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APPLIED HYDRODYNAMICS

APPLIED

HYDRODYNAMICS

H. R. V ALLENTINE

Professor of Civil Engineering

University of Newcastle, New South Wales

SECOND EDITION

Springer Science+Business Media, LLC

1967

First published by Butterworth & Co. (Publishers) Ltd.

First Edition, 1959 Second lmpression, 1961 Third Impression, 1963 Fourth Impression, 1965

Second Edition, 1967

© Springer Science+Business Media New York

1967

Original1y published by Butterworth & Co. (Pub1ishers) Ltd. in 1967.

Library ofCongress Catalog Gard number 67-29085

Set in M onotype Baskerville type London and Colchester

ISBN 978-1-4899-6270-6 ISBN 978-1-4899-6586-8 (eBook) DOI 10.1007/978-1-4899-6586-8

Softcover reprint of the hardcover 2nd edition 1967

PREFACE TO THE SECOND EDITION

The continuing demand for this book by students of Engineering, Science and Applied Mathematics appears to justify this new edition. In its general revision, numerous minor changes have been made to the text, a new chapter devoted to Vortex Motion has been added and Chapters 4 and 5 have been extended by the inclusion of treat-ment of the Method of Images and its applications. The tr~atment of some aspects of the flow of real fluids in Chapter 2 is purposely brief since it is assumed that the Engineering and Science students, for whom the chapter is primarily intended, will have had an introduction to fluid mechanics.

This edition incorporates a number of suggestions made by re-viewers and by colleagues in various countries, to all of whom the author is particularly grateful.

Newcastle H. R. v ALLENTINE

PREFACE TO THE FIRST EDITION

DEVELOPMENTS over the past few decades in the science of fluid mechanics have provided scientists and engineers with the means of predicting the behaviour of fluids with a precision far greater than was considered possible at the turn of the century. In particular, the principles and methods of classical hydrodynamics, once neglected as unpractical, have been shown to be usefully applicable, within defined limits, to many practical flow problems. The study of this branch of applied mathematics provides not only a broad funda-mental basis for the scientist's and the engineer's knowledge of fluid mechanics, but also the means of attacking problems of flow in two, and in many instances, three dimensions.

This book is intended for use in universities as an introduction to Hydrodynamics for students of Applied Mathematics, and as a course in Fluid Dynamics for senior and post-graduate students in Civil, Mechanical and Aeronautical Engineering. In lts _(lpproach to the subject, the book has a bias towards practical application and, in this respect, it differs from those established hydrodynamic works which are essentially mathematical treatises. For the student who finds difficulty in mastering mathematical abstractions, particular attention has been paid to the provision of detailed physical ex-planations of such concepts as stream function, potential function and conformal transformation, as a prelude to the more formal mathematical treatment of these topics.

The practical approach has made it seem desirable to depart from the tradition of hydrodynamics texts in other ways. Since the object is to demonstrate the methods of adapting non-viscous flow theory to the analysis of real, or viscous, fluid flow, it is necessary that the student be forewarned regarding possible sources of error· in the application of the methods. For this reason, following upon the presentation of the fundamentals of non-viscous flow in Chapter 1, there is provided, in Chapter 2, a brief description of the distinguish-ing characteristics of real fluid flow and of the exact and approximate methods available for its analysis. The chapter provides a justifi-cation for the application of the methods of classical hydrodynamics under certain conditions and defines, in general terms, the permissible limits of such application.

Other departures from tradition are evident in the presentation of the subject-matter chapter by chapter as a series of more or less

vii

PREFACE

alternative methods of approach; the inclusion of the useful yet simple methods of graphical and numerical analysis; and the order of presentation, which has been so arranged that, for Engineering students, the first three chapters are suitable for inclusion in under-graduate lecture courses in Fluid Mechanics whilst the balance of the book can be used in elective, honours or graduate courses; for Mathematics and Science students, a selection can be made of those chapters appropriate to their particular courses of study.

The book is based upon the author's courses oflectures in Hydro-dynamics presented over the past five years to final year and graduate students in Civil Engineering at the University of New South Wales. Material has been drawn from numerous sources as the reference lists show, but particular acknowledgment is made to Lamb's Hydrodynamics (Cambridge University Press, 1932), Streeter's Fluid Dynamics (McGraw-Hill, 1948), to the lectures on Hydrodynamics given in 1952 at the Iowa Institute ofHydraulic Research of the State University oflowa by Professor C. S. Yih and to the various published works of the Director of that Institute, Professor Hunter Rouse.

The author is particularly grateful to his colleagues, Professor L. C. Woods, Professor A. S. Hall, Mr. P. S. Barna and to fellow members of the staff of the University's Hydraulic Research Station for many valuable suggestions.

H. R. VALLENTINE Sydney

viii

CONTENTS

PREFACE TO THE SECOND EDITION

PREFACE TO THE FIRST EDITION •

1. FLOW OF AN IDEAL FLUID Introduction. Fluid properties, viscosity; the ideal fluid. Pressure at a point. Equation of continuity. Boundary con-ditions. Streamlines. Two-dimensional flow patterns. Rota-tional and irrotational flow, rotation, vorticity. Stream functions, definition and properties. Velocity potential func-tions; definition and properties; use of polar co-:ordinates. Flow nets; definition and characteristics. Euler's equations of motion. The Bernoulli equation, general and restricted forms. Velocity and pressure distributions; the effect of gravity; piezo-metric head; flow with a free surface. Energy considerations; non-viscous flow; irrotational flow; irrotational flow theorems. Determination of flow patterns.

Page

v

vii

2. FLOW OF A REAL FLUID • 55 The effects of viscosity; laminar and turbulent flow. Laininar flow; pressure at a point; the Navier-Stokes equations, exact solutions, approximate solutions for flows at low and at high Reynolds numbers. Turbulent flow and the boundary layer; laininar and turbulent boundary layers, boundary layer thick-ness, pressure distribution along a boundary. Velocities in the boundary layer; the laininar sub-layer, the Karman-Prandtl equations for flow past smooth and rough boundaries. Bound-ary layer separation, streamlining, submerged and free surfaces 'of separation. The Bernoulli equation in real fluid flow. Flow pattern analysis.

3. GRAPHICAL FLOW NETS, NUMERICAL ANALYSIS AND ExPERIMENTAL ANALOGIES

Graphical flow nets; principle; method of construction; seepage flow nets, confined and unconfined flow. Numerical analysis; principle; method. Experimental analogies; the membrane analogy; the electrical analogy; theviscous flow analogy.

77

4. STANDARD PATTERNS OF FLOW. 102 Uniform flow. Source. Irrotational vortex; circulation. Doublet. Graphical addition of patterns. Source and sink. Vortex pair. Source and vortex-spiral flow. Source and uni-form flow-flow past a half-body. Doublet and uniform flow-flow past a cylinder; virtual mass. Doublet, vortex and uniform

I* ix

CONTENTS

flow-flow past a cylinder with circulation; Magnus effect; coefficient of lift. Source, sink and uniform flow-flow past a Rankine body. Method of images.

5. CONFORMAL TRANSFORMATION-I Complex numbers. Operations with complex numbers. Func-tions of a complex variable. Analytic functions; singularities; the Cauchy-Riemann equations; complex potential. Signifi-cance of dw/dz: (I) The complex operator; (2) The complex velocity. Inverse transformations. Successive transformations. Some simple transformations: ( 1) Uniform flow. (2) ( i) Source at z=a; (ii) Vortex at z=a; (iii) Spiral vortex at z=a. (3) Doublet. (4) Source and sink. (5) Flow at a wall angle. (6) Flow through an aperture. (7) Flow into a rectangular channel. (8) Flow past a cylinder: (i) without circulation; ( ii) with circulation. Transformations of the circle, J oukowski profiles. ( 1) Flow parallel to a flat plate. (2) Flow normal to a flat plate. (3) Flow past an ellipse. (4) Flow past a streamlined strut. (5) Flow past a circular arc. (6) Flow past an aerofoil. Lift of an infinite aerofoil; Joukowski's hypothesis. Use of images.

136

6. CoNFORMAL TRANSFORMATION-I! }89 Simple closed polygons. The Schwarz-Christoffel theorem. Semi-infinite and infinite strips. Flow out of the end of a chan-nel. Flow out of the side of a channel. Flow past a flat plate without separation. Flow into a rectangular channel. Free streamline theory. Flow through a plane slot. Flow through a 90° slot. Flow through a nozzle. Borda's mouthpiece. Flow past a flat plate with separation.

7. THREE-DIMENSIONAL IRROTATIONAL FLOW . Introduction. Equation of continuity in cartesian, cylindrical and spherical co-ordinates. Stokes's Stream Function for axi-symmetric flow; relationship between ,P and Q; irrotational flow equations. Velocity potential function for three-dimen-sional flow; axisymmetric flow. Standard flow patterns; uni-form flow, source, doublet, line source. Source and uniform flow-flow past a half-body. Doublet and uniform flow-flow past a sphere. Source, sink and uniform flow-flow past a Rankine body. Source, line sink and uniform flow-flow past a streamlined body. Numerical analysis.

8. VoRTEX MoTio::>~ . Introduction. Circulation and vorticity; vortex lines and tubes; persistence of vortex strength; vortex sheet; vortex street; motion due to two-dimensional vortices; vortex rings; the finite wing.

