AERO 301 NotesFundamentals of Aerodynamics

Pressure is the normal force per unit area exerted on a surface due to the time rate of change of momentum of the gas molecules impacting on or crossing that surface.

Density is defined as the mass per unit volume.

The temperature of a gas is directly proportional to the molecular kinetic

energy of the molecules of the fluid.

Temperature and density are point properties that can vary from point to point

in a fluid. Flow velocity is the velocity of flowing gas at any fixed point B in space is the

velocity of an infinitesimally small fluid element as it sweeps through B. The flow velocity V has both magnitude and direction and is therefore a vector quantity.

The aerodynamic forces and moments exerted on a body are due to only two basic sources:o Pressure distribution over the body surface

Acts normal to the surfaceo Shear stress distribution over the body surface

Acts tangential to the surface Relationship between lift and drag…

where is the force normal to the chord, is the force parallel to the chord, and

is the angle of attack. Aerodynamic force on an element of the body surface…

o Upper:

o Lower:

o The total normal and axial forces per unit span are obtained by integrating the

equations above from the leading edge (LE) to the trailing edge (TE). Shear stress in a streamline…

The value of the shear stress at a point on a streamline is proportional to the spatial rate of change of velocity normal to the streamline at that point.

The TOTAL normal and axial forces per unit span are obtained by integrating from the leading edge (LE) to the trailing edge (TE).

The moment per unit span about the leading edge due to and on the elemental area on…o The upper surface…

o The lower surface…

The moment about the leading edge per unit span…

The sources of the aerodynamic lift, drag, and moments on a body are the pressure and shear stress distributions integrated over the body.

Free stream dynamic pressure has the units of pressure.

The dimensionless force and moment coefficients are defined as follows…

Integral form for the force and moment coefficients…

Lift and drag coefficients…

The aerodynamic force and moment coefficients can be obtained by integrating

the pressure and skin friction coefficients over the body. The speed of sound…

The drag coefficient for a cone is equal to its surface pressure coefficient.

Center of Pressure

The standard convention is that aerodynamic moments are positive if they tend to

increase the angle of attack. The center of pressure is the location where the resultant of a distributed load

effectively acts on the body.o If moments were taken about the center of pressure, the integrated effect of

the distributed loads would be zero.o Alternate definition: the point on the body about which the aerodynamic

moment is zero. In cases where the angle of attack of the body is small, and , which

leads

As and decrease, increases. As the forces approach zero, the center of pressure moves to infinity.

To define the force-and-moment system due to a distributed load on a body, the resultant force can be placed at any point on the body, as long as the value of the moment about that point is also given.

The resultant aerodynamic force…

Force coefficient…

Freestream Reynolds number…

o The Reynolds number is physically a measure of the ratio of inertia forces to viscous forces in a flow and is one of the most powerful parameters in fluid dynamics.

Mach number…

o The Mach number is the ratio of the flow velocity to the speed of sound.

Through dimensional analysis, it is seen that can be expressed in terms of a

dimensionless force coefficient and is a function of only .

Different flows are dynamically similar if…o The streamline patterns are geometrically similar.

o The distributions of , etc., throughout the flow field are the same when plotted against common nondimensional coordinates.

o The force coefficients are the same. Two flows will be dynamically similar if…

o The bodies and any other solid boundaries are geometrically similar for both flows.

o The similarity parameters are the same for both flows. In a limited sense, but applicable to many problems, we can say that flows over

geometrically similar bodies at the same Mach and Reynolds numbers are dynamically similar, and hence the lift, drag, and moment coefficients will be identical for the bodies.

As a consequence of being similar flows, we know that…o The streamline patterns are geometrically similar.o The nondimensional pressure, temperature, density, velocity, etc., distributions

are the same.o The drag coefficients are the same.

In order to sustain an airplane in level flight, the lift must be equal to the force exerted on the plane by gravity.

For steady, unaccelerated flight, the thrust must be equal to the drag. Typically, for conventional cruising flight, .

