FM Notes Vimp.

8/13/2019 FM Notes Vimp.

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Origin of Fluid Forces

We are often particularly interested in the forces and moments applied to bodies moving through

the fluid. These can be divided into just two types: pressures and shears.

Pressuresare created at the surface of a body due

to (nearly) elastic collisions between molecules of

the fluid and the surface of the body.

Shearing forcesare produced by fluid viscosity. This quantity is a measure of how well

momentum is transferred between adjacent layers of the fluid.

Although both types of forces are important in applied aerodynamics, pressures are usually the

dominant type of force.

Each of the shapes below, drawn to scale, have the same drag. The reason that the streamlined

airfoil can be so much larger is that most of the force is due to shear stress, not pressure forces.

The cylinder, with its separated airflow, has large pressure forces that give rise to high drag.

In fact, it is quite amazing how much force can be generated by differences in pressure: Many

airliners have wing loadings (weight / wing area) of over 100 lbs/ ft2(4.8 KPa). This means that

it takes a section of wing only as large as a book to lift a large dog (for example).

This is possible because the normal atmospheric pressure is 2116 lb/ft2(or 101 KPa) at

sea level. So, in fact 100 psf (4.8 KPa) represents only a 5% change in the pressure on

the upper side of the wing.
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At 68,000 ft (20.7 km) you would have to create a complete vacuum on the wing upper surface

to lift that much weight.

The Origin Of Pressure Forces

Pressure arises because each molecule that bounces off the surface transfers momentum to the

body. If a particle of mass, m, hits the body "straight-on" and bounces off, it transfers momentum

of the amount 2mc where c is the speed of the molecule. The pressure is then proportional to the

number of molecules striking a unit area of the surface per unit time, (Number density*c), times

the momentum transfer per particle, (~mc) or: p = k1 c2

Note that since temperature is defined as proportional to the mean kinetic energy of the

molecules, T = k2 c2. So we expect: p = k T, the perfect gas relation.

The component of molecular velocity normal to the surface is what is really needed in the above

expression, and if the body is moving, we must add its velocity to the molecular velocity

measured in the "fluid-fixed" reference frame.

Typically, we do not consider these direct interactions, but rather model the molecules as a

continuous fluid. This works well for most flows of interest. However, for very rarefied flows

such as those associated with initial re-entry of space vehicles, it is sometimes possible to

analyze aerodynamics with kinetic theory, keeping track of the molecular interactions. The figure


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below shows the results of one such calculation.

The Origin Of Shear Forces

As molecules in adjacent layers with different average

velocities collide, they transfer momentum between the

layers. The rate of change of momentum produces a shear

stress in the fluid. At the surface of a body, molecules

transfer momentum to the surface as they collide, resulting in a tangential, shear force.

When molecules hit the surface of a body, they bounce around among the surface molecules and

finally leave with a tangential velocity which is, on average, that of the surface itself. Thus, the

average tangential velocity near the surface of a body is zero with respect to the body. This is the

so-called no-slip condition.

This layer of slow moving fluid near the body surface is called the

boundary layer, and the viscosity of the fluid causes a distribution of

tangential velocity above the surface as shown here. As the tangentialmomentum of the air molecules is transferred to the surface, a shear

stress is produced.

This shear stress is related to the viscosity and velocity gradient by the expression:

= dU/dy


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We can see more quantitatively how the transfer of tangential momentum between fluid layers

leads to this relation by considering a small section in the boundary layer.

Molecules starting near the top of the box and moving to the bottom, lose momentum in the

amount:

m h dU/dy

Since the shear stress, , is the rate of change of momentum:

= n m h dU/dywhere n is the number of molecules passing through this area per unit time.

Now n is related to the average molecular velocity, c, and the density, so:

m n = c

and the shear stress is:

= c h dU/dy

so that: = c h

The height, h, over which molecules transfer their momentum is related to the mean free path, ,

with more detailed calculations showing that:

= 0.49 c

Now the mean free path decreases almost in proportion to density and the average molecular

speed varies with T, so we expect that varies with T and does not depend on pressure. This is

approximately true for most fluids.

Dimensionless Groups

The forces on a body, moving through a fluid, depend on the body velocity (V), the fluid density

(), temperature, and viscosity (), the size of the body (l), and its shape. Using the speed of

sound, a, rather than temperature (they are directly related) we can then make the following tablethat shows the units associated with each of the parameters. Here the numbers indicate the power

to which the mass, length, or time units are raised:


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F = f (V, , a, , l, shape)

mass 1 0 1 0 1 0 0

length 1 1 -3 1 -1 1 0

time -2 -1 0 -1 -1 0 0

The Buckingham pi theorem states that the number of dimensionless parameters is equal to the

number of parameters minus the rank of the above matrix. In this case 7 - 3 = 4. So, there exists a

functional relationship among the four dimensionless groups. We can express the force on a

body, for instance, by a relationship between the following four dimensionless parameters:

F / ( V l ) V l / V / a shape

Dimensionless Force Coefficient Reynolds Number Mach Number Geometry

The relation is: F / ( V2

l2

) = f (Vl/, V/a, shape)

We will discuss each of these dimensionless groups in a moment, but let's first look at thefunctional relationship between them. Much of applied aerodynamics involves finding the

function f, but there is a great deal we can say, even without knowing it. For example, we can see

that a wide variety of similar flows exist. The forces on a large, slow-moving body could bepredicted from tests of a small higher-speed model as long as the speed of sound were

sufficiently high. Also, the flow around a small insect could be represented by a large model in a

very viscous fluid. The idea behind model testing is to simulate the flow over one body by

matching the dimensionless parameters of another.

This is not always easy -- or possible. The followingfigure, from J. McMasters of Boeing shows the Mach

and Reynolds number range of several wind tunnels.

Why can't wind tunnels be designed to more fully coverthis range of parameters? What alternatives exist to

wind tunnel tests? (See assignments.)

Subsequent pages consider each of the dimensionlessgroups in a bit more detail. First note that we could

have included other fluid properties such as specific

heats. This would lead to additional dimensionlessparameters such as the Prandtl number which isimportant in the study of compressible boundary layerswith heat conduction. We have also left out gravity

which is often important in the flow of water around ships. This would lead to an additionaldimensionless parameter called the Froude number. There are often several ways of combining

the parameters to form dimensionless groups, but these are commonly used in aerodynamics.
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Dimensionless Forces

We use dimensionless force and moment coefficients defined by:

L = 0.5 V2S CL(Re, M, shape)

M = 0.5 V2S c Cm(Re, M, shape)

CLis called the lift coefficient, Cmthe moment coefficient. The length2term in our first

dimensionless parameter has been replaced by the area, S. This area could be anything we

choose (the contact area of the nose wheel, the wing planform area, the fuselage cross-sectional

area). In a particular application people generally agree on a reference area. For car dragcoefficients the frontal area is often used. For aircraft the wing area is a common reference area.