X

231

256

CONTENTS

APPENDIX A The Theorems of Green, Stokes, Cauchy and Blasius . 274

APPENDIX B A summary of trigonometric, exponential and hyperbolic functions 286

APPENDIX C A summary of hydrodynamic equations for incompressible non-viscous flow 289

INDEX • 293

xi

1

FLOW OF AN IDEAL FLUID

1.1 Introduction

HYDRAULIC and aerodynamic engineering analysis and design involve predictions of patterns of fluid motion and of fluid forces associated with those patterns. The selection of the appropriate method for making the predictions is determined by the nature of the particular problem and by the precision desired. In some cases the task is quite simple, as, for example, the precise determination, by the principles of hydrostatics, of pressure forces exerted by a stationary body of water. Certain cases of viscous flow can be solved, either precisely or with negligible error, by means of the viscous flow equations due to Navier and Stokes. In the great majority of fluid flow problems, however, precise analytical determinations of forces and .velocities are not possible, owing to the complex effects upon the flow of fluid viscosity. In such cases, recourse must be had to simplifying assumptions in order that approximate analytical solu-tions may be obtained.

The simplest and most common approximation is the method of analysis in one dimension, which yields entirely adequate solutions to many problems requiring the determination of total forces rather than pressure distributions, or of average velocities rather than velocity distributions. In a one-dimensional analysis, velocities and pressures are assumed to vary with distance only in the general direction of flow, and mean values of the velocity on planes normal to this flow direction are adopted for purposes of calculation. On these planes, velocity variations due to the effects of viscosity and to changes in boundary alignment are ignored (Fig. 1.1).

Pressure and velocity distributions in general cannot be determined by means of the one-dimensional approach. In certain cases, they can be determined with good precision by analysis in two or three dimensions and, where such methods are inadequate, laboratory model tests must be undertaken.

Most analytical approaches are based upon the methods of classical hydrodynamics, the branch of applied mathematics which treats of the perfect or ideal fluid. This hypothetical fluid is, by definition, incompressible and non-viscous, so that it experiences no shearing stresses and its elements in contact with solid boundaries

l


slip tangentially, without resistance, along those boundaries. The fluid force on any element of the boundary surface is normal to that surface.

In the case of motion of a real fluid, shearing stresses are always present and there is no slip at the boundary. The fluid particles in contact with the boundary adhere to it and have no tangential motion along it. These fundamental differences between ideal and real fluids with respect to slip and shear stresses account for the divergence of theoretical predictions from experimental fact in many cases of real fluid flow. According to the ordinary theory of ideal fluid flow, for example, a body which moves through a fluid ofinfinite extent experiences no drag force (d'Alembert's paradox). In a real fluid, the condition of no slip between the surface of the body and the

~Turbulent flow ~;,.,."'"""""~ Irrotabonal flow

(a)

1"111~. f::::

---::r-4~

(b)

Figure J.J-Velocity distribution in a two-dimensional contraction. (a) Irrotational flow and turbulent flow, (b) one-dimensional

approximation

fluid elements in contact with it, and the existence of viscous tan-gential stresses in the neighbouring fluid layers together account for the ever-present drag on the moving body.

There are, nevertheless, many practical problems in fluid dynamics which can be solved with fair precision by the methods of classical hydrodynamics, in particular, by the potential flow theory. They include such cases as the distributions of velocity and pressure around the leading portions of streamlined objects and in flow in converging passages, the form and motion of surface waves on a liquid, the profiles offree jets and weir nappes, pressure distributions resulting from impulsive actions (prior to the occurrence of appreci-able fluid motion) and, paradoxically for an ideal fluid theory, certain classes of essentially viscous motion, such as percolation through granular materials.

A sound appreciation of the possibilities and limitations of classical hydrodYIJamic theory as a practical engineering design tool requires a knowledge of elementary hydraulics, which it is assumed the reader

2

FLUID PROPERTIES

possesses, and some familiarity with modern developments in the science of the mechanics of fluids. Therefore, following the treatment of the fundamentals of ideal fluid flow in this chapter, attention is directed, in the next, to the characteristics of real fluid flow which distinguish it to a greater or lesser extent from ideal fluid flow. The remainder of the book is devoted to patterns of ideal fluid flow and methods of predicting those patterns, and it should be read with the nature of real fluid flow borne in the mind, so that the possibilities and limitations of ideal flow theory as a practical design approach may be assessed correctly.

1.2 Fluid properties A fluid is a material which flows. Its capacity to flow, which

distinguishes it from a solid, results from the fact that its particles can be readily displaced under the action of shearing stres~es. Like solids, the fluids known as the liquids offer considerable resistance to compression and tensile stresses but all fluids deform readily and continuously under the action of a shearing stress, so long as that · stress operates. If the rate of shear deformation is small, the fluid offers negligible resistance. With increased rates of deformation, it offers an increased resistance, which disappears, howe\rer, once the deforming motion ceases. The resistance arises from the existence of viscosity in the fluid.

A Newtonian fluid is one in which the shearing stress, in one directional flow, is proportional to the rate of deformation as meas-ured by the velocity gradient across the flow. Thus,

du T = /L- •. • • (1.1)

dy

where -r is the shearing stress in the x-direction on planes normal to they-axis; u is the velocity in the x-direction, varying only withy; and IL• the constant of proportionality, is the dynamic viscosity or coefficient of viscosity, which for any particular fluid varies sig-nificantly only with change in temperature (Fig. 1.2). The common fluids, such as air, water and light petroleum oils, are Newtonian fluids. Non-Newtonian fluids, whose behaviour does not conform to Eq. 1.1, are not usually met in engineering practice and will not be considered further.

A liquid is a fluid which, at a given temperature, occupies a definite volume which is little affected by change in pressure. Poured into a stationary container, the liquid will occupy the lower part of the container and form a free level surface. In most hydraulic calculations, liquids can be regarded as incompressible. In fact,

3


under many conditions, it is convenient and reasonable to regard even a gas such as atmospheric air as incompressible, provided that the velocities involved are small compared with the velocity of sound in the gas.

The mathematical approach to the study of fluid flow is greatly simplified if attention is restricted to a hypothetical 'ideal', or 'perfect' fluid, which is assumed to possess zero viscosity. In common with real fluids, it has density and it flows, but it offers no resistance to shearing deformations so that there can be no shear stress in an ideal fluid. It is assumed to exhibit no surface tension effects and it does not vaporize, so that cavitation, or the formation oflow pressure vapour pockets, does not occur. The density of the fluid and the

1

!Y!..- 0 dx-(a)

dp -ve dx

(b) Figure 1.2-Relationship between shear stress and velocity gradient in one-

PRESSURE AT A POINT

'point' as a very small volume which is yet very large compared with the spacing of fluid molecules. Alternatively, the fluid is assumed to be a continuous medium in which point values correspond to the average values in the small volumes mentioned above.

The forces acting upon a small element in a mass of fluid are classified for convenience as body forces and surface forces. A body force is an external force, which is proportional to the volume of the element and acts 'at a distance' through its centre of mass. Gravity forces are the only body forces with which the hydraulic engineer is

0

y

Figure !.~Forces acting on an element of fluid in two-dimensional non-viscous flow

concerned. A surface force is an internal force with a magnitude proportional to the area upon which it acts. Internal pressure and viscous shear forces are the surface forces involved in fluid flow. In an ideal fluid, since there is no tangential or shearing stress, the only surface forces are pressure forces normal to the areas they act upon The interrelationship of the stresses at a point resulting from these forces is developed for two-dimensional ideal fluid flow, with reference to Fig. 1.3 in which for purposes of generality, they-axis is inclined at some angle ,8 to the line of action of gravity. The lengths of the sides of the triangular element are respectively a, b and c and the corre-sponding normal pressures are Px• py, and Pa.· The fluid density is p and the acceleration due to gravity is g.

The sum of the x-direction components of the forces acting upon the element within the fluid equals the product of the mass of the

5


element and its acceleration, ax, in the x-direction, in accordance with Newton's second law of motion.

Pxa-p,_csinor.+tpabgsinfJ = lpabax

:. Px-Prr.~+ipbgsinfJ = ipbax a

In order to consider the stress at a point, let a and b approach zero. The equation reduces to

Px-Prr. = 0

Since a similar treatment of the force components acting in the y-direction shows that py-p,_ = 0, it follows that

Px = PY .... (l.2a) For the three-dimensional case, it can be shown that

.... (l.2b)

so that, in an ideal fluid, even if the fluid is accelerating, the pressure at a point is the same in all directions.

1.4 Equation of continuity The velocity components and the density of the fluid at a point

are related through the requirement that the fluid must be con-tinuous, both in space (that is, no voids occur in the fluid) and in time (that is, fluid mass is neither .created nor destroyed). The relationship holds for both viscous and non-viscous fluids. Consider an elemental parallelepiped of dimensions Sx, Sy, Sz, through which fluid is flowing (Fig. 1.4). If the centre of the element is at (x,y, z) and the velocity components at the timet at this point are respectively u, v and w, then the mass flow rate past the centre, through the element in the x-direction is (mass-per-unit-volume) x velocity x area, pu Sy Sz. The mass flow rate in through the face nearer the origin a distance !Sx is (pu- 0~:) tsx) SySz and the mass flow rate out through the face farther from the origin a distance tSx is

(pu+0~:) tsx) SySz. The net gain in mass per unit time, within

the element from these two faces is - 0~:) Sx8y8z. Similarly, the gains in mass per unit time from the other two pairs of faces are

- o(pv) Sx8y8z and - o(pw) Sx8y8z ay az

6

EQUATION OF CONTINUITY

The total gain in mass per unit time from all faces is

- [o(pu) + o(pv) + o(pw)] 3x3y3z ax ay oz

which must equal the time rate of increase in mass :t (p3x3ydz) or ~ 3x3y8;:; whence

.... ( 1.3)

which is the Equation of Continuity. It is applicable throughout all fluids, except at isolated singularities, to be discussed in later chapters.

