1

MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS

Finite Wings: General Lift Distribution Summary

April 18, 2011

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

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SUMMARY: PRANDTL’S LIFTING LINE THEORY (1/2)

2

20

0

2

20

0

2

20

00

00

4

1

4

1

4

1

b

bi

b

b

b

bL

dyyy

dyd

Vy

dyyy

dyd

yw

dyyy

dyd

VycV

yy

Fundamental Equation of Prandtl’s Lifting Line Theory

Geometric angle of attack, , is equal to sum of effective angle of attack, eff, plus induced angle of attack, i

Equation gives value ofDownwash, w, at y0

Equation for induced angle of attack, i, along finite wing

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SUMMARY: PRANDTL’S LIFTING LINE THEORY (2/2)

dyyySV

C

dyyyVD

dyySVSq

LC

dyyVL

yVyL

b

biiD

i

b

bi

b

bL

b

b

2

2

,

2

2

2

2

2

2

00

2

2

Lift distribution per unit span given by Kutta-Joukowski theorem

Total lift, L

Lift coefficient, CL

Induced drag, Di

Induced drag coefficient, CD,i

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PRANDTL’S LIFTING LINE EQUATION

• Fundamental Equation of Prandtl’s Lifting Line Theory

– In Words: Geometric angle of attack is equal to sum of effective angle of attack plus induced angle of attack

– Mathematically: = eff + i

• Only unknown is (y)

– V∞, c, , L=0 are known for a finite wing of given design at a given a

– Solution gives (y0), where –b/2 ≤ y0 ≤ b/2 along span

2

20

000

00 4

1b

bL dy

yy

dyd

Vy

ycV

yy

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WHAT DO WE GET OUT OF THIS EQUATION?

1. Lift distribution

2. Total Lift and Lift Coefficient

3. Induced Drag

dyyySVSq

DC

dyyyVdyyyLD

LD

dyySVSq

LC

dyyVL

dyyLL

yVyL

b

bi

iiD

i

b

bi

b

bi

iii

b

bL

b

b

b

b

2

2

,

2

2

2

2

2

2

2

2

2

2

00

2

2

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GENERAL LIFT DISTRIBUTION (§5.3.2)• Circulation distribution

• Transformation

– At =0, y=-b/2

– At =, y=b/2

• Circulation distribution in terms of suggests a Fourier sine series for general circulation distribution

• N terms

– now as many as we want for accuracy

• An’s are unkowns, however must satisfy fundamental equation of Prandtl’s lifting-line theory

2

20

000

00

1

0

2

0

4

1

sin2

sin

cos2

21

b

bL

N

n

dyyy

dyd

Vy

ycV

yy

nAbV

by

b

yy

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GENERAL LIFT DISTRIBUTION (§5.3.2)

N

nL

N

n

N

n

L

N

n

N

n

b

bL

N

n

nnAnA

c

b

dnnA

nAc

b

dy

dnnAbV

dy

d

d

d

dy

d

dyyy

dyd

Vy

ycV

yy

nAbV

1 0

000

10

00

0 0

100

10

00

1

2

20

000

00

1

sin

sinsin

2

coscos

cos1

sin2

cos2

4

1

sin2

• General circulation distribution

• Lifting line equation

• Finding d/dy

• Transform to

• Last integral has precise form for simplification

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GENERAL LIFT DISTRIBUTION (§5.3.2)

• Evaluated at a given spanwise location, 0 is specified• Givens:

– b: wingspan– c(0): chord at the given location for evaluation– The zero lift angle of attack, L=0(0), for the airfoil at this specified location

• Note that the airfoil may vary from location to location, and hence the zero lift angle of attack may vary from location to location

• Can put twist into the wing– Geometric twist– Aerodynamic twist

• This is one algebraic equation with N unknowns written at 0

• Must choose N different spanwise locations to write the equation to give N independent equations

N

nL

N

n

nnAnA

c

b

1 0

000

10

00 sin

sinsin

2

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WING TWIST

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GENERAL LIFT DISTRIBUTION (§5.3.2)

ARAS

bAC

dn

dn

dnAS

bC

dyySV

C

L

N

nL

b

b

L

1

2

1

0

0

01

2

2

2

1nfor 0sinsin

1nfor 2

sinsin

sinsin2

2

• General expression for lift coefficient of a finite wing

• Substitution of expression for () and transformation to

• Integral may be simplified

• CL depends only on leading coefficient of the Fourier series expansion (however must solve for all An’s to find leading coefficient A1)

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GENERAL LIFT DISTRIBUTION (§5.3.2)

