5.5 Laws of Vortex MotionIn Sect ion 5.4, we saw that a point vortex can be considered as a string of rotat ing part iclessurrounded by fluid at large moving irrotat ionally. Further, the flow invest igat ion was confinedto a plane sect ion normal to the length or axis of the vortex. A more general definit ion is that avortex is a flow system in which a finite area in a plane normal to the axis of a vortex containsvorticity. Figure 5.10 shows a sect ional area S in the plane normal to the axis of a vortex. Theaxis of the vortex is clearly, always normal to the two-dimensional flow plane considered andthe influence of the so-called line vortex is the influence, in a sect ion plane, of an infinitely longstraight line vortex of vanishingly small area.

Figure 5.10 The vort icity of a sect ion of vortex tube.

The axis of a vortex, in general, is a curve in space, and area S is a finite size. It is convenientto consider that the area S is made up of several elemental areas. In other words, a vortexconsists of a bundle of elemental vortex lines or filaments. Such a bundle is termed a vortextube, being a tube bounded by vortex filaments.

The vortex axis is a curve winding about within the fluid. Therefore, it can flexure andinfluence the flow as a whole. The est imat ion of its influence on the fluid at large is somewhatcomplex. In our discussions here the vort ices considered are fixed relat ive to some axes in thesystem or free to move in a controlled manner and can be assumed to be linear. Furthermore,the vort ices will not be of infinite length, therefore, the three-dimensional or end influence mustbe accounted for.

In spite of the above simplificat ions, the vort ices conform to laws of mot ion appropriate totheir behavior. A rigorous treatment of the vort ices, without the simplificat ions imposed in ourtreatment here can be found in Milne-Thomson (1952) [2] and Lamb (1932) [3].

5.6 Helmholtz's Theorems

The four fundamental theorems governing vortex mot ion in an inviscid flow are calledHelmholtz's theorems (named after the author of these theorems). The first theorem refers toa fluid part icle (or element) in general mot ion possessing all or some of the following:

Linear velocity.Vort icity.Distort ion.

This theorem has been discussed in part in Sect ion 5.3, where the vorticity was explainedand its expression in Cartesian or polar coordinates were derived. Helmholtz's first theoremstates that:

“the circulation of a vortex tube is constant at all cross-sections along the tube.”The second theorem demonstrates that:

“the strength of a vortex tube (that is, the circulation) is constant along its length.”This is somet imes referred to as the equat ion of vortex continuity. It can be shown that the

strength of a vortex cannot grow or diminish along its axis or length. The strength of a vortex isthe magnitude of the circulat ion around it , and is equal to the product of vort icity ζ and area S.Thus:

It follows from the second theorem that, ζ S is constant along the vortex tube (or filament), sothat if the cross-sect ional area diminishes, the vort icity increases and vice versa. Since infinitevort icity is unacceptable, the cross-sect ional area S cannot diminish to zero. In other words, avortex cannot end in the fluid. In reality the vortex must form a closed loop, or originate (orterminate) in a discont inuity in the fluid such as a solid body or a surface of separat ion. In adifferent form it may be stated that a vortex tube cannot change its strength between twosections unless vortex filaments of equivalent strength join or leave the vortex tube, as shown inFigure 5.11.

Figure 5.11 Vortex-tube fragmentat ion.

It is seen that at sect ion A the vortex tube strength is Γ. Downstream of sect ion A anopposite vortex filament of strength −ΔΓ joins the vortex tube. Therefore, at sect ion B, thestrength of the vortex tube is:

as shown in Figure 5.11. This is of great importance to the vortex theory of lift .The third theorem demonstrates that a vortex tube consists of the same part icles of fluid,

that is:

“there is no fluid interchange between the vortex tube and surrounding fluid.”The fourth theorem states that:

“the strength of a vortex remains constant in time.”

5.7 Vortex TheoremsNow let us have a closer look at the theorems governing vortex mot ion. Consider thecirculat ion of a closed material line. By definit ion (Equat ion 5.3), we have the circulat ion as:

The t ime rate of change of Γ can be expressed as:

(5.7) since ds/dt = V, where V is the velocity, s is length and t is t ime. The second integral in Equat ion(5.7) vanishes, since is the total different ial of a single valued funct ion, andthe start ing point of integrat ion coincides with the end point .

By Euler equat ion, we have:

where FB is the body force. From Equat ion (5.7) and the Euler equat ion, we obtain the rate ofchange of the line integral over the velocity vector in the form:

(5.8) In Equat ion (5.8), DΓ/Dt vanishes if (FB · ds) and p/ρ can be writ ten as total different ials. Whenthe body force FB has a potent ial (that is, when the body force is a conservat ive force field);implying that the work done by the weight in taking a body from a point P to another point Q isindependent of the path taken from P to Q, and depends only on the potent ial, the first closedintegral in Equat ion (5.8) becomes zero because:

(5.9) For a homogeneous density field or in barotropic flow, the density depends only on pressure,that is ρ = f(p). For such a flow, the second term on the right-hand side of Equat ion (5.8), canbe expressed as:

(5.10) Therefore, for barotropic fluids, the second integral also vanishes in Equat ion (5.8).

Equat ion (5.8) to (5.10) form the content of Thompson's vortex theorem or Kelvin'scirculation theorem. This theorem states that:

“in a flow of inviscid and barotropic fluid, with conservative body forces, the circulationaround a closed curve (material line) moving with the fluid remains constant with time,” if themotion is observed from a nonrotating frame.The vortex theorem can be interpreted as follows:

“The position of a curve c in a flow field, at any instant of time, can be located by following themotion of all the fluid elements on the curve.”

That is, Kelvin's circulat ion theorem states that, the circulation around the curve c at the twolocations is the same. In other words:

(5.11) where D/Dt(≡ ∂/∂ t + ·) has been used to emphasize that the circulat ion is calculated arounda material contour moving with the fluid.

With Kelvin's theorem as the start ing point , we can explain the famous Helmholtz's vortextheorem, which allows a vivid interpretat ion of vortex mot ions which are of fundamentalimportance in aerodynamics. Before venturing to explain Helmholtz's vortex theorems, it wouldbe beneficial if we consider the origin of the circulat ion around an aerofoil, in a two-dimensionalpotent ial flow, because Kelvin's theorem seems to contradict the formulat ion of this circulat ion.

It is well known that, the force on an aerofoil in a two-dimensional potent ial flow isproport ional to the circulat ion. Also, the lift , namely the force perpendicular to the undisturbedincident flow direct ion, experienced by the aerofoil is direct ly proport ional to the circulat ion, Γ,

incident flow direct ion, experienced by the aerofoil is direct ly proport ional to the circulat ion, Γ,around the aerofoil. The lift per unit span of an aerofoil can be expressed as:

where ρ and V, respect ively, are the density and velocity of the freestream flow.Now let us examine the flow around a symmetrical and an unsymmetrical aerofoil in ident ical

flow fields, as shown in Figure 5.12. As seen from Figure 5.12(a), the flow around thesymmetrical aerofoil at zero angle of incidence is also symmetric. Therefore, there is no netforce perpendicular to the incident flow direct ion. The contribut ion of the line integral ofvelocity about the upper-half of the aerofoil to the circulat ion has exact ly the same magnitudeas the contribut ion of the line integral of velocity about the lower-half, but with opposite sign.Therefore, the total circulat ion around the symmetric aerofoil is zero.

Figure 5.12 (a) Symmetrical and (b) unsymmetrical aerofoil in uniform flow.

The flow around the unsymmetrical aerofoil, as shown in Figure 5.12(b), is asymmetric. Thecontribut ion of the line integral of velocity about upper-half of the aerofoil has an absolutevalue larger than that of the contribut ion about the lower-half. Therefore, the circulat ionaround the unsymmetrical aerofoil is nonzero. By Bernoulli theorem it can be inferred that thevelocity along a streamline which runs along the upper-side of the aerofoil is larger on thewhole than the velocity on the lower-side. Therefore, the pressure on the upper side is lessthan the pressure on the lower side. Thus there is a net upward force act ing on the aerofoil.

For an unsymmetrical aerofoil the flow velocity over the upper and lower surfaces aredifferent even when it is at zero angle of incidence to the freestream flow. Because of this thepressure on either side of the dividing streamline, shown in Figure 5.13, are different. Also, thevelocit ies on either side of the separat ion surface are different, as shown in the figure. Thisimplies that the pressure on either side of the separat ion surface are different. It is well knownthat the separat ion surface, which is also called slipstream, cannot be stable when thepressures on either side are different [4]. The slipstream will assume a shape in such a mannerto have equal pressure on either side of it . Here the pressure at the lower side is higher thanthat at the upper side. Thus, the slipstream bends up, as shown in Figure 5.14(a).

Figure 5.13 Velocity on either side of separat ion surface behind the aerofoil.

Figure 5.14 Flow past an aerofoil at start-up.

At the first instant of start-up, the flow around the trailing edge of the aerofoil is at very highvelocit ies. Also, the flow becomes separated from the upper surface. Flow field around anaerofoil at different phases of start-up is shown in Figure 5.14. The separat ion at the uppersurface is caused by the very large decelerat ion of the flow from the maximum thicknesslocat ion to the separat ion point S, which is formed on the upper surface since the flow is st illcirculat ion-free flow [Figure 5.14(a)]. This flow separates from the upper surface even with verylit t le viscosity (that is, μ → 0) and forms the wake, which becomes the discont inuity surface inthe limit ing case of μ = 0. The flow is irrotat ional everywhere except the wake region. Soonafter start-up, the separat ion point is dipped to the trailing edge, as per Kutta hypothesis, andthe slipstream rolls-up as shown in Figure 5.14(b). The vortex thus formed is pusheddownstream and posit ioned at a locat ion behind the aerofoil, as shown in Figure 5.14(c). Thisvortex is called starting vortex. The start ing vortex is essent ially a free vortex because it isformed by the kinematics of the flow and not by the viscous effect .

By Kelvin's circulat ion theorem, a closed curve which surrounds the aerofoil and the vortexst ill has zero circulat ion. In other words, the circulat ion of the start ing vortex and the boundvortex (this is due to the boundary layer at the surface of the aerofoil in viscous flow) are ofequal magnitude, as shown in Figure 5.15.

Figure 5.15 Circulat ion of start ing and bound vort ices.

A closed line which surrounds only the vortex has a fixed circulat ion and must necessarilycross the discont inuity surface. Therefore, Kelvin's circulat ion theorem does not hold for thisline. A curve which surrounds the aerofoil only has the same circulat ion as the free vortex, butwith opposite sign, and therefore the aerofoil experiences a lift . The circulat ion about theaerofoil with a vortex lying over the aerofoil, due to the boundary layer at the surface, is calledthe bound vortex.

In the above discussion, we used the obvious law that the circulat ion of a closed loop isequal to the sum of the circulat ion of the meshed network bounded by the curve, as shown inFigure 5.16.

(5.12) That is, the sum of the circulat ions of all the areas is the neighboring circulat ion of the circuitas a whole. This is because, as the ΔΓ of each element is added to the ΔΓ of the neighboringelement, the contribut ion of the common sides (Figure 5.16) disappears. Applying thisargument from one element to the neighboring element throughout the area, the only sides

contribut ing to the circulat ion when the ΔΓs of all elemental areas are summed together arethose sides which actually form the circuit itself. This means, that for the circuit as a whole, thecirculat ion is:

In this relat ion, the surface integral implies that the integrat ion is over the area of the meshednetwork, and the cyclic integral implies that the integrat ion is around the circuit of the meshednetwork.

Figure 5.16 Circulat ion of meshed network.

For discussing the physics of Helmholtz's theorem, we need to make use of Stoke's integraltheorem.

5.7.1 Stoke's TheoremStoke's theorem relates the surface integral over an open surface to a line integral along thebounded curve. Let S be a simply connected surface, which is otherwise of arbit rary shape,whose boundary is c, and let u be any arbit rary vector. Also, we know that any arbit rary closedcurve on an arbit rary shape can be shrunk to a single point . The Stoke's integral theoremstates that:

“The line integral ∫u · dx about the closed curve c is equal to the surface integral over any surface of arbitrary shape which has c as its boundary.”

That is, the surface integral of a vector field u is equal to the line integral of u along thebounding curve:

(5.13) where dx is an elemental length on c, and n is unit vector normal to any elemental area on ds,as shown in Figure 5.17.

Figure 5.17 Sign convent ion for integrat ion in Stoke's integral theorem.

Stoke's integral theorem allows a line integral to be changed to a surface integral. Thedirect ion of integrat ion is posit ive counter-clockwise as seen from the side of the surface, asshown in Figure 5.17.

Helmholtz's first vortex theorem states that:

“the circulation of a vortex tube is constant along the tube.”A vortex tube is a tube made up of vortex lines which are tangent ial lines to the vort icity

vector field, namely curl u (or ζ). A vortex tube is shown in Figure 5.18. From the definit ion ofvortex tube it is evident that it is analogous to the streamtube, where the flow velocity istangent ial to the streamlines const itut ing the streamtube. A vortex line is therefore related tothe vort icity vector in the same way the streamline is related to the velocity vector. If ζx, ζy andζz are the Cartesian components of the vort icity vector ζ, along x- , y- and z-direct ions,respect ively, then the orientat ion of a vortex line sat isfies the equat ion:

which is analogous to:

along a streamline. In an irrotat ional vortex (free vortex), the only vortex line in the flow field isthe axis of the vortex. In a forced vortex (solid-body rotat ion), all lines perpendicular to theplane of flow are vortex lines.

Figure 5.18 A vortex tube.

Now consider two closed curves c1 and c2 in a vortex tube, as shown in Figure 5.19.

Figure 5.19 Two loops on a vortex tube.

According to Stoke's theorem, the two line integrals over the closed curves in Figure 5.19vanish, because the integrand on the right-hand side of Equat ion (5.13) is zero, since curl u is,by definit ion, perpendicular to n. The contribut ion to the integral from the infinitely closesegments c3 and c4 of the curve cancel each other, leading to the equat ion:

(5.14) since the distance between the segments c3 and c4 are infinitesimally small, we ignore thatand treat c1 and c2 to be closed curves. By changing the direct ion of integrat ion over c2, thuschanging the sign of the second integral in Equat ion (5.14), we obtain Helmholtz's first vortextheorem:

(5.15) Derivat ion of this equat ion clearly demonstrates the kinematic nature of Helmholtz's firstvortex theorem. Another approach to the physical explanat ion of this theorem stems from thefact that the divergence of the vort icity vector vanishes. That is, the vort icity vector field curl ucan be considered as analogous to an incompressible flow (for which the divergence of velocityis zero). In other words, the vortex tube becomes the streamtube of the new field. Nowapplying the cont inuity equat ion in its integral form (that is, ∫ ∫ sρuini ds = 0) to a part of thisstreamtube, and at the same t ime replacing u by curl u, we get:

Since ρ is a constant, we can write this as:

(5.16) that is, for every closed surface s, the flux of the vort icity is zero. Applying Equat ion (5.16) to apart of the vortex tube whose closed surface consists of the surface of the tube and twoarbit rarily oriented cross-sect ions A1 and A2, we obtain:

(5.17) since the integral over the tube surface vanishes. The integral:

is called the vortex strength. It is ident ical to the circulat ion. From Equat ion (5.17) it is evidentthat:

“the vortex strength of a vortex tube is constant.”Noting the sense of integrat ion of the line integral, Stoke's theorem transforms Equat ion (5.17)into Helmholtz's first theorem [Equat ion (5.15)]. From this representat ion it is obvious that, justlike the streamtube, the vortex tube also cannot come to an end within the fluid, since theamount of fluid which flows through the tube (in unit t ime) cannot simply vanish at the end ofthe tube. The tube must either reach out to infinity (that is, should extend to infinity), or end atthe boundaries of the fluid, or close around into itself and, in the case of a vortex tube, form avortex ring.

A very thin vortex tube is referred to as a vortex filament. The vortex filaments are ofpart icular importance in aerodynamics. For a vortex filament the integrand of the surfaceintegral in Stoke's theorem [Equat ion (5.13)]:

(5.18) can be taken in front of the integral to obtain:

(5.19) or

(5.20) where ω is the angular velocity. From this it is evident that the angular velocity increases withdecreasing cross-section of the vortex filament.

It is a usual pract ice to idealize a vortex tube of infinitesimally small cross-sect ion into avortex filament. Under this idealizat ion, the angular velocity of the vortex, given by Equat ion(5.20), becomes infinitely large. From the relat ion:

(5.21) we have ω→ ∞, for Δs → 0.

The flow field outside the vortex filament is irrotat ional. Therefore, for a vortex of strength Γat a part icular posit ion, the spat ial distribut ion of curl u is fixed. In addit ion, if div u is also given(for example, div u = 0 in an incompressible flow), then according to the fundamental theoremof vortex analysis, the velocity field u (which may extend to infinity) is uniquely determinedprovided the normal component of velocity vanishes asymptot ically sufficient ly fast at infinityand no internal boundaries exist .

The fundamental theorem of vector analysis is also essent ially purely kinematic in nature.Therefore, it is valid for both viscous and inviscid flows, and not restricted to inviscid flows only.Let us split the velocity vector u into two parts, namely due to potent ial flow and rotat ionalflow. Therefore:

(5.22) where uIR is velocity of irrotat ional flow field and uR is velocity of rotat ional flow field. Thus, uIRis velocity of an irrotat ional flow field, that is:

(5.23) The second is a solenoidal (coil like shape) flow field, thus:

(5.24) Note that Equat ion (5.23) is the statement that “the vort icity of a potent ial flow is zero” andEquat ion (5.24) is the statement of cont inuity equat ion of incompressible flow.

The combined field is therefore neither irrotat ional nor solenoidal. The field uIR is a potent ialflow, and thus in terms of potent ial funct ion ϕ, we have uIR = ϕ. Let us assume that thedivergence u to be a given funct ion g(x). Thus:

that is:

(5.25) since · uR = 0. Also, uIR = ϕ. Therefore:

(5.26) This is an inhomogeneous Laplace equation, also called Poisson's equat ion. The theory of thispart ial different ial equat ion is the subject of potential theory which plays an important role inmany branches of physics as in fluid mechanics. It is well known from the results of potent ialtheory that the solut ion of Equat ion (5.26) is given by:

(5.27) where x is the place where the potent ial ϕ is calculated, and x ' is the abbreviat ion for theintegrat ion variables x ' 1, x ' 2 and x ' 3, and is a different ial volume. Thedomain ∞ implies that the integrat ion is to be carried out over the ent ire space.

5.8 Calculation of uR, the Velocity due toRotational Flow

We see that Equat ion (5.24) is sat isfied if uR is represented as the curl of a new, yet unknown,vector field a. Thus:

(5.28) We know that the divergence of the curl always vanishes.3 Therefore:

(5.29)

Now let us form the curl of u and, from Equat ion (5.23), obtain the equat ion:

(5.30) But using the vector ident ity:

We can express Equat ion (5.30) as:

(5.31) Up to now the only condit ion on vector a is to sat isfy Equat ion (5.28). But this condit ion doesnot uniquely determine this vector, because we can always add the gradient of some otherfunct ion f to a without changing Equat ion (5.28), since × f ≡ 0. If, in addit ion, we want thedivergence of a to vanish (that is, · a = 0), we obtain from Equat ion (5.31) the simplerequat ion:

(5.32) In this equat ion, let us consider × u as a given vector funct ion b(x), which is determined bythe choice of the vector filament and its strength (that is, circulat ion). Thus, the Cartesiancomponent form of the vector Equat ion (5.32) leads to three Poisson's equat ions, namely:

(5.33) For each of these component equat ions, we can apply the solut ion [Equat ion (5.27)] ofPoisson's equat ion. Now, vectorially combining the result , we can write the solut ion for a, fromEquat ion (5.32), in short as:

(5.34) Thus, calculat ion of the velocity field u(x) for a given distribut ion g(x) ≡ div u and b(x) = curl u isreduced to the following integrat ion processes, which may have to be done numerically:

(5.35) Now, let us calculate the solenoidal term of the velocity uR, using Equat ion (5.35). This is theonly term in incompressible flow without internal boundaries. Consider a field which isirrotat ional outside the vortex filament, shown in Figure 5.20.

Figure 5.20 A vortex filament.

The velocity field outside the filament is given by:

(5.36) The integrat ion is carried out only over the volume of the vortex filament, whose volumeelement is:

(5.37) where dx ' = n ds ', is the vectorial element of the vortex filament, ds is the cross-sect ional areaand n is the unit vector.

Also, the unit vector n can be expressed as:

Therefore, Equat ion (5.37) becomes:

or

Subst itut ing this into Equat ion (5.36), we get:

(5.38) since:

First let us integrate over a small cross-sect ion surface ΔS. For ΔS → 0, the change of thevector x ' over this surface can be neglected. Thus, taking in front of the surface integral,we obtain:

(5.39) From Stokes theorem, the surface integral is equal to the circulat ion Γ. By Helmholtz's firstvortex theorem Γ is constant along the vortex filament, and therefore independent of x '. Thus,from Equat ion (5.39) we get:

(5.40) In index notat ion, the right-hand side of Equat ion (5.40) can be writ ten as:

The operator4 can direct ly be taken into the integral. The term (with ri = (xi − xi ') and r= |r|) becomes:

In vector form, this is simply:

Therefore, subst itut ing the above into Equat ion (5.40), we get the famous Biot-Savart law:

(5.41) where r = (x − x '). The Biot-Savart law is a useful relat ion in aerodynamics.

5.9 Biot-Savart LawBiot-Savart law relates the intensity of magnitude of magnet ic field close to an electric currentcarrying conductor to the magnitude of the current. It is mathematically ident ical to theconcept of relat ing intensity of flow in the fluid close to a vort icity-carrying vortex tube to thestrength of the vortex tube. It is a pure kinematic law, which was originally discovered throughexperiments in electrodynamics. The vortex filament corresponds there to a conduct ing wire,the vortex strength to the current, and the velocity field to the magnet ic field. Theaerodynamic terminology namely, “induced velocity” stems from the origin of this law.

Now let us calculate the induced velocity at a point in the field of an elementary length δs ofa vortex of strength Γ. Assume that a vortex tube of strength Γ, consist ing of an infinitenumber of vortex filaments, to terminate in some point P, as shown in Figure 5.21.

Figure 5.21 Vortex-tube discharging into a sphere.

The total strength of the vortex tube will be spread over the surface of a spherical boundary

of radius R. The vort icity in the spherical surface will thus have the total strength of Γ. Becauseof symmetry the velocity of flow at the surface of the sphere will be tangent ial to the circularline of intersect ion of the sphere with a plane normal to the axis of the vortex tube. Such planewill be a circle ABC of radius r subtending a conical angle 2θ at P, as shown in Figure 5.22.

Figure 5.22 Vortex-tube discharged into an imaginary sphere.

If the velocity on the sphere at (R, θ) from P is , then the circulat ion round the circuit ABCis Γ ', where:

The radius of the circuit is r = R sin θ, therefore, we have:

(5.42) But the circulat ion round the circuit is equal to the strength of the vort icity in the containedarea. This is on the cap ABCD of the sphere. Since the distribut ion of the vort icity is constantover the surface, we have:

that is:

(5.43) From Equat ions (5.42) and (5.43), we obtain the induced velocity as:

(5.44) Now, assume that the length of the vortex decreases unt il it becomes very short , as shown

(P1P) in Figure 5.23. The circle ABC is influenced by the opposite end P1 also (that is, both theends P and P' of the vortex influence the circle). Now the vortex elements entering the sphereare congregat ing on P1. Thus, the sign of the vort icity is reversed on the sphere of radius R1.The velocity induced at P1 becomes:

(5.45)

Figure 5.23 A short vortex tube discharged into an imaginary sphere.

The net velocity on the circuit ABC is the sum of Equat ions (5.44) and (5.45), therefore, wehave:

As the point P1 approaches P,

and

Thus, at the limit ing case of P1 approaching P, we have the net velocity as:

(5.46) This is the velocity induced by an elementary length δs of a vortex of strength Γ whichsubtends an angle δθ at point P located by the ordinate (R, θ) from the element. Also, r = R sinθ and R δθ = δs sin θ, thus we have:

(5.47) It is evident from Equat ion (5.47) that to obtain the velocity induced by a vortex this equat ionhas to be integrated. This t reatment of integrat ion varies with the length and shape of thefinite vortex being studied. In our study here, for applying Biot-Savart law, the vort ices ofinterest are all nearly linear. Therefore, there is no complexity due to vortex shape. The

vort ices will vary only in their overall length.

5.9.1 A Linear Vortex of Finite LengthExamine the linear vortex of finite length AB, shown in Figure 5.24. Let P be an adjacent pointlocated by the angular displacements α and β from A and B respect ively. Also, the point P hascoordinates r and θ with respect to an elemental length δs of AB. Further, h is the height of theperpendicular from P to AB, and the foot of the perpendicular is at a distance s from δs.

Figure 5.24 A linear vortex of finite length.

The velocity induced at P by the element of length δs, by Equat ion (5.47), is:

(5.48) The induced velocity is in the direct ion normal to the plane ABP, shown in Figure 5.24.5

The velocity at P due to the length AB is the sum of induced velocit ies due to all elements,such as δs. However, all the variables in Equat ion (5.48) must be expressed in terms of a singlevariable before integrat ing to get the effect ive velocity. A variable such as ϕ, shown in Figure5.24 may be chosen for this purpose. The limits of integrat ion are:

since ϕ passes through zero while integrat ing from A to B. Here we have:

Thus, we have the induced velocity at P due to vortex AB, by Equat ion (5.48), as:

(5.49) This is an important result of vortex dynamics. From this result we obtain the following specificresults of velocity in the vicinity of the line vortex.

5.9.2 Semi-Infinite VortexA vortex is termed semi-infinite vortex when one of its ends stretches to infinity. In our case letthe end B in Figure 5.24 stretches to infinity. Therefore, β = 0 and cos β = 1, thus, fromEquat ion (5.49), we have the velocity induced by a semi-infinite vortex at a point P as:

(5.50)

5.9.3 Infinite VortexAn infinite vortex is that with both ends stretching to infinity. For this case we have α = β = 0.Thus, the induced velocity due to an infinite vortex becomes:

(5.51) For a specific case of point P just opposite to one of the ends of the vortex, say A, we have α =π/2 and cos α = 0. Thus, the induced velocity at P becomes:

(5.52) This amounts to precisely half of the value for the infinitely long vortex filament [Equat ion(5.51)], as we would expect because of symmetry.

While discussing Figure 5.15, we saw that the circulat ion about an aerofoil in two-dimensional flow can be represented by a bound vortex. We can assume these bound vort icesto be straight and infinitely long vortex filaments (potent ial vort ices). As far as the lift isconcerned we can think of the whole aerofoil as being replaced by the straight vortex filament.

The velocity field close to the aerofoil is of course different from the field about a vortexfilament in cross flow, but both fields become more similar when the distance of the vortexfrom the aerofoil becomes large.

In the same manner, a start ing vortex can be assumed to be a straight vortex filament whichis at tached to the bound vortex at plus and minus infinity. The circulat ion of the vortexdetermines the lift , and the lift formula which gives the relat ion between circulat ion, Γ, and liftper unit width, l, in inviscid potent ial flow is the Kutta-Joukowski theorem,6 namely:

(5.53) where l is the lift per unit span of the wing, Γ is circulat ion around the wing, U∞ is thefreestream velocity and ρ is the density of the flow.

It is important to note that the lift force on a wing sect ion in inviscid (potent ial) flow isperpendicular to the direct ion of the undisturbed stream and thus an aerofoil experiences onlylift and no drag. This result is of course contrary to the actual situat ion where the wingexperiences drag also. This is because here in the present approach the viscosity of air isignored whereas in reality air is a viscous fluid. The Kutta-Joukowski theorem in the form ofEquat ion (5.53) with constant Γ holds only for wing sect ions in a two-dimensional plane flow. Inreality all wings are of finite span and hence the flow essent ially becomes three-dimensional.But as long as the span is much larger than the chord of the wing sect ion, the lift can beest imated assuming constant circulat ion Γ along the span. Thus, the lift of the whole wingspan 2b is given by:

(5.54) But in reality there is flow communicat ion from the bottom to the top at the wing t ips, owing tohigher pressure on the lower surface of the wing than the upper surface. Therefore, by Eulerequat ion, the fluid flows from lower to upper side of the wing under the influence of thepressure gradient, in order to even out the pressure difference. In this way the magnitude ofthe circulat ion on the wing t ips tends to become zero. Therefore, the circulat ion over the wingspan varies and the lift is given by:

(5.55) where the origin is at the middle of the wing, x is measured along the span, and b is the semi-span of the wing.

According to Helmholtz's first vortex theorem, being purely kinematic, the above relat ions forlift are also valid for the bound vortex. Thus, isolated pieces of a vortex filament cannot exist .Also, it cannot cont inue to be straight along into infinity, where the wing has not cut throughthe fluid and thus no discont inuity surface has been generated as is necessary for theformat ion of circulat ion. Therefore, free vort ices, Γt, which are carried away by the flow must beattached at the wing t ips. Together with the bound vortex, Γb, and the start ing vortex, Γs, they(the t ip vort ices) form a closed vortex ring frame in the fluid region cut by the wing, as shown inFigure 5.25.

Figure 5.25 Simplified vortex system of a finite wing.

If a long t ime has passed since start-up, the start ing vortex is at infinity (far downstream ofthe wing), and the bound vortex and the t ip vort ices together form a horseshoe vortex.

Even though the horseshoe vortex system represents only a very rough model of a finitewing, it can provide a qualitat ive explanat ion for how a wing experiences a drag in inviscid flow,as already ment ioned. The velocity induced at the middle of the wing by the two t ipvort ices accounts for double the velocity induced by a semi-infinite vortex filament at distanceb. Therefore, by Equat ion (5.50), we have:

(5.56)

This velocity is directed downwards and hence termed induced downwash. Thus, the middle ofthe wing experiences not only the freestream velocity U∞, but also a velocity u, which arisesfrom the superposit ion of U∞ and downwash velocity , as shown in Figure 5.26.

Figure 5.26 Illustrat ion of induced drag.

In inviscid flow, the force vector is perpendicular to the actual approach direct ion of the flowstream, and therefore has a component parallel to the undisturbed flow, as shown in Figure5.26, which manifests itself as the induced drag Di, given by:

(5.57) It is important to note that Equat ion (5.57) holds if the induced downwash from both vort ices isconstant over the span of the wing. However, the downwash does change since at a distancex from the wing center, one vortex induces a downwash of:

whereas the other vortex induces:

Both the downwash are in the same direct ion, therefore adding them we get the effect ivedownwash as:

From this it can be concluded that the downwash is the smallest at the center of the wing(that is, Equat ion (5.57) underest imates the induced drag) and tends to infinity at the wing t ips.The unrealist ic value there (at wing t ips) does not appear if the circulat ion distribut iondecreases towards the wing t ips, as in deed it has to. For a semi-ellipt ical circulat iondistribut ion over the span of the wing, the downwash distribut ion becomes constant andEquat ion (5.57) is applicable. Helmholtz first vortex theorem st ipulates that for an infinitesimalchange in the circulat ion in the x-direct ion of:

and a free vortex of the same infinitesimal strength must leave the trailing edge. This processleads to an improved vortex system, as shown in Figure 5.27.

Figure 5.27 Simplified vortex system of a wing.

The free vort ices form a discont inuity surface in the velocity components parallel to thetrailing edge, which rolls them into the kind of vort ices, as shown in Figure 5.28.

Figure 5.28 Vort ices format ion due to rollup of the discont inuity.

These vort ices must be cont inuously renewed as the wing moves forward. This calls forcont inuous replenishment of kinet ic energy in the vortex. The power needed to do this is thework done per unit t ime by the induced drag.

The manifestat ion of Helmholtz's first theorem can be encountered in daily life. Recall thedimples formed at the free surface of coffee in a cup when a spoon is suddenly dipped into it .The format ion process of dimples looks like that shown schematically in Figure 5.29.

Figure 5.29 Vortex format ion due to dipping of a spoon.

As the fluid flows together from the front and back, a surface of discont inuity forms alongthe rim of the spoon. The discont inuity surface rolls itself into a bow-shaped vortex whoseendpoints form the dimples on the free surface, as shown in Figure 5.29.

The flow outside the vortex filament is a potent ial flow. Thus, by incompressible Bernoulliequat ion, we have:

This is valid both along a streamline and between any two points in the flow field.7 Also, at thefree surface the pressure is equal to the ambient pressure pa. Further, at some distance awayfrom the vortex the velocity is zero and there is no dimple at the free surface, and hence z = 0.Thus, the Bernoulli constant is equal to pa and we have:

Near the end points of the vortex the velocity increases by the formula given by Equat ion(5.52), and therefore z must be negat ive, that is, a depression of the free surface. In reality, thecross-sect ional surface of the vortex filament is not infinitesimally small, therefore we cannottake the limit h → 0 in Equat ion (5.52), for which the velocity becomes infinite. However, theinduced velocity due to the vortex filament is so large that it causes a not iceable format ion ofdimples.

It should be noted that an infinitesimally thin filament cannot appear in actual flow becausethe velocity gradient of the potent ial vortex tends to infinity for h → 0, so that the viscousstresses cannot be ignored even for very small viscosity. Also, it is well known that the viscousstresses make no contribut ion to part icle accelerat ion in incompressible potent ial flow, butthey do deformat ion work and thus provide a contribut ion to the dissipat ion. The energydissipated in heat stems from the kinet ic energy of the vortex.

5.9.4 Helmholtz's Second Vortex TheoremThe second vortex theorem of Helmholtz's states that:

“a vortex tube is always made up of the same fluid particles.”In other words, a vortex tube is essent ially a material tube. This characterist ic of a vortex tubecan be represented as a direct consequence of Kelvin's circulat ion theorem. Let us consider avortex tube and an arbit rary closed curve c on its surface at t ime t0, as shown in Figure 5.30. ByStokes integral theorem, the circulat ion of the closed curve c is zero (that is, DΓ/Dt = 0). Thecirculat ion of the curve, which is made up of the same material part icles, st ill has the same(zero) value of circulat ion at a lat ter instant of t ime t.

Figure 5.30 A closed curve on a vortex ring at t imes t0 and t.

By invert ing the above reasoning, it follows from Stokes integral theorem that these materialpart icles must be on the outer surface of the vortex tube.

If we examine smoke-rings, it can be seen that the vortex tubes are material tubes. Thesmoke will remain in the vortex ring and will be t ransported with it , so that it is the smoke itselfwhich carries the vort icity. This statement holds under the restrict ions of barotropy (that is, ρ =ρ(p), the density is a funct ion of pressure only) and zero viscosity. The slow disintegrat ion seenin smoke-rings is due to frict ion and diffusion. A vortex ring which consists of an infinitesimallythin vortex filament induces an infinitely large velocity on itself (similar to the horseshoevortex), so that the ring would move forward with infinitely large velocity. The induced velocityat the center of the ring remains finite (as in horseshoe vortex). From Biot-Savart law, theinduced velocity becomes:

This velocity becomes infinitely large (that is, unrealist ic) when the cross-sect ion of the vortexring is assumed to be infinitesimally small. For finite cross-sect ion, the velocity induced by thering on itself, that is, the velocity with which the ring moves forward remains finite. But in realitythe actual cross-sect ion of the ring is not known, and probably depends on how the ring was

formed.In pract ice we not ice that the ring moves forward with a velocity which is slower than the

induced velocity in the center. Also, it is well known that two rings moving in the same direct ioncont inually overtake each other whereby one slips through the other in front. Thisphenomenon, illustrated in Figure 5.31, is explained by mutually induced velocit ies on the ringsand formula given above for the velocity at the center of the ring.

Figure 5.31 Two vortex rings passing through one another.

In a similar manner it can be explained why a vortex ring towards a wall becomes larger indiameter and at the same t ime its velocity gets reduced. Also, the diameter decreases and thevelocity increases when a vortex ring moves away from a wall, as illustrated in Figure 5.32.

Figure 5.32 Kinematics of a vortex ring near a wall.

To work out the mot ion of vortex rings the cross-sect ion of vortex must be known. Further,for infinitesimally thin rings the calculat ion fails because vortex rings, such as curved vortexfilaments, induce large velocit ies on themselves. However, for straight vortex filaments, that is,for vortex filaments in two-dimensional flows, a simple descript ion of the “vortex dynamics” forinfinitesimally thin filaments is possible, since for such a case the self-induced translat ionalvelocity vanishes. We know that vortex filaments are material lines, therefore it is sufficient tocalculate the paths of the fluid part icles which carry the rotat ion in xy-plane perpendicular tothe filaments, using:

that is, to determine the paths of the vortex centers. The induced velocity which a straightvortex filament at posit ion xi induces at posit ion x is known from Equat ion (5.49), that is:

As we have seen, the induced velocity is perpendicular to the vector hi = ri = (x − xi), andtherefore has the direct ion , so that the vectorial form of Equat ion (5.41) reads as:

For x → xi the velocity tends to infinity, but because of symmetry the vortex cannot be movedby its own velocity field, that is, the induced translat ional velocity is zero. The induced velocityof n vort ices with the circulat ion Γi (i = 1, 2, . . . n) is:

If there are no internal boundaries, or if the boundary condit ions are sat isfied by reflect ion, as inFigure 5.32, the last equat ion describes the ent ire velocity field, and using dx/dt = u(x, t) or dxi/dt= ui(xi, t), the “equat ion of mot ion” of the k th vortex becomes:

(5.58) For i = k, the induced translat ional velocity becomes zero, owing to symmetry, and henceexcluded from the summation. Equat ion (5.58) gives the 2n relat ions for the path coordinates.

The dynamics of vortex mot ion have invariants which are analogous to the invariants of apoint mass system on which no external forces act . The conservat ion of strengths of thevort ices by Helmholtz's theorem (∑Γk = constant) corresponds to mass conservat ion of totalmass of the point mass system. When the Equat ion (5.58) is mult iplied by Γk, summed over kand expanded, we get:

In the above equat ion, the terms on the right-hand side cancel out in pairs, and the equat ionreduces to:

On integrat ion this results in:

(5.59) The integrat ion constants are writ ten as xg, which is like a center of gravity coordinate (this isdone here for dimensional homogeneity). Equat ion (5.59) states that:

“the center of gravity of the strengths of the vortices is conserved.”For a point mass system, by conservat ion of momentum, we have the corresponding law,

namely:

“the velocity of the center of gravity is a conserved quantity in the absence of external forces.”For ∑ Γk = 0, the center of gravity lies at infinity, so that, for example, two vort ices with Γ1 = −

Γ2 must take a turn about a center of gravity point Pg which is at a finite distance, as shown inFigure 5.33.

Figure 5.33 Pathlines of a pair of straight vort ices.

The paths of the vortex pairs are determined by numerical integrat ion of Equat ion (5.58).The paths will look like those shown in Figure 5.34.

Figure 5.34 Pathlines of two straight vortex pairs.

5.9.5 Helmholtz's Third Vortex TheoremThe third vortex theorem of Helmholtz's states that:

“the circulation of a vortex tube remains constant in time.”Using Helmholtz's second theorem and Kelvin's circulat ion theorem, the above statement canbe interpreted as “a closed line generat ing the vortex tube is a material line whose circulat ionremains constant.”

Helmholtz's second and third theorems hold only for inviscid and barotropic fluids.

5.9.6 Helmholtz's Fourth Vortex TheoremThe fourth theorem states that:

“the strength of a vortex remains constant in time.”This is similar to the fact that the mass flow rate through a streamtube is invariant as the

tube moves in the flow field. In other words, the circulat ion distribut ion gets adjusted with thearea of the vortex tube. That is, the circulat ion per unit area (that is, vort icity) increases withdecrease in the cross-sect ional area of the vortex tube and vice versa.

5.10 Vortex MotionVortex is a fluid flow in which the streamlines are concentric circles. The vortex mot ionsencountered in pract ice may in general be classified as free vortex or potential vortex andforced vortex or flywheel vortex. The streamline pattern for a vortex may be represented asconcentric circles, as shown in Figure 5.35.

Figure 5.35 Streamline pattern of a vortex.

When a fluid flow is along a curved path, as in a vortex, the velocity of the fluid elementsalong any streamline will undergo a change due to its change of direct ion, irrespect ive of anychange in magnitude of the fluid stream. Consider the streamtube shown in Figure 5.36.

Figure 5.36 Fluid element in a vortex.

As the fluid flows round the curve there will be a rate of change of velocity, that is, anaccelerat ion, towards the curvature of the streamtube. The consequent rate of change ofmomentum of the fluid must be due to a force act ing radially across the streamlines result ingfrom the difference of pressure between the sides BC and AD of the streamline element, asper Newton's second law. The control volume ABCD in Figure 5.36 subtends an angle δθ at thecenter of curvature O and has length δs in the direct ion of flow. Let the thickness of ABCDperpendicular to the plane of diagram be ‘b'. For the streamline AD, the radius of curvature is rand that for BC is (r + δr). The pressure and velocity at AD and BC are p, V, (p + δp) and (V +δV), as shown in Figure 5.36. Thus, the change of pressure in the radial direct ion is δp.

The change of velocity in the radial direct ion (as shown in the velocity diagram in Figure 5.36)is:

But δθ = δs/r. Thus, the radial change of velocity between AB and CD is:

(5.60) This rate of change of momentum is produced by the force due to the pressure differencebetween BC and AD of the control volume, given by:

(5.61) According to Newton's second law, Equat ion (5.60) = Equat ion (5.61). Thus:

or

(5.62) For an incompressible fluid, density ρ is constant and Equat ion (5.62) can be expressed interms of pressure head h. Pressure is given by:

Therefore,

Subst itut ing this into Equat ion (5.62), we get:

or

In the limit δr → 0, this gives the rate of change of pressure head in the radial direct ion as:

(5.63) The curved flow shown in Figure 5.36 will be possible only when there is a change of pressurehead in a radial direct ion, as seen from Equat ion (5.63). However, since the velocity V alongstreamline AD is different from the velocity (V + δV) along BC, there will also be a change in thevelocity head from one streamline to another. Such a change of velocity head in the radialdirect ion is given by:

(5.64) For a planar flow (say in the horizontal plane), the changes in potent ial head is zero. Therefore,the change of total head H, that is, the total pressure energy per unit weight, in a radialdirect ion, δH/δr, is given by:

Subst itut ing Equat ions (5.63) and (5.64) into the results of the above equat ion, we get:

(5.65) The term is also known as vorticity of the flow.

In obtaining Equat ion (5.65), it is assumed that the streamlines are horizontal. But thisequat ion also applies to cases where the streamlines are inclined to horizontal, since the fluidin a control volume is in effect weight less, being supported vert ically by the surrounding fluid.

5.11 Forced VortexForced vortex is a rotat ional flow field in which the fluid rotates as a solid body with a constantangular velocity ω, and the streamlines form a set of concentric circles. Because the fluid in aforced vortex rotates like a rigid body, the forced vortex is also called flywheel vortex. Thechange of total energy per unit weight in a vortex mot ion is governed by Equat ion (5.65).

The velocity at any radius r is given by:

From this we have:

and

Subst itut ing dV/dr and V/r into Equat ion (5.65), we get:

Integrat ing this we get:

(5.66) where c is a constant. By Bernoulli equat ion, at any point in the fluid, we have:

Note that, in the above equat ion and Equat ion (5.66), the unit of the total head is meters.Subst itut ion of this into Equat ion (5.66) results in:

If the rotat ing fluid has a free surface, the pressure at the surface will be atmospheric;therefore, the pressure at the free-surface will be zero.

Replacing with 0 in the above equat ion, the profile of the free surface is obtained as:

(5.67) Thus, the free surface of a forced vortex is in the form of a paraboloid.

Similarly, for any horizontal plane, for which z will be constant, the pressure distribut ion will begiven by:

(5.68) The typical shape of the free surface and the velocity variat ion along a radial direct ion of a

forced vortex are shown in Figure 5.37.

5.12 Free VortexFree vortex is an irrotat ional flow field in which the streamlines are concentric circles, but thevariat ion of velocity with radius is such that there is no change of total energy per unit weightwith radius, so that dH/dr = 0. Since the flow field is potent ial, the free vortex is also calledpotential vortex.

For a free vortex, from Equat ion (5.65), we have:

Integrat ing, we get:

or

where c is a constant known as the strength of the vortex at any radius r.The tangent ial velocity becomes:

(5.69) This shows that in the flow around a vortex core the velocity is inversely proport ional to theradius (see Sect ion 5.4). When the core is small, or assumed concentrated on a line axis, it isapparent from the relat ion V = c/r that when r is small V can be very large. However, within thecore the fluid behaves as though it were a solid cylinder and rotates at an uniform angularvelocity. Figure 5.38 shows the variat ion of velocity with radius for a typical free vortex. Thesolid line represents the idealized case, but in reality it is not so precise, and the velocity peakis rounded off, as shown by the dashed lines.

Figure 5.38 Velocity distribut ion in a free vortex core.

At any point in the flow field, by Bernoulli equat ion, we have:

Subst itut ing Equat ion (5.69), we get:

At the free surface, . Thus, the profile of the free surface is given by:

(5.70) This is a hyperbola asymptot ic to the axis of rotat ion and to the horizontal plane through z =H, as shown in Figure 5.39.

Figure 5.39 Free vortex: (a) shape of free surface, (b) velocity variat ion.

For any horizontal plane, z is constant and the pressure variat ion is given by:

(5.71) Thus, in a free vortex, pressure decreases and circumferent ial velocity increases as we movetowards the center, as shown in Figure 5.39.

The free vortex discussed above is essent ially a free cylindrical vortex. The fluid movesalong streamlines that are horizontal concentric circles; there is no variat ion of total energywith radius. Combinat ion of a free cylindrical vortex and radial flow will result in a free spiralvortex.

5.12.1 Free Spiral VortexA free spiral vortex is essent ially the combinat ion of a free cylindrical vortex and a radial flow.Before gett ing into the physics of free spiral vortex, let us see what is a radial flow.

Radial FlowExamine the flow between two parallel planes as shown in Figure 5.40. In the flow thestreamlines will be radial straight lines and the streamtube will be in the form of sectors. Thiskind of flow in which the fluid flows radially inwards, or outwards from a center is called a radialflow. The area of the flow will therefore increase as the radius increases, causing the velocityto decrease. The flow pattern is symmetrical and therefore, the total energy per unit weight Hwill be the same for all streamlines and for all points along each streamline if we assume thatthere is no loss of energy.

Figure 5.40 A radial flow: (a) streamlines, (b) pressure variat ion.

If Vr is the radial velocity and p is the pressure at any radius r, then the total energy per unitweight H becomes:

(5.72) Assuming the flow to be incompressible (as would be the case of a liquid), by cont inuity, wehave the volume flow rate as:

where b is the distance between the planes. Thus,

(5.73) Subst itut ing this into Equat ion (5.72), we get:

The plot of pressure p at any radius will be, as shown in Figure 5.40(b), parabolic and issometimes referred to as Barlow's curve.

If the radial flow discharges to atmosphere at the periphery, the pressure at any pointbetween the two plates will be below atmospheric (that is, subatmospheric); there will be aforce tending to bring the plates together and so shut-off the flow. This phenomenon can beobserved in the case of a disc valve. Radial flow under the disc will cause the disc to be drawnonto the valve seat ing. This will return to atmospheric and the stat ic pressure of the fluid onthe upstream side of the disc will push it off its seat ing again. The disc will tend to vibrate onthe seat ing and the flow will be intermit tent .

Now, let us find an expression for the pressure difference between two points on the samehorizontal plane in a free vortex. For a free cylindrical vortex, the streamlines are concentriccircles and there is no variat ion of the total energy with radius, that is:

Also, by Equat ion (5.69), we have:

Let p1 and p2 be the pressures in two concentric streamlines of radii r1 and r2 which havevelocit ies V1 and V2, respect ively. Since there is no change of total energy with radius, for thesame horizontal plane, by Bernoulli equat ion:

But V1 = c/r1 and V2 = c/r2. Thus:

(5.74) Now, let us obtain an expression for the pressure difference between two points at radii R1

Now, let us obtain an expression for the pressure difference between two points at radii 1and R2, on a radial line, when a fluid flows radially inward or outward from a center, neglect ingfrict ion. Flow is radial and therefore in straight line so that r, the radius of curvature of thestreamlines, is infinite, dH/dr = 0, and for all streamlines:

If p1 and p2 are the pressures at radii R1 and R2, respect ively, where the velocit ies are V1 andV2:

By volume conservat ion:

where is the volume flow rate and t is the distance between the radial passage boundaries.That is:

Thus,

(5.75) It is evident from Equat ions (5.73) and (5.74) that the relat ion between pressure and radiusand between velocity and radius is similar for both free vortex and radial flow. Both types ofmot ion may therefore occur together. The fluid rotates and flows radially forming a free spiralvortex in which a fluid element will follow a spiral path, as shown in Figure 5.41.

Figure 5.41 A free spiral vortex.

5.13 Compound VortexIn the free vortex, and thus, theoret ically, the velocity becomes infinite at the center. Thevelocit ies near the axis would be very high and, skin frict ion losses vary as the square of thevelocity, they will cease to be negligible. Also, the assumption that the total head H remainsconstant will cease to be true. The port ion of fluid around the axis tends to rotate as a solidbody. Thus, the central port ion essent ially forms a forced vortex. The free surface profile ofsuch a compound vortex and the pressure variat ion with radius on any horizontal plane in thevortex is shown in Figure 5.42.

Figure 5.42 Compound vortex.

The velocity at the common radius R must be the same for the two vort ices. For the freevortex, if y1 = depression of the free surface at radius R below the level of the surface atinfinity, then:

For the forced vortex, if y2 is the height of the free surface at radius R above the center of thedepression:

Thus, the total depression is:

(5.76) For the forced vortex, the velocity at radius R is ωR, while for the free vortex, from Equat ion(5.69), the velocity at radius R is c/R. Therefore, the common radius at which these twovelocity will be the same is given by:

5.14 Physical Meaning of CirculationIn Equat ion (5.3) the vector field is taken as velocity V. But the vector field need not be the flowvelocity alone. The vector field can be force, mass flow rate of a fluid, etc. Therefore, in general,the circulation may be defined as the line integral of a vector field around a closed plane curvein a flow field. If the vector is a force F, then the integral cF · ds is equal to the work done bythe force. Taking the vector as ρV, the mass flow rate per unit area, we can get a physicalmeaning of the circulat ion in the following way. Imagine a t iny paddle-wheel probe is placed[Figure 5.43(c)] in any of the flow patterns shown in Figure 5.43.

Figure 5.43 Streamlines in: (a) a vortex, (b) parallel flow with constant velocity, (c) parallel flowwith variable velocity, (d) flow around a corner.

When the flow velocity on one side is greater than the other side, as in Figure 5.43(c), thewheel will turn. If the mass flow rate per unit area ρV is larger on one side of the wheel than theother, then the circulat ion is different from zero, but if ρV is the same on both sides as in Figure5.43(b), then the circulat ion is zero. We shall show that the component of curl along theaxis of the paddle wheel equals:

(5.77) where dA is the area enclosed by the curve along which we calculate circulat ion. The paddlewheel then acts as a “curl meter” to measure curl ; when the wheel does not rotate, curl

. In Figure 5.43(c), curl in spite of the fact that the streamlines are parallel. In Figure5.43(d), it is possible to have curl even though the streamlines go around a corner. In fact ,for the flow of water around a corner curl . We must realize that the value of curl at apoint depends upon the circulat ion in the neighborhood of the point and not on the overall flowpattern.

Example 5.2A cylindrical tank of 1 m diameter and 0.75 m height is filled with a liquid of relat ive density 0.9up to 0.5 m from the bottom of the tank and the rest of the tank contains atmospheric air. Thetank revolves about its vert ical axis at a speed such that the liquid begins to uncover the base.(a) Calculate the speed of rotat ion and (b) the upward force on the cover.

SolutionThe flow field described is shown in Figure 5.44.

Figure 5.44 Flow field in a rotat ing tank.

(a) When the tank is stat ic, the volume of oil is:

While rotat ing, a forced vortex is formed and the free surface will be a paraboloid CGD.