Figure 1.4-Equation of Continuity. Mass flows in the x-direction across the faces of a parallelepiped in three-dimensional flow

For incompressible fluids, with p constant, the Equation of Continuity reduces to

.... (1.4)

If the velocity component is constant in one direction say, the ;:;-direction, the corresponding term disappears from the continuity equation which reduces to the two-dimensional form

~u + ~v = 0 .... (1.5) vx oy

7


One-dimensional considerations lead to the simpler form of the continuity equation of elementary hydraulic texts. The tube-shaped volume in Fig. 1.5 is so located in the flow that fluid flows only axially along it; flow does not occur inwards or outwards through its sides but only across its ends. If its cross-sectional area and the mean velocity of flow midway along its length are respectively A and V, both

Figure 1.5-Equation of Continuity. Mass flows across the faces of a stream tube

of these being functions of the distance, s, along the axis of the tube, an approach similar to that adopted above yields the relationship

o(pA) o(pAV) O at+ ----as- =

For incompressible fluids with p constant and A a function of s and t, as in unsteady open-channel flow

oA + o(AV) = 0 ae as o d(AV)

and for steady flow, at = 0, hence dS = 0 A V = constant .... (1.6)

In this form the continuity equation for steady flow of an incompres-sible fluid relates the mean velocity in a given direction to the flow cross-sectional area normal to that direction. Eqs. 1.3, 1.4 and 1.5, on the other hand, show the interrelationship of the velocity com-ponents at any point in the flow.

1.5 Boundary conditions The data necessary for an analysis of a fluid flow problem must

include sufficient information concerning all of the boundaries including any arbitrarily located fluid inflow and outflow boundaries.

8

BOUNDARY CONDITIONS

The analysis consists of the application of the principles of fluid mechanics so as to predict the behaviour of the fluid when subjected to these boundary conditions.

The possible conditions specified for a boundary may include its nature, whether solid, fluid or free surface, its geometrical form, the pressure distribution on it or the velocity distribution along or across it.

An ideal fluid flowing along a solid boundary is assumed to remain in contact with the boundary without penetrating it, so that a fluid particle on a stationary boundary can have no component of velocity normal to the boundary. If the boundary is moving, the velocity normal to the boundary of a particle at a point on it must be equal to the velocity of the boundary normal to itself at that point. If l, m and n are the direction cosines of the inward normal to the surface at a point (that is, the cosines of the angles between the

z

of boundary

Figure 1.6-Velocity relationships at a solid boundary

particle's velocity components, u, v and w, and the normal drawn into the fluid) the components of u, v and w, along the normal are lu, mv, and nw, respectively. The particle velocity along the inward normal is the arithmetic sum of these components (Fig. 1.6). If the boundary velocity normal to itself at the point is vn then

lu+mv+nw = vn .... (1.7a) and if the boundary is stationary,

lu+mv+nw = 0 .... (1.7b)

Use will be made of these equations in the consideration of energy relationships in Section 1.15.

9


It is evident that fluid particles on a stationary or moving boundary must remain in contact with it, their movements being wholly tangential to it. (However, there may be isolated points or lines of discontinuity, to be discussed later, where the fluid particles do, in fact, leave the boundary.) If the equation of the boundary surface is F( x, y, z, t) = 0 the co-ordinates of any fluid particle on the boundary must continuously satisfy this equation.

Suppose that, in a small element of time, St, a particle moves along the boundary through a short distance whose components are Sx, Sy, Sz. Since its new position must satisfY the equation of the boundary surface, the change SF must be zero.

oF oF oF oF SF= -Sx+-Sy+-Sz+-St = 0

ox oy oz ot

SF_ oFSx oF~ oFSz oF= O St- oxSt+oy St+ozSt+ot

and, therefore, as St -? 0

dF oF oF oF oF - = u-+v-+w-+- = 0 dt ox oy oz ot

.... (I. 7c)

where u, v and ware the velocity components of the particle. This equation must be satisfied at all points on a boundary surface

(except at points of discontinuity in the flow pattern).

ExaDlple ·

1.1. The parabolic profiley= kx112 moves in the negative x-direction with a velocity U, through a fluid which was initially stationary. If u and v are the instantaneous velocity components of a fluid particle on the boundary,

v k2 show that -- = -

u-U 2y

Solution.-The equation of the moving profile is y=k(x-Ut)112 or y2=k2(x-Ut) or

F = y 2 -k2(x- Ut) = 0

with t measured from the instant the profile is tangential to they axis ( U

being negative). Since':::= 0 for all particles on the profile

oF oF dF u-+v-+- = -uk2 +2r:Y+ Uk2 - 0 ox oy dt

•. 2r:Y= (u-U)kZ

v k2 ·· u- U == 2)

10

STREAMLINES

If the co-ordinate axes are considered to move with the profile, U becomes zero and the undisturbed fluid has a steady velocity of magnitude U in the positive x-direction. The profile is now tangential to they-axis and, for fluid particles in contact with it, the above equation becomes

v k1 u = 2Y

that is, the slope, ~, of the velocity vector equals the slope of the profile, u

since dy = _

In addition to the above kinematical condition for solid boundaries, certain other physical boundary conditions can be stated briefly. Stationary fluids and non-viscous fluids exert pressure forces which are wholly normal to the elements of solid boundaries, while forces exerted by moving viscous fluids have tangential (shearing) as well as normal components. When two different fluids are in contact, the pressure must be the same in each fluid at a point in the surface of contact, that is, the pressure cannot suddenly increase ~long a line passing through the surface. In the case of motion of a body under the action of a finite force through a fluid which extends to infinity, an essential condition is for the velocity of the fluid at infinity to remain unchanged by the body's motion. Otherwise the action of a finite force would be imparting kinetic energy to an infinite mass of fluid in a finite time, which is impossible.

1.6 StreaJnlines Analysis of a particular condition of flow, in order to determine

the velocities and pressures, involves the determination of the pattern of flow. The flow rna y be two-dimensional, in which case its character-istics, such as velocity and pressure, vary only in two co-ordinate directions, say, the x- andy-directions. There is no flow variation in the z-direction so that the patterns of flow on all planes in the fluid normal to the z-axis are the same.

In three-dimensional flow, the flow characteristics vary in the x-, y- and z-directions. A special class of three-dimensional flow is axially symmetrical (or 'axisymmetric') flow in which the pattern of flow in all planes containing the axis is the same (Figs. 1.7, 7.13, 7.14).

A line which is at all points tangential to the velocity vectors at a given instant is a streamline. For example, in the two-dimensional flow pattern in Fig. 1.8, the streamline passing through the point

11


P(x,y) is tangential to the velocity vector Vat P. If u and v are the x- andy-components of V,

or

or

(a)

v dy -=tan8=-u dx

dx dy u v

udy-vdx = 0

.... (1.8a)

.... (1.8b)

.... (1.8c)

(b)

Figure 1.7-Examples of three-dimensional axisymmetric flow (see also Figs. 7.13, 7.14)

y

Figure 1.8-Definition of streamlines

For three-dimensional flow, the corresponding relationship is

dx dy dz u v w

.... (1.8d)

It is evident that no flow occurs across a streamline. A surface across which no flow occurs, in three-dimensional flow, is a stream-surface and a streamsurface in the form of a tube is a stream tube.

12

TWO-DIMENSIONAL FLQW PATIERNS

1. 7 Two-dimensional flow patterns For the visual representation of a flow pattern, the configuration

of an arbitrary number of selected streamlines is established either analytically, graphically or experimentally. The streamlines are selected so that they divide the flow into a number of channels with equal flow rates. If the total flow rate is Q per unit depth normal to the plane of the streamlines, and the number of channels is n, the

flow throughout the length of each channel is q = q. The mean n

velocity, V, in any channel, at a section where the channel width is

A.

c

Distribution of velocity along CD

Figure 1.9-Two-dimensional flow at a conduit entrance. Velocity distributions detennined from relative spacings of streamlines

b, is V = i and hence, throughout the pattern, the velocity at any point is inversely proportional to the streamline spacing at that point.

In Fig. 1.9, two-dimensional flow at the entrance to a conduit is represented by four streamlines, two of which coincide with solid .boundaries, and the total flow is divided into three channels. If the reference section is the region of uniform flow within the conduit, where the velocity is V0 and the channel width is b0, then for any other point P,

13


By measurement of streamline spacings normal to the direction of flow, the distribution of velocity within the flow is readily established. Since a velocity so determined is an average value for the width of the channel, an increase in the number of streamlines, within practical limits, results in an increase in precision in the estimate of the velocity at a point.

Some of the characteristics of these streamline flow patterns deserve brief consideration.

(1) Since a streamline is tangential to the velocity vector at all points, there can be no finite component of velocity normal to it, that is, there can be no flow across a streamline.

Irrotational flow

Turbulent flow

1·0

Figure 1.10-Effects of abrupt changes in boundary alignment on the velocity

(2) The streamline spacing varies inversely as the velocity, so that relatively narrow spacings indicate relatively high velocities. Streamlines converging in the direction of flow indicate an increase in velocity with respect to distance, that is, a convective acceleration.

(3) Streamlines do not cross. They intersect only at points of theoretically infinite velocity (Fig. 4.2) and at isolated points of zero velocity. The latter are called stagnation points (for example, the pointS in Fig. 1.1Jb).

