3D Lift Distribution

3D Aerodynamics and Finite Wings

AA200

Lecture 11

March 6, 2011

AA200 - Applied Aerodynamics 1

AA200 - Applied Aerodynamics Lecture 11

Basic 3D Theory

We start our discussion of three dimensional aerodynamics by consideringthe simple case of irrotational, inviscid flow. When compressibility can alsobe neglected, we consider solutions to Laplace’s equation in 3D:

∇2φ = 0

Just as in the 2-D case, since this equation is linear, we might constructsolutions by superimposing known solutions. In 2-D we used sources andvortices extensively to construct the flow over airfoils. In 3-D we also usesources and vortices to model the flow over wings and bodies.

This section shows how this is done, starting with some results for somefundamental 3-D singularities.

2


Fundamental Singularities in 3D Potential Flow

One may derive fundamental solutions to Laplace’s equation in 3-D, justas we did in 2-D (although complex variables are not quite so useful).

3-D Source

It was easily discovered that the potential: φ = −kr satisfied Laplace’sequation in 3-D. Since V = ∇φ, the velocity associated with this solutionis directed radially with a magnitude: V = k

r2. It is easily shown that the

constant k is related to the volume flow rate, S by: k = S4π, so:

V =S

4πr2

. The velocity distribution associated with this 3-D source dies off as r2

rather than r as in the 2-D case.

3


Point Doublet

Another basic solution, that has been used with some success insupersonic aerodynamics programs is the point doublet, obtained by movinga point source and sink together while keeping the product of their strength,S, and separation, L, constant. With µ = SL, the velocity associated withthe point doublet is:

~V =µ cos θ2πr3

r +µ sin θ4πr3

θ

Vortex Filament

One of the most useful fundamental solutions to the 3-D Laplaceequation is that of a vortex filament. A vortex filament may be visualized asa thin tube in which the flow has vorticity, ω. In the limit as the diameterof the tube is made small, but the circulation, Γ, is held fixed, this region ofvorticity is called a vortex filament. We often represent regions of vorticityas vortex filaments or discrete vortex elements.

4


Helmholtz Vortex Theorems

Helmholtz summarized some of the properties of vortex filaments, orvortices, in 1858 with his vortex theorems. These three theorems governthe behavior of inviscid three-dimensional vortices:

1. Vortex strength is constant

2. Vortices are forever (end on boundaries or form a closed path)

3. Vortices move with the flow

5


Vortex strength is constant: A vortex line in a fluid has constantcirculation.

This can be proved by imagining a closed 3-D loop around the vortexline as shown:

The integral around the closed loop from a to b to c to d to a cutsthrough no vorticity so from Stokes’ theorem the integral is zero. But asthe slit is made very small the integral approaches the sum of the integralfrom b to c and the integral from d to a. These are the local circulationsaround the vortex line and so, the circulations must be constant along theline.

Since the vortex strength is constant along the vortex line the strength

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cannot suddenly go to zero. Thus, a vortex cannot end in the fluid. It canonly end on a boundary or extend to infinity. Of course in an real, viscousfluid, the vorticity is diffused through the action of viscosity and the widthof the vortex line can become large until it is hardly recognized as a vortexline. A tornado is an interesting example. One end of the twister is on aboundary; but at the other end, the vortex diffuses over a large area withvorticity.

As discussed in the section on sources and vortices, singularities suchas vortices in the flow move along with the local flow velocity. Here,interactions of the vortices in the trailing wake, cause them to curve aroundeach other and to form the nonplanar wake shown below.

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Image from Head 1982 in van Dyke, An Album of Fluid Motion, usedwith permission.

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Biot-Savart Law

The Biot-Savart law relates the velocity induced by a vortex filamentto its strength and orientation. The expression, used frequently inelectromagnetic theory, can be derived from the basic equations for the3D potential. The result is:

In the simple case of an infinite vortex we obtain the 2-D result:

||V || = Γ2πr

In the case of a horseshoe vortex, the two trailing legs contribute:

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In the web notes you will find a simple subroutine to compute thevelocity components due to a vortex filament of length Gx, Gy, Gz with thestart of the vortex rx, ry, rz from the point of interest.

