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M&AE 305 October 24, 2006

Wings with Elliptic Span Loading

D. A. Caughey

Sibley School of Mechanical & Aerospace Engineering

Cornell University

Ithaca, New York 14853-7501

These notes provide, as a supplement to our textbook [2], a description of the analysis used todemonstrate the properties of elliptic span loading for wings of finite span.

1 The Elliptical Load Distribution

For a wing with an elliptical spanwise load distribution the sectional lift, or lift per unit span , asa function of the spanwise coordinate y can be written

(y) = 0

1

2y

b

2, (1)

where b is the wing span and 0 is the maximum sectional lift (at the center of the wing). The valueof 0 can be related to the wing lift coefficient by noting that the integral across the span of thesection lift is the total lift L, or

L =b/2b/2 0

12y

b2

dy . (2)

This integral can be evaluated using the trigonometric substitution

cos = 2y

b, (3)

which gives

L =0b

2

0

sin2 d =0b

4. (4)

Note that this integral could also have been evaluated using the fact that the area of an ellipse issimply times the product of its semi-minor (0) and semi-major (b/2) axes. The lift coefficient isthus given by

CL =

L

12U2S =

0b

2U2S . (5)

Now, the downwash velocity wi(y) (taken positive downward) is related to the distribution of vor-ticity () across the span by

wi(y) = 1

4

b/2b/2

()

yd , (6)

where the strength of the vortex sheet is related to the spanwise loading by

() =d

d=

1

U

d

d=

40Ub2

1

2b

2 . (7)

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REFERENCES 2

Thus, combining Eqs. (5), (6), and (7), we can write

wi(y) =2US

2b3CL

b/2b/2

( y)

1

2b

2d (8)

Introducing the trigonometric substitution of Eq. (3) and, correspondingly,

cos = 2

b,

Eq.(8) can be written in the form

wi() =US

2b2CL

0

cosd

cos cos . (9)

The integral appearing here is a standard Glauert integral, which can be evaluated from the generalformula (see, e.g., [1])

0

cosn d

cos cos =

sinn

sin . (10)

This gives the value of the integral appearing in Eq. (9) as , so we have

wi =U

ARCL , (11)

where we have introduced the wing aspect ratio AR = b2/S.

Thus, the induced angle of attack is seen, for the elliptical span loading, to be given by

= sin1wiU

CL

AR. (12)

Thus, for the elliptical span loading, the induced angle of attack is seen to be:

1. Constant across the span;

2. Proportional to the wing lift coefficient; and

3. Inversely proportional to the wing aspect ratio AR = b2/S.

Since, for the elliptical distribution of lift, the induced angle of attack is constant, independent ofspanwise position, the induced drag Di is simply equal to

Di = L sin . (13)

Thus, the coefficient of induced drag is given by

CDi = CL =C2LAR

. (14)

Although not demonstrated here, it can also be shown (see, e.g., [1]) that the elliptical span loadingproduces the minimum induced drag for a given total lift and wing span.

References

[1] L. M. Milne-Thompson, Theoretical Aerodynamics, Dover, New York, 1958.

[2] Richard S. Shevell, Fundamentals of Flight, Second Edition, Prentice-Hall, New York, 1989.