Since the volume of the paraboloid is equal to half the volume of the circumscribing cylinder, nooil is spilled out from the cylinder, therefore:

For the free surface, by Equat ion (5.67), we have:

Between C and G, taking G is the datum level, we have:

and

Thus:

(b) The top cover annular area from r = r1 to r = R is in contact with the oil. If p is the pressureat any radius r, the force on an annular of radius r and width dr is given by:

Integrat ing from r = r1 to r = R we get the force F act ing on the top cover as:

The pressure, given by Equat ion (5.68) is:

since the pressure at r1 is atmospheric, p = 0 (gauge), where r = r1, thus:

Therefore:

Subst itut ing this in the force equat ion above, we have:

Example 5.3Show that a free vortex is an irrotat ional mot ion. A hollow cylinder of diameter 1 m, open at thetop, spins about its axis which is vert ical, thus generat ing a forced vortex mot ion of the liquidcontained in it . Calculate the height of the vessel so that the liquid just reaches the top whenthe minimum depth of the free surface of the liquid (from the top) is 25 cm at 200 rpm.

SolutionThe tangent ial velocity in a free vortex (excluding the core) is given by Equat ion (5.69) is:

where c is a constant. This field is potent ial if the vort icity content in the field is zero. Thevort icity ζ in polar coordinates is [Equat ion (5.2)]:

In free vortex, the normal component of velocity qn = 0. Thus:

We know that:

and

Hence:

Thus, the mot ion in a free vortex is irrotat ional.The spinning cylinder described in the problem is shown schematically in Figure 5.45.

Figure 5.45 A spinning cylinder containing water.

For a forced vortex, the free surface ABC is a paraboloid, as shown in the figure. The shape isgiven by Equat ion (5.67) as:

At r = 0 on the free surface, zB = 25 cm = 0.25 m. Thus:

At r = R, zA = h. Therefore, we have the expression for free surface as:

Given, ω = 200 rpm = 200/60 rps = 200/60 × 2π radian/second. Therefore:

5.15 Rectilinear VorticesA rect ilinear vortex is a vortex tube whose generators are perpendicular to the plane of mot ion.Now, let us have a closer look at some aspects of two-dimensional vortex mot ion. We knowthat the circulat ion in an infinitesimal plane circuit is proport ional to the area of the circuit . In atwo-dimensional mot ion the vort icity vector ζ at any point P, which is perpendicular to theplane of mot ion and whose magnitude is equal to the limit of the rat io of the circulat ion is aninfinitesimal circuit embracing P to the area of the circuit . That is, the vort icity vector is bydefinit ion perpendicular to the plane of the mot ion, so that the vortex lines are straight andparallel. All vortex tubes are therefore cylinders whose generators are perpendicular to theplane of mot ion. Such vort ices are known as rectilinear vortices. For our discussions in thissect ion, let us consider the fluid between parallel plates at unit distance apart and parallel tothe plane of the mot ion, which is half-way between them.

5.15.1 Circular VortexA circular vortex is that with the shape of its cross-sect ion normal to its axis of rotat ion ascircular. For example a single cylindrical vortex tube, whose cross-sect ion is a circle of radius ‘a’,surrounded by unbounded fluid, as shown in Figure 5.46 is a circular vortex.

Figure 5.46 A cylindrical vortex tube.

The sect ion of this cylindrical vortex by the plane of mot ion is a circle, as shown in Figure5.47.

Figure 5.47 Sect ion of cylindrical vortex tube.

Let us assume that the vort icity ζ over the area of the circle is a constant. Outside the circlethe vort icity is zero. Consider concentric circles of radii r ' and r, where r ' < a < r, as shown inFigure 5.47. Let the tangent ial speeds of the fluid mot ion on the circles of radii r ' and r are V 'and V, respect ively.

We know that “the circulat ion in a closed circuit is the line integral of the tangent ialcomponent of the velocity taken round the circuit in the sense in which the arc length(elemental length along the circuit ) increases.” Thus, the circulat ion around the circles of radii rand r ', respect ively, are:

where ds is an elemental arc length. Also, V and V ' are constants. Therefore:

where ω is the angular velocity. Thus:

when r ' = r = a we have so that the velocity is cont inuous as we pass through the circle.From the above discussions, it can be inferred that the existence of a vortex implies the co-

existence of certain distribut ion of velocity field. This velocity field which co-exists with thevortex is known as the induced velocity field and the velocity at any point of it is called theinduced velocity. It is important to note that it is customary to refer to the velocity at a point inthe field as the velocity induced by the vortex, but it is merely a convenient abbreviat ion of thecomplete statement that were the vortex to be alone in the otherwise undisturbed field thevelocity at the point would have the value in quest ion. In other words, when several vort icesare present in the field, each will contribute to the velocity at a point .

For circular vortex the induced velocity at the extremity of any radius vector r joining thecenter of the vortex to a point of the fluid external to the vortex is of magnitude inverselyproport ional to r and is perpendicular to r. Thus the induced velocity tends to zero at greatdistances. The fluid inside the vortex will have velocity of magnitude proport ional to r andtherefore the fluid composing the vortex moves like a rigid body rotat ing about the center Owith angular velocity . The velocity at the center is zero. That is:

“a circular vortex induces zero velocity at its center.”Thus, a circular vortex alone in the otherwise undisturbed fluid will not tend to move.

The velocit ies at the extremit ies of oppositely directed radii are of the same magnitude butof opposite sense so that the mean velocity of the fluid within the vortex is zero. Thus, if acircular vortex of small radius be “placed” at a point in a flow field where the velocity is u, themean velocity at its center will st ill be u and the fluid composing the vortex will move withvelocity u. That is, the vortex will move with the stream carrying its vort icity with it .

Naturally occurring t ropical cyclone, hurricane and typhoon which at tains a diameter of from150 to 800 kilometers, and travels at a speed seldom exceeding 25 kilometer per hour arecircular vort ices on a large scale. Within the area the wind can reach hurricane force, whilethere is a central region termed “the eye of the storm,” of diameter about 15–30 kilometerswhere condit ions may be completely calm.

5.16 Velocity DistributionConsider a circular vortex of strength γ, defined by:

Thus,

Therefore, the velocity induced by this circular vortex is:

The velocit ies at all points of a diameter are perpendicular to that diameter, hence theextremit ies of the velocity vectors at the different points of the diameter will lie on a curvewhich gives the velocity distribut ion as we go along the diameter from −∞ to +∞, as shown inFigure 5.48.

Figure 5.48 Velocity distribut ion along a diameter of a circular vortex.

From points between C and D the velocity variat ion (along a diameter) is a straight line EAF,the variat ion from C to −∞ and D to +∞ is part of a rectangular hyperbola whose asymptotesare the diameter CD and the perpendicular diameter through the center A. The ordinates DE

and CF, each represent the velocity γ/a. Thus, if for a circular vortex of constant strength γ, asthe radius a decreases, DE will increase. Therefore, in the limit when a → 0 the velocityvariat ion will consist of the rectangular hyperbola with the asymptote perpendicular to CD.

Now let us study the field of two ident ical circular vort ices of radius a but with oppositevort icity (ζ and −ζ) at a finite distance apart , as shown in Figure 5.49. If the distance betweentheir center BA is sufficient ly large compared to a, as a first approximat ion, we can supposethat the vort ices do not interfere and remain circular. For such a case their velocity fields maybe compounded by the ordinary law of vector addit ion. The effect on the velocity distribut ionplot of A will reduce all the velocit ies at points near A on the diameter CD (see Figure 5.48) byapproximately . The general shape of the velocity distribut ion plot for the pair of vort ices willbe as shown in Figure 5.49.

Figure 5.49 General shape of the velocity distribut ion for a pair of vort ices.

It can be seen that the center of each vortex is now in the field of velocity induced by theother and would therefore move with velocity in the direct ion perpendicular to AB. Thusthe vort ices are no longer at rest , but move with equal uniform velocity, remaining at aconstant distance apart . This is an applicat ion of the theorem that “a vortex induces novelocity on itself.” The vort ices and velocity field shown in Figure 5.49 has its applicat ion to thestudy of the induced velocity due to the wake of a monoplane aerofoil at a distance behind thetrailing edge.

5.17 Size of a Circular VortexIt can be shown that the pressure at the center of a circular vortex of strength γ and radius a isthe lowest pressure in the field of the vortex and the value is (p∞ − γρ2/a2), where p∞ is thepressure at infinity and ρ is the local density. Therefore, if the pressure in the field to beposit ive everywhere, a2 ≤ γ2ρ/p∞. That is, the radius of the vortex should be greater than orequal to γ2ρ/p∞. But in many occasions we are concerned with the case of a → 0. In such acase the result ing point vortex must be regarded as an abstract ion. However, we can make aas small as we wish by making γ small enough, or p∞ large enough, but we shall st ill have acircular vortex and the induced velocity will be everywhere finite. The apparent ly infinitevelocit ies which occur are due to the over-simplificat ion of taking a = 0. Note that a similarlower limit is encountered for the size of a point source in two-dimensional mot ion, and is givenby the same relat ion if γ is taken as the strength of the source.

5.18 Point Rectilinear VortexIt is the limit ing case of a circular vortex of constant strength γ with radius a tending to zero.We have seen that any point outside a circular vortex of strength γ, at distance r from thecenter, the velocity γ/r is at right angles to r. If we let the radius a of the vortex tend to zero,the circle shrinks to a point . This limit ing vortex of zero radius is called a point rect ilinear vortex,or simply a point vortex of strength γ. When the radius tends to zero, the cylindrical vortex tubeshown in Figure 5.46 shrinks to a straight line and the vortex becomes a single rect ilinearvortex represented by a point in the plane of mot ion, as shown in Figure 5.50.

Figure 5.50 Point rect ilinear vortex at origin in xy-plane.

If we take the origin at the vortex, the velocity at the point P(r, θ) is represented by acomplex number:

We can relate this to the complex potent ial as follows:

But z = x + iy, therefore:

since u = ∂ ϕ/∂ x, .Thus,

Integrat ing this, we get the complex potent ial as:

Here, the constant is irrelevant and hence can be ignored, then:

Note that the mot ion is irrotat ional except at the origin O where the vortex is posit ioned andso a complex potent ial exists, with a logarithmic singularity at the vortex.

If the vortex were at the point z0 instead of at the origin, the complex potent ial would be:

It is essent ial to note that the velocity derived from the complex potent ial is the velocityinduced by the vortex.

5.19 Vortex PairTwo vort ices of equal strength γ but opposite nature (one rotat ing clockwise and the otherrotat ing counterclockwise) from a vortex pair, as shown in Figure 5.51.

Figure 5.51 A vortex pair.

Each vortex in the pair induces a velocity γ/AB on the other, in the direct ion perpendicular toAB and in the same sense. Thus the vortex pair moves in the direct ion perpendicular to AB,remaining at the constant distance AB apart . The fluid velocity at O, the mid-point of AB, is:

which is four t imes the velocity of each vortex (see Figure 5.49).Taking O as the origin and the x-axis along OA, if AB = 2a, we have the complex potent ial, at

the instant when the vort ices are on the x-axis, as:

(5.78) Thus,

With y = 0, this gives the velocity distribut ion along the x-axis as:

Thus u = 0, . The plot of against x is shown in Figure 5.52.

Figure 5.52 Variat ion of with x.

The curve is as per the equat ion , so that the asymptotes are the straight port ions ofFigure 5.52 go over into the asymptotes x ± a and thus the velocity of vortex A cannot bereached in Figure 5.52, although this is st ill one-quarter of the velocity at O.

5.20 Image of a Vortex in a PlaneFor a vortex shown in Figure 5.51, because of symmetry there will not be any flux across yy ',the perpendicular bisector of AB. Thus yy ' can be regarded as a streamline and couldtherefore be replaced by a rigid boundary. Hence the mot ion due to a vortex at A in thepresence of this boundary is the same as the mot ion that would result if the boundary wereremoved and an equal vortex of opposite rotat ion were placed at B. The vortex at B is calledt he image of the actual vortex at A with respect to the plane boundary and the complexpotent ial is st ill given by Equat ion (5.78).

5.21 Vortex between Parallel PlatesLet us consider a vortex of strength γ midway between the planes y = ± a/2 and at the origin,as shown in Figure 5.53.

Figure 5.53 (a) A vortex between two plates, (b) a vortex and its image.

The transformat ion ζ = ieπz/a would map the strip between the planes on the upper half oft he ζ-plane (the thick and thin lines in Figures 5.53(a) and 5.53(b) indicate which parts of theboundaries correspond) as follows:

For z = x + iy = 0, ζ = i. That is:

Thus, by the image system, we have vort ices of strength γ at ζ = i and −γ at ζ = − i, asillustrated in Figure 5.53(b). Therefore, by Equat ion (5.78):

But:

Thus, we have:

Thus when y = 0:

and the velocity at different points on the x-axis is given by this relat ion.When there are no walls present, on the x-axis, . Thus:

Therefore, the walls reduce the velocity at points on the x-axis. For example, for x = a:

The streamlines of a vortex at the origin between two parallel plates would be as shown inFigure 5.54.

Figure 5.54 Streamlines of a vortex at origin, bounded by two parallel planes.

Note that the walls increase the velocity component u when x = 0 and decrease when y= 0. In other words, the walls make the vortex to stretch along the x-direct ion and shrink alongthe y-direct ion.

5.22 Force on a VortexA rect ilinear vortex may be regarded as the limit of a circular vortex which rotates about itscenter as if rigid. Consider a circular vortex inserted in a steady flow field as shown in Figure5.55, so that its center is at the point whose velocity is before the vortex is inserted. Thevortex would then move with the fluid with velocity soon after insert ing, so that the flowmotion would no longer be steady. Let us imagine the vortex to be held fixed by the applicat ionof a suitable force (in the form of pressure distribut ion). This force would be equal but oppositeto that exerted by the fluid on the vortex.

Figure 5.55 Circular vortex in a flow field.

When the mot ion is steady, the force exerted by the fluid is the Kutta-Joukowski lift which isindependent of the size and shape of the vortex. This force, being independent of the size, isalso the force exerted by the fluid on a point vortex. The direct ion of the force (shown in Figure5.55) is obtained by rotat ing the velocity vector through a right angle in the direct ion oppositeto that of the circulat ion (vort icity).

5.23 Mutual action of Two VorticesConsider two vort ices of strength γ and γ ' located at (0, 0) and (0, h), as shown in Figure 5.56.These two vort ices repel one another if γ and γ ' have the same sign, and at t ract if the signsare opposite. This result has its applicat ion to the act ion between the vort ices shed by thewings of a biplane.

Figure 5.56 Two vort ices at a finite distance between them.

5.24 Energy due to a Pair of VorticesConsider two circular vort ices of equal radius a and equal strength γ placed as shown in Figure5.57 with the distance 2b between their centers very large compared to a, so that their circularform is preserved.

Figure 5.57 Two small circular vort ices at a finite distance apart .

Neglect ing the interact ion between them, we can write the vort icity as:

The stream funct ion is:

where r1, r2 are the distance of the point z from the vort ices, as shown in Figure 5.57.For the region external to the vort ices the kinet ic energy of the fluid is:

Now in terms of stream funct ion ψ:

But the region outside the vort ices is irrotat ional and hence vort icity:

Thus,

Therefore, we have:

By Stokes theorem:

Thus:

The integrat ion is taken posit ively (in the counterclockwise direct ion) round c, and thecircumference of the vortex at z = b. The factor 2 is to account for the two vort icescontribut ing the same amount to the energy.

Now:

where Vs is the speed tangent ial to contour c and ds is arc length along c. Therefore:

Also, on c, r1 = a, and r2 = 2b (approximately), so that we may express the KEo as:

The fluid inside the contour c is rotat ing (Figure 5.48) with angular velocity γ/a2 and moving asa whole with velocity γ/2b induced by the other vortex. Thus the KE inside c is:

where the first term is the contribut ion due to the whole mot ion and the second term is due tothe angular velocity (rω). But a2/b2 is small and hence can be neglected. Hence:

Thus the total KE is:

5.25 Line VortexConsider a cont inuous distribut ion of vort ices on a straight line AB stretching from (− c/2, 0) to(c/2, 0), as shown in Figure 5.58.

Figure 5.58 Cont inuous line of vort ices.

Let the elements dξ of the line at point (ξ, 0) behave like a point rect ilinear vortex of strengthγdξ, where γ may be constant or a funct ion of ξ. This element taken by itself will induce at thepoint P(x, 0) a velocity of dvx, in the negat ive y-direct ion, as shown in Figure 5.58, given by:

Thus the whole line of vort ices will induce at P the velocity:

(5.79) Note that in Equat ion (5.79), ξ is a variable and x is fixed. When ξ = x, the integrand is infinite.On the other hand, using the principle that a vortex induces no velocity at its own center, thepoint x must be omit ted from the range of variat ion of ξ. To do this we define the “improper”integral Equat ion (5.79) by its “principal value,” namely:

(5.80) In this way the point (x, 0) is always the center of the omit ted port ion between (x − ) and (x + ).

In the theory of aerofoil the type of integral (Equat ion 5.80) in which we shall be interested isthat for which and γ = γn sin nϕ where γn is independent of ϕ.

Now let , where θ, like x, is fixed. We get from Equat ion (5.79):

In this relat ion, we have integral of the type:

It can be shown that:

Therefore:

5.26 SummaryThe following are the three possible ways in which a fluid element can move.

1. Pure translation –the fluid elements are free to move anywhere in space but cont inueto keep their axes parallel to the reference axes fixed in space.2. Pure rotation –the fluid elements rotate about their own axes which remain fixed inspace.3. The general motion in which translat ion and rotat ion are compounded.

A flow in which all the fluid elements behave as in item (a) above is called potential orirrotational flow. All other flows exhibit , to a greater or lesser extent, the spinning property ofsome of the const ituent fluid elements, and are said to posses vorticity, which is theaerodynamic term for elemental spin. The flow is then termed rotational flow.

The angular velocity is given by:

The quant ity 2ω is the elemental spin, also referred to as vorticity, ζ. Thus:

The units of ζ are radian per second. It is seen that:

that is, the vorticity is twice the angular velocity.In the polar coordinates, the vort icity equat ion can be expressed as:

where r and θ are the polar coordinates and qt and qn are the tangent ial and normalcomponents of velocity, respect ively.

Circulation is the line integral of a vector field around a closed plane curve in a flow field. Bydefinit ion:

Circulat ion implies a component of rotation of flow in the system. This is not to say that thereare circular streamlines, or the elements, of the fluid are actually moving around some closedloop although this is a possible flow system. Circulat ion in a flow means that the flow systemcould be resolved into an uniform irrotat ional port ion and a circulat ing port ion.

If circulat ion is present in a fluid mot ion, then vort icity must be present, even though it maybe confined to a restricted space, as in the case of the circular cylinder with circulat ion, wherethe vort icity at the center of the cylinder may actually be excluded from the region of flowconsidered, namely that outside the cylinder.

The sum of the circulat ions of all the elemental areas in the circuit const itutes the circulat ionof the circuit as a whole:

A line vortex is a string of rotat ing part icles. In a line vortex, a chain of fluid part icles arespinning about their common axis and carrying around with them a swirl of fluid part icles whichflow around in circles.

Vort ices can commonly be encountered in nature. The difference between a real (actual)vortex and theoret ical vortex is that the real vortex has a core of fluid which rotates like a solid,although the associated swirl outside is the same as the flow outside the point vortex. Thestreamlines associated with a line vortex are circular, and therefore, the part icle velocity at anypoint must be only tangent ial.

The stream funct ion for a vortex is:

The potent ial funct ion ϕ for a vortex is:

A vortex is a flow system in which a finite area in a plane normal to the axis of a vortexcontains vorticity. The axis of a vortex, in general, is a curve in space, and area S is a finite size.It is convenient to consider that the area S is made up of several elemental areas. In otherwords, a vortex consists of a bundle of elemental vortex lines or filaments. Such a bundle istermed a vortex tube, being a tube bounded by vortex filaments.

The four fundamental theorems governing vortex mot ion in an inviscid flow are calledHelmholtz's theorems.

Helmholtz's first theorem states that:

“the circulation of a vortex tube is constant at all cross-sections along the tube.”The second theorem demonstrates that:

“the strength of a vortex tube (that is, the circulation) is constant along its length.”This is somet imes referred to as the equat ion of vortex continuity. It follows from the second

theorem that ζ S is constant along the vortex tube (or filament), so that if the cross-sect ionalarea diminishes, the vort icity increases and vice versa. Since infinite vort icity is unacceptable,the cross-sect ional area S cannot diminish to zero. In other words, a vortex cannot end in thefluid. In reality the vortex must form a closed loop, or originate (or terminate) in a discont inuity inthe fluid such as a solid body or a surface of separat ion. In a different form it may be statedthat a vortex tube cannot change its strength between two sections unless vortex filaments ofequivalent strength join or leave the vortex tube.

The third theorem demonstrates that a vortex tube consists of the same part icles of fluid,that is:

“there is no fluid interchange between the vortex tube and surrounding fluid.”The fourth theorem states that:

“the strength of a vortex remains constant in time.”By definit ion (Equat ion 5.3), we have the circulat ion as:

The t ime rate of change of Γ can be expressed as:

By Euler equat ion, we have:

Thompson's vortex theorem or Kelvin's circulation theorem states that:

“in a flow of inviscid and barotropic fluid, with conservative body forces, the circulationaround a closed curve (material line) moving with the fluid remains constant with time,” if themotion is observed from a nonrotating frame.The vortex theorem can be interpreted as follows:

“The position of a curve c in a flow field, at any instant of time, can be located by following themotion of all the fluid elements on the curve.”

That is, Kelvin's circulat ion theorem states that the circulation around the curve c at the twolocations is the same. In other words:

where D/Dt(≡ ∂/∂ t + ·) has been used to emphasize that the circulat ion is calculated arounda material contour moving with the fluid.

With Kelvin's theorem as the start ing point , we can explain the famous Helmholtz's vortextheorem, which allows a vivid interpretat ion of vortex mot ions which are of fundamentalimportance in aerodynamics.

The lift per unit span of an aerofoil can be expressed as:

where ρ and V, respect ively, are the density and velocity of the freestream flow.A closed line which surrounds only the vortex has a fixed circulat ion and must necessarily

cross the discont inuity surface. Therefore, Kelvin's circulat ion theorem does not hold for thisline. A curve which surrounds the aerofoil only has the same circulat ion as the free vortex, butwith opposite sign, and therefore the aerofoil experiences a lift . The circulat ion about theaerofoil with a vortex lying over the aerofoil, due to the boundary layer at the surface, is calledthe bound vortex.

The Stoke's integral theorem states that:

“The line integral ∫u · dx about the closed curve c is equal to the surface integral over anysurface of arbitrary shape which has c as its boundary.”

That is, the surface integral of a vector field u is equal to the line integral of u along thebounding curve:

where dx is an elemental length on c, and n is unit vector normal to any elemental area on ds.Helmholtz's first vortex theorem states that:

“the circulation of a vortex tube is constant along the tube.”A vortex tube is a tube made up of vortex lines which are tangent ial lines to the vort icity

vector field, namely curl u (or ζ).In an irrotational vortex (free vortex), the only vortex line in the flow field is the axis of the

vortex. In a forced vortex (solid-body rotation), all lines perpendicular to the plane of flow arevortex lines.

The integral:

is called the vortex strength. It is ident ical to the circulat ion.A very thin vortex tube is referred to as a vortex filament. The vortex filaments are of

part icular importance in aerodynamics. The angular velocity increases with decreasing cross-sect ion of the vortex filament.

is an inhomogeneous Laplace equation, also called Poisson's equat ion. The theory of thispart ial different ial equat ion is the subject of potential theory which plays an important role inmany branches of physics as in fluid mechanics.

Biot-Savart law relates the intensity of magnitude of magnet ic field close to an electriccurrent carrying conductor to the magnitude of the current. It is mathematically ident ical to theconcept of relat ing intensity of flow in the fluid close to a vort icity-carrying vortex tube to thestrength of the vortex tube. It is a pure kinematic law, which was originally discovered throughexperiments in electrodynamics. The vortex filament corresponds there to a conduct ing wire,the vortex strength to the current, and the velocity field to the magnet ic field. The

aerodynamic terminology namely, “induced velocity” stems from the origin of this law.The induced velocity is given by:

The velocity induced at P by the element of length δs is:

The induced velocity at a point P above vortex AB is:

where α = ∠ PAB and β = ∠ PBA.A vortex is termed semi-infinite vortex when one of its ends stretches to infinity. The velocity

induced by a semi-infinite vortex at a point P:

where α = ∠ PAB and end B tends to infinity.An infinite vortex is that with both ends stretching to infinity. For this case we have α = β = 0.

Thus, the induced velocity due to an infinite vortex becomes:

For a specific case of point P just opposite to one of the ends of the vortex, say A, we have α =π/2 and cos α = 0. Thus, the induced velocity at P becomes:

The circulat ion of the vortex determines the lift , and the lift formula which gives the relat ionbetween circulat ion, Γ, and lift per unit width, l, in inviscid potent ial flow is the Kutta-Joukowskitheorem.

The Kutta-Joukowski theorem states that “the force per unit length act ing on a rightcylinder of any cross sect ion whatsoever is equal to ρ∞V∞Γ∞, and is perpendicular to thedirect ion of V∞,” namely:

where l is the lift per unit span of the wing, Γ is circulat ion around the wing, U∞ is thefreestream velocity and ρ is the density of the flow.

The lift of the whole wing span 2b is given by:

If a long t ime has passed since start-up, the start ing vortex is at infinity (far downstream of thewing), and the bound vortex and the t ip vort ices together form a horseshoe vortex.

The velocity induced at the middle of the wing by the two t ip vort ices accounts todouble the velocity induced by a semi-infinite vortex filament at distance b. Therefore:

This velocity is directed downwards and hence termed induced downwash.The induced drag Di, given by:

The flow outside the vortex filament is a potent ial flow. Thus, by incompressible Bernoulliequat ion, we have:

The second vortex theorem of Helmholtz's states that:

“a vortex tube is always made up of the same fluid particles.”In other words, a vortex tube is essent ially a material tube. This characterist ic of a vortex tubecan be represented as a direct consequence of Kelvin's circulat ion theorem.

The third vortex theorem of Helmholtz's states that:

“the circulation of a vortex tube remains constant in time.”

Using Helmholtz's second theorem and Kelvin's circulat ion theorem, the above statement canbe interpreted as “a closed line generat ing the vortex tube is a material line whose circulat ionremains constant.”

Helmholtz's second and third theorems hold only for inviscid and barotropic fluids.The fourth theorem states that:

“the strength of a vortex remains constant in time.”This is similar to the fact that the mass flow rate through a streamtube is invariant as the

tube moves in the flow field. In other words, the circulat ion distribut ion gets adjusted with thearea of the vortex tube. That is, the circulat ion per unit area (that is, vort icity) increases withdecrease in the cross-sect ional area of the vortex tube and vice versa.

Vortex is a fluid flow in which the streamlines are concentric circles. The vortex mot ionsencountered in pract ice may in general be classified as free vortex or potential vortex andforced vortex or flywheel vortex.

Forced vortex is a rotat ional flow field in which the fluid rotates as a solid body with aconstant angular velocity ω, and the streamlines form a set of concentric circles. Because thefluid in a forced vortex rotates like a rigid body, the forced vortex is also called flywheel vortex.The free surface of a forced vortex is in the form of a paraboloid.

Free vortex is an irrotat ional flow field in which the streamlines are concentric circles, but thevariat ion of velocity with radius is such that there is no change of total energy per unit weightwith radius, so that dH/dr = 0. Since the flow field is potent ial, the free vortex is also calledpotential vortex.

For a free vortex, the tangent ial velocity becomes:

This shows that in the flow around a vortex core the velocity is inversely proport ional to theradius.

In a free vortex, pressure decreases and circumferent ial velocity increases as we movetowards the center.

A free spiral vortex is essent ially the combinat ion of a free cylindrical vortex and a radial flow.Flow in which the fluid flows radially inwards, or outwards from a center is called a radial flow.A free spiral vortex is that in which a fluid element will follow a spiral path.A rect ilinear vortex is a vortex tube whose generators are perpendicular to the plane of

mot ion.A circular vortex is that with the shape of its cross-sect ion normal to its axis of rotat ion as

circular.The existence of a vortex implies the co-existence of certain distribut ion of velocity field.

This velocity field which co-exists with the vortex is known as the induced velocity field andthe velocity at any point of it is called the induced velocity.

A circular vortex induces zero velocity at its center. Thus, a circular vortex alone in theotherwise undisturbed fluid will not tend to move.

Naturally occurring t ropical cyclone, hurricane and typhoon which at tains a diameter of from150 to 800 kilometers, and travels at a speed seldom exceeding 25 kilometer per hour arecircular vort ices on a large scale. Within the area the wind can reach hurricane force, whilethere is a central region termed “the eye of the storm,” of diameter about 15–30 kilometerswhere condit ions may be completely calm.

Point rect ilinear vortex is the limit ing case of a circular vortex of constant strength γ withradius a tending to zero.

Two vort ices of equal strength γ but opposite nature (one rotat ing clockwise and the otherrotat ing counterclockwise) from a vortex pair.

Exercise Problems1. Evaluate the vort icity of the following two-dimensional flow.

a. u = 2xy, .

b. u = x2, .c. ur = 0, uθ = r.d. ur = 0, .

2. If the velocity induced by a rect ilinear vortex filament of length 2 m, at a point equidistantfrom the extremit ies of the filament and 0.4 m above the filament is 2 m/s, determine thecirculat ion around the vortex filament.

[Answer: 5.414 m2/s]

3. A point P in the plane of a horseshoe vortex is between the arms and equidistant from allthe filaments. Prove that the induced velocity at P is:

where Γ is the intensity and AB is the length of the finite side of the horseshoe.

4. If the velocity induced by an infinite line vortex of intensity 100 m2/s, at a point above thevortex is 40 m/s, determine the height of the point above the vortex line.

[Answer: 0.398 m]

5. If a wing of span 18 m has a constant circulat ion of 150 m2/s around it while flying at 400km/h, at sea level, determine the lift generated by the wing.

[Answer: 367.5 kN]

6. If the tangent ial velocity at a point at radial distance of 1.5 m from the center of a circularvortex is 35 m/s, determine (a) the intensity of the vortex and (b) the potent ial funct ion of thevortex flow.

[Answer: (a) 329.87 m2/s, (b) 52.5 θ]

7. Show that a circular vortex ring of intensity Γ induces an axial velocity at the center ofthe ring, where R is the radius of the vortex.

8. A rotat ing device to sprinkle water is shown in Figure 5.59. Water enters the rotat ingdevice at the center at a rate of 0.03 m3/s and then it is directed radially through threeident ical channels of exit area 0.005 m2 each, perpendicular to the direct ion of flow relat ive tothe device. The water leaves at an angle of 30 relat ive to the device as measured from theradial direct ion, as shown. If the device rotates clock wise with a speed of 20 radians/srelat ive to the ground. Compute the magnitude of the average velocity of the fluid leavingthe vane as seen from the ground.

[Answer: 9.16 m/s, at an angle of 79 with respect to the ground (horizontal)]

Figure 5.59 A sprinkler.

9. Determine an expression for the vort icity of the flow field described by:

Is the flow irrotat ional?[Answer: . The flow is not irrotat ional, since the vort icity is not zero.]

10. When a circulat ion of strength Γ is imposed on a circular cylinder placed in an uniformincompressible flow of velocity U∞, the cylinder experiences lift . If the lift coefficient CL = 2,

calculate the peak (negat ive) pressure coefficient on the cylinder.[Answer: − 4.373]

11. A wing with an ellipt ical planform and an ellipt ical lift distribut ion has aspect rat io 6 andspan of 12 m. The wing loading is 900 N/m2, when flying at a speed of 150 km/hr at sea level.Calculate the induced drag for this wing.

[Answer: 969.44 N]

12. A free vortex flow field is given by , for r > 0. If the flow density ρ = 103 kg/m3 and thevolume flow rate q = 20π m2/s, express the radial pressure gradient, ∂p/∂ r, as a funct ion ofradial distance r, and determine the pressure change between r1 = 1 m and r2 = 2 m.

[Answer: , 37.5 kPa]

13. A viscous incompressible fluid is in a two-dimensional mot ion in circles about the originwith tangent ial velocity:

where ν is kinematic velocity and t is t ime. Find the vort icity ζ.[Answer:

14. A circular cylinder of radius a is in an otherwise uniform stream of inviscid fluid but with aposit ive circulat ion round the cylinder, as shown in Figure 5.60. Find the lift and drag per unitspan of the cylinder. Also, sketch the streamlines around the cylinder if the circulat ion issubcrit ical.

[Answer: lift = ρV∞Γ, drag = 0]

Figure 5.60 A circular cylinder in a uniform stream with circulat ion.

15. A square ring vortex of side 2a. If each side has strength Γ, calculate the velocity inducedat the center of the ring.

[Answer:

Notes

1. Note that in Equat ion (5.3) both V and ds are vector quant it ies. Therefore, their “dot”product has become 2. From Equat ion (5.5), it is seen that the circulat ion is undefined for r ≤ r0, that is, around theaxis of rotat ion the circulat ion is not defined.3. Indeed, this is t rue for any vector, for example, if a and b are vectors,a · (a × b) = [aab] = 0 .Therefore, in general, it can be expressed,[ a] = 0,where and a are vectors. The representat ion “[]” is termed “box” notat ion in vector algebra.4. ijk is a tensor and is posit ive when the subscripts i, j, k are expressed in cyclic order. Thatis,

ijk = jki = kij .But, for noncyclic orders of i, j, k

ijk = − jki

and so on.5. The induced velocity for the circulat ion shown (that is, clockwise when viewed from right toleft ), is into the page. When the circulat ion direct ion is reversed (that is, counter clockwise)

left ), is into the page. When the circulat ion direct ion is reversed (that is, counter clockwise)the induced velocity will be from the page to upwards.6. The Kutta-Joukowski theorem is a fundamental theorem of aerodynamics. It is namedafter the German Mart in Wilhelm Kutta and the Russian Nikolai Zhukovsky (or Joukowski)who first developed its key ideas in the early 20th century. The theorem relates the liftgenerated by a right cylinder to the speed of the cylinder through the fluid, the density of thefluid, and the circulat ion. The circulat ion is defined as the line integral, around a closed loopenclosing the cylinder or aerofoil, of the component of the velocity of the fluid tangent to theloop. The magnitude and direct ion of the fluid velocity change along the path.

The flow of air in response to the presence of the aerofoil can be treated as thesuperposit ion of a t ranslat ional flow and a rotat ional flow. It is, however, incorrect to thinkthat there is a vortex like a tornado encircling the cylinder or the wing of an airplane in flight . Itis the integral's path that encircles the cylinder, not a vortex of air. (In descript ions of theKutta-Joukowski theorem the aerofoil is usually considered to be a circular cylinder or someother Joukowski aerofoil.)

The theorem refers to two-dimensional flow around a cylinder (or a cylinder of infinite span)and determines the lift generated by one unit of span. When the circulat ion Γ∞ is known, thelift L per unit span (or l) of the cylinder can be calculated using the following equat ion:l = ρ∞V∞Γ∞,where ρ∞ and V∞ are the density and velocity far upstream of the cylinder, and Γ∞ is thecirculat ion defined as the line integral,Γ∞ = cV cos θ dsaround a path c (in the complex plane) far from and enclosing the cylinder or airfoil. This pathmust be in a region of potent ial flow and not in the boundary layer of the cylinder. The V cosθ is the component of the local fluid velocity in the direct ion of and tangent to the curve c,and ds is an infinitesimal length on the curve c. The above equat ion for lift l is a form of theKutta-Joukowski theorem.

The Kutta-Joukowski theorem states that, “the force per unit length act ing on a rightcylinder of any cross sect ion whatsoever is equal to ρ∞V∞Γ∞, and is perpendicular to thedirect ion of V∞.”7. It would be of value to note that, for a steady, incompressible viscous flow, the Bernoulliequat ion can be applied between any two points along a streamline only. But for a steady,incompressible and inviscid (that is, potent ial) flow, the Bernoulli equat ion can be appliedbetween any two points, in the ent ire flow field. That is, the two points between which theBernoulli equat ion is applied need not lie on a streamline.

References1. Rathakrishnan, E., Fluid Mechanics –An Introduction, 3rd edn. PHI Learning, New Delhi, 2012.2. Milne-Thomson, L.M., Theoretical Aerodynamics, 2nd edn. Macmillan & Co., Ltd, London,1952.3. Lamb, H., Hydrodynamics, 6th edn. Dover Publicat ions, 1932.4. Rathakrishnan, E., Applied Gas Dynamics, John Wiley, NJ, 2010.

6

Thin Aerofoil Theory

6.1 IntroductionThe main limitat ion of Joukowski's theory is that it is applicable only to the Joukowski family ofaerofoil sect ions. Similar is the case with aerofoils obtained with other t ransformat ions. Theseaerofoils do not permit a sat isfactory solut ion of the reverse problem of aerofoil design, that is,to start with the loading distribut ion and from the loading, obtain the necessary aerofoil profile.For the indirect or reverse solut ion to be possible, a theory which consists of more localrelat ionships is required. That is:

The overall lift ing property of a two-dimensional aerofoil depends on the circulat ion itgenerates and this, for the far-field or overall effects, has been assumed to beconcentrated at a point within the aerofoil profile, and to have a magnitude related to theincidence, camber and thickness of the aerofoil.The loading on the aerofoil, or the chordwise pressure distribut ion, follows as aconsequence of the parameters, namely the incidence, camber and thickness. But thecamber and thickness imply a characterist ic shape which depends in turn on theconformal t ransformat ion funct ion and the basic flow to which it is applied.The profiles obtained with Joukowski t ransformat ion do not lend themselves to modernaerofoil design.However, Joukowski t ransformat ion is of direct use in aerofoil design. It introduces somefeatures which are the basis to any aerofoil theory, such as:

1. The lift generated by an aerofoil is proport ional to the circulat ion around the aerofoilprofile, that is, L ∝ Γ.2. The magnitude of the circulat ion Γ must be such that it keeps the velocity finite inthe vicinity of t railing edge.

It is not necessary to concentrate the circulat ion in a single vortex, as shown in Figure6.1(a), and an immediate extension to the theory is to distribute the vort icity throughoutthe region surrounded by the aerofoil profile in such a way that the sum of the distributedvort icity equals that of the original model, as shown in Figure 6.1(b), and the vort icity atthe trailing edge is zero.

Figure 6.1 Vortex distribut ion over an aerofoil region: (a) single vortex around an aerofoil, (b)distribut ion of smaller vort ices equivalent to the single vortex.

This mathematical model may be simplified by distribut ing the vort ices on the camber lineand disregarding the effect of thickness. In this form it becomes the basis for the classical “thinaerofoil theory” of Munk and Glauert .

Considering the fact that the t ransformat ion applied to a circle in an uniform streamgives a straight line aerofoil (that is, a flat plate), the theory assumed that the general thinaerofoil could be replaced by its camber line, which is assumed to be only a slight distort ion of astraight line. Consequent ly the shape from which the camber line has to be transformed wouldbe a similar distort ion from the original circle. The original circle could be found by t ransformingthe slight ly distorted shape, shown in Figure 6.2.

Figure 6.2 Transformat ion of a distorted circle into a camber line.

This t ransformat ion funct ion defines the distort ion, or change of shape, of the circle, andhence by implicat ion, the distort ion (or camber) of the straight line aerofoil. As shown in Figure6.2, the circle (z = aeiθ) in the z-plane is t ransformed to the “S” shape in the z '-plane using thet ransformat ion z ' = f(z), and then the S shape to a cambered profile using the followingtransformat ion:

It is evident that z ' = f(z) defines the shape of the camber and Glauert used the seriesexpansion:

for this. Using potent ial theory and Joukowski hypothesis, the lift and pitching moment act ingon the aerofoil sect ion were found in terms of the coefficients Ax, that is in terms of the shapeparameter.

The usefulness or advantage of the theory lies in the fact that the aerofoil characterist icscould be quoted in terms of the coefficient Ax, which in turn could be found by graphicalintegrat ion method from any camber line.

6.2 General Thin Aerofoil TheoryThis theory is based on the assumption that the aerofoil is thin so that its shape is effect ivelythat of its camber line and the camber line shape deviates only slight ly from the chord line. Inother words, the theory should be restricted to low angles of incidence. The modificat ion to theaerofoil theory with the above simplificat ions are the following:

Replacement of the camber line by a string of line vort ices of infinitesimal strengths, asshown in Figure 6.3(a).

Figure 6.3 (a) Replacement of the camber line by a string of line vort ices, (b) an element oflength δs on the chord.

The camber line is replaced by a line of variable vort icity so that the total circulat ion aboutthe chord is the sum of the vortex elements. Thus, the circulat ion around the camberbecomes:

(6.1) where k is the vort icity distribut ion over the element of camber line, δs, circulat ion is takenposit ive (+ve) in the clockwise direct ion, as shown in Figure 6.3(a), and c is the chord ofthe profile.

Following Glauert , the leading edge is taken as the origin, ox along the chord and oy normalto it . The basic assumptions of the theory permit the variat ion of vort icity along the camber linebe assumed to be the same as the variat ion along ox-axis, that is, δs differs negligibly from δx.Therefore, the circulat ion can be expressed as:

(6.2) Hence the lift per unit span is given by:

(6.3) With ρUk = p, Equat ion (6.3) can be writ ten as:

(6.4) Again for unit span, p has the units of force per unit area or pressure and the moment of thesechordwise pressure forces about the leading edge or origin of the system is:

(6.5) The negat ive sign for Mle in Equat ion (6.5) is because it is convent ional to take the nose-downmoment as negat ive and nose-up moment as posit ive. For an aircraft in normal flight with liftact ing in the upward direct ion, the moment about the leading edge of the wing will be nose-down. Thus, the thin wing sect ion has been replaced by a line discont inuity in the flow in theform of a vort icity distribut ion. This gives rise to an overall circulat ion, as does the aerofoil, andproduces a chordwise pressure variat ion. The stat ic pressures p1 and p2 above and below the

element δs at a locat ion with velocit ies (U + u1), (U + u2), respect ively, over the upper andlower surfaces, are as shown in Figure 6.3(b). The overall pressure difference is (p2 − p1). Byincompressible Bernoulli equat ion, we have:

where p∞ is the freestream pressure. Therefore the overall pressure difference becomes:

For a thin aerofoil at small incidence, the perturbat ion velocity rat ios u1/U and u2/U will be verysmall, and therefore, higher order terms can be neglected. Therefore, the overall pressuredifference simplifies to:

(6.6) The equivalent vort icity distribut ion indicates that the circulat ion due to the element δs is kδx(δs is replaced with δx because the camber line deviates only slight ly from the ox-axis).

Evaluat ing the circulat ion around δs and taking clockwise circulat ion as posit ive in this caseand by taking the algebraic sum of the flow of fluid along the top and bottom of δs, we get:

(6.7) From Equat ions (6.6) and (6.7), it is seen that p = ρUk, as introduced earlier.

The flow direct ion everywhere on the aerofoil must be tangent ial to the surface and makesan angle tan −1(dy/dx), since the aerofoil is thin, the angle dy/dx from ox-axis is as shown inFigure 6.4.

Figure 6.4 Flow direct ion at a general point on the aerofoil.

Resolving the vert ical velocity components, we have:

since both dy/dx and α are small, approximated as and sin α is approximated as α.Ignoring the second-order quant it ies, we can express the above equat ion as:

(6.8) The induced velocity is found by considering the effect of the elementary circulat ion kδx atx, a distance (x − x1) from the point considered.

Circulat ion kδx induces a velocity at the point x1 equal to:

The effect of all such elements of circulat ion along the chord is the induced velocity ,where:

Using Equat ion (6.8) this becomes:

(6.9) The solut ion of kdx which sat isfies Equat ion (6.9) for a given shape of camber line (definingdy/dx) and incidence can be introduced in Equat ion (6.4) and (6.5) to obtain the lift L andpitching moment M for the aerofoil shape. That is, using this circulat ion distribut ion (kdx), thelift and pitching moment can be calculated with Equat ions (6.4) and (6.5), respect ively. Fromthese L and Mle, the lift coefficient CL and pitching moment coefficient can be determined.Thus, once the circulat ion distribut ion is known, the characterist ic lift coefficient CL andpitching moment coefficient follow direct ly and hence the center of pressure coefficient ,kcp and the angle of zero lift .

6.3 Solution of the General EquationThe problem now is to express the vort icity distribut ion k as some expression in terms of thecamber line shape. Another method of finding kdx is to ut ilize the method of Equat ion (6.9),where simple expressions can be found for the velocity distribut ion around skeleton aerofoils.The present approach is to work up to the general case through part icular skeleton shapesthat do provide such simple expressions, and then apply the general case to some pract icalconsiderat ions.

6.3.1 Thin Symmetrical Flat Plate AerofoilFor a flat plate, dy/dx = 0. Therefore, the general equat ion [Equat ion (6.9)] simplifies to:

(6.10) It is convenient to express the variable x in terms of θ, through:

and x1 in terms of θ1 as:

The integrat ion limits of Equat ion (6.10) become:

and

Equat ion (6.10) becomes:

(6.11) A value of k which sat isfies Equat ion (6.11) is:

Therefore:

A more direct method for gett ing the vort icity distribut ion k is found as follows.Transformat ion of the circle z = aeiθ, through the Joukowski t ransformat ion , to a lift ing

flat plate sect ion at an incident angle α requires by the Joukowski hypothesis that sufficientcirculat ion be imposed to bring the rear stagnat ion point down to m on the cylinder, as shownin Figure 6.5(a), where the velocity at a given point P(a, θ) is:

Figure 6.5 Transformat ion of a circle to a flat plate through Joukowski t ransformat ion.

Equat ing this to zero at m gives the circulat ion as:

Therefore,

The velocity at point P(ξ, 0) (ξ = 2a cos θ, η = 0) on the flat plate in Figure 6.5(b) is:

where qc and qa refer to velocit ies on the circle and aerofoil, respect ively. The term is:

Therefore,

that is:

(6.12) For small α this simplifies to:

(6.13) The variable θ in Equat ion (6.13) is the same as that used in the general equat ion [Equat ion(6.11)]. This can be easily shown by shift ing the axes in Figure 6.5(b) to the leading edge andmeasuring x rearward. When the chord c = 4a, the distance x becomes:

Then taking:

where qa1 is the velocity at the point where θ = θ1 and qa2 is the velocity at the same point onthe other side of the aerofoil where θ = − θ1. Therefore:

Thus, in general, the elementary circulat ion at any point on the flat plate is:

(6.14)

Example 6.1(a) Find the circulat ion at the mid-point of a flat plate at 2 to a freestream of speed 30 m/s.(b) Will this be greater than or less than the circulat ion at the quarter chord point?

Solution(a) Given, α = 2 , U = 30 m/s.At the mid-point of the plate θ = π/2.Circulat ion around a plate at a small angle of incidence, by Equat ion (6.14), is:

At θ = π/2, the circulat ion is:

(b) The coordinate along the plate is:

At the quarter chord point , ξ = 3b/4, since chord is 4b and the chord is measured from θ = π,that is, from the trailing edge. Thus:

Therefore, the circulat ion at the quarter chord point becomes:

The circulat ion at the quarter chord point is greater than that at the middle of the plate.

6.3.2 The Aerodynamic Coefficients for a Flat PlateThe expression for k can be put in the equat ions for lift and pitching moment, by using thepressure:

(6.15) It is to be noted that:

Full circulat ion is involved in k.The circulat ion k vanishes at the t railing edge, where x = c and θ = π. This must

necessarily be so for the velocity at the t railing edge to be finite.The lift per unit span is given by:

Subst itut ing:

we get:

Lift per unit span is also given by:

Therefore, the lift coefficient CL becomes:

(6.16) The pitching moment about the leading edge is:

The pitching moment per unit span is also given by:

Therefore, the pitching moment coefficient becomes:

that is:

(6.17) From Equat ions (6.16) and (6.17), we get:

(6.17a) For small values of angle of at tack, α, the center of pressure coefficient , kcp, (defined as therat io of the center of pressure from the leading edge of the chord to the length is chord), isgiven by:

(6.18) This shows that the center of pressure, which is a fixed point , coincides with the aerodynamiccenter. This is t rue for any symmetrical aerofoil sect ion. The resultant force at the leadingedge, at the quarter chord point and the center of pressure of a symmetrical aerofoil areshown in Figure 6.6.

Figure 6.6 Resultant force on a symmetrical aerofoil.

By inspect ion, the quant itat ive relat ion between the three cases shown in Figure 6.6 can beexpressed as:

(6.19) Thus, the momentum coefficient about the quarter-chord point is:

But:

Therefore:

(6.20) This is the theoret ical result that “the center of pressure is at the quarter-chord point for asymmetrical aerofoil.”

By definit ion the point on the aerofoil where the moments are independent of angle of

attack is called the aerodynamic center. The point from the leading edge of the aerofoil atwhich the resultant pressure acts is called the center of pressure. In other words, center ofpressure is the point where line of act ion of the lift L meets the chord. Thus the posit ion of thecenter of pressure depends on the part icular choice of chord.

The center of pressure coefficient is defined as the ratio of the center of pressure from theleading edge of the aerofoil to the length of chord. This is represented by the symbol kcp. Oneof the desirable propert ies of an aerofoil is that the t ravel of center of pressure in the workingrange of incidence (that is from zero-lift incidence to the stalling incidence) should not be large.As incidence increases the center of pressure moves towards the quarter-chord point .

From the above result , it is seen that the moment about the quarter-chord point is zero forall values of α. Hence, for a symmetrical aerofoil, we have the theoret ical result that “thequarter-chord point is both the center of pressure and the aerodynamic center.” In otherwords, for a symmetrical aerofoil the center of pressure and aerodynamic center overlap. Thatis, for a symmetrical aerofoil the center of pressure and aerodynamic center coincide.

The relat ive posit ion of center of pressure cp and aerodynamic center ac plays a vital role inthe stability and control of aircraft . Let us have a closer look at the posit ions of cp and ac,shown in Figure 6.7.

Figure 6.7 Relat ive posit ions of cp and ac.

We know that the aerodynamic center is located around the quarter chord point , whereasthe center of pressure is a moving point , strongly influenced by the angle of at tack. When thecenter of pressure is aft of aerodynamic center, as shown in Figure 6.7(a), the aircraft willexperience a nose-down pitching moment. When the center of pressure is ahead ofaerodynamic center, as shown in Figure 6.7(b), the aircraft will experience a nose-up pitchingmoment. When the center of pressure coincided with the aerodynamic center, as shown inFigure 6.7(c), the aircraft becomes neutrally stable.

From our discussions on center of pressure and aerodynamic center, the following can beinferred:

“Center of pressure” is the point at which the pressure distribut ion can be considered toact–analogous to the “center of gravity” as the point at which the force of gravity can beconsidered to act .The concept of the “aerodynamic center” on the other hand, is not very intuit ive. Becausethe lift and locat ion of the center of pressure on an airfoil both vary linearly (more or less)with angle of at tack, α, at least within the unstalled range of α. That is we can define apoint on the chord of the airfoil at which the pitching moment remains “constant,”regardless of the α. That point is usually near the quarter-chord point and for asymmetrical airfoil the constant pitching moment would be zero. For a cambered airfoil thepitching moment about the aerodynamic center would be nonzero, but constant. Theusefulness of the aerodynamic center is in stability and control analysis where the aircraftcan be defined in terms of the wing and tail aerodynamic centers and the required lift andmoments calculated without worrying about the shift in center of pressure with α.

The horizontal posit ion of the center of gravity has a great effect on the stat ic stability ofthe wing, and hence, the stat ic stability of the ent ire aircraft . If the center of gravity issufficient ly forward of the aerodynamic center, then the aircraft is stat ically stable. If the centerof gravity of the aircraft is moved toward the tail sufficient ly, there is a point –the neutral point–where the moment curve becomes horizontal; this aircraft is neutrally stable. If the center ofgravity is moved farther back, the moment curve has posit ive slope, and the aircraft islongitudinally unstable. Likewise, if the center of gravity is moved forward toward the nose toofar, the pilot will not be able to generate enough force on the tail to raise the angle of at tack toachieve the maximum lift coefficient .