The c (for chord) in the moment coefficient definition is similarly agreed upon. This "agreement"on reference area is very important as can be seen in advertisements for cars. (Automobile dragcoefficients are usually based on frontal area and numbers like 0.4 are sometimes mentioned in

car ads. But, the drag coefficient means nothing by itself. If we chose the reference area to be the

floor area of the Fremont GM plant, we would have very low drag coefficients.)

Reynolds Number

The quantity: V l / is called the Reynolds number.

is the fluid density, V is the speed, is the fluid viscosity, and l is some characteristic length.

This length is, like the areas in the definition of dimensionless force coefficients, agreed on as a

standard by whoever is using it. So, we have chord Reynolds numbers which are based on wing

chord lengths or Reynolds numbers based on the diameter of a sphere, or any other characteristic

length that can be devised.

The Reynolds number is one of the most important and strange dimensionless numbers. It varies

over many orders of magnitude and expresses the importance of viscosity: high Reynolds

numbers can be achieved by decreasing the viscosity or making the length or speed very large.

The Reynolds number, in a sense, represents a ratio of pressure to shear forces:

V l / = V2/ (V/l)

V2is related to the pressure while V/l is related to dU/dy, the shear stress.


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The range of Reynolds number -- from McMasters.

Viscosity, and hence Reynolds number, strongly affects the performance of wings and airfoils,

making it an important parameter to match in wind tunnel tests. It is often not possible to match

these dimensionless parameters precisely.

The plot below shows the effect of Reynolds number on maximum lift to drag ratio for two

dimensional airfoil sections. Note the plight of insects.

The plot here shows the effect of Reynolds number on the maximum section lift coefficient of a

few typical airfoil sections. Note that these are not necessarily the best sections for high lift,

though.

Recent studies have shown that substantial changes in CLmaxare seen even at quite high Reynolds

numbers, making it difficult to extrapolate data on small wind tunnel models.


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Mach Number

The Mach number is the ratio of flow speed, V, to the speed of sound, a. It reflects the

importance of the compressibility of the fluid. This ratio is important because pressure

disturbances propagate in a fluid at the local speed of sound and the compressibility of a fluid

permits a sound wave to travel. The speed of sound in a fluid is related to the way in which

density and pressure vary: a2= dp/d

Assuming isentropic flow and a perfect gas: a2= R T

The flow pattern and pressures can change dramatically with Mach number as the applicable

differential equation changes form. (See later sections.) At low subsonic speeds, the effect of

compressibility is not large and the flow behaves almost as if pressure disturbances traveled with

infinite speed. As the flow speed is increased, but remains subsonic the effects of compressibilitystart to appear slowly. As the flow velocity approaches Mach 1 (transonic flow) more significant

compressibility effects appear quickly. Because of the increase in local velocity over parts of an

airfoil, the local Mach number can be much higher than the freestream Mach number. In fact,

compressibility effects can be important for high-lift sections at freestream Mach numbers as low

as 0.3. As the flow velocity increases beyond Mach 1.0, it becomes supersonic and its

characteristics change greatly. Very high velocity flows (usually above Mach 5 or 6) are called

hypersonic. These types of flow are of great importance in the aerodynamics of rockets and re-

entry vehicles, which achieve Mach numbers as high as 25.

Conservation Laws

To derive the equations of motion for fluid particles we rely on various conservation principles.

These principles are entirely intuitive. They are a statement of the fact that the rate of change ofmass, momentum, or energy in a certain volume is equal to the rate at which it enters the borders


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of the volume plus the rate at which it is created inside. The first two of these will be used

extensively here.

These integral expressions are combined with the divergence theorem and the fact that they hold

over arbitrary volumes to obtain the differential form of the equations:

We can use the momentum theorem by itself to obtain useful results. In this example, we apply

the momentum theorem to relate the force on a body to the properties of the flow some distancefrom the body. This technique is useful in wind tunnel tests and is the basis of several

fundamental theorems related to lift and induced drag of wings.

We take the control volume shown below, bounded by the single surface, S which we divide into

3 parts: the outer surface (Souter), the inner surface (Sinner), and the pieces of the surfaceconnecting the two (S*). We can write the integral form of the momentum equation for steady

flow with no body forces as shown.


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Note that the contribution from the part of the surface connecting Sinnerand Souterto the integrals

is zero because as the two pieces of S* are made close together, the unit normals point in

opposite directions while p and V are equal.

Simplifying Approximations

The equations of motion for a general fluid are extremely complex and even if the problem could be

formulated it would be impractical to solve. Thus, from the outset, certain simplifying approximations

that are often very accurate, are made. These may include the following assumptions.

Continuity and Homogeneity

We assume that the fluid is composed of particles which are so small and plentiful that the

statistically-averaged properties of interest are the same at any scale. This works well for gases

and fluids under most conditions. It does not work for studying the flow of sand. It does not workwhen the fluid is so rarefied that the mean free path is of the same order as the dimensions of

interest in the problem. The mean free path varies with altitude as shown in the plot. We further

assume that the medium can be treated as a single type of fluid -- no suspensions of oil and


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water.

Inviscid

The effect of viscosity may sometimes be neglected or modeled indirectly. For manyaerodynamic flows of interest, the region of high shear and vorticity is confined to a thin layer of

fluid. Outside this layer, the fluid behaves as if it were inviscid. Thus the simpler equations of an

inviscid fluid are often solved outside of the shear layers.

There are some fluids which seem to be almost completely inviscid. Tests in superfluid heliumhave given results similar to inviscid calculations.

Incompressible (constant density)

When the fluid density does not change with changes in pressure, the fluid is incompressible.Water density changes very little with changes in pressure and is generally treated as an

incompressible fluid. Air is compressible, but if pressure changes are small in comparison with

some nominal value, the corresponding changes in density are small also and incompressibleequations work quite well in describing the flow. The degree to which the fluid density changes

with pressure is related to the speed of sound in the fluid. Thus, assuming that the flow is

incompressible is equivalent to assuming that the speed of sound is infinite. When the local

Mach number is less than 0.2 to 0.5 compressibility effects can often be ignored. The reason forthis is discussed further in the chapter on compressibility, but one can see qualitatively that in

order to make an appreciable change to the nominal 2116 lb/ft^2 air pressure at sea level,

substantial speeds are required.


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Irrotational

Circulation is defined as:

It is a measure of the rotation of an area of fluid. As the integration contour is shrunk down to apoint, the ratio of circulation to the area enclosed by the curve is called the vorticity.

Fluid that starts out without rotational motion will not develop it unless there has been some

shear stress acting on it*. And if the shear is confined to a small region, the vorticity will be also.Thus, for many cases, especially in inviscid flow, much of the flow field may be treated as

irrotational:curl V = 0

When this is the case, the vector field, V, may be written as the gradient of a scalar field, :V = grad

where is the called the potential. This simplifies many of the equations discussed in subsequent

sections. The velocity components are then: u = d / dx and v = d / dy

Steady

When the variables describing the fluid properties at a given point do not change in time, theflow may be treated as steady and the time derivatives in the equations of motion are zero. This

condition depends on the chosen coordinate system. If the system is at rest with respect to a bodyin uniform motion through a fluid the equations in that system are steady, but expressed in a

system fixed with respect to the undisturbed fluid, the flow is unsteady. It is often convenient to

transform the coordinate system to one in which the flow is steady. This is, of course, not always

possible. We will assume that the flow is steady in most of the discussions in this course butunsteady effects are often important in the study of bird flight, propellers, aircraft gust response,

dynamics, and aeroelasticity as well as in the study of turbulence.

