Finite Wing Sumary Wph

Finite Wings Summary MAE 5510

W. F. Phillips

Spring 2007

For an arbitrary finite wing, the circulation distribution can be written as N

θ = cos −1 (− 2 z b)

Γ (θ ) = 2bV∞ ∑ An sin(nθ ); n =1

(1.8.3)

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

⎡

⎤ 4b + n ⎥ sin( nθ ) = α (θ ) − α L 0 (θ ) ⎣ L ,α c(θ ) sin(θ ) ⎦

∑ An ⎢ C~

n =1

(1.8.4)

The lift coefficient and induced drag coefficient for a finite wing can be expressed as C L = π R A A1 ,

2 RA = b Sw

C L2 C L2 (1 + σ ) = , π RA π R A es

N

C Di = π R A ∑ n An2 = n =1

σ =

(1.8.5) N

⎛A ⎞

∑ n ⎜⎝ A1n ⎟⎠

n=2

2

, es =

1 1+ σ

(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, c( z ) =

4b 1 − (2 z b )2 , π RA

CL =

~ C L ,α (α − α L 0 ) , ~ 1 + C L ,α π R A

C Di =

C L2 π RA

For an arbitrary finite wing with geometric and/or aerodynamic twist, we can obtain the coefficients from An ≡ an (α − α L 0 )root − bn Ω

(1.8.19)

where ⎡

⎤ 4b + n ⎥ sin(nθ ) = 1 ⎣ L,α c(θ ) sin(θ ) ⎦

(1.8.20)

⎤ 4b + n ⎥ sin(nθ ) = ω (θ ) θ sin( ) ⎣ L ,α c(θ ) ⎦

(1.8.21)

N

∑ an ⎢ C~

n =1 N

⎡

∑ bn ⎢ C~

n =1

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

2 C L = C L ,α [(α − α L 0 )root − ε Ω Ω ] C Di =

(1.8.24)

C L2 (1 + κ D ) − κ DL C L C L ,α Ω + κ DΩ (C L ,α Ω ) 2 πR A

(1.8.25)

where C L,α = πR A a1 =

κL ≡

~ C L ,α ~ (1 + C L,α π R A )(1 + κ L )

(1.8.26)

~ 1 − (1 + π R A C L ,α )a1 ~ (1 + π R A C L ,α )a1

(1.8.27)

b1 a1

(1.8.28)

εΩ κD ≡

κ DL ≡ 2

b1 a1

N

a2

n=2

1

∑ n an2

N

a ⎛b

(1.8.29)

a ⎞

∑ n a1n ⎜⎝ b1n − a1n ⎟⎠

(1.8.30)

n=2

b1 ⎞ ⎟ ⎝ a1 ⎠

κ DΩ ≡ ⎛⎜

≡

2 N

a ⎞ ⎛b ∑ n⎜⎝ b1n − a1n ⎟⎠ n=2

2

(1.8.31)

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

0.04

RA=4 8 12

4 8

16 20

0.03

κL 0.02

0.01

0.00 0.0

0.2

0.4 0.6 Taper Ratio

0.8

1.0

3

0.48 0.46

RA=20 12

elliptic planform (εΩ = 4/3π)

8 4

0.44

εΩ

16

0.42 0.40 0.38 0.36 0.34 0.0

0.2

0.4

0.6

0.8

1.0

RT

0.20

RA=20 18 16

0.15

14

κD 0.10

12 10 8

0.05

6 4

0.00 0.0

0.2

0.4 0.6 Taper Ratio

0.8

1.0

4

RA=20 18 16 14 12

0.3 0.2

10 8 6 4

0.1

κDL

0.0 elliptic planform (κDL = 0)

-0.1 -0.2 -0.3 0.0

0.2

0.4

0.6

0.8

1.0

RT

RA=20 18 16 14

0.20

12 10

0.15

κDΩ

8 6

0.10

0.05 0.0

4

0.2

0.4

0.6 RT

0.8

1.0

5

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

ε d ( x , y ,0) =

κ v κ p C Lw κ b R Aw

(4.5.5)

where ∞

An sin(n π 2) n = 2 A1

κ v = 1+ ∑ ∞

κb =

κp

π+ ∑ 4

n=2

nAn cos(nπ 2) (n 2 − 1) A1

(4.5.4)

∞

An sin(nπ 2) A n=2 1

1+ ∑

⎡ x ( x 2 + 2 y 2 + κ b2 ) ⎢1 + = 2 2 π ( y + κ b2 ) ⎢ ( x 2 + y 2 ) x 2 + y 2 + κ 2 b ⎣ 2 κ b2

x =

(4.5.3)

x , bw 2

y =

⎤ ⎥ ⎥ ⎦

y bw 2

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

(4.5.6)

6

1.2 16

12

20

8

1.1

RAw=4

κv 1.0 4

elliptic wing (κv = 1.0)

8

0.9

12 16 20

0.8 0.0

0.2

0.4 0.6 Wing Taper Ratio

0.8

1.0

1.0 20 16

0.9

12 8

elliptic wing (κb = π/4)

4

κb 0.8

0.7

RAw=4 8 12

0.6 0.0

16 20

0.2

0.4 0.6 Wing Taper Ratio

0.8

1.0

7

0.70 0.65 0.60 0.55

y = 0.0 κbb/2 0.1 0.2

κp 0.50 0.45 0.40

0.3 0.4 0.5

0.35 0.30 0.4

0.6

0.8

1.0

1.2

1.4 1.6 x κbb/2

1.8

2.0

2.2

2.4