The horizontal tail is the main controllable moment contributor to the complete aircraft

moment curve. A larger horizontal tail will give a more stat ically stable aircraft than a smallertail (assuming, as is the normal case, that the horizontal tail lies behind the center of gravity ofthe aircraft ). Of course, its distance from the center of gravity is important. The farther awayfrom the center of gravity it is, the more it enhances the stat ic stability of the aircraft . The tailefficiency factor depends on the tail locat ion with respect to the aircraft wake and slipstreamof the engine, and power effects. By design it is made as close to 100% efficiency as possiblefor most stat ic stability.

Finally, with respect to the tail, the downwash from the wing is of considerable importance.Air is deflected downward when it leaves a wing, and this deflect ion of air results in the wingreact ion force or lift . This deflected air flows rearward and hits the horizontal-tail plane. If theaircraft is disturbed, it will change its angle of at tack and hence the downwash angle. Thedegree to which it changes direct ly affects the tail's effect iveness. Hence, it will reduce thestability of the airplane. For this reason, the horizontal tail is often located in a locat ion suchthat it is exposed to as lit t le downwash as possible, such as high on the tail assembly.

6.4 The Circular Arc AerofoilFollowing the same procedure as before for finding the distribut ion of k, it can be shown thatfor a circular arc aerofoil at an angle of at tack α to the flow k can be expressed as:

(6.21) From Equat ions (6.12) and (6.14) is seen that the effect of camber is to increase k distribut ionby (2U × 2β sin θ) over that of the flat plate. Thus:

where:

arises from the incidence of the aerofoil alone and:

(6.22) which is due to the effect of the camber alone. Note that this distribut ion sat isfies the Kutta-Joukowski hypothesis by allowing k to vanish at the t railing edge of the aerofoil where θ = π.

Example 6.2If the maximum circulat ion caused by the camber effect of a circular arc aerofoil is 2 m2/s, whenthe freestream velocity is 500 km/h, determine the percentage camber.

SolutionGiven, kb = 2 m2/s, U = 500/3.6 = 138.9 m/s.By Equat ion (6.22), the circulat ion due to camber is:

This circulat ion will be maximum when sin θ = 1, thus:

Therefore:

The % camber becomes:

6.4.1 Lift, Pitching Moment, and the Center of PressureLocation for Circular Arc Aerofoil

We know that the lift L, pitching moment about the leading edge of the aerofoil Mle and thepressure p act ing on the aerofoil can be expressed as:

Now, subst itut ing:

the pressure becomes:

Also,

Therefore, the lift becomes:

that is:

(6.23) The lift coefficient is:

This gives:

(6.23a) Thus the lift -curve slope is:

From the above relat ions for CL and dCL/dα, it is evident that :at α = 0, CL = 2πβat α = − β, CL = 0

and the lift -curve slope is independent of camber.For a cambered aerofoil, we have:

For CL = 0, α = − β or −αL=0 = β. Thus:

(6.24) The pitching moment is:

Therefore, the pitching moment coefficient becomes:

(6.25) In terms of CL, the can be expressed as follows. By Equat ion (6.23a), we have the CL as:

The expression for , in Equat ion (6.25), can be arranged as:

But , thus:

or

(6.26)

The center of pressure coefficient , kcp, becomes:

(6.27) Thus, the effect of camber is to set back the center of pressure by an amount whichdecreases with increasing incidence or lift . At zero lift , the center of pressure is an infinitedistance behind the aerofoil, which means that there is a moment on the aerofoil even whenthere is no resultant lift force.

Comparing this with Equat ion (6.17a) ( ) for flat plate we see that the camber of circulararc decreases the moment about the leading edge by πβ/2.

Example 6.3(a) A flat plate is at an incidence of 2 in a flow; determine the center of pressure. (b) If acircular arc of 3% camber is in the flow at the same incidence, where will be center ofpressure?

Solution(a) Given, α = 2 .For a flat plate, by Equat ion (6.16), the lift coefficient is:

By Equat ion (6.17):

The center of pressure, By Equat ion (6.18), is:

(b) Given α = 2 , , therefore:

By Equat ion (6.23a):

By Equat ion (6.26):

The center of pressure, by Equat ion (6.27), is:

Aliter:Note that the kcp is also given by Equat ion (6.27), as:

This is the same as that given by dividing with CL.

6.5 The General Thin Aerofoil SectionIn Sect ion 6.4, we saw that the general camber line can be replaced by a chordwise distribut ionof circulat ion. That is:

where ka is the same as the distribut ion over the flat plate but must contain a constant (A0) toabsorb any difference between the equivalent flat plate and the actual chord line. Therefore:

(6.28) Note that this ka distribut ion sat isfies the Kutta-Joukowski distribut ion, since ka = 0 when θ =

π, that is, at x = c.The corresponding kb is represented by a Fourier series. Providing 0 < θ < π, the end

condit ions are sat isfied, and any variat ion in shape is accommodated if it is a sine series. Thus:

(6.29) Thus, k = ka + kb becomes:

(6.30) Note that, for circular arc aerofoil, we have kb = 2UA1 sin θ.

The coefficients A0, A1, A2, · · · · , An can be obtained by subst itut ing for k in the generalEquat ion (6.30), suitably converted with regard to units, that is:

Subst itut ing:

we get:

Using Equat ion (6.30), we get:

At the point x1 (or θ1) on the aerofoil:

Expressing ∑An sin nθ sin θ as:

we have:

where Gn signifies the integral:

which has the solut ion:

Therefore:

For the general point on the aerofoil, we get:

that is:

(6.31) On integrat ing from θ = 0 to π, the third term on the right-hand-side of Equat ion (6.31)vanishes. Therefore, we have:

This simplifies to:

(6.32) Mult iplying Equat ion (6.32) by cos mθ, where m is an integer, and integrat ing with respect to θ,we get:

(6.33)

The integral:

for all values of n except at n = m. Therefore, the first term on the right-hand-side of Equat ion(6.33) vanishes, and also the second term, except for n = m becomes:

Thus:

(6.34)

6.6 Lift, Pitching Moment and Center of PressureCoefficients for a Thin Aerofoil

From Equat ion (6.30), the circulat ion distribut ion is:

Therefore, the lift becomes:

since:

The lift is also given by:

Therefore, the lift coefficient becomes:

(6.35) The pitching moment is given by:

since:

Therefore:

(6.36) that is:

The center of pressure coefficient is:

(6.37) From Equat ion (6.37) it is seen that for this case also the center of pressure moves as the liftor incidence is changed. We know that the kcp is also given by [Equat ion (6.19)]:

Comparing Equat ions (6.36) and (6.37), we get:

(6.38) This shows that, theoret ically, the pitching moment about the quarter chord point for a thinaerofoil is a constant, depending on the camber parameters only, and the quarter chord pointis therefore the aerodynamic center.

Example 6.4The camberline of a thin aerofoil is given by:

where x and y are in terms of unit chord and the origin is at the leading edge. If the maximumcamber is 2.2% of chord, determine the lift coefficient and the pitching moment coefficientwhen the angle of at tack is 4 .

SolutionGiven, camber is 0.022 and α = 4 .At the maximum camber locat ion, let x = xm. At the maximum camber:

that is:

Out of the above two values of 1.577 and 0.423, the second one is the only feasible solut ionfor xm. Therefore, the maximum camber is at xm = 0.423. Subst itut ing this we have:

But:

Subst itut ing this, we get:

By Equat ion (6.32):

By Equat ion (6.34):

By Equat ion (6.35):

By Equat ion (6.38):

Note: The moment is given with a negat ive sign because this is a nose-down moment.

Example 6.5A sail plane of wing span 18 m, aspect rat io 16 and taper rat io 0.3 is in level fight at an alt itudewhere the relat ive density is 0.7. The true air speed measured by an error free air speedindicator is 116 km/h. The lift and drag act ing on the wing are 3920 N and 160 N, respect ively.The pitching moment coefficient about the quarter chord point is −0.03. Calculate the meanchord and the lift and drag coefficients, based on the wing area and mean chord. Also,calculate the pitching moment about the leading edge of the wing.

SolutionGiven 2b = 18 m, ARlig;++ = 16, λt/λr = 0.3, σ = 0.7, L = 3920 N, D = 160 N, Vr = 116 km/h, .The relat ive density is:

where ρ0 is the sea level density, equal to 1.225 kg/m3. Therefore:

Equivalent air speed is:

The mean chord is:

The wing area is:

The lift coefficient is:

The drag coefficient is:

By Equat ion (6.38):

By Equat ion (6.36):

The pitching moment about the leading edge is:

The negat ive sign to the moment implies that it is a nose-down moment.

6.7 Flapped AerofoilThe flap at the t railing edge of an aerofoil is essent ially a high-lift ing device, which whendeflected down causes increase of lift , essent ially by increasing the camber of the profile. Thethin aerofoil discussed in the previous sect ions of this chapter can readily be applied toaerofoils with variable camber such as flapped aerofoils. It has been found that the circulat iondistribut ion along the camber line for the general aerofoil can comfortably be split into thecirculat ion due to a flat plate at an incidence and the circulat ion due to the camber line.

It is sufficient for the assumptions in the theory to consider the influence of a flap deflect ionas an addit ion to the above two components. Figure 6.8 illustrates how the three contribut ionsto lift generat ion can be combined to get the resultant effect .

Figure 6.8 Linear split t ing of the effect of incidence, camber of the profile and camber effectdue to flap deflect ion, (a) camber line of profile with a flap at an angle of at tack, (b) camber lineeffect , (c) flap deflect ion effect , (d) angle of at tack effect .

Indeed the deflect ion of the flap about a hinge in the camber line effect ively alters thecamber of the profile so that the contribut ion due to flap deflect ion is an addit ion to the effectof the camber line shape. In this manner, the problem of a cambered aerofoil with flap isreduced to the general case of fit t ing a camber line made up of the chord of the aerofoil andthe flap deflected through an angle η, as shown in Figure 6.9.

Figure 6.9 An aerofoil chord at an angle of incidence, with deflected flap.

The thin aerofoil theory st ipulates that the slope of the aerofoil surface is small and that thedisplacement from the x-axis is small. In other words, the leading and/or t railing edges need notbe on the x-axis.

Let us define the camber as hc so that the slope of the part AB of the aerofoil is zero, andthe slope of the flap h/F. To find the coefficient of the circulat ion k for the flap camber, let ussubst itute these values of slope in Equat ions (6.32) and (6.34) but confining the limits ofintegrat ion to the parts of the aerofoil over which the slopes occur. Thus:

(6.39)

where ϕ is the value of θ at the hinge, that is:

hence cos ϕ = 2F − 1. Integrat ing Equat ion (6.39), we get:

that is:

(6.40) Similarly, from Equat ion (6.34):

This gives:

(6.41) Thus:

The chordwise circulat ion distribut ion due to flap deflect ion becomes:

(6.42) For constant α, Equat ion (6.42) is a linear funct ion of η, as is the lift coefficient , for examplefrom Equat ion (6.35):

giving:

(6.43) Likewise, the pitching moment coefficient from Equat ion (6.36) is:

that is:

(6.44) In Equat ions (6.43) and (6.44)ϕ is given by:

Note that a posit ive (that is, downward) deflect ion of the flap decreases the momentcoefficient , tending to pitch the aerofoil nose down and vice versa.

6.7.1 Hinge Moment CoefficientThe characterist ic of a flapped aerofoil, which is of great importance in stability and control ofthe aircraft , is the aerodynamic moment about the hinge line, H, shown in Figure 6.10.

Figure 6.10 Illustrat ion of a flap and hinge moment.

Taking moments of elementary pressures p act ing on the flap, about the hinge:

where:

and:

Now putt ing:

and k from Equat ion (6.42), we get the hinge moment as:

Subst itut ing:

and simplifying, we obtain:

(6.45) where:

In the convent ional notat ion:

where:

From Equat ion (6.45):

That is:

(6.46) Similarly from Equat ion (6.45):

This will reduce to:

(6.47) The parameter a1 = ∂ CL/∂ α is 2π and a2 = ∂ CL/∂ η from Equat ion (6.43) becomes:

(6.48) Thus thin aerofoil theory provides an est imate of all the parameters of flapped aerofoil.

6.7.2 Jet FlapTreat ing the jet flap as a high-velocity sheet of air issuing from the trailing edge of an aerofoilat some downward angle θ to the chord line of the aerofoil, as shown in Figure 6.11, an analysiscan be made by replacing the jet stream as well as the aerofoil by a vortex distribut ion [1, 2].

Figure 6.11 Illustrat ion of a jet flap as a band of spanwise vortex filaments.

The flap contributes to the lift in the following two ways:1. The downward deflect ion of the efflux produces a lift ing component of react ion.2. The jet affects the pressure distribut ion on the aerofoil in a similar manner to thatobtained by an addit ion to the circulat ion round the aerofoil.

The jet is shown to be equivalent to a band of spanwise vortex filaments which for smalldeflect ion angles θ can be assumed to be on the ox-axis, shown in Figure 6.11.

Considering both the contribut ions ment ioned above, it can be shown that the lift coefficientcan be expressed as:

(6.49) where A0 and B0 are the init ial coefficients in the Fourier series associated with the deflect ionof the jet and the incidence of the aerofoil, respect ively, and which can be obtained in terms of

the momentum of the jet .

6.7.3 Effect of Operating a FlapLet us consider an aerofoil whose rear part is movable about a hinge at P on the camber line,as shown in Figure 6.12(a). Essent ially the rear part PH is the flap, which can be raised orlowered from the neutral posit ion shown in the figure.

Figure 6.12 (a) An aerofoil with flap, (b) simplified representat ion of the aerofoil and flap.

In an aerofoil of finite aspect rat io, the flap movement affects only part of a wing. Ailerons areflaps near the wing t ips and are arranged so that the port and starboard ailerons move inopposite senses (that is, one up and one down). In our discussions here let us consider a two-dimensional problem for simplicity and assume that the aerofoil is thin, and the port ion PH ofthe camber line is straight, and the angle ξ through which the flap is rotated is small.

For the thin aerofoil shown in Figure 6.12(b) the eccentric angle which defines the posit ion ofhinge P is η. In the neutral posit ion PH of the flap we can express the lift coefficient , [seeEquat ion (6.23a)], as:

where y denotes the yc. The effect of raising the flap is to decrease to on the raisedpart PH ' and to leave it unaltered on the part PA. The lift coefficient is thereby altered to CL 'where:

(6.50) Thus:

(6.51) Thus the effect of raising the flap is to decrease the lift coefficient , the effect of lowering theflap is to increase the lift coefficient . Therefore, in part icular, when the flaps are lowered justbefore landing, increased lift is obtained (and also increased drag). In the case of ailerons, if theport aileron is raised and the starboard aileron depressed, the lift on the port wing is decreasedand that on the starboard wing is increased, causing a rolling moment which tends to raise thestarboard wing t ip.

6.8 SummaryThe overall lift ing property of a two-dimensional aerofoil depends on the circulat ion itgenerates and this, for the far-field or overall effects, has been assumed to be concentrated ata point within the aerofoil profile, and to have a magnitude related to the incidence, camberand thickness of the aerofoil.

The loading on the aerofoil, or the chordwise pressure distribut ion, follows as a consequenceof the parameters, namely the incidence, camber and thickness. But the camber and thicknessimply a characterist ic shape which depends in turn on the conformal t ransformat ion funct ionand the basic flow to which it is applied.

The profiles obtained with Joukowski t ransformat ion do not lend themselves to modernaerofoil design. However, Joukowski t ransformat ion is of direct use in aerofoil design. Itintroduces some features which are the basis to any aerofoil theory, such as (a) the liftgenerated by an aerofoil is proport ional to the circulat ion around the aerofoil profile, L ∝ Γ. (b)The magnitude of the circulat ion Γ must be such that it keeps the velocity finite in the vicinityof t railing edge. It is not necessary to concentrate the circulat ion in a single vortex, the vort icitycan be distributed throughout the region surrounded by the aerofoil profile in such a way thatthe sum of the distributed vort icity equals that of the original model, and the vort icity at the

t railing edge is zero. This mathematical model may be simplified by distribut ing the vort ices onthe camber line and disregarding the effect of thickness. In this form it becomes the basis forthe classical “thin aerofoil theory” of Munk and Glauert .

The usefulness or advantage of the theory lies in the fact that the aerofoil characterist icscould be quoted in terms of the coefficient Ax, which in turn could be found by graphicalintegrat ion method from any camber line.

General thin aerofoil theory is based on the assumption the aerofoil is thin so that its shapeis effect ively that of its camber line and the camber line shape deviates only slight ly from thechord line.

The camber line is replaced by a line of variable vort icity so that the total circulat ion aboutthe chord is the sum of the vortex elements. Thus, the circulat ion around the camber becomes:

The lift per unit span is given by:

Again for unit span, the moment of pressure forces about the leading edge is:

For a flat plate, dy/dx = 0. Therefore, the general equat ion [Equat ion (6.9)] simplifies to:

The elementary circulat ion at any point on the flat plate is:

Lift per unit span is given by:

The lift coefficient CL becomes:

The pitching moment per unit span is:

The pitching moment coefficient becomes:

For small values of angle of at tack, α, the center of pressure coefficient , kcp, (defined as therat io of the center of pressure from the leading edge of the chord to the length is chord), isgiven by:

This shows that the center of pressure, which is a fixed point , coincides with the aerodynamiccenter. This is t rue for any symmetrical aerofoil sect ion.

The center of pressure is at the quarter-chord point for a symmetrical aerofoil.By definit ion the point on the aerofoil where the moments are independent of angle of

at tack is called the aerodynamic center. The point from the leading edge of the aerofoil atwhich the resultant pressure acts is called the center of pressure. In other words, center ofpressure is the point where line of act ion of the lift L meets the chord. Thus the posit ion of thecenter of pressure depends on the part icular choice of chord.

The center of pressure coefficient is defined as the ratio of the center of pressure from theleading edge of the aerofoil to the length of chord.

The moment about the quarter-chord point is zero for all values of α. Hence, for asymmetrical aerofoil, we have the theoret ical result that “the quarter-chord point is both thecenter of pressure and the aerodynamic center.”

The aerodynamic center is located around the quarter chord point . Whereas the center ofpressure is a moving point , strongly influenced by the angle of at tack.

Center of pressure is the point at which the pressure distribut ion can be considered to act –analogous to the “center of gravity” as the point at which the force of gravity can beconsidered to act .

The horizontal posit ion of the center of gravity has a great effect on the stat ic stability ofthe wing, and hence, the stat ic stability of the ent ire aircraft . If the center of gravity issufficient ly forward of the aerodynamic center, then the aircraft is stat ically stable. If the centerof gravity of the aircraft is moved toward the tail sufficient ly, there is a point –the neutral point–where the moment curve becomes horizontal; this aircraft is neutrally stable. If the center ofgravity is moved farther back, the moment curve has posit ive slope, and the aircraft islongitudinally unstable.

For a circular arc aerofoil at an angle of at tack α to the flow k can be expressed as:

The effect of camber is to increase k distribut ion by (2U × 2β sin θ) over that of the flat plate.Thus:

where:

arises from the incidence of the aerofoil alone and:

which is due to the effect of the camber alone.The lift L act ing on the aerofoil can be expressed as:

Thus the lift -curve slope is:

From the above relat ions for CL and dCL/dα, it is evident that :at α = 0, CL = 2πβat α = − β, CL = 0

and the lift -curve slope is independent of camber.For a cambered aerofoil, we have:

For CL = 0, α = − β or −αL=0 = β. Thus:

The pitching moment is:

Therefore, the pitching moment coefficient becomes:

or

The center of pressure coefficient , kcp, becomes:

Thus, the effect of camber is to set back the center of pressure by an amount whichdecreases with increasing incidence or lift .

The general camber line can be replaced by a chordwise distribut ion of circulat ion. That is:

where

Note that this ka distribut ion sat isfies the Kutta-Joukowski distribut ion, since ka = 0 when θ =π, that is, at x = c.

The corresponding kb is represented by a Fourier series. Providing 0 < θ < π, the endcondit ions are sat isfied, and any variat ion in shape is accommodated if it is a sine series. Thus:

Thus, k = ka + kb becomes:

The coefficients A0, A1, A2, · · · · , An can be obtained by subst itut ing for k in the generalEquat ion (6.30), suitably converted with regard to units.

For a thin aerofoil, the circulat ion distribut ion is:

The lift is also given by:

Therefore, the lift coefficient becomes:

The pitching moment is given by:

The pitching moment coefficient is:

The center of pressure coefficient is:

Flap at the t railing edge of an aerofoil is a high-lift ing device, which when deflected downcauses increase of lift , essent ially by increasing the camber of the profile. The deflect ion of theflap about a hinge in the camber line effect ively alters the camber of the profile so that thecontribut ion due to flap deflect ion is an addit ion to the effect of camber line shape.

The chordwise circulat ion distribut ion due to flap deflect ion becomes:

The lift coefficient is:

Likewise, the pitching moment coefficient is:

The characterist ic of a flapped aerofoil, which is of great importance in stability and controlof the aircraft , is the aerodynamic moment about the hinge line, H:

In an aerofoil of finite aspect rat io, the flap movement affects only part of a wing. Ailerons areflaps near the wing t ips and are arranged so that the port and starboard ailerons move inopposite senses (that is, one up and one down). The effect of raising the flap is to decreasethe lift coefficient , the effect of lowering the flap is to increase the lift coefficient . Therefore, inpart icular, when the flaps are lowered just before landing, increased lift is obtained (and alsoincreased drag). In the case of ailerons, if the port aileron is raised and the starboard ailerondepressed, the lift on the port wing is decreased and that on the starboard wing is increased,causing a rolling moment which tends to raise the starboard wing t ip.

Exercise Problems1. Determine the maximum circulat ion due to camber of a circular arc aerofoil of percentagecamber 0.05, in a flow of velocity 200 km/h.

[Answer: 0.2224 m2/s]

2. If the lift coefficient and lift curve slope of an aerofoil of percentage camber 0.6 are 1.02and 2, respect ively, determine (a) the pitching moment about the leading edge and (b) thecenter of pressure coefficient .

[Answer: (a) −0.254, (b) 0.2685]

3. If the elemental circulat ion at 30% chord of a flat plate in a flow at 40 m/s is 24 m2/s,determine the angle of at tack.

[Answer: 11.25 ]

4. A two-dimensional wing of NACA 4412 profile flies at an incidence of 4 . Determine the liftcoefficient of the wing.

[Answer: 1.024]

5. An aerofoil of average chord 1.2 m, at an angle of at tack 2 to a flow at 45 m/s at sea level,experiences a lift of 500 N per unit area. Determine the pitching moment about the leadingedge, (a) assuming the profile to be symmetrical and (b) assuming the profile is camberedwith 3% camber.

[Answer: (a) −178.6 N-m, (b) −434 N-m]

6. A Joukowski profile of 3.3% camber is in an air stream of speed 60 m/s at an angle ofat tack of 4 . Determine the circulat ion around the maximum thickness locat ion.

[Answer: 28.22 m2/s]

7. A parabolic camber line of unit chord length is at an incidence of 3 in a uniform flow ofvelocity 20 m/s. If the camber line is given by:

determine the velocity induced at the mid-chord locat ion, assuming the incidence as an idealangle of at tack.

[Answer: −3.14 m/s]

8. An aircraft of wing area 42 m2 and mean chord 3 m flies at 120 m/s at an alt itude wherethe density is 0.905 kg/m3. The center of pressure is at 0.28 t imes mean chord behind theleading edge of the wing when the wing lift coefficient is 0.2. (a) If the lift on the tail planeacts through a point 8 m horizontally behind the center of pressure, determine the tail liftrequired to t rim the aircraft . (b) Assuming the wing profile as a circular arc, find thepercentage camber. Assume the pitching moments on the tail plane, fuselage and nacellesare negligibly small.

[Answer: (a) 5201 N, (b) 0.191]

9. A thin aerofoil of 3% camber in a freestream has a lift coefficient of 1.2. (a) If the liftcoefficient has to be increased by 10% of the init ial value, what should be the increase in theangle of at tack required? (b) Find the percentage change in the pitching moment coefficientcaused by this change in the angle of at tack.

[Answer: (a) 1 . 15 , (b) 7.88%]

10. A flat plate of length 1.2 m and width 1 m, in a uniform air stream of pressure 1 atm,temperature 30 C and velocity 30 m/s, experiences a lift of 1500 N. Determine the liftcoefficient , angle of at tack, pitching moment about the leading edge and the locat ion of

center of pressure.[Answer: CL = 2.384, α = 21 . 71 , , kcp = 0.25]

References1. Spence, D.A., The lift coefficient of a thin, jet flapped wing, Proc. Roy. Soc. A. , 1212,December 1956.2. Spence, D.A., The lift on a thin aerofoil with jet augmented flap, Aeronautical Quarterly,August 1958.

7

Panel Method

7.1 IntroductionPanel method is a numerical technique to solve flow past bodies by replacing the body withmathematical models; consist ing of source or vortex panels. Essent ially the surface of body tobe studied will be represented by panels consist ing of sources and free vort ices. These arereferred to as source panel and vortex panel methods, respect ively. If the body is a liftgenerat ing geometry, such as an aircraft wing, vortex panel method will be appropriate forsolving the flow past, since the lift generated is a funct ion of the circulat ion or the vort icityaround the wing. If the body is a nonlift ing structure such as a pillar of a river bridge, the sourcepanel method might be employed for solving the flow past that .

7.2 Source Panel MethodConsider the source sheet of finite length along the s-direct ion and extending to infinity in thedirect ion normal to s, as shown in Figure 7.1.

The source strength per unit length along s-direct ion of the panel, shown in Figure 7.1, is λ =λ(s). Also, the small length segment ds is t reated as a dist inct source of strength λ ds.

Figure 7.1 A source sheet.

Let us consider a point P as shown in Figure 7.1. The small segment of the source sheet ofstrength λ ds induces an infinitesimally small velocity potent ial dϕ at point P. That is:

The complete velocity potent ial at point P, due to source sheet from a to b, is given by:

In general, the source strength λ(s) can change from posit ive (+ve) to negat ive (−ve) along thesource sheet. That is, the ‘source’ sheet can be a combinat ion of line sources and line sinks.

Next, let us consider a given body of arbit rary shape shown in Figure 7.2.

Figure 7.2 An arbit rary body (a) covered with source sheet, (b) the flow past it .

Let us assume that the body surface is covered with a source sheet, as shown in Figure7.2(a), where the strength of the source λ(s) varies in such a manner that the combined act ionof the uniform flow and the source sheet makes the aerofoil surface and streamlines of theflow, as shown in Figure 7.2(b).

The problem now becomes that of finding appropriate distribut ion of λ(s), over the surface ofthe body. The solut ion to this problem is carried out numerically, as follows:

Approximate the source sheet by a series of straight panels, as shown in Figure 7.3.Let the source strength λ(s) per unit length be constant over a given general panel, butallow it to vary from one panel to another. That is, for n panels the source strengths areλ1, λ2, λ3, . . . . . , λn.The main object ive of the panel technique is to solve for the unknowns λj, j = 1 to n, suchthat the body surface becomes a stream surface of the flow.This boundary condit ion is imposed numerically by defining the mid-point of each panel tobe a control point and by determining the λjs such that the normal component of the flowvelocity is zero at each point .

Figure 7.3 A body of arbit rary shape covered with a series of straight source panels.

Let P be a point at (x, y) in the flow, and let rpj be the distance from any point on the j th panelto point P, as shown in Figure 7.3. The velocity at P due to the j th panel, Δϕj, is:

where the source strength λj is constant over the j th panel, and the integrat ion is over the j th

panel only.The velocity potent ial at point P due to all the panels can be obtained by taking the

summation of the above equat ion over all the panels. That is:

where the distance:

where (xj, yj) are the coordinates along the surface of the j th panel.Since P is just an arbit rary point in the flow, it can be taken at anywhere in the flow including

the surface of the body, which can be regarded as a stream surface (essent ially the stagnat ionstream surface). Let P be at the control point of the i th panel. Let the coordinates of thiscontrol point be given by (xi, yi), as shown in Figure 7.3. Then:

(7.1) where

Equat ion (7.1) is physically the contribut ion of all the panels to the potent ial at the controlpoint on the i th panel.

The boundary condit ion is at the control points on the panels and the normal component offlow velocity is zero. Let n be the unit vector normal to the i th panel, directed out of the body.

The slope of the i th panel is (dy/dx)i. The normal component of velocity V∞ with respect tothe i th panel is:

where βi is the angle between V∞ and ni. Note that V∞,n is posit ive (+ve) when directed awayfrom the body.

The normal component of velocity induced at (xi, yi) by the source panel, from Equat ion (7.1),is:

(7.2) When the different iat ion in Equat ion (7.2) is carried out, rij appears in the denominator.Therefore, a singular point arises on the i th panel because at the control point of the panel, j =i and rij = 0. It can be shown that when j = i, the contribut ion to the derivat ive is λi/2. Therefore:

(7.3) where λi/2 is the normal velocity induced at the i th control point by the i th panel itself.

The normal component of flow velocity is the sum of the normal components of freestreamvelocity V∞,n and velocity due to the source panel Vn. The boundary condit ion states that:

Therefore, the sum of Equat ion (7.3) and V∞,n results in:

(7.4) This is the heart of source panel method. The values of the integral in Equat ion (7.4) dependsimply on the panel geometry, which are not the propert ies of the flow.

Let Ii,j be the value of this integral when the control point is on the i th panel and the integralis over the j th panel. Then, Equat ion (7.4) can be writ ten as:

(7.5) This is a linear algebraic equat ion with n unknowns λ1, λ2, . . . . . . , λn. It represents the flowboundary condit ion evaluated at the control point of the i th panel.

Now let us apply the boundary condit ion to the control points of all the panels, that is, inEquat ion (7.5), let i = 1, 2, 3, . . . . , n. The results will be a system of n linear algebraic equat ionswit h n unknowns (λ1, λ2, . . . . . . , λn), which can be solved simultaneously by convent ionalnumerical methods:

After solving the system of equat ions represented by Equat ion (7.5) with i = 1, 2, 3, . . . . , n,we have the distribut ion source panel strength which, in an approximate fashion, causesthe body surface to be a streamline of the flow.This approximat ion can be made more accurate by increasing the number of panels,hence more closely represent ing the source sheet of cont inuously varying strength λ(s).

7.2.1 Coefficient of PressureOnce source strength distribut ions λi are obtained, the velocity tangent ial to the surface ateach control point can be calculated as follows.

Let s be the distance along the body surface, as shown in Figure 7.3, measured posit ive(+ve) from front to rear. The component of freestream velocity tangent ial to the surface is:

The tangent ial velocity Vs at the control point of the i th panel induced by all the panels isobtained by different iat ing Equat ion (7.1) with respect to s. That is:

(7.6) Note that the tangent ial velocity Vs on a flat source panel induced by the panel itself is zero;hence in Equat ion (7.6), the term corresponding to j = i is zero. This is easily seen by intuit ion,because the panel can emit volume flow only in a direct ion perpendicular to its surface and notin the direct ion tangent ial to its surface.

The surface velocity Vi at the control point of the i th panel is the sum of the contribut ionV∞,s from the freestream and Vs given by Equat ion (7.6).

(7.7)

The pressure coefficient Cp at the i th control point is:

(7.8) Note: It is important to note that the pressure coefficient given by Equat ion (7.8) is valid onlyfor incompressible flows with freestream Mach number less than 0.3. For compressible flowsthe pressure coefficient becomes:

where pi is the stat ic pressure at the i th panel and p∞, ρ∞ and V∞, respect ively, are thepressure, density and velocity of the freestream flow. The dynamic pressure can be expressedas:

since by thermal state equat ion:

Dividing the numerator and denominator of the right-hand side by γ, we have the dynamicpressure as:

since . This simplifies to:

Thus the pressure coefficient for compressible flows becomes:

(7.9)

7.2.1.1 Test on AccuracyLet sj be the length of the j th panel and λj be the source strength of the j th panel per unitlength. Hence, the strength of the j th panel is λjsj. For a closed body, the sum of the strengthsof all the sources and sinks must of zero, or else the body itself would be adding or absorbingmass from the flow. Hence, the values of the λjs obtained above should obey the relat ion:

(7.10) This equat ion provides an independent check on the accuracy of the numerical results.

7.3 The Vortex Panel MethodThis method is analogous to the source panel method studied earlier. The source panelmethod is useful only for nonlift ing cases since a source has zero circulat ion associated with it .But vort ices have circulat ion, and hence vortex panels can be used for lift ing cases. It is onceagain essent ial to note that the vort ices distributed on the panels of this numerical method areessent ially free vort ices. Therefore, as in the case of source panel method, this method is alsobased on a fundamental solut ion of the Laplace equat ion. Thus this method is valid only forpotent ial flows which are incompressible.

7.3.1 Application of Vortex Panel MethodConsider the surface of an aerofoil wrapped with vortex sheet, as shown in Figure 7.4.

Figure 7.4 An aerofoil wrapped with vortex sheet.

We wish to find the vortex distribut ion γ(s) such that the body surface becomes a streamlineof the flow. There exists no closed-form analyt ical solut ion for γ(s); rather, the solut ion must be

obtained numerically. This is the purpose of the vortex panel method.The procedure for obtaining solut ion using vortex panel method is the following:

Approximate the vortex sheet shown in Figure 7.4 by a series of straight panels.Let the vortex strength γ(s) per unit length be constant over a given panel, but allow it tovary from one panel to the next.That is, the vortex strength per unit length of the n panels are γ1, γ2, γ3, . . . . , γn. Thesepanel strengths are unknowns. Therefore, the main object ive of the vortex paneltechnique is to solve for γj, j = 1 to n, such that the body surface becomes a streamline ofthe flow and the Kutta condit ion is also sat isfied.The mid-point of each panel is a control point at which the boundary condit ion is applied;that is, at each control point , the normal component of flow velocity is zero.

Let P be a point located at (x, y) in the flow, and let rpj be the distance from any point on thej th panel to P. The radial distance rPj makes an angle θPj with respect to x-axis. The velocitypotent ial induced at P due to the j th panel [Equat ion (2.42)] is:

(7.11) The component of velocity normal to the i th panel is given by:

(7.12) The normal component of velocity induced at (xi, yi) by the vortex panels is:

(7.13) From Equat ions (7.11) and (7.13), we get the normal component of velocity induced as:

(7.14) By the boundary condit ions, at the control point of the i th, we have:

(7.15) that is:

(7.16) This equat ion is the crux of the vortex panel method. The values of the integrals in Equat ion(7.16) depend simply on the panel geometry; they are propert ies of the flow.

Let Jij be the value of this integral when the control point is on the i th panel. Now, Equat ion(7.16) can be writ ten as:

(7.17) Equat ion (7.17) is a linear algebraic equat ion with n unknowns, γ1, γ2, γ3, . . . . , γn. It representsthe flow boundary condit ions evaluated at the control point of the j th panel. If Equat ion (7.13) isapplied to the control points of all the panels, we obtain a system of n linear equat ions with nunknowns.

The discussion so far has been similar to that of the source panel method. For source panelmethod, the n equat ions for the n unknown source strength are rout inely solved, giving theflow over a nonlift ing body.

For a lift ing body with vortex panels, in addit ion to the n equat ions given by Equat ion (7.17)applied at all the panels, we must also ensure that the Kutta condit ion is sat isfied. This can bedone in many ways. For example, consider the t railing edge of an aerofoil, as shown in Figure7.5, illustrat ing the details of vortex panel distribut ion at the t railing edge.

Figure 7.5 Details of vortex panels at a t railing edge.

Note that the length of each panel can be different, their length and distribut ion over thebody is at our discret ion. Let the two panels at the t railing edge be very small. The Kutta

condit ion is applied at the t railing edge and is given by:

To approximate this numerically, if points i and (i − 1) are close enough to the trailing edge, wecan write:

(7.18) such that the strength of the two vortex panels i and (i − 1) exact ly cancel at the point wherethey touch at the t railing edge. Thus, the Kutta condit ion demands that Equat ion (7.18) mustbe sat isfied.

Note that Equat ion (7.17) is evaluated at all the panels and Equat ion (7.18) const itutes anover-determined system of n unknowns with (n + 1) equat ions. Therefore, to obtain adetermined system, Equat ion (7.17) is evaluated at one of the control points. That is, wechoose to ignore one of the control points, and evaluate Equat ion (7.17) at the other (n − 1)control points. This, on combinat ion with Equat ion (7.18), gives n linear algebraic equat ionswith n unknowns.

At this state, conceptually we have obtained γ1, γ2, γ3, . . . . . . , γn which make the bodysurface a streamline of the flow and which also sat isfy the Kutta condit ion. In turn, the flowvelocity tangent ial to the surface can be obtained direct ly from γ. To see this more clearly,consider the aerofoil shown in Figure 7.6.

Figure 7.6 Details of vortex panels and the velocity components at two specified points on anaerofoil.

The velocity just inside the vortex sheet on the surface is zero. This corresponds to u2 = 0.Hence:

Therefore, the local velocit ies tangent ial to the aerofoil surface are equal to the local values ofγ. In turn the local pressure distribut ion can be obtained from Bernoulli's equat ion. The totalcirculat ion around the aerofoil is:

(7.19) Hence, the lift per unit span is:

(7.20)

7.4 Pressure Distribution around a CircularCylinder by Source Panel Method

Let us consider a circular cylinder with a distribut ion of source panels on its circumference, asshown in Figure 7.7.

Figure 7.7 Details of source panels on a circular cylinder.

To evaluate the integral Iij, let us consider the i th and j th panels, as illustrated in Figure 7.8.

Figure 7.8 Details of source panels i and j on the circular cylinder.

The control point on the i th panel is (xi, yi). Coordinates of boundary points of i th panel are(xi, yi) and (xi+1, yi+1). An elemental length segment dsj on the j th panel is at a distance of rijfrom (xi, yi), as shown in Figure 7.8. The point (xj, yj) is the running point on the j th panel. The

boundary points of the jth panel are (xj, yj) and (xj+1, yj+1). The integral Iij is given by:

where

Therefore:

Let ϕi and ϕj be the angles measured in the counter-clockwise direct ion from the x-axis to thebottom of each panel. Thus:

Also, from the geometry shown in Figure 7.8, we have:

where Xj and Yj are the projected length of the j th panel along x-and y-direct ions, respect ively.Subst itut ing the above expressions in Iij, we get:

where

Subst itut ing:

we get:

(7.21) This is the general expression for two arbit rarily oriented panels; it is not restricted to circularcylinder only.

Now, let us apply Equat ion (7.21) to the circular cylinder. Let us assume panel 4 as the i th

panel and panel 2 as the j th panel. That is, let us calculate I4,2. Assume a unit radius for thecylinder, we have:

Subst itut ing these numbers, we get:

and

Similarly:

Now, returning to the equat ion:

which is evaluated for the i th panel, considering panel 4 as the i th panel, the above equat ionbecomes (after mult iplying each term by 2 and not ing that βi = 45 for panel 4), to result in:

(7.22) Evaluat ing Equat ion (7.21) for each of the seven other panels, simultaneously, we obtain atotal of 8 equat ions. Solving them we get:

Note that the symmetric distribut ion of these values of m which is to be expressed for thenonlift ing cylinder, account ing:

The velocity at the control point of the i th panel can be obtained from:

In this expression, the integral over the j th panel is a geometrical quant ity which is evaluated ina similar manner as before. The result is:

(7.23) The pressure coefficient for the i th panel is given by:

The distribut ion of this coefficient of pressure around the circular cylinder is as shown in Figure7.9.

Figure 7.9 Theoret ical distribut ion of Cp around a circular cylinder.

7.5 Using Panel MethodsThe major steps to be followed in the use of panel methods are:

Vary the size of panels smoothly.Concentrate panels where the flow field and/or geometry is changing rapidly.Don't spend more money and t ime (that is, numbers of panels) than required.

Panel placement and variat ion of panel size affect the quality of the solut ion. However,extreme sensit ivity of the solut ion to the panel layout is an indicat ion of an improperly posedproblem. If this happens, the user should invest igate the problem thoroughly. Panel methodsare an aid to the aerodynamicist . We must use the results as a guide to help us and developour own judgement. It is essent ial to realize that the panel method solut ion is an approximat ionof the real life problem; an idealized representat ion of the flow field. An understanding ofaerodynamics that provides an intuit ive expectat ion of the types of results that may beobtained, and an appreciat ion of how to relate your idealizat ion to the real flow is required toget the most from the methods.

7.5.1 Limitations of Panel Method1. Panel methods are inviscid solut ions. Therefore, it is not possible to capture the viscouseffects except via user “modeling” by changing the geometry.2. Solut ions are invalid as soon as the flow develops local supersonic zones [that is, ].For two-dimensional isentropic flow, the exact value of Cp for crit ical flow is:

7.5.2 Advanced Panel MethodsSo-called “higher-order” panel methods use singularity distribut ions that are not constant onthe panel, and may also use panels which are nonplanar. Higher order methods were found tobe crucial in obtaining accurate solut ions for the Prandt l-Glauert Equat ion at supersonicspeeds. At supersonic speeds, the Prandt l-Glauert equat ion is actually a wave equat ion(hyperbolic), and requires much more accurate numerical solut ion than the subsonic case inorder to avoid pronounced errors in the solut ion (Magnus and Epton). However, subsonichigher order panel methods, although not as important as the supersonic flow case, have been

studied in some detail. In theory, good results can be obtained using far fewer panels withhigher order methods. In pract ice the need to resolve geometric details often leads to the needto use small panels anyway, and all the advantages of higher order panelling are notnecessarily obtained. Nevertheless, since a higher order panel method may also be a newprogram taking advantage of many years of experience, the higher order code may st ill be agood candidate for use.

Example 7.1Discuss the the main differences of panel method compared to thin aerofoil theory and compilethe essence of panel method.

SolutionAlthough thin airfoil theory provides invaluable insights into the generat ion of lift , the Kutta-condit ion, the effect of the camber distribut ion on the coefficients of lift and moment, and thelocat ion of the center of pressure and the aerodynamic center, it has several limitat ions thatprevent its use for pract ical applicat ions. Some of the primary limitat ions are the following:

1. It ignores the effects of the thickness distribut ion on lift (Cl) and mean aerodynamicchord (mac).2. Pressure distribut ions tend to be inaccurate near stagnat ion points.3. Aerofoils with high camber or large thickness violate the assumptions of airfoil theory,and, therefore, the predict ion accuracy degrades in these situat ions even away fromstagnat ion points.

To overcome the limitat ions of thin airfoil theory the following alternat ives many beconsidered:

1. In addit ion to sources and vort ices, we could use higher order solut ions to Laplace'sequat ion that can enhance the accuracy of the approximat ion (doublet , quadrupoles,octupoles, etc.). This approach falls under the denominat ion of mult ipole expansions.2. We can use the same solut ions to Laplace's equat ion (sources/sinks and vort ices) butplace them on the surface of the body of interest , and use the exact flow tangencyboundary condit ions without the approximat ions used in thin airfoil theory.

This lat ter method can be shown to t reat a wide range of problems in applied aerodynamics,including mult i-element aerofoils. It also has the advantage that it can be naturally extended tothree-dimensional flows (unlike stream funct ion or complex variable methods). The distribut ionof the sources/sinks and vort ices on the surface of the body can be either cont inuous ordiscrete.

A cont inuous distribut ion leads to integral equat ions similar to those we saw in thin airfoiltheory which cannot be t reated analytically.

If we discret ize the surface of the body into a series of segments or panels, the integralequat ions are t ransformed into an easily solvable set of simultaneous linear equat ions. Thesemethods are called panel methods.

There are many choices as to how to formulate a panel method (singularity solut ions,variat ion within a panel, singularity strength and distribut ion, etc.). The simplest and first t rulypract ical method was proposed by Hess and Smith, Douglas Aircraft , in 1966. It is based on adistribut ion of sources and vort ices on the surface of the geometry. In their method:

where, ϕ is the total potent ial funct ion and its three components are the potent ialscorresponding to the free stream (ϕ∞), the source distribut ion (ϕs), and the vortex distribut ion (

). These last two distribut ions have potent ially locally varying strengths q(s) and γ(s), wheres is an arc-length coordinate which spans the complete surface of the airfoil in any way we

want.The potent ials created by the distribut ion of sources/sinks and vort ices are given by:

Note that in these expressions, the integrat ion is to be carried out along the completesurface of the airfoil. Using the superposit ion principle, any such distribut ion of sources/sinksand vort ices sat isfies Laplaces equat ion, but we will need to find condit ions for q(s) and γ(s)such that the flow tangency boundary condit ion and the Kutta condit ion are sat isfied. Not icethat we have mult iple opt ions. In theory:

We could use the source strength distribut ion to sat isfy flow tangency and the vortexdistribut ion to sat isfy the Kutta condit ion.Use arbit rary combinat ions of both sources/sinks and vort ices to sat isfy both boundarycondit ions simultaneously.

Hess and Smith made the following valid simplificat ion:“take the vortex strength to be constant over the whole airfoil and use the Kutta condit ion to

fix its value, while allowing the source strength to vary from panel to panel so that, togetherwith the constant vortex distribut ion, the flow tangency boundary condit ion is sat isfiedeverywhere.”

Alternat ives to this choice are possible and result in different types of panel methods.

Example 7.2Calculate the pressure coefficient over an NACA0012 aerofoil with the source panel method,where the freestream attack angle is zero. Show the aerofoil shape, list the code for this, plotthe source strength variat ion, tangent ial flow speed distribut ion over the aerofoil surface andthe pressure coefficient distribut ion over the aerofoil.

SolutionNACA0012 profile used is shown in Figure 7.10.

Figure 7.10 NACA0012 aerofoil.

The FORTRAN program to calculate the pressure distribut ion over an aerofoil is given below:

The source strength over the aerofoil surface is plot ted in Figure 7.11.

Figure 7.11 Source strength over the aerofoil.

The tangent ial speed variat ion is shown in Figure 7.12.

Figure 7.12 Tangent ial speed variat ion over the aerofoil.

The Cp distribut ion over the aerofoil is shown in Figure 7.13.

Figure 7.13 The Cp distribut ion over the aerofoil.

Example 7.3Calculate the pressure coefficient over a cylinder with unit diameter using the source panelmethod. List the program, compute and compare the pressure coefficient variat ion over thecylinder, represent ing it with 8 panels and with 180 panels. Also, show the source strengthvariat ion over the cylinder front to rear end.

SolutionThe program in FORTRAN is given below.

Schematic of the cylinder with 8 panels is shown in Figure 7.14.Nondimensional pressure distribut ion over the cylinder, computed with 8 panels and 180

panels are compared in Figure 7.15. Variat ion of source strength from the front end to the rearend of the cylinder is shown in Figure 7.16.

Figure 7.14 Cylinder with 8 panels.

Figure 7.15 Variat ion of pressure coefficient over the cylinder.

Figure 7.16 Source strength variat ion from the front to rear end of the cylinder.

Example 7.4Write and list the code for solving flow past a NACA0012 aerofoil, using vortex panel method.Compute and plot lift and drag coefficients and the aerodynamic efficiency for angle of at tackrange from −5 to +20 .

SolutionThe FORTRAN program for vortex panel method is listed below:

The lift and drag coefficient variat ion with the angle of at tack for NACA0012 aerofoil,computed by vortex panel method, are shown in Figures 7.17 and 7.18, respect ively. Variat ionof the aerodynamic efficiency of the aerofoil, calculated from the lift and drag computed withthe vortex panel method, is shown in Figure 7.19.

Figure 7.17 Lift coefficient variat ion with angle of at tack.

Figure 7.18 Drag coefficient variat ion with angle of at tack.

Figure 7.19 Aerodynamic efficiency variat ion with angle of at tack.

7.6 SummaryPanel method is a numerical technique to solve flow past bodies by replacing the body withmathematical models; consist ing of source or vortex panels. These are referred to as sourcepanel and vortex panel methods, respect ively.

In general, the source strength λ(s) can change from posit ive (+ve) to negat ive (−ve) alongthe source sheet. That is, the “source” sheet can be a combinat ion of line sources and linesinks.

The velocity potent ial at point P due to all the panels can be obtained by taking thesummation of the above equat ion over all the panels. That is:

where the distance:

where (xj, yj) are the coordinates along the surface of the j th panel.The boundary condit ion is at the control points on the panels and the normal component of

flow velocity is zero.The boundary condit ion states that:

Therefore:

This is the heart of source panel method. The values of the integral in this equat ion dependsimply on the panel geometry, which are not the propert ies of the flow.

Once source strength distribut ions λi are obtained, the velocity tangent ial to the surface ateach control point can be calculated. The pressure coefficient Cp at the i th control point is:

For compressible flows the pressure coefficient becomes:

The vortex panel method is analogous to the source panel method studied earlier. Thesource panel method is useful only for nonlift ing cases since a source has zero circulat ionassociated with it . But vort ices have circulat ion, and hence vortex panels can be used for lift ingcases. It is once again essent ial to note that the vort ices distributed on the panels of thisnumerical method are essent ially free vort ices. Therefore, as in the case of source panelmethod, this method is also based on a fundamental solut ion of the Laplace equat ion. Thusthis method is valid only for potent ial flows which are incompressible.

The mid-point of each panel is a control point at which the boundary condit ion is applied;that is, at each control point , the normal component of flow velocity is zero.

This equat ion is the crux of the vortex panel method. The values of the integrals in thisequat ion depend simply on the panel geometry; they are propert ies of the flow.Pressure distribut ion around a body, given by source panel method is:

This is the general expression for two arbit rarily oriented panels; it is not restricted to circularcylinder only.Exercise Problems

1. Using vortex panel method, compute and plot (a) the pressure coefficient distribut ion overa NACA0012 aerofoil and (b) the variat ion of drag coefficient with lift coefficient , for Cl in therange from −0.15 to 0.55, if the aerofoil is at an angle of at tack of 8 in a uniform freestream.

2. List the procedure steps, along with equat ions, involved in the FORTRAN code vortexpanel method given in Example 7.4.

3. Compute and plot the pressure coefficient variat ion over a NACA0012 aerofoil in a uniformfreestream at angles of at tack 0, 2, 5 and 8 degrees. Also plot the vortex distribut ion over theaerofoil profile for these angles.

Reference1. Magnus, A.E., and Epton, M.A., “PAN AIR -A computer program for predict ing subsonic orsupersonic linear potent ial flows about arbit rary configurat ions using a higher order panelmethod”, Volume I –Theory Document (Version 1.0), NASA CR 3251, April 1980.

8

Finite Aerofoil Theory

8.1 IntroductionThe vortex theory of a lift ing aerofoil proposed by Lancaster and the subsequent developmentby Prandt l made use of for calculat ing the forces and moment about finite aerofoils. The vortexsystem around a finite aerofoil consists of the start ing vortex, the t railing vortex system andthe bound vortex system, as illustrated in Figure 8.1.

Figure 8.1 Vortex system around an aerofoil.

The horseshoe vortex system around an aerofoil, consist ing of the bound and trailingvort ices, can be simplified as shown in Figure 8.2.

Figure 8.2 Simplified horseshoe vortex system around an aerofoil.

8.2 Relationship between Spanwise Loading andTrailing Vorticity

From Helmholtz's second theorem, the strength of the circulat ion around any sect ion of abundle of vortex tubes is the sum of the strength of the vortex filaments cut by the sect ionplane. As per this theorem, the spanwise variat ion of the strength of the combined boundvortex filaments may be shown as illustrated in Figure 8.3.

Figure 8.3 Spanwise distribut ion of bound vortex filaments.

If the circulat ion curve can be described as some funct ion of y, say f(y), then the strength ofthe circulat ion shed by the aerofoil becomes:

that is:

(8.1) Now at a sect ion of the aerofoil the lift per unit span is given by:

where ρ and U are the density and velocity of the freestream. Thus, for a given flight speedand flow density, the circulat ion strength k is proport ional to l. From the above discussion, itcan be inferred that:

The trailing filaments are closer showing that the vort icity strength is larger near the wingt ips than other locat ions. That is, near the wing t ips, the vort icity content of the vort icesshed are very strong.Aerofoils with infinite span (b→ ∞) or two-dimensional aerofoils will have constantspanwise loading.

8.3 DownwashLet us consider the aerofoil with hypothet ical spanwise variat ion of circulat ion due to thecombined bound vortex filaments as shown in Figure 8.4. At some point y along the span, aninduced velocity equal to:

will be felt in the downward direct ion. All elements shed vort icity along the span and add theircontribut ion to the induced velocity at y1 so that the total influence of the t railing system at y1is:

Figure 8.4 Spanwise variat ion of the strength of the combined bound vortex filaments.

that is:

(8.2) The induced velocity at y1, in general, is in the downward direct ion and is called downwash.