For an airplane of a given shape at given Mach and Reynolds number, and are simply functions of the angle of attack, of the airplane.

will be the smallest when is a maximum. Hence, the stalling velocity for a

given airplane is determined by …

The maximum velocity for a given airplane with a given maximum thrust from the

engine is determined by the value of minimum drag coefficient, .

For level flight where

For level flight where

Lift to drag ratio…

Hydrostatic equation…

o Relates the change in pressure in a fluid with a change in vertical height.

Buoyancy force…o The buoyancy force on a body = weight of fluid displaced by the body.

Chapter 2

Dot Product:

Cross Product: Cartesian Coordinate System:

o

o

Cylindrical Coordinate System:

o o Transformation between Cartesian and cylindrical:

o Inversely:

Spherical Coordinate System:

o o Transformation between Cartesian and spherical:

o Inversely:

Scalar fields:

Vector fields:

Consider a scalar field:

o The gradient of , , at a given point in space is defined as a vector such that: Its magnitude is the maximum rate of change of per unit length of the

coordinate space at the given point. Its direction is that of the maximum rate of change in at the given point.

o Directional derivative in the s direction: Gradient in different coordinate systems:

o Cartesian:

o Cylindrical:

o Spherical:

Divergence of a vector field: Consider a vector field…

o can represent any quantity, but think of it as a flow velocity.o Visualize a small fluid element of fixed mass moving along a streamline at

velocity .o The time rate of change of the volume of a moving fluid element of fixed mass,

per unit volume of that element, is equal to the divergence of , .o The divergence of a vector is a scalar quantity, and is one of two ways that the

derivative of a vector field can be defined.

Cartesian:

Cylindrical:

Spherical:

Curl of a vector field: Consider a vector field…

o Think of as a flow velocity once again.o Visualize a small fluid element moving along a streamline. It is possible for this

fluid element to be rotating with an angular velocity as it translates along the streamline. Cartesian:

Cylindrical:

Spherical:

Relations between line, surface, and volume integrals:o The line integral of over is related to the surface integral of over by

Stokes’ Theorem:

o Divergence Theorem:

o Gradient Theorem:

Development of basic aerodynamic equations:1. Invoke three fundamental physical principles that are deeply entrenched in our

macroscopic observations of nature, namely,a. Mass is conserved (i.e., mass can neither be created nor destroyed).b. Newton’s second law: c. Energy is conserved: it can only change from one form to another.

2. Determine a suitable model of the fluid. Remember that a fluid is a squishy substance, and therefore it is usually more difficult to describe than a well-defined solid body. Hence, we have to adopt a reasonable model of the fluid to which we can apply the fundamental principles stated in item 1.

3. Apply the fundamental physical principles listed in item 1 to the model of the fluid determined in item 2 in order to obtain mathematical equations which properly describe the physics of the flow. In turn, use these fundamental equations to analyze any particular aerodynamic flow problem of interest.

There are three different models that have been used successfully throughout the modern evolution of aerodynamics for fluids:o Finite control volume

The control volume is defined as a closed volume drawn within a finite region of the flow.

The control surface is defined as the closed surface which bounds the control volume.

The control volume is a reasonably large, finite region of the flow. The fundamental physical principles are applied to the fluid inside the control

volume, and to the fluid crossing the control surface, thus limiting our attention to just the fluid in the finite region of the volume itself.

o Infinitesimal fluid element Consider an infinitesimally small fluid element in the flow, with a differential

volume, . It is infinitesimal, but it is large enough to contain a huge number of molecules so that it can be viewed as a continuous medium.

The fluid element may be fixed in space with the fluid moving through it, or it may be moving along a streamline with velocity equal to the flow velocity at each point.

The fundamental physical principles are applied to just the fluid element itself, and not the whole field flow.

o Molecular The motion of a fluid is a ramification of the mean motion of its atoms and

molecules. Therefore, the fundamental laws of nature can be applied directly to the atoms and molecules.

Physical Meaning of the Divergence of Velocityo is physically the time rate of change of the volume of a moving fluid

element of fixed mass per unit volume of that element.

o A moving control volume that is essentially becoming an infinitesimal moving fluid element is described as…

The equation below states that is physically the time rate of change of the volume of a moving fluid element per unit volume.