We will always apply the first of these assumptions and will sometimes adopt one or more of thelatter in the following discussions.

*Some important exceptions to the idea that without viscosity irrotational flow remainsirrotational:Vorticity can be created in a gravitational field when density gradients exist or in a rotating


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system (such as the earth) due to Coriolus forces. These are important sources of vorticity in

meteorology.

Equations of Fluid Flow

The conservation laws may be used to derive the equations of fluid flow. These are

supplemented with constitutive relations such as the perfect gas law:

p = R T

or the isentropic relation between pressure and density:p2/ p1= (2/ 1)

Some of the most commonly-solved equations are shown in the following table along with the

corresponding assumptions.

Equation Inviscid IrrotationalSmall

PerturbationsIncompressible Notes

Navier-Stokes - - - - Homogeneous

Reynolds-

AveragedNavier-Stokes

- - - -

Modeled

Turbulence

Euler X - - -

Full Potential X X - -

TransonicSmall

Disturbance

X X X -

Prandtl-Glauert X X X - Linearized

Acoustic X X X - Linearized

Laplace X X - X

Navier-Stokes Equations
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The Navier-Stokes equations describe the flow of a continuous, Newtonian fluid. They may be

derived from the principal of conservation of momentum. (For more details seenoteor Kuethe

and Chow Appendix B, Moran Ch. 6, Anderson Ch. 15).

where X, Y, Z are the body forces per unit mass in each direction and t is the stress tensor. X,Y,

and Z are often associated with gravitational forces and are often neglected.

The equations become more usable when the stress tensor is expressed in terms of viscosity andpressure. The pressure and shear forces may be expanded so that the NS equations are:

is the "bulk viscosity" , relating the normal stress to the rate of change of volume, div(V). Ifpressure is a function only of density and not of the rate of change of density, then:

= - 2/3

In the simplest case, with no body force, these equations become:

Solutions of the full Navier-Stokes equations show the onset of turbulence, the interaction of

shear layers, and almost all of the interesting aerodynamic phenomena (with the exception of

interacting or rarefied gas flows).

Unfortunately, the equations are very difficult to solve. As the Reynolds number is increased, the

scale of the interesting dynamics gets smaller so that most solutions of the full NS equations are

done at Reynolds numbers of 1 to 1000. One of the most recent solutions of a flat plate boundary

layer pushed the calculations to a Reynolds number of 1410 based on boundary layer thickness.
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These calculations took hundreds of hours on the Cray computer at NASA Ames.

Thevideoshows the results of a direct simulation of turbulence with color-coded vorticity

contours, from S. Robinson, NASA.

Even at very small Reynolds numbers, the geometries which can be analyzed using the full NS

equations are quite simple and it currently does not make sense to consider solving these

equations for realistic aircraft configurations. One reason that this is the case is that many of the

approximate equations work quite well in such cases and are much more easily solved.

When the time averaged Navier-Stokes equations are not a sufficient description of the problem,

one may resort to "large eddy simulations". This is a numerical solution of the time-dependent

Navier-Stokes equations, with only the smaller scales of turbulence modeled in an averaged way.

Larger scale turbulent motion can be included in this way. While this is faster than solving the

full equations, it is still very slow. The figure below shows results from a large eddy simulationof the flow over a 2D circular cylinder. Each simulation required approximately 300 CPU hours

and about 10 megawords of core memory on the Cray C-90. Figure from NASA / Parviz Moin.

Navier-Stokes Derivation
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This is the basic idea. We look at the momentum equation in the x direction:

where Fxrepresents a general force on a fluid element that might include viscous forces or

gravity.

Noting that:

conservation of mass permits us to substitute:

so:

The momentum equation in the x-direction may then be written:

or:

Substituting:

and canceling terms leads to:

Reynolds Averaged Navier-Stokes Equations

One of the most popular simplifications made to the Navier-Stokes Equations is "Reynolds

Averaging". This simplification to the full Navier-Stokes equations involves taking time

averages of the velocity terms in the equations.

Writing: u = + u ', v = + v', etc. (where represents a time average)

with the fluctuations having zero mean value:


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we have: =

2+


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From NAS Technical Summaries, High-Lift Configuration CFD, Karlin Roth, NASA Ames

Research Center

Euler Equations

The momentum equation is sometimes called Euler's equation. (There are lots of equations called

Euler equations!) But when people talk of solving the Euler equations these days, they arereferring to the inviscid equations of motion given by:

With some work*, the equation in the x direction becomes:

or in vector notation:

These are combined with the equations of energy and continuity.The equations are often solved by finite differences whereby the values of each velocity

component, the density, and the internal energy are computed at each point in the flow. From

these quantities constitutive relations such as the perfect gas law or the isentropic pressurerelation are used to find pressure. Since Euler equations permit rotational flow and enthalpy

losses (through shock waves), they are very useful in solving transonic flow problems, propeller


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or rotor aerodynamics, and flows with vortical structures in the field.

* Recall DF/Dt, the substantial, or particle derivative of F is defined by:

DF/Dt = dF/dt + V grad F

Also see the note in the derivation of the NS equations. It looks as though we have assumed

constant density, but this is not the case.

Full Potential Equation

The full potential equation is derived from the assumption of irrotational flow and the equations

of continuity and momentum. The pressure and density terms in the Euler equations can becombined when use is made of the perfect gas law and the isentropic relation between pressure

and density.

Ashley and Landahl show how we mayderivethe following vector form of the unsteady fullpotential equation:
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This may be simplified for the case of steady flow in 2-D to:

About the notation:

When flow is irrotational curl V = 0 and by definition of curl and gradient:curl (grad ) = 0

where is a scalar field.

Thus we can define a nonphysical scalar potential, , that describes the velocity field. is related

to the velocities by the relation:

V = grad

The equations can thus be written in terms of the unknown scalar rather than the 3 components

of the velocity. This simplifies their solution.

In the above expressions: a is the local speed of sound, x is the streamwise coordinate, and V is

the vector velocity. Subscripts denote partial derivatives with respect to the subscripted variables(e.g. Ux= du/dx)

Derivation of the Potential Equation


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The derivation of the full potential equation is easily seen in the case of 2-D steady flow. In this

case, the continuity equation is:

Also, for steady, inviscid flow, Euler's relation between p and is:dp = - V dV = - d(u

2+ v

2)/2

If the flow is also isentropic, then another relation between p and is:

dp = a2d

Combining the last two expressions:

We can thus write:

Then, substituting into the continuity equation, we obtain:

Note that the local speed of sound may be written in terms of some constants and the local

velocities:

Transonic Small Disturbance Equation

When the full potential equation is simplified by assuming that perturbation velocities are small

and we relate the local speed of sound to the freestream value by making use of the isentropic

relations we obtain the small disturbance equation (derivation):

When we let the freestream Mach number go to one and ignore the last term, the equation

becomes the classic transonic small disturbance equation:
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A great deal has been written about this nonlinear equation and its variants. (See Nixon.) It isused less frequently these days since finite difference methods can be used to solve the full

potential equation directly.