The downwash has the following two important consequences which modify the flow aboutthe aerofoil and alter its aerodynamic characterist ics:

The downwash at y1 is felt to a lesser extent ahead of y1 and to a greater extent behind,and has the effect of t ilt ing the resultant wind at the aerofoil through an angle:

(8.3) The downwash around an aerofoil will be as illustrated in Figure 8.5.

The downwash reduces the effect ive incidence so that for the same lift as theequivalent infinite or two-dimensional aerofoil at incidence α, an incidence of α = α∞ + isrequired at that sect ion of the aerofoil. Variat ion of downwash in front of and behind anaerofoil will be as shown in Figure 8.5. As illustrated in Figure 8.5, the downwash willdiminish to zero at locat ions far away from the leading edge and will become almost twiceof its magnitude at the center of pressure, downstream of the t railing edge.

In addit ion to this mot ion of the air stream, a finite aerofoil spins the air flow near the t ipsinto what eventually becomes two trailing vort ices of considerable core size. Thegenerat ion of these vort ices requires a quant ity of kinet ic energy. This constantexpenditure of energy appears to the aerofoil as the trailing vortex drag.

Figure 8.5 Variat ion of downwash caused by the vortex system around an aerofoil.

Figure 8.6 shows the two velocity components of the relat ive wind superimposed on thecirculat ion generated by the aerofoil. In Figure 8.6, L∞ is the two-dimensional lift , VR is theresultant velocity and V is the freestream velocity. Note that the two-dimensional lift is normalt o VR and the actual lift L is normal to V. The two-dimensional lift is resolved into theaerodynamic forces L and , respect ively, normal and against the direct ion of the velocity Vof the aerofoil. Thus, an important consequence of the downwash is the generat ion of drag

. Also, as illustrated in Figure 8.6, the vortex system inducing downwash t ilts theaerofoil in the nose-up direct ion. In Figure 8.6, V is the forward speed of aerofoil, VR is theresultant velocity at the aerofoil, α is the incidence, is the downwash angle, α∞ = (α − ), theequivalent two-dimensional incidence and is the trailing vortex drag. The trailing vortexdrag is also referred to as vortex drag or induced drag.

Figure 8.6 Lift and drag caused by the downwash around an aerofoil.

The forward wind velocity generates lift and the downwash generates the vortex drag :

(8.4) This shows that there is no vortex drag if there is no trailing vort icity.

As a consequence of the t railing vort ices, which are produced by the basic lift ing act ion of a(finite span) wing, the wing characterist ics are considerably modified, almost always adversely,from those of the equivalent two-dimensional wing of the same sect ion. A wing whose flowsystem is closer to the two-dimensional case will have better aerodynamic characterist ics thanthe one where the end effects are conspicuous. That is, large aspect rat io aerofoils are betterthan short span aerofoils.

8.4 Characteristics of a Simple SymmetricalLoading –Elliptic Distribution

The spanwise variat ion in circulat ion is taken to be represented by a semi-ellipse having thespan (2b) as the major axis and the circulat ion at the mid-span (k0) as the semi-minor axis, asshown in Figure 8.4, will have the lift and induced drag act ing on it as shown in Figure 8.7.

Figure 8.7 An aerofoil with ellipt ic distribut ion of circulat ion.

From the general expression for an ellipse, shown in Figure 8.4, we have:

or

(8.5) This is the expression k = f(y) which can now be subst ituted in expression for L, and .

8.4.1 Lift for an Elliptic Distribution

The general expression for the lift of an aerofoil of span 2b is:

Subst itut ing Equat ion (8.5), we have:

(8.6) Therefore:

But the lift is also given by:

where S = ( span × chord) is the projected area of the wing, in the direct ion normal to thefreestream velocity U. Therefore:

(8.6a)

8.4.2 Downwash for an Elliptic DistributionThe circulat ion for ellipt ical distribut ion is:

Different iat ing with y, this gives:

The downwash becomes:

Adding and subtract ing y1 to the numerator, this can be expressed as:

Evaluat ing the first integral which is in a standard form, and writ ing I for the second integral, weget:

Now, as this is a symmetric flight case, the vort icity shed is the same from each side of thewing and the value of the downwash at point y1 is ident ical to that at the corresponding point−y1 on the other wing. Therefore, subst itut ing for ±y1 for y1 in the above equat ion andequat ing them, we get:

This is sat isfied only if I = 0. Therefore:

(8.7) This is an important result , which implies that the downwash is constant along the wing span.

Example 8.1If a wing of span 20 m and chord 2.5 m has ellipt ical load distribut ion. If the downwash is 0.4m/s, find the expression for circulat ion around the wing.

SolutionGiven, 2b = 20 m, c = 2.5 m, m/s.

By Equat ion (8.7):

Therefore:

By Equat ion (8.5), the circulat ion is:

Therefore:

Note that this circulat ion distribut ion is ellipt ical in nature.

8.4.3 Drag due to Downwash for Elliptical DistributionThe drag caused by the downwash is:

On integrat ion this results in:

(8.4a) By Equat ion (8.6a):

Therefore:

The drag can also be writ ten as:

Therefore, the drag coefficient becomes:

(8.8) where is the aspect rat io, defined as the rat io of the span to chord of the wing, given by:

From the above expression for it is seen that, when the lift becomes zero, becomeszero.

8.5 Aerofoil Characteristic with a More GeneralDistribution

A more general distribut ion must sat isfy the end condit ions, namely, at the wing t ips thevort icity should be zero. That is:

It is found that, for plain rectangular or slight ly tapered aerofoils, the spanwise distribut ion doesnot depart drast ically from ellipt ic distribut ion. The modified ellipt ic loading can sat isfy thissituat ion. Let:

The constant a can vary posit ively or negat ively and therefore, can change the shape, but theend condit ions are sat isfied, as illustrated in Figure 8.8. In this figure, the area enclosed by thecurves for a < 0, a = 0 and a > 0 are the same. That is, the total lift for ellipt ic and modifiedellipt ic loading in this figure are the same.

Figure 8.8 Comparison of ellipt ic and modified ellipt ic loading.

The lift for the modified ellipt ic loading becomes:

But the distribut ion is symmetrical, therefore:

Now, let :

Therefore, dy = b cos ϕ dϕ and the limits become 0 and π/2.Subst itut ing these, the lift becomes:

Now, writ ing a = 4λ, we get:

These are standard forms integrable by Walls' rule to give:

or

(8.9) The lift coefficient becomes:

(8.9a) Comparing this with the equat ion for the same lift , Equat ion (8.6), from the equivalentellipt ically loaded aerofoil with mid-span circulat ion, for example, kE, we get:

Therefore:

(8.10) Thus a distribut ion which diminishes fairly rapidly from the mid-span sect ions would have a (orλ) negat ive, while for flat distribut ion a is posit ive.

8.5.1 The Downwash for Modified Elliptic LoadingThe downwash is given by:

Let a = 4λ and y = − b cos θ. Therefore:

The new limits of integrat ion are:

Therefore, becomes:

and

But:

Therefore:

Subst itut ing this we get the downwash as:

Integrals of the type:

can be solved by integrat ing over two ranges, namely, 0 to (θ1 − ) and (θ1 + ) to π and taking

the limits as tends to zero. Thus:

Integrat ing, we get:

for all values of n > 0. When n = 0, G0 = 0, these integrals become the integrals for the case ofellipt ic loading.

Introducing this solut ion into the downwash expression, we get:

We can express as:

Therefore:

For the point y1 = − b cos θ1:

For any point y along the span the downwash is

(8.11) From Equat ion (8.11), we can infer the following:

The downwash in the general case will vary in magnitude along the span, and may evenbecome negat ive (−ve) and give an upwash near the t ips if λ < − 0.1.If the downwash is negat ive (−ve) near the t ips the induced (vortex) drag is negat ive(−ve) and that region of wing gives a thrust . This is, however, compensated by a greaterdrag grading over the central regions of the wing.

8.6 The Vortex Drag for Modified LoadingThe vortex drag, by Equat ion (8.4), is:

But for modified loading, the downwash, by Equat ion (8.11), is:

and the vortex distribut ion is:

Subst itut ing for and k, we get the vortex drag as:

The load is symmetrical about the mid plane, therefore the limits can be writ ten as:

Let y = b sin ϕ, therefore, dy = b cos ϕ dϕ and the limits become 0 and π/2. Hence:

These are standard integrals which result as:

This simplifies to:

(8.12) But drag is also given by:

Therefore, the drag coefficient becomes:

Subst itut ing for , from Equat ion (8.12), we have:

Subst itut ing for , from Equat ion (8.9a), the drag coefficient becomes:

since:

Writ ing:

the drag coefficient can be expressed as:

(8.13) This drag coefficient for the modified loading is more than that for ellipt ical loading by anamount δ, which is always posit ive since it contains λ2 terms only.

8.6.1 Condition for Vortex Drag MinimumFor the vortex drag to be minimum, the δ in Equat ion (8.13) must be zero. That is, λ (or a)must be zero so that the distribut ion for minimum becomes:

This is an ellipse, thus the vortex distribut ion for drag minimum is semi-ellipse. The minimumdrag distribut ion produces a constant downwash along the span while all other distribut ionsproduce a spanwise variat ion in induced velocity, as illustrated in Figure 8.9.

Figure 8.9 (a) Constant downwash due to ellipt ic distribut ion, (b) varying downwash due tonon-ellipt ic distribut ion, (c) equivalent variat ion of downwash.

It is seen that the ellipt ic distribut ion gives constant downwash and minimum drag, non-ellipt ic distribut ion gives varying downwash.

If the lift for the aerofoils with ellipt ical and non-ellipt ical distribut ion is the same under givencondit ions, the rate of change of vert ical momentum in the flow is the same for both. Thus, forellipt ical distribut ion the lift becomes:

For non-ellipt ic distribut ion the lift is:

where is a representat ive mass flow meet ing unit span. But lift L is the same on each wing,therefore:

Now the energy transfer or rate of change of the kinet ic energy of the representat ive massflow is the vortex drag (or induced drag). Thus, for ellipt ical distribut ion the vortex drag is:

For non-ellipt ic distribut ion the vortex drag is:

But:

Therefore:

or

(8.14) and since:

and f1(y) is an explicit funct ion of y:

since is always posit ive whatever be the sign of f1(y). Hence the induced drag Dvb for non-ellipt ic loading is always greater than the induced drag Dva for ellipt ic loading.

8.7 Lancaster –Prandtl Lifting Line TheoryIt is a representat ion to improve on the accuracy of the horseshoe vortex system. In lift ing linetheory, the bound vortex is assumed to lie on a straight line joining the wing t ips (known aslift ing line). Now the vort icity is allowed to vary along the line. The lift ing line is generally takento lie along the line joining the sect ion quarter-chord points of the wing. The results obtainedusing this representat ion is generally good provided that the aspect rat io of the wing ismoderate or large, generally not less than 4.

Consider the lift ing line as shown in Figure 8.10. At any point on the lift ing line, the boundvortex is Γ(y), and there is consequent ly t railing vort icity of strength dΓ/dy per unit length shed.Note that Γ(y) is used to represent vortex distribut ion, instead of k(y). This is because it is acommon pract ice to use both k(y) and Γ(y) to represent the vortex distribut ion.

Figure 8.10 A representat ive lift ing line.

The velocity induced by the elements of t railing vort icity of strength (dΓ/dy) . dy at point P1 isgiven by:

Total downwash at point P1 is:

(8.15) The assumption in this analysis is that the downwash velocity is small compared to thefreestream velocity V, so that is equal to the downwash angle .

Le t αe(y1) be the effect ive angle of incidence of the wing sect ion at point P1. Thegeometrical incidence of the same sect ion be α(y). Let both these angles be measured fromthe local zero-lift angle. Then:

If a∞ is the lift -curve slope of the wing sect ion at point P1, which may also vary across the span,the local lift coefficient CL is given by:

The lift per unit span becomes:

Therefore:

Thus:

where the local chord c may vary across the span.

But , so that:

Equat ion (8.15) may be writ ten as:

or

(8.16) This is an integral equat ion from which the bound vort icity distribut ion may be determined.

Let us introduce θ such that y = − b cos θ, so that θ = 0 at the port wing t ip, θ = π at thestarboard wing t ip, and θ = π/2 at the center line of the wing, that is, in the plane of symmetry.Now, the circulat ion Γ can be expressed as a Fourier series:

(8.17) Note that Γ = 0 at both the t ips. Different iat ing with respect to y, we get:

The left -hand side of Equat ion (8.16) gives:

Using Glauert 's integral formula, this can be expressed as:

(8.18) From Equat ion (8.16), omit t ing the subscript 1, we get:

Writ ing μ = a∞c/8b, we get, mult iplying by μ sin θ:

(8.19) From Equat ion (8.19), the Fourier coefficients An may be determined if α and μ, which aregeneral funct ions of θ, are known, that is, the wing geometry is fully specified.

8.7.1 The LiftThe lift generated by the wing, by Equat ion (8.6), is:

Subst itut ing Equat ion (8.17), we have:

But y = − b cos θ and dy = b sin θ dθ, thus:

At y = b, θ = 0 and at y = − b, θ = π, therefore:

That is:

(8.20) since all other terms vanish.

The wing area is:

since span/chord = 2b/c = .The lift coefficient becomes:

or

(8.20a) Thus, the lift coefficient CL depends on A1, which in turn depends on the values anddistribut ion of α and μ [Equat ion (8.19)].

8.7.2 Induced DragThe induced drag, by Equat ion (8.4), is:

By Equat ion (8.17):

and by Equat ion (8.18):

Also, dy = b sin θ dθ, therefore:

that is:

(8.21) In this equat ion, all the terms involving a product An, . . . . . , Am, where n ≠ m, vanish whenintegrated, and the integral becomes:

This can be demonstrated by mult iplying, say, the first three odd harmonics, thus:

On integrat ion from 0 to π all terms other than the squared terms vanish leaving:

Thus:

or

(8.22) where:

and is usually very small. Also, A1 = CL/π , so that:

That is:

(8.22a) where , is the induced drag factor, and hence Γ depends on the values of the Fouriercoefficients, and hence on the wing geometry, especially on the planform.

Note that:

is always a posit ive quant ity because it consists of squared terms which must be posit ive. Theinduced drag coefficient can be a minimum only when δ = 0. That is, when A3 = A5 = A7 =

= 0 and the only term remaining in the series is A1 sin θ.

8.8 Effect of Downwash on IncidenceFor an aerofoil:

Geometrical incidence is the angle between the chord of the profile and the direct ion ofmot ion of the aerofoil.Absolute incidence is the angle between the axis of zero lift of the profile and thedirect ion of mot ion of the aerofoil.

When the axes of zero lift of all the profiles of the aerofoil are parallel, each profile meets thefreestream wind at the same absolute incidence, the incidence is the same at every point onthe span of the aerofoil, and the aerofoil is said to be aerodynamically untwisted.

An aerofoil is said to have aerodynamic twist when the axes of zero lift of its individualprofiles are not parallel. The incidence is then variable across the span of the aerofoil.

To dist inguish between the lift and induced drag coefficients and the incidences of theaerofoil (that is, wing) as a whole and its individual profiles, let us use CL, , α for the aerofoiland CL ', , α ' for an individual profile. The coefficients CL ', and α ' for a profile arefunct ions of the coordinate y which defines the posit ion of the profile. For a symmetricalaerofoil they are even funct ions of y, that is, to say that they are the same for +y and −y.

The lift and induced drag coefficients of the profile can be expressed as:

(8.22)

(8.23) Therefore the drag and lift rat io becomes:

(8.24) For a properly designed profile, the rat io of induced drag to lift is always small in the working

range of incidence, and therefore ', which is called the angle of downwash, is a small angle. Itfollows that if then V ' = V, neglect ing the second order of small quant it ies.

As shown in Figure 8.11, the resultant aerodynamic force on the strip is perpendicular to thedirect ion of V ' and not to the direct ion of V. Since ' is a small angle, the coefficient of thisforce is CL '. Therefore in respect to lift the strip (profile) behaves like a strip of a two-dimensional aerofoil in a relat ive wind in the direct ion of V ', that is at incidence α0 ', where:

(8.25) The angle α0 ' is called the effective incidence. Thus the effect ive downwash is the downwashvelocity that combines with the actual relat ive wind of speed V to produce an effect ive relat ivewind in the direct ion of V '.

Figure 8.11 A profile of an aerofoil moving horizontally.

The lift coefficient CL ' variat ion with incidence will be as shown by curve a in Figure 8.12.This curve is the graph proper to the profile operat ing as a two-dimensional aerofoil and the liftcurve slope is given by:

But when the profile is operat ing as a part of the actual wing (that is, a three-dimensionalaerofoil), the variat ion of CL ' with incidence will be as shown by curve b in Figure 8.12. The liftcurve slope is given by:

Figure 8.12 Profile lift coefficient variat ion with incidence (a) for a two-dimensional aerofoil, (b)for a three-dimensional aerofoil.

It is seen that the graphs of CL ' against incidence are straight lines. In Figure 8.12, thegraphs a and b are drawn with the assumption that the angle of downwash vanishes when the

wind is along the axis of zero lift , that is, the axis of zero lift is assumed to be the same in thetwo-dimensional and three-dimensional aerofoils. With this assumption we have:

(8.26) We know that for an actual aerofoil in a subsonic flow the main components of the drag are

the profile drag and the skin friction drag. The induced drag caused by the downwash is anaddit ional component of drag. Therefore, the total drag coefficient of the strip (profile), usingEquat ion (8.24), is:

(8.27) where is the coefficient of profile drag for the profile.

It may be noted that the profile drag is largely independent of incidence in the working range.Profile drag is the sum of the skin frict ion due to viscosity, and form drag due to the shape. Thepart due to skin frict ion is due to the no-slip caused by the viscosity of the air in the boundarylayer at the surface of the body. This viscous effect is always present, but can be reduced bysmoothening the surface and reducing the surface area.

The form drag due to the shape is owing to the high pressure at the leading edge and lowpressure at the t railing edge (that is the low pressure in the wake). By shaping the body toreduce the wake to be of negligible thickness, that is by streamlining, the form drag can bealmost eliminated.

8.9 The Integral Equation for the CirculationFor a profile of chord c ' at distance y from the plane of symmetry, the lift coefficient of theprofile is given by:

Therefore:

since CL ' = a0 ' α0 ', by Equat ion (8.26). But by Equat ion (8.25):

Therefore:

(8.28) or

But by Equat ion (8.15):

Therefore:

(8.29) This is the integral equation from which circulat ion k(y) is to be determined. Using this k(y), thelift , drag, and downwash can be determined.

Note that in general α ', a0 ', c ' are funct ions of y, since incidence, chord and profile may varyfrom sect ion to sect ion. If the profiles are similar curves, α0 ' is the same at every sect ion, but α 'is not the same unless the sect ions are also similarly situated (untwisted aerofoil).

For a given wing α0 ' a0 ', c ' are known funct ions of y, and in part icular for thin wings we mayassume a0 ' = 2π.

The following are the two problems associated with aerofoils:For a given circulat ion k(y), the form of the aerofoil and the induced drag are to bedetermined.

For a given form of aerofoil, the distribut ion of circulat ion and the induced drag are to bedetermined.

8.10 Elliptic LoadingFor an ellipt ical load, as shown in Figure 8.4, the (k(y), y) curve is an ellipse. If P is a point on thespan whose eccentric angle is θ, we have y = − b cos θ and therefore:

(8.30) where k0 is the value of k(y) at y = 0. It is easily seen that the eliminat ion of θ gives:

which is an ellipse.Subst itut ing y = − b cos ϕ in Equat ion (8.15), the downwash velocity at the t railing edge is:

(8.31) Thus for ellipt ic loading, the downwash velocity is the same at every point on the trailing edge.

Now, by subst itut ing Equat ion (8.28) into Equat ion (8.29), we get:

(8.32) where a0 ' and α ' refer to the sect ion at distance y from the plane of symmetry.

The chord c ' and incidence α ' depend, in general, on the part icular profile sect ionconsidered, that is on θ. Also k0/V depends on the incidence of the aerofoil. If we increase theincidence of the aerofoil by β, the incidence of each profile sect ion will also increase by β. Thus:

(8.33) where denotes the new value of .

From Equat ions (8.32) and (8.33), we get:

The only term containing θ is , therefore, for the loading to remain elliptic at all incidence,we shall have:

(8.34) where c0 is the chord of the middle sect ion of the aerofoil (that is, at y = 0), and a0 ' will besame at every sect ion and Equat ion (8.34) becomes:

(8.35) This implies that the plot of chord c ' against y is also an ellipse. This situat ion can be realizedby an aerofoil so constructed that its planform is bounded by two half ellipses whose majoraxis is equal to the span of the aerofoil. This can be proved by using the ellipses:

It follows that:

and if c ' = x1 ± x2, c0 = a1 ± a2, Equat ion (8.35) is sat isfied.Finally, it is evident from Equat ions (8.32) and (8.34) that for ellipt ic loading, which remains

ellipt ic for all incidences, the incidence is the same at every profile sect ion, and k0 isproport ional to the incidence, and therefore, from Equat ion (8.31), the downwash isproport ional to the incidence.

Another case arises for an aerofoil of rectangular planform. Here the chord c ' may be takenas constant and equal to c0. Retaining the hypothesis that a0 ' = a0, which will be t rue if thesect ions are similar, or if they are thin, Equat ion (8.32) becomes:

(8.36) This shows that the incidence at each sect ion is different, so that the aerofoil is twisted. Theincidence at the middle sect ion will be α, got by assigning θ = π/2 in Equat ion (8.36), andtherefore:

(8.37) Equat ion (8.37) shows that if the loading is ellipt ic at the incidence α, it ceases to be ellipt ic ata different incidence.

8.10.1 Lift and Drag for Elliptical LoadingThe lift coefficient for an aerofoil in terms of the circulat ion k0 around it , by Equat ion (8.6a), is:

The aspect rat io of the aerofoil is:

Therefore:

By Equat ion (8.7):

Thus, the lift coefficient in terms of constant downwash velocity, at the t railing edge, is:

By Equat ion (8.3), and by Equat ion (8.25):

Therefore:

(8.38) By Equat ion (8.8), the induced drag coefficient is:

The variat ion of CL with is called the polar curve of the aerofoil.Equat ion (8.8) shows that the polar curve of an ellipt ically loaded aerofoil is a parabola,

provided the only source of the drag is the induced velocity.The polar curve is as shown in Figure 8.13. The polar curve can be graduated in incidence as

indicated in Figure 8.13. Since α is proport ional to the lift coefficient CL, equal increments ofincidence gradients of the polar correspond to equal increment of CL.

Figure 8.13 Variat ion of lift coefficient with induced drag coefficient for ellipt ical loading.

In pract ice, in addit ion to induced drag there is profile drag due to skin frict ion and wake. Thecoefficient of profile drag is indicated by . Thus the complete drag coefficient is:

Variat ion of drag coefficient CD is shown in dotted line in Figure 8.13.

8.10.2 Lift Curve Slope for Elliptical LoadingIf a0 is the lift curve slope in two-dimensional mot ion and a is the lift curve slope for an aerofoil

of finite aspect rat io with ellipt ical loading, by Equat ion (8.38), we have:

Different iat ing with respect to CL, we get:

where a and a0 are the lift curve slope. If we take the theoret ical value of a0 as 2π, then:

Thus the lift curve slope becomes:

(8.39) For incidence below the stall, the CL verses α curves are straight lines whose slopes

increase as the aspect rat io increases, as shown in Figure 8.14.

Figure 8.14 Variat ion of lift coefficient with incidence.

8.10.3 Change of Aspect Ratio with IncidenceBy Equat ion (8.38), we have:

Also the induced drag coefficient , by Equat ion (8.8), is:

Hence, if the aspect rat io is reduced to ' and if the ‘primes' refer to the new aerofoil withthe same incidence, we have:

(8.40)

(8.41) Thus for a given lift curve, decrease of aspect rat io increases both the geometrical incidenceand the induced drag coefficient .

8.10.4 Problem IIIn problem I for a given circulat ion k(y), the form of the aerofoil has been found. Problem II is aninverse problem in which the form of the aerofoil is known and the circulat ion has to bedetermined. To do this we must solve the integral equat ion [Equat ion (8.29)], not ing that thesymmetry with respect to the median plane of the aerofoil demands that k(y) = − k(y). In termsof the eccentric angle θ we can therefore write the Fourier sine series, since k(y) vanishes atthe t ips of the aerofoil, that is at θ = 0 and θ = π.

(8.42) Note that n must be an odd integer to ensure the equality of sin nθ and sin n(π − θ). Thus:

(8.43) This formula for k(y) is unchanged when (π − θ) is writ ten for θ, and the value at the centergiven by θ = π/2 is:

(8.44) Subst itut ion of Equat ion (8.43) into (8.29) gives:

(8.45) At this stage it will be useful to understand the integral on the right-hand side of this

equat ion. The integral on the right-hand side of Equat ion (8.45) is of the type:

where n is an integer.In terms of its principal value, let us define In as:

This is physically equivalent to omit t ing the vort icity between (θ − ) and (θ + ) and thentaking the limit → 0 so that θ remains the center of the omit ted port ion.

If n = 0 we have, by different iat ion:

Hence:

It follows that we may write:

For n = 1, we have I1 = π. Now:

and

Therefore, we have:

To solve this let In = xn, which gives:

so that:

Thus:

Since I0 = 0, we have B = 0, and since I1 = π we have A = π cosec θ and therefore:

This is valid for all integer values of n including zero.Now, let us write αθ for α ' to convey that α ' is a funct ion of θ and put:

(8.46) Then from Equat ion (8.45) we get:

(8.47) To find the coefficients of A2n+1 in Equat ion (8.47), we need to expand each side, and eachterm on the left hand side in a Fourier series, thus leading to infinite number of equat ions andinfinite number of unknowns. To overcome this difficulty and solve Equat ion (8.47), we shouldresort to a pract ical method of solut ion, due to Glauert .

Let us replace the infinite series of Equat ion (8.43) by a finite series of, say, (m + 1) terms,where m is a given integer, thus giving:

(8.48) This equat ion cannot be sat isfied ident ically. However, if we take a part icular value of θ we geta linear equat ion in the coefficients A1, A3, . . . A2m+1. If (m + 1) part icular values are assigned toθ we get (m + 1) linear equat ions from which the coefficients A2n+1 can be calculated, and thevalues so obtained will sat isfy Equat ion (8.48), not ident ically, but only at the selected points.The solut ion will be sat isfactory if the coefficients so determined sat isfy Equat ion (8.48) atother points within the standard of accuracy required for any part icular case.

Since Equat ion (8.48) is sat isfied in any case when θ = 0 or π, we have (m + 1) points other

than these points. The chosen points are usually taken as equally spaced in θ over the half-span. Thus if m = 3 we should take:

and with these values we could determine four coefficients:

(8.49) A rough approximat ion is obtained by taking m = 1, and θ = π/2, π/4. This will determine twocoefficients A1 ' , A3 ' but it must not be inferred that, comparing with Equat ion (8.49), A1 ' = A1,A3 ' = A3.

If the incidence αθ has the same value α at each point on the span, Equat ion (8.47) showst hat A2n+1 is proport ional to α, and if we write A2n+1 = α B2n+1, the coefficients B2n+1 areindependent of incidence and may therefore be determined once for all.

8.10.5 The Lift for Elliptic LoadingFrom Equat ion (8.43), we have:

(8.50) since y = − b cos θ; dy = b sin θ. Also:

For n = 0:

The lift is given by:

With Equat ion (8.50), the lift becomes:

For n = 0:

Thus the lift coefficient for the whole aerofoil is:

That is:

(8.51) Thus the coefficient A1 = CL/(π ), and this gives a check on the theoret ical value of A1, withwhich CL can be determined by wind tunnel measurements.

For ellipt ical loading all the A2n+1 are zero except A1, therefore, Equat ion (8.45) gives thecirculat ion as:

(8.52) From Equat ions (8.51) and (8.52), we get the lift coefficient as:

If the incidence α is the same at every point of the wing span, by Equat ion (8.46), A1 isproport ional to α.

From Equat ion (8.51), we have:

Dividing and mult iplying by α:

Therefore, the lift curve slope a becomes:

If a0 and α0 are the corresponding slope and incidence in two-dimensional mot ion, we have:

Therefore:

Subst itut ing for a, we get:

(8.53) For a wing with aerodynamic twist , the incidence α becomes a variable across the span. We

can express the incidence as:

where αm is the incidence at the middle sect ion. For this case A2n+1 in Equat ion (8.47) can beexpressed as:

rendering Equat ion (8.47) as equivalent to the two equat ions:

and all the numbers so determined are independent of incidence. Thus:

where a (= π A1 ') and b (= π B1) are constants for the aerofoil.

8.10.6 The Downwash Velocity for Elliptic LoadingAt the point of the t railing edge of an aerofoil, whose eccentric angle is θ, by Equat ion (8.2):

Using Equat ion (8.43), we can express this as:

Using the values of integrals given in Subsect ion (8.5.1), we have:

(8.54) For ellipt ic loading this becomes:

By Equat ion (8.7), we have:

Therefore:

which is constant across the wing span.

8.10.7 The Induced Drag for Elliptic LoadingThe induced drag due to circulat ion is:

where is the downwash.From Equat ions (8.50) and (8.54) we have:

But

where r is any integer other than n, but if r = n the value of the integral is π/2, so that:

Therefore:

This can be expressed as:

(8.55) where:

(8.56) Note that δ is never negat ive, and is zero only in the case of ellipt ic loading.

The total drag coefficient is:

(8.57) where is the profile drag coefficient , caused by the skin frict ion and the wake. This can beexpressed, using Equat ion (8.55), as:

But, π A1 = CL, therefore:

(8.58) Equat ion (8.58) is the drag polar for the aerofoil. A typical drag polar is shown in Figure 8.15.

Figure 8.15 Drag polar of a wing with ellipt ic loading.

For wings with loading other than ellipt ic, the drag polar becomes:

(8.58a) where e is known as the Oswald wing efficiency and for ellipt ic loading e = 1.

For ellipt ic loading, δ = 0 and e = 1, therefore, the drag polar becomes:

(8.58b) where . When the lift becomes zero, that is, for CL = 0, Equat ion (8.58b), reduces to .Thus is referred to as zero-lift drag coefficient.

If the profile drag coefficient for each profile sect ion is a funct ion of the posit ion of thesect ion, the profile drag coefficient of the aerofoil is:

where the superscript ' refers to the sect ion at a distance y from the plane of symmetry.

8.10.8 Induced Drag MinimumFor a wing of span 2b, we have the lift and induced drag as:

For a given lift L, the induced drag will be minimum when δ = 0. By Equat ion (8.56):

Thus for minimum the condit ion is:

This implies that A3 = A5 = A7 = = 0. Therefore the loading is ellipt ic. Thus, of all the wingsof given span and lift , the ellipt ically loaded wing gives the least induced drag.

Example 8.2

For a wing of ellipt ic loading, in a straight level flight , find the condit ion for drag minimum.

SolutionThe Drag D can be expressed as:

For level flight , L = W, therefore:

Thus for a wing of given weight, the minimum drag occurs when (D/L) is a minimum or (L/D) is amaximum.

Now

For ellipt ic loading, by Equat ion (8.58b), the drag polar is:

Therefore:

For drag minimum, the condit ion is:

That is:

That is:

This simplifies to:

That is:

Thus for minimum drag, the zero-lift drag coefficient is equal to the lift dependent dragcoefficient , and the actual drag forces associated with these coefficients are equal.

8.10.9 Lift and Drag Calculation by Impulse MethodLet us consider an aerofoil, regarded as a lifting line AB, started from rest and which moves in astraight line. Let the velocity be V at t ime t. Let at t ime t the start ing vortex be assumed to beA0B0, as shown in Figure 8.16.

Figure 8.16 A lift ing line moving straight.

Let us assume that the wake ABA0B0 remains as a rectangular sheet, as shown in Figure8.16. If P is the point (0, y, 0) and Q is the point (− l, y, z). The flow from point P reaching the lineA0B0 experiences a circulat ion of −dk(y) [Equat ion (8.1)]. Therefore the whole wake may beregarded as result ing from the superposit ion of vortex rings, a typical one being PP ' QQ ' ofcirculat ion −dk(y).

Let PQ = h. The area of the ring is 2yh. If n is the unit normal to a small area dS separat ingpoints P and Q, the impulse on this area due to impulsive pressure is:

where:

where c is any circuit joining points P and Q and not intersect ing the plane containing P and Q.Thus the resultant linear impulse on the system is the vector:

Thus the impulse for the area (2yh) of the vortex ring, with n as the unit normal to the planerectangle, is:

since −hn = kl + iz, by geometry. The impulse I of the whole wake is:

Thus:

The t ime rate of change of the impulse gives the force. Therefore the aerodynamic force isdI/dt and dl/dt = V, while , the normal velocity at Q, which by symmetry of the ring is equal tothe downwash velocity at P. Thus the aerodynamic force is:

which consists of the same lift and induced drag as were calculated by Prandt l's hypotheses.The method of the impulse is applicable whatever the form of the wake.

8.10.10 The Rectangular AerofoilThis is an aerofoil whose planform is a rectangle. An aerofoil whose shape is that of a cylindererected on an aerofoil profile sat isfies this requirement.

8.10.11 Cylindrical Rectangular AerofoilThis is the simplest type, of span 2b and chord c, which is constant at all sect ions. All thesect ions are similar and similarly situated.

8.11 Aerodynamic Characteristics of AsymmetricLoading

The vortex distribut ion k(y) for symmetrical loading falls symmetrically about the mid-spansect ion of an aerofoil, involving only the odd terms, and producing vortex drag and downwashvariat ion which are also symmetrical about the centerline. In the general case, where theloading or lift distribut ion is not symmetrical about mid-span sect ion, even terms appear in thedistribut ion, and as a consequence of the asymmetry other characterist ics of aerofoil appear.

When the lift distribut ion is not symmetrical about the centerline, one wing will have higherlift than the other. This will cause a rolling moment about the longitudinal axis passing throughthe mid-span of the wing.

Further, as the lift is not symmetric nor is the spanwise distribut ion of circulat ion, thedownwash will vary across the span without being symmetrical about the centerline and so willbe the vortex drag grading. Hence, more drag will be experienced on one wing (the one withmore lift ) than on the other and a net yawing moment will result about the vert ical (normal) axisthrough the mid-span sect ion. In addit ion to these there will be the overall lift and vortex dragforce normal and parallel to the plane of the aerofoil in the plane of symmetry.

8.11.1 Lift on the AerofoilFollowing a similar procedure we used for determining the lift and vortex drag associated withsymmetrical loading, we can show that:

giving the lift coefficient as:

the same as Equat ion (8.20a).

8.11.2 DownwashAs given by Equat ion (8.18), the downwash for asymmetrical loading also becomes:

But this will no longer be symmetrical as it contains even harmonics.

8.11.3 Vortex DragAs in the case of symmetrical loading, integrat ing from 1 to ∞, the drag becomes:

Thus the drag coefficient becomes:

By Equat ion (8.20a), A1 = CL/(π ), thus:

where:

and δ > 0.

Example 8.3A monoplane weighing 73575 N has ellipt ic wing of span 15 m. When it flies at 300 km/h at sealevel, determine the circulat ion around sect ions half-way along the wings.

SolutionGiven, W = 73575 N, 2b = 15 m, V = 300/3.6 = 83.33 m/s.The air density at sea level is 1.225 kg/m3 and in level flight , L = W.The lift for ellipt ical loading, by Equat ion (8.6), is:

Therefore, the circulat ion at the mid-span becomes:

The circulat ion for ellipt ical distribut ion, by Equat ion (8.5), is:

Therefore, the circulat ion around sect ions half-way along the wings, that is, at b/4, becomes:

8.11.4 Rolling MomentLet us consider a rectangular wing with an asymmetrical lift grading and the correspondingdrag grading, as shown in Figure 8.17.

Figure 8.17 Schematic of asymmetrical lift and drag grading on a rectangular wing withasymmetrical vortex loading.

The lift act ing on any sect ion of spanwise length δy at a distance y from the centerline (ox-axis) will produce a negat ive increment of rolling moment equal to:

(8.59) where l is the lift grading given by l = ρVk.

The total moment becomes:

(8.60) Subst itut ing k = 4bV ∑ An sin nθ and expressing y = b cos θ, we get:

or:

(4.60a) The rolling moment is also given by:

where is the rolling moment coefficient . Therefore:

But , therefore:

(8.61)

8.11.5 Yawing MomentThe asymmetrical drag grading across the span, shown in Figure 8.17, gives rise to yawingmoment N. The contribut ion of the vortex drag of an element of span dy, at a distance y fromthe oz-axis is:

(8.62) where is the vortex drag per unit span and . Integrat ion over the whole span gives theyawing moment as:

Insert ing the series expressions for the circulat ion k and downwash , and changing thelimits and variables from Cartesian to polar, we get:

The yawing moment can also be expressed as:

(8.63) where CN is the yawing moment coefficient . Thus:

Mult iplying these series for a few terms, we can express the general solut ion as:

(8.63a) since all terms other than those with coefficients which are products of A1A2, A2A3, A3A4, etc.vanish on integrat ion.

Example 8.4A symmetrical profile of aspect rat io 7 and chord 1.5 m flying at 200 km/h at sea level issuddenly subjected to a downwash of 2.4 m/s. If the horizontal tail of span 3 m and chord 0.6 mis 5 m aft of the aerodynamic center of the profile, determine the tail deflect ion required to

counter the pitching moment caused by the sudden downwash.

SolutionGiven, = 7, c = 1.5 m, V = 200/3.6 = 55.56 m/s, m/s, lt = 5 m.The planform areas of the profile and horizontal tail are:

For the symmetrical profile, α0 = 0, therefore the lift coefficient , by Equat ion (8.38), is:

The angle of at tack, by Equat ion (8.3), is:

Thus:

The lift generated by the profile is:

For symmetrical profile, the aerodynamic center is at c/4 from the leading edge, therefore thepitching moment about the leading edge becomes:

This is a nose-down moment, therefore, the horizontal tail should generate a moment of−10608.8 N-m to counter it . Thus:

The tail lift coefficient becomes:

Therefore:

Note that the pitching moment caused by the downwash is nose-down (negat ive). Therefore,the horizontal tail has to generate a downward lift , result ing in a pitching moment which isposit ive (to counter the nose-down moment). This calls for an upward deflect ion of thehorizontal tail by 14 . 28 .

8.12 Lifting Surface TheoryLift ing surface theory is a method which treats the aerofoil as a vortex sheet over whichvort icity is spread at a given rate. In other words, the aerofoil is regarded as a surfacecomposed of lift ing elements. This is different from the lift ing line theory, discussed in Sect ion8.7. The essent ial difference between the lift ing surface theory and lift ing line theory is that inthe former the aerofoil is t reated as a vortex sheet, whereas in the lat ter, the aerofoil isrepresented by a straight line joining the wing t ips, over which the vort icity is distributed.

8.12.1 Velocity Induced by a Lifting Line ElementLet us consider a horseshoe vortex of infinitesimal span ds and circulat ion Γ, as shown in Figure8.18. The origin is at the mid-point of the span and the x-axis parallel, but opposed in sense, tothe arms I, I ' of the horseshoe. Let us calculate the induced velocity at the point A (x, y, z). Forthis let us first consider a single semi-infinite vortex OK of circulat ion Γ, in the same sense asthe circulat ion about arm I. Let the velocity induced by vortex OK at A be . If this vortex OKwere shifted to coincide with I the induced velocity would be, by Taylor's theorem:

because the effect on the velocity is the same as if no shift were made, and the y-coordinateof A were increased by .

Figure 8.18 A horseshoe vortex of infinitesimal span.

The vort icity direct ion on arm I ' is opposite to that on arm I, therefore the velocity inducedby arm I ' at A would be:

Therefore the total velocity induced at A by the pair I, I ' is:

Project ing A on the plane x = 0, the plane y = 0 and the z-axis we get the points B, N, M shownin Figure 8.18. Let OB = n, OA = r.

The vortex OK induces a velocity q1 perpendicular to plane OAB. Thus:

where by Equat ion (5.50):

Hence:

Thus the induced velocity at A due to I and I ' has components:

In addit ion the velocity induced at A by the vortex ds has to be taken into account. Thisvelocity (du2, dv2, dw2) is of magnitude:

where α = ∠ OAN, and is perpendicular to OAN and therefore parallel to the plane OMN. Thus:

where γ = ∠ MON. Now:

Simplifying the above relat ions, the components of velocity induced at A, can be expressedas:

(8.64a)

(8.64b)

(8.64c)

8.12.2 Munk's Theorem of StaggerMunk's theorem of stagger states that “the total drag of a mult iplane system does not changewhen the elements are t ranslated parallel to the direct ion of the wind, provided that thecirculat ions are left unchanged.” Thus the total induced drag depends only on the frontalaspect. To illustrate this, let us consider a lift ing element of length ds1 placed at O, andanother lift ing element of length ds2 placed at A in a plane parallel to x = 0, as shown in Figure8.19.

Figure 8.19 Two parallel lift ing elements on the lift ing surface.

Let the circulat ions of the elements ds1 and ds2 are Γ1 and Γ2, respect ively. The normal AH

drawn in the plane parallel to x = 0 makes an angle ϕ2 with OB and:

If dwn is the induced velocity at A along AH, using Equat ion (8.64), we can express dwn as:

If there is a flow of velocity V along xO, the drag induced by ds1 on ds2 is:

To get the drag induced on ds1 by ds2 let us replace θ with −θ, the angle of stagger. Then:

Thus the total drag mutually induced on the pair of lift ing elements becomes:

which is independent of the angle of stagger. This yields Munk's theorem of stagger, that is:

“the total drag of a multiplane system does not change when the elements are translatedparallel to the direction of the wind, provided that the circulations are left unchanged.”When the system is unstaggered (that is, when θ = 0):

and thus if the lift ing systems are in the same plane normal to the wind, the drag induced inthe first by the second is equal to the drag induced in the second by the first . This resultconst itutes Munk's reciprocal theorem.

The total mutual induced drag is:

where ϕ1 is the angle between the plane containing the normals to the element ds1 and theproject ion of the line joining the elements on the plane normal to the wind and ϕ2 is the anglebetween the plane containing the normals to the element ds2 and the project ion of the linejoining the elements on the plane normal to the wind.

8.12.3 The Induced LiftLet us consider two vortex elements with circulat ions Γ1 and Γ2, as shown in Figure 8.20.

Figure 8.20 Two vortex elements.

The velocity induced at ds2 by ds1 along Ox is:

This induced velocity is against the wind. This induces a lift in the element ds2, given by:

The velocity induced at ds1 by ds2 is with the wind, and the induced lift is:

Resolving along n, the project ion of the line joining the elements on a plane normal to the wind,we get:

Resolving perpendicular to n, we get:

This vanishes when ϕ1 = ϕ2 and is small in general.

8.12.4 Blenk's MethodThis method is meant for wings of finite aspect rat io and is based on the lift ing line theory ofPrandt l, discussed in Sect ion 8.7, hence limited to aerofoils moving in the plane of symmetryand with a t railing edge which could be regarded as approximately straight. This methodconsiders the wing as a lift ing surface, that is to say the wing is replaced by a system of boundvort ices distributed over its surface rather than along a straight line coinciding with the span.However, this method has the limitat ion that the wing is assumed to be thin and pract icallyplane. The shapes considered are shown in Figure 8.21

Figure 8.21 Some wing shapes meant for Blenk's method; (a) rectangular wing moving in theplane of symmetry, (b) a skew wing in the shape of a parallelogram moving parallel to itsshorter sides, (c) a symmetrical arrow-shaped wing (sweep-back), (d) rectangular wing side-slipping.

In all cases the arrow indicates the direct ion of mot ion. The angle β, which is considered tobe small, is the angle between a leading edge and the normal, in the plane of the wing, to thedirect ion of mot ion.

In all the cases in Figure 8.21 it is assumed that the bound vort ices are parallel to the leadingedge, so that in part icular for wing shape (c) the bound vortex lines are also arrow-shaped.

The following are the two main approaches employed in Blenk's method:1. Given the load distribut ion and the plan, find the profiles of the sect ions.2. Given the plan and the profiles, find the load distribut ion (that is the vort icitydistribut ion).

8.12.5 Rectangular AerofoilLet us assume that the load distribut ion is given for the rectangular planform of Figure 8.21(a).Now the task is to find the profiles at different sect ions.

Let us assume the profile to be thin so that the whole aerofoil may be considered to lie in thexy plane, as shown in Figure 8.22.

Figure 8.22 A rectangular wing moving in the plane of symmetry.

Let γ1(x, y) be the circulat ion per unit length of chord at the point (x, y, 0) so that thecirculat ion round the profile at distance y from the plane of symmetry is:

(8.65) Introducing the dimensionless coordinates:

[where c is the chord and b is the span (note that here b is taken as the span, instead of 2b, forconvenience)] we get the circulat ion as:

(8.66) For γ(ξ, η), let us choose the following ellipt ic distribut ion over the span:

and for γ0(ξ), let us consider the following three different funct ions:

(8.67a)

(8.67b)

(8.67c)

where a0, b0, c0 are arbit rary constants. Note that Equat ion (8.67a) is the distribut ion for a thinflat aerofoil in two-dimensional mot ion. The most general distribut ion considered here will thenbe of the form:

(8.68)

8.12.6 Calculation of the Downwash VelocityConsider first the velocity induced at P(x, y, 0) by a vortex MM ', shown in Figure 8.22, parallel tothe span (lift ing line) and the trailing vort ices which spring from it . Let Q(x ' , y ' , 0) be a point onthe vortex MM '. The circulat ion at Q is then γ1(x ' , y ') dx ' and from Q there t rails a vortex ofcirculat ion:

The downwash velocity induced by the trailing vortex caused by MM ' (see Chapter 5) atpoint P is:

(8.69) The downwash velocity induced at P by the whole aerofoil is:

(8.70) In terms of ξ and η, this becomes:

(8.71) Thus from Equat ion (8.69) we get:

(8.72) where, being the aspect rat io:

(8.73) Note that, if ξ ' = ξ Equat ion (8.72) reduces to its first term and if we put , the ellipt ic

distribut ion across the span, we get:

(8.74) Subst itut ing Equat ion (8.74) into Equat ion (8.71) we get the downwash velocity.

The induced drag is given by:

which includes the suct ion force at the leading edge and hence the leading edge should not berounded.

The integrals in Equat ion (8.74) are ellipt ic type and cannot be evaluated in terms ofelementary funct ions. Blenk therefore adopted an ingenious method of approximat ion, eventhough it is lengthy. However, this approximat ion is valid only to the middle part of the wing sothat the end effects are uncertain. The approximat ion is better suited for larger aspect rat ios.The method leads to replacing Equat ion (8.74) by:

(8.75) where the coefficients A1, B1, C1, D1 and E1 are funct ions of η which depend on the part icularcase among the four wing shapes considered. For the rectangular wing moving in the plane ofsymmetry:

The downwash may be calculated from Equat ion (8.71) with the aid of Equat ion (8.75)To determine the profile of the sect ion at distance y from the center, let us assume the

relat ive wind to blow along the x-axis. Since we consider the perturbat ions in the freestream tobe small and the air must flow along the profile, we have:

Therefore:

Comparison of this result with the theory of the lift ing line gives the following mean addit ions tothe incidence and curvature for the rectangular wing:

In the case of sweep-back wing, the mean increase of incidence, according to Blenk, should be percent of the absolute incidence without sweep-back.

8.13 Aerofoils of Small Aspect RatioFor aerofoils with aspect rat io less than about unity, the agreement between theoret ical andexperimental lift distribut ion breaks down. The reason for this breakdown is found to be theconsequence of Prandt l's hypothesis that the free vortex lines leave the trailing edge in thesame line as the main stream. This assumption leads to a linear integral equat ion for thecirculat ion. Let us consider a port ion of a flat rectangular aerofoil whose chord c is largecompared to the span 2b, as shown in Figure 8.23. Let us take the chord axes with the origin atthe center of the rectangle.

Figure 8.23 A port ion of a flat rectangular aerofoil of small aspect rat io.

The bound vort icity γ(x) is assumed to be independent of y, that is to say is constant acrossa span such as PQ but is variable along the chord. The downwash is also assumed to beindependent of y and may therefore be calculated at the center of each span. The main pointof the theory developed for aerofoils of small aspect rat io is that the t railing vort ices, whichleave the t ips of each span such as PQ, make an angle θ with the chord which is different fromα, the incidence, since the trailing vort ices follow the fluid part icles which leave the edges ofthe aerofoil at angle θ will presumably be a funct ion of x. To a first approximat ion it is assumedto be constant.

8.13.1 The Integral EquationLet us begin with the calculat ion of the velocity induced at the center C(x, 0, 0) of the span PQ,shown in Figure 8.24.

Figure 8.24 A representat ive wing surface.

Consider then the span RS, center D(ξ, 0, 0). The bound vortex associated with RS is ofcirculat ion γ(ξ) dξ and induces at C a velocity, in the z-direct ion:

This gives the downwash due to the whole set of bound vort ices as:

(8.76) This is the velocity induced in the z-direct ion. If u1 is the velocity induced in the x-direct ion, itcan be shown that:

(8.77) To find the velocity induced at C by the vort ices t railing from R and S, let T, M, U be the

project ions of P, C, Q on the plane of these vort ices. Then the vortex t railing from R induces atC a velocity of magnitude dq perpendicular to the plane RCT, and the vortex t railing from Sinduces at C a velocity of the same magnitude perpendicular to the plane SUC. Let dqn be theresultant induced velocity and its direct ion is along CM. Then, if the angle ∠TCM = δ, we have:

Therefore, for all the t railing vort ices, the resultant induced velocity becomes:

(8.78) If u2, are the components of qn in the x-and z-direct ions:

(8.79) The boundary condit ion is that there shall be no flow through the aerofoil, this implies that thenormal induced velocity just cancels the normal velocity due to the stream. Therefore:

(8.80) The required integral equat ion can be obtained by subst itut ing the values from Equat ion (8.76)to Equat ion (8.79). At this stage it will be useful to employ “dimensionless” coordinates. For thisproblem, the dimensionless coordinates are 2x/c and 2ξ/c. Also, b/c = . In terms of thedimensionless coordinates we can cast Equat ion (8.78) as:

(8.81) This is a nonlinear equat ion since θ itself is a funct ion of γ(ξ).

The method proposed for the tentat ive solut ion of Equat ion (8.81) is to put:

(8.82) which is valid for large aspect rat ios and then to use Equat ion (8.80) to determine γ0 in termso f θ and then approximate to a suitable mean value for θ. This is a laborious exercise, sowithout venturing into this let us examine the variat ion of the normal force coefficient on anaerofoil CN, defined as:

where N is the normal force. Variat ion of CN for = 1/30 with incidence angle α is shown inFigure 8.25.

Figure 8.25 Variat ion of normal force coefficient with incidence.

For the same profile, the variat ion of CN, calculated with lift ing line theory, with α will be asshown by the line of dashes. From Figure 8.25, it is seen that for aerofoils with very smallaspect rat io the stalling incidence is very high and hence they can fly at high values of α,without stalling.

8.13.2 Zero Aspect RatioFor the limit ing case of a profile with zero aspect rat io (c→ ∞), it has been found that:

This is the same as the behavior predicted by Isaac Newton for a flat plate which experiencesa normal force proport ional to the t ime rate of change of momentum in inelast ic(incompressible) fluid part icles impinging on it . In fact here we should have the normal force:

which gives the above CN.