An unsteady flow is one where the flow field variables at any given point are changing with time.

o If the flow is unsteady, by watching only point 1, the pressure, density, etc. will fluctuate with time.

o A steady flow is one where the flow field variables at any given point are invariant with time.

CONTINUITY EQUATION

Mass Flow –

oo Mass flow through area A is the mass crossing A per second.

Mass Flux – o Max flux is the mass flow per unit area.

The continuity equation is the application of the physical principle of conservation of mass to a finite control volume fixed in space.

o The continuity equation has the advantage of relating aerodynamic phenomena over a finite region of space without being concerned about the details of precisely what is happening at a given distinct point in the flow.

o Below is the continuity equation in the form of a partial differential equation, relating the flow field variables at a point in the flow, as opposed to the previous equation which deals with a finite space.

* The continuity equations above make the assumption that the nature of the fluid is a continuum. Therefore, they will generally hold for three-dimensional, unsteady flow of any type of fluid, inviscid or viscous, compressible or incompressible.

Unsteady Flowo The flow field variables are a function of both spatial location and time.

Steady Flow

o The flow field variables are a function of spatial location only.

o For steady flow, , the continuity equations reduce to…

MOMENTUM EQUATION Force is the time rate of change of momentum.

o Force comes from two sources:

o Body forces: gravity, electromagnetic forces, or any other forces which “act at a

distance” on the fluid inside .o Surface forces: pressure and shear stress acting along the control surface .

Body force is defined as…

The pressure force is…

In viscous flow, the shear and normal viscous stresses also exert a surface force. Therefore, the total force experienced by the fluid as it is sweeping through the

fixed control volume is given by…

Looking at the initial force equation, o Net flow of momentum out of control volume across surface

The flow has a certain momentum as it enters the control volume, and, in general, it has a different momentum as it leaves the control volume due in

part to the force that is exerted on the fluid as it is sweeping through . The net flow of momentum out of the control volume across the surface is simply the outflow minus the inflow of momentum across the control surface.

o Time rate of change of momentum due to unsteady fluctuations of flow

properties inside control volume .

o Combining the two equations, we obtain an expression for the total time rate of

change of momentum of the fluid as it sweeps through the fixed control volume…

Thus, from substitution, applied to fluid flow is…

o This equation has the advantage of relating aerodynamic phenomena over a finite region of space without being concerned with the details of precisely what is happening at a given distinct point in the flow.

Momentum equation in differential form (Navier-Stokes equations)…

o These are the partial differential equations that relate flow-field properties at any point in the flow.

Specialized to a steady , inviscid flow with no body forces , the equations become (Euler equations)…

The drag of a body in terms of the known freestream velocity and the flow-

field properties across a vertical station downstream of the body…

o Drag can be thought of as a wake momentum decrement from the flow velocity

decrease that occurs from the drag on the body.

SUBSTATIAL DERIVATIVE

The substantial derivative is the instantaneous time rate of change of a property of a fluid element as it passes through a fixed point. The time rate of change due to the movement of the fluid element from one location to another in the flow field where the flow properties are spatially different. (Locked on fluid element.)

The partial derivative is the time rate of change of a property at a fixed point. (Locked on fixed point.)

So…

Continuity Equation – in terms of the substantial derivative

Momentum Equation – in terms of the substantial derivative

Energy Equation – in terms of the substantial derivative

Pathlines, Streamlines, and Streaklines of a Flow

o A pathline is the path an element takes while moving from one point to another point later in time.

o In general, for unsteady flows, the pathlines for different fluid elements passing through the same point are not the same.

o A streamline is a curve whose tangent at any point in the direction of the velocity vector at that point.

o You can visualize a pathlines as a time-exposure photograph of a given fluid element, whereas the streamline pattern is like a single frame of a motion picture of the flow. In a steady flow, the magnitude and direction of the velocity vectors at all

points are fixed, invariant with time. Therefore, the pathlines for different fluid elements going through the same point are the same. Moreover, the pathlines and streamlines are identical. Therefore, in steady flow there is no distinction between pathlines and streamlines; they are the same curves in space.

