Lifting Line Theory

Louis Marion

December 13, 2013

Contents

1 Introduction 2

2 Introduction to Lifting Line Theory 2

2.1 Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Circulation and the Kutta-Joukowski Theorem . . . . . . . . . . . . . . . . . . . . . 3

2.3 Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Numerical implementation 6

3.1 Modelling the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Implementation of lifting line theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Conclusion 11

A Results with alternative integration method 12

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1 Introduction

The focus of my project is the aerodynamic problem of determining the lift distribution on the

wing of an aircraft. Lift is a fundamental force of aerodynamics and understanding lift is therefore

a key issue in aerodynamics. Moreover, considering that we will analyse the effect of the nature

of the wing on the lift distribution, our study will shed light on the design of aircraft wings. How

can we numerically model the problem and the aircraft’s wing in order to show the impact of the

shape of the wing on the lift distribution across the wing?

To answer this question, we will firstly introduce the theory behind our numerical method, Prandtl

and Lancaster’s Lifting Line Theory. We will then present our numerical implementation and the

results we obtained.

2 Introduction to Lifting Line Theory

2.1 Lift

Firstly, let us recall the four fundamental forces on an aircraft during flight.

Vertically, the plane is subject to two forces, lift pointing up and weight pointing down. Horizontally,

it is subject to drag pointing in the direction of the incoming flow, and thrust pointing towards the

incoming flow.

Figure 1: Four forces on an aircraft during flight

The lift on an aircraft with wing surface area S flying in a flow with velocity v and density ρ is

(note that the velocity of the aircraft is equal to the velocity of the incoming flow in the aircraft’s

reference frame) :

L =1

2CLρv

2S

where CL is a constant, the lift coefficient.

Now, if we consider the lift not on the entire wing but on a differential section of the wing instead,

we get the lift per unit of span. If the width of the wing (distance between the leading edge and

the trailing edge of the wing), or chord, at that section is denoted by c, the lift per unit of span is:

L′ =1

2cLρv

2c

where cL is the section lift coefficient.

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Figure 2: Cross-sectional view of a wing

2.2 Circulation and the Kutta-Joukowski Theorem

We define the circulation Γ as the line integral of a velocity field around a closed curve. The

circulation gives the strength of a circulatory flow around a given path.

From the definition of Γ and from Bernoulli’s equation for incompressible flow, we can derive the

Kutta-Joukowski theorem:

L′ = ρvΓ

Hence, once we have the circulation across our wing, we can compute the lift distribution across

the wing. In my project, I will sent constants equal to 1, since I am interested in the behaviour of

the distribution rather than the actual values of lift along the wing; thereby, I will simply compute

the circulation.

Moreover, recall that L′ = 12cLρv

2c. Hence,

ρvΓ =1

2cLρv

2c

=⇒ Γ =1

2vcLc

2.3 Lifting Line Theory

Lifting line theory is a mathematical model developed in the early twentieth century by Prandtl

and Lancaster to predict the lift distribution over a wing.

Firstly, when an airplane is flying, and when a wing creates aerodynamic lift, free vortices are

generated at the wingtips. These two vortices are parallel to the direction of flow. Around these

vortices, we have an inward circulation (towards the aircraft). Thereby, the free vortices induce

a downward flow along the wing, the downwash w (and an upward flow outside the span of the

wing).

Moreover, along the wing, a circulation Γ is created around the vortex bound by the wing.

The above description can be illustrated by the following diagram:

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Figure 3: Trailing vortices and downwash

Prandtl replaced the wing and two free vortices by a bound vortex and two free vortices, defining

a horseshoe vortex. The bound vortex is the lifting line.

Figure 4: Replacing the wing by a horseshoe vortex

However, there are not two free vortices, but an infinity of free vortices bound by the same

lifting line. Then, since each horseshoe vortex defines a circulation Γ along the lifting line, an

infinity of horseshoe vortices will define a continuous circulation distribution along the wing, as we

can see below:

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Figure 5: Vortex sheet and circulation distribution

This mathematical model of a vortex sheet is the core of lifting line theory. Once we have this

circulation distribution, using the Kutta-Joukowski Theorem, we get the lift distribution.

Before proceeding in our discussion, let us first define a few terms.

Angles of attack

The angle of attack is the angle that the wing makes with the direction of the incoming flow.

Moreover, because of downwash, the actual flow on the wing may not be in the same direction

as the incoming flow. The downwards flow due to downwash gives a downward component to the

flow; as a result, we define the induced angle of attack αi as the angle between the incoming flow

and the actual flow perceived by the wing.

Figure 6: Angles of attack during flight

Then, the effective angle of attack is given by:

αeff = α− αi

where α is given.