8.13.3 The Acceleration PotentialLet us consider an aerofoil placed in a uniform flow of velocity −V in the negat ive direct ion of x-axis. We can as usual consider the aerofoil replaced by bound vort ices at its surface enclosingair at rest and accompanied by a wake of free trailing vort ices. Outside the region consist ing ofthe bound vort ices and the wake the flow is irrotat ional, and hence the flow field can berepresented by a velocity potent ial ϕ such that the air velocity is given by:

(8.83) If the velocity induced by the vortex system is , then:

(8.84) The mot ion is steady, therefore the accelerat ion is:

(8.85) If we assume1 that the magnitude of the induced velocity is small compared to the mainflow velocity V, Equat ion (8.85) can be expressed, using Equat ion (8.83), as:

(8.86) Thus we have:

(8.87) where Φ is the acceleration potential. Since the velocity potent ial ϕ sat isfies Laplace'sequat ion 2ϕ, it follows from Equat ion (8.87) that :

(8.88) Assuming the flow to be incompressible and neglect ing external forces, the accelerat ion canbe expressed as:

(8.89) This shows that an accelerat ion potent ial always exists. However, only with our assumption ofsmall magnitude of induced velocity this sat isfies Laplace's equat ion. Comparing Equat ions(8.87) and (8.89) we see that Φ and p/ρ can differ only by a constant, and we can take:

(8.90) where I is the pressure at infinity.

8.14 Lifting SurfaceFor thin aerofoils, which can be approximated by replacing them by their plan areas in the xy-plane, the accelerat ion potent ial can be applied comfortably. Let us consider such an aerofoil,shown in Figure 8.26(a), and replace it by its plan area represented by its sect ion AB, shown inFigure 8.26(b).

Figure 8.26 (a) A thin aerofoil, (b) plan area of the thin aerofoil.

I f pu is the pressure at a point on the upper surface of AB and pl is the pressure at acorresponding point at the lower surface, then it can be shown that:

(8.91) where Φl and Φu are the corresponding values of the accelerat ion potent ial. Thus we have thelift and pitching moment as:

(8.92)

(8.93) where S is the surface area of the aerofoil planform. The center of pressure is at a distance xp= M/L from the origin.

The downwash velocity is obtained by equat ing the values of the z-component of theaccelerat ion, Equat ions (8.86) and (8.87). Thus:

But the downwash vanishes at x =∞, therefore:

(8.94) The induced drag is given by:

(8.95) For a given y, the profile z = z(x, y) is determined by:

(8.96)

Example 8.5A wing with ellipt ical loading, with span 15 m, planform area 45 m2 is in level flight at 750 km/h,at an alt itude where density is 0.66 kg/m3. If the induced drag on the wing is 3222 N, (a)determine the lift coefficient , (b) the downwash velocity, and (c) the wing loading.

SolutionGiven, 2b = 15 m, S = 45 m2, ρ = 0.66 kg/m3, V = 750/3.6 = 208.33 m/s.

The aspect rat io of the wing is:

Given that the induced drag is 3222 N, therefore, the induced drag coefficient becomes:

(a) By Equat ion (8.8), the induced drag coefficient is:

(b) By Equat ion (8.7), the downwash is:

But k0, by Equat ion (8.6a) is:

Therefore:

(c) In level flight , L = W, therefore:

The wing loading is:

8.15 SummaryThe vortex theory of a lift ing aerofoil proposed by Lancaster and the subsequent developmentby Prandt l made use of for calculat ing the forces and moment about finite aerofoils. The vortexsystem around a finite aerofoil consists of the start ing vortex, the t railing vortex system and

the bound vortex system.From Helmholtz's second theorem, the strength of the circulat ion round any sect ion of a

bundle of vortex tubes is the sum of the strength of the vortex filaments cut by the sect ionplane.

If the circulat ion curve can be described as some funct ion of y, say f(y), then the strength ofthe circulat ion shed by the aerofoil becomes:

At a sect ion of the aerofoil the lift per unit span is given by:

The induced velocity at y1, in general, is in the downward direct ion and is called downwash.The downwash has the following two important consequences which modify the flow about

the aerofoil and alter its aerodynamic characterist ics.The downwash at y1 is felt to a lesser extent ahead of y1 and to a greater extent behind,and has the effect of t ilt ing the resultant wind at the aerofoil through an angle,

The downwash reduces the effect ive incidence so that for the same lift as the equivalentinfinite or two-dimensional aerofoil at incidence α, an incidence of α = α∞ + is required atthat sect ion of the aerofoil.

In addit ion to this mot ion of the air stream, a finite aerofoil spins the air flow near the t ipsinto what eventually becomes two trailing vort ices of considerable core size. Thegenerat ion of these vort ices requires a quant ity of kinet ic energy. This constantexpenditure of energy appears to the aerofoil as the trailing vortex drag.

The forward wind velocity generates lift and the downwash generates the vortex drag .

This shows that there is no vortex drag if there is no trailing vort icity.The expression k = f(y) which can be subst ituted in expression for L, and is:

The lift of an aerofoil of span 2b is:

The circulat ion for ellipt ical distribut ion is:

The downwash becomes:

This is an important result , which implies that the downwash is constant along the wing span.The drag caused by the downwash is:

Therefore, the drag coefficient becomes:

For modified ellipt ical loading:

The lift coefficient becomes:

The downwash for modified ellipt ic loading at any point y along the span is:

The vortex drag for modified loading is:

The drag coefficient becomes:

This drag coefficient for the modified loading is more than that for ellipt ical loading by anamount δ, which is always posit ive since it contains λ2 terms only.

If the lift for the aerofoils with ellipt ical and non-ellipt ical distribut ion is the same under givencondit ions, the rate of change of vert ical momentum in the flow is the same for both. Thus, forellipt ical distribut ion the lift becomes:

For non-ellipt ic distribut ion, the lift is:

where is a representat ive mass flow meet ing unit span. But lift L is the same on each wing,therefore:

Now the energy transfer or rate of change of the kinet ic energy of the representat ive massflow is the vortex drag (or induced drag). Thus, for ellipt ical distribut ion the vortex drag is:

For non-ellipt ic distribut ion the vortex drag is:

Lancaster–Prandt l lift ing line theory is a representat ion to improve on the accuracy of thehorseshoe vortex system. In lift ing line theory, the bound vortex is assumed to lie on a straightline joining the wing t ips (known as lift ing line). Now the vort icity is allowed to vary along theline. The lift ing line is generally taken to lie along the line joining the sect ion quarter-chordpoints. The results obtained using this representat ion is generally good provided that theaspect rat io of the wing is moderate or large –generally not less than 4.

The integral equat ion from which the bound vort icity distribut ion may be determined is:

The lift generated by the wing is:

The lift coefficient is:

Thus, the lift coefficient CL depends on A1, which in turn depends on the values anddistribut ion of α and μ.

The induced drag is:

where:

and is usually very small. Also, A1 = CL/π , so that:

where , is the induced drag factor, and hence Γ depends on the values of the Fouriercoefficients, and hence on the wing geometry, especially on the planform.

For an aerofoil:Geometrical incidence is the angle between the chord of the profile and the direct ion ofmot ion of the aerofoil.Absolute incidence is the angle between the axis of zero lift of the profile and thedirect ion of mot ion of the aerofoil.

When the axes of zero lift of all the profiles of the aerofoil are parallel, each profile meets thefreestream wind at the same absolute incidence, the incidence is the same at every point onthe span of the aerofoil, and the aerofoil is said to be aerodynamically untwisted.

An aerofoil is said to have aerodynamic twist when the axes of zero lift of its individualprofiles are not parallel. The incidence is then variable across the span of the aerofoil.

The drag and lift rat io can be expressed as:

For an actual aerofoil in a subsonic flow the main components of the drag are the profiledrag and the skin friction drag. The induced drag caused by the downwash is an addit ionalcomponent of drag. Therefore, the total drag coefficient of the strip (profile), using Equat ion(8.24), is:

where is the coefficient of profile drag for the profile.It may be noted that the profile drag is largely independent of incidence in the working range.

Profile drag is the sum of the skin frict ion due to viscosity and form drag due to the shape.The form drag due to the shape is owing to the high pressure at the leading edge and low

pressure at the t railing edge (that is the low pressure in the wake).The following are the two problems associated with aerofoils:

For a given circulat ion k(y), the form of the aerofoil and the induced drag are to bedetermined.For a given form of aerofoil, the distribut ion of circulat ion and the induced drag are to bedetermined.

In pract ice, in addit ion to induced drag there is profile drag due to skin frict ion and wake. Thecoefficient of profile drag is indicated by . This the complete drag coefficient is:

The lift curve slope for an aerofoil of finite aspect rat io with ellipt ical loading is:

If the aspect rat io is reduced to ' and if the ‘primes' refer to the new aerofoil with thesame incidence, we have:

In problem I the aerofoil shape is found for a given circulat ion k(y). Problem II is an inverseproblem in which, for a given aerofoil geometry the circulat ion is determined.

At the point of the t railing edge of an aerofoil, whose eccentric angle is θ, we have:

For ellipt ic loading this becomes:

By Equat ion (8.7), we have:

Therefore:

which is constant across the wing span.For wings with loading other than ellipt ic, the drag polar becomes:

where e is known as the Oswald wing efficiency and for ellipt ic loading e = 1.For ellipt ic loading, δ = 0 and e = 1, therefore, the drag polar becomes:

For drag minimum:

Rectangular aerofoil is an aerofoil whose planform is a rectangle. An aerofoil whose shape isthat of a cylinder erected on an aerofoil profile sat isfies this requirement.

Cylindrical rectangular aerofoil is the simplest type, of span 2b and chord c, which is constantat all sect ions. All the sect ions are similar and similarly situated.

In the general case, where the loading or lift distribut ion is not symmetrical about mid-spansect ion, even terms appear in the distribut ion, and as a consequence of the asymmetry othercharacterist ics of aerofoil appear.

When the lift distribut ion is not symmetrical about the centerline, one wing will have higherlift than the other and a net rolling moment about the longitudinal axis through the mid-spanwill result .

Further, as the lift is not symmetric nor is the spanwise distribut ion of circulat ion, thedownwash will vary across the span without being symmetrical about the centerline and so willbe the vortex drag grading. Hence, more drag will be experienced on one wing (the one withmore lift ) than on the other and a net yawing moment will result about the vert ical (normal) axisthrough the mid-span sect ion. In addit ion to these there will be the overall lift and vortex dragforce normal and parallel to the plane of the aerofoil in the plane of symmetry.

The lift act ing on any sect ion of spanwise length δy at a distance y from the centerline (ox-axis) will produce a negat ive increment of rolling moment equal to:

where l is the lift grading given by l = ρVk.The total moment becomes:

The asymmetrical drag grading across the span, gives rise to yawing moment N.The yawing moment can be expressed as:

where CN is the yawing moment coefficient .Lift ing surface theory is a method which treats the aerofoil as a vortex sheet over which

vort icity is spread at a given rate. In other words, the aerofoil is regarded as a surfacecomposed of lift ing elements. This is different from the lift ing line theory. The essent ialdifference between the lift ing surface theory and lift ing line theory is that in the former theaerofoil is t reated as a vortex sheet, whereas in the lat ter, the aerofoil is represented by astraight line joining the wing t ips, over which the vort icity is distributed.

Munk's theorem of stagger states that “the total drag of a mult iplane system does notchange when the elements are t ranslated parallel to the direct ion of the wind, provided thatthe circulat ions are left unchanged.” Thus the total induced drag depends only on the frontalaspect.

The total drag mutually induced on the pair of lift ing elements becomes:

which is independent of the angle of stagger. This yields Munk's theorem of stagger, that is:

“the total drag of a multiplane system does not change when the elements are translatedparallel to the direction of the wind, provided that the circulations are left unchanged.”When the system is unstaggered (that is, when θ = 0):

and thus if the lift ing systems are in the same plane normal to the wind, the drag induced inthe first by the second is equal to the drag induced in the second by the first . This resultconst itutes Munk's reciprocal theorem.

The total mutual induced drag is:

Blenk's method is meant for wings of finite aspect rat io and is based on the lift ing line theoryof Prandt l, hence limited to aerofoils moving in the plane of symmetry and with a t railing edgewhich could be regarded as approximately straight. This method considers the wing as a lift ingsurface, that is to say the wing is replaced by a system of bound vort ices distributed over itssurface rather than along a straight line coinciding with the span. However, this method hasthe limitat ion that the wing is assumed to be thin and pract ically plane.

The following are the two main approaches employed in Blenk's method:1. Given the load distribut ion and the plan, find the profiles of the sect ions.2. Given the plan and the profiles, find the load distribut ion (that is the vort icitydistribut ion).

For aerofoils with aspect rat io less than about unity, the agreement between theoret ical andexperimental lift distribut ion breaks down. The reason for this break down is found to be theconsequence of Prandt l's hypothesis that the free vortex lines leave the trailing edge in thesame line as the main stream. This assumption leads to a linear integral equat ion for thecirculat ion.

Exercise Problems1. An ellipt ical wing of aspect rat io 8 and span 20 m is in steady level flight , at sea level at aspeed of 300 km/h. If the induced drag on the wing is 400 N, determine the lift act ing on thewing.

[Answer: 46.146 kN]

2. A wing of aspect rat io 7.5 and span 15 m has a wing loading of 1100 N/m 2, while flying at210 km/h at sea level. Determine the induced drag act ing on the wing. Assume both the wingplanform and lift distribut ion to be ellipt ical.

[Answer: 739.5 N]

3. If the resultant wind over an aerofoil flying at 300 km/h is t ilted by 1.2 , determine thedownwash.

[Answer: 1.746 m/s]

4. An aerofoil with ellipt ical load distribut ion flying at 60 m/s has lift coefficient as 1.2, at sealevel condit ion. If the mean chord is 2.4 m, determine the circulat ion about the mid-span.

[Answer: 110 m2/s]

5. For an aerofoil with ellipt ical loading show that:

6. A wing of ellipt ical loading has zero lift angle as −1 . 8 . The lift coefficient of the wing atangle of at tack of 3 is 0.9. If the lift coefficient has to be increased by 10%, (a) what shouldbe the angle of at tack, (b) how much will be the percentage change in induced drag due tothis change in angle of at tack?

[Answer: (a) 3 . 48 , (b) 21%]

7. An aircraft of allup weight 160 kN is in level flight at an alt itude where the air density is 0.1kg/m3. The span and average chord of the wing with ellipt ic loading are 24 m and 3 m,respect ively. If the flight speed is 620 km/h and profile drag coefficient is 0.1, determine theaerodynamic efficiency.

[Answer: 7.93]

8. A wing of aspect rat io 4 and efficiency of 0.8 has a profile drag coefficient of 0.12. If thetotal drag coefficient experienced by the wing is 0.5, determine the lift coefficient .

[Answer: 1.95]

9. A wing of aspect rat io 4 is designed for ellipt ic loading with a design lift coefficient of 1.2.But in actual flight the lift coefficient is found to be 10% less than the design value. If theprofile drag coefficient of the design and actual wing is 0.1 and their aerodynamic efficienciesare equal, determine (a) the percentage change of their drag and (b) the efficiency of theactual wing.

[Answer: (a) 10%, (b) 0.993]

10. Consider a tapered wing kept in a uniform stream of 50 m/s. The spanwise variat ion ofchord is given by:

where λ is taper rat io which is 0.4, cr is root chord equal to 2 m and 2b is span of the wingequal to 12 m. The lift coefficient at a sect ion for which spanwise locat ion is 1.2 m from theroot end of the wing is found to be 0.4. Assuming standard atmospheric condit ions, est imatethe lift at the sect ion based on Prandt l's lift ing line theory.

[Answer: 1.078 kN/m]

11. An aircraft weighing 70 kN and having a wing span of 16 m, flies straight and level at sealevel. If the flight speed is 90 m/s, find (a) the circulat ion around the mid-span and (b) theinduced drag.

[Answer: (a) 50.52 m2/s, (b) 1227.78 N]

12. The drag polar of an ellipt ic wing is:

where K is a constant slight ly more than unity. (a) Show that when the wing is in level flightthe drag will be minimum when the speed is:

where is the wing loading. (b) Find the aerodynamic efficiency for this minimum drag.[Answer: (b) ]

13. A wing of ellipt ic planform, of aspect rat io 7, wing area 26 m2, in level flight at an alt itudeof 3000 m with a speed of 88 m/s, supports a weight of 38 000 N. Determine (a) the liftcoefficient , (b) the circulat ion at the mid-span, (d) the induced drag coefficient , and (d) thedownwash induced by the trailing vortex.

[Answer: (a) 0.415, (b) 35.19 m2/s, (c) 0.00783, (d) 1.66 m/s]

14. Show that in ellipt ic loading k0/U is a linear funct ion of incidence.

15. An ellipt ic wing of aspect rat io 5 has the lift and drag coefficients as 0.914, 0.0588,respect ively, at an angle of at tack of 6.5 . (a) Determine the zero lift incidence. (b) What willbe the CL and CD for an ident ical wing of aspect rat io 8, at the same incidence, if the profiledrag coefficient is the same as that of aspect rat io 5. (c) Determine the percentage changein aerodynamic efficiency with increase of aspect rat io from 5 to 8.

[Answer: (a) 3.17 , (b) CL = 1.463, CD = 0.0908, (c) 3.67%]

16. A rectangular wing of span 0.75 m and chord 0.1 m at an angle of at tack 7 , in a flow of 30m/s and 1.23 kg/m3 experiences a lift of 33 N and drag of 2 N. Assuming the wing loading asellipt ic, (a) calculate the coefficients of lift , drag and induced drag coefficients. (b) Find thecorresponding angle of at tack, and drag coefficient for a wing of the same profile, the samelift coefficient and profile drag coefficient (as for aspect rat io 7.5), but with aspect rat io 5.

lift coefficient and profile drag coefficient (as for aspect rat io 7.5), but with aspect rat io 5.[Answer: (a) CL = 0.795, CD = 0.0482, ; (b) α ' = 7 . 967 , CD ' = 0.0616]

17. A wing of aspect rat io 9 and mean chord 1.5 m flies at sea level, with a speed of 200 km/hand with an angle of at tack of 3 . (a) If the aerodynamic efficiency is 6, determine the dragforce, assuming the loading as ellipt ic. (b) Find the profile drag coefficient . Assume theeccentricity and camber to be small, and the loading as ellipt ic.

[Answer: (a) 2099.425 N, (b) 0.051]

18. An ellipt ic wing of aspect rat io 7 is in a minimum drag flight mode. If the profile dragcoefficient is 0.01, determine the angle of at tack.

[Answer: 4.27 ]

19. An aircraft of = 4 flies at an alt itude, when the angle of at tack is 4 . If an aircraft of = 8 has to fly at the same alt itude with the same lift coefficient , what should be angle of

at tack?[Answer: 3.33 ]

20. A rectangular wing of span 9 m and chord 1.2 m with an asymmetrical lift gradingexperiences a rolling moment coefficient of 0.7 while flying at sea level with a velocity of 300km/h, determine the rolling moment.

[Answer: 144648.1 N-m]

21. An aircraft with wing span 8 m and wing area 15 m2, flying at sea level at 200 km/h, isyawed. If the yawing moment is 2000 N-m, find the yawing moment coefficient and thedeflect ion angle of the vert ical tail, assuming that the ent ire vert ical tail is deflected as oneunit .

[Answer: 0.0176, 0.16 ]

22. An aerofoil is so shaped that the velocity along the upper and lower surfaces arerespect ively 25% greater than, and 25% smaller than, the velocity of the incoming stream ofvelocity 320 km/h and density 1.25 kg/m3. (a) What is the lift force on the wing, if the span is15 m and mean chord is 3 m? (b) Also, find the lift coefficient . Assume the flow to beincompressible.

[Answer: (a) L = 222.23 kN, (b) CL = 1]

Note

1. This assumption will fail at a stagnat ion point , for then . However, this will not causeany subsequent difficulty.

9

Compressible Flows

9.1 IntroductionOur discussions so far were on incompressible flow past lift ing surfaces. That is, the effect ofcompressibility of the air has been ignored. But we know that the incompressible flow is thatfor which the Mach number is zero. This definit ion of incompressible flow is only ofmathematical interest , since for Mach number equal to zero there is no flow and the state is

essent ially a stagnat ion state. Therefore, in engineering applicat ions we treat the flow withdensity change less than 5% of the freestream density as incompressible [1]. This correspondsto M = 0.3 for air at standard sea level state. Thus, flow with Mach number greater than 0.3 istreated compressible. Compressible flows can be classified into subsonic, supersonic andhypersonic, based on the flow Mach number. Flows with Mach number from 0.3 to around 1 istermed compressible subsonic, flows with Mach number greater than 1 and less than 5 arereferred to as supersonic and flows with Mach number in the range from 5 to 40 is termedhypersonic. In our discussions here, only subsonic and supersonic flows will be considered. InChapter 2, we discussed some aspects of compressible flows only briefly. Therefore, it will be ofgreat value to read books specializing on gas dynamics and its applicat ion aspects, such asRathakrishnan (2010) [1], before gett ing into this chapter.

In our discussion in this chapter, the air will be t reated as a perfect , compressible and inviscidfluid. In other words, the important consequence of viscosity, namely, the skin friction drag dueto the viscous effects in the boundary layer will not be considered in our discussions.

9.2 Thermodynamics of Compressible FlowsIn Chapter 2, we saw that a perfect gas has to be thermally as well as calorically perfect ,sat isfying the thermal state equat ion and at least two calorical state equat ions. Thus for aperfect gas:

(9.1a) or

(9.1b) where p is the pressure and R is the gas constant, given by:

where Ru is the universal gas constant equal to 8314 J/(kg K) and is the molecular weightof the gas. Thus, of the four quant it ies p, ρ, T, R, in Equat ion (9.1b), only two are independent.

Taking log on both sides and different iat ing Equat ion (9.1b), we get:

(9.2) Let us assume unit mass of a gas receiving a small quant ity of heat q. By the first law of

thermodynamics, we know that heat is a form of energy [2]. Thus the quant ity of heat q isequivalent to q units of mechanical energy. Hence addit ion of heat q will supply energy to thegas, result ing in the increase of its specific volume from to . Thus, the heat q addeddoes a mechanical work of pdv. Let us assume that the expansion is taking place very slowly,so that no kinet ic energy is developed. For this process, we can write:

(9.3) where du is the increase in the internal energy of the gas. It is essent ial to note from Equat ion(9.3) that only a part of the heat q supplied is converted to mechanical work pdv and the restof the heat is dumped into the internal energy of the gas mass. This demonstrates that theenergy conversion is 100% efficient . The work pdv is referred to as flow work. Thus, for doing,say, 1 unit of work (pdv) we need to supply q/η amount of heat, where η is the efficiency of thework producing cycle and η is always less than 1. For example, the work producing devices,such as spark ignit ion (SI) engine, compression ignit ion (CI) engine and gas turbine (jet) engineshas efficiencies of 40%, 60% and 30%, respect ively.

For a perfect gas, the internal energy u is a funct ion of the absolute temperature T alone.This hypothesis is a generalizat ion for experimental results. It is known as Joule's law. Thus:

(9.4) Subst itut ing this into Equat ion (9.3), we have:

(9.5) For a constant volume (isochoric) process, dv = 0. Thus for a constant volume process,Equat ion (9.5) reduces to:

We can express this as:

where is called the specific heat at constant volume. It is the quant ity of heat required toraise the temperature of the system by one unit while keeping the volume constant. Thus fromEquat ion (9.5), with dv = 0, we get:

Similarly the specific heat at constant pressure, cp, defined as the quant ity of heat required toraise the temperature of the system by one unit while keeping the pressure constant. Now, forp = constant, Equat ion (9.2) simplifies to:

But , therefore:

Subst itut ing in Equat ion (9.5), we get:

For p = constant, q = cpdT, therefore:

or

(9.6) This relat ion is popularly known as Mayer's Relation, in honor of Julius Robert von Mayer(November 25, 1814–March 20, 1878), a German physician and physicist and one of thefounders of thermodynamics. He is best known for enunciat ing in 1841 one of the originalstatements of the conservat ion of energy or what is now known as one of the first versions ofthe first law of thermodynamics, namely, “energy can be neither created nor destroyed.”

Another parameter of primary interest in thermodynamics is entropys. The entropy,temperature and heat q are related as:

(9.7) Subst itut ing for q from Equat ion (9.5), we get:

But from state equat ion:

Thus:

Replacing R with , we get the different ial change of entropy as:

This can be expressed as:

But . Thus:

This shows that ds is an exact different ial.When the state changes from to , the entropy increase is given by:

or

(9.8) The second law of thermodynamics assumes that the entropy of an isolated system can neverdecrease, that is

When the entropy remains constant throughout the flow, the flow is termed isentropic flow.Thus, for an isentropic flow, ds = 0. For isentropic flows, by Equat ion (9.8), we have:

This can be generalized as:

(9.9) This relat ion is known as the isentropic process relation.

The cont inuity and momentum equat ions for a steady flow of air, respect ively, are:

(9.10)

(9.11) There are three unknowns; the pressure p, density ρ and velocity V in Equat ions (9.10) and(9.11). Therefore these two equat ions alone are insufficient to determine the solut ion. To solvethis mot ion, we can make use of the process Equat ion (9.9) as the third equat ion, presumingthat the mot ion is isentropic.

9.3 Isentropic FlowThe fundamental equat ions for isentropic flows can be derived by considering a simplifiedmodel of a one-dimensional flow field, as follows.

Consider a streamtube different ial in equilibrium in a one-dimensional flow field, asrepresented by the shaded area in Figure 9.1. p is the pressure act ing at the left face of thestreamtube and is the pressure at the right face. Therefore, the pressure force in posit ives-direct ion, Fp, is given by:

Figure 9.1 Forces act ing on streamtube.

For equilibrium, dm (dV/dt) = sum of all the forces act ing on the streamtube different ial, wheredm is the mass of fluid in the streamtube element considered, and dV/dt is the substant ialaccelerat ion.

In the above equat ion for substant ial accelerat ion, ∂V/ ∂ t is the local accelerat ion oraccelerat ion at a point , that is, change of velocity at a fixed point in space with t ime. Theconvect ive accelerat ion is the accelerat ion between two points in space, that is, change ofvelocity at a fixed t ime with space. It is present even in a steady flow.

The substant ial derivat ive is expressed as:

(9.12) Therefore, the equilibrium equat ion becomes:

But dm = ρ dA ds. Subst itut ing this into the above equat ion, we get:

that is:

(9.13) Equat ion (9.13) is applicable for both compressible and incompressible flows; the onlydifference comes in solut ion. For steady flow, Equat ion (9.13) becomes:

(9.14) Integrat ion of Equat ion (9.14) yields:

(9.15) This equat ion is often called the compressible form of Bernoulli's equat ion for inviscid flows. If ρis expressible as a funct ion of p only, that is, ρ = ρ(p), the second expression is integrable.Fluids having these characterist ics, namely the density is a funct ion of pressure only, are calledbarotropic fluids. For isentropic flow process:

(9.16)

(9.17) where subscripts 1 and 2 refer to two different states. Therefore, integrat ing dp/ρ betweenpressure limits p1 and p2, we get:

(9.18) Using Equat ion (9.18), Bernoulli's equat ion can be writ ten as:

(9.19) Equat ion (9.19) is a form of energy equat ion for isentropic flow process of gases.

For an adiabat ic flow of perfect gases, the energy equat ion can be writ ten as:

(9.20a) or

(9.20b) or

(9.20c) Equat ions (9.20) are more general in nature than Equat ion (9.19), the restrict ions on Equat ion(9.19) are more severe than those of Equat ion (9.20).

Equat ions (9.20) can be applied to shock, but not Equat ion (9.19), as the flow across theshock is non-isentropic. With Laplace equat ion a2 = γp/ρ, Equat ion (9.20c) can be writ ten as:

(9.20d) or

(9.20e) The subscript “0” refers to stagnat ion condit ion when the flow is brought to rest isentropicallyor when the flow is connected to a large reservoir. All these relat ions are valid only for perfectgas.

9.4 Discharge from a ReservoirConsider a reservoir as shown in Figure 9.2, containing air at high pressure p0. Let the density,temperature, speed of sound and velocity of air be ρ0, T0, a0 and V0, respect ively.

Figure 9.2 Discharge of high pressure air through a small opening.

Because of the large volume of the reservoir, the velocity of air inside is V0 = 0. Let the high

pressure air be discharged to ambient atmosphere at pressure pa and velocity V = 0, throughan opening as shown in Figure 9.2. Now the velocity V at the opening, with which the air isdischarged, can be obtained by subst itut ing V1 = 0, p1 = p0, ρ1 = ρ0 and p2 = pa into Equat ion(9.19) as:

(9.21) For discharge into vacuum, that is, if pa = 0, Equat ion (9.21) results in the maximum velocity:

(9.22) V max is the limit ing velocity that may be achieved by expanding a gas at any given stagnat ioncondit ion into vacuum. For air at T0 = 288 K, V max = 760.7 m/s = 2.236 a0. This is the maximumvelocity that can be obtained by discharge into vacuum in a frict ionless flow. From Equat ion(9.22), we can see that V max is independent of reservoir pressure but it depends only on thereservoir temperature. For incompressible flow, by Bernoulli's equat ion:

(9.23) Therefore:

(9.23a) In this relat ion ρ is replaced by ρ0, because ρ is constant for incompressible flow. CombiningEquat ions (9.22) and (9.23a), we get:

For air, with γ = 1.4:

That is, the error involved in t reat ing air as an incompressible medium is 90%.For the case when the flow is not into vacuum, pa/p0 ≠ 0 and Equat ions (9.21) and (9.23) may

be expressed by dividing them by a0 as:

(9.24)

(9.25) In the course of discussion in this sect ion, we came across three speeds namely, V max, a0

and V*( = a*) repeatedly. These three speeds serve as standard reference speeds for gasdynamic study. We know that for adiabat ic flow of a perfect gas, the velocity can be expressedas:

where T0 is the stagnat ion temperature. Since negat ive temperatures on absolute scales arenot at tainable, it is evident from the above equat ion that there is a maximum velocitycorresponding to a given stagnat ion temperature. This maximum velocity, which is often usedfor reference purpose, is given by:

Another useful reference velocity is the speed of sound at the stagnat ion temperature, givenby:

Yet another convenient reference velocity is the crit ical speed V*, that is, velocity at MachNumber unity, or:

This may also be writ ten as:

This results in:

Therefore, in terms of stagnat ion temperature, the crit ical speed becomes:

From this equat ion, we may get the following relat ions between the three reference velocit ies(with γ = 1.4):

9.5 Compressible Flow EquationsThe one-dimensional analysis given in Sect ion 9.3 is valid only for flow through infinitesimalstreamtubes. In many real flow situat ions, the assumption of one-dimensionality for the ent ireflow is at best an approximat ion. In problems like flow in ducts, the one-dimensional t reatmentis adequate. However, in many other pract ical cases, the one-dimensional methods are neitheradequate nor do they provide informat ion about the important aspects of the flow. Forexample, in the case of flow past the wings of an aircraft , flow through the blade passages ofturbine and compressors, and flow through ducts of rapidly varying cross-sect ional area, theflow field must be thought of as two-dimensional or three-dimensional in order to obtain resultsof pract ical interest .

Because of the mathematical complexit ies associated with the treatment of the mostgeneral case of three-dimensional mot ion –including shocks, frict ion and heat t ransfer, itbecomes necessary to conceive simple models of flow, which lend themselves to analyt icalt reatment but at the same t ime furnish valuable informat ion concerning the real and difficultflow patterns. We know that by using Prandt l's boundary layer concept, it is possible to neglectfrict ion and heat t ransfer for the region of potent ial flow outside the boundary layer.

In this chapter, we discuss the different ial equat ions of mot ion for irrotat ional, inviscid,adiabat ic and shock-free mot ion of a perfect gas.

9.6 Crocco's TheoremConsider two-dimensional, steady, inviscid flow in natural coordinates (l, n) such that l is alongthe streamline direct ion and n is perpendicular to the direct ion of the streamline. Theadvantage of using natural coordinate system –a coordinate system in which one coordinateis along the streamline direct ion and other normal to it –is that the flow velocity is always alongthe streamline direct ion and the velocity normal to streamline is zero.

Though this is a two-dimensional flow, we can apply one-dimensional analysis, byconsidering the port ion between the two streamlines 1 and 2 (as shown in Figure 9.3) as astreamtube and taking the third dimension to be ∞.

Figure 9.3 Flow between two streamlines.

Let us consider unit width in the third direct ion, for the present study. For this flow, theequat ion of cont inuity is:

(9.26) The l-momentum equat ion is1:

The l-momentum equat ion can also be expressed as:

(9.27) The n-momentum equat ion is:

But there will be centrifugal force act ing in the n-direct ion. Therefore:

(9.28) The energy equat ion is:

(9.29) The relat ion between the entropy and enthalpy can be expressed as [1]:

Different iat ion of Equat ion (9.29) gives dh + VdV = dh0. Therefore, the entropy equat ionbecomes:

This equat ion can be split as follows:i.

because along the streamlines.ii.

Introducing ∂p/∂ l from Equat ion (9.27) and ∂p/ ∂ n from Equat ion (9.28) into the above twoequat ions, we get:

(9.30a)

(9.30b) But, is the vort icity of the flow. Therefore:

(9.31) This is known as Crocco's theorem for two-dimensional flows. From this, it is seen that therotat ion depends on the rate of change of entropy and stagnat ion enthalpy normal to thestreamlines.

Crocco's theorem essent ially relates entropy gradients to vort icity, in steady, frict ionless,nonconduct ing, adiabat ic flows. In this form, Crocco's equat ion shows that if entropy (s) is aconstant, the vort icity (ζ) must be zero. Likewise, if vort icity ζ is zero, the entropy gradient inthe direct ion normal to the streamline (ds/dn) must be zero, implying that the entropy (s) is aconstant. That is, isentropic flows are irrotational and irrotational flows are isentropic. Thisresult is t rue, in general, only for steady flows of inviscid fluids in which there are no body forcesact ing and the stagnat ion enthalpy is a constant.

From Equat ion (9.30a) it is seen that the entropy does not change along a streamline. Also,Equat ion (9.30b) shows how entropy varies normal to the streamlines.

The circulat ion is:

(9.32) By Stokes theorem, the vort icity ζ is given by:

(9.33)

where ζx, ζy, ζz are the vort icity components. The two condit ions that are necessary for africt ionless flow to be isentropic throughout are:

1. h0 = constant, throughout the flow.

2. ζ = 0, throughout the flow.From Equat ion (9.33), ζ = 0 for irrotat ional flow. That is, if a frict ionless flow is to be isentropic,the total enthalpy should be constant throughout and the flow should be irrotat ional.

the total enthalpy should be constant throughout and the flow should be irrotat ional.It is usual to write Equat ion (9.33) as follows:

When ζ ≠ 0Since h0 = constant, T0 = constant (perfect gas). For this type of flow, we can show that:

(9.34) From Equat ion (9.34), it is seen that in an irrotat ional flow (that is, with ζ = 0), stagnat ionpressure does not change normal to the streamlines. Even when there is a shock in the flowfield, p0 changes along the streamlines at the shock, but does not change normal to thestreamlines.

Let h0 = constant(isoenergic flow). Then Equat ion (9.31) can be writ ten in vector form as:

(9.35a) where grad s stands for increase of entropy s in the n-direct ion. For a steady, inviscid andisoenergic flow:

(9.35b) If s = constant, V × curl V = 0. This implies that (a) the flow is irrotat ional, that is, curl V = 0, or(b) V is parallel to curl V.Irrotat ional flowFor irrotat ional flows ( curl V = 0), a potent ial funct ion ϕ exists such that:

(9.36) On expanding Equat ion (9.36), we have:

Therefore, the velocity components are given by:

The advantage of introducing ϕ is that the three unknowns Vx, Vy and Vz in a general three-dimensional flow are reduced to a single unknown ϕ. With ϕ, the irrotat ionality condit ionsdefined by Equat ion (9.33) may be expressed as follows:

Also, the incompressible cont inuity equat ion · V = 0 becomes:

or

This is Laplace's equat ion. With the introduct ion of ϕ, the three equat ions of mot ion can bereplaced, at least for incompressible flow, by one Laplace equat ion, which is a linear equat ion.

9.6.1 Basic Solutions of Laplace's EquationWe know from our basic studies on fluid flows [2] that :

1. For uniform flow (towards posit ive x-direct ion), the potent ial funct ion is:

2. For a source of strength Q, the potent ial funct ion is:

3. For a doublet of strength μ (issuing in negat ive x-direct ion), the potent ial funct ion is:

4. For a potent ial (free) vortex (counterclockwise) with circulat ion Γ, the potent ial funct ion

is:

9.7 The General Potential Equation for Three-Dimensional Flow

For a steady, inviscid, three-dimensional flow, by cont inuity equat ion:

that is:

(9.37) Euler's equat ions of mot ion (neglect ing body forces) are:

(9.38a)

(9.38b)

(9.38c) For incompressible flows, the density ρ is a constant. Therefore, the above four equat ions aresufficient for solving the four unknowns Vx, Vy, Vz and p. But for compressible flows, ρ is also anunknown. Therefore, the unknowns are ρ, Vx, Vy, Vz and p. Hence, the addit ional equat ion,namely, the isentropic process equat ion, is used. That is, p/ργ = constant is the addit ionalequat ion used along with cont inuity and momentum equat ions.

Introducing the potent ial funct ion ϕ, we have the velocity components as:

(9.39) Equat ion (9.37) may also be writ ten as:

(9.37a) From isentropic process relat ion, ρ = ρ(p). Hence:

because from Equat ion (9.38a):

Similarly:

With the above relat ions for , and , Equat ion (9.37a) can be expressed as:

But the velocity components and their derivat ives in terms of potent ial funct ion can beexpressed as:

Therefore, in terms of potent ial funct ion ϕ, the above equat ion can be expressed as:

(9.40) This is the basic potential equation for compressible flow and it is nonlinear.

The difficult ies associated with compressible flow stem from the fact that the basic equat ion

is nonlinear. Hence the superposit ion of solut ions is not valid. Further, in Equat ion (9.40) thelocal speed of sound ‘a’ is also a variable. By Equat ion (2.9e) of Reference 1, we have:

(9.41) To solve a compressible flow problem, we have to solve Equat ion (9.40) using Equat ion (9.41),but this is not possible analyt ically. However, numerical solut ion is possible for given boundarycondit ions.

9.8 Linearization of the Potential EquationThe general equat ion for compressible flows, namely Equat ion (9.40), can be simplified for flowpast slender or planar bodies. Aerofoil, slender bodies of revolut ion and so on are typicalexamples for slender bodies. Bodies like wing, where one dimension is smaller than others, arecalled planar bodies. These bodies introduce small disturbances. The aerofoil contour becomesthe stagnat ion streamline.

For the aerofoil shown in Figure 9.4, with the except ion of nose region, the perturbat ionvelocity is small everywhere.

Figure 9.4 Aerofoil in an uniform flow.

9.8.1 Small Perturbation TheoryAssume that the velocity at any point in the flow field is given by the vector sum of thefreestream velocity V∞ along the x-axis, and the perturbat ion velocity components u, and

along x, y and z-direct ions, respect ively. Consider the flow around an aerofoil shown inFigure 9.4. The velocity components around the aerofoil are:

(9.42) where Vx, Vy, Vz are the main flow velocity components and u, , are the perturbat ion(disturbance) velocity components along x, y and z direct ions, respect ively.

The small perturbat ion theory postulates that the perturbat ion velocit ies are smallcompared to the main velocity components, that is:

(9.43a) Therefore:

(9.43b) Now, consider a flow at small angle of at tack or yaw as shown in Figure 9.5. Here:

Figure 9.5 Aerofoil at an angle of at tack in an uniform flow.

Since the angle of at tack α is small, the above equat ions reduce to:

Thus, Equat ion (9.42) can be used for this case also.With Equat ion (9.42), linearizat ion of Equat ion (9.40) gives:

(9.44) neglect ing all higher order terms, where M is the local Mach number. Therefore, Equat ion (9.41)should be used in solving Equat ion (9.44).

The perturbat ion velocit ies may also be writ ten in potent ial form, as follows:

Let ϕ = ϕ∞ + ϕ, where:

Therefore, ϕ may be called the disturbance (perturbat ion) potent ial and hence theperturbat ion velocity components are given by:

(9.45) With the assumptions of small perturbat ion theory, Equat ion (9.41) can be expressed as:

(9.46)

Using Binomial theorem, (a∞/a)2 can be expressed as:

(9.47) Subst itut ing the above expression for (a∞/a) in the equat ion:

the relat ion between the local Mach number M and freestream Mach number M∞ may beexpressed as (neglect ing small terms):

(9.48) The combinat ion of Equat ions (9.48) and (9.44) gives:

(9.49) Equat ion (9.49) is a nonlinear equat ion and is valid for subsonic, t ransonic and supersonic flowunder the framework of small perturbat ions with and . It is, however, not valid forhypersonic flow even for slender bodies (since u ≈ V∞ in the hypersonic flow regime). Theequat ion is called the linearized potential flow equation, though it is not linear.

Equat ion (9.49) may also be writ ten as:

(9.50) Further linearizat ion is possible if:

(9.51) With this condit ion Equat ion (9.50) results in:

(9.52) This is the fundamental equat ion governing most of the compressible flow regime. Equat ion(9.52) is valid only when Equat ion (9.51) is valid and Equat ion (9.51) is valid only when thefreestream Mach number M∞ is sufficient ly different from 1. Hence, Equat ion (9.51) is valid forsubsonic and supersonic flows only. For t ransonic flows, Equat ion (9.49) can be used. For M∞ ≈1, Equat ion (9.49) reduces to:

(9.53) The nonlinearity of Equat ion (9.53) makes transonic flow problems much more difficult thansubsonic or supersonic flow problems.

Equat ion (9.52) is ellipt ic (that is, all terms are posit ive) for M∞ < 1 and hyperbolic (that is, notall terms are posit ive) for M∞ > 1. But in both the cases, the governing different ial equat ion islinear. This is the advantage of Equat ion (9.52).

9.9 Potential Equation for Bodies of RevolutionFuselage of airplane, rocket shells, missile bodies and circular ducts are the few bodies ofrevolut ions which are commonly used in pract ice. The general three-dimensional Cartesian

equat ions can be used for these problems. But it is much simpler to use cylindrical polarcoordinates than Cartesian coordinates. Cartesian coordinates are x, y, z and thecorresponding velocity components are Vx, Vy, Vz. The cylindrical polar coordinates are x, r, θand the corresponding velocity components are Vx, Vr, Vθ. For axisymmetric flows withcylindrical coordinates, the equat ions will be independent of θ. Thus, mathematically, cylindricalcoordinates reduce the problem to become two-dimensional. However, for flows which are notaxially symmetric (e.g., missile at an angle of at tack), θ will be involved. The cont inuity equat ionin cylindrical coordinates is:

(9.54) Expressing the velocity components in terms of the potent ial funct ion ϕ as:

(9.55) The potent ial Equat ion (9.50) can be writ ten, in cylindrical polar coordinates, as:

(9.56) Also,

(9.57) The small perturbat ion assumptions are:

where Vx, Vr, Vθ are the mean velocity components and u, , are the perturbat ionvelocity components along the x- , r- and θ-direct ion, respect ively. Introduct ion of theserelat ions in Equat ion (9.56) results in:

(9.58) where M is the local Mach number after Equat ion (9.48). The relat ions for u, , in polarcoordinates, under small perturbat ion assumption are:

With these expressions for u, and , Equat ion (9.49) can be writ ten as:

(9.59) This equat ion corresponds to Equat ion (9.49) with the same term on the right hand side.Therefore, with:

Equat ion (9.59) simplifies to:

(9.60) This is the governing equat ion for subsonic and supersonic flows in cylindrical coordinates. Fortransonic flow, Equat ion (9.59) becomes:

(9.61) For axially symmetric, subsonic and supersonic flows, ϕθθ = 0. Therefore, Equat ion (9.60)reduces to:

(9.62) Similarly, Equat ion (9.61) reduces to:

(9.63) Equat ion (9.63) is the equat ion for axially symmetric t ransonic flows. All these equat ions arevalid only for small perturbat ions, that is, for small values of angle of at tack and angle of yaw (<15 ).

ConclusionsFrom the above discussions on potent ial flow theory for compressible flows, we can draw thefollowing conclusions:

1. The small perturbat ion equat ions for subsonic and supersonic flows are linear, but fort ransonic flows the equat ion is nonlinear.2. Subsonic and supersonic flow equat ions do not contain the specific heats rat io γ, butt ransonic flow equat ion contains γ. This shows that the results obtained for subsonic andsupersonic flows, with small perturbat ion equat ions, can be applied to any gas, but thiscannot be done for t ransonic flows.3. All these equat ions are valid for slender bodies. This is t rue of rockets, missiles, etc.4. These equat ions can also be applied to aerofoils, but not to bluff shapes like circularcylinder, etc.5. For nonslender bodies, the flow can be calculated by using the original nonlinearequat ion.

9.9.1 Solution of Nonlinear Potential Equationi. Numerical methods:The nonlinearity of Equat ion (9.49) makes it tedious to solve the equat ion analyt ically.However, solut ion for the equat ion can be obtained by numerical methods. But anumerical solut ion is not a general solut ion, and is valid only for a specific configurat ion offlow field with a fixed Mach number and specified geometry.ii. Transformation (Hodograph) methods:When one velocity component is plot ted against another velocity component, theresult ing curve may be linear, whereas in the physical plane, the relat ion may be nonlinear.This method is used for solving certain t ransonic flow problems.3. Similarity methods:In these methods, the boundary condit ions need to be specified for solving the equat ion.Detailed discussion of this method can be found in Chapter 6 on Similarity Methods.

9.10 Boundary ConditionsExamine the streamlines around an aerofoil kept in a flow field as shown in Figure 9.6.

Figure 9.6 Cambered aerofoil at an angle of at tack.

In inviscid flow, the streamline near the boundary is similar to the body contour. The flowmust sat isfy the following boundary condit ions (BCs):Boundary condit ion 1 –Kinetic flow conditionThe flow velocity at all surface points are tangent ial on the body contour. The component ofvelocity normal to the body contour is zero.Boundary condit ion 2A t z→ ± ∞, perturbat ion velocit ies are zero or finite. The kinematic flow condit ion for theaerofoil shown in Figure 9.6, with small perturbat ion assumptions, may be writ ten as follows.Body contour: f = f(x, y, z)The velocity vector V at any point on the body is tangent ial to the surface. Therefore, on thesurface of the aerofoil, (V · f) = 0, that is:

(9.64)

But u/V∞ < 1, therefore, Equat ion (9.64) simplifies to:

(9.65) For two-dimensional flows, ; ∂f/∂ y = 0. Therefore, Equat ion (9.64) reduces to:

(9.66) where the subscript “c” refers to the body contour and (∂ z/∂ x) is the slope of the body, and uand are the tangent ial and normal components of velocity, respect ively. Expressing u and

as power series of z, we get:

The coefficients a 's and b 's in these series are funct ions of x. If the body is sufficient ly slender:

that is, for sufficient ly slender bodies, it is not necessary to fulfill the boundary condit ion on thecontour of the aerofoil. It is sufficient if the boundary condit ion on the x−axis of the body issat isfied, that is, on the axis of a body of revolut ion or the chord of an aerofoil. With u/V∞ < 1,the above condit ion becomes:

(9.67) For planar bodies: ∂f/∂ y = 0 and therefore:

(9.68) that is, the condit ion is sat isfied in the plane of the body. In Equat ions (9.67) and (9.68), theelevat ion of the body above the x−axis is neglected.

9.10.1 Bodies of RevolutionFor bodies of revolut ion, the term present in the cont inuity Equat ion (9.54) becomes finite.Because of this term, the perturbat ions near the body become significant. Therefore, a powerseries for velocity components is not possible. However, we can apply the followingapproximat ion to express the perturbat ion velocity as a power series. For axisymmetric bodies:

when or rvr = a0(x). Thus, even though the radial component of velocity on the axis ofa body of revolut ion is of the order of 1/r, it can be est imated near the axis similar to a potent ialvortex. For a potent ial vortex, the radial velocity is:

Now, can be expressed in terms of a power series as:

For the axisymmetric body with its surface profile contour given by the funct ion R(x), we have:

The simplified kinematic flow condit ion for the body in Figure 9.7 is:

(9.69) where subscript “0” refers to the axis of the body.

Figure 9.7 An axisymmetric body in a flow.

Equat ion (9.69) is called the simplified kinematic flow condit ion in the sense that the kinematicflow condit ion is fulfilled on the axis, rather than on the surface of the body contour.

On the axis of the body, Equat ion (9.69) gives:

(9.70)

From the above discussions, it may be summarized that the boundary condit ions for this kindof problem are the following.

For two-dimensional (planar) bodies:

(9.71) For bodies of revolut ion (elongated bodies):

(9.72)

9.11 Pressure CoefficientPressure coefficient is the nondimensional difference between a local pressure and thefreestream pressure. The idea of finding the velocity distribut ion is to find the pressuredistribut ion and then integrate it to get lift , moment, and pressure drag. For three-dimensionalflows, the pressure coefficient Cp given by (Equat ion (2.54) of Reference 1) is:

or

where M∞ and V∞ are the freestream Mach number and velocity, respect ively, u, and are the x, y and z-components of perturbat ion velocity and γ is the rat io of specific heats.Expanding the right-hand side of this equat ion binomially and neglect ing the third and higher-order terms of the perturbat ion velocity components, we get:

(9.73) For two-dimensional or planar bodies, the Cp simplifies further, result ing in:

(9.73a) This is a fundamental equat ion applicable to three-dimensional compressible (subsonic andsupersonic) flows, as well as for low speed two-dimensional flows.

9.11.1 Bodies of RevolutionFor bodies of revolut ion, by small perturbat ion assumption, we have u < V∞, but and are not negligible. Therefore, Equat ion (9.73) simplifies to:

(9.74) The above equat ion, which is in Cartesian coordinates, may also be expressed as:

(9.75) Combining Equat ions (9.72) and (9.75), we get:

(9.76) where R is the expression for the body contour.

9.12 Similarity RuleFrom Sect ion 9.8, it is seen that the governing equat ion for compressible flow is ellipt ic forsubsonic flows (that is, for M∞ < 1) and becomes hyperbolic for supersonic flows (that is, for M∞> 1). This change in the nature of the part ial different ial equat ion, upon going from subsonic tosupersonic flow, indicates the possibility of deriving similarity relat ionships between subsoniccompressible flow and the corresponding incompressible flow, and the importance of Machwave in a supersonic solut ion. In this chapter we shall derive an expression which relates the

subsonic compressible flow past a certain profile to the incompressible flow past a secondprofile derived from the first principles through an affine transformation. Such an expression iscalled a similarity law.

If the governing equat ions of mot ion could be solved easily, the solut ion themselves wouldindicate quite clearly the nature of any similarit ies which might exist among members of afamily of flow patterns. Then there is no need for a separate derivat ion of similarity laws.

But in the majority of situat ions, we are unable to solve the equat ions of mot ion. However,even though solut ions are lacking, we may use our knowledge of the forms of the different ialequat ions and the related boundary condit ions to derive the similarity laws.

9.13 Two-Dimensional Flow: Prandtl-Glauert Rulefor Subsonic Flow

9.13.1 The Prandtl-Glauert TransformationsPrandt l and Glauert have shown that it is possible to relate the solut ion of compressible flowabout a body to incompressible flow solut ion.

The transformat ion from one to another is achieved in the following manner: Laplaceequat ion for two-dimensional compressible and incompressible flows, respect ively, are:

(9.77)

(9.78) where x coordinate is along the flow direct ion, z coordinate is normal to the flow, M∞ is thefreestream Mach number and ϕ is the velocity potent ial funct ion. These equat ions, however,are not the complete descript ion of the problem, since it is also necessary to specify theboundary condit ions.

Equat ions(9.77) and (9.78) can be transformed into one another by the followingtransformat ion:

(9.79a)

(9.79b) I n Equat ion (9.79), the variables with subscript “inc” are for incompressible flow and thevariables without subscript are for compressible flow. Combining Equat ions (9.77) and (9.79),we get:

that is:

This is ident ical to the incompressible potent ial Equat ion (9.78) if:

(9.80) Now, K2 is to be determined from the boundary condit ions. For slender bodies, by smallperturbat ion theory [Equat ion (9.71)], we have:

(9.81) since u/V∞ < 1. Equat ion (9.81) can be expressed in terms of the potent ial funct ion as:

(9.82a)

(9.82b) Also, by Equat ion (9.79):

With this relat ion and Equat ions (9.82), we get:

(9.83a)

(9.83b) From Equat ion (9.83b), it is seen that K2 can be determined from the boundary condit ions.

Equat ion (9.83b) simply means that the slope of the profile in the compressible flow patternis t imes the slope of the corresponding profile in the related incompressible flow pattern.