Equation for a streamline…

Angular Velocity, Vorticity, and Strain

Angular velocity…

Vorticity…

o In a velocity field, the curl of the velocity is equal to the vorticity.

If at every point in a flow, the flow is called rotational. This implies that the fluid elements have a finite angular velocity.

If at every point in a flow, the flow is called irrotational. This implies that the fluid elements have no angular velocity; rather, their motion through space is pure translation.

o The condition of irrotationality for two-dimensional flow.

In general, viscous flows are rotational.

In a uniform flow, In an inviscid flow, there is no internal friction and no shear stress at the wall to

introduce vorticity into the flow. A flow field that is originally irrotational, without any internal

mechanisms such as frictional shear stress to generate vorticity, will remain irrotational throughout.

Circulation is considered to be positive if the circulation is clockwise.

The circulation about a curve C is equal to the vorticity integrated over any open

surface bounded by C. This leads that if the flow is irrotational everywhere within the contour of integration, then .

At a point P in a flow, the component of vorticity normal to dS is equal to the

negative of the “circulation per unit area” where the circulation is taken around the boundary dS.

STREAM FUNCTION The mass flow between two streamlines is defined per unit depth perpendicular to

the page.

For a steady flow, the mass flow inside a given streamtube is constant along the

tube; the mass flow across any cross section of the tube is the same. is constant for a given streamtube.

If is known for a given flow field, then at any point in the flow the products

For incompressible flow only…

VELOCITY POTENTIAL

For an irrotational flow, there exists a scalar function such that the velocity is

given by the gradient of which is the velocity potential.

Distinct differences between and :

o The flow-field velocities are obtained by differentiating in the same direction

as the velocities, whereas is differentiated normal to the velocity direction.o The velocity potential is defined for irrotational flow only. In contrast, the

stream function can be used in either rotational or irrotational flows.o The velocity potential applies to three-dimensional flows, whereas the stream

function is defined for two-dimensional flows only.

Because irrotational flows can be described by the velocity potential such flows are called potential flows.

The slope of = constant line is the negative reciprocal of the slope of a = constant line, meaning that streamlines and equipotential lines are mutually perpendicular.

Analytical solutions has three advantages:o The act of developing these solutions puts you in intimate contact with all the

physics involved in the problem.o The results, usually in closed-form, give you direct information on what are the

important variables, and how the answers vary with increases or decreases in these variables.

o The results in closed-form provide simple tools for rapid calculations, making possible the proverbial “back of the envelope calculations” so important in the preliminary design process and in other practical applications.

CFD solutions are completely numerical solutions and can only be carried out by a computer.

CHAPTER 3 – Fundamentals of Inviscid, Incompressible Flow Bernoulli’s Equation

o For an inviscid flow with no body forces, the x-component…

or

o Euler’s Equation – Applies to an inviscid flow with no body forces, and it relates

the change in velocity along a streamline to the change in pressure along the same streamline.

For a steady flow,

For an incompressible flow, the density is constant. Therefore…

o Euler’s equation holds for roatational and irrotational flows. However…

For a general, rotational flow, the value of the constant will change from one streamline to the next.

For an irrotational flow, the constant is the same for all streamlines.

The physical significance of Bernoulli’s equation is that when the velocity increases, the pressure decreases, and when the velocity decreases, the pressure increases.o It is also a relation for mechanical energy in an incompressible flow. It states

that the work done on a fluid by pressure forces is equal to the change in kinetic energy of the flow.

The strategy for solving most problems in inviscid, incompressible flow is as follows:o Obtain the velocity field from the governing equations.o Once the velocity field is known, obtain the corresponding pressure field from

Bernoulli’s equation.

In a duct, the variation of the area is often moderate, and for such cases it is reasonable to assume that the flow-field properties are uniform across any cross section, and hence only vary in the x direction. This is called quasi-one-dimensional flow.

In a convergent/divergent duct…

or, if the density is constant…

For incompressible flow, the dynamic pressure is precisely the difference

between total and static pressure.