Skipping the derivation, we can obtain the following equation relating the induced angle of

attack to the circulation:

αi(y0) =1

4πv

∫ b2

− b2

(dΓ/dy)

y0 − ydy

Then, we can get the effective angle of attack.

With the latter it is possible to obtain the lift distribution. Indeed, for regular flying conditions,

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the section lift coefficient is, according to aerodynamics:

cL = 2π(αeff − αL=0)

where αL=0 is the angle of attack for zero lift; it is a constant relevant to the airplane’s structure.

3 Numerical implementation

3.1 Modelling the problem

We want to compute the circulation distribution along the wing, for different wing planforms (top-

view shape). Note that lifting line theory is limited to straight wings, wings with a planform

symmetrical about their span. We will therefore consider the elliptical wing, the rectangular wing

and the tapered wing.

Figure 7: Different straight wing planforms (top view wings)

Let us first model our wing. We discretise our wingspan into a mesh with N + 1 points with

spacing ∆y between them.

Figure 8: Discretised wing model (top view)

We can now define the different wing planforms. The shape of our wing is entirely determined

by the length of the chord at each section of the wing. In the above figure then, the shape is

determined by an array of chord lengths which correspond to the lengths of the dotted lines for

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each i ∈ [0, N + 1], i ∈ N.

Our different wing shapes are thereby defined as, with wingspan b and y = − b/2+ i∆y where i is

the point or section of the wing as shown above (note that we normalise our wing such that the

maximal chord length is 1):

i. for the elliptical wing: c(y) =√

1− yb/2

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ii. for the rectangular wing: c(y) = 1

iii. for the tapered wing: c(y) = 2S(1+λ)b [1 −

2(1−λ)b y] for the positive half of the wingspan

and c(−y) for the other half. S is the surface area of the wing and λ is the taper ratio, the

ratio of the chord length at the tip ctip of the wing and the maximal chord length, the chord

length in the middle croot. Note that the surface area is S =ctip+croot

2 b.

Since we are dealing with a finite wing, the lift distribution should go to zero at the tips of the

wing, since there is no lift outside of the wingspan and we do not want any discontinuities at the

wing boundaries. Moreover, we have some intuition that the lift should be maximal in the middle

of the wingspan, where the fuselage is. We can therefore assume an elliptical distribution as an

initial distribution for our circulation.

Then, using this initial circulation, we can generate a new circulation. Doing this process iteratively,

convergence will determine our circulation distribution.

3.2 Implementation of lifting line theory

With our initial circulation, we can generate a new circulation using the following algorithm. We

run the script until we reach convergence between the precedent and current circulation.

while Γold and Γnew don’t converge do

Γold = Γ;

compute dΓ/dy;

compute αi;

compute αeff ;

compute cL;

compute a new circulation Γnew;

generate a new input circulation Γinput;

Γ = Γinput;

endAlgorithm 1: Iterative method to compute circulation

Let us now go through the details of our method.

We first compute the induced angle of attack with the integro-differential equation given by lifting

line theory:

αi(y0) =1

4πv

∫ b2

− b2

(dΓ/dy)

y0 − ydy

Note that the integral above has bounds − b2 and b

2 . Since this is a very large interval, we must

divide it into small intervals.

In our method, we used Simpson’s rule, which is given by:∫ b

af(x)dx =

b− a6

[f(a) + 4f(a+ b

2) + f(b)]

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We thus need to evaluate the integrand in the first, middle and last point of the interval. We can

therefore divide the interval into intervals of spacing 2∆y; then, we evaluate the integrand at mesh

points i − 1, i and i + 1. We get the following equation for the induced angle of attack (where

yi = − b2 + i∆y):

αi(yi) =1

4πv

2∆y

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N∑j=1,3,5,...

(dΓ/dy)j−1

yi − yj−1+ 4

(dΓ/dy)jyi − yj

+(dΓ/dy)j+1

yi − yj+1

Note that for i = j−1, i = j and i = j+1, we have singularities, as the denominator in the sum

terms go to zero. When such singularities occur, we replace the problematic term by the average

of the corresponding terms for the two neighbouring points. For example, for i = j we replace(dΓ/dy)jyi−yj by 1

2((dΓ/dy)j−1

yi−yj−1+

(dΓ/dy)j+1

yi−yj+1).

If we do not have preceding or succeeding terms (like at the beginning and end of our array), we

use extrapolation to substitute the problematic terms. It turns out that this method does not seem

to solve the problem of discontinuity that we have for rectangular and tapered wings, as the chord

of those wings suddenly go to zero at the tips, as we will note in the results.

(Note: we also performed this integral with trapezoidal integration, which yielded similar results.)

Once we have the induced angle of attack, we can compute the effective angle of attack and

the section lift coefficient using the relevant equations mentioned earlier. We then obtain a new

circulation distribution Γnew.