N

ni

N

ni

N

ni

b

bi

i

N

niD

b

biiD

nnA

nnA

dn

nAy

dyyy

dyd

Vy

dnAS

bC

dyyySV

C

1

1 0

00

1 0 00

2

20

0

0 1

2

,

2

2

,

sin

sin

sin

sin

coscos

cos1

4

1

sinsin2

2

• General expression for induced drag coefficient

• Substitution of () and transformation to

• Expression contains induced angle of attack, i()

• Expression for induced angle of attack

• Can be mathematically simplified

• Since 0 is a dummy variable which ranges from 0 to across the span of wing, it can simply be replaced with

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GENERAL LIFT DISTRIBUTION (§5.3.2)

Nn

iD

N

niD

N

n

N

niD

N

n

N

niD

N

ni

i

N

niD

A

AnARAC

nAAARC

nAARnAS

bC

km

km

dnnAnAS

bC

nnA

dnAS

bC

2

2

1

21,

2

221,

1

2

1

22

,

0

0

10 1

2

,

1

0 1

2

,

1

2

2

kmfor 2

sinsin

kmfor 0sinsin

sinsin2

sin

sin

sinsin2

• Expression for induced drag coefficient

• Expression for induced angle of attack

• Substitution of i() in CD,i

• Mathematical simplification of integrals

• More simplifications leads to expression for induced drag coefficient

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GENERAL LIFT DISTRIBUTION (§5.3.2)

eAR

CC

e

A

An

AR

CC

ARAS

bAC

A

AnARAC

LiD

Nn

LiD

L

Nn

iD

2

,

2

2

1

2

,

1

2

1

2

2

1

21,

1

1

1

1

• Repeat of expression for induced drag

coefficient

• Repeat of expression for lift coefficient

• Substituting expression for lift coefficient into expression for induced drag coefficient

• Define a span efficiency factor, e, and note that e ≤ 1

– e=1 for an elliptical lift distribution

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VARIOUS PLANFORMS FOR STRAIGH WINGS

Elliptic Wing

Rectangular Wing

Tapered Wing

cr ct

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INDUCED DRAG FACTOR, (e=1/(1+))

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SPECIAL CASE:Elliptical Wings → Elliptical Lift Distribution

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ELLIPTICAL LIFT DISTRIBUTION• For a wing with same airfoil shape across span and no twist, an elliptical

lift distribution is characteristic of an elliptical wing planform

AR

CC

AR

C

LiD

Li

2

,

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SUMMARY: ELLIPTICAL LIFT DISTRIBUTION (1/2)

2

0

2

0

21

21

b

yVyL

b

yy

Points to Note:

1. At origin (y = 0) = 0

2. Circulation and Lift Distribution vary elliptically with distance, y, along span, b

3. At wing tips (-b/2) = (b/2) = 0

– Circulation and Lift → 0 at wing tips

y/b

/

0

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SPECIAL SOLUTION:ELLIPTICAL LIFT DISTRIBUTION

Elliptic distribution

Equation for downwash

Coordinate transformation →

See reference for integral

bVV

wb

w

db

w

db

dyb

y

dy

yyby

y

byw

by

y

bdy

d

i

b

b

2

2

coscos

cos

2

sin2

;cos2

41

41

4

0

00

0 0

00

2

20

21

2

22

00

2

220

Downwash is constant over span for an elliptical lift distribution

Induced angle of attack is constant along spanNote: w and i → 0 as b → ∞

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SUMMARY: ELLIPTICAL LIFT DISTRIBUTION

l

LiD

Li

i

cq

yLc

AR

CC

AR

C

bVL

bV

bw

2

,

0

0

0

4

2

2Downwash is constant over span for an elliptical lift distribution

Induced angle of attack is constant along span for an elliptical lift distribution

Total lift

Alternate expression for induced angle of attack, expressed in terms of lift coefficient

Induced drag coefficient

For an elliptic lift distribution, the chord must vary elliptically along the span

→ the wing planform is elliptical in shape

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SPECIAL SOLUTION:ELLIPTICAL LIFT DISTRIBUTION

AR

CC

dyySV

C

AR

CS

bAR

b

SC

bVdy

b

yVL

LiD

b

b

iiD

Li

Li

b

b

2

,

2

2

,

2

2

0

2

2

21

2

2

0

2

4

41

CD,i is directly proportional to square of CL

Also called ‘Drag due to Lift’

We can develop a moreuseful expression for i

Combine L definition for elliptic profile with previous result for i

Define AR because itoccurs frequently

Useful expression for i

Calculate CD,i