TSD Derivation

We begin with the 2-D full potential equation:

Ignoring terms of second order in the perturbation velocities

and with:

The local speed of sound, a, may be related to the local velocity by isentropic relations. After

some algebra, and again dropping terms of second order in the perturbations:

Substituting:

The final term is sometimes neglected.


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Prandtl-Glauert Equation

The Prandtl-Glauert equation is a linearized form of the full potential equation.

Full potential:

If the velocity perturbations are much smaller than the freestream velocity, this expression

becomes:

or in the unsteady case:

The 3-D version is easily constructed with the addition of z derivatives corresponding to the y

derivatives shown here.

Note that this linearized form of the equation does not hold near the nose of an airfoil where thevelocity perturbation is of the same order as the freestream, unless the freestream Mach number

is itself small. Also note that this expression holds for subsonic and supersonic flow (but not

transonic flow). It forms the basis for many aerodynamic analysis methods.

Analysis of P51 Mustang from Analytical Methods, Inc. using VSAERO, a code that solves the

Prandtl-Glauert equations.

Acoustic Equation


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The acoustic equation may be obtained from the full potential equation by assuming that there is

no freestream velocity, and that all perturbation velocities are small.

Or, by changing from a coordinate system fixed to the body to one fixed with respect to the

undisturbed fluid, the Prandtl-Glauert equation may be transformed to the acoustic equation.

This equation is often used in the study of sound propagation and sometimes for rotor

aerodynamics; thus the name.

Laplace's Equation

Laplace's equation is the Prandtl-Glauert equation in the limit as the freestream Mach number

goes to zero. It was actually first derived by Euler. The derivation is very simple, requiring only

the equation of continuity, and the assumptions of irrotational and constant density flow.

The continuity equation becomes then:

Since the flow is irrotational:

Substitution into the continuity equation yields:It is interesting to note that Laplace's equation does not require the assumption of small

perturbations, while the Prandtl-Glauert equation does. In fact, near the stagnation point of an

airfoil where velocities become small, the full potential equation reduces to Laplace's equation,not the Prandtl-Glauert equation.

Note also that all of the time dependent terms in the full potential equation are multiplied by 1/a2

so that this form of the equation holds for unsteady phenomena as well.

Bernoulli Equations

The Equations

Some of the equations we have discussed are posed in terms of state variables that do not include


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pressures. In these cases (e.g. the potential flow equations) the differential equations and

boundary conditions allow one to compute the local velocities, but not the pressures.

Once the velocities are known, however, the momentum equation can be used to find the local

pressure. Such equations are known as Bernoulli equations and they come in various forms,

depending on the assumptions that can be made about the flow.

The conservation of momentum principle is the source of the relation between pressure and

velocity. It can be used very simply toderive the Bernoulli equation.

To illustrate the basic physics behind the Bernoulli equations, we can derive a simple form: that

for steady, incompressible flow.

In this case we show that along a streamline:

When the flow is not steady, the Euler equations can be integrated to obtain a more general form

of this result: Kelvin's equation, the Bernoulli equation for irrotational flow.

Where f is a body force per unit mass (such as gravity) and F is an arbitrary function of time.

If we do not assume that the flow is irrotational, we cannot introduce the potential and the

expression is not so nicely integrable. If, however, we assume that the flow is steady with no

"body forces", but not necessarily irrotational we can write the following expression that holds

along a streamline:

While the above equations hold for steady flows along a streamline, for irrotational flows theyhold throughout the fluid.

We can derive a more useful form of the Bernoulli equation by starting with the expression for

steady flow without body forces shown just above.

If the flow is assumed to be isentropic flow (no entropy change or heat addition): p = constant *

Substitution yields the compressible Bernoulli equation:
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This actually works for adiabatic (no heat transfer) flows as well as isentropic flows.

In summary, we often deal with one of two simple forms of the Bernoulli equation shown below.

The Pressures

In both the incompressible and compressible forms of Bernoulli's equation shown above there

are 3 terms. The quantity pTis the total or stagnation pressure. It is the pressure that would be

measured at points in the flow where V = 0. The other p in the above expressions is the staticpressure.

Note that in incompressible flow, the speed is directly related to the difference in total and static

pressure. This can be measured directly with a pitot-static probe shown below.

The dynamic pressure is defined as:

The static pressure coefficient is defined as:

where p is the freestream static pressure.

In incompressible flow, the expression for Cpis especially simple:

If the local velocity is expressed as a small perturbation in the freestream:

Then the incompressible Cprelation can be written:Be careful with this expression! It is often not a good approximation and the correct expression isnot very difficult.


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The expression for Cpin compressible isentropic flow (sometimes called the isentropic pressure

rule) is derived from the compressible Bernoulli equation along with the expression for the speed

of sound in a perfect gas. In terms of the local Mach number the expression is:

In air with gamma = 1.4

Some interesting results follow from this expression...

We can tell if the flow is supersonic, just by looking at the value of Cp. The critical value of Cp,

denoted Cp* is found by setting M = 1 in the above expression:

Also, we see that there is a minimum value of Cp, corresponding to a complete vacuum. Setting

the local Mach number to infinity yields:

Cpcannot be any more negative than this. Experiments show that airfoils can get to about 70% of

vacuum Cp. This can limit the maximum lift of supersonic wings.

Solution Methods

This chapter is a brief overview of methods used to investigate fluid flows.

It includes a bit of discussion on the role of experimental, analytical, and computational methodsand outlines the basic ideas behind the computational approaches.

1. The Role ofTheory and Experiment

2.Analytic Methods

3.CFD Overview
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4.Panel Methods

5.Nonlinear CFD

6.References

Panel Methods -- Introduction

Since the equations solved by panel methods are linear, we can multiply a known solution by a

scalar and add these results together to form more general solutions. This can be made to work in

both subsonic and supersonic cases.

Panel methods may be based on one or more fundamental solutions to the Prandtl-Glauert

equation or Laplace's equation. These commonly include source, vortex, and doublet flows,

discussed in the section on potential theory.

The basic idea is to add up known solutions...... such as a uniform flow...

...and a point source....

... to produce a streamline pattern that matches the flow of interest.

Here we add a freestream, a source, and a sink (negative source strength) to produce the flow
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over an oval (called a Rankine Oval).

We could superimpose many sources and sinks to get nearly any flow pattern we desired:

Panel methods are based on this idea. Sources (or doublets or vortices) of some strength are

located in the flow such that their combined solutions satisfy the boundary conditions of the

problem. The boundary conditions are typically that the combined flow does not go through the

surface, and that far from the body, the flow approaches the freestream solution.

Panel Methods -- Introduction

Since the equations solved by panel methods are linear, we can multiply a known solution by a

scalar and add these results together to form more general solutions. This can be made to work in

both subsonic and supersonic cases.

