Single Horseshoe Vortex Wing Model

Lift due to a horseshoe vortex Kutta-Joukowsky Theorem

bVdyVL

b

b

Γ=Γ= ∞∞

−

∞∞ ∫ ρρ2

2

Sb

bVC

SV

bV

SV

LC

L

L

2

22

221

21

∞

∞∞

∞∞

∞∞

Γ=

Γ==

ρ

ρ

ρ

AbV

C L∞

Γ=

2

b

S

b trailing vortices

bound vortex

Γ

Γ

Γ

Single Horseshoe Vortex Wing Model

16.100 2002 2

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 ( 0=y ).

iieffieffi

effieff

ieff

LLLDLLL

ααα

α

ααα

=≈=

≈=

−= ∞

sincos

For small iαα &∞

∞

−≈

Vwi

iα

To calculate downwash, we apply Biot-Savart:

∫×Γ

=filament r

rdzyxw 34),,(

vlv

v

π

L effL

iα iD

∞V

iw− iα

∞α effα x

b

y

x Γ

Γ

Γ

rv

+∞→x

+∞→x)0,

2,( bx −′

ld

iDLVL

⊥⊥ ∞

Single Horseshoe Vortex Wing Model

16.100 2002 3

At 0=== zyx (i.e. wire root), the bound vortex does not produce a downwash,

so, we only have the 2 trailing vortices at 2by ±= .

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

Γ= ∫∫

+=−=22

4)0,0,0(

byby

wπ

v

Let’s do the 2by −= integral first:

∫

∫ ∫

∞+

=

+∞= +∞=

⎥⎦⎤

⎢⎣⎡ +′

′−=

⎥⎦⎤

⎢⎣⎡ +′

⎥⎦⎤

⎢⎣⎡ ++′−×′−

=×

0

23

22

0 0

23

22

22

3

)2

(

2

)2

(

)2

()()(

bx

kxdb

bx

jbixixd

rrdx

x x

v

vvvv

lv

b

wπΓ

−=)0,0,0(

Then, combining this we can find:

∞

∞

∞

Γ=

Γ−−=

−=≅

bVLD

bVLD

Vw

LLD

i

i

iii

π

π

α

)(

But bV

LbVL∞∞

∞∞ =Γ⇒Γ=ρ

ρ

⇒ 22

2

bVLDi

∞∞

=πρ

Single Horseshoe Vortex Wing Model

16.100 2002 4

⇒ 22

22 2)

21

(

21 b

S

SV

L

SV

DC i

Di πρρ ∞∞∞∞

==

ok?? thisis Hmmm

2

2

AC

C LDi π=

Recall: Lifting line:

1,2

≤= eeA

CC LDi π