Finite Wings Summary

MAE 5510 W. F. Phillips Spring 2007 For an arbitrary finite wing, the circulation distribution can be written as

)2(cos;)sin(2)( 1

1

bznAbV

N

n

n −== −

=∞∑ θθθΓ (1.8.3)

where the coefficients An are determined by forcing the following equation to be satisfied at N specific sections

along the wing;

)()()sin()sin()(

~4

0

,1

θαθαθθθαL

L

N

n

n nn

cC

bA −=⎥

⎦

⎤⎢⎣

⎡+∑

= (1.8.4)

The lift coefficient and induced drag coefficient for a finite wing can be expressed as

w

AALS

bRARC

2

1, == π (1.8.5)

σσπσππ

+=⎟

⎠

⎞⎜⎝

⎛==+== ∑∑== 1

1,,)1(

2

2

1

22

1

2s

N

n

n

sA

L

A

LN

n

nAD eA

An

eR

C

R

CAnRC

i (1.8.6)

The finite elliptic wing with no geometric and no aerodynamic twist will result in the minimum possible induced

drag for a given lift coefficient and aspect ratio. The chord length, lift coefficient, and induced drag coefficient for a finite elliptic wing are,

( )2214

)( bzR

bzc

A

−= π , AL

LL

L

RC

CC

παα

α

α

,

0,

~1

)(~

+−

= , A

LD

R

CC

i π2

=

For an arbitrary finite wing with geometric and/or aerodynamic twist, we can obtain the coefficients from

( ) Ωαα nLnn baA −−≡root0 (1.8.19)

where

1)sin()sin()(

~4

1 ,

=⎥⎦

⎤⎢⎣

⎡+∑

=

N

n L

n nn

cC

ba θθθα

(1.8.20)

)()sin()sin()(

~4

1 ,

θωθθθα=⎥

⎦

⎤⎢⎣

⎡+∑

=

N

n L

n nn

cC

bb (1.8.21)

The lift coefficient and induced drag coefficient for a finite wing with geometric and/or aerodynamic twist can be

expressed as

2

( )[ ]Ωαα Ωα ε−−= root0, LLL CC (1.8.24)

A

LDLLDLDL

DiR

CCCCC π

ΩκΩκκ αΩα2

,,2 )()1( +−+

= (1.8.25)

where

)1)(

~1(

~

,

,1,

LAL

L

AL

RC

CaRC

κππ

α

αα

++== (1.8.26)

1,

1,

)~

1(

)~

1(1

aCR

aCR

LA

LA

L

α

α

ππκ

++−

≡ (1.8.27)

1

1

a

b≡Ωε (1.8.28)

∑=

≡N

n

nD

a

an

2

2

1

2

κ (1.8.29)

∑=

⎟⎠

⎞⎜⎝

⎛ −≡N

n

nnnDL

a

a

b

b

a

an

a

b

2 1111

12κ (1.8.30)

∑=

⎟⎠

⎞⎜⎝

⎛ −⎟⎠

⎞⎜⎝

⎛≡N

n

nnD

a

a

b

bn

a

b

2

2

11

2

1

1Ωκ (1.8.31)

For tapered wing with linear geometric and aerodynamic twist, the wing coefficients are plotted in the following

figures.

Taper Ratio

0.0 0.2 0.4 0.6 0.8 1.0

κL

0.00

0.01

0.02

0.03

0.04R

A=4

8

16

12

20

4

8

3

RT

0.0 0.2 0.4 0.6 0.8 1.0

εΩ

0.34

0.36

0.38

0.40

0.42

0.44

0.46

0.48

RA=20

8

4

1216

elliptic planform (εΩ = 4/3π)

Taper Ratio

0.0 0.2 0.4 0.6 0.8 1.0

κD

0.00

0.05

0.10

0.15

0.20

RA=20

18

12

16

14

10

8

6

4

4

RT

0.0 0.2 0.4 0.6 0.8 1.0

κDL

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3R

A=20

8

6

18 16

14 12

10

4

elliptic planform (κDL

= 0)

RT

0.0 0.2 0.4 0.6 0.8 1.0

κDΩ

0.05

0.10

0.15

0.20

RA=20

12

16

14

8

6

4

18

10

5

Downwash The downwash angle aft of an unswept finite wing, can be estimated from

w

w

A

L

b

pvd

R

Cyx κ

κκε =)0,,( (4.5.5)

where

)2sin(12 1

πκ nA

A

n

n

v ∑∞

=+= (4.5.3)

∑

∑

∞

=

∞

=

+

−+

=

2 1

2 1

2

)2sin(1

)2cos()1(4

n

n

n

n

b

nA

A

n

An

nA

π

ππ

κ (4.5.4)

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

+++

+++

+=

22222

222

222

2

)(

)2(1

)(

2

b

b

b

bp

yxyx

yxx

y κ

κκπ

κκ (4.5.6)

2wb

xx = ,

2wb

yy =

Figure 4.5.1. Vortex model for estimating the downwash on an aft tail behind an unswept wing.

6

Wing Taper Ratio

0.0 0.2 0.4 0.6 0.8 1.0

κv

0.8

0.9

1.0

1.1

1.2

RAw

=4

8

1216

20

12

8

4

16

20

elliptic wing (κv = 1.0)

Wing Taper Ratio

0.0 0.2 0.4 0.6 0.8 1.0

κb

0.6

0.7

0.8

0.9

1.0

RAw

=4

8

12 16

20

128

4

16

20

elliptic wing (κb = π/4)

7

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

κp

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.1

0.2

0.3

0.4

0.5

x

κbb/2

= 0.0y

κbb/2