( 4) A point in the flow where the direction of boundary streamline changes abruptly is a stagnation point if the included angle, measured within the fluid, is less than 180°; and a point of theoretically infinite velocity if the angle exceeds 180°. In Fig. 1.10, the point B is a stagnation point and the point C is a point at which the velocity is theoretically infinite.

14

TWO-DIMENSIONAL FLOW PATTERNS

(5) In steady flow, the configuration of the pattern of streamlines does not change with time. In unstea4J! flow with a free surface or with a moving internal or external boundary, the flow pattern referred to a stationary origin does change with time. In the case of unsteadiness resulting from a variation with time of the total flow rate between fixed and solid boundaries, the general configuration of the pattern does not change, but the flow rate in each channel, and hence the velocity at each point, does.

(6) Solid, stationary boundaries are streamlines provided that separation of the flow from the boundary does not occur, (the pheno-menon of separation is described in Section 2.5); steady-flow, free-surface profiles such as that of gradually varied flow in an open channel and those of a sharp-crested weir nappe are streamlines.

(a) {b) Figure 1.11-Motion of a body through a fluid. (a) Unsteady pattern, as would be recorded by a stationary camera with a brief time exposure, (b) steady pattern as would be recorded by a camera moving with the body

However, in unsteady flow patterns, such as the instantaneous pattern presented to a stationary observer by the movement of a submerged object through a fluid (Fig. l.lla) or by the passage of a surface wave (Fig. 1.12a), the moving boundary, whether a solid or a free surface, is not a streamline.

(7) Certain cases of unsteady flow patterns resulting from the movement of a solid or free surface boundary at a constant velocity relative to the observer can be transformed into steady flow patterns by the superposition of a pattern of constant velocity in the direction opposite to that of the boundary motion. In effect, the boundary is brought to rest relative to the observer by the movement of the observer with the velocity of the boundary. The procedure results in an entirely altered streamline pattern and, by eliminating local acceleration effects simplifies, the study of the fluid motion (Figs.l.Jl

15


and 1.12). Analytical and graphical methods of combining flow patterns are presented in Chapter 4.

(8) The distinction between streamlines, pathlines and streak-lines is important. A pathline is the track actually followed by a particle of fluid. A streakline is a line joining the instantaneous

~ (a)

(b) Figure 1.12-Streamlines of surface waves. (a) Unsteady

pattern, (b) steady pattern

B

Figure 1.13--Streamlines, streaklines and pathlines

positions of a succession of particles which have issued from the one source or passed through one point and its form varies with time in unsteady flow. In steady flow, streamlines, pathlines and streaklines coincide. In unsteady flow, they do not, asmaybeseenfromFig.l.l3, which shows instantaneous views of a streakline consisting of a wisp

16

ROTATIONAL AND IRROTATIONAL FLOW

of smoke, at time tl> and a short interval later, at time t2• A varying lateral wind is represented by the instantaneous streamlines wl at time t1 and W2 at time t2• The smoke particles are blown along the lines a-b, which are pathlines. The distinction between these terms is of particular importance in the use of time exposure photography for the determination of flow patterns in the laboratory.

1.8 Rotational and irrotational flow

A fluid particle in flow in a straight or curved path may suffer distortion or rotation, or both, in the course of its motion (Fig. 1.14). If none of the particles in a region of fluid suffers rotation, the flow is said to be irrotational in that region. Since both the rotational and

(a) (b) Figure 1.14--Fiow around a curved path. (a) Distortion without rotation, (b) rotation without appreciable distortion ·

the irrotational forms of ideal fluid flow have their approximate counterparts in real fluid behaviour, and since the characteristics of the two forms are quite distinct, it is profitable to consider the distinction from a mathematical viewpoint.

A particle is said to have zero rotation in a plane if the average of the angular velocities of two mutually perpendicular linear elements of the particle in that plane is zero. For example, if one line rotates in an anticlockwise direction at the same rate as the other rotates in a clockwise direction (Fig. 1.14a), the particle is distorting, but not rotating. In Fig. 1.15, which shows a rectangular element in two-dimensional flow, the broken lines indicate the displacement of the element relative to one of its points, A, in the period ot. The angular velocity of AB about the z-axis is

and of AD

ov -oxot lim ofh = lim ~ ov

Bt-+0 ot Bt-+0 OXOt = OX

lim 082 = ou ar-+o ot - oy

17


The average of the angular velocities of these two line elements is known as the rotation, w.

w = t(av- au) .... (1.9) ax ay When w is not zero at a point or region, the flow is rotational and

'vorticity', {, which is numerically equal to 2w, is said to exist at that point or region. '= ov- au ax ay

- .E!:L 6 y~yt ay __ .., --- I D -----

I I I I I

au 6 u • ay 'Y I

\ vI

I 6 o,

I I I I

c

v + 8 " 6x 8x

a" 6x 1St ax

.... (1.10)

-L--~A~~~~~u~--6x---------1~~--------.. x

Figure 1.15-Definition of rotation

The condition for two-dimensional flow to be irrotational is that the rotation, and hence the vorticity, is everywhere zero, i.e.,

ov au ox ay .... (1.11)

In the case of three-dimensional flow, rotation is possible about each of three axes which are parallel to the x-,y- and z-axes respectively.

There are then three possible components of rotation, wx, wy, and wz, and three corresponding components of vorticity, ~' 7J and {.

Wy = 2 OZ-: OX = 27] . • •. (1.12) I (au ow) 1 }

18

STREAM FUNCTIONS

and the condition for irrotationality in that, throughout the flow,

three-dimensional flow is

ow OV - = -; oy oz

ou ow oz = ox;

ov ou ox= oy .... (1.13)

There may be isolated points or lines in an otherwise irrotational flow where these conditions are not satisfied. Such points or lines are known as singularities. They are points or lines where the velocity is zero or theoretically infinite.

Section 4.3 introduces the subject of circular or vortex motion, in which the flow may be rotational throughout the region, or pre-dominantly irrotational with a central core of rotational flow. The circulation concept is defined and an alternative method of evalua-ting vorticity in terms of circulation is presented. A more compre-hensive treatment of vortex motion is given in Chapter 8.

Exa!nple 1.2. Show that the following velocity field is a possible case ofirrotational

flow of an incompressible fluid

u- yzt

v- zxt

w- xyt

Solution.-The equations satisfy the equation of continuity, for

ou ov ow ou ov ow - = - = - = 0 so that- +- + - - 0 ox oy oz ox oy oz

and therefore, the field is a possible case of fluid flow. The components of rotation are

w - !(ow- ov) - t(xt-xt) - 0 x 2 oy oz

I (ou ow) w = - -- - = i(yt-yt) - 0 y 2 oz ox

w = !(~-~) = t(zt-zt) = 0 z 2 ox oy

hence vorticity is zero and the field could represent irrotational flow.

1.9 StreaJD functions It is convenient to have some means of describing, in a concise

manner, the form of any particular pattern of flow. An adequate description should convey the notion of the shape of the boundaries,

19


the shapes of the streamlines and the scale or magnitude of the flow, or of the velocity components at one or more representative points in the flow. It would take several lines to convey such a description in words even for a pattern so elementary as flow at a corner (Fig. 1.16).

y

6 5

4

3 2

X

0 2 3 4 5 6 Figure 1.16---Irrotational flow at a 90° corner

A mathematical device which serves the above purpose with accuracy, completeness and conciseness is called the Stream Function and each pattern of flow for which a stream function can be found is effectively described by that function. In steady two-dimensional flow in the x-y plane the stream function (r/J) is a function of the variables x andy,

r/1 = f(x,y) and it has the following convenient properties:

(i) When the stream function of a particular flow pattern is equated to a constant, there results the general equation for the streamlines of that pattern, different constants defining different streamlines.

(ii) When the stream function is differentiated with respect toy and to x, in order, the general equations for the velocity components u and v are obtained.

(iii) In a flow pattern, the volume flow rate from left to right between any two streamlines r/J = C1 and r/J = C2 is oQ = orp = C2 - C1•

(iv) The effect of combining different flow patterns is easily determined, for the stream function of the resulting pattern is simply the sum of the stream functions of the component patterns.

As an example, the characteristics of the stream function for steady, two-dimensional, irrotational flow at a 90° corner can be stated (Fig. 1.16). It will be seen from subsequent considerations that, for this pattern, the stream function is

r/J = axy 20

STREAM FUNCTIONS

and the general equation for the streamlines is t/J = constant, that is

axy = C

the streamlines being rectangular hyperbolae. The coefficient, a, determines the scale or magnitude of the flow and different values of C define different streamlines. Further, the velocity components, u and v, at any point (x,y) are given by the following partial differentiations,

oifl u =-=+ax oy

oifl v = -ox= -ay

If a is unity, the velocity components, u and v, and total velocity V, at any point, say P(3, 4), in Fig. 1.16 are therefore respectively,

U = +X= 3; v = -y = -4

v = v(u2 +v2) = 5; 8 = tan-1 ; = tan-1 ( -~) The flow rate between any two streamlines, say,

o/2 = xy = 10 and o/3 = xy = 15 is 8Q = 8ifl = r/13 -r/12 = 15-10 = 5 ft. 3fsec. per foot depth normal to the plane of the flow.

The question of addition of different flow patterns will be treated in Chapter4.