10


Finite Wing Models

One may apply the results of 3-D potential theory in several ways. Wefirst consider the theory of finite wings.

We might start out by saying that each section of a finite wing behavesas described by our 2-D analysis. If this were true then we would stillfind that the lift curve slope was 2π per radian, that the drag was0, and the distribution of lift would vary as the distribution of chord.Unfortunately, things do not work this way. There are several reasons for this:

One explanation is that the high pressure on the lower surface of thewing and the low pressure on the upper surface causes the air to leak aroundthe tips, causing a reduction in the pressure difference in the tip regions.

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In fact, the lift must go to zero at the tips because of this effect. We willnext see how and why we must model the 3-D wing differently from 2-D.

If we were to take the naive view that the 2-D model would work in3-D, we might have the picture shown on the right. If each section had thedistribution of vorticity along its chord that it had in 2-D, the lift would beproportional to the chord, and would not drop off at the tips as we know itmust.

This sort of model does not conform to our physical picture of whathappens at the wing tips. And indeed, it does not satisfy the equations of

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3-D fluid flow. The reason that this does not work is that in this case thestreamlines are not confined to a plane. They move in 3-D and the flowpattern is quite different.

We could go back to the governing equations and startsimply with the linear Laplace equation. By superimposingknown solutions we could obtain a simple model of a 3-D wing.We might start by superimposing vortices on the wing itself:

But this is no more than the strip theory model that did not work.The reason that this model (which seems just to be a superposition ofknown solutions) is not adequate is that it violates the governing equationsin certain regions. The model does not satisfy the Helmholtz laws sincevorticity ends in the flow near the tips. Some additional requirements must

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be imposed on the model. The requirements for such a model are just theHelmholtz vortex theorems, discussed previously. Our simple 3-D modelabove may be modified as shown below to satisfy the first of the Helmholtztheorems.

In fact, as can be seen from the picture here, this vortex model is nottoo far from reality.

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The downwash field and the existence of trailing vorticesare not just some strange mathematical result. They arenecessary for the conservation of mass in a 3-D flow.

Air is pushed downward behind the wing, but this downward velocitydoes not persist far from the wing. Instead it must move outward. Theoutward-moving air is then squeezed upward outboard of the wing and theflow pattern shown above develops.

The trailing vortex is visualized by NASA engineers byflying an agricultural airplane through a sheet of smoke.

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The main effect of this vortex wake is to produce a downwash field onthe wing.

This downwash field has several very significant effects:

1. It changes the effective angle of attack of the airfoil section. Thischanges the lift curve slope and has many implications.

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2. Induced drag: Lift acts normal to flow in 2D. This accounts for about40% of the fuel used in a commercial airplane, and as much as 80

3. It produces interference effects that are important in the analysis ofstability and control.

The magnitude of the downwash can be estimated using the Biot-Savartlaw, discussed previously.

When applied to our simple model with two discrete trailing vortices,the equation predicts infinite downwash at the wing tips, a result that isclearly wrong. In fact, the induced downwash is not even very large.

The failure of this simple model led Prandtl to develop a slightly moresophisticated one in 1918. Rather than representing the wing with justone horseshoe-shaped vortex, the wing is represented by several of them:

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In this way the circulation on the wing can vary from the root to the tip.The strength of the trailing vortex filaments is related to the circulation onthe wing then by:

Γwake = ∆Γwing

A vortex is shed from the wing whenever the circulation changes.

In the limit as the number of horseshoe vortices goes to infinity, thetrailing wake is a sheet of vorticity.

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The trailing vortex strength per unit length in the y direction (vorticity)is the derivative of the total circulation on the wing at that station. Fromthis model, we can derive the basic relations for finite wings.