For further t reatment of similarity law, let us consider the three specific versions of theproblems, namely, the direct problem (Version I), in which the body profile is t reated asinvariant, the indirect problem (Version II), which is the case of equal potent ials (the pressuredistribut ion around the body in incompressible flow and compressible flow are taken as thesame), and the streamline analogy (Version III), which is also called Gothert's rule.

9.13.2 The Direct Problem-Version IConsider an invariant profile. In this case, there is no transformat ion of geometry at all. For theprofile to be invariant, from Equat ion (9.83b), we have the condit ion:

(9.84) Therefore, Equat ion (9.83b) reduces to:

(9.85) Equat ion (9.85) contradicts the original t ransformat ion equat ions (9.79). However, the errorinvolved in this contradict ion is not large since the Prandt l-Glauert t ransformat ion is valid onlyfor small perturbat ions.

By Equat ion (9.79), we have:

(9.86) Equat ion (9.79) is valid only for streamlines away from the body. Since the Prandt l-Glauertt ransformat ion is based on small perturbat ion theory, the error increases with increasingthickness of the body. In addit ion to this, some error is introduced by the above contradict ion[see Equat ion (9.85)].

Equat ion (9.86) shows that the streamlines around a body in a compressible flow are moreseparated than those around a body in incompressible flow by an amount given by . Inother words, by the existence of body in the flow field, the streamlines are more displaced in acompressible flow than in an incompressible flow, as shown in Figure 9.8, that is thedisturbances introduced by an object are larger in compressible flow than in incompressibleflow and they increase with the rise in Mach number. This is so because in compressible flowthere is density decrease as the flow passes over the body due to accelerat ion, whereas inincompressible flow there is no change in density at all. That is to say, across the body there isa drop in density, and hence by streamtube area-velocity relat ion (Sect ion 2.4,Reference 1),the streamtube area increases as the density decreases. At M∞ = 1, this disturbance becomesinfinitely large and this t reatment is no longer valid.

Figure 9.8 Aerofoil in an uniform flow.

The potent ial funct ion for compressible flow given by Equat ion (9.79) is:

(9.87a) By Equat ions (9.45) and (9.73), we have the velocity u and the pressure coefficient Cp as:

Using Equat ion (9.87a), the perturbat ion velocity and the pressure coefficient may beexpressed as follows:

Therefore:

(9.87b) Since the lift coefficient CL and pitching moment coefficient CM are integrals of CP, they can

be expressed following Equat ion (9.87b) as:

(9.87c)

(9.87d) For a flat plate in compressible flow:

(9.87e)

(9.87f) Similarly, we can express the circulat ion in compressible flow in terms of circulat ion inincompressible flow as:

(9.87g) From the discussion on version I of the Prandt l-Glauert t ransformat ion, the following two

statements can be made:1. Streamlines for compressible flow are farther apart from each other by than inincompressible flow.2. The rat io between aerodynamic coefficients in compressible and incompressible flowsis also .

From Equat ions (9.87c) and (9.87f), we infer that the locat ions of aerodynamic center andcenter of pressure do not change with the freestream Mach number M∞, as they are rat iosbetween CM and CL.

The theoret ical lift -curve slope and drag coefficient from the Prandt l-Glauert rule and themeasured CL and CD versus Mach number for symmetrical NACA-profiles of differentthickness are shown in Figure 9.9.

Figure 9.9 Variat ion of (a) lift -curve slope and (b) drag coefficient with Mach number ( -measured).

From this figure it is seen that the thinner the aerofoil the better is the accuracy of the P-Grule. For 6% aerofoil there is good agreement up to M∞ = 0.8; for 12% aerofoil also, theagreement is good up to M∞ = 0.8; thus 12% may be taken as the limit of applicability of thePrandt l-Glauert (P-G) rule. For 15% aerofoil, there is good agreement up to M∞ = 0.6. Butabove Mach 0.6, there is no more agreement. However, for supersonic aircraft the profiles usedare very thin; so from a pract ical point of view, the P-G rule is very good even with thecontradict ing assumptions involved.

Beyond a certain Mach number, there is decrease in lift . This can be explained by Figure9.9(b). There is sudden increase in drag when the local speed increases beyond sonic speed.This is because at sonic point on the profile, there is a λ−shock which gives rise to separat ionof boundary layer, as shown in Figure 9.10.

Figure 9.10 Flow separat ion caused by λ-shock.

The freestream Mach number which gives sonic velocity somewhere on the boundary iscalled crit ical Mach number . The crit ical Mach number decreases with increasing thicknessrat io of profile. The P-G rule is valid only up to about .

9.13.3 The Indirect Problem (Case of Equal Potentials): P-GTransformation – Version II

In the indirect problem, the requirement is to find a t ransformat ion, for the profile, by which wecan obtain a body in incompressible flow with exact ly the same pressure distribut ion, as in thecompressible flow.

For two-dimensional or planar bodies, the pressure coefficient Cp is given by Equat ion(9.73a) as:

and the perturbat ion velocity component, u, is:

But in this case, ; therefore, from the above expressions for Cp and u, we have:

For this situat ion the transformat ion Equat ion (9.79) gives:

(9.88) From Equat ion (9.83b) with K2 = 1, we get:

(9.89) Equat ion (9.89) is the relat ion between the geometries of the actual profile in compressibleflow and the transformed profile in the incompressible flow, result ing in same pressuredistribut ion around them.

From Equat ion (9.89), we see that in a compressible flow, the body must be thinner by thefactor than the body in incompressible flow as shown in Figure 9.11. Also, the angle ofat tack in compressible flow must be smaller by the same factor than in incompressible flow.

Figure 9.11 Aerofoils in (a) incompressible and (b) compressible flows.

From the above relat ion for Cp, we have:

(9.90) That is, the lift coefficient and pitching moment coefficient are also the same in both theincompressible and compressible flows. But, because of decreased α in compressible flow:

This is so because of the fact that the disturbances introduced in a compressible flow arelarger than those in an incompressible flow and, therefore, we must reduce α and the geometryby that amount (the difference in the magnitude of disturbance in a compressible and anincompressible flow). In other words, because of Equat ion (9.79) (z = K1z inc), every dimension inthe z-direct ion must be reduced and so the angle of at tack α should also be transformed.

9.13.4 The Streamline Analogy (Version III): Gothert's RuleGothert 's rule states [3] that the slope of a profile in a compressible flow pattern is larger bythe factor than the slope of the corresponding profile in the related incompressible flowpattern. But if the slope of the profile at each point is greater by the factor , it is also t ruethat the camber (f) rat io, angle of at tack (α) rat io, the thickness (t) rat io, must all be greater for

the compressible aerofoil by the factor .Thus, by Gothert 's rule we have:

Compute the aerodynamic coefficients for this t ransformed body for incompressible flow. Theaerodynamic coefficients of the given body at the given compressible flow Mach number aregiven by:

(9.91) The applicat ion of Gothert 's rule is much more complicated than the applicat ion of version I

of the P-G rule. This is because, for finding the behavior of a body with respect to M∞, we haveto calculate for each M∞ at a t ime, whereas by the P-G rule (version I) the complete variat ion isobtained at a t ime. However, only the Gothert rule is exact with the framework of linearizedtheory and the P-G rule is only approximate because of the contradict ing assumptionsinvolved.

Now, we can see some aspects about the pract ical significance of these results. A fairlygood amount of theoret ical and experimental informat ion on the propert ies of classes ofaffinely related profiles in incompressible flow, with variat ions in camber, thickness rat io, andangle of at tack is available. If it is necessary to find the CL of one of these profiles at a finiteMach number M∞, either theoret ically or experimentally, we first find the lift coefficient inincompressible flow of an affinely related profile. The camber, thickness and angle of at tack aresmaller than the corresponding values for the original profile by the factor . Then, bymult iplying this CL for incompressible flow profile by , we find the desired lift coefficient forthe compressible flow.

This method of collect ing data for incompressible flow is cumbersome, since the data isrequired for a large number of thickness rat ios. It would be more convenient in many respectsto know how Mach number affects the performance of a profile of fixed shape. The directproblem, discussed in Subsect ion 9.13.2, yields informat ion of this type.

9.14 Prandtl-Glauert Rule for Supersonic Flow:Versions I and II

In Sect ion 9.13, we have seen the similarity rules for subsonic flows. Now let us examine thesimilarity rules for supersonic flows. We can visualize from our previous discussions on similarityrule for subsonic compressible flows that the factor K1 in the t ransformat ion Equat ion (9.79)should have the following relat ions depending on the flow regime:

Therefore, in general, we can write:

(9.92) However, there is one important difference between the treatment of supersonic flow andsubsonic flow, that is, we cannot find any incompressible flow in the supersonic flow regime.

9.14.1 Subsonic FlowWe know that for subsonic flow the transformat ion relat ions are given by Equat ion (9.79) as:

The transformed equat ion is:

and the condit ion to be sat isfied by this equat ion in order to be ident ical to Equat ion (9.78) is:

For this case the above transformed equat ion becomes Laplace equat ion.

9.14.2 Supersonic FlowThe transformat ion relat ions for supersonic flow are:

where the variables with “prime” are the transformed variables. The aim in writ ing thesetransformat ions is to make the Mach number M∞ in the governing Equat ion (9.77) to vanish.

With the above transformat ion relat ions, the governing equat ion becomes:

For supersonic flow, M∞ > 1, therefore the above equat ion becomes:

By inspect ion of this equat ion, we can see that the Mach number M∞ can be eliminated fromthe above equat ion with:

The equat ion becomes:

(9.93a) Now we must find out as to which supersonic Mach number this flow belongs.

The original form of the governing different ial equat ion for this kind of flow, given by Equat ion(9.77), is:

(9.93b) For Equat ions (9.93a) and (9.93) to be ident ical, it is necessary that:

By following the arguments of P-G rule for subsonic compressible flow, we can show thefollowing results for versions I and II of the Prandt l-Glauert rule for supersonic flow.

9.14.2.1 Analogy Version IFor this case of invariant profile in supersonic flow:

Compute the flow around the given body at . For any other supersonic Mach number, theaerodynamics coefficients are given by:

(9.94a) where Cp, CL and CM are at and Cp ', CL ' and CM ' are at any other supersonic Machnumber.

9.14.2.2 Analogy Version IIHere the requirement is to find a t ransformat ion for the profile, by which we can obtain a body,for which the governing equat ion is Equat ion (9.93a) with exact ly the same pressuredistribut ion as the actual body for which the governing equat ion is Equat ion (9.93b). For this:

The derivat ion of the above two results are left to the reader as an exercise. From the aboveresults, we see that in supersonic flow plays the same role as M∞ = 0 in subsonic flow.

For version II, we can write:

(9.94b)

9.14.2.3 Analogy Version III: Gothert RuleFor any given body, at given Mach number M∞, find the transformed shape by using the rule:

(9.95) where α is the angle of at tack, f and t are the camber and thickness of the given body,respect ively. The primed quant it ies are for the t ransformed body and unprimed ones are forthe actual body.

Compute the aerodynamic coefficients of the t ransformed body for . The aerodynamiccoefficients of the given body at the given Mach number M∞ follow from:

(9.96) We can state the Gothert rule for subsonic and supersonic flows by using a modulus: .

From the discussion on similarity rules for compressible subsonic and supersonic flows, it isclear that , in subsonic flow, there is a ready made linearized solut ion for M∞ = 0. Hence, forsuch cases we can use the Prandt l-Glauert rule. But for supersonic flow the linear theoryequat ions are very simple and, therefore, we can convenient ly use the Gothert rule.

Example 9.1A given profile has, at M∞ = 0.29, the following lift coefficients:

where α is the angle of at tack. Plot the relat ion showing dCL/dα vs. M∞ for the profile for valuesof M∞ up to 1.0.

SolutionAt M∞ = 0.29:

By the Prandt l-Glauert rule:

Therefore:

For any other subsonic Mach number, by the Prandt l-Glauert rule:

Therefore, we have the following variat ion:

9.15 The von Karman Rule for Transonic FlowThe potent ial Equat ion (9.49), for the present case of two-dimensional t ransonic flow, reducesto:

(9.97) Equat ion (9.97) results in a form due to Sprieter (see also Liepmann and Roshko, 1963 [4]) forM∞ ≈ 1, as:

(9.97a) where

(9.97b) and is the similarity pressure coefficient . It follows from Equat ion (9.97a) that the lift anddrag coefficients are given by:

(9.97c)

(9.97d) Equat ions(9.97a), (9.97c) and (9.97d) are valid for local as well as for total values. Sometimes,instead of thickness rat io t/c, ‘fineness rat io’ defined as in Figure 9.12 is used.

Figure 9.12 Wedge at an angle of at tack.

For the wedge shown in Figure 9.12:

The rat io t/c is called the fineness rat io (at angle of at tack = 0).

(9.98) where the ‘plus' sign is for the upper surface and the ‘minus' sign is for the lower surface. Forfinding the local values of Cp, CL and CD, we must use fineness rat io defined by theseequat ions.

9.15.1 Use of Karman RuleIf we know the solut ion for one profile, we can find solut ions for other affinely related profiles.For example, the NACA profiles designated by 8405, 8410, 8415 all have the same distribut ion,same nose radius etc.; only the absolute magnitude of t/c is different. This rule can beextended to t ransonic flow range also. From Figure 9.13, it is seen that in the t ransonic range,the aerodynamic coefficients change very quickly with Mach number, so that the proper valuesto be considered are not M∞, CL, CD and Cp; instead they are χ, , and .

Figure 9.13 The transonic similarity rule.

From the discussion made so far, we can make the following remarks:1. For subsonic and supersonic flows, the governing equat ion is independent of γ, sothat the results from similarity rules can be applied to any gas; but for t ransonic flow, thepotent ial equat ions are not independent of γ. Therefore, the results have to be properlyapplied to different gases, with suitable correct ion for γ, for example, a probe used for air intransonic range can be calibrated for steam.2. For t ransonic flow:

For subsonic flow:

For supersonic flow:

Transonic flow is characterized by the occurrence of shock and boundary layer separat ion.This explains the steep increase in CD at t ransonic range. We should also recall that the shockshould be sufficient ly weak for small perturbat ion. For circular cylinder this theory cannot beapplied, because the perturbat ions are not small.

9.16 Hypersonic SimilarityThe linear theory is not valid at high supersonic Mach numbers, since:

Even slender bodies produce large disturbances in hypersonic flow. The original nonlinearequat ions have to be used for such flows. So, mathematically hypersonic flow is similar totransonic flow. In supersonic flow, slender bodies produce weak shocks and so these can beconsidered as Mach lines.2 But in hypersonic flow, even slender bodies produce strong shocksand, therefore, in hypersonic flow we can no more deal with Mach lines and must deal with theactual shock waves. At high Mach numbers, the Mach angle μ may be of the same order orless than the maximum deflect ion angle θ of the body.

From these considerat ions, the similarity rule can be obtained for hypersonic flow. The Machangle μ is given by the relat ion:

For the present case of flow shown in Figure 9.14:

where θ is the half angle of the wedge in the figure, that is, for hypersonic flow:

(9.99) But in hypersonic flows even for small disturbances, there are shock lines and the angle ofshock is always less than the angle of Mach line. Therefore, in reality the inequality in Equat ion(9.99), obtained with the approximat ion that Mach angle μ is of the same order or less than theflow turning angle θ, has to be modified since the shock angle is always less than μ. In otherwords, it can be stated that M∞θ is greater than some quant ity K, whose numerical value canbe less than unity also.

Figure 9.14 Slender body in hypersonic flow.

It is a common pract ice to express:

(9.100)

(9.101) where K is called the Hypersonic similarity parameter.

Example 9.2For θ = 10 (≈ 0.174 radian), M∞ = 4; the hypersonic similarity parameter K = M∞θ = 0.7. For θ =20 and M∞ = 2:

That is, for a wedge with half-angle 20 , M∞ = 2 should be considered as hypersonic. Thisimplies that M ≥ 5 for hypersonic flow is only a crude limit . For θ = 5 and M∞ = 8:

Thus, a wedge with half-angle 5 in a flow with M∞ = 8 produces shocks as strong as a wedgewith half-angle 20 in a flow with M∞ = 2.

Also, by Equat ion (9.98):

(9.102) Whenever M∞θ is the same for a number of bodies, the flow about them will be dynamicallysimilar, that is, to invest igate the hypersonic flow about a wedge with half-angle 5 and M∞ = 8,

similar, that is, to invest igate the hypersonic flow about a wedge with half-angle 5 and ∞ = 8,we can use a supersonic tunnel with M∞ = 2 and θ = 20 . This is of paramount importance intest ing; of course the two bodies should be affinely related (geometrically similar). Consider twomodels, 1 and 2:

This condit ion for dynamic similarity will be sat isfied only when:

That is, these two condit ions should be sat isfied for dynamic similarity, when there is geometricsimilarity:

(9.103) The total lift and drag coefficients are given by:

(9.104)

(9.105) Equat ions (9.103)– (9.105) give the funct ional dependence of various aerodynamic coefficientsfor hypersonic flow.

A plot like the one shown in Figure 9.15 gives the correct representat ion of the differentparameters. This similarity rule is valid for axially symmetric bodies like rockets and missiles,also.

Figure 9.15 Variat ion of CL/(t/c)2 with α/(t/c).

The transonic and hypersonic similarity rules discussed here are just a few glimpses,highlight ing some of the vital features associated with them. Those who are looking for adeeper understanding of these problems should consult standard books on these topics.

9.17 Three-Dimensional Flow: The Gothert Rule

9.17.1 The General Similarity RuleThe Prandt l-Glauert rule is approximate because it sat isfies the boundary condit ions only onthe axis and not on the contour. But Gothert rule is exact and valid for both two-dimensionaland three-dimensional bodies. The potent ial equat ion is (for M∞ < 1 or > 1):

(9.106) For M∞ < 1, the equat ion is ellipt ic in nature and for M∞ > 1, it is hyperbolic. Here also, we maketransformat ion by which the transformed equat ion does not contain M∞ explicit ly any more.Let:

With the above new variables, Equat ion (9.106) t ransforms into:

M∞ vanishes from the above equat ion for:

(9.107) With Equat ion (9.107), the result ing potent ial flow equat ion for subsonic flow is:

and for supersonic flow:

Again, for subsonic flow, the equat ion is exact ly the same as the Laplace equat ion. Forsupersonic flow, the equat ion is ident ical with the compressible flow equat ion [Equat ion(9.106)] with .

Now:

(9.108a)

(9.108b)

(9.108c)

(9.109) and

(9.110) with the assumption that V∞ = V ' ∞. This assumption really does not impose any restrict ion onthe rule, because in supersonic flow, the velocity itself is not important (that is, V/a is morerelevant than V). Introduct ion of Equat ion (9.108a) into Equat ion (9.110) results in:

that is:

(9.111) The kinematic flow condit ion [Equat ion (9.82a)] states that:

(9.112a)

(9.112b) Combining Equat ions (9.112a) and (9.112b), we get:

since x ' = x and z '/z = K1. But by Equat ion (9.108c); therefore:

that is:

(9.113) Therefore:

(9.114) Equat ion (9.114) is valid (exact ly) at any point on the boundary of the body, as well as in theflow field. Therefore:

(9.115) Equat ion (9.115) is an important equat ion, relat ing the aerodynamic coefficients for the actualand transformed bodies.

9.17.2 Gothert RuleThe aerodynamic coefficients of a body in three-dimensional compressible flow are obtainedas follows.

The geometry of the given body is t ransformed in such a way that its lateral and normaldimensions (both in y and z direct ions) are mult iplied by . If the flow is subsonic, computethe incompressible flow about the t ransformed body; if the flow is supersonic, compute thefield with about the t ransformed body. The aerodynamic coefficients of the given body in

given flow, follow from transformed flow with Equat ion (9.115).Gothert rule can be applied to two-dimensional flows also (stated as version III of the

Prandt l-Glauert rule).It is exact in the framework of linear theory, whereas the Prandt l-Glauert rule is only

approximate. For thicker bodies, when there is doubt about the accuracy with P-G rule, Gothertrule should be used even though it is tedious.

The coefficient of pressure is:

The error involved in the pressure coefficients rat io is:

That is why the P-G rule, though approximate, can be used quite sat isfactorily up to t/c = 15%(because the error is less). Gothert rule is st ill superior and is applicable not only to flow pastbodies but also to flow through ducts where the diameter is small.

9.17.3 Application to Wings of Finite SpanConsider a wing planform transformat ion described here.Planform

(9.116)

(9.117) For subsonic flow, the t ransformat ion decreases A and for supersonic flow, the transformat ionincreases A. Note that ϕ is sweep angle here.ProfileThe profile is given by the relat ions:

(9.118) Thus, for wings (three-dimensional bodies), the Gothert rule is st ill more complicated; we haveto t ransform not only the profile but also the planform, for each M∞. But this is the onlyreasonable method for wing analysis. In subsonic flow, these similarity rules are of greatimportance; but in supersonic flow, they are not that much important because even in two-dimensional subsonic flow, the ellipt ical equat ion is very difficult to solve, but in supersonic flow,the hyperbolic equat ion can be easily solved.

After making the transformat ions with Equat ions (9.116) and (9.118), find CL, CM, etc. for theincompressible case and then the corresponding coefficients for compressible case will bedetermined by the relat ions [Equat ion (9.115)]:

But it is tedious to find the variat ion of Cp, CL, CM with M∞ because for each M∞ we have tomake the above transformat ions.

9.17.4 Application to Bodies of Revolution and FuselageThe general, three-dimensional equat ions can be applied to these shapes. But it is moreconvenient to use polar coordinates for bodies of revolut ion and fuselage.

The potent ial equat ion in cylindrical polar coordinates, for incompressible flow is:

(9.119) where x, r and θ are the axial, radial and angular (circumferent ial) coordinates, respect ively. For

compressible flow, the equat ion is:

(9.120) The transformat ion is:

(9.121) where the primed parameters are the transformed ones. Equat ion (9.120) is independent of M∞for:

From the streamline analogy:

Here again, as in Cartesian coordinates, transform the geometry and then calculate theaerodynamic coefficients for incompressible case and then the values for compressible caseare given by Equat ion (9.115). If f = 0, the only t ransformat ion required will be . Thevariat ions of , and with M∞ are shown in Figures 9.16(a)–9.16(c), respect ively.

In Figure 9.16(a), it is seen that beyond the chain line the results cannot be applied becauseonce the speed of sound is reached locally, there will be shock somewhere and this is certainlya nonlinear effect . Though the plot is for a sphere, which is not a slender body, the results ofGothert rule are quite good (at M∞ = 0.5, the error is only ~5%). For slender bodies, Gothert 'srule applies very well.

Figure 9.16 Results of Gothert 's rule for 3-D subsonic flow.

In Figure 9.16(b), the results for NACA 0012 profile with Aspect Rat io (A ') 1.15 are shown.For those Mach numbers for which locally speed of sound is not reached anywhere on theprofile, Gothert 's rule agrees very well with experimental values. The Prandt l-Glauert rule for A '=∞ shows that for large A ', the dCL/dα obtained is much higher.

The three-dimensional relief effect is shown in Figure 9.16(c). For an infinitely long circularcylinder in a stream of velocity V∞, u max = V∞, but for a sphere u max = 0.5V∞. From the plot , the3-D relief effect increases with increase in M∞. A slender body (small A ') introduces smallerperturbat ions, that is, the disturbances produced by wings are much more as compared tofuselage. This difference in disturbances of wings and fuselage is greater at larger M∞. So,locally, speed of sound is reached first on wings and not on fuselage. That is, we should findout the crit ical Mach number for wings and not for the fuselage, since only the former issignificant. The crit ical Mach number Mcr for the fuselage will be much higher than the Mcr forthe wing.

Comparison of Two-Dimensional Symmetric Body and Axially SymmetricBodyFor an axisymmetric body, in any cross-sect ion the flow will be same. But this will not be so fora two-dimensional body. Also, at any cross-sect ion, the disturbances produced by anaxisymmetric body will be much smaller, that is, the accelerat ion of flow will be much less andhence the drop in the pressure coefficient Cp is much smaller compared to a two-dimensionalbody.

9.17.5 The Prandtl-Glauert RuleThis is only an approximat ion and a greater simplificat ion compared to Gothert 's rule. Here weneed not effect any transformat ion in the z-direct ion at all. That means Equat ion (9.118) is nomore necessary. Only Equat ion (9.116) which gives t ransformat ion to planform alone isnecessary.

General considerat ionsThe P-G rule introduces the concept of affinely related profiles in incompressible flow. Affinelyrelated profiles are those for which, for example, the t/c rat io alone is different and α and f aresame, that is, all the ordinates of the two profiles are related simply by a constant.

Similarly, we can obtain affinely related profiles by changing α alone or f alone. In general,affinely related profiles as shown in Figure 9.17, can be obtained by:

(9.122)

Figure 9.17 Affinely related aerofoils.

We should effect only one of these parameters in Equat ion (9.122), in order to get affinelyrelated profiles. For such profiles, it follows from theory and experiment that :

(9.123) This can be thought as: if α for one wing is K t imes α for the second wing, then the CL, Cp

and CM for the first wing should be correspondingly K t imes larger than those for the secondwing. This is so because of the linearity of lift curve, shown in Figure 9.18.

Figure 9.18 Lift coefficient variat ion with angle of at tack.

These relat ionships hold only for the linear port ion, because of the linearity involved in thetheory.

P-G Rule for Two-Dimensional Flow, using Equations (9.122) and (9.123)We have to use Equat ion (9.122) with (9.118) and, (9.123) with (9.115), and set inEquat ions (9.122) and (9.123). What we have to prescribe now is our postulat ion for P-G ruleversions I and II:Version I:M∞ = 0, for subsonic flow and, therefore,

(9.124) where the prime refers to incompressible case.Version II:

for supersonic flow and

Therefore,

(9.125) where the double prime refers to t ransformed profile.

Application to WingsThe general relat ion between the pressure coefficients of closely related wing profiles[Equat ion (9.115)] is:

where “s” is the semi-span of wing. This t ransformed pressure coefficient rat io corresponds toM∞ = 0 (Version I of P-G rule), for subsonic flow.

For (supersonic flow), by Equat ion (9.125) of Version II, we get:

(9.126)

(9.127)

By Gothert rule [Equat ion (9.115)], we have:

(9.118) By similarity rule for affinely related profiles in incompressible flow [Equat ion (9.122)], if:

then

(9.123) This is an empirical rule. For low speed flows, this can be explained with respect to α. But theseequat ions are only approximate. Actually, for supersonic flow, CL does not depend on t at all. Itdepends only on f and α. We relate the given profile in compressible flow (unprimed) to thetransformed profile (double primed) by:

(9.124a) With Equat ions (9.124a) and (9.118), we find that:

Then the aerodynamic coefficients of the given profile in compressible flow are related tothose of the t ransformed profile (which has the same geometry) in incompressible flow or at

by:

(9.125a) because

Application to Wings of Finite SpanThe Gothert 's rule [Equat ion (9.115)] states that:

and by P-G rule, we have:

(9.126a) Equat ion (9.126a) is only an approximate relat ion. Further:

(9.127a) The P-G rule is only approximate, but the Gothert 's rule, though exact, is very tedious,especially in three-dimensions, because here we have to t ransform the profile also. For P-Grule, only the planform has to be transformed.

From the P-G rule, for three-dimensional wings we obtain a similarity rule in the followingway: if the relat ion:

(9.128) for a wing is known at M∞ = 0 and , then it follows for an arbit rary Mach number fromEquat ions (9.116), (9.117) and (9.126), that :

(9.129a)

(9.129b)

(9.129c) where λ is the taper rat io.

In Equat ion (9.129a): θ means α or f/c or t/c.In Equat ion (9.129b): θ means α or f/c or t/c, but t/c only in subsonic flow.In Equat ion (9.129c): θ means either t/c or f/c.

In Equat ions (9.128) and (9.129), ϕ is the angle of sweep for the wing.

Application to Bodies of RevolutionThe applicat ion of P-G rule to bodies of revolut ion is similar to that for aerofoils (2-D), that is, notransformat ion of the body is necessary. The aerodynamic coefficients in compressible flow arethe same as in incompressible flow or at . Hence, there is no Mach number effect at all andthe results are same as those for slender body theory.

This contradicts the more exact Gothert rule. A closer examinat ion shows that the P-G rulefor bodies of revolut ion is valid only for very slender and extremely pointed (sharp-nosed)bodies. This theory is applied to rockets, very small aspect rat io wings, etc. Of course, wavedrag is influenced by M even for slender bodies. We can use the results of incompressible flowfor calculat ion of pressure distribut ion, etc.

From Figure 9.16(c), it is seen that for very small aspect rat io, the effect of Mach number isvery small, and at A = 0 the Mach number effect vanishes.

9.17.6 The von Karman Rule for Transonic Flow

Application to WingsFor M∞ = 1:

(9.130a)

(9.130b)

(9.130c) Mathematically, these can be derived from the nonlinear different ial equat ion (9.49). Theselaws are also approximately valid in the vicinity of M∞ = 1. The main advantage of thesesimilarity rules is that we have to invest igate the influence of λ, A tan ϕ, Aθ1/3 only and not theinfluence of λ, A, ϕ and θ separately, which is very tedious. Thus, the rules are very importantfor experimental invest igat ions.

Application to Bodies of RevolutionThe pressure distribut ion of a body (unprimed) is related to the pressure distribut ion of anaffinely related body (primed) at M∞ = 1, by the relat ion:

(9.131) where the subscript f stands for fuselage. This rule was derived by von Karman, but later on itwas shown that a correct ion factor should be applied.

9.18 Moving DisturbanceThe presence of a small disturbance is felt throughout the field by means of disturbancewaves traveling at the local velocity of sound relat ive to the medium. Let us examine thepropagat ion of pressure disturbance created by a moving object shown in Figure 9.19. Thepropagat ion of disturbance waves created by an object moving with velocity V = 0, V = a/2, V =a and V > a is shown in Figures 9.19(a), (b), (c), (d), respect ively. In a subsonic flow, thedisturbance waves reach a stat ionary observer before the source of disturbance could reachhim, as shown in Figures 9.19(a) and 9.19(b). But in supersonic flows it takes considerableamount of t ime for an observer to perceive the pressure disturbance, after the source haspassed him. This is one of the fundamental differences between subsonic and supersonicflows. Therefore, in a subsonic flow the streamlines sense the presence of any obstacle in the

flow field and adjust themselves well ahead of the obstacles and flow around it smoothly.

Figure 9.19 Propagat ion of disturbance waves.

But in a supersonic flow, the streamlines feel the obstacle only when they hit it . The obstacleacts as a source and the streamlines deviate at the Mach cone as shown in Figure 9.19(d).That is in a supersonic flow the disturbance due to an obstacle is sudden and the flow behindthe obstacle has to change abrupt ly.

Flow around a wedge shown in Figures 9.20(a) and 9.20(b) illustrate the smooth and abruptchange in flow direct ion for subsonic and supersonic flow, respect ively. For M∞ < 1, the flowdirect ion changes smoothly and the pressure decreases with accelerat ion. For M∞ > 1, there isa sudden change in flow direct ion at the body and the pressure increases downstream of theshock.

In Figure 9.19(d), it is shown that for supersonic mot ion of an object there is a well-definedconical zone in the flow field with the object located at the nose of the cone and thedisturbance created by the moving object is confined only to the field included inside the cone.The flow field zone outside the cone does not even feel the disturbance. For this reason, vonKarman termed the region inside the cone as the zone of action and the region outside thecone as the zone of silence. The lines at which the pressure disturbance is concentrated andwhich generate the cone are called Mach waves or Mach lines. The angle between the Machline and the direct ion of mot ion of the body is called the Mach angle μ. From Figure 9.19(d), wehave:

that is:

(9.132)

Figure 9.20 Flow around a wedge.

From the disturbance waves propagat ion shown in Figure 9.19, we can infer the followingfeatures of the flow regimes:

When the medium is incompressible (M = 0, Figure 9.19(a)) or when the speed of themoving disturbance is negligibly small compared to the local sound speed, the pressurepulse created by the disturbance spreads uniformly in all direct ions.When the disturbance source moves with a subsonic speed (M < 1, Figure 9.19(b)), thepressure disturbance is felt in all direct ions and at all points in space (neglect ing viscousdissipat ion), but the pressure pattern is no longer symmetrical.For sonic velocity (M = 1, Figure 9.19(c)) the pressure pulse is at the boundary betweensubsonic and supersonic flow and the wave front is a plane.For supersonic speeds (M > 1, Figure 9.19(d)) the disturbance wave propagat ionphenomenon is totally different from those at subsonic speeds. All the pressuredisturbances are included in a cone which has the disturbance source at its apex and theeffect of the disturbance is not felt upstream of the disturbance source.

9.18.1 Small DisturbanceWhen the apex angle of wedge δ is vanishingly small, the disturbances will be small and we canconsider these disturbance waves to be ident ical to sound pulses. In such a case, the deviat ionof streamlines will be small and there will be infinitesimally small increase of pressure across theMach cone shown in Figure 9.21.

Figure 9.21 Mach cone.

9.18.2 Finite DisturbanceWhen the wedge angle δ is finite the disturbances introduced are finite, then the wave is notcalled Mach wave but a shock or shock wave (see Figure 9.22). The angle of shock β is alwayssmaller than the Mach angle. The deviat ion of the streamlines is finite and the pressureincrease across a shock wave is finite.

Figure 9.22 Shock wave.

9.19 Normal Shock WavesIn Sect ion 2.13 of Chapter 2, we briefly discussed about the compression and expansionwaves. Now, let us have a closer look at these waves and the flow process across them. Shockis a compression front across which the flow propert ies jump. Shock may also be described ascompression front in a supersonic flow field and the flow process across the front results in anabrupt change in fluid propert ies. In other words, shock is a thin region where large gradients intemperature, pressure and velocity occur, and where the transport phenomena of momentumand energy are important. The thickness of the shocks is comparable to the mean free path ofthe gas molecules in the flow field.

9.19.1 Equations of Motion for a Normal Shock WaveFor a quant itat ive analysis of changes across a normal shock wave, let us consider anadiabat ic, constant-area flow through a nonequilibrium region, as shown in Figure 9.23(a). Letsect ions 1 and 2 be sufficient ly away from the non-equilibrium region so that we can defineflow propert ies at these stat ions, as shown in Figure 9.23(a). Now we can write the equat ionsof mot ion for the flow considered as follows:

Figure 9.23 Flow through a normal shock.

By cont inuity

(9.133) The momentum equat ion is

(9.134) The energy equat ion is

(9.135) Equat ions (9.133)–(9.135) are general –they apply to all gases. Also, there is no restrict ion

on the size or details of the nonequilibrium region as long as the reference sect ions 1 and 2 areoutside of it . The solut ion of these equat ions gives the relat ions that must exist between theflow parameters at these two sect ions.

Since there are no restrict ions on the size or details of the nonequilibrium region, it may beidealized as a vanishingly thin region, as shown in Figure 9.23(b), across which the flowparameters jump. The control sect ions 1 and 2 may also be brought arbit rarily close to the thinregion. Such a compression front across which the flow propert ies change suddenly is called ashock wave. Heat is neither added to nor taken away from the flow as it t raverses the shockwave; hence the flow process across the shock wave is adiabat ic.

In many text books shock is defined as a discont inuity. From our discussions above, the

quest ion obviously arises; is it possible to have a discont inuity in a cont inuum flow field of a realfluid? We should realize that the above considerat ion is only an idealizat ion of the very highgradients of flow propert ies that actually occur in a shock wave, in the t ransit ion from state 1to state 2. These large gradients produce viscous stress and heat t ransfer, that is,nonequilibrium condit ions inside the shock. The processes taking place inside the shock waveitself are extremely complex, and cannot be studied on the basis of equilibriumthermodynamics. Temperature and velocity gradients inside the shock provide heatconduct ion and viscous dissipat ion that render the flow process inside the shock internallyirreversible. In most pract ical applicat ions, primary interest is not generally focused on theinternal mechanism of the shock wave, but on the net changes in fluid propert ies taking placeacross the wave. However, there are situat ions where the detailed informat ion about the flowmechanism inside the shock describing its structure is essent ial for studying pract icalproblems. But since such condit ions occur only in flow regimes like rarefied flow fields, it is notof any interest for the present study. Thus, shock is not a discont inuity but an act ivecont inuum compression front causing sudden changes to the flow propert ies.

9.19.2 The Normal Shock Relations for a Perfect GasFor a calorically perfect gas, we have the equat ion of state, viz.

(9.136) and the enthalpy is given by:

(9.137) Equat ions (9.133) – (9.137) form a set of five equat ions with five unknowns: p2, ρ2, T2, V2 andh2. Hence, they can be solved algebraically. In other words, Equat ions (9.133)– (9.135) are thegeneral equat ions for a normal shock wave and for a perfect gas, it is possible to obtain explicitsolut ions in terms of Mach number, M1, ahead of the shock using Equat ions (9.136) and (9.137)along with Equat ions (9.133)– (9.135), as follows: Dividing Equat ion (9.134) by Equat ion (9.133),we get:

(9.138) Recalling that the speed of sound , Equat ion (9.138) becomes:

(9.139) Now, and in Equat ion (9.139) may be replaced with energy equat ion for a perfect gasas follows.

By energy equat ion, we have:

From the above relat ion, and can be expressed as:

Because the flow process across the shock wave is adiabat ic, a* in the above relat ions for and has the same constant value.

Subst itut ing these relat ions into Equat ion (9.139), we get:

Dividing this equat ion by (V2 − V1), we obtain:

This may be solved to result in:

(9.140) which is called the Prandtl relation.

In terms of the speed rat io M* = V/a*, Equat ion (9.140) can be expressed as:

(9.141) Equat ion (9.141) implies that the velocity change across a normal shock must be fromsupersonic to subsonic and vice versa. But, it will be shown later in this sect ion that only theformer is possible. Hence, the Mach number behind a normal shock is always subsonic. This isa general result , not limited just to a calorically perfect gas.

The relat ion between the characterist ic Mach number M* and actual Mach number M isgiven [Equat ion (2.25) of Rathakrishnan (2010) [1]] as:

(9.142) Using Equat ion (9.142) to replace and in Equat ion (9.141), we get:

(9.143) Equat ion (9.143) shows that, for a perfect gas, the Mach number behind the shock is afunct ion of only the Mach number M1 ahead of the shock. It also shows that when M1 = 1, M2 =1. This is the case of an infinitely weak normal shock, which is ident ical to a Mach wave. It isessent ial to realize that the Mach waves in a supersonic flow field are at an angle μ = sin−1(1/M), which is always less than π/2. In other words, a Mach wave is essent ially an isentropicwave degenerated to a level that the flow across it will not experience any significant changeof property. But, as M1 increases above 1, the normal shock becomes stronger and M2becomes progressively less than 1, and in the limit , as M1→ ∞, M2 approaches a finite minimumvalue, , which for air (at standard condit ions), with γ = 1.4 is 0.378.

The rat io of velocit ies may also be writ ten as:

(9.144) Equat ions (9.142) and (9.144) are useful for the derivat ion of other normal shock relat ions.

From Equat ion (9.133), we can write:

(9.145) To obtain pressure relat ion, consider the momentum Equat ion (9.134):

which, combined with Equat ion (9.133), gives:

Dividing throughout by p1, we get:

Now, recalling , we obtain:

(9.146) Subst itut ing for V2/V1 from Equat ion (9.145), we get:

(9.147) Equat ion (9.147) may also be writ ten as:

(9.148) The rat io (p2 − p1)/p1 = Δp/p1 is called the shock strength.

The state equat ion p = ρRT can be used to get the temperature rat io. With the stateequat ion, we can write:

(9.149) Subst itut ing Equat ions (9.148) and (9.145) into Equat ion (9.149) and rearranging, we get:

(9.150)

The entropy change in terms of pressure and temperature rat ios across the shock can beexpressed as:

From Equat ions (9.148) and (9.150):

(9.151) From Equat ions (9.143), (9.145), (9.148), (9.150) and (9.151), it is obvious that, for a perfect gaswith a given γ, variables M2, ρ2/ρ1, p2/p1, T2/T1 and (s2 − s1) are all funct ions of M1 only. Thisexplains the importance of Mach number in the quant itat ive governance of compressible flows.At this stage, we should realize that the simplicity of the above equat ions arises from the factthat the gas is assumed to be perfect . For high-temperature gas dynamic problems, closedform expressions such as Equat ions (9.143)– (9.150) are generally not possible and the normalshock propert ies must be computed numerically. The results of this sect ion hold reasonablyaccurately up to about M1 = 5 for air at standard condit ions. Beyond Mach 5, the temperaturebehind the normal shock becomes high enough that the specific heats rat io γ is no longer aconstant.

The limit ing case of M1→ ∞ can be considered either as V1→ ∞, where, because of hightemperatures the perfect gas assumption becomes invalid, or as a1 → 0 where, because ofextremely low temperatures the perfect gas assumption becomes invalid. That is, when M1→∞ (either by V1→ ∞ or by a → 0), the perfect gas assumption is not valid. But, it is interest ing toexamine the variat ion of propert ies across the normal shock, for this limit ing case. When M1→∞, we find, for γ = 1.4:

At the other extreme case of an infinitely weak normal shock degenerat ing into a Mach wave,that is, at M1 = 1, Equat ions (9.143), (9.145), (9.148) and (9.150) yield M2 = ρ2/ρ1 = p2/p1 = T2/T1= 1. That is, when M1 = 1, no finite changes occur across the wave.

Equat ion (9.151) just ifies the statement we made earlier in this sect ion: “from Prandt lequat ion, although it is possible for the flow to decelerate from supersonic to subsonic and viceversa across a normal shock wave, only the former is physically feasible.” From Equat ion(9.151), if M1 = 1, then Δs = 0; if M1 < 1, Δs < 0; and if M1 > 1, Δs > 0. Therefore, since it isnecessary that Δs ≥ 0 for a physically possible process, from the second law ofthermodynamics, M1 must be greater than or equal to 1. When M1 is subsonic, the entropyacross the wave decreases, which is impossible. Therefore, the only physically possible flow isM1 > 1, and from the above results we have M2 < 1, ρ2/ρ1 > 1, p2/p1 > 1 and T2/T1 > 1.

The changes in flow propert ies across the shock take place within a very short distance, ofthe order of 10−5 cm. Hence, the velocity and temperature gradients inside the shock structureare very large. These large gradients result in increase of entropy across the shock. Also, thesegradients internal to the shock provide heat conduct ion and viscous dissipat ion that renderthe shock process internally irreversible.

9.20 Change of Total Pressure across a ShockThere is no heat added to or taken away from the flow as it t raverses a shock wave; that is,the flow process across the shock wave is adiabat ic. Therefore, the total temperature remainsthe same ahead of and behind the wave:

(9.152) Now, it is important to note that Equat ion (9.152), valid for a perfect gas, is a special case ofthe more general result that the total enthalpy is constant across a normal shock, as given byEquat ion (9.135). For a stat ionary normal shock, the total enthalpy is always constant across

the wave which, for calorically or thermally perfect gases, t ranslates into a constant totaltemperature across the shock. However, for a chemically react ing gas, the total temperature isnot constant across the shock. Also, if the shock wave is not stationary (that is, for a movingshock), neither the total enthalpy nor the total temperature are constant across the shock wave.

For an adiabat ic process of a perfect gas, we have:

In the above equat ion, all the quant it ies are expressed as stagnat ion quant it ies. It is seen fromthe equat ion that the entropy varies only when there are losses in pressure. It is independentof velocity and hence there is nothing like stagnat ion entropy. Therefore, the entropydifference between states 1 and 2 is expressed, without any reference to the velocity level, as:

(9.153) The exact expression for the rat io of total pressure may be obtained from Equat ions (9.153)and (9.151) as:

(9.154) Equat ion (9.154) is an important and useful equat ion, since it connects the stagnat ionpressures on either side of a normal shock to flow Mach number ahead of the shock. Also, wecan see the usefulness of Equat ion (9.154) from the applicat ion aspect. When a pitot probe isplaced in a supersonic flow facing the flow, there would be a detached shock standing aheadof probe nose and, therefore, the probe measures the total pressure behind that detachedshock. However, the port ion of the shock ahead of a pitot probe mouth can be approximatedas a normal shock. Thus, what a pitot probe facing a supersonic flow measures is the totalpressure p02 behind a normal shock. Knowing the stagnat ion pressure ahead of the shock,which is the pressure in the reservoir, for isentropic flow up to the shock, we can determine theflow Mach number ahead of the shock with Equat ion (9.154).

9.21 Oblique Shock and Expansion WavesThe normal shock wave, a compression front normal to the flow direct ion. However, in a widevariety of physical situat ions, a compression wave inclined at an angle to the flow occurs. Sucha wave is called an oblique shock. Indeed, all naturally occurring shocks in external flows areoblique.

In steady subsonic flows, we generally do not think in terms of wave mot ion. It is usuallymuch simpler to view the mot ion from a frame of reference in which the body is stat ionary andthe fluid flows over it . If the relat ive speed is supersonic, the disturbance waves cannotpropagate ahead of the immediate vicinity of the body and the wave system travels with thebody. Thus, in the reference frame in which the body is stat ionary, the wave system is alsostat ionary; then the correspondence between the wave system and the flow field is direct .

The normal shock wave is a special case of oblique shock waves, with shock angle β = 90 .Also, it can be shown that superposit ion of a uniform velocity, which is normal to the upstreamflow, on the flow field of the normal shock will result in a flow field through an oblique shockwave. This phenomenon will be employed later in this sect ion to get the oblique shockrelat ions. Oblique shocks are usually generated when a supersonic flow is turned into itself.The opposite of this, that is, when a supersonic flow is turned away from itself, results in theformat ion of an expansion fan. These two families of waves play a dominant role in all flowfields involving supersonic velocit ies. Typical flows with oblique shock and expansion fan areillustrated in Figure 9.24.

Figure 9.24 Supersonic flow over compression and expansion corners.

I n Figure 9.24(a), the flow is deflected into itself by the oblique shock formed at the

compression corner, to become parallel to the solid wall downstream of the corner. All thestreamlines are deflected to the same angle θ at the shock, result ing in uniform parallel flowdownstream of shock. The angle θ is referred to as flow deflection angle. Across the shockwave, the Mach number decreases and the pressure, density and temperature increase. Thecorner which turns the flow into itself is called compression or concave corner. In contrast , in anexpansion or convex corner, the flow is turned away from itself through an expansion fan, asillustrated in Figure 9.24(b). All the streamlines are deflected to the same angle θ after theexpansion fan, result ing in uniform parallel flow downstream of the fan. Across the expansionwave, the Mach number increases and the pressure, density and temperature decrease. FromFigure 9.24, it is seen that the flow turns suddenly across the shock and the turning is gradualacross the expansion fan and hence all flow propert ies through the expansion fan changesmoothly, with the except ion of the wall streamline which changes suddenly.

Oblique shock and expansion waves prevail in two-and three-dimensional supersonic flows,in contrast to normal shock waves, which are one-dimensional. In this chapter, we shall focusour at tent ion only on steady, two-dimensional (plane) supersonic flows.

9.21.1 Oblique Shock RelationsThe flow through an oblique shock is illustrated in Figure 9.25(b). The flow through a normalshock (Figure 9.24(a)) has been modified to result in flow through an oblique shock, bysuperimposing a uniform velocity Vy (parallel to the normal shock) on the flow field of thenormal shock (Figure 9.25(a)).

Figure 9.25 Flow through an oblique shock wave.

The resultant velocity upstream of the shock is and is inclined at an angle β = tan−1(Vx1/Vy) to the shock. This angle β is called shock angle. The velocity component Vx2 isalways less than Vx1; therefore, the inclinat ion of the flow ahead of the shock and after theshock are different. The inclinat ion ahead is always more than that behind the shock wave,that is, the flow is turned suddenly at the shock. Because Vx1 is always more than Vx2, theturning of the flow is always towards the shock. The angle θ by which the flow turns towardsthe shock is called flow deflect ion angle and is posit ive as shown in Figure 9.25. The rotat ion ofthe flow field in Figure 9.25(a) by an angle β results in the field shown in Figure 9.25(b), with V1in the horizontal direct ion. The shock in that field inclined at an angle β to the incomingsupersonic flow is called the oblique shock.

The relat ions between the flow parameters upstream and downstream of the flow fieldthrough the oblique shock, illustrated in Figure 9.25(b), can be obtained from the normal shockrelat ions, since the superposit ion of uniform velocity Vy on the normal shock flow field in Figure9.25(a) does not affect the flow parameters (e.g., stat ic pressure) defined for normal shock.The only change is that in the present case the upstream Mach number is:

The component of the upstream Mach number M1 normal to the shock wave is:

(9.155) Thus, replacement of M1 with M1 sin β in normal shock relat ions given by Equat ions (9.145),(9.148), (9.150) and (9.151) results in the following relat ions for an oblique shock:

(9.156)

(9.157)

(9.158)

(9.159) The normal component of Mach number behind the shock Mn2 is given by:

(9.160) From the geometry of the oblique shock flow field shown in Figure 9.25, it is seen that theMach number behind the oblique shock, M2, is related to Mn2 by:

(9.161) In the above equat ions, M2 = V2/a2 and Mn2 = Vx2/a2. The Mach number M2 after a shock canbe obtained by combining Equat ions (9.160) and (9.161).

Numerical values of the oblique shock relat ions for a perfect gas, with γ = 1.4, are presentedin graphical form. The same in tabular form is given in Table 3 of the Appendix ofRathakrishnan (2010) [1].

It is seen from the oblique shock relat ions given by Equat ions (9.155)– (9.159) that the rat ioof thermodynamic variables depends only on the normal component of velocity (M1 sin β)ahead of the shock. But, from normal shock analysis we know that this component must besupersonic, that is, M1 sin β ≥ 1. This requirement imposes the restrict ion on the wave angle βthat it cannot go below a limit ing minimum value for any given M1. At this minimum limit ingvalue of shock angle, the shock gets degenerated to an isentropic wave (also called Machwave) across which the change of flow propert ies become negligibly small. Such a weakisentropic wave is termed Mach wave. The maximum value of β is that for a normal shock, β =π/2. Thus for a given init ial Mach number M1, the possible range of wave angle is:

(9.162) The limit ing values of the wave angle in Equat ion (9.162) are of special significance. Thelimit ing minimum value is is the Mach angle μ and the maximum value corresponds tonormal shock. Thus, the strongest wave possible in a given supersonic flow is the normal shockcorresponding to the given M1. The weakest wave is the Mach wave corresponding to thegiven M1. It is essent ial to note that the shock wave format ion is not mandatory in asupersonic flow. For example, in uniform supersonic streams such as the flow in a supersonicwind tunnel test-sect ion, no shocks are formed when the test-sect ion is empty, whereas theweakest limit ing isentropic waves, namely the Mach waves, are always present in allsupersonic flows. Even in the empty test-sect ion of a supersonic tunnel the Mach waves arepresent. But we know that the waves in a supersonic flow are due to perturbat ions in the flowfield. Therefore, it is natural to ask, “in an undisturbed uniform supersonic flow why shouldthere be Mach waves present?” The answer to this quest ion is the following. In a uniformsupersonic flow such as that in a wind tunnel test-sect ion, if the test-sect ion walls areabsolutely smooth there will not be any Mach wave present in the flow. However, absolutesmooth surface is only a theoret ical assumption. For instance, even surfaces such as that of agood quality Schlieren mirror has a finish of only about λ/20, where λ is the wavelength of light .Thus, any pract ical surface is with some roughness and not absolutely smooth. Therefore, anysupersonic flow field generated by a pract ical device is bound to possess Mach waves. Indeed,the size of the gas molecules are enough to cause Mach wave generat ion. Therefore, even ina free supersonic flow without any solid confinement Mach waves will be present.

An important feature to be inferred here is that the Mach waves, like characterist ics will berunning to the left and right in the flow field. Because of this the Mach waves of oppositefamilies prevailing in the flow field cross each other. But being the weakest degenerat ion ofwaves, the Mach waves would cont inue to propagate as linear waves even after passingthrough a number of Mach waves. In other words, the Mach waves would cont inue to be simple

waves even after intersect ing other Mach waves. Because of this nature of the Mach waves, aflow region traversed by the Mach waves is simple throughout.