The above characteristics can be established by defining a stream function so that it possesses any one of the characteristics and then by showing that it must possess the others also. Thus, a stream function is required to have the property of yielding velocity components on differentiation. Therefore, to begin with, it is dejined as a function of x andy (and t, for the general case of unsteady flow) such that when differentiated with respect toy, it yields u; and when differentiated with respect to x, it yields - v. This sign convention is adopted so as to give to t/J, in addition, the characteristic (i) above. Hence, by definition, for all stream functions,

or/J

} u =-oy

.... (1.14) or/J v = --OX

2 21


If these values of u and v be substituted in the streamline equation (Eq. 1.8c)

udy-vdx = 0

it follows that, along a streamline

oifl dy+ oifl dx = o oy ox

and therefore the total differential

d!fo = oifl dx+ oifl dy = o ox oy

:. 1/J = constant, along a streamline .... (1.15)

In other words, the general equation for the streamlines in a flow pattern is obtained by equating the stream function of the pattern to

y

,=6 ,, JhO

X 0 1t

(a) (b) Figur1 1.17-Physical significance otrfs in two-dimensional flow

a constant, different constants defining different streamlines. This is the first of the consequences of the above definition of 1/J.

The physical significance of 1/J follows from consideration of the difference in the values of 1/J for two streamlines !fo1 and !fo2 (Fig. 1.17a).

d!fo = !fos - !fo1 If the volumetric flow rate from left to right across any straight line oflength 81 joining the two streamlines be 8Q, then

8Q = (usin8-vcos8)81

= (u2-v~)81 81 81 = u8y-v8x

= oifl 8y+ oifl 8x oy ox

8Q = 81/J •••• (1.16) 22

STREAM FUNCTIONS

Hence the flow rate between any pair of streamlines in two-dimen-sional flow is numerically equal to the difference in their 1/J-values. If the streamline passing through the origin is 1/J = 0 then the flow from left to right between any other streamline and the origin equals the 1/J-value for that streamline (Fig. 1.17b). This physical significance is the second consequence of the method of defining 1/J.

The above characteristics of 1/J apply both in rotational and in irrotational flows. In addition, for irrotational flow, it has been

shown that ~;- ~~ = 0 and substitution of - ~: for v and ~; for u yields

()21/J ()2lfl ox2+ ()y2 = 0 .... (1.17)

Th . r ;;z F ()2 F 0 . th d' . 1 . . e equation 10rm OX2 + oy2 = lS e two- rmensmna carteSian co-ordinate form of what is known as the Laplace equation. The

d. h d' . lr . ()2 F ()2 F (}2 F 0 d h correspon mgt ree- 1mensmna _10rmis ox2 + oy2 + oz2 = an t e general form, in vector notation, is V2 F = 0. This Lapla~ equation is met in several other fields of physical science, for example, in electrostatics, as well as in hydrodynamics.

Whilst it is evident that only certain functions of x andy satisfy Eq. 1.17, the above reasoning indicates that the stream functions, 1/J, of all possible two-dimensional irrotational flow patterns satisfy it or, in other words, are solutions of the Laplace equation. Since there is an infinite variety of possible boundary conditions and hence of irrotational flow patterns, there is an infinite number of 1/J-functions which have this common characteristic of being solutions to the Laplace equation. The value of 1/J varies, in general, from point to point in a flow and the Laplace equation sets a limitation on the way it may varyt.

t If ordinates with magnitudes r/1 be erected normal to the x-y plane for each point x.,y, in a two-dimensional irrotational flow pattern, the tops of the ordinates define a curved r/J-surface. The radii of curvature of this r/J-surface in the x andy directions at any point are, approximately,

1 1 R,- o•rfJ and Rg- o•rfJ

ox• oy1 d · o•rfJ o•rfJ 0 R R h ' h ·1• urf: ' h .. an , smce ox• + 0_,1 = , "- - ,, t at 1s, t e 't'-s ace at any pomt as a positive

curvature in one direction and an equal negative curvature in the other, after the style of a horse-saddle.

23


It will be shown in Section LISe ( ii) that only one pattern of irrotational flow results from a given set of boundary conditions. Hence only one lft-function satisfies both the Laplace equation and the boundary requirements of a particular flow pattern. This characteristic of lft in irrotational flow forms the basis of several relatively simple methods of determining the pattern of flow once the form of the boundaries is known. These methods are dealt with in Chapter 3.

Examples of stream functions which will be developed in the following pages for some elemerhary two-dimensional steady flow patterns, are:

(i) Uniform flow with a velocity U from left to right, parallel to the x-axis ... .o/ = Uy

(ii) Radial flow to a point outlet at the origin, the flow rate Q being Q .... lft = - 277 8

(iii) Flow past a cylinder of radius, a, at the origin, of a fluid of infinite extent whose undisturbed velocity is U

.... lft = uy(1-~) X +y

Elaunple 1.3. Determine the stream function for parallel flow with a velocity V,

inclined at an angle octo the x-axis (Fig. 1.18).

y

Figure 1.18---Parallel flow inclined at an angle ex to the x-axis

Solution.-By definition

a.p = u- Vcosot cry

.. .p- JiJrcosot+A(x) 24

Also

VELOCITY POTENTIAL FUNCTIONS

a.p- -v-- Vsincx ax •• t/s = - Vxsin ex+ j 8(y)

Hence .p - Vy cos ex- Vx sin ex+ A where A is a constant. If the value of .p for the streamline passing through the origin is zero,

then A= 0 and .p = V(ycoscx-xsincx) = '4)1-vx

The streamline equation is .p - C i.e •.

or

V(ycoscx-xsincx) - C

c y- xtancx+ Vcoscx

which is a straight-line equation in the standard form,y- mx+ b.

For flow parallel to the x-axis, ex- 0 and .p = Vy. Then u = :; - + V;

v - - ~~ = 0 and the flow from left to right between the origin and any streamline, .Pu aty- y1 is Q- t/11 - JY1, which is self-evident for this simple case.

1.10 Velocity potential functions The stream function for a two-dimensional flow pattern, when

equated in turn to each member of a sequence of constants, yields the equations of the streamlines of the pattern. A useful, complementary function of a similar nature, for irrotational flow only, is the Velocity Potential Function, cfo, which is also a function of x, y and t. When equated in turn to each member of a sequence of constants it yields the equations of a family of velocity potential lines, each of which crosses the streamlines at right angles. The streamlines and potential lines form a mesh, or grid of lines, all intersections being at right angles. An important difference between cfo- and tfo-functions lies in the fact that c/J-functiolis exist only for irrotational flows, as will be seen below, while tfo-functions are not restricted to irrotational flows.

The velocity potential function is defined as a function of x,y and t such that, when differentiated with respect to distance in any particular direction, it yields the velocity in that direction. Hence, for any direction s, in which the velocity is v8

aq, os = v8 •... (1.18)

so that the velocity potential c/J increases in the direction of flow. 25


For the x- andy-directions, therefore,

()~ U=-

OX

()~ v =-oy

} .... (l.I9)

As one consequence of this definition, lines of constant ~-value can be shown to be perpendicular to lines of constant rfi-value at all points, that is, ~-lines intersect rfi-lines normally. At any particular instant,~ is a function only of x andy, even though, over an interval

Figure 1.19-Streamlines and equi-poten tiallilies

of time, it varies also with t in unsteady flow. At any instant ~ is constant along any ~-line

d~ = 0~ dx+ 0~ dy = 0 ox oy

along a ~-line. Substitution from Eq. 1.19 for~: and~; yields the relationship

along a ~-line whence

udx+vdy = 0

u

v

The slope of the ~-line at any point is seen to be equal to the negative

reciprocal of the slope (~) of the r/J-line at that point, that is, the line 26


of constant ,P-value intersects the streamline at the point at right angles (Fig. 1.19).

The second consequence of the definition of ,P follows from substi-• r d"th · · · OU OV 0 hih"ld tution 10r U an V m e COntinuity equation OX+ oy = , W C fle S

o2,P o2,P ox2 + oy2 = 0

showing that the ,P-function, like the ~/~-function for irrotational flow, is a solution of the Laplace equation.

Finally, upon substitution for u and v in the irrotational flow

equation (Eq. 1.11), ~=-:; = 0, it is seen that o2,P o2,P ---=0 oxoy oxoy .... (1.20)

which shows that ,P satisfies the conditions for irrotational flow, or, in other words, the existence of a velocity potential implies that flow is irrotational. It can be shown that the converse is truo, that is, that the condition of irrotationality implies the existence of'a velocity potential. On the other hand, stream functions are not restricted to irrotational flow.

The interrelationship of ,p, .p and the velocity components u and v at any point (x, y) in the cartesian co-ordinate system can be sum-marized in the two equationst

o,P o.p

I u =- = -· ox oy

.... (1.21) o,P o.p

v=- = -OX oy

If v is zero, ~; = 0, that is ,P depends only on x. Also ~~ = 0, that is, .p depends only on y, so that

d,P d.fo U=-=-

dx dy

t Some authors prefer to use an alternative sign convention according to which a.p otf> or/J otf> u= --- --· v= +-- -- and3r/J- -3Q. oy ox' ox oy

27


In terms of natural co-ordinates s and n, where s is the distance measured in the direction of flow along a streamline, and n is the distance measured across the flow along an equipotential line, the velocity V at a point will be directed wholly along the streamline and

v = d,P = dl/J ds dn

.... (1.22)

Use will be made ofthis relationship in the discussion of flow nets in the next section.