The vorticity strength in the trailing vortex sheet is given by:

γ =dΓdy

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and since the wing circulation changes most quickly near the tips, thetrailing vorticity is strongest in this region. This is why we see tip vortices,and not a complete vortex sheet, as in this NASA photo of an F-111 in a4-g turn. The vortices are visible in this picture because the low pressure inthis region lowers the temperature and we see the condensed water vapor.

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Lifting Line Theory

These are the basic concepts of finite wing theory. Mathematicalmodelling of a lifting wing via the superposition of a number of horseshoevortices that obey Helmholz’s laws yields a reasonable approximation to theflow around a wing of high apect ratio. The key elements of this theory aredescribed as follows:

“The main problem of finite-wing theory is the determination of thedistribution of airloads on a wing of given geometry flying at a given speedand orientation in space. The analysis is based on the assumption that atevery point along the span the flow is essentially two-dimensional; thus theresultant force per unit span at any point is that calculated for the airfoilusing two-dimensional thin airfoil theory, but at an angle of attack correctedfor the influence of the trailing vortex configuration that itself depends onthe spanwise variation of the lift distribution on the wing.”

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Figure 1 presents a good summary of the various elements involved inthis theory that we have discussed so far. The key thing to understand isthat there is strong feedback between a lift distribution and its wake: thespanwise derivative of the wing lift distribution determines the local strengthof the wake, which, in turn, determines the distribution of downwash at thewing, which, again, determines the lift distribution along the span of thewing. The key is to determine a distribution of lift/circulation along thespan of the wing which is in harmony with the downwash created by itswake. Once this is accomplished all aerodynamic performance parameterscan be derived, including the total lift and drag of the wing, the downwashdistribution and the deviation of the produced lift distribution from theoptimum one. More importantly, lifting line theory can tell us about theeffect of variations in the wing planform and twist distributions on theperformance of the actual wing.

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Figure 1: Summary of Key Elements of Lifting Line Theory

Consider the unknown spanwise distribution of circulation to be givenby Γ(y). The strength of the circulation at any point of the wake is then

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given by

γx(y) =dΓ(y)dy

.

The net effect of this trailing vortex system on any point on the boundvortex is a downwash, w(y), whose magnitude at at each point is given bythe integrated effect of the continuous distribution of semi-infinite vorticityover the range −b/2 < y < b/2. Notice from the Figure that resultantvelocity at the wing has two components V∞ and w, at each point, whichdefine the induced angle of attack

αi(y) = tan−1 w(y)V∞

(1)

Using the Kutta-Joukwoski law, the force per unit span on the bound vortexhas magnitude ρV Γ and it is normal to V , that is, it is inclined with respectto the z-axis at an angle αi. Using small angle approximations, these

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various quantities can be shown to be

αi(y) =w

V∞

L′(y) = ρV∞Γ

D′i(y) = −L′αi = −ρwΓ

We have shown that the net effect of the wake on the wing is the downwashvelocity which can be calculated as

αi(y) =w(y)V∞

= − 14πV∞

P

∫ b2

−b2

dΓ(t)dt

y − tdt (2)

We had also seen that

cl(y) = clα(α(y)− αL0(y)− αi(y)) (3)

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where clα is the lift curve slope (which could also be a function of y), α(y)is the geometric angle of attack, and αL0 is the zero-lift angle of attack(due to camber) of that particular cross-section. Since:

cl =L′

12ρ∞V

2∞c(y)

where c(y) is the local chord, then

Γ(y) =12clαV∞c(α− αL0 − αi) (4)

and plugging in Equation 4 into Equation 2 we obtain the governing equationfor Γ(y) as

Γ(y) =12V∞c clα

[α− αL0 −

14πV∞

P

∫ b2

−b2

Γ′(t)dty − t

](5)

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Elliptical Lift Distribution

Equation 5 is easily solved for c(y) if we were given Γ(y). This procedureinvolves the solution of an algebraic equation, rather than an integralequation. The most important case of this type of solution procedure isthat of the elliptical lift distribution. Suppose that Γs represents the value ofthe circulation at the plane of symmetry. Then, an elliptical lift distributioncan be written as