Panel methods may be based on one or more fundamental solutions to the Prandtl-Glauert


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equation or Laplace's equation. These commonly include source, vortex, and doublet flows,

discussed in the section on potential theory.

The basic idea is to add up known solutions...

... such as a uniform flow...

...and a point source....

... to produce a streamline pattern that matches the flow of interest.

Here we add a freestream, a source, and a sink (negative source strength) to produce the flow

over an oval (called a Rankine Oval).

We could superimpose many sources and sinks to get nearly any flow pattern we desired:


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Panel methods are based on this idea. Sources (or doublets or vortices) of some strength are

located in the flow such that their combined solutions satisfy the boundary conditions of the

problem. The boundary conditions are typically that the combined flow does not go through the

surface, and that far from the body, the flow approaches the freestream solution.

Panel Methods -- AIC Matrix

The next step, after dividing the geometry into panels is to compute the flow pattern at each

panel, i, associated with a source or doublet or vortex of unit strength at panel j. The component

of the velocity normal to the panel is denoted AIC(i,j), an element of the aerodynamic influence

coefficient matrix.

The flow at panel i associated with a singularity of unit strength at panel j can be computed from

the basic singularity solution. The result depends on the vector distance between the panels, R.

More specifically, the vector from the jth singularity to the control point of panel i (often at the

panel centroid).

The fundamental solution for the flow field some distance from the singularity is discussed in

following sections, but the jth panel usually contains not just a single point source or vortex or

doublet, but several vortex lines or a distribution of such singularities. The form of this

distribution is one of the things that differentiates various panel methods.

The distribution of singularity strength over a panel may be a constant value, or may vary

linearly or quadratically in both directions. The list below shows some of the panel codes and the

choice of singularity types and distributions.


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Panel Methods -- Boundary Conditions

After the AIC matrix is computed, we specify the boundary conditions.

The total normal velocity at panel i is then given by the expression below.

This must be zero if the flow is tangent to the surface of the body and constitutes the boundary

conditions of the problem.

Here, {n} is a vector of surface unit normals {sigma} represents the unknown singularity

strengths. Each element of these vectors is associated with one panel of the geometry.

The boundary conditions for panel methods must express the requirement that streamlines follow

the surface contour. But they do not have to explicitly set Vn = 0. In fact, the method currently

more in vogue is to specify the B.C.'s in terms of the potential. This is called a Dirichlet (as

opposed to the von Neumann) type of boundary condition. It works as follows on the doublet

panel method.

The total potential in the interior of the section is set to 0. If the total potential is 0 everywhere

inside the body (in practice it is set to 0 just inside at each panel control point) then the velocity

there is 0 also. In particular the velocity normal to the panel, on the inside of the panel is 0. Since

doublets produce no jump in normal velocity (see next section) then Vn = 0 in the external flowas well. This form of the B.C.'s is often better behaved (numerically) than the direct (Neumann)

type of B.C..


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Nonlinear Methods

Nonlinear CFD methods can be used to predict complex flow fields such as those associated withtransonic or separated flows. They have been used recently for predicting flows from air over

hypersonic aircraft to blood through artificial hearts. The list below includes links to internet

sites with example applications:NAS Technical Summaries (NASA Ames)

A gallery of CFD examples from Fluent, Inc.

Analytical Methods, Inc.

Here are some suggestions about using these general methods, which seem in principle to be

capable of doing anything:

Use the simplest method or model adequate for the job.One need not always use the most sophisticated method to solve fluid mechanics

problems. It is sometimes fashionable to use the latest technology, but one need notsubmit a Cray job to find the roots of a quadratic. Sometimes the "outdated" 1960's

technology is just what is needed.

Evaluate CFD results critically before accepting them.Know assumptions, limitations of the method.Do an order of magnitude analysis.

Do results make sense qualitatively?
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Neither computation nor experiment is infallible."Nobody believes theoretical predictions but the engineer who computed them;

everybody believes experimental results but the engineer who conducted the test."

The basis of modern CFD techniques for the solution of the nonlinear equations of fluid flow isillustrated here.

We start with the differential equation such as:

This partial differential equation may be solved in two fundamentally different ways:

Finite Difference: Discretization of differential form of equations

Solutions for all unknowns computed at node points

Finite Volume:

Discretization of integral form of equations

Solution computed at cell centroids

To do this requires that the flow field be first divided into a grid. This is often difficult as the grid

must not only conform to the body but also be dense in regions of large flow gradients. The two

grid directions must be relatively orthogonal so that the difference equations are goodapproximations to the real PDE. The process of generating such a grid is one of the more

difficult aspects of nonlinear CFD. The following examples illustrate some of the approaches.

The intersection of the grid with the surface of a wing/nacelle is shown here. The grid is divided

into a number of pieces to better accommodate the complex geometry. (From NAS Summary:Flow Simulation for Subsonic Transports, Pieter G. Buning, NASA Langley Research Center)

Although grids with a fixed topological structure are often used, unstructured adaptive grids are
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also used, especially over complex, multiply-connected domains as might be found on a multi-

element airfoil.

From NAS Summaries: Adaptive Unstructured Flow Computations, Dimitri J. Mavriplis, NASA

Langley

Shown below is the computer power required to solve various non-linear problems in a

reasonable period of time (~15 min). Some of the points are a bit optimistic.


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The vast amounts of data generated by these codes are often displayed using computer graphics.

Sometimes it is illuminating to use simulated experimental techniques such as oil flow patterns,tufts, or smoke particle traces in order to visualize the results.

From: NAS Summaries -- Viscous Unstructured-Grid Computations, William K. Anderson,

NASA Langley

Click here for a short Quicktimevideo clip of a streamline simulationover a wing-body model.Here is another clip from NASA Ames of the flow over avertical take-off aircraft in ground-

effect.

Note that a 128 x 128 x 128 grid requires 2.1 M grid points (25 MB just for velocities with 4

bytes per number). A wing with 20 x 40 panels might be modeled with a 60x80x80 gridrequiring only 384K points.

Finite Difference Methods
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A matrix with these equations (tridiagonal system) is then solved at each time step. The matrix is

huge (perhaps a million by a million) but sparse.

Finite Volume Methods


2-D Potential Flow

This chapter starts the description of solution methods in detail. Beginning with the simplest

flows: two-dimensional, inviscid and irrotational, the chapter describes the basic theoreticalresults. These are applied to airfoil problems in later chapters and then modified to include the

effects of compressibility and viscosity.

1. Basic Theory2. Sources and Vortices3. Interactive Calculations4. References5.Basic 2-D Potential Theory
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6.7. We outline here the way in which the "known" solutions used in panel methods can be

generated and obtain some useful solutions to some fundamental fluid flow problems.Often the known solutions just come out of thin air and can be applied, but sometimes

other approaches are possible.

The simplest case, two-dimensional potential flow illustrates this process. We shalldiscuss 2-D incompressible potential flow and just mention the extension to linearized

compressible flow.

For this case the relevant equation is Laplace's equation:

There are several ways of generating fundamental solutions to this linear, homogeneous,

second order differential equation with constant coefficients. Two methods areparticularly useful: Separation of variables and the use of complex variables.