In many cases of flow it is more convenient to work with the polar

Figure 1.20-The relationship of streamfunction to velocity in two-dimensional flow-polar co-ordinates

co-ordinates, rand 8, than with x andy. From Fig. 1.20 it is evident that, for any point C (r, fJ)

X= rcos8; y = rsin8; r = y(x2+y2); 8 =tan+~ X

. ( Sr 38) If the stream functiOn has the values 1/J .4 at A r-2", 8- 2 and lfB at B(r+i' fJ+ s:). the difference in the 1/J-values, dl/J = 1/Js-!f.-t, equals the rate of flow from left to right across the line AB, as a consequence of the definition of 1/J. This flow rate can be considered as the total of two flow rates, that in the outward radial direction, with the velocity v, and that in the clockwise circumferential or tangential direction with the velocity, - vo. The radial flow rate is (radial velocity) x (width DE), that is v,d(r8) or v,rdO; and the tan-gential flow rate is (tangential velocity) x (width FB), that is, - vodr.

do/ = v,rdfJ-vodr .... (1.23) 28


At any instant ifl is a function of rand 8, therefore

.... (1.24)

From a comparison of Eqs. 1.23 and 1.24 it is clear that,

.... (1.25)

For the potential function, it follows from the definition of cp that oc/J

v, = or' oc/J 1 or/J

V(J = o(r(}) = r o(} .... (1.26)

The polar co-ordinate equations corresponding to Eq. 1.21 are therefore

oc/J 1 oifl v =- = --r or roO .... (1.27a)

1 or/J oifl V(J = -- = --r o8 or .... (1.27b)

The equivalence of the cartesian and polar co-ordinate forms, including the signs, can be seen for any point G on the x-axis, where dx = dr, dy = rd8, u = v, and v = V(J.

Stream functions and velocity potential functions are frequently expressed in polar co-ordinates. For example, for the pattern of flow past a cylinder of radius, a, of a fluid of infinite extent whose undis-turbed velocity is U,

= u 1---( a2 ) ifl ~ x2+y2 = Ursine( I-::) or u(r- ~)sinO and

cp= ux(l+~) Urcose( 1 +~) or u(r+~) cosO X +y

Exa~nples

1.4. Given the function r/J = xy, determine the flow pattern. Show that the flow is irrotational and determine the -function.

29


Solution.-The equation of the streamlines is .p - C or xy - C, which equation represents a family of rectangular hyperbolae. Considering the first quadrant only, the pattern represents flow at a 90° corner (Fig. 1.16). The pattern of the first and second quadrants together represents flow towards a flat plate, or stagnation flow (Fig. 1.21).

y

----~~~~~--~~~--X

Figure 1.21-Fiow towards a flat plate

Since ~- au - a• .p + a• .p - 0 the flow is irrotational and therefore a ax ay ax• ay•

~-function will exist for the pattern.

a; a.p --u---% ax ay

.. ., - ixl+ft{y) .... (a) Also

a~ v-a.p

ay- -ax- -y

.. r/>- -!y•+J.(x) .... (b) The value .p- i(x'-y 1) +a constant, satisfies both (a) and (b). The lines ~ - constant plot as a family of rectangular hyperbolae orthogonal to the lines .p - constant. These rf>-lines are represented by the broken lines in Fig.1.21.

1.5. Show that the two-dimensional irrotational flow stream function

rp = u(r- ~) sin 8 represents the pattern of steady flow in the x-direction, past a cylinder of radius a, of an infinite fluid whose undisturbed velocity is U. Determine the potential function and find the distribution of velocity on the boundary of the cylinder.

Solution.-(a) The Pattern-For the streamline .p- 0, either 8- 0 or 11, that is, this streamline coincides with the x-axis; or r- a, that is, the streamline is a circle of radius a, with its centre the origin (Fig.1.22). The other stream-lines plot as shown. Streamlines inside the circle can have no influence on the pattern outside the circle and this circle can be regarded therefore as the solid boundary of a cylinder of radius a, lying acroSs the flow.

30


(b) The Potential Function-The !/>-function is found by integration of the velocity functions determined from the r/s-functions

Also

o!fo Iar/s ar - v,- r o9

- u( 1- ~) cos 9 •. "'- u(r+~) cos9+ft(9)

104> arts r o9- v,- -or --u( 1 + ~) sin 9

.. "'- u(r+ ~) cos9+J.(r)

' Figure 1.22-Irrotational flow past a circular cylinder

.... (a)

•••• (b)

Comparison of the two values for 4> indicates that 4>- u(r+ ~) cos9+C is the required function, the additive term being a constant which does not affect the pattern of the !/>-lines.

(c). Veloci~ Distribution-The components of velocity, v, and v8 for any point in the flow are as determined above. At very large distances from the cylinder, afr approaches zero and

v,~Ucos9, v8~- Usin9

.. v- v(v~+v:>~u On the cylinder boundary, which is a streamline,

v,- 0 and r- a

•• v8 - -2Usin9 31


the negative sign resulting from the sign convention for e and v8 (- ve clock-wise). Denoting the velocity at the boundary by V',

V' = - v8 for 0< 0< 1T

V' = 2Usin0

at 0 = 0, 1r, the stagnation points,

and I 1T

Vmax = 2U at 0 - 2

1.6. In the two-dimensional contraction in Fig. l.lb, if the upstream width is 8ft. and the downstream width is 4ft., estimate roughly the change in velocity potential between cross-sections 10 ft. upstream and downstream of the mid-section for a flow of 16 ft. 3 /sec. per foot of depth.

Solution.-With the one-dimensional approximation, u = ~' 8 = u 8x approximately

Taking u as 3ft. per sec., the average of the upstream and downstream velocities, and 8x as 20 ft., the distance between the two cross-sections, the required difference in velocity potential is approximately 8 = 3 x 20 = 60ft. 2 per second.

1.11 Flow nets

It has been shown that a two-dimensional flow pattern can be represented visually by a scale drawing of certain streamlines chosen so as to divide the flow into a number of channels conveying equal flow rates, 8Q. Since 8Q = 8!f;, the change in if from one streamline to the next, it is clear that 8lf; is constant throughout the pattern. If the lateral spacing of the streamlines in the region of a point is 8n, then the velocity at that point is approximately

v = 8Q = 8!f; 8n 8n

.... (1.28)

which is in accord with Eq. 1.22. Since 8Q is constant, the velocity is inversely proportional to 8n, the lateral spacing of the streamline throughout the flow pattern.

If the flow is irrotational the pattern can be represented with equal precision by means of selected lines of constant potential, chosen so that the change, 8if>, from one equipotential line to the next is con-stant. From Eq. 1.22, it follows that, at any point

v = 84> 8s

32

.... (1.29)

FLOW NETS

approximately, where 8s is the spacing of the equipotential lines, s being measured along the streamline passing through the point.

If the family of streamlines and the family of equipotential lines are both represented on the one figure, the resulting network of lines is called a flow net and, in the region of any selected point, the velocity is given approximately by both Eqs. 1.28 and 1.29

.•.. (1.30)

If, as is usual, the


The principal value of the flow net concept lies in the fact that it is the basis of an extremely simple, trial-and-error graphical method of determining the flow pattern for any given set of boundary conditions. This method is dealt with in detail in Chapter 3.

Figure 1.23---Irrotational flow in a curved path. Vr- constant

1.12 Euler's equations of motion for a non-viscous fluid Fluid motion can be treated mathematically by either of two

approaches, both of which are due to Euler. The first, known as the Lagrange or 'historical' approach, deals with the positions, velocities and accelerations of individual particles, the co-ordinates, x,y and z, of a particle being variables which are functions of the initial position of the particle and of time. The second, known as the Euler approach, is the one adopted in the present treatment. In this method, x,y and z define a general point in space and they do not vary with time. Consideration is given to the velocities and accelerations of particles as they pass through the general point rather than to the variations of the velocities and accelerations of particles as they follow their various paths.

If u, v and w are, in order, the components of velocity in the x-, y-and z-directions at the point (x,y, z) at time t, then u, v and ware functions of position (x,y, z) and time (t). For a particular value of t, they define the motion at all points in the fluid; and for a particular point, (x,y, z), they are simply functions of time, providing a history of the velocity variations at the point. Except where otherwise stated, it is assumed that u, v and w are finite and continuous functions

f d d th th . d . . au av aw 0 x, y an z an at elr space envatlves, ax, ax, ax . . . are ever-yWhere finite. First, the components of acceleration at a point

34

EULER'S EQ.UATIONS OF MOTION FOR A NON-VISCOUS FLUID

are considered and then Newton's second law of motion, incorporat-ing these components of acceleration, is applied to a fluid mass.

A particle at the point (x,y, z) at timet, will move, in the time 8t, a distance ax = u8t in the x-direction, ay = vat in the y-direction and az = wat in the z-direction. The change, au, in the particle's u-component of velocity will be the sum total of the convectional changes, due to the changes ax, ay and az in position (see Fig. 1.24),

z 'J"'\\'J

u+ff6x+/f6y+J:iz (x+ox,.r+8 y, z+liz J

Figure 1.24---Acceleration. Convectional components of the change in velocity in the .¥·direction of a fluid particle as it moves from P 1 toP 1 in the interval 8t.

There is, in addition, the local component, :~ 81

and the local change, due to change with the passage of time 8t, at the point x,y, z. Mathematically,

u = A(x,y, z, t) au au au au au = -at+-ax+- ay+-az at ax ay az

au au auax auay auaz at = at+ axat + ayat + azat

If at be considered to approach zero, the total acceleration in the x-direction is obtained

du au au au au - = -+u-+v-+w-dt at ax ay az

35

.... (1.3la)


Simifarly, the total accelerations in they- and z-directions, respec-tively, are

dv ov ov ov ov - =- +u-+v -+w-dt ot ox oy oz

dw ow ow ow ow - = -+u-+v-+w-dt ot ox oy oz

.... (1.3lbf

.... (l.3lc)

In these equations, the first terms on the right-hand sides are known as 'local' accelerations, since they arise from changes in velocity with time at the point x,y, z; and the remaining terms are known as 'convectional' acceleration since they arise from changes of velocity with change of position. In steady Bow, that is, flow in which velocities and accelerations do not vary with time, the local acceleration is zero but fluid particles will possess convective accelera-tions if the flow is non-uniform, as in flow in a converging passage. In uniform flow, that is, flow in which the velocities and accelerations do not vary with position, the convective acceleration is zero but the fluid particles will possess local acceleration if the flow is unsteady, as in flow with an increasing discharge.