We could then use this newly obtained circulation to compute a new set of angles of attacks and

in turn generate a new circulation. However, using Γnew as the input circulation did not yield any

convergence.

We therefore use a more general form of input in order to have more control over the input and

the convergence:

Γinput = Γold + h(Γnew − Γold)

where h is a real number between 0 and 1.

Note here that for h = 1, we recover Γinput = Γnew. However, we get the best results for a small h;

indeed, I ran most of my convergence schemes with h ∈]0, 0.1].

Convergence

Note that we discretise all functions of span (circulation, angles of attack, ...) and that the circu-

lation is defined by an array of N + 1 elements, for each mesh point.

Then, the convergence of the circulation Γ is determined by the distance between the two arrays

Γold and Γnew as: √√√√ N∑i=0

(Γnew[i]− Γold[i])2

We choose an arbitrary distance as the convergence threshold; we took a distance between 0 and 1

as our threshold.

3.3 Results

We ran our program for the three types of straight wings, the elliptical, rectangular and tapered

wing. The following results are those obtained by using Simpson’s rule in the integral.

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(Note: we normalised all equations so that our maximal circulation is 1. Moreover, note that the

wingspan is shown in function of the corresponding mesh point i)

Let us first look at how the circulations converge to the final circulation.

For the elliptical wing, we get the following plot:

Figure 9: Convergence for elliptical wing planform

We only see one distribution on this plot. Thus, we see that the initial elliptical distribution

we assumed matches the circulation we get for an elliptical wing. This result can be analytically

shown in lifting line theory.

Now, consider the rectangular wing. We get the following:

Figure 10: Convergence for rectangular wing planform

Here, we clearly see how we start with the elliptical distribution. The latter becomes more and

more “rectangular” with each iteration.

When we reach convergence, we see that we have a close to constant distribution along the central

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portion of the wing. However, as we move towards the tips, the circulation decreases.

As mentioned earlier, the circulation should go to zero at the tips. Yet, as we see in this plot, our

results do not match this prediction accurately. This is most likely due to the discontinuity of the

wing planform as well as the limitations of the numerical integration we used.

Finally, for the tapered wing with a taper ratio (defined above) of 0.6, we obtain:

Figure 11: Convergence for tapered wing planform (taper ratio = 0.6)

Again, we observe the convergence from an elliptical distribution towards the final distribution,

influenced by wing planform. Moreover, as with the rectangular wing, we have inconsistent results

at the tips, since the distribution does not go to zero as it should, in theory.

We can then compare the three different distributions:

Figure 12: Comparison of circulation distribution for different wing planforms

Here we can clearly see the influence of the wing planform on the circulation distribution and

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therefore on the lift distribution along the wing span.

However, we also see how the rectangular and tapered wings yield a distribution that does noticeably

decrease close to the tips (and eventually go to zero at the tips).

Yet, if we reduce the convergence threshold to a distance of 1 between the two arrays (which is still

a very small distance), we get:

Figure 13: Comparison of circulation distribution for different wing planforms for reduced conver-

gence threshold

These results are closer to the theoretical results of lifting line theory. Moreover, if we take into

account the fact that in reality the distributions should all vanish at the tips, we see that elliptical

distributions are a good approximation for straight wing planforms’ distributions.

4 Conclusion

In our project, we succeeded in modelling the circulation (or lift) distribution for various wing

planforms. We discretised the problem in order to implement Prandtl’s mathematical model nu-

merically. For all wings, we saw that the distribution is indeed maximal in the middle of the wing

while it decreases at the tips. Our results were satisfying, although because of the discontinuities

in the wing planforms as well as the limitations of the numerical integration scheme at the wing

tips, we did not get results which perfectly match our predictions.

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A Results with alternative integration method

As mentioned above, we also computed the induced angle of attack using the trapezoidal rule for

integration.

We got similar results, although trapezoidal integration yields worse results at the wingtips. We

get the following:

Figure 14: Convergence for elliptical wing planform (using trapezoidal rule)

Figure 15: Convergence for rectangular wing planform (using trapezoidal rule)

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Figure 16: Convergence for tapered wing planform with taper ratio 0.6 (using trapezoidal rule)

For the comparative plot, we have, with the same convergence threshold we used earlier in the

stricter case:

Figure 17: Comparison of circulation distribution for different wing planforms

In the above plot, we see how the distributions do not have a noticeable decrease when they

get closer to the wingtip, as they did when we used Simpson’s rule.

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References

[1] Professor M. Pavone. Lectures and personal communication. 2013.

[2] J. D. Anderson. Fundamentals Of Aerodynamics. Chapter 5. 2001.

[3] R. Shevell. Fundamentals Of Flight. 1989.

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