Complex variables are especially useful in solving Laplace's equation because of thefollowing:

We know, from the theory of complex variables, that in a region where a function of the

complex variable z = x + iy is analytic, the derivative with respect to z is the same in anydirection. This leads to the famous Cauchy-Riemann conditions for an analytic function

in the complex plane.

Consider the complex function: W = + i

The Cauchy-Riemann conditions are:

Differentiating the first equation with respect to x and the second with respect to y andadding gives:

Thus, analytic function of a complex variable is a solution to Laplace's equation and may

be used as part of a more general solution.

W = + i is called the complex potential.

It consists of the usual velocity potential as the real part and the stream function as its

imaginary part.

The flow velocities can be then be written as a single complex number:dW/dz = u - iv (Try deriving this.)

We consider some simple analytic functions for W that are of great use in applied

aerodynamics:

Uniform flow:
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Line Source or Vortex:

Doublet:

Uniform Flow

If U is real the flow is in the x direction with a speed U. The flow direction can be adjusted by

changing real and imaginary parts.

This is a good example of the fact that the potential is not defined apart from an arbitrary

constant. Although the flow is uniform everywhere, the potential depends on our choice of the

origin. Differences in the potential are physically meaningful, though and do not depend on the

choice of the origin.

Line Source or Vortex

The same expression describes a "point" source or vortex in 2D (which can be thought of as a

vortex line or line of sources in 3-D). When K is real the expression describes a source with

radially directed induced velocity vectors; imaginary values lead to vortex flows with induced

velocities in the tangential direction. Further discussion of these flows is given in the next

section.
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Doublet

A doublet is formed by superimposing a source and a sink along the x-axis. The doublet strength

is given by S dx. The fundamental doublet singularity with the potential shown above is formed

by taking the limit as dx goes to zero and S goes to infinity while keeping the product constant.

The doublet is commonly used as one of the fundamental singularities in many panel methods.

Sources and Vortices

Notice that many of the solutions to the 2-D potential equation that we proposed are singular. In

fact, the source solution seems the ultimate way of violating continuity while the vortex is the

essence of rotational (not irrotational as we assumed) flow.


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These solutions are indeed singular at a point and do not satisfy the differential equation at that

point. Away from the singularity, however, they are perfectly adequate solutions as can be seen

by evaluating the integral forms of the continuity and irrotationality conditions.

Why the flow field near a source satisfies continuity:

Why the flow field near a vortex satisfies irrotationality:

The solutions are singular at a point, but even near the singular point strange things happen: the

velocity gets very large. In real life, the large velocities in this region give rise to compressibility

effects; viscous effects smear the discrete vortex into a distribution of vorticity in a viscous core.

The actual velocity distribution near the core of a free vortex behaves more like a solid body

with a velocity distribution V(R) = kR. (This is the result obtained by assuming a Gaussian

distribution of distributed vorticity in the core region. The size of the viscous core depends on

the Reynolds number, often taken as /.)

This 1/r behavior of the vortex induced velocity is not just a mathematical result. It is essential

for the flow to exist in equilibrium. We can easily see that the velocity must vary as 1/r for the

pressure gradients to balance the centrifugal force acting on the fluid. Thederivationis given
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here.

We can combine these singularities in different locations to produce the desired flow pattern.

Since the solution to Laplace's equation is uniquely determined in regions without singularities

when the solution on the boundaries is specified, we can use combinations of singularities to

model many flows of interest.

Method of Images:Ground Effects, Wall Interference

Source Doublets, Circular Cylinder, Ellipses, Blasius Theorem

Groups of Vortices, Far Field Flow, Stokes Theorem

Free Vortex Motion

Method of Images

The flow field created by singularities in the presence of solid boundaries can be simulated by

superimposing "image vortices".

This works because the symmetry of the problem on the right ensures that there is no flow

through the plane of symmetry. The boundary does the same thing for the problem on the left.

Since both of these problems have the same boundary conditions and satisfy the same linear

differential equation, the flow must be the same.
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This technique is useful for simulating the effects of the ground on the aerodynamics of cars or

airplanes at low altitude.

It can also be used in more complex situations. Here, three images are required to simulate the

boundary conditions associated with a corner.

This technique is used to predict the effects of wind tunnel walls on the flow field of models

being tested. Imagine the system of image vortices that would be required to simulate wall

effects on a 2D airfoil test. Yes, more than 2 images are required. The 3-D situation cannot in

general be solved with images.

Cylinders

The flow on a circular cylinder may be computed from a uniform stream and a doublet. (See

previous section.)

Some interesting conclusions and generalizations follow from the expressions for the velocity

and the potential on a circular cylinder shown above.


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Note that on the surface of the cylinder, the tangential velocity is: V = 2U sin ,

so the maximum velocity is twice the freestream value.

The more general forms of these results hold for all ellipsoids:

Vmax = V (1 + t/c) and V at surface = n x (n x Vmax))

Notice that this holds exactly in incompressible potential flow, even if the ellipse has a t/c much

larger than 1. Of course, in such a case, the real flow will probably look quite different from the

potential flow solution.

The force on a general 2-D cylinder can be computed by calculating the velocities, using

Bernoulli's law to compute pressures, then integrating the surface pressures. However, the total

forces and moments can be derived directly from the complex potential. The result is called the

Blasius theorem.

It is not derived here, but the result follows from the theory of residues, the complex potential,

and the incompressible Bernoulli equation. (Or one might just use the momentum equation and

compute the net force by far field integrals.)

where is the total circulation and S is the net source strength. In the case of no net source

strength, the net force exerted on a collection of sources and vortices in a flow with freestream

velocity U is perpendicular to the freestream and proportional to U and the total circulation.

Circulation, Vorticity, and Stokes Theorem

Stokes' theorem is an integral identity that may be written:

When the vector function F is taken to be the velocity field, V, then this relation in 2-D may be

restated as:


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This result implies that the circulation around a contour that contains a group of vortices is just

equal to the sum of the enclosed vortex strengths.

This allows application of the Blasius theorem to find the force acting on a group of vortices.

The result is sometimes called the Kutta-Joukowski law:

We can also treat the flow field far from a group of vortices as if it were created by a single

vortex with a strength equal to the sum of the individual vortices. Such far field solutions can be

especially simple and useful as a check of more complex results. Far field solutions can also beused as boundary conditions for the more complex near field solution, reducing the required

extent of computational grids.

We should note here that just because we find a superposition of singularities that satisfies the

boundary conditions and the differential equation, it does not mean that we have found the only

solution to the problem. For example, we could add a vortex to the doublet that was used to

model the circular cylinder, and we would still find that the flow went around the cylinder. These

non-unique solutions are problemsome and we appeal to additional considerations to find the

one(s) that actually will appear in nature. Just such an auxiliary condition, the Kutta condition, is

provided by viscous effects which then determine the value of circulation.

Free Vortices

Singularities that are free to move in the flow do not behave in response to F = ma (what is m?).

Rather they move with the local flow velocity. Thus, vortices and sources are convecteddownstream with the flow. And interacting singularities can produce complex motions due to

their mutual induced velocities.