Applications of Newton's second law of motion in the three co-ordinate directions yield the equations known as Euler's Equations of motion for a non-viscous fluid. Let p be the pressure and p the density at the centre point P(x,y, z), of the element whose dimen-sions are Sx, Sy, Sz (Fig. 1.25); and let X, Y and Z be the components of body force per unit mass in the x,;y, z, directions at the time t. Since force-per-unit-mass is dimensionally an acceleration and since, in hydraulic engineering, the only common body force is the weight force, X, Y and Z can be regarded as the components of g. The product of the mass of the element and its total acceleration in the x-direction must equal the sum of the components of force acting on the element in that direction.

pSxSySz :~ = pSxSySzX + (P- ~~~) SySz- (P+ ~S2x) SySz

Similarly

du = ou + u ou + v ou + w ou = X_ ~ op dt ot ox oy oz pox

.... (1.32a)

.... (1.32b)

dw ow ow ow ow 1 op - = -+u-+v-+w- = Z--- .... (l.32c) dt ot ox oy oz poz

36

THE BERNOULLI EQUATION

These are the Euler equations of motion for a non-viscous fluid. Each term has the dimensions of force per unit mass, or acceleration, and the total acceleration in a given direction is seen to be equal to the sum of the gravitational component and the component due to the existence of a pressure gradient in that direction.

Integration of these equations between the limits represented by any two points, (x1,y1, z1,) and (x2,y2, z2), in the flow might be expected to yield (force-per-unit-mass) x (distance) terms represent-ing the energy changes which would be involved in the movement of unit mass of fluid from one point to the other. In fact, it is possible

z

X

6 W =,o6xfJy& zg Figure 1.25-Equation of motion. Forces acting in the x-direction

on an element of fluid in three-dimensional non-viscous flow

to integrate the equations to produce this general result, provided that the flow is irrotational. The integrations yield the well-known equation of Bernoulli.

1.13 The Bernoulli equation In the Euler equations the gravity force components X, Y, and Z

are expressible in terms of a 'gravity force potential'. This is a mathematical device which is analogous to the velocity potential, for it is a function of x, y and z such that, when differentiated with respect to distance in any given direction it yields minus the com-ponent of gravity force per unit mass in that direction.

If the direction vertically upward, that is, opposite to the direction of gravity, is represented by the h-axis, then the potential energy or force potential, with respect to some selected datum level, of unit

37


mass at a height h above datum, is

n = +gh .... (1.33) The force per unit mass in the positive h-direction is the negative

derivative with respect to h of the force potential

on --= -g oh

Similarly the forces per unit mass, X, Y and Z, in the x-, y- and z-directions taken in order are

Also

u=oq,, OX

.... (1.34)

oq, v = -, oy

Substitution of these relationships and those of Eq. 1.13, which represent the conditions for flow to be irrotational, into the Euler equations yields

o2 q, +uo"+vov+wow = _on_~op ox ot ox ox ox ox P ox o2 q, ou ov ow. on 1 op -+u-+v-+w- = -----oyot oy oy oy oy P oy o2 q, ou ov OW on 1 op --+u-+v-+w- = -----ozot oz oz oz oz p oz

If p be considered constant, integration with respect to x,y and z respectively, yields the following equations:

i(uz+vz+w2) + ~q, +n+e = F1 (y, z, t) ut p

i(ull+vz+w2) + ~q, +n+e = F11 (z, x, t) ut p

i(ull+vll+w2) + ~q, +n+.e = F3 (x,y, t) ut p

The left-hand sides of these equations are identical, the first term equalling i VJI, where Vis the velocity whose components are u, v and

38

THE BERNOULLI EQUATION

w. The right-hand sides of the equations must therefore be equal, and, since they do not each contain x,y and z, they must be independent of x, y and z, being either functions of time or constants. For the three identical equations therefore, one equation can be written

.... (1.35)

whkh is the Bernoulli equation for the unsteady, irrotational flow of a non-viscous, incompressible fluid.

For steady flow, t. disappears from the equation and, with gh substituted for 0, the Bernoulli equation for steady irrotational flow is obtained.

JT2 p JT2 p -+-+gh = C or -+-+h = H 2 p 2g y

.... (1.36)

where His a constant throughout the region of irrotational flow. If consideration is restricted to the movement of a particle along

its streamline in steady flow, a simple form of the Euler equation results and it can be integrated along the streamline without the requirement of irrotationality. If s is the distance co-ordinate, measured along the streamline, Vis the particle velocity at the time t, and S is the component of gravity in the instantaneous direction of motion (Fig. 1.26), then

V =f(s, t)

ov ov SV = -St+-Ss ot os sv oV oVSs Yt = ae+ os & dV _ ov vov dt - ot + os

Application of Newton's second law yields

ov + vov = s-~ op ot os P os .... (1.37)

At points where the streamline is curved, the particle will experience a pressure gradient and an acceleration normal to the direction of motion. If the inward normal direction, in the plane of curvature is represented by n, and if the component of gravity in this direction is

39


N, the total acceleration will consist of the local acceleration °;" and the convective acceleration V2 (centripetal acceleration). The

r equation of motion for the n direction becomes

oVn + V2 = N _! op ot r pon

.... (1.38)

an equation which establishes the variation of pressure with radius in curved flow.

6W=p6A6sg Figure 1.26-Equation of Itlotion. Forces acting in the direction of motion on an element of fluid· in three-dimensional non-

viscous flow

Integration of Eq. 1.37 with respect to s, the distance measured along the pathline, is possible in steady flow, for which the pathline

. ml' d av. lS a strea me an at 1s zero.

Substitution of - ~~ for S gives

vav +an+~ ap = 0 as as Pas and integration with respect to s, without the restriction that flow be irrotational, results in the equation,

!V2+gh+e = C' or p

40

.... (1.39)

VELOCITY AND PRESSURE DISTRIBUTIONS

which is the restricted form of the Bernoulli equation, relating to points on any one streamline in steady flow of a non-viscous incom-pressible fluid.

The terms of the Bernoulli equation, 2V2, !!_ and h, have the dimen-g y sions of energy per unit weight of fluid, for example, foot pounds per

pound weight or, simply, feet. In fact, :; is the kinetic energy per

pound and his the gravitational potential energy per pound referred

to some datum level. The pressure term, 1!._ does not represent, in y

itself, a pressure energy content per unit weight of fluid. (The fluid under consideration being incompressible, the concept of elastic strain-energy due to compression is not involved.) However, the

difference p B-p A , between the pressure terms on each side of the y y

Bernoulli equation expressed in the form

does represent the energy expended by pressure forces in moving unit weight of fluid from point A to point B.

The three terms are known as the velocity head, the pressure head and the elevation head, respectively, and their sum, H or H' is the total head. It is evident that, in steady non-viscous flow, the total head, H', is constant along a streamline and that if, in addition, the flow is irrotational, the total head, H, is constant throughout the flow.

1.14 Velocity and pressure distributions

Distributions of velocity can be represented either by means of a series of individual arrows, the length of each representing the velocity at its rear tip, or by means of velocity distribution curves. The arrow lengths and curve ordinates may indicate either absolute velocities or velocities relative to a reference velocity, which is usually selected in a region of uniform flow (Figs. 1.9, 1.10). The use of relative velocities yields a dimensionless velocity diagram, readily applicable to various rates of flow and various scales of magnitude.

Distributions of pressure in steady irrotational flow are determined

41


from the velocity distributions through application of the Bernoulli Equation, the elevation term being written henceforth as ~.

V2 P vg Po -+-+~ = -+-+~o = H 2gy 2gy •... (1.40)

In this equation the first three terms refer to any point in the flow and those with the subscript to an arbitrary reference point.

If gravity effects are absent, as in a pattern offlow in a horizontal plane, the elevation head (~) terms disappear. Multiplication by y removes the gravity terms y and g to yield the pressure-velocity relationship

lpVll+P = lpfl+Po .... (1.4la)

P =Po+lp(fl-V2) =Po+lpfl[t-(~)] .... (1.4lb)

:. ~;t; or ~t: = 1- (~r .... (1.4lc) Since ; equals~, the ratio of the spacing of the streamlines

0 which can be readily determined from the flow pattern, Eq. 1.4lb enables the pressure at any point to be determined in terms of the reference quantities Po and V0 • Eq. 1.4lc provides a dimensionless relationship, which is conveniently plotted by subtracting ( ~) 2 from unity, and which is independent of the extent of the flow and the absolute magnitudes of the velocities and pressures. At stagnation

points, Vis zero, hence !~to equals unity and the stagnation pressure is

Pst = Po+lpV: .... (1.42)

Esam.ple

1.7. The velocity at the boundary of a cylinder immersed in a fluid was shown (Example 1.5) to be

V'- 2Usin8

where U is the undisturbed velocity of the fluid and 8 is measured from the direction of flow. If~e pressure in the undisturbed flow isp0, determine the pressure distribution around the cylinder, the location of the stagnation points; and the stagnation pressure.

