Single Horseshoe Vortex Wing Model
Γ
S
Γ
b
b
bound
vortex
Γ
Lift due to a horseshoe vortex
Kutta-Joukowsky Theorem
b
2
L = ρ ∞V∞ ∫ Γdy =ρ ∞V∞ Γb
−
CL =
b
2
L
1
ρ ∞V∞2 S
2
2Γ b 2
CL =
V∞ b S
CL =
2Γ
A
V∞ b
=
ρ ∞V∞ Γb
1
ρ ∞V∞2 S
2
trailing
vortices
Single Horseshoe Vortex Wing Model
Induced Drag
To estimate the induced drag using this simple model, we will assume that the
3-D lift is tilted by the downwash occurring at the wing root ( y = 0 ).
Leff
L
αi
αi
α eff
Di
V∞
L ⊥ V∞
− wi
L ⊥ Di
α∞
x
α eff = α ∞ − α i
L = Leff cos α i ≈ Leff
Di = Leff sin α i ≈ Leff α i = Lα i
For small α ∞ & α i
αi ≈
− wi
V∞
To calculate downwash, we apply Biot-Savart:
y
Γ
x → +∞
b
Γ
x
v
r
Γ
Γ
v
w( x, y, z ) =
4π
16.100 2002
b
( x ′,− ,0)
2
x → +∞
dl
v v
dl × r
∫ r3
filament
2
Single Horseshoe Vortex Wing Model
At x = y = z = 0 (i.e. wire root), the bound vortex does not produce a downwash,
so, we only have the 2 trailing vortices at y = ± b .
2
⎡
⎤
Γ ⎢
v
⎥
+ ∫ ⎥
w(0,0,0) =
∫
⎢
4π
b
⎢ y = − b2
y =+ ⎥
2⎦
⎣
Let’s do the y = −
b
integral first:
2
v v
dl × r
∫ r3 =
x = +∞
x =0
v ⎡
v
b v⎤
(−dx ′i ) × ⎢(− x ′) 2 i + (+ ) 2 j ⎥
2
⎣
⎦
0
∫
3
x = +∞
0
=
∫
3
⎡ ′2 b 2 ⎤ 2
⎢⎣ x + ( 2 ) ⎥⎦
+∞
w(0,0,0) = −
⎡ ′2 b 2 ⎤ 2
⎢⎣ x + ( 2 ) ⎥⎦
v
b
− dx ′k
2
Γ
πb
Then, combining this we can find:
Di ≅ Lα i = − L
wi
V∞
Di = − L(
Di =
But
−Γ
)
πbV∞
LΓ
πbV∞
L = ρ ∞V∞ Γb ⇒ Γ =
⇒ Di =
16.100 2002
L
ρ ∞V∞ b
L2
πρ ∞V∞2 b 2
3
Single Horseshoe Vortex Wing Model
⇒ C Di =
C Di =
Di
1
ρ ∞V∞2 S
2
=(
L
1
ρ ∞V∞2 S
2
)2
S
2πb 2
C L2
2πA
Hmmm is this ok??
Recall: Lifting line:
C Di =
16.100 2002
C L2
,e ≤ 1
πAe
4