BERNOULLI’S EQUATION HOLDS FOR INCOMPRESSIBLE FLOW ONLY. The pressure coefficient at a stagnation point for incompressible flow is equal to

1.0 The pressure coefficient is…

o Top equation holds for all types of flow.o Bottom equation holds only for incompressible flow.

Laplace’s Equation – The combination of irrotational and incompressible flow.

When examining a flow over different shaped objects, all are governed by the

same equation – In order to differentiate between these flows, we apply boundary conditions.

o Infinity Boundary Conditions – These are the boundary conditions on velocity at infinity. They apply at an infinite distance from the body in all directions, above and below, and to the left and right of the body.

o Wall Boundary Conditions – If the body has a solid surface, it is impossible

for the flow to penetrate it. If the flow is viscous, the influence of friction between the fluid and the solid

surface creates a zero velocity at each surface. If the flow is inviscid, the velocity at the surface can be finite, but because

the flow cannot penetrate the surface, the velocity vector must be tangent to the surface.

If we are dealing with then the wall boundary condition is the following, where is the distance measured along the body surface…

When dealing with either the velocity potential or the stream function, but rather the velocity components themselves, then…

*Boxed equation can be used for all inviscid flows, regardless of compressibility.

UNIFORM FLOW

The velocity potential for a uniform flow with velocity oriented in the +x-direction and applies to any uniform flow regardless of compressibility.

The stream function for an incompressible uniform flow in the +x-direction.

Circulation around any closed curve in a uniform flow is zero.

SOURCE/SINK FLOW A source flow is a two-dimensional, incompressible flow where all the streamlines

are straight lines emanating from a central point O.

o In a source flow, the streamlines are directed away from the origin.o In a sink flow, the streamlines are directed toward the origin.

For a source flow, everywhere except at the origin, where it is infinite. The origin is known as a singular point. The velocity is inversely proportional to the radial distance.

The total mass flow rate across the surface of a cylinder is…

Therefore, the rate of volume flow across the surface of the cylinder is…

o Denote the volume rate per unit length as…

o from the source, per

unit depth perpendicular to the page. The velocity potential for a source is…

The stream function for a source is…

There is no circulation associated with the source flow.

COMBINATION OF A UNIFORM FLOW WITH A SOURCE AND SINK

The velocity field for the figure above…

o The stagnation points in the flow can be determined by setting the equations

above equal to zero and solving for . The stagnation point is some distance directly upstream from the source.

o If the source strength is increased, keeping the same, the stagnation point will be blown further upstream.

o If is increased, keeping the source strength the same, the stagnation point will be blown further downstream.

If we want to construct the flow over a solid, semi-finite body, then all we need to

do is take a uniform stream with velocity and add it to a source of strength at a point D.

o The stream function for the combined flow at any point P is…

o The equation of the specific streamline going through the stagnation points

yields a value of zero for the constant. Thus, the stagnation streamline is given

by …

o All the flow from the source is consumed by the sink and is contained entirely

inside the oval, whereas the flow outside the oval has originated with the uniform stream only. Therefore, the region inside the oval can be replaced by a

solid body with the shape given by and the region outside the oval can be interpreted as the inviscid, potential (irrotational), incompressible flow over the solid body.

DOUBLET FLOW

A doublet is a special, degenerate case of a source-sink pair that leads to a singularity called a doublet.

The strength of a doublet is .

The stream function for a doublet…

The velocity potential for a doublet...

The streamlines for a doublet…

Streamlines flow from the source to the sink. Source on left, sink on right.o A doublet has a sense of direction associated with it and is the direction with

which the flow moves around the circular streamlines.o By convention, we designate the direction of the doublet by an arrow drawn

from the sink to the source.o As the source and sink become closer and closer, At this point, they fall

on top of eachother, but are indistinguishable due to the absolute magnitude of

their strengths becomes infinitely large in the limit resulting in a singularity of

strength – an indeterminate form that can have a finite value.o Think of a doublet flow as being induced by a discrete doublet of strength

placed at the origin. Therefore, a doublet is a singularity that induces about it a double-lobed circular flow pattern.