42


Solution.-The pressure pat any point on the boundary is given by Eq. l.4lc

-- = 1- - ~ l-4sm2 8 P-P0 (V') 2 • ipU2 U

.... (a)

whence

.... (b)

The distribution is plotted non-dimensionally, in termsofp- Po, in Fig.l.27. ipU2

From Eq. (a) it is evident that the stagnation points, where V' is zero, are

at 8- 0 and 8- 1r and that, at these points,~- ~0 equals unity. ~pU

!E.. =-3 1hpU2

___ ........ Figure 1.27-Irrotational flow past a circular cylinder. Distribution of pressure on the cylinder

The points on the boundary where the pressure is the same as that of the undisturbed flow are obtained by setting p equal to Po· It follows that l-4sin2 8 equals zero, so that sinO- ±!and therefore 8-30°, 150°, 210° and 330°.

The minimum pressure occurs where 1- 4sin2 8 has its minimum value, that is, where 8 = 90° and 270°, and, at these points, it follows from Eq. (a)

P-Po that !pU2 - - 3.

In Fig. 1.28 are shown the distributions of pressure along the boundary and the centre-line of a two-dimensional contraction in a horizontal plane. Of interest are the locations of points of maximum pressure, at the stagnation point S, and of minimum pressure, at X on the curved boundary outside the region of uniform low pressure

43


and high velocity. The x- andy-components of the thrust on the curved portion of the boundary can be determined from the areas of the subsidiary pressure diagrams, in which the pressure magnitudes are plotted undiminished on the respective projections of the curved boundary. Total thrusts obtained from such pressure distribution curves frequently can be checked by application of the impulse-momentum equation in a one-dimensional analysis.

If gravity effects are present, as in flow in a vertical plane, the elevation term (z) of the Bernoulli equation must be taken into

Figure 1.28-Distribution of pressure along' the boundary and the centre-line of a two-dimensional contraction

account. If the flow is enclosed and has no free surface, it is con-venient to combine the pressure and elevation head terms to form the piezometric head, h, where

h = .e+z y

The piezometric head at a point in the flow is the height above the datum to which a fluid would rise in an open pressure tube, or piezo-meter, inserted at that point (Fig. 1.29). The substitution of h in the Bernoulli equation yields

.... (1.43a)

.... (1.43b)


It is apparent from the similarity of Eqs. 1.4lc and 1.43b that a dimensionless pressure distribution curve for flow in a horizontal plane would equally well serve as a piezometric head distribution curve for flow with similar closed boundaries in a vertical plane. The flow pattern in both cases is determined solely by the boundary form and the only effect of gravity is to increase the pressure linearly along the downward vertical.

Stagnation tube

Figure 1.29-Variation of velocity head and piezometric head in a two· dimensional contraction {after Rouse)

Steady flow with a free surface is essentially a gravity flow and, since the pressure along the free surface is constant with the value zero (gauge pressure), the free surface velocity will vary in accordance with the Bernoulli equation with the pressure terms omitted.

V2 v~ -+z = ___!i+zo = H 2g 2g .... (1.44a)

where V0 and Zo refer to a free surface reference point and H is the total head, which is constant in irrotational flow. The free surface velocity at any poirit such as on a jet or weir nappe is therefore

V = y'2g(H- z) 45

.... (1.44b)


Within the jet or nappe, the pressures will not be zero and therefore Eq. 1.44 will not apply, unless the streamlines are straight and parallel (Fig. 1.30).

Datum

Figure 1.30---Distributions of velocity and pressure in flow over a sharp-crested weir

1.15 Energy considerations (a) Energy Equationfor Non-viscous Fluids

At a point in the flow where the velocity is V, the kinetic energy of an element of fluid, of dimensions Sx, Sy, Sz, is one half the product of the mass and the square of the vel~city

ST = ipV2 SxSySz and the total kinetic energy in the fluid is

T = t I I I pV2 dxdydz •... (1.45) If the potential energy per unit mass at the point is .0 = gh, the total potential energy in the fluid is

'I

v =I I I p.Odxdydz .... (1.46) From the Euler equations and Eqs. 1.45 and 1.46, it can be shown

that, for incompressible fluids

~ (T + V) = I Ip(lu+mv+nw) dS .... (1.47) The first term of this equation represents the time rate of change

of the kinetic plus potential energies; the second, since (lu+mv+nw) is the velocity of the boundary normal to itself in the direction of the

46

ENERGY CONSIDERATIONS

fluid, equals JfPVndS and represents the rate at which the pressure forces pdS exerted from without are doing work on the fluid. Hence the total increase in kinetic plus potential energy in an incompressible non-viscous fluid equals the work done by the pressures on its surface. This statement is valid for both rotational and irrotational flow.

(b) Kinetic Energy in I"otational Flow A general theorem, due to Green, t has several useful applications

in irrotational flow theory. It is expressed in the equation

J J (lP+mQ+nR) dS = - J J J (~: + ~; + ~~) dxdydz .... (1.48) in which P, Q and R are any finite single-valued differentiable functions in a connectedt region completely bounded by one or more closed surfaces, S, of which 8S is an element; and l, m, n are the direction cosines of the normal to the 8S, drawn inwards. For example, P, Q and R may be the x-,y- and ;:-components of velocity or of momentum, or they may represent, in order, the products rpu, rpv, and rpw, in a fluid contained within a closed spherical or other surface; or in the space between two closed surfaces, one of which lies inside the other. The surfaces are not necessarily solid boundaries.

If P, Q and R are velocity components, the equation states that the rate of flow into an enclosed region equals minus the rate of expansion, i.e. the rate of contraction of the fluid in the region.

A useful expression for the kinetic energy can be obtained if P, Q and R are defined as

p = rpu = rp ~:; where rp is the velocity potential. The velocity along the inward normal, n, to an element of surface is§ Vn = ~: and it equals the sum of the components of u, v and w in the n-direction

t See Appendix A (i).

aq, aq, aq, aq, - = 1-+m-+n-on OX oy oz

~ A region of space is said to be • connected' if it is possible to pass from any point to any other point in the region without leaving the region. A connected region is said to be 'simply' connected if all possible closed curves within the region can be contracted to zero without leaving the region. Examples are the interior of a sphere, the region exterior to a sphere or between two concentric spheres. A connected region which does not conform to this definition, for example, the region between two concentric cylinders of infinite length, is said to be • multiply connected'.

§ The use of n to represent distance along the normal and also one of the direction cosines fortunately does not lead to confusion.

47


Similarly the sum of the components of P, Q and R along the inward normal is lP + mQ + nR

lP+mQ+nR = l~ 0~ +m~ 0~ +~ 0~ ox oy oz

= ~ (ta~ +m a~ +n a~) ox oy oz

= .1. 0~ (1 50) 't' on ..... which enables P, Q and R to be eliminated from the left-hand side of Green's equation.

In Eq. 1.49, differentiation of the product~~~ yields

oP = (o~)~ +~ o2~ ox OX ox2

and similar expressions result for ~; and ~: By addition

oP + aQ +oR = (o~) 2 + (o~) 2 + (o~) 2 +~V2~ ox oy oz ox oy oz . = u2+v2+w2+~V2~

= VZ+~V~~ .... (1.51)

Eq. 1.51 enables P, Q and R to be eliminated from the right-hand side of Green's equation, which now becomes after multiplication

throughout by -~

-~I I~~: dS = ; I I I VZdxdydz+; I I I ~V2~dxdydz In irrotational flow, V2~ is zero, hence the last term disappears.

The second term is the total kinetic energy, T, (Eq. 1.45), and hence

T = -- ~-dS PII 0~ 2 on .... (1.52)

which expresses the total kinetic energy of a flow in terms of con-ditions over its surface. Eq. 1.52 can be shown to hold for an infinite

48

DETERMINATION OF FLOW PATTERNS

fluid at rest at infinity with an internal solid boundary, such as a moving solid body.

(c) Irrotational Flow Theorems Kinetic energy considerations lead to several important con-

clusions regarding irrotational flow. ( i) Irrotational motion is impossible if all qf the boundaries are solid and

at rest-Since the velocity normal to the boundary, ~:. must be zero over the whole solid boundary surface, the total kinetic energy, T, must be zero by virtue ofEq. 1.52, and hence the velocity, V, must be everywhere zero. It can also be shown that, in an infinite fluid at rest at infinity, if the interior boundary is at rest, irrotational motion is impossible.

(ii) The pattern qf irrotationaljlow which satisfies the Laplace equation and prescribed boundary conditions is unique, that is, there can be no more than one such pattern-Suppose cf>t and c/>2 both satisfy V2cf> = 0 and the boundary conditions. Since the Laplace equation is_ a linear, homo-geneous, differential equation, the sum or difference of any two solutions to it is also a solution. Hence c/> 3 = c/>1 -c/>2 is a solution. The velocity normal to the surface at any point must be the same for each pattern, hence

and therefore

ocf>a = o(cf>t-cf>s) = ocf>t_ocf>s = 0 on on on on

c/>3 = constant

that is, cf>t and c/>2 can differ only by a constant amount. The patterns of flow given by cf>t and c/>2 are therefore identical.

In a similar manner it can be shown that the pattern ofirrotational flow of a fluid at rest at infinity is uniquely determined by the motion of the interior solid boundaries.

1.16 Determination of :ftow patterns The problem of determining a particular pattern of irrotational

flow amounts, in mathematical terms, to finding a stream function, or a potential function, which satisfies both the Laplace equation

49


and the boundary conditions of the flow in question. These boundary conditions relate to the form and nature of the boundaries, including those across which the fluid flows. For example, in steady flow, solid boundaries must be streamlines and free surface boundaries must be streamlines at constant pressure. The inflow and outflow boun-daries fr

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