Γ = Γs

√1−

(y

b/2

)2

. (6)

The induced angle of attack created by this elliptical load distribution canbe computed by subtituting Equation 6 into Equation 2. One can thenshow that

αi(y) = − Γs4πV∞

P

∫ b2

−b2

ddt

√1−

(tb/2

)2

y − tdt

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Using the Glauert trigonometric substitution that we have seen several timesby now, we can show that

αi(θ0) =Γs

2πbV∞P

∫ π

0

cos θdθcos θ0 − cos θ

which can be shown to be equal to a constant

αi = − Γs2bV∞

(7)

Therefore, the angle of attack of every airfoil section on the wing is constant(assuming no geometric twist). If we further assume that the sectional lift-curve slope is independent of y, both the sectional lift and drag coefficientwill also be independent of y. Therefore

cl = clαα0

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cdi =D′iq∞c

= −clαi

Using these conditions, the product of clαα0c must vary elliptically, since

L′ = ρV∞Γs

√1−

(y

b/2

)2

= clαα012ρV 2∞c

clαα0c =2ΓsV∞

√1−

(y

b/2

)2

Assuming that clαα0 is a constant (untwisted wing of identical cross-sectional properties) this indicates that in order to obtain an ellipticallyloaded wing, we must have an elliptical planform. If the planform werenot elliptical, an elliptical load distribution can still be achieved by makingsure that the wing be twisted in such a way that the product of clαα0c still

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varied elliptically. Notice that although this is fine in theory, extreme taperratios may lead to excessively high sectional lift coefficients in the outboardwing sections that may exceed the local clmax and lead to stall.

The wing properties of this elliptical lift distribution can be computedby integrating the spanload variations as follows

CL =1

12ρV

2∞S

∫ b2

−b2L′dy =

Γsπb2V∞S

The wing lift coefficient and the sectional lift coefficient are equal whenthe sectional lift coefficients are constant along the span, and therefore,CL = cl.

Solving for Γs in the expression above, and substituting into the

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expression for the induced angle of attack we find that

αi = − CLπAR

= − clπAR

and then, the wing induced drag coefficient is given by

CDi = cdi = −CLαi =C2L

πAR

The expression for the lift curve slope is then given by

CLα =clα

1 + clα/πAR

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Γ(y) Solution for Arbitrary Lift Distributions

Assume that you are given the geometry of the wing and the two-dimensional lift curve slope of each of its cross sections. That is, thequantities αL0, c(y), and clα(y) are known. What is the best approachto solve for Γ(y)? The solution procedure should bring about a sense ofdeja-vu from thin airfoil theory.

Let us transform the spanwise coordinate y into a polar variable θaccording to

t =b

2cos θ

y =b

2cos θ0

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Then,

Γ(y) = g(θ0) =12clαscsV∞

∑n=1

An sinnθ0 (8)

where the constant outside the summation is included for convenience andthe s subscripts indicate values at the symmetry plane. Note that the Ancoefficients are all non-dimensional, since the circulation Γ has units ofL L/T . In addition

Γ′(t)dt =dΓdθ

dθ

dtdt = g′(θ)dθ

=12clαscsV∞

∑n=1

nAn cosnθdθ

and

P

∫ b2

−b2

Γ′(t)dty − t

=12clαscsV∞

∑n=1

nAn P

∫ 0

π

cosnθdθb2(cos θ0 − cos θ)

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You may remember the value of the following principal value integral fromthin airfoil theory

P

∫ π

0

cosnθcos θ − cos θ0

=π sinnθ0

sin θ0

which can be used to evaluate the integral and, substituting the result andEquation 8 into Equation 5 we get

∑n=1

An sinnθ0 =c clαcsclαs

(α− αL0)− c clα4b

∑n=1

nAnsinnθosin θ0

(9)

which must be solved for the values of An. This is typically done withthe collocation method by truncating the series after a certain value of nand evaluating the expression at n different values of θ0 to create a linearsystem of n equations in n unknowns An. Unfortunately, the only way tocheck if your truncation was too crude is to repeat the procedure with ahigher value of n and compute the difference between the results.