NON-LIFTING FLOW OVER A CIRCULAR CYLINDER

Stream function also written as…

The velocity field is obtained by differentiating the stream function…

o Set these equations equal to zero to find the stagnation points.

o Located at The equation of the streamline is found by setting the stream function to zero. The inviscid, irrotational, incompressible flow over a circular cylinder of radius R

can be characterized by…

o The entire flow field is symmetrical about both the horizontal and vertical axes

through the center of the cylinder.o Because the pressure distribution is symmetrical about both axes, then there is

no differential resulting in no lift and no drag. Despite the result above, we know that when an aerodynamic body is immersed in

a real flow, it will experience drag. This is known as d’Alembert’s Paradox.o This drag is due to viscous effects which generate frictional shear stress at the

body surface and which cause the flow to separate from the surface on the back of the body, thus creating a large wake downstream of the body and destroying the symmetry of the flow about the vertical axis through the cylinder.

o However, such viscous effects are not included in our present analysis of the inviscid flow over the cylinder. As a result, the inviscid theory predicts that the flow closes smoothly and completely behind the body. It predicts no wake, and no asymmetries, resulting in the theoretical result of zero drag.

Velocity distribution on the surface of the cylinder:

The surface pressure coefficient over a circular cylinder is…

VORTEX FLOW Vortex flow is…

o Vortex flow is a physically possible incompressible flow, that is, at every point.

o Vortex flow is irrotational, that is, at every point except the origin.

Velocity field for a vortex: Therefore, for vortex flow, the circulation taken about all streamlines is the same

value, being . By convention, is called the strength of the vortex flow.o A vortex of positive strength rotates in the clockwise direction.

Vortex flow is irrotational everywhere except at the point The velocity potential for a vortex…

The stream function for a vortex…

HANDY TABLE

LIFTING FLOW OVER A CYLINDER

The stream function for lifting flow of a cylinder…

o This is valid for inviscid, incompressible flow over a circular cylinder of radius R.o The streamlines are asymmetrical about the horizontal axis of the origin, and

therefore the cylinder will experience lift.o The streamlines are still symmetrical about the vertical axis through the origin

so there is still theoretically no drag experienced by the cylinder.o Because a vortex has been added, the circulation of the cylinder is equal to .

For a lifting flow over a cylinder of radius R, the flow velocity is…

*Although a stagnation point falls within the cylinder, we are concerned with external flow.*

Stagnation point obtained by setting to zero: The pressure coefficient for the cylindrical lift case…

THE KUTTA-JOUKOWSKI THEOREM AND THE GENERATION OF LIFT

The lift per unit span…

The Kutta-Joukowski theorem states that lift per unit span on a two-dimensional body is directly proportional to the circulation around the body.

*Remember that all three elementary flows are irrotational at all points, except for the vortex which has infinite vorticity at the origin.

Lift is caused by the net imbalance of the surface pressure distribution, and circulation is simply a defined quantity determined from the same pressure.o In the theory of incompressible, potential flow, it is generally much easier to

determine the circulation around the body rather than calculate the detailed surface pressure distribution, which is why the circulation theory of lift is useful.

CHAPTER 4 – Thin Airfoil Theory The aerodynamic consideration of wings can be split into two parts:

o The study of the section of a wing – an airfoil.o The modification of such airfoil properties to account for the complete, finite

wing.

At low-to-moderate angles of attack, the lift-coefficient varies linearly with the

angle of attack. The slope of this straight line is denoted by and is called the lift slope.o In this region, the flow moves smoothly over the airfoil and is attached over

most of the surface. As the angle of attack becomes large, the flow tends to separate from the top

surface of the airfoil, creating a large wake of relatively “dead air” behind the airfoil.

o Inside this separated region, the flow is recirculating, and part of the flow is actually moving in a direction opposite to the freestream – called reversed flow – and is due to viscous effects.

o The consequences of this separated flow at high angles of attack is a precipitous decrease in lift and a large increase in drag, resulting in the airfoil being stalled.

The max value occurs just prior to the stall, and is denoted by . It is one of the most important aspects of airfoil performance because it determines the stalling speed of an airplane.

o The higher is , the lower the stalling speed. F