A pair of counter-rotating vortices moves

downward because of their mutual induced

velocities.

Co-rotating vortices orbit each other under theinfluence of their mutual induced velocities.


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Streamlines Past Sources and Vortices

Drag any of the singularities from the well on the right into the main computation area.

Set the freestream speed (the flow is from left to right), then click Compute.

The marks on the page simulate small tufts and indicate the direction of the local flow.

Experiment with multiple singularities to simulate a pair of wing trailing vortices, a source/sink

doublet, or a spinning baseball.

Airfoils, Part I: Introduction

In this chapter, an introduction to airfoils and airfoil theory is followed by the application of potential

flow methods to the analysis of airfoils.

The purpose of this section is to discuss the relation between airfoil geometry and airfoil performance.

To do this we will discuss the methods that are used to compute the distribution of pressures over the

airfoil surface. Then we will discuss the relation between these pressures and the airfoil performance.

Outline of this Chapter

The chapter is divided into several sections. The first of these consist of an introduction to airfoils: some

history and basic ideas. The latter sections deal with simple analyses and results that relate the airfoil

geometry to its basic aerodynamic characteristics.

History and Development Airfoil Geometry Pressure Distributions Relation between Cp and Performance Relating Geometry and Cp
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Methods of Airfoil Analysis References History of Airfoil Development

The earliest serious work on the development ofairfoil sections began in the late 1800's. Although

it was known that flat plates would produce lift

when set at an angle of incidence, some

suspected that shapes with curvature, that moreclosely resembled bird wings would produce

more lift or do so more efficiently. H.F. Phillips

patented a series of airfoil shapes in 1884 after

testing them in one of the earliest wind tunnels inwhich "artificial currents of air (were) produced

from induction by a steam jet in a wooden trunk

or conduit." Octave Chanute writes in 1893, "...itseems very desirable that further scientific

experiments be be made on concavo-convex

surfaces of varying shapes, for it is notimpossible that the difference between success

and failure of a proposed flying machine will

depend upon the sustaining effect between a

plane surface and one properly curved to get amaximum of 'lift'."

At nearly the same time Otto Lilienthal had similar ideas.

After carefully measuring the shapes of bird wings, hetested the airfoils shown here (reproduced from his 1894

book, "Bird Flight as the Basis of Aviation") on a 7m

diameter "whirling machine". Lilienthal believed that thekey to successful flight was wing curvature or camber. He

also experimented with different nose radii and thickness

distributions.

Airfoils used by the Wright Brothers closely resembled Lilienthal's sections: thin and

highly cambered. This was quite possibly because early tests of airfoil sections were doneat extremely low Reynolds number, where such sections behave much better than thicker
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ones. The erroneous belief that efficient airfoils had to be thin and highly cambered was

one reason that some of the first airplanes were biplanes.

The use of such sections gradually diminished over the next decade.

A wide range of airfoils was developed, based primarily on trial and error. Some of themore successful sections such as the Clark Y and Gottingen 398 were used as the basis

for a family of sections tested by the NACA in the early 1920's.

In 1939, Eastman Jacobs at the NACA in Langley, designed and tested the first laminarflow airfoil sections. These shapes had extremely low drag and the section shown here

achieved a lift to drag ratio of about 300.

A modern laminar flow section, used on sailplanes, illustrates that the concept is practical

for some applications. It was not thought to be practical for many years after Jacobsdemonstrated it in the wind tunnel. Even now, the utility of the concept is not wholly

accepted and the "Laminar Flow True-Believers Club" meets each year at the homebuiltaircraft fly-in.


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One of the reasons that modern airfoils look quite different from one another and

designers have not settled on the one best airfoil is that the flow conditions and design

goals change from one application to the next. On the right are some airfoils designed forlow Reynolds numbers.

At very low Reynolds numbers (


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Airfoil geometry can be characterized by the coordinates of the upper and lower surface.

It is often summarized by a few parameters such as: maximum thickness, maximumcamber, position of max thickness, position of max camber, and nose radius. One can

generate a reasonable airfoil section given these parameters. This was done by Eastman

Jacobs in the early 1930's to create a family of airfoils known as the NACA Sections.

The NACA 4 digit and 5 digit airfoils were created by superimposing a simple meanline

shape with a thickness distribution that was obtained by fitting a couple of popularairfoils of the time:

y = (t/0.2) * (.2969*x0.5

- .126*x - .3537*x2+ .2843*x

3- .1015*x

4)

The camberline of 4-digit sections was defined as a parabola from the leading edge to theposition of maximum camber, then another parabola back to the trailing edge.

NACA 4-Digit Series: 4 4 1 2 max camber position max thickness in % chord of max camber in % of chord in 1/10 of c After the 4-digit sections came the 5-digit sections such as the famous NACA 23012.

These sections had the same thickness distribution, but used a camberline with morecurvature near the nose. A cubic was faired into a straight line for the 5-digit sections.

NACA 5-Digit Series: 2 3 0 1 2 approx max position max thickness camber of max camber in % of chord in % chord in 2/100 of c The 6-series of NACA airfoils departed from this simply-defined family. These sections

were generated from a more or less prescribed pressure distribution and were meant to

achieve some laminar flow. NACA 6-Digit Series: 6 3, 2 - 2 1 2 Six- location half width ideal Cl max thickness


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Series of min Cp of low drag in tenths in % of chord in 1/10 chord bucket in 1/10 of Cl After the six-series sections, airfoil design became much more specialized for the

particular application. Airfoils with good transonic performance, good maximum lift

capability, very thick sections, very low drag sections are now designed for each use.

Often a wing design begins with the definition of several airfoil sections and then theentire geometry is modified based on its 3-dimensional characteristics.

Airfoil Pressure Distributions

The aerodynamic performance of airfoil sections can be studied most easily by reference to the

distribution of pressure over the airfoil. This distribution is usually expressed in terms of the

pressure coefficient:

Cpis the difference between local static pressure and freestream static pressure,

nondimensionalized by the freestream dynamic pressure. (See discussions of Cpand the

Bernoulli equation.)

What does an airfoil pressure distribution look like? We generally plot Cpvs. x/c.

x/c varies from 0 at the leading edge to 1.0 at the trailing edge. Cpis plotted "upside-down" withnegative values (suction), higher on the plot. (This is done so that the upper surface of a

conventional lifting airfoil corresponds to the upper curve.)

The Cpstarts from about 1.0 at the stagnation point near the leading edge...

It rises rapidly (pressure decreases) on both the upper and lower surfaces...

...and finally recovers to a small positive value of Cpnear the trailing edge.

Various parts of the pressure distribution are depicted in the figure below and are described inthe following sections.
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We can get a more intuitive picture of the pressure distribution by looking at someexamples and this is done in the follo

Airfoil Pressures and Performance

The shape of the pressure distribution is directly related to the airfoil performance asindicated by some of the features shown in the figure below.

Most of these considerations are related to the airfoil boundary layer characteristics

which we will take up later, but even in the inviscid case we can draw some conclusions.We may compute the maximum local Mach numbers and relate those to lift and

thickness; we can compute the pitching moment and decide if that is acceptable.