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Assuming that we are able to solve the Equation 9 for the unknowncoefficients of the circulation distribution An, we can now use these valuesto compute the lift and drag coefficients of the wing. Firstly, the sectionallift and drag coefficients can be found to be

cl =ρV∞Γq∞c

=clαscs

c

∞∑m=1

An sinnθ (10)

cdi = −clαi =c2lαsc

2s

4bc

( ∞∑n=1

An sinnθ

)( ∞∑k=1

kAksin kθsin θ

)

which can be integrated to yield

CL =∫ b

2

−b2

clq∞cdy

q∞S=clαscs

S

∫ π

0

∞∑n=1

An sinnθb

2sin θdθ

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The integration is carried out easily by switching the orders of integrationand summation and noting that

∫ π

0

sinnθ sin kθdθ ={

0 for n 6= kπ2 for n = k

Since k = 1 in all of the terms of the interal above, all terms vanish exceptfor the n = 1 term and therefore we have

CL =clαscsπb

4SA1 (11)

and therefore, the wing lift coefficient for an arbitrary symmetrical circulationdistribution is proportional to A1 and independent of all other Fouriercoefficients. For an elliptical lift distribution CL was proportional to Γswhich is the only surviving term in the Fourier expansion.

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Similarly, the coefficient of drag of the wing can be computed as follows

CDi =∫ b

2

−b2

cdiq∞cdy

q∞S=c2lαsc

2s

8S

∫ π

0

∞∑n=1

∞∑k=1

kAnAk sinnθ sin kθdθ

which can be shown to be

CDi =c2lαsc

2sπ

16S

∞∑n=1

nA2n

which proves that, for a given lift coefficient (A1 6= 0), the optimum liftdistribution (the one that yields minimum induced drag) is the one thatminimizes CDi. In other words, all Ai except for A1 must be equal to zero.This corresponds to the elliptic lift distribution that we have been discussingearlier.

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Interestingly, using the result for CL in Equation 11, one can show thatthe induced drag coefficient for the wing is given by

CDi =C2L

πAR

∞∑n=1

n

(AnA1

)2

which is simply the result for the elliptically loaded wing with a correctionfactor that is sometimes called the span efficiency factor, e, defined as

1e

=∞∑n=1

n

(AnA1

)2

In the case of symmetric loadings, the situation is simplified considerably.If we assume that Γ(y) = −Γ(−y), then all of the even coefficients of theseries expansions, A2, A4, ... are zero.

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Comparison with Experiment

Classical results on rectangular planforms reported by Prandtl can beseen in the following Figures. Seven wings of aspect ratios between 1 and7 were tested in a wind tunnel. Figure 2 shows the variation of CL vs thegeometric angle of attack. In Figure 2 the results have also been correctedby the induced angle for AR = 5

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Figure 2: CL vs. α and α′ for 1 ≤ AR ≤ 7

α′ = α+CLπ

(15− 1AR

)40


Figure 3 shows the drag polars for each of these wings, where

CD = CD0 + CDi

In Figure 3 these graphs have also been corrected to AR = 5 so the newdrag coefficient, CD′ becomes

CD′ = CD +C2L

π

(15− 1AR

)41


Figure 3: CL vs. CD and C ′Dfor 1 ≤ AR ≤ 7

The corrected Figures show that there is indeed significant departurefrom the theory for aspect ratios under AR = 3. If AR is larger thanthis value, significant departures from elliptical wing loading can still betolerated by this theory. In fact, it is typical to introduce the span efficiency

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factor, e, to account for some of these deviations so that

CDi =C2L

πARe

where e = 1.0 for a perfectly elliptically loaded wing, while it is smaller(typically in the range 0.9 ≤ e ≤ 1.0) for wings with lift distributions thatdeviate from the elliptical load.

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