9.21.2 Relation between β and θIt is seen from Equat ion (9.161) that for determining M2 the flow deflect ion angle θ must beknown. Further, for each value of shock angle β at a given M1 there is a corresponding flowturning angle θ. Therefore, θ can also be expressed as a unique funct ion of M1 and β. FromFigure 9.25, we have:

(9.163)

(9.164) Combining Equat ions (9.163) and (9.164), we get:

(9.165) By cont inuity:

Now, subst itut ing for ρ1/ρ2 from Equat ion (9.156), we get:

(9.166) Equat ion (9.166) is an implicit relat ion between θ and β, for a given M1. With sometrigonometric manipulat ion, this expression can be rewrit ten to show the dependence of θ onMach number M1 and shock angle β, as:

(9.167) Equat ion (9.167) is called the θ–β–M relat ion. This relat ion is important for the analysis ofoblique shocks. The expression on the right-hand side of Equat ion (9.167) becomes zero at β= π/2 and , which are the limit ing values of β, defined in Equat ion (9.162). The deflect ionangle θ is posit ive in this range and must therefore have a maximum value. The resultsobtained from Equat ion (9.167) are plot ted in Figure 9.26, for γ = 1.4. From the plot of θ–β–M(Figure 9.26) curves, the following observat ions can be made:

1. For any given supersonic Mach number M1, there is a maximum value of θ. Therefore, ata given M1, if θ > θ max, then no solut ion is possible for a straight oblique shock wave. Insuch cases, the shock will be curved and detached, as shown in Figure 9.27.2. When θ < θ max, there are two possible solut ions, for each value of θ and M, having twodifferent wave angles. The larger value of β is called the strong shock solution and thesmaller value of β is referred to as the weak shock solution. For strong shock solut ion, theflow behind the shock becomes subsonic. For weak shock solut ion, the flow behind theoblique shock remains supersonic, except for a small range of θ slight ly smaller than θ max,the zone bounded by the M2 = 1 curve and θ = θ max curve shown in Figure 9.26.

3. If θ = 0, then β = π/2, giving rise to a normal shock, or β decreases to the limit ing value μ,that is, shock disappears and only Mach waves prevail in the flow field. That is, when theflow turning angle θ is zero, the following two solut ions are possible for the shock angle β,for a given M1. (a) Either β = π/2 giving rise to a normal shock which does not cause anyflow deflect ion, but would decelerate the flow to subsonic level, or (b) β = sin −1(1/M1) = μcorresponding a Mach wave, which even though inclined to the upstream flow, would notcause any flow deflect ion, being the limit ing case of the weakest isentropic wave for agiven M1.

Figure 9.26 Oblique shock solut ion.

Figure 9.27 Detached shocks.

A very useful form of θ–β–M relat ion can be obtained by rearranging Equat ion (9.166) in thefollowing manner: Dividing the numerator and denominator of the right-hand side of Equat ion(9.166) by and solving, we obtain:

This can be simplified further to result in:

(9.168) For small deflect ion angles θ, Equat ion (9.168) may be approximated as:

(9.169) If M1 is very large, then β < 1, but M1β > 1 and Equat ion (9.169) reduces to:

(9.170) It is important to note that oblique shocks are essent ially compression fronts across which

the flow decelerates and the stat ic pressure, stat ic temperature and stat ic density jump tohigher values. If the decelerat ion is such that the Mach number behind the shock cont inues tobe greater than unity, the shock is termed weak oblique shock. If the downstream Machnumber becomes less than unity then the shock is called strong oblique shock. It is essent ial tonote that only weak oblique shocks are usually formed in any pract ical flow and it calls forspecial arrangement to generate strong oblique shocks. One such situat ion where strongoblique shocks are generated with special arrangements is the engine intakes of supersonicflight vehicles, where the engine has provision to control its backpressure. When thebackpressure is increased to an appropriate value, the oblique shock at the engine inlet wouldbecome a strong shock and decelerate the supersonic flow passing through it to subsoniclevel.

9.21.3 Supersonic Flow over a WedgeFrom studies on inviscid flows, we know that any streamline can be regarded as a solidboundary. In our present study, we treat the supersonic flow as inviscid and, therefore, herealso the streamlines can be assumed as solid boundaries. Thus the oblique shock flow results,already described, can be used for solving pract ical problems like supersonic flow over acompression corner, as shown in Figure 9.28. For any given values of M1 and θ, the values ofM2 and β can be determined from oblique shock charts or table (oblique shock charts and tableare given in the Appendix of Rathakrishnan (2010)).

Figure 9.28 Supersonic flow over a compression corner.

In a similar fashion, problems like supersonic flow over symmetrical and unsymmetricalwedges (Figure 9.29) and so on also can be solved with oblique shock relat ions, assuming thesolid surfaces of the objects as streamlines in accordance with nonviscous (or inviscid) flowtheory.

Figure 9.29 Flow past (a) symmetrical and (b) unsymmetrical wedges.

In Figure 9.29(b), the flow on each side of the wedge is determined only by the inclinat ion ofthe surface on that side. If the shocks are at tached to the nose, the upper and lower surfacesare independent and there is no influence of wedge on the flow upstream of the shock waves.

In our discussion on shock angle β and flow turning angle θ, we have seen that when θdecreases to zero, β decreases to the limit ing value μ giving rise to Mach waves in the

supersonic flow field (see Figure 9.30(b)), which is given from Equat ion (9.168) as:

(9.171)

Figure 9.30 Waves in a supersonic stream.

Also, the pressure, temperature and density jump across the shock (p2 − p1, T2 − T1 and ρ2 −ρ1) given by Equat ions (9.156)– (9.158) become zero. There is, in fact , no finite disturbance inthe flow. The point P in Figure 9.30(b) may be any point in the flow field. Then the angle μ issimply a characterist ic angle associated with the Mach number M by the relat ion:

(9.172) This is called Mach angle-Mach number relation. These lines which may be drawn at any pointin the flow field with inclinat ion μ are called Mach lines or Mach waves. It is essent ial tounderstand the difference between the Mach waves and Mach lines. Mach waves are theweakest isentropic waves in a supersonic flow field and the flow through them will experienceonly negligible changes of flow propert ies. Thus, a flow traversed by the Mach waves do notexperience change of Mach number. Whereas the Mach lines, even though are weak isentropicwaves will cause small but finite changes to the propert ies of a flow passing through them. Inuniform supersonic flows, the Mach waves and Mach lines are linear and inclined at an anglegiven by . But in nonuniform supersonic flows the flow Mach number M varies from point topoint and hence the Mach angle μ, being a funct ion of the flow Mach number, varies with Mand the Mach lines are curved.

In the flow field at any point P (Figure 9.30(c)), there are always two lines which are inclinedat angle μ and intersect the streamline, as shown in Figure 9.30(c). In a three-dimensional flow,the Mach wave is in the form of a conical surface, with vertex at P. Thus, a two-dimensionalflow of supersonic stream is always associated with two families of Mach lines. These arerepresented with plus and minus sign. In Figure 9.30(c), the Mach lines with ‘+’ sign run to theright of the streamline when viewed through the flow direct ion and those lines with “−” sign runto the left . These Mach lines which introduce an infinitesimal, but finite change to flowpropert ies when a flow passes through them are also referred to as characteristics, which arenot physical unlike the Mach lines and Mach waves. But the mathematical concept ofcharacterist ics (taken as ident ical to the Mach lines), even though not physical forms the basisfor the numerical method termed method of characteristics, used to design contoured nozzlesto generate uniform and unidirect ional supersonic flows.

At this stage it is essent ial to note the difference between the Mach waves, characterist icsand expansion waves. Even though all these are isentropic waves, there is a dist inct differencebetween them. Mach waves are weak isentropic waves across which the flow experiencesinsignificant change in its propert ies. Whereas, the expansion waves and characterist ics areisentropic waves which introduce small, but finite property changes to a flow passing them.Thus, even though we loosely state that the Mach lines and Mach waves are isentropic wavesin a supersonic flow, inclined at angle μ to the freestream direct ion, in reality they are dist inct lydifferent. Mach waves are the weakest degenerat ion of isentropic waves to the limit ing case ofzero strength that a flow across which will not experience any change of property. Whereas, aMach line is a weak isentropic wave in a supersonic flow field, causing small but finite change ofpropert ies to the flow passing through it .

The characterist ic lines play an important role in the compression and expansion processesin the sense that it is only through these lines that it is possible to retard or accelerate asupersonic flow isentropically. Also, this concept will be employed in designing supersonicnozzles with Method of Characterist ics.

9.21.4 Weak Oblique Shocks

We have seen that the compression of supersonic flow without entropy increase is possibleonly through the Mach lines. In the present discussion on weak shocks also, it will be shownthat these weak shocks, which result when the flow deflect ion angle θ is small and Machnumber downstream of shock M2 > 1, can also compress the flow with entropy increase almostclose to zero. It is important to note that, when we discussed about flow through obliqueshocks, we considered the shock as weak when the downstream Mach number M2 issupersonic (even though less than the upstream Mach number M1). When the flow traversedby an oblique shock becomes subsonic (that is, M2 < 1), the shock is termed strong. But whenthe flow turning θ caused by a weak oblique shock is very small, then the weak shock assumesa special significance. This kind of weak shock with both decrease of flow Mach number (M1 −M2) and flow turning angle θ, which is small, can be regarded as isentropic compression waves.

For small values of θ, the oblique shock relat ions reduce to very simple forms. For this case:

Therefore, Equat ion (9.168) simplifies to:

Also, M2 > 1 for weak oblique shocks. Therefore, we may approximate this weak shock withboth (M1 − M2) and θ extremely small as a Mach line. Thus, the shock angle β can be regardedas almost equal to the Mach angle μ. With this approximat ion, we can express tan β as follows:

Subst itut ing for tan β in preceding equat ion, we get:

(9.173) Equat ion (9.173) is considered to be the basic relat ion for obtaining all other appropriateexpressions for weak oblique shocks since all oblique shock relat ions depend on M1 sin β,which is the component of upstream Mach number normal to the shock.

It is seen from Equat ions (9.157) and (9.173) that the pressure change across a shock ,termed the shock strength can be easily expressed as:

(9.174) Equat ion (9.174) shows that the strength of the shock wave is proport ional to the flowdeflect ion angle θ.

Similarly, it can be shown that the changes in density and temperature are also proport ionalt o θ. But the change in entropy, on the other hand is proport ional to the third power of shockstrength as shown below. By Equat ion (9.159), we have:

(9.175) where [Note that for weak oblique shocks under considerat ion, that is, for weak obliqueshocks with (M1 − M2) < 1 and θ very small, is approximated as .] For values of M1 closeto unity, m is small and the terms within the parentheses are like 1 + ε, with ε < 1. Expandingthe terms as logarithmic series, we get:

or

(9.176) Because the entropy cannot decrease in an adiabat ic flow, Equat ion (9.176) st ipulates that M1≥ 1. Thus, the increase in entropy is of third order in . This may be writ ten in terms of shockstrength, Δp/p, as:

(9.176) But by Equat ion (9.174), the shock strength is proport ional to θ and hence:

(9.177) Thus, a small but finite change of pressure across a weak oblique shock, for which there arecorresponding first-order changes of density and temperature, gives only a third-order changeof entropy, that is, a weak shock produces a nearly isentropic change of state.

Now, let the wave angle β for the weak shock be different from the Mach angle μ by a smallangle ε. That is:

where ε < μ. Therefore, sin β = sin (μ + ε) = sin μ + ε cos μ. Also, sin μ = 1/M1 and . Thus:

(9.178) or

(9.179) From Equat ions (9.173) and (9.179), we obtain:

(9.180) That is, for a finite flow deflect ion angle θ, the direct ion of weak oblique shock wave differsfrom the Mach wave direct ion μ by an amount ε, which is of the same order as θ.

9.21.5 Supersonic CompressionCompressions in a supersonic flow are not usually isentropic. Generally, they take placethrough a shock wave and hence are nonisentropic. But there are certain cases, for which thecompression process can be regarded isentropic. A compression process which can be treatedas isentropic is illustrated in Figure 9.31, where the turning of the flow is achieved through largenumber of weak oblique shocks. These kinds of compression through a large number of weakcompression waves is termed continuous compression. These kinds of corners are calledcont inuous compression corners. Thus, the geometry of the corner should have cont inuoussmooth turning to generate large number of weak (isentropic) compression waves.

Figure 9.31 Smooth cont inuous compression.

The weak oblique shocks divide the field near the wall into segments of uniform flow. Awayfrom the wall the weak shocks might coalesce and form a strong shock as illustrated in Figure9.31. We have seen that the entropy increase across a weak wave is of the order of thirdpower of deflect ion angle θ. Let the flow turning through an angle, shown in Figure 9.31, betaking place through n weak compression waves, each wave turning the flow by an angle Δθ.The overall entropy change for this compression process is:

Thus, if the compression is achieved through a large number of weak compression waves, theentropy increase can be reduced to a very large extent, as compared to a single shock causingthe same net deflect ion. When Δθ is made vanishingly small, a smooth cont inuous turning ofthe flow as shown in Figure 9.31 is achieved. The entropy increase associated with such acont inuous smooth compression process is vanishingly small, that is, the compression can betreated as isentropic.

At this stage it is natural to ask, whether this kind of isentropic compression is only oftheoret ical interest or it is used in pract ical devices too? The answer to this quest ion is that itis used in pract ical devices too. For example, in the gas turbine engines used to propelsupersonic aircraft such as fighters, the freestream supersonic air stream entering the engineintake needs to be decelerated to incompressible Mach numbers (of the order of 0.2) beforereaching the combust ion chamber, because with the present technology cont inuous andstable combust ion is possible only at low incompressible Mach numbers. This can be achieved

by a single normal shock or even with a strong oblique shock to decelerate the supersonicstream to a subsonic Mach number and then the subsonic stream can be decelerated furtherin a diffuser to reach the required incompressible Mach number before entering thecombust ion chamber. But both these decelerat ions will result in a large increase of entropy andthe associated large pressure loss. This kind of large increase of entropy is desirable for anefficient mixing of fuel and air in the combust ion chamber, but the severe pressure loss withthe nonisentropic compression through the shocks is undesirable. We know that the engine isused to generate thrust by react ion. The momentum thrust produced by an engine is:

where is the mass flow rate of the combust ion products of the fuel-air mixture burnt in thecombust ion chamber, expanded through the nozzle of the engine, and Vj is the flow velocity atthe nozzle exit . By Bernoulli principle it is known that a large velocity Vj can be generated byexpanding a gas at high stagnat ion pressure p0. Thus, the aim of the process through theengine is to achieve high p0. If possible we can use a compressor to achieve the desired levelof p0. But carrying a compressor in a gas turbine engine is not a pract ically possible solut ion,mainly due to the weight penalty and the need for addit ional source of energy to run thecompressor. Therefore, as an alternat ive, the high pressure required is achieved throughcombust ion where liberat ion of thermal energy by burning a fuel–air mixture results in a largeincrease of total temperature T0 and the associated increase of total pressure p0. Now, we willnot ice an interest ing point if we keenly observe the process involved. The vehicle is flying at asupersonic Mach number. Because of the skin frict ion, shock and expansion waves around thevehicle and other drag producing causes the vehicle encounters drag. This drag has to becompensated with thrust to maintain the supersonic flight speed. Thus, the basic work of theengine is to supply the required momentum to compensate the momentum loss due to thedrag. In other words, basically the loss caused by the drag can be viewed as loss of totalpressure p0. Therefore, the engine must compensate the pressure loss in order to maintain theconstant p0 required for the supersonic flight at the given alt itude. Instead of adding thestagnat ion pressure equivalent to compensate for the pressure loss due to drag, we are doingthe same thing in an indirect manner. This is done through combust ion. For performingcombust ion, the supersonic air entering the engine is decelerated to low incompressible Machnumber, fuel is mixed with the air and combust ion is performed at such a low Mach number toincrease p0 through the increase of stagnat ion temperature T0. The combust ion products atlow-Mach number is accelerated through the engine nozzle to achieve the required jet velocityat the nozzle exit . In the decelerat ion process through shock/shocks at the engine intake,considerable total pressure is lost . Therefore, it would be appropriate and beneficial if the fuelis added to the air entering the engine with supersonic speed and the combust ion is performedat the same freestream supersonic Mach number. But even though this is the most suitableand efficient situat ion, we are not in a posit ion to do so. This is because the technology forperforming stable combust ion at supersonic Mach number is not yet established. Manyresearch groups in various countries are working on establishing combust ion at supersonicMach numbers. Indeed, stable combust ion at Mach number around 2 is reported by fewadvanced countries, such as USA, China, Britain, France and Japan. Once the technology forsupersonic combust ion is established, the pressure lost in decelerat ing the supersonic airstream to the incompressible Mach number to enable combust ion with the present technologycan be eliminated to a large extent. This will result in a significant increase of the engineefficiency. In other words, the pressure loss associated with the decelerat ion of supersonic orhypersonic flow entering the engine to the required incompressible Mach number for stablecombust ion with the present technology can be completely eliminated if technology isdeveloped to perform stable combust ion at supersonic/hypersonic Mach numbers.

9.21.6 Supersonic Expansion by Turning

Consider the turning of a two-dimensional supersonic flow through a finite angle at a convexcorner, as illustrated in Figure 9.32. Let us assume that the flow is turned by an oblique shockat the corner, as shown in the figure.

Figure 9.32 Supersonic flow over a convex corner.

The flow turning shown in Figure 9.32 is possible only when the normal component ofvelocity V2n after the shock is greater than the normal component V1n ahead of the shock,since V1t and V2t on either side of the shock must be equal. Although this would sat isfy theequat ions of mot ion, it would lead to a decrease of entropy across the shock. Therefore, thisturning process is not physically possible. From the geometry of the flow shown in Figure 9.32,it follows that V2n must be greater than V1n. The normal momentum equat ion yields:

Combining this with cont inuity equat ion:

we obtain:

Because V2n > V1n, it follows that the pressure downstream of the corner should be less thanthe pressure upstream of the corner (p2 < p1). For this, the flow should pass through anexpansion fan at the corner. Thus, the wave at the convex corner must be an expansion fan,causing the flow to accelerate. In other words, the shock wave shown at the convex corner inFigure 9.32 is a physically impossible solut ion.

In an expansion process, the Mach lines are divergent, as shown in Figure 9.33 and,consequent ly, there is a tendency to decrease the pressure, density and temperature of theflow passing through them. In other words, an expansion is isentropic throughout.

Figure 9.33 Centered and cont inuous expansion processes.

It is essent ial to note that the statement “expansion is isentropic throughout” is not t ruealways. To gain an insight into the expansion process, let us examine the centered andcont inuous expansion processes illustrated in Figures 9.33(a) and 9.33(b). We know that theexpansion rays in an expansion fan are isentropic waves across which the change of pressure,temperature, density and Mach number are small but finite. But when such small changescoalesce they can give rise to a large change. One such point where such a large change offlow propert ies occurs due to the amalgamation of the effect due to a large number ofisentropic expansion waves is point P, which is the vertex of the centered expansion fan inFigure 9.33(a). As illustrated in Figures 9.33(a), the pressure at the wall suddenly drops from p1t o p2 at the vertex of the expansion fan. Similarly, the temperature and density also dropsuddenly at point P. The Mach number at P suddenly decreases from M1 to M2. The entropychange across the vertex of the expansion fan is:

It is seen that entropy change associated with the expansion process at point p is finite. Thus,the expansion process at point P is nonisentropic. Therefore, it is essent ial to realize that acentered expansion process is isentropic everywhere except at the vertex of the expansionfan, where it is nonisentropic.

But for the cont inuous expansion illustrated in Figure 9.33(b), there is no sudden change offlow propert ies. Even at the wall surface the propert ies change gradually as shown in thefigure, due to the absence of any point such as P in Figure 9.33(a), where all the expansionrays are concentrated. Therefore, the cont inuous expansion is isentropic everywhere.

The expansion at a corner (Figure 9.33(a)) occurs through a centered wave, defined by a“fan” of straight expansion lines. This centered wave, also called a Prandt l-Meyer expansionfan, is the counterpart , for a convex corner, of the oblique shock at a concave corner.

A typical expansion over a cont inuous convex turn is shown in Figure 9.33(b). Since the flowis isentropic, it is reversible.

9.21.7 The Prandtl-Meyer FunctionIt is known from basic studies on fluid flows that a flow which preserves its own geometry inspace or t ime or both is called a self-similar flow. In the simplest cases of flows, such mot ionsare described by a single independent variable, referred to as similarity variable. The Prandt l-Meyer funct ion is such a similarity variable.

The Prandt l-Meyer funct ion in terms of the Mach number M1 just upstream of the expansionfan can be writ ten as:

(9.181) Equat ion (9.181), expressing the Prandt l-Meyer funct ion ν in terms of the Mach number, is avery important result of supersonic flow. From this relat ion, it is seen that for a given M1, thereis a fixed ν. For a detailed discussion about Prandt l-Meyer expansion process seeRathakrishnan (2010) [1].

9.21.8 Shock-Expansion TheoryThe shock and expansion waves discussed in this chapter are the basis for analyzing largenumber of two-dimensional, supersonic flow problems by simply “patching” togetherappropriate combinat ions of two or more solut ions. That is, the aerodynamic forces act ing on abody present in a supersonic flow are governed by the shock and expansion waves formed atthe surface of the body. This can be easily seen from the basic fact that the aerodynamicforces act ing on a body depend on the pressure distribut ion around it and in supersonic flow,the pressure distribut ion over an object depends on the wave pattern on it , as shown in Figure9.34.

Figure 9.34 Wave pattern over objects.

Consider the two-dimensional diamond aerofoil kept at zero angle of at tack in a uniformsupersonic flow, as shown in Figure 9.34(a). The supersonic flow at M1 is first compressed anddeflected through an angle ε by the oblique shock wave at the leading edge, forcing the flow totravel parallel to the wedge surface. At the shoulder located at mid-chord, the flow is expandedthrough an angle 2ε by the expansion fan. At the t railing edge, the flow is again deflectedthrough an angle ε, in order to bring it back to the original direct ion. Therefore, the surfacepressures on the wedge segments ahead and after the shoulder, will be at a constant levelover each segment for supersonic flow, according to oblique shock and the Prandt l-Meyerexpansion theory.

On the diamond aerofoil, at zero angle of at tack, the lift is zero because the pressuredistribut ions on the top and bottom surfaces are the same. Therefore, the only aerodynamicforce act ing on the diamond aerofoil is due to the higher-pressure on the forward face andlower-pressure on the rearward face. The drag per unit span is given by:

that is:

(9.182) Equat ion (9.182) gives an expression for drag experienced by a two-dimensional diamondaerofoil, kept at zero angle of at tack in an inviscid flow. This is in contrast with the familiar

result from studies on subsonic flow that, for two-dimensional inviscid flow over a wing ofinfinite span at a subsonic velocity, the drag force act ing on the wing is zero –a theoret icalresult called d’Alembert 's paradox. In contrast with this, for supersonic flow, drag exists even inthe idealized, nonviscous fluid. This new component of drag encountered when the flow issupersonic is called wave drag, and is fundamentally different from the skin-frict ion drag andseparat ion drag which are associated with boundary layer in a viscous fluid. The wave drag isrelated to loss of total pressure and increase of entropy across the oblique shock wavesgenerated by the aerofoil.

For the flat plate at an angle of at tack α0 in a uniform supersonic flow, shown in Figure9.34(c), from the uniform pressure on the top and bottom sides, the lift and drag are computedvery easily, with the following equat ions:

(9.183)

where c is the chord.

Example 9.3A flat plate is kept at 15 angle of at tack to a Mach 2.4 air stream, as shown in Figure 9.35.Solve the flow field around the plate and determine the inclinat ion of slipstream to thefreestream direct ion using shock-expansion theory.

Figure 9.35 A flat plate in supersonic flow.

SolutionUsing the shock and expansion wave propert ies, Table 9.1 can be formed.

T able 9.1

Table 9.1 lists the flow propert ies around the flat plate. Slip-surface inclinat ion relat ive tofreestream is negligibly small. The velocity jump across the slip-surface is found to be 1 m/s.

9.22 Thin Aerofoil TheoryWe saw that the shock-expansion theory gives a simple method for comput ing lift and dragact ing over a body kept in a supersonic stream. This theory is applicable as long as the shocksare at tached. This theory may be further simplified by approximat ing it by using theapproximate relat ions for the weak shocks and expansion, when the aerofoil is thin and is keptat a small angle of at tack, that is, if the flow inclinat ions are small. This approximat ion will resultin simple analyt ical epressions for lift and drag.

At this stage, we may have a doubt about the difference between shock-expansion theoryand thin aerofoil theory. The answer to this doubt is the following:

“In shock-expansion theory, the shock is essentially a non-isentropic wave causing a finiteincrease of entropy. Thus, the total pressure of the flow decreases across the shock. But inthin aerofoil theory even the shock is regarded as an isentropic compression wave.Therefore, the flow across this compression wave is assumed to be isentropic. Thus thepressure loss across the compression wave is assumed to be negligibly small.”From our studies on weak oblique shocks, we know that the basic approximate expression

[Equat ion (9.174)] for calculat ing pressure change across a weak shock is:

Because the wave is weak, the pressure p behind the shock will not be significant ly differentfrom p1, nor will the Mach number M behind the shock be appreciably different from thefreestream Mach number M1. Therefore, we can express the above relat ion for pressurechange across a weak shock, without any reference to the freestream state (that is, withoutsubscript 1 to the pressure and Mach number) as:

Now, assuming all direct ion changes to the freestream direct ion to be zero and freestreampressure to be p1, we can write:

where θ is the local flow inclinat ion relat ive to the freestream direct ion.The pressure coefficient Cp is defined as:

where p is the local stat ic pressure and p1 and q1 are the freestream stat ic pressure anddynamic pressure, respect ively. In terms of freestream Mach number M1, the pressurecoefficient Cp can be expressed as:

Subst itut ing the expression for (p − p1)/p1 in terms of θ and M1, we get:

(9.184) The above equat ion, which states that the pressure coefficient is proportional to the local flowdirection, is the basic relat ion for thin aerofoil theory.

9.22.1 Application of Thin Aerofoil TheoryApplying the thin aerofoil theory relat ion, Equat ion (9.184), for the flat plate shown in Figure9.34(c) at a small angle of at tack α0, the Cp on the upper and lower surfaces of the plate canbe expressed as:

(9.185) where the minus sign is for Cp on the upper surface and the plus sign is for Cp on the lowersurface. The lift and drag coefficients are respect ively given by:

In the above expressions for CL and CD, cos α0 ≈ 1 and sin α0 ≈ α0, since α0 is small and thesubscripts l and u refer to the lower and upper surfaces, respect ively and c is the chord.Therefore:

Using Equat ion (9.185), the CL and CD of the flat plate at a small angle of at tack may beexpressed as:

(9.186)

Now, consider the diamond sect ion aerofoil shown in Figure 9.34(a), with nose angle 2ε, atzero angle of at tack. The pressure coefficient Cp on the front and rear faces are given by:

where the + sign is for the front face where the pressure p2 is higher than p1 and the − sign isfor the rear face with pressure p3 less than p1. This can be rewrit ten in terms of pressuredifference to give:

Therefore, the drag is given by:

where q1 is the freestream dynamic pressure and c is the chord of the aerofoil.In terms of the drag coefficient , the above drag equat ion becomes:

(9.187a) or

(9.187b) In the above two applicat ions, the thin aerofoil theory was used for specific profiles to getexpressions for CL and CD. A general result applicable to any thin aerofoil may be obtained asfollows. Consider a cambered aerofoil with finite thickness at a small angle of at tack t reated bylinear resolut ion into three components, each of which contribut ing to lift and drag, as shown inFigure 9.36.

Figure 9.36 Linear resolut ion of aerofoil into angle of at tack, camber and thickness.

By thin aerofoil theory, the Cp on the upper and lower surfaces are obtained as:

(9.188)

where yu and yl are the upper and lower profiles of the aerofoil. The profile may be resolvedinto a symmetrical thickness distribut ion h(x) and a camber line of zero thickness yc(x). Thus,we have:

(9.189a)

(9.189b) where α(x) = α0 + αc(x) is the local angle of at tack of the camber line, and α0 is the angle ofat tack of the freestream and αc is the angle at tack due to the camber. The lift and drag aregiven by:

(9.190a)

(9.190b) Subst itut ing Equat ions (9.188) and (9.189) into Equat ions (9.190a) and (9.190b), we get:

The integrals may be replaced by average values, for example:

Also, not ing that by definit ion , we get:

Similarly:

Using the above averages in the lift and drag expressions, we obtain the lift and dragcoefficients as:

(9.191a)

(9.191b) Equat ions (9.191) give the general expressions for lift and drag coefficients of a thin aerofoil in

a supersonic flow. In thin aerofoil theory, the drag is split into drag due to lift, drag due tocamber and drag due to thickness, as given by Equat ion (9.191b). But the lift coefficientdepends only on the mean angle of at tack.

Example 9.4A symmetric diamond aerofoil of sides 1 m and maximum thickness 150 mm is in a Mach 1.6 airstream at zero angle of at tack. Determine the drag coefficient using (a) shock-expansiontheory and (b) thin aerofoil theory. Also, est imate the percentage error involved in assumingthe aerofoil as thin.

SolutionGiven, l = 1 m, t = 0.15 m, M1 = 1.6, α = 0 , p1 = 50 kPa.The aerofoil and the waves over that are as shown in Figure 9.37.

Figure 9.37 A symmetric diamond aerofoil in a supersonic flow.

The semi-angle at the nose is:

(a) For M1 = 1.6 and θ = 4 . 3 , from oblique shock chart 1:

Therefore:

For M1n = 1.11, from normal shock table:

Therefore:

For M2 = 1.41, from isentropic table:

The flow from zone 2 to zone 3 is expanded by 2θ = 8 . 6 . Therefore:

For ν3 = 17 . 876 , from isentropic table, .The drag per unit span of the wing, by Equat ion (9.182), is:

The drag coefficient is:

where S = c × 1 = c is the planform area per unit span. The dynamic pressure q1 can beexpressed as:

Therefore:

(b) The drag coefficient given by thin aerofoil theory, Equat ion (9.187b), is:

The thin aerofoil theory underest imates the drag. The error commit ted in assuming the aerofoilas thin is:

9.23 Two-Dimensional Compressible FlowsThe equat ions of mot ion in terms of velocity potent ial for steady, irrotat ional isentropic mot ion,as derived in Sect ion 9.7, turn out to be nonlinear part ial different ial equat ions. Although theequat ions were derived somewhat easily, exact solut ions of these equat ions for part icular flowproblems often involve tedious mathematical procedures; in many cases, solut ions are notpossible. To solve this problem, the following two courses of act ion seem to be open:

1. Find exact solut ions for a simplified problem with the hope of obtaining a qualitat iveunderstanding of the nature of other flow patterns for which solut ions are not available.2. Find simple, though approximate, solut ions suitable for pract ical applicat ions. Bothmethods of approach yield useful informat ion and in a sense complement each other, asthe few exact solut ions serve as a check to the validity and reliability of the approximatemethods. In this chapter, we shall see how the second method may be applied to someimportant problems of two-dimensional flow.

The assumption of two-dimensionality itself serves as a first approximat ion to the flow pastthe wings of airplane, the flow through the blade system of propellers and of axial-flow incompressors and turbines. In many such applicat ions the velocity of perturbat ions produced bythe body immersed in the flow are small, because the bodies are very thin. In this fact lies theessence of the linearized method –that the flow pattern may be thought of as thecombinat ion of a uniform, parallel velocity on which small perturbat ion velocit ies superposed.

The advantage of making such an assumption lies in the fact that the governing equat ion ofmot ion is great ly simplified and also becomes linear. Further, it is shown that, from thislinearized theory or small perturbat ion theory, we can draw useful approximate informat ion asto the effect of Mach number for subsonic flow. The linearized theory also makes evident, anapproximate similarity law for different flow fields.

9.24 General Linear Solution for Supersonic FlowThe fundamental equat ion governing most of the compressible flow regime, within the frame ofsmall perturbat ions is [Equat ion (9.52)]:

(9.192) Equat ion (9.192) is ellipt ic for M∞ < 1 and hyperbolic for M∞ > 1. There is hardly any methodavailable for obtaining the analyt ical solut ion of the above equat ion for M∞ < 1. But for M∞ > 1,analyt ical solut ions are available for Equat ion (9.192).

Solutions of Equation (9.192) for M∞ > 1For M∞ > 1, Equat ion (9.192) is of the hyperbolic type, with the form being similar to that of thewave equat ion. The general solut ion to this equat ion can be writ ten as the sum of twoarbit rary funct ions f and g such that:

(9.193) where μ is the Mach angle and:

(9.194) The arbit rary funct ions f and g are to be determined from the boundary condit ions for thespecific problems.

ProofTo show that Equat ion (9.193) is the solut ion to Equat ion (9.192) when M∞ > 1, rewriteEquat ion (9.193) as:

where ξ and η are the new variables, defined as:

Therefore,

On simplificat ion this yields:

(9.195a)

Different iat ion of the above expression for ϕz with respect to z gives:

(9.195b) Subst itut ing Equat ions (9.195) into (9.192), we get:

This equat ion is sat isfied for tan μ from Equat ion (9.194). That is, Equat ion (9.193) is thegeneral solut ion of Equat ion (9.192). However, the funct ion f and g differ from problem toproblem. Instead of Equat ion (9.193), solut ion to Equat ion (9.192) can also be writ ten as:

(9.196) where

(9.197) On inspect ion of the solut ion Equat ion (9.193) or (9.196), it is seen that ϕ and hence, all the

flow propert ies are constant along the straight lines given by the equat ion:

This equat ion gives two families of straight lines as shown in Figure 9.38, one family running tothe left of the object and the other family running to the right , when viewed in the flowdirect ion.

Figure 9.38 Flat plate in a supersonic stream.

These are called Mach lines or characterist ics. The lines of constant f that make a posit iveangle with the flow direct ion and run to the left of the disturbance (object) are called left -running characterist ics and lines of constant g, making a negat ive angle with the flow direct ionand running to the right of object are called right-running characterist ics. Depending on thegeometry of the object , there will only be left -running or right-running or both thecharacterist ics present in the field as shown in Figure 9.39.

Figure 9.39 Characterist ics on different objects in supersonic flow.

9.24.1 Existence of Characteristics in a Physical ProblemFrom the above discussions it is observed that:

1. Disturbances and Mach lines can be produced only by boundaries.2. Disturbances can travel only in the downstream direct ion.

In Figure 9.39, we have shown that the characterist ics of two families are independent ofeach other. This is because the geometries chosen are such that on one side of the boundary

there is only one family of Mach lines. This is not the case always. In fact , in many situat ions ofpract ical importance, the opposite characterist ics will intersect each other as shown in Figure9.40.

Figure 9.40 Coexistence of left -running (l − r) and right-running (r − r) characterist ics.

By knowing the type of Mach lines present in the problems, the equat ions can be suitablytaken.

From Equat ion (9.196), we have the potent ial funct ion as:

where f represents the left -running Mach lines, on which g = 0 and g represents the right-running Mach lines, on which f = 0. The perturbat ion velocit ies are:

(9.198a)

(9.198b) Then the pressure coefficient is given by Equat ion (9.73a) as:

(9.199) That is, to compute the pressure distribut ion, we need to know only the derivat ives of f and g.There is no need to know the funct ions f and g themselves.

9.24.2 Equation for the Streamlines from Kinematic FlowCondition

From Sect ion 9.10, by kinematic flow condit ion we know that:

To make the integrat ion of this equat ion easier, we write the equat ion as follows:

where and the denominator has been writ ten as:

This is possible because and so . Hence, the error introduced by this change is notsignificant. Rearranging the above equat ion, we get:

Subst itut ing for u and from Equat ion (9.198), we obtain:

since:

Hence:

Integrat ing, we get the result :

(9.200) This is the general solut ion of supersonic flow. Once the geometry is known, Equat ion (9.200)gives g and f and then from Equat ion (9.199)Cp and hence the lift and drag can be calculated.Therefore, in any problem if we are not interested in the geometry of the body present, then itis not necessary to find f and g. It is sufficient if f ' and g ' are found, to get the Cp, which is verymuch simpler.

Example 9.5The upper and lower surfaces of a symmetrical two-dimensional aerofoil are given by z = ± εx(1− x/c)2, where c is the chord and ε < 1. The aerofoil is at zero incidence in a steady supersonicstream of Mach number M∞ in posit ive x−direct ion. (a) Find the velocity components accordingto the linear theory in the upper region of disturbance. (b) Show that the drag coefficient of theaerofoil is given by:

Solution(a) Given:

(i.) The governing equat ion is:

where . On the upper surface, the boundary condit ion is:

With Equat ion (i), the boundary condit ion becomes:

Therefore:

(b)

Subst itut ing in the equat ion for CD and simplifying, we get:

9.25 Flow over a Wave-Shaped WallConsider a uniform flow of velocity V∞ over a two-dimensional wave-shaped wall, as shown inFigure 9.41, with wavelength L and amplitude h.

Figure 9.41 Flow past a wave-shaped wall.

Let the wall shape be defined by the equat ion:

(9.201) In Equat ion (9.201), subscript stands for wall and λ = 2π/L. Let us assume h < L, so thatlinear theory can be applied. By kinematic flow condit ion [Equat ion (9.68)], for z → 0, we have:

(9.202) Now, with this background, let us t ry to solve the governing equat ion for incompressible flow,compressible subsonic flow and supersonic flow.

9.25.1 Incompressible FlowThe governing equat ion for incompressible flow is the Laplace equat ion:

This can be solved by expressing the potent ial funct ion as:

Solving by separat ion of variables, we get:

(9.203) The potent ial funct ion given by Equat ion (9.203) is only the perturbat ion potent ial. Obtainingthe expression for ϕ, given by Equat ion (9.203), is left as an exercise to the reader.

Using Equat ion (9.203), we can easily get the resultant velocity U and perturbat ion velocity as:

(9.204a)

(9.204b)

9.25.2 Compressible Subsonic FlowThe governing equat ion for this flow is:

Solving as before, we get the result :

(9.205) Hence, we have:

(9.206a)

(9.206b)

9.25.3 Supersonic FlowFor supersonic flow the governing potent ial equat ion is:

(9.207) For this equat ion, by Equat ion (9.196), we have the solut ion as:

where .From the geometry of the problem under considerat ion, since the disturbances can move

only in the direct ion of flow, there can be only left -running Mach lines, as shown in Figure9.41(c). Therefore:

Hence, the perturbat ion velocity on the wall is

Equat ing this to given by Equat ion (9.202), we get:

This is only on the wall. In general:

(9.208) that is:

(9.209) Therefore:

(9.210a)

(9.210b) The ϕ here is only the disturbance potent ial, and if the total potent ial is required, add (V∞x) toϕ.

9.25.4 Pressure CoefficientThe fundamental form of expression for the coefficient of pressure applicable to two-dimensional compressible flow, with the frame of small perturbat ions, given by Equat ion (9.73),is:

Therefore, in the present problem:

On the surface of the wall (z = 0) and above results reduce to:

(9.211)

(9.212)

(9.213) In the above solut ion we did not get f direct ly. The results are obtained from f '. If only Cp on thewall is needed, it is not necessary to find f, since the Cp on the wall is given by Equat ion (9.199).

Usually, for aerodynamic applicat ions, only Cp on the wall is necessary.From the plots of incompressible, compressible subsonic and supersonic flow over wave-

shaped wall, shown in Figure 9.41, the following observat ions can be made:1. For M∞ = 0, the disturbances die down rapidly because of the e−λz term in Cpexpression.2. For M∞ < 1, larger the M∞, the slower is the dying down of disturbances in thetransverse direct ion to the wall.3. For M∞ = 1, the disturbances do not die down at all (of course the equat ions derived inthis chapter cannot be used for t ransonic flows).4. For M∞ > 1, the disturbances do not die down at all. The disturbance can be felt even at∞ (far away from the wall) if the flow is inviscid.

Further, for equal perturbat ions, we have:

As z→ ∞,for M∞ < 1, the disturbances vanish.for M∞ > 1, the disturbances are finite and they do not die down at all.

Equat ion (9.212) is symmetric with respect to wall geometry and Equat ion (9.213) isasymmetric with respect to wall geometry. Therefore, when Cp is integrated along x, for M∞ <1, Cp becomes zero and for M∞ > 1, the magnitude of Cp is >0. In other words, in subsonic flow,the pressure coefficient is in phase with the wall shape so that there is no drag force on thewall, but in supersonic flow, the pressure coefficient is out of phase with the wall shape andhence there is drag force act ing on the wall.

9.26 Summary

The incompressible flow is that for which the Mach number is zero. This definit ion ofincompressible flow is only of mathematical interest , since for Mach number equal to zero thereis no flow and the state is essent ially a stagnat ion state. Therefore, in engineering applicat ionswe treat the flow with density change less than 5% of the freestream density asincompressible. This corresponds to M = 0.3 for air at standard sea level state. Thus flow withMach number greater than 0.3 is t reated compressible. Compressible flows can be classifiedinto subsonic, supersonic and hypersonic, based on the flow Mach number. Flows with Machnumber from 0.3 to around 1 is termed compressible subsonic, flows with Mach number greaterthan 1 and less than 5 are referred to as supersonic and flows with Mach number in the rangefrom 5 to 40 is termed hypersonic.

A perfect gas has to be thermally as well as calorically perfect , sat isfying the thermal stateequat ion and at least two calorical state equat ions.

For a perfect gas, the internal energy u is a funct ion of the absolute temperature T alone.This hypothesis is a generalizat ion for experimental results. It is known as Joule's law. We canexpress this as:

where is called the specific heat at constant volume. It is the quant ity of heat required toraise the temperature of the system by one unit while keeping the volume constant.

Similarly the specific heat at constant pressure, cp, defined as the quant ity of heat requiredto raise the temperature of the system by one unit while keeping the pressure constant. For p= constant, q = cpdT, therefore:

or

This relat ion is popularly known as Mayer's Relation.Another parameter of primary interest in thermodynamics is entropy s. The entropy,

temperature and heat q are related as:

The second law of thermodynamics assumes that the entropy of an isolated system can neverdecrease, that is .

When the entropy remains constant throughout the flow, the flow is termed isentropic flow.Thus, for an isentropic flow, ds = 0.

This equat ion is often called the compressible form of Bernoulli's equat ion for inviscid flows.The Bernoulli's equat ion can be writ ten as:

This is a form of energy equat ion for isentropic flow process of gases.For an adiabat ic flow of perfect gases, the energy equat ion can be writ ten as:

For air, with γ = 1.4:

This maximum velocity, which is often used for reference purpose, is given by:

Another useful reference velocity is the speed of sound at the stagnat ion temperature, givenby:

Yet another convenient reference velocity is the crit ical speed V*, that is, velocity at Mach

Number unity, or:

The one-dimensional analysis is valid only for flow through infinitesimal streamtubes. In manyreal flow situat ions, the assumption of one-dimensionality for the ent ire flow is at best anapproximat ion. In problems like flow in ducts, the one-dimensional t reatment is adequate.However, in many other pract ical cases, the one-dimensional methods are neither adequatenor do they provide informat ion about the important aspects of the flow. For example, in thecase of flow past the wings of an aircraft , flow through the blade passages of turbine andcompressors, and flow through ducts of rapidly varying cross-sect ional area, the flow field mustbe thought of as two-dimensional or three-dimensional in order to obtain results of pract icalinterest .

Because of the mathematical complexit ies associated with the treatment of the mostgeneral case of three-dimensional mot ion –including shocks, frict ion and heat t ransfer, itbecomes necessary to conceive simple models of flow, which lend themselves to analyt icalt reatment but at the same t ime furnish valuable informat ion concerning the real and difficultflow patterns. We know that by using Prandt l's boundary layer concept, it is possible to neglectfrict ion and heat t ransfer for the region of potent ial flow outside the boundary layer.

Crocco's theorem for two-dimensional flows is:

It is seen that the rotat ion depends on the rate of change of entropy and stagnat ion enthalpynormal to the streamlines. Crocco's theorem essent ially relates entropy gradients to vort icity,in steady, frict ionless, nonconduct ing, adiabat ic flows. In this form, Crocco's equat ion showsthat if entropy (s) is a constant, the vort icity (ζ) must be zero. Likewise, if vort icity ζ is zero, theentropy gradient in the direct ion normal to the streamline (ds/dn) must be zero, implying thatthe entropy (s) is a constant. That is, isentropic flows are irrotational and irrotational flows areisentropic. This result is t rue, in general, only for steady flows of inviscid fluids in which there areno body forces act ing and the stagnat ion enthalpy is a constant.

The circulat ion is:

By Stokes theorem, the vort icity ζ is given by:

where ζx, ζy, ζz are the vort icity components. The two condit ions that are necessary for africt ionless flow to be isentropic throughout are:

1. h0 = constant, throughout the flow.

2. ζ = 0, throughout the flow.From Equat ion (9.33), ζ = 0 for irrotat ional flow. That is, if a frict ionless flow is to be isentropic,the total enthalpy should be constant throughout and the flow should be irrotat ional.

For irrotat ional flows ( curl V = 0), a potent ial funct ion ϕ exists such that:

The advantage of introducing ϕ is that the three unknowns Vx, Vy and Vz in a general three-dimensional flow are reduced to a single unknown ϕ.

The incompressible cont inuity equat ion · V = 0 becomes:

or

This is a Laplace equat ion. With the introduct ion of ϕ, the three equat ions of mot ion can bereplaced, at least for incompressible flow, by one Laplace equat ion, which is a linear equat ion.

The basic studies on fluid flows (Rathakrishnan, 2012) [2] say that:1. For uniform flow (towards posit ive x-direct ion), the potent ial funct ion is:

2. For a source of strength Q, the potent ial funct ion is:

3. For a doublet of strength μ (issuing in negat ive x-direct ion), the potent ial funct ion is:

4. For a potent ial (free) vortex (counterclockwise) with circulat ion Γ, the potent ial funct ionis:

For a steady, inviscid, three-dimensional flow, by cont inuity equat ion:

that is:

Euler's equat ions of mot ion (neglect ing body forces) are:

The basic potential equation for compressible flow is:

The difficult ies associated with compressible flow stem from the fact that the basic equat ion isnonlinear.

The general equat ion for compressible flows can be simplified for flow past slender or planarbodies. Aerofoil, slender bodies of revolut ion and so on are typical examples for slender bodies.Bodies such as a wing, where one dimension is smaller than others, are called planar bodies.These bodies introduce small disturbances. The aerofoil contour becomes the stagnat ionstreamline.

The small perturbat ion theory postulates that the perturbat ion velocit ies are smallcompared to the main velocity components, that is:

Therefore,

The equat ion valid for subsonic, t ransonic and supersonic flow under the framework of smallperturbat ions with and .

This equat ion is called the linearized potential flow equation, though it is not linear.For t ransonic flows (M∞ ≈ 1), the governing equat ion is:

The nonlinearity of this equat ion makes transonic flow problems much more difficult thansubsonic or supersonic flow problems.

Fuselage of airplane, rocket shells, missile bodies and circular ducts are the few bodies ofrevolut ions which are commonly used in pract ice. The governing equat ion for subsonic andsupersonic flows in cylindrical coordinates is:

For t ransonic flow, this becomes:

For axially symmetric, subsonic and supersonic flows, ϕθθ = 0. Therefore, the governingequat ion for subsonic and supersonic flows reduces to:

Similarly, the t ransonic equat ion reduces to:

The small perturbat ion equat ions for subsonic and supersonic flows are linear, but fort ransonic flows the equat ion is nonlinear. Subsonic and supersonic flow equat ions do notcontain the specific heats rat io γ, but t ransonic flow equat ion contains γ. This shows that theresults obtained for subsonic and supersonic flows, with small perturbat ion equat ions, can beapplied to any gas, but this cannot be done for t ransonic flows. All these equat ions are valid forslender bodies. This is t rue of rockets, missiles, etc. These equat ions can also be applied toaerofoils, but not to bluff shapes like circular cylinder, etc. For nonslender bodies, the flow canbe calculated by using the original nonlinear equat ion.

Pressure coefficient is the nondimensional difference between a local pressure and thefreestream pressure. The idea of finding the velocity distribut ion is to find the pressuredistribut ion and then integrate it to get lift , moment, and pressure drag. For three-dimensionalflows, the pressure coefficient Cp is given by:

or

Expanding the right-hand side of this equat ion binomially and neglect ing the third and higher-order terms of the perturbat ion velocity components, we get:

For two-dimensional or planar bodies, the Cp simplifies further, result ing in:

This is a fundamental equat ion applicable to three-dimensional compressible (subsonic andsupersonic) flows, as well as for low speed two-dimensional flows.

For bodies of revolut ion, by small perturbat ion assumption:

where R is the expression for the body contour.An expression which relates the subsonic compressible flow past a certain profile to the

incompressible flow past a second profile derived from the first principles through an affinetransformation. Such an expression is called a similarity law.

Prandt l and Glauert have shown that it is possible to relate the solut ion of compressible flowabout a body to incompressible flow solut ion.

The direct problem (Version I), in which the body profile is t reated as invariant, the indirectproblem (Version II), which is the case of equal potent ials (the pressure distribut ion around thebody in incompressible flow and compressible flow are taken as the same), and the streamlineanalogy (Version III), which is also called Gothert's rule.

Streamlines for compressible flow are farther apart from each other by than inincompressible flow.

The rat io between aerodynamic coefficients in compressible and incompressible flows is also.

The freestream Mach number which gives sonic velocity somewhere on the boundary iscalled crit ical Mach number . The crit ical Mach number decreases with increasing thicknessrat io of profile. The P-G rule is valid only up to about .

In the indirect problem, the requirement is to find a t ransformat ion, for the profile, by whichwe can obtain a body in incompressible flow with exact ly the same pressure distribut ion, as inthe compressible flow. For this case:

That is, the lift coefficient and pitching moment coefficient are also the same in both theincompressible and compressible flows. But, because of decreased α in compressible flow:

Gothert 's rule states that the slope of a profile in a compressible flow pattern is larger by thefact or than the slope of the corresponding profile in the related incompressible flowpattern. But if the slope of the profile at each point is greater by the factor , it is also t ruethat the camber (f) rat io, angle of at tack (α) rat io, the thickness (t) rat io, must all be greater forthe compressible aerofoil by the factor .

Thus, by Gothert 's rule we have:

Compute the aerodynamic coefficients for this t ransformed body for incompressible flow. Theaerodynamic coefficients of the given body at the given compressible flow Mach number aregiven by:

Compute the flow around the given body at . For any other supersonic Mach number, theaerodynamics coefficients are given by:

where Cp, CL and CM are at and Cp ', CL ' and CM ' are at any other supersonic Machnumber.

For version II, we can write:

Gothert rule for subsonic and supersonic flows gives:

We can state the Gothert rule for subsonic and supersonic flows by using a modulus: .For t ransonic flow:

For subsonic flow:

For supersonic flow:

Transonic flow is characterized by the occurrence of shock and boundary layer separat ion.This explains the steep increase in CD at t ransonic range. We should also recall that the shockshould be sufficient ly weak for small perturbat ion. For circular cylinder this theory cannot beapplied, because the perturbat ions are not small.

where K is called the Hypersonic similarity parameter.The presence of a small disturbance is felt throughout the field by means of disturbance

waves traveling at the local velocity of sound relat ive to the medium. The lines at which thepressure disturbance is concentrated and which generate the cone are called Mach waves orMach lines. The angle between the Mach line and the direct ion of mot ion of the body is calledthe Mach angle μ.

Shock may be described as compression front in a supersonic flow field and the flow processacross the front results in an abrupt change in fluid propert ies. In other words, shock is a thinregion where large gradients in temperature, pressure and velocity occur, and where thetransport phenomena of momentum and energy are important. The thickness of the shocks iscomparable to the mean free path of the gas molecules in the flow field.

is called the Prandtl relation.In terms of the speed rat io M* = V/a*, we have:

This implies that the velocity change across a normal shock must be from supersonic tosubsonic and vice versa. But, it can be shown that only the former is possible. Hence, the Machnumber behind a normal shock is always subsonic. This is a general result , not limited just to acalorically perfect gas.

