finite wing sumary wph
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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
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