Whether we use the inviscid pressures to form qualitative conclusions about the section,

or use them as input to a more detailed boundary layer calculation, we must firstinvestigate the close relation between the airfoil geometry to these pressures.

Relating Airfoil Geometry and Pressures

Before discussing in detail the methods used to predict airfoil pressure distributions let's

consider, more intuitively the relationship between airfoil geometry and airfoil pressure

distributions.

We first look at the effect of changes in surface curvature (Click on figure to look inmore detail.)


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The figure below shows how the airfoil pressures vary with angle of attack. Note that the"nose peak" becomes more extreme as the angle increases.

To make this a bit more clear, you may use the small java program below to change the

angle of attack and see its effect on Cp, Cl, and Cm. Click on the upper half of the plot toincrease the angle of attack, alpha, and on the lower portion to decrease it.

Let's consider, in more detail the relationship between airfoil geometry and airfoilpressure distributions. The next few examples show some of the effects of changes in

camber, leading edge radius, trailing edge angle, and local distortions in the airfoil

surface.

A reflexed airfoil section has reduced camber over the aft section producing less lift over

this region. and therefore less nose-down pitching moment. In this case the aft section is

actually pushing downward and Cm at zero lift is positive.

A natural laminar flow section has a thickness distribution that leads to a favorable
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pressure gradient over a portion of the airfoil. In this case, the rather sharp nose leads to

favorable gradients over 50% of the section.

This is a symmetrical section at 4 angle of attack.Note the pressure peak near the nose. A thicker section would have a less prominent

peak.

Here is a thicker section at 0. Only one line is shown on the plot because at zero lift, the

upper and lower surface pressure coincide.

A conventional cambered section.


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An aft-loaded section, the opposite of a reflexed airfoil carries more lift over the aft part

of the airfoil. Supercritical airfoil sections look a bit like this.

The best way to develop a feel for the effect of the airfoil geometry on pressures is to

interactively modify the section and watch how the pressures change. A Program for

ANalysis and Design of Airfoils (PANDA) does just this and is available fromDesktopAeronautics.A very simple version of this program, is built into this text and allows you

to vary airfoil shape to see the effects on pressures. (Go toInteractive Airfoil Analysis

page by clicking here.) The full version of PANDA permits arbitrary airfoil shapes,permits finer adjustment to the shape, includes compressibility, and computes boundary

layer properties.

Interactive Airfoil Analysis

Introduction

The program built into this page allows you to experiment with the effect of airfoil shape and

angle of attack on the pressure distribution.

Instructions

Click on the top part of the plot to increase the angle of attack; clicking on the lower portionreduces alpha. Drag the handles shown on the upper or lower surfaces to modify the shape of the

section and watch the effects on Cp.

Suggested Exercises

Change the airfoil thickness and note the effect on upper and lower surface pressures. Notice

how thickness affects the Cpat the trailing edge. Create a pressure peak near the nose on the

upper or lower side by changing the angle of attack. Change the camber near the nose to removethe pressure peak. Try to create a positive pitching moment section, a very thin, highly cambered

section, and a symmetrical section.
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Technical Details

This program uses a combination of thin airfoil theory and conformal mapping to very quicklycompute pressures on an airfoil. A method like this was used in the 1950's to compute airfoil

pressure distributions before Java was invented. The section shape is very simple as well: upper

and lower surfaces consist of a quadratic in sqrt(x) and a quadratic in (1-x), patched together atthe control points. This provides just 4 degrees of freedom, but does lead to curves that look like

airfoils.

Conformal Mapping

Any analytic function of a complex variable satisfies the equation for incompressible,irrotational flow:Why?

We can, therefore, relate one flow field to another by setting:

where z' is related to z by an analytic function of z, z' = f(z). (Recall z = x + iy.)

The idea behind airfoil analysis by conformal mapping is to relate the flow field around one

shape which is already known (by whatever means) to the flow field around an airfoil. Mostoften a circle is used as the first shape. The problem is to find an analytic function that relates

every point on the circle to a corresponding point on the airfoil.

Joukowski found that the simple function: z' = z + 1/ztransforms a circle to a shape which looks a bit like an airfoil. By taking the origin of the

circle at various points, different airfoil-like shapes are produced.

With just 2 degrees of freedom (the coordinates of the circle origin), the number of airfoil

shapes that can be represented with the simple Joukowski transformation is limited.
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Furthermore, the thickness is greatest at 25% chord -- rather far forward.

This constraint has led to a number of generalizations of the Joukowski mapping that producemore practical shapes. But all of these methods are based on the same basic idea which is

illustrated in the simple mapping.

We begin with a circle. The center of the circle is at Xc, Yc.

Every point z is mapped to the point z' by the relation: z' = z + 1/z.

If W(z) = F(z) + i G(z) is the complex potential function, then the velocity is given by:

w = u - iv = dW/dz.

We set W(z') = W(z) so that the velocity on the airfoil may be related to the velocity on the

cylinder by:w(z') = dW/dz' = (dW/dz) (dz/dz') = w(z) / ( 1-z

-2).

Notice that this relates the velocity on the airfoil directly to that on the circle, but the relationblows up when z

2= 1.

We know that the velocity on the airfoil does not go to infinity anywhere. The reason that themapping does not work at these points is that the mapping is not analytic here. This does not

mean it cannot be used, it just means that we must make sure that such points are not in the

flow: they must be inside or on the airfoil. Here we choose the circle so that it encloses the

point -1,0, and we choose the circulation so that the velocity is 0 at the point 1,0. The point

1,0 maps to the airfoil trailing edge.

If we are to have a stagnation point on the cylinder at 1,0 we must have a certain amount of

circulation. The origin of the circle thus determines the lift on the airfoil at a given angle ofattack. (Also note that the point -1,0 must be enclosed.)

One of the troubles with conformal mapping methods is that parameters such as xc and yc are

not so easily related to the airfoil shape. Thus, if we want to analyze a particular airfoil, wemust iteratively find values that produce the desired section. A technique for doing this was

developed by Theodorsen.

Another technique involves superposition of fundamental solutions of the governingdifferential equation. This method, discussed in subsequent sections, is called thin airfoil

theory


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Thin Airfoil Theory Derivation

We start with the analysis of a very thin cambered plate and will build up the solution to a

more arbitrary airfoil.

A distribution of vorticity on the airfoil will be a solution to Laplace's equation. It will satisfy

the boundary conditions if the combination of the velocity induced by the vortices cancels the

component of the freestream normal to the plate:

(where small angle approximations have been introduced)

The basic approximation of thin airfoil theory is that the velocity induced

at some point x due to the vorticity at x'...

... may be approximated by the velocity induced at the same x position on the x axis due to avortex on the x axis:

The velocity induced by this bit of vorticity is computed from the basic vortex singularity.

The formula is known as the Biot-Savart Law and in 2-D for the element of vorticity at x', itreads:

So, the total induced velocity at the point x is given by:

Combining this expression with the flow-tangency boundary condition, we have the basic


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