The relat ion between the characterist ic Mach number M* and actual Mach number M is:

The Mach number behind is normal shock, M2, is:

The density rat io across a normal shock is:

The pressure rat io across a normal shock is:

The rat io (p2 − p1)/p1 = Δp/p1 is called the shock strength.The entropy change in terms of pressure and temperature rat ios across the shock can be

expressed as:

The changes in flow propert ies across the shock take place within a very short distance, of theorder of 10−5 cm. Hence, the velocity and temperature gradients inside the shock structure arevery large. These large gradients result in increase of entropy across the shock. Also, thesegradients internal to the shock provide heat conduct ion and viscous dissipat ion that renderthe shock process internally irreversible.

The flow process across the shock wave is adiabat ic, therefore:

For a stat ionary normal shock, the total enthalpy is always constant across the wave which, forcalorically or thermally perfect gases, t ranslates into a constant total temperature across theshock. However, for a chemically react ing gas, the total temperature is not constant across theshock. Also, if the shock wave is not stationary (that is, for a moving shock), neither the totalenthalpy nor the total temperature are constant across the shock wave.

For an adiabat ic process of a perfect gas, we have:

Therefore, the entropy difference between states 1 and 2 is expressed, without any referenceto the velocity level, as:

The rat io of total pressure may be obtained as:

A compression wave inclined at an angle to the flow occurs. Such a wave is called an obliqueshock. Indeed, all naturally occurring shocks in external flows are oblique.

The normal shock wave is a special case of oblique shock waves, with shock angle β = 90 .

Also, it can be shown that superposit ion of a uniform velocity, which is normal to the upstreamflow, on the flow field of the normal shock will result in a flow field through an oblique shockwave. All the streamlines are deflected to the same angle θ at the shock, result ing in uniformparallel flow downstream of shock. The angle θ is referred to as flow deflection angle. Acrossthe shock wave, the Mach number decreases and the pressure, density and temperatureincrease. The corner which turns the flow into itself is called compression or concave corner. Incontrast , in an expansion or convex corner, the flow is turned away from itself through anexpansion fan. All the streamlines are deflected to the same angle θ after the expansion fan,result ing in uniform parallel flow downstream of the fan. Across the expansion wave, the Machnumber increases and the pressure, density and temperature decrease.

Oblique shock and expansion waves prevail in two-and three-dimensional supersonic flows,in contrast to normal shock waves, which are one-dimensional.

The Mach number behind the oblique shock, M2, is related to Mn2 by:

For a given init ial Mach number M1, the possible range of wave angle is:

An important feature to be inferred is that the Mach waves, like characterist ics will berunning to the left and right in the flow field. Because of this the Mach waves of oppositefamilies prevailing in the flow field cross each other. But being the weakest degenerat ion ofwaves, the Mach waves would cont inue to propagate as linear waves even after passingthrough a number of Mach waves. In other words, the Mach waves would cont inue to be simplewaves even after intersect ing other Mach waves. Because of this nature of the Mach waves, aflow region traversed by the Mach waves is simple throughout.

The θ–β–M relat ion of oblique shock is:

Oblique shocks are essent ially compression fronts across which the flow decelerates andthe stat ic pressure, stat ic temperature and stat ic density jump to higher values. If thedecelerat ion is such that the Mach number behind the shock cont inues to be greater thanunity, the shock is termed weak oblique shock. If the downstream Mach number becomes lessthan unity then the shock is called strong oblique shock. It is essent ial to note that only weakoblique shocks are usually formed in any pract ical flow and it calls for special arrangement togenerate strong oblique shocks. One such situat ion where strong oblique shocks aregenerated with special arrangements is the engine intakes of supersonic flight vehicles, wherethe engine has provision to control its backpressure. When the backpressure is increased to anappropriate value, the oblique shock at the engine inlet would become a strong shock anddecelerate the supersonic flow passing through it to subsonic level.

The angle μ is simply a characterist ic angle associated with the Mach number M by therelat ion:

This is called Mach angle–Mach number relation. These lines which may be drawn at any pointin the flow field with inclinat ion μ are called Mach lines or Mach waves. It is essent ial tounderstand the difference between the Mach waves and Mach lines. Mach waves are theweakest isentropic waves in a supersonic flow field and the flow through them will experienceonly negligible changes of flow propert ies. Thus, a flow traversed by the Mach waves does notexperience a change of Mach number, whereas the Mach lines, even though they are weakisentropic waves, they will cause small but finite changes to the propert ies of a flow passingthrough them. In uniform supersonic flows, the Mach waves and Mach lines are linear andinclined at an angle given by . But in nonuniform supersonic flows the flow Mach number Mvaries from point to point and hence the Mach angle μ, being a funct ion of the flow Machnumber, varies with M and the Mach lines are curved.

Even though all these are isentropic waves, there is a dist inct difference between them.

Mach waves are weak isentropic waves across which the flow experiences insignificantchange in its propert ies, whereas the expansion waves and characterist ics are isentropicwaves which introduce small but finite property changes to a flow passing them.

It is important to note that, when we discussed about flow through oblique shocks, weconsidered the shock as weak when the downstream Mach number M2 is supersonic (eventhough less than the upstream Mach number M1). When the flow traversed by an obliqueshock becomes subsonic (that is, M2 < 1), the shock is termed strong. But when the flowturning θ caused by a weak oblique shock is very small, then the weak shock assumes aspecial significance. These kinds of weak shocks with both decrease of flow Mach number (M1− M2) and small flow turning angle θ can be regarded as isentropic compression waves.

This is considered to be the basic relat ion for obtaining all other appropriate expressions forweak oblique shocks since all oblique shock relat ions depend on M1 sin β, which is thecomponent of upstream Mach number normal to the shock.

Similarly, it can be shown that the changes in density and temperature are also proport ionalto θ.

Compressions in a supersonic flow are not usually isentropic. Generally, they take placethrough a shock wave and hence are nonisentropic. But there are certain cases for which thecompression process can be regarded as isentropic. This kind of compression through a largenumber of weak compression waves is termed continuous compression and these kinds ofcorners are called cont inuous compression corners. Thus, the geometry of the corner shouldhave cont inuous smooth turning to generate a large number of weak (isentropic) compressionwaves.

In an expansion process, the Mach lines are divergent, consequent ly, there is a tendency todecrease the pressure, density and temperature of the flow passing through them. In otherwords, an expansion is isentropic throughout.

It is essent ial to note that the statement “expansion is isentropic throughout” is not t ruealways. To gain an insight into the expansion process, let us examine the centered andcont inuous expansion processes. We know that the expansion rays in an expansion fan areisentropic waves across which the change of pressure, temperature, density and Mach numberare small but finite. But when such small changes coalesce they can give rise to a largechange. Therefore, it is essent ial to realize that a centered expansion process is isentropiceverywhere except at the vertex of the expansion fan, where it is nonisentropic.

The expansion at a corner occurs through a centered wave, defined by a “fan” of straightexpansion lines. This centered wave, also called a Prandt l-Meyer expansion fan, is thecounterpart , for a convex corner, of the oblique shock at a concave corner.

It is known from basic studies on fluid flows that a flow which preserves its own geometry inspace or t ime or both is called a self-similar flow. In the simplest cases of flows, such mot ionsare described by a single independent variable, referred to as similarity variable. The Prandt l-Meyer funct ion is such a similarity variable.

The Prandt l-Meyer funct ion in terms of the Mach number M1 just upstream of the expansionfan as:

The shock and expansion waves discussed in this chapter are the basis for analyzing largenumber of two-dimensional, supersonic flow problems by simply “patching” togetherappropriate combinat ions of two or more solut ions. That is, the aerodynamic forces act ing on abody present in a supersonic flow are governed by the shock and expansion waves formed atthe surface of the body.

The only aerodynamic force act ing on the diamond aerofoil is due to the higher-pressure onthe forward face and lower-pressure on the rearward face. The drag per unit span is given by:

that is:

This gives the drag experienced by a two-dimensional diamond aerofoil, kept at zero angle ofat tack in an inviscid flow. This is in contrast with the familiar result from studies on subsonicflow that for two-dimensional inviscid flow over a wing of infinite span at a subsonic velocity,the drag force act ing on the wing is zero –a theoret ical result called d’Alembert 's paradox. Incontrast with this, for supersonic flow, drag exists even in the idealized, nonviscous fluid. Thisnew component of drag encountered when the flow is supersonic is called wave drag, and isfundamentally different from the skin-frict ion drag and separat ion drag which are associatedwith boundary layer in a viscous fluid. The wave drag is related to loss of total pressure andincrease of entropy across the oblique shock waves generated by the aerofoil.

For the flat plate at an angle of at tack α0 in a uniform supersonic flow, the lift and drag arecomputed very easily, with the following equat ions:

where c is the chord.We saw that the shock-expansion theory gives a simple method for comput ing lift and drag

act ing over a body kept in a supersonic stream. This theory is applicable as long as the shocksare at tached. This theory may be further simplified by approximat ing it by using theapproximate relat ions for the weak shocks and expansion, when the aerofoil is thin and is keptat a small angle of at tack, that is, if the flow inclinat ions are small. This approximat ion will resultin simple analyt ical expressions for lift and drag.

At this stage we may have a doubt about the difference between shock-expansion theoryand thin aerofoil theory. The answer to this doubt is the following:

“In shock-expansion theory, the shock is essentially a non-isentropic wave causing a finiteincrease of entropy. Thus, the total pressure of the flow decreases across the shock. But inthin aerofoil theory even the shock is regarded as an isentropic compression wave.Therefore, the flow across this compression wave is assumed to be isentropic. Thus thepressure loss across the compression wave is assumed to be negligibly small.”

The above equat ion, which states that the pressure coefficient is proportional to the local flowdirection, is the basic relat ion for thin aerofoil theory.

The CL and CD of the flat plate at a small angle of at tack may be expressed as:

For a diamond wedge of chord c:

or

In the above two applicat ions, the thin aerofoil theory was used for specific profiles to getexpressions for CL and CD. A general result applicable to any thin aerofoil may be obtained asfollows. Consider a cambered aerofoil with finite thickness at a small angle of at tack t reated bylinear resolut ion into three components, each of which contribut ing to lift and drag.

By thin aerofoil theory, the Cp on the upper and lower surfaces are obtained as:

where yu and yl are the upper and lower profiles of the aerofoil. The profile may be resolvedinto a symmetrical thickness distribut ion h(x) and a camber line of zero thickness yc(x). Thus,we have:

where α(x) = α0 + αc(x) is the local angle of at tack of the camber line and α0 is the angle ofat tack of the freestream and αc is the angle at tack due to the camber. The lift and drag aregiven by:

The above two expressions give the general expressions for lift and drag coefficients of a thinaerofoil in a supersonic flow. In thin aerofoil theory, the drag is split into drag due to lift, drag dueto camber and drag due to thickness. But the lift coefficient depends only on the mean angle ofat tack.

The fundamental equat ion governing most of the compressible flow regime, within the frameof small perturbat ions is:

This is ellipt ic for M∞ < 1 and hyperbolic for M∞ > 1. There is hardly any method available forobtaining the analyt ical solut ion of the above equat ion for M∞ < 1. But for M∞ > 1, analyt icalsolut ions are available.

The governing equat ion for compressible subsonic flow is:

Solving as before, we get the result :

Hence, we have:

For supersonic flow the governing potent ial equat ion is:

For this equat ion, we have the solut ion as:

where .The fundamental form of expression for the coefficient of pressure applicable to two-

dimensional compressible flow, with the frame of small perturbat ions, is:

1. For M∞ = 0, the disturbances die down rapidly because of the e−λz term in Cpexpression.2. For M∞ < 1, larger the M∞, the slower is the dying down of disturbances in thetransverse direct ion to the wall.3. For M∞ = 1, the disturbances do not die down at all (of course the equat ions derived inthis chapter cannot be used for t ransonic flows).4. For M∞ > 1, the disturbances do not die down at all. The disturbance can be felt even at∞ (far away from the wall) if the flow is inviscid.

Further, for equal perturbat ions, we have:

As z→ ∞,for M∞ < 1, the disturbances vanish.for M∞ > 1, the disturbances are finite and they do not die down at all.

Exercise Problems1. A flat plate aerofoil in a Mach 2 freestream experiencing a lift coefficient of 0.16 has anaerodynamic efficiency of 14.65, determine the drag coefficient and angle of at tack.[Answer: 0.0109, 3 . 97 ]2. Show that for compressible flow of a perfect gas, the variat ion of total pressure acrossa streamline is given by:

where n is the direct ion normal to the streamline.3. The nose of a cylindrical body has the profile R = εx3/2, 0 ≤ x ≤ 1. Show that thepressure distribut ion on the body is given by:

Est imate the drag coefficient for and ε = 0.1.(Hint : For obtaining CD, use , where S(x) is the cross-sect ional area of the body at xand L is the length of the body.)

[Answer: CD = 0.0786]

4. A slender model with semi-vertex angle θ = 3 has to operate at M∞ = 10 with angle ofat t ack α = 3 . What are the respect ive angles of at tack required to simulate thecondit ions if a wind tunnel test has to be carried out at (a) M∞ = 3.0, θ = 12 and (b) M∞ =3.0, θ = 3 ?

[Answer: (a) 7 . 3 , (b) 16 . 3 ]5. A missile has a conical nose with a semi-vertex angle of 4 and is subjected to a Machnumber of 12 under actual condit ions. A model of the missile has to be tested in asupersonic wind tunnel at a test sect ion Mach number of 2.5. Calculate the semi-vertexangle of the conical nose of the model.

[Answer: 19 . 2 ]6. Show that the results of the linearized supersonic theory for flow past a wedge of semi-wedge angle θ may be put into the following similarity form:

where

7. A shallow irregularity of length l, in a plane wall, shown in Figure 9.42, is given by theexpression y = kx(1 − x/l), where 0 < x < l and k < 1. A uniform supersonic stream withfreestream Mach number M∞ is flowing over it . Using linearized theory, show that thevelocity potent ial due to disturbance in the flow is:

where .

Figure 9.42 A shallow irregularity in a plane wall.

8. A two-dimensional wing profile shown in Figure 9.43 is placed in a Mach 2.5 air streamat an incidence of 2 . Using linearized theory, calculate the lift coefficient CL and the drag

coefficient CD.[Answer: CL = 0.06096 and CD = 0.04372].

Figure 9.43 A two-dimensional wing profile in Mach 2.5 air stream.

9. A two-dimensional thin aerofoil shown in Figure 9.44 is placed in Mach 3.0 air stream atan angle of at tack of 2 . Using linearized theory, est imate and Cpl.

[Answer: Cpu = 0.211, Cpl = 0.046 (0 ≤ x ≤ 0.3c) ; Cpu = − 0.1258, Cpl = − 0.0551 (0.3c ≤ x ≤c)]

Figure 9.44 A two-dimensional thin aerofoil in Mach 3.0 air stream.

10. The two-dimensional aerofoil shown in Figure 9.45 is t raveling at a Mach number of 3and at an angle of at tack of 2 . The thickness to chord rat io of the aerofoil is 0.1 and themaximum thickness occurs at 30 percent of the chord downstream from the leading edge.Using the linearized theory, show that the moment coefficient about the aerodynamiccenter is −0.035, the center of pressure is at 1.217c and the drag coefficient is 0.0354.Show also that the angle of zero lift is 0 .

Figure 9.45 A two-dimensional aerofoil in a Mach 3 stream.

11. A two-dimensional wedge shown in Figure 9.46 moves through the atmosphere atsea-level, at zero angle of at tack with M∞ = 3.0. Calculate CL and CD using shock-expansion theory.

[Answer: CL = − 0.0388, CD = 0.02265]

Figure 9.46 A two-dimensional wedge in a supersonic flow.

12. Calculate the lift and drag coefficient experienced by a flat plate kept at an angle ofat tack of 5 to an air stream at Mach 2.3 and pressure 101 kPa, using (a) shock–expansion theory and (b) Ackeret 's theory.

[Answer: (a) CL = 0.1735, CD = 0.0152, (b) CL = 0.1685, CD = 0.0147]

13. Calculate the CL and CD for a half-wedge of wedge angle 5 kept in an air stream atMach 2 and 101 kPa at (a) 0 angle of at tack, (b) at 3 angle of at tack.

[Answer: (a) CL = − 0.054, CD = 0.00497, (b) CL = 0.1778, CD = 0.01554]

14. If gives the local speed of sound, obtain the following forms of Bernoulli's equat ion.a. .b. .c. .

Notes

1. Momentum equat ion. For incompressible flow, ∑Fi = ρ ∫ VxdQ, where Q is the volume flowrate. For compressible flow, ∑Fi = ∫ ρVx dQ .2. It is essent ial to note that Mach lines are weak isentropic waves across which the changesin the flow propert ies are small but finite, thus, Mach lines are different from the weakisentropic waves termed Mach waves across which the changes in the flow propert ies arenegligible.

References1. Rathakrishnan, E., Applied Gas Dynamics, John Wiley, NJ, 2010.2. Rathakrishnan, E., Fluid Mechanics -An Introduction, 3rd edn. PHI Learning, Delhi, India, 2012.3. Shapiro, A.H., The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. I , TheRonald Press Company, New York, 1953.4. Liepmann H.W and Roshko A., Elements of Gas Dynamics, John Wiley & Sons, Inc. New York,1957.

10

Simple Flights

10.1 IntroductionSo far we were focusing on the wing and its sect ional profiles, considering the geometricalparameters of the aerofoil (wing) and the parameters of the flow to which it is exposed. Thepressure loading, lift and drag associated with aerofoil were discussed for both two-dimensional (infinite) and three-dimensional wings. In this chapter, let us consider a completeflying machine and study some of the basic flights associated with it . A flying machine and itscontrol surfaces are schematically shown in Figure 10.1

Figure 10.1 Schematic of a monoplane and its control surfaces.

When the control surfaces are in their neutral posit ions the aircraft , like the aerofoil, has amedian plane of symmetry, and when properly located the center of gravity G lies in this plane.

The fixed fin of the aircraft is in the plane of symmetry, as shown in Figure 10.1. For simplicity,let us assume the aircraft to be in straight level (horizontal) flight , with all the control surfacesin their neutral posit ions. The lift is generated by the wings (port and starboard wings) and tail(the lift associated with the body of the aircraft is ignored). The thrust produced by the engineovercomes the drag, that is the thrust is assumed to be horizontal.

In Figure 10.1, the lateral axis Gy is perpendicular to the plane of symmetry and posit ive tostarboard (that is to right). The symmetry will not be disturbed if the elevators are deflected.Raising the elevators will decrease the lift on the tail, and will cause a pitching moment, posit ivewhen the nose tends to be lifted. Moving the rudder to starboard will cause a yawing moment,tending to deflect the nose to the starboard, the posit ive sense.

The ailerons move in opposite senses, one up, one down, by a single mot ion by the controlcolumn. If we depress the port aileron and therefore simultaneously raise the starboard one,the lift on the port wing will increase and that on the starboard wing will decrease so that therolling moment will be caused tending to dip the starboard wing, and this sense will be posit ive.This movement also causes a yawing moment, for the drag on the two wings will likewise bealtered. To minimize this the ailerons are generally geared to move different ially so that onemoves through a greater angle than the other.

Mot ion of ailerons or rudder will disturb the symmetry of the aircraft . A single-engine aircraftalso has a dynamical asymmetry (tendency to t ilt ).

10.2 Linear FlightWhen the aircraft velocity V is in a fixed straight line the flight is termed linear. When V is in theplane of symmetry the flight is termed symmetric. There are three types of linear symmetricalflight ; gliding, horizontal, and climbing, as illustrated in Figure 10.2. Among these, gliding is theonly flight possible without use of the engine.

Figure 10.2 Illustrat ion of gliding, horizontal and climbing flight .

The flights can be steady (constant V) or accelerated. In the case of steady flight theresultant force on the aircraft must be zero. The forces are: (i) engine thrust (under control ofthe pilot), (ii) weight (not under control of the pilot), and (iii) aerodynamic force (in somemeasure under control of the pilot by use of ailerons, rudder, and elevators).

10.3 StallingWe know that the lift coefficient CL is a funct ion of the absolute angle of incidence α, andstrongly influenced by the Reynolds number Re = ρVl/μ. For a given aircraft we could thereforedraw a surface which is the locus of the point (CL, α, Re) which is the characteristic lift surfacefor that aircraft . Since the aircraft is given, l is known, and for the freestream flow the state ofthe air is given, so that in this case CL is a funct ion of incidence α and of the forward speed V.From this point of view the characterist ic surface may then be regarded as the locus of thepoint (CL, α, V). Let us consider three points on this surface , , , where let us supposeV1 < V2 < V3 and α1 < α2 < α3. The variat ion of CL with α corresponding to velocit ies V1, V2, V3are shown in Figure 10.3.

Figure 10.3 Lift curve slope variat ion with incidence for different velocit ies.

It is seen that the straight port ions of CL versus α graph corresponding to values of V1, V2,V3 of V, the straight port ions are pract ically in the same line. This plot may be thought of asshowing sect ions of the characterist ic surface by planes V = V1, V = V2, V = V3. In all ourdiscussions in the previous chapters, we considered CL to be direct ly proport ional to α, that is,we have restricted ourselves to the linear part of the graph about which pure theory can makestatements. Plots of the type shown in Figure 10.3 must necessarily be obtained fromexperimental measurements, and the graph shows that, with increasing incidence, CL rises toa maximum value and then decreases.

It is generally, but not always, the case that for a given V increases as V increases. If thesect ions of the characterist ic surface by the planes α = α1, α = α2, α = α3, we get the variat ionof CL with V as shown in Figure 10.4.

Figure 10.4 Lift curve slope variat ion with V in the sect ions of the characterist ic surface cut bythe planes α = α1, α = α2, α = α3.

The stalled state is that in which the airflow on the suct ion side of the aerofoil is turbulent. Itis found that, just before the stalled state sets in, the lift coefficient at tains its maximum value,and the corresponding speed is called the stalling speed. Thus the stalling speedcorresponding to a given can be read from the cont inuous curve of Figure 10.4.

Stalling speed is a funct ion of incidence. Variat ion of α with V in the sect ions of thecharacterist ic surface by planes , , is as shown in Figure 10.5.

Figure 10.5 Variat ion of α with V in some sect ions of the characterist ic surface.

In Figure 10.5 the cont inuous line shows the stalling speed as a funct ion of the stallingincidence αS. Any point (α, V) above this curve corresponds to stalling flight , any point below itwith normal flight.

It should be noted that the foregoing discussion only applies to speeds V such that the flowspeed over the aerofoil nowhere approaches the speed of sound, that is, we neglect variat ionwith Mach number.

The graphs of the above type are all cases deduced from experiments, generally in windtunnels for condit ion corresponding to linear flight at constant speed. When the aircraft flies ina curved path the graph will differ slight ly from the above, but invest igat ions made byWieselsberger [Reference 9] show that the changes are of the order of the square of the rat ioof the span to radius of curvature of the path and may therefore, in general, be neglected.

Moreover, for most calculat ions it is sufficient to subst itute one of the CL graphs in Figure10.3 for the whole graph, namely the one which corresponds to the landing speed, because thedanger of stalling is generally greatest when the aircraft is about to land and is therefore flyingnear to the stalling incidence and at a low speed. When we subst itute Reynolds number Re forvelocity throughout, the foregoing condit ions may be held to apply to a family of geometricallysimilar aircraft . For such a family there will be one characterist ic lift surface.

10.4 GlidingFor an aircraft gliding steadily with the engine off, as shown in Figure 10.6, the resultantaerodynamic force F ad balances the weight W, that is:

Figure 10.6 An aircraft in steady glide.

Thus, if L and D, respect ively, are the lift and drag act ing on the aircraft :

where γ is the angle which the direct ion of mot ion makes with the horizontal, called the glidingangle. Therefore:

This equat ion expresses the gliding angle γ in terms of CL and CD. It should be noted that CLand CD here are the lift and drag coefficients for the whole aircraft . The attitude of the aircraftis the angle which a line fixed in the aircraft makes with the horizontal, as shown in Figure 10.6.If we measure the incidence α and at t itude θ from the same line we have θ = α − γ. Note that θand α can be negat ive, as shown in Figure 10.6, but the glide angle γ is necessarily posit ive.Also note that the direct ion of the glide does not, in general, coincide with any fixed direct ion inthe aircraft , in other words the at t itude is a funct ion of the incidence α.

The extreme att itude is that assumed when the aircraft is diving vert ically, the terminalvelocity dive. In this case the lift vanishes, the incidence is that of zero lift , and if the dive isundertaken from sufficient ly great height, the weight just balances the drag, the speed beingthe terminal speed, may be five or six t imes the stalling speed. The at t itude will then be about−90°.

Example 10.1A glider of aspect rat io 6 has a drag polar of:

Find the change in minimum angle of glide if the aspect rat io is increased to 10.

SolutionLet subscripts 1 and 2 refer to aspect rat io 6 and 10, respect ively.Given, .Therefore, the wing efficiency is:

For minimum glide angle, the drag should be minimum. For drag minimum, , therefore,

For = 10:

Therefore,

The drag coefficients for aspect rat ios 6 and 10 are:

Therefore, the gliding angles for these cases become:

The difference between the gliding angles is:

10.5 Straight Horizontal FlightThe forces act ing on an aircraft in straight horizontal flight is shown in Figure 10.7. There arethree forces; the thrust T, weight W, and aerodynamic force F ad act ing on the aircraft . Thecomponents of F ad in the direct ion of V and normal to V, respect ively, are the lift L and drag D.

Figure 10.7 Forces act ing on an aircraft in steady level flight .

By proper choice of chord the incidence may be taken equal to the at t itude and

In pract ice θ is small, so that T = D, L = W, and

(10.1) where is called the wing loading, that is, the average load per unit area of wing plan. When

, ρ (that is alt itude), and V are given, CL is determined, and therefore incidence from the CLversus α graph (Figure 10.3).

At the stalling speed VS, Equat ion (10.1) becomes:

(10.2) The stalling speed VS can be determined by plot t ing a graph of versus V and CL versus Vtogether, as shown in Figure 10.8.

Figure 10.8 Plots of CL and CLV2 variat ions with V.

It is seen that the stalling speed VS increases with alt itude (that is, with decrease of flow

density ρ). From Equat ions (10.2) and (10.2), we have:

(10.3) The air-speed indicator measures but is graduated to read V. It is, therefore, correct onlyfor the part icular value of the density ρ for which it is graduated, but, if we neglect variat ions of

, it follows from Equat ion (10.3) that the aircraft will stall always at the same indicatedairspeed when it is in straight horizontal flight , whatever the height.

10.6 Sudden Increase of IncidenceLet us assume the aircraft to be flying steadily and horizontally, so that if CL ' is the liftcoefficient :

a sudden increase of incidence will increase the lift coefficient to:

and the aircraft will acquire an upward accelerat ion f given by:

so that it will begin to describe a curved path of radius of curvature r given by f = V2/r, where:

In this analysis we ignore the change in drag. If the speed is high, CL ' is small and CL cannotexceed for speed V. Thus the absolute minimum value of r is given by:

(10.4) where VS is the appropriate stalling speed. Since is accompanied by a rather large drag,the theoret ical value of r min in Equat ion (10.4) cannot be at tained.

Example 10.2An aircraft weighing 200 kN, wing span 12 m and mean chord 2 m is in steady level flight at sealevel, at a speed of 120 m/s. If the lift coefficient is suddenly increased by 10%, determine theupward accelerat ion causing the lift increase.

SolutionGiven, W = 200 kN, 2b = 12 m, c = 2 m, V = 120 m/s.In level flight :

At sea level, ρ = 1.225 kg/m3. Therefore:

The new lift coefficient is:

The expression for upward accelerat ion f is:

Hence:

10.7 Straight Side-SlipConsider an aircraft flying steadily and horizontally, as shown in Figure 10.9(a), to be rolledthrough an angle ϕ from the vert ical, as shown in Figure 10.9(b) and held in this posit ion bycontrols. The lift will no longer balance the weight.

Figure 10.9 An aircraft (a) in steady level flight and (b) rolled through an angle ϕ.

If the aircraft is supposed to be flying towards us so that the starboard wing is dipped, theaircraft will accelerate in the direct ion of the resultant of L and W, and will cont inue toaccelerate unt il a steady state is reached owing to the wind blowing across the body andproducing a side force in the direct ion of the span. The direct ion of mot ion is now inclined tothe plane of symmetry at an angle β, say, measured posit ively when the direct ion of mot ion isto starboard. The aircraft is now moving crab-wise in the straight path and is said to besideslipping. If V is the speed, the component of V sin β perpendicular to the plane ofsymmetry is called the velocity of side-slip.

Side-slip will neither diminish the drag nor increase the lift as compared to symmetrical flightat the same speed. If D ' and L ' are the drag and lift , respect ively, in the steady side-slipinduced by the above maneuver then L ' < L, D ' > D. The gliding angle γ ' will be given by:

or γ ' > γ. The effect of side-slip is therefore to increase the gliding angle without reducing thespeed.

10.8 Banked TurnThis is a steady mot ion in a horizontal circle with the plane of symmetry inclined to the vert ical,as shown in Figure 10.10. The direct ion of mot ion is longitudinal and there is no side-force.

Figure 10.10 An aircraft in a banked turn.

If ϕ is the angle of bank and r is the radius of the turn, then:

If the differences due to the difference in speed at the outer and inner wing t ips are ignored,then:

and therefore, as in Figure 10.8, the stalling speed VS ' is determined by the intersect ion of thecurves CLV2 versus V with versus V graph. From the relat ion:

(10.5) it appears that banking increases the stalling speed, and if we treat as a constant, theincrease is in the rat io:

10.9 Phugoid MotionA phugoid is the path of a part icle which moves under gravity in a vert ical plane and which isacted upon by a force L normal to the path and proport ional to V2.

Since no work is done by force L, it follows that , the total energy of the part icle (per unitmass), is constant, z being the depth of the part icle below horizontal line, when speed is V, as

shown in Figure 10.11.

Figure 10.11 An object in phugoid mot ion.

We can choose the posit ion of this line so that the constant energy is zero, and then weshall have:

(10.6) If θ is the inclinat ion of the path to the horizontal, as in Figure 10.11, then:

(10.7) where R is the radius of curvature.

If we could imagine an aircraft flying at constant incidence, and so arrange that the thrustexact ly balances the drag, the center of gravity of the aircraft would describe a phugoid, forsuch a case and CL is constant for an incidence (if we neglect the effect of curvature ofthe path on lift coefficient).

Now let us assume that V1 is the speed at which the aircraft would fly in steady straighthorizontal flight at the same incidence as in the phugoid. Then:

so that Equat ion (10.7) will give:

by Equat ion (10.6), V2 = 2g z, , therefore the above equat ion becomes:

(10.8) Now if ds is an element of the arc of the path in Figure 10.11, then:

(10.9) Therefore Equat ion (10.8) can be writ ten in the equivalent form as follows:

Now let us assume that the “constant” to be . Therefore:

(10.10) Let us different iate this with respect to θ:

From Equat ion (10.9), we have , therefore:

or

But from Equat ion (10.9), , therefore:

or

(10.11) It can be shown that in Equat ion (10.10), cos θ > 1 if C > 2/3, so that no phugoid is possible.If C = 2/3, Equat ion (10.10) gives cos θ = 1 so that θ = 0 and R =∞. For this condit ion the

phugoid is along a horizontal straight line, at depth z, below the datum line.If C = 0, Equat ion (10.11) gives R = 3z1, and the phugoid reduces to a set of semicircles of

radius 3z1. The cusps are on the datum line and the paths correspond to unsuccessfulat tempts at “loop the loop,” as shown in Figure 10.12.

For 0 < C < 2/3, the flight path will become trochoidal-like paths, as shown in Figure 10.13.

Figure 10.13 An aircraft in t rochoidal-like paths.

If z1 and the init ial values of z and θ are prescribed, Equat ion (10.11) shows that for a givenvalue of C there are two possible radii of curvature owing to the ambiguity sign of the squareroot. If there is a sudden gust, an aircraft describing a t rochoidal-like paths as in Figure 10.13will get into a loop, as shown in Figure 10.14.

Figure 10.14 An aircraft describing a loop.

10.10 The Phugoid OscillationLet an aircraft describing straight line phugoid, as shown in Figure 10.11, corresponding to C =3/2, z = z1, cos θ = 1, have its path slight ly disturbed, say by a gust. It may then begin todescribe a sinuous path of small slope, as in Figure 10.12, having the straight line as mean. Thismot ion is called phugoid oscillation. Since the vert ical upward accelerat ion is −d2z/dt2, andsince cos θ = 1 to the first order, we have, for the vert ical mot ion:

and therefore from Equat ion (10.10):

This is simple harmonic mot ion whose period is 2π/(z1/g), showing that the disturbed mot ion isstable. In terms of the speed V∞ the period is .

Example 10.3An aircraft weighing 105600 N flies at an alt itude where the air density is 0.16 kg/m3. The wingarea, aspect rat io and efficiency are 28 m2, 6 and 0.95, respect ively. If the drag polar is:

determine the speed and Mach number at which the aerodynamic efficiency will be maximum.

SolutionGiven, W = 105600 N, S = 28 m2, ARlig;++ = 6.Therefore:

For maximum aerodynamic efficiency, the drag has to be the minimum. For minimum drag, .Therefore, the lift coefficient becomes:

At level flight , L = W. The speed for drag minimum becomes:

From standard atmospheric table, for the alt itude with density 0.16 kg/m3, the pressure is11145.75 Pa. The corresponding speed of sound is:

The Mach number corresponding to minimum drag is:

Example 10.4An aircraft of mass 30 000 kg, with an ellipt ical wing of area 225 m2, aspect rat io 7 deliveringconstant thrust of 53 kN, is taking off at sea level. The maximum possible lift coefficient is 2.0and the profile drag coefficient while lift -off is 0.02 and the lift -off speed is 1.2 t imes the stallingspeed. Assuming the rolling resistance to be negligible, calculate the lift -off distance required.

SolutionGiven m = 30000 kg, S = 225 m2, T = 53000 N, , , ARlig;++ = 7.The stalling speed Vs is the speed in level flight , with . Thus:

Therefore, the lift -off speed is:

The lift coefficient at lift -off becomes:

The drag coefficient at lift -off becomes:

At lift -off, by force balance we have:

We can write as:

where s is the distance along the run way, that is the distance travelled by the aircraft fromstart ing to any instantaneous state. Therefore:

This gives:

Integrat ing from s = 0 (V = 0) to s = slo (V = V), we have the lift -off distance as:

10.11 SummaryFor a flying machine, when the control surfaces are in their neutral posit ions the aircraft , likethe aerofoil, has a median plane of symmetry, and when properly located the center of gravityG lies in this plane.

For an aircraft in straight level (horizontal) flight , the lift generated by the wings balances theweight and the thrust produced by the engine overcomes the drag.

Raising the elevators will decrease the lift on the tail, and will cause a pitching moment andmoving the rudder will cause a yawing moment.

The ailerons move in opposite senses, one up, one down, by a single mot ion by the controlcolumn. If we depress the port aileron and therefore simultaneously raise the starboard one,the lift on the port wing will increase and that on the starboard wing will decrease so that therolling moment will be caused tending to dip the starboard wing, and this sense will be posit ive.This movement also cause a yawing moment, for the drag on the two wings will likewise bealtered.

When the aircraft velocity V is in a fixed straight line the flight is termed linear. When V is inthe plane of symmetry the flight is termed symmetric. There are three types of linearsymmetrical flight ; gliding, horizontal, and climbing.

The flights can be steady (constant V) or accelerated. In the case of steady flight theresultant force on the aircraft must be zero. The forces are: (i) engine thrust , (ii) weight, (iii)aerodynamic force.

The lift coefficient CL is a funct ion of the absolute angle of incidence α, and stronglyinfluenced by the Reynolds number Re = ρVl/μ.

The stalled state is that in which the airflow on the suct ion side of the aerofoil is turbulent.Just before the stalled state sets in, the lift coefficient at tains its maximum value, and thecorresponding speed is called the stalling speed. Thus the stalling speed corresponds to agiven . Stalling speed is a funct ion of incidence.

For an aircraft gliding steadily with the engine off the resultant aerodynamic force F adbalances the weight W, that is:

Thus, if L and D, respect ively, are the lift and drag act ing on the aircraft :

where γ is the angle which the direct ion of mot ion makes with the horizontal, called the glidingangle.

The forces act ing on an aircraft in straight horizontal flight are the thrust T, weight W, andaerodynamic force F ad.

By proper choice of chord the incidence may be taken equal to the at t itude and

In pract ice θ is small, so that T = D, L = W, and

where is called the wing loading, that is, the average load per unit area of wing plan.At the stalling speed VS, CL becomes:

For aircraft flying steadily and horizontally, if CL ' is the lift coefficient :

A sudden increase of incidence will increase the lift coefficient to:

and the aircraft will acquire an upward accelerat ion f given by:

so that it will begin to describe a curved path of radius of curvature r given by f = V2/r, where:

In this analysis we ignore the change in drag. If the speed is high, CL ' is small and CL cannotexceed for speed V. Thus the absolute minimum value of r is given by:

where VS is the appropriate stalling speed.When an aircraft flying steadily and horizontally is rolled through an angle ϕ from the vert ical,

the lift will no longer balance the weight. If the aircraft is supposed to be flying towards us sothat the starboard wing is dipped, the aircraft will accelerate in the direct ion of the resultant ofL and W, and will cont inue to accelerate unt il a steady state is reached owing to the windblowing across the body and producing a side force in the direct ion of the span. The direct ionof mot ion is now inclined to the plane of symmetry at an angle β, say, measured posit ively

when the direct ion of mot ion is to starboard. The aircraft is now moving crab-wise in thestraight path and is said to be sideslipping. If V is the speed, the component of V sin βperpendicular to the plane of symmetry is called the velocity of side-slip.

Side-slip will neither diminish the drag nor increase the lift as compared to symmetrical flightat the same speed.

A banked turn is a steady mot ion in a horizontal circle with the plane of symmetry inclined tothe vert ical. The direct ion of mot ion is longitudinal and there is no side-force.

A phugoid is the path of a part icle which moves under gravity in a vert ical plane and which isacted upon by a force L normal to the path and proport ional to V2.

Exercise Problems1. If the lift coefficient of an aircraft is given by:

find the maximum value of the lift coefficient , for b = 2, a = 2π.[Answer: 2]

2. Show that the minimum radius of a t rue banked turn, for a given angle of bank ϕ, is:

3. Calculate the minimum init ial radius of curvature of the path when an aircraft whosestalling speed is 110 km/h in straight flight has the incidence suddenly increased.

[Answer: 95.2 m]4. For small angle of bank, ϕ, show that the angle of side-slip is given by β = − 2.5 CL ϕ,assuming the coefficient of side force as cy = − 0.4 β.

5. An aircraft of wing loading 20 N/m2 makes a banked turn at an alt itude where thefreestream flow density is 0.175 kg/m3. (a) If the minimum radius turn is 98 m and the bankangle is 20°, determine the . (b) What will be the stalling speed at that alt itude if theaircraft is in level flight without bank?

[Answer: (a) 0.253, (b) 30.06 m/s]6. The aerodynamic efficiency of a sail plane of weight 3150 N and wing area 10 m2 is 30.If it is in level flight at sea level with a speed of 170 km/h, determine the drag coefficient .

[Answer: 0.0077]7. An aircraft weighing 23000 N has a span of 14 m and average chord of 1.2 m. If it flieswith a velocity of 90 m/s and angle of at tack 4° in sea level (a) determine the lift curveslope. (b) If the aircraft glides steadily with the same speed at a glide angle of 6°, what willbe the lift curve slope? (c) If the lift coefficient during glide has to be the same as that inthe level flight what should be the flight speed?

[Answer: (a) 3.954, (b) 2.65, (c) 90.24 m/s]8. An aircraft flies at a t rue speed of 350 m/s at an alt itude where the pressure andtemperature are 18.25 kPa and 216.5 K, respect ively. (a) If a one-fourteenth scale modelof the aircraft is to be tested, under dynamically similar condit ions, in a wind tunnel withtest-sect ion temperature 288 K, what should be the pressure in the test-sect ion?Assume the viscosity of air varies with temperature as T3/4 approximately. (b) Show thatthe forces on the model will be about 10% of the corresponding forces on the prototype.

[Answer: (a) 365.26 kPa]9. An aircraft wing of span 10 m and mean chord 2 m is designed to develop 45 kN lift atfreestream velocity 400 km/h and density 1.2 kg/m3. A 1/20 scale model of the wingsect ion is tested in a wind tunnel at velocity 500 m/s and density 5.33 kg/m3. The totaldrag measured is 400 N. Assuming the wind tunnel data refer to a sect ion of infinite span,

calculate the total drag and aerodynamic efficiency of the aircraft wing, assuming the loaddistribut ion to be ellipt ic.

[Answer: D = 2.649 kN, L/D = 16.99]

Further ReadingsAbbott , I.H. and Von Doenhoff, A.E., Theory of Wing Sections, Published by the authors, 1959.Bert in, J.J. and Smith, M.L., Aerodynamics for Engineers, Prent ice-Hall Internat ional, Inc. NJ,1988.Boas, M.L., Mathematical Methods in the PhysicalSciences, 3rd edn. John Wiley & Sons, Inc.,New York. 2006.Hoerner, S.F., Fluid Dynamic Drag, Published by the author, 1975.Houghten E.L and Carruthers N.R, Aerodynamics for Engineering Students, 3rd edn. EdwardArnold, London, 1982.Kuthe, A.M. and Chow, C.y., Foundations of Aerodynamics –Bases of Aerodynamic Design , 4thedn, John Wiley & Sons, Inc., New York, 1986.Lamb, H., Hydrodynamics, 6th edn, Dover Publicat ions, 1932.Liepmann H.W and Roshko A., Elements of Gas Dynamics, John Wiley & Sons, Inc. New York,1957.Milne-Thomson, L.M, Theoretical Aerodynamics, 2nd edn. Macmillan & Co., Ltd, London, 1952.Rathakrishnan, E., Applied Gas Dynamics, John Wiley & Sons, Inc., NJ, 2010.Rathakrishnan, E., Fluid Mechanics –An Introduction, 3rd edn. PHI Learning, Delhi, India, 2012.Schlicht ing, H., Boundary Layer Theory, 4th edn. McGraw-Hill Book Co., New York, 1960.Shapiro, A.H., The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. I , TheRonald Press Company, New York, 1953.

IndexAbsolute incidenceAccelerat ion

localmaterialpotent ial

AerodynamicsAerodynamic centerAerodynamic efficiencyAerodynamic force

coefficient ofparameters governing

Aerodynamic force andmoment coefficients

Aerodynamic twistAerofoil

cambereddefinit ion of

mean camber linenomenclaturepolar curvesect ionsymmetricalthickness distribut iontrailing-edge angle

Aerofoil characterist icswith general distribut ionof asymmetrical loading

Aerofoil geometryNACA 4-digit seriesNACA 5-digit seriesNACA 6-digit seriesNACA 7-digit seriesNACA 8-digit series

Aerofoils of smallaspect rat io

AileronAnalysis of fluid flowAngle of at tackAngle of downwashArea-Mach number relat ionArgand diagramAspect rat ioAverage chord

Banked turnBarlow's curveBarotropic fluidBasic and subsidiary lawsBernoulli equat ionBiot-Savart lawBiplaneBlenk's methodBluff bodyBodies of revolut ionBoundary layer

definit ion ofequat ionsthickness

Bound vortexBoyle's lawBulk modulus of elast icity

Calculat ion ofdownwash velocity

Calorically perfect gasCalorical state equat ions

Capillary act ionCamber

of circular arclowermeanupper

Camber lineCauchy-Riemann equat ionsCenter of pressureCentered expansionChange of aspect rat io

with incidenceCharacterist icsCharles' lawChord

of an aerofoillineof a profile

Circular arc aerofoilcenter of pressure

locat ion ofCircular vortex

size ofvelocity distribut ion

Circulat ioncrit icaldefinit ion ofillustrat ion ofphysical meaning ofsubcrit icalsupercrit icaltheorem

Climbing flightCoefficient of pressureComplex number

argument ofconjugate ofprincipal value of

Compound vortexCompressibilityCompressible Bernoulli

equat ionCompressible flows

thermodynamics ofCompressible flow

basic potent ialequat ion of

equat ions

fundamental equat ionCompressible subsonic flowCondit ion for vortex

drag minimumConformal t ransformat ionCont inuity equat ionCont inuumConservat ion of energyConservat ion of matterControl mass systemControl surfaceControl volumeConvect ion effectCrit ical stateCrocco's theoremCylindrical rectangular

aerofoilCylindrical wing

Darcy frict ion factord’Alembert 's paradoxDensityDetached shockDifferent ial analysisDihedral angleDimensionless velocityDisplacement thicknessDoublet

axis ofstream funct ion ofstrength of

Downwashanglefor ellipt ic loadingfor modified ellipt ic loading

Dragcoefficientinducedof bodiesof a supersonic profilepolarpressureprofile

EccentricityEffect of downwash

on incidenceEffect of operat ing

a flap

Effect ive incidenceElevatorEllipt ical distribut ion

characterist icsdownwash fordrag due to downwashlift for

Ellipt ic loadingdownwash fordownwash velocity forinduced drag forlift forlift and drag forlift curve slope for

Energy due to apair of vort ices

Energy thicknessEnthalpyEntrance length

in a pipeEntropy

change across a shockEuler's accelerat ion formulaEuler's equat ionEulerian descript ionExact Joukowski

pressure distribut iontransformat ionvelocity distribut ion

Expansion waves

Fanno flowFinFineness rat ioFinite aerofoil theory

spanwise loading and trailing vort icityFirst law of thermodynamicsFlapFlapped aerofoilFlow deflect ion angleFlow development lengthFlow over a

wave-shaped wallFlow past a

circular cylinderwith circulat ionwithout circulat ion

Flow past a half-bodyFlow through pipes

entrance lengthfully developedhead loss for

Flow with area changeFlow with frict ionFlow with T0-changeFluids

definit ion ofNewtoniannon-Newtonianpropert ies of

Force and momentcoefficients

Forced vortexForce on a vortexForm dragFree spiral vortexFree vortex

strength ofFrict ion coefficient

definit ion ofFrict ional dragFully developed regionFuselage

Gas constantGas dynamics

definit ion ofGeneral linear solut ion

for supersonic flowGeneral mot ionGeneral thin aerofoil

theoryGeneral thin aerofoil

sect ionL, M, kcp, of

Geometric twistGeometrical angle of at tackGeometrical incidenceGlidingGliding angleGothert 's rule

for 3-D flowto bodies of revolut ion and fuselageto wings of finite span

Gradient operatorGraphical descript ion

of fluid mot ionHelmholtz's theorems

firstfourthsecondthird

Hinge moment coefficientHorizontal flightHydrostat ic pressure

distribut ionHypersonic regimeHypersonic similarity

parameter

Ideal gasImage of a vortexImpact pressureIncidenceIncompressible flow

definit ion ofIncrease of entropy

principleInduced drag

for ellipt ic loadingminimum

Induced downwashInduced liftInduced velocityInfinite vortexIntegral and different ial analysisIntegral analysisIntegral equat ion

for circulat ionIrrotat ional flowIsentropic flowIsentropic process relat ionIsentropic relat ions

Jet flapJoukowski hypothesisJoukowski t ransformat ionJoule's law

Kelvin's circulat iontheorem

Kinematics of fluid flowKinematic viscosity

coefficientKutta condit ion

applied to airfoilsin aerodynamics

Kutta-Joukowski

theoremtransformat ion

funct ion

Lagrangian descript ionLaminar aerofoilLaminar flowLaminar sublayerLancaster–Prandt l

lift ing line theoryLaplace equat ion

basic solut ions ofLateral axisLaws of vortex mot ionLift

coefficientellipt ic loadingon the aerofoilof Joukowski aerofoil

Lift and drag by impulse methodLift ing line theoryLift ing surfaceLift ing surface theoryLine vortexLinear flightLinear vortex of finite lengthLocal rates of changeLongitudinal axis

Mach angleMach angle-Mach number

relat ionMach lineMach number

definit ion ofMach waveMagnus effectMass

definit ion ofMaterial rates of changeMayer's relat ionMean aerodynamic chordMean camber lineMoment coefficientMomentum equat ionMomentum thicknessMonoplane

main features oftypes of

Moving disturbanceMunk's reciprocal theoremMunk's theorem of staggerMutual act ion of two vort ices

NASA aerofoilsNavier-Stokes equat ionsNewtonian fluidNon-Newtonian fluidNormal axisNormal shock

relat ionstotal pressure across

Oblique shockrelat ionsθ-β-M relat ionstrongweak

Oswald wing efficiency

Panel methodParameters governing

the aerodynamic forcesPathlinePercentage camberPerfect gasPhugoid mot ionPhugoid oscillat ionPhysical meaning of circulat ionPhysical planePitching moment

coefficientPlane of symmetryPoint rect ilinear vortexPoisePoisson's equat ionPotent ial flowPotent ial equat ion for bodies of revolut ionPotent ial funct ionPrandt l-Glauert rule

for subsonic flowsupersonic flow

Prandt l-Glauertt ransformat ions

Prandt l lift ingline theory

Prandt l-Meyer funct ionPrandt l relat ionPressure

definit ion ofstat ictotal

Pressure coefficientdefinit ion of

Pressure dragPressure lossPressure distribut ion on Joukowski aerofoilPressure-hillProfileProfile dragPure rotat ionPure translat ion

Quarter chord pointRadial flowRankine's half-body

pressure distribut ion overRankine's theoremRarefied flowRat io of specific heatsRayleigh flowRectangular aerofoilRect ilinear vort icesReynolds number

crit icallowerupper

definit ion ofReynolds stressRolling momentRoot chordRotat ional and irrotat ional mot ionRotat ional flowRudder

Scale effectSect ion lift coefficientSemi-infinite vortexSeparat ion pointSeparat ionSeparat ion process

illustrat ion ofShock

angledefinit ion ofdetachednormaloblique

strengthShock-expansion theorySide-slipSimilarity ruleSimple flightsSimple vortexSingularit iesSingular pointsSkin frict ionSkin frict ion coefficient

definit ion ofdrag

Small perturbat ion theorySolut ion of nonlinear

potent ial equat ionSource panel method

governing equat ion ofpressure distribut ion

around cylinderaccuracy of

SpanSpecific heat

at constant pressureat constant volume

Speed of soundStallingStart ing vortexSteady flowStoke's theoremSource and sinkSource-sink pairState equat ion

caloricalthermal

Straight horizontal flightStreaklineStreamlineStreamline analogyStreamlined bodyStream funct ion

for simple vortexfor sinkfor source

StreamtubeSubsonic flowSupersonic flow

over a wedgeSudden increase in

incidenceSupersonic compressionSupersonic expansionSupersonic flowSupersonic regimeSurface tensionSweep angleSystem and control volume

Taper rat ioTemperature

stat icThermal conduct ivity of airThermal equat ion of stateThermally perfect gasThermodynamic propert iesThickness-to-chord rat io

of cambered aerofoilThin aerofoil theory

applicat ion ofThompson's vortex theoremTimelinesTip chordTip vortexTrailing vortex dragTransformat ion of circle

to camberedaerofoil

to circular arcto ellipseto straight lineto symmetrical

aerofoilTransformat ion of flow patternTransformat ion funct ionTransformed planeTransit ion pointTransonic flowTrimTrim dragTriplaneTurbulence

descript ion ofnumber

Turbulent boundary layerTwisted wingTwo-dimensional

compressible flows

Universal gas constantUnsteady flow

Velocity distribut ionon Joukowski aerofoil

Velocity potent ialVelocity of soundViscosity

absolute coefficient ofdynamic coefficient ofkinematic coefficient ofMaxwell's equat ion forNewton's law ofSutherland's relat ion

Viscous flowsvon Karman rule for

t ransonic flowapplicat ion to wings

Vortex betweenparallel plates

Vortex dragcondit ion for minimumfor modified loading

Vortex mot iondefinit ion of

Vortex pairVortex panel method

applicat ion ofgoverning equat ion of

Vortex theoremsVortex theoryVortex tubeVort icity

equat ion inpolar coordinatesrectangular coordinates

WakeWall shear stressWash-inWash-outWave dragWeak oblique shockWing

cylindricalgeometry oflow-winghigh-wingmid-wing

parasol-wingplanform areaportprofileshoulder wingspanstarboardt ipstwist

Wing geometricalparameters

Wing loading

Yawing moment

Zero aspect rat ioZone of act ionZone of silence