Thin Airfoil Theory Summary

• Replace airfoil with camber line (assume small cτ )

• Distribute vortices of strength )(xγ along chord line for 0 x c≤ ≤ .

• Determine )(xγ by satisfying flow tangency on camber line.

0

( ) 02 ( )

cdZ dVdx x

γ ξ ξαπ ξ∞

− − = − ∫

• The pressure coefficient can be simplified using Bernoulli & assuming small

perturbation:

x

z

c

τ(x) = thickness

z(x) = camber line

x

z

c

z(x) = camber line

x

z

c

γ(x)dx

Thin Airfoil Theory Summary

16.100 2002 2

{ }

2

2 2 2

2 2

22

2 2 2

2

2 2

2

higher order

12

1 1( )2 2

( )112

21

2

pp pcV

p V u V p V

p p V u VVV

V V u u VV

u u VV V

ρ

ρ ρ

ρ

∞

∞

∞ ∞ ∞

∞ ∞

∞∞

∞ ∞

∞

∞ ∞

−=

+ + + = +

− + +⇒ = −

+ + += −

+= − −

2puCV∞

⇒ = −

• It can also be shown that

( ) ( ) ( )2 ( )

lower upper

upper lower

p p p upper lower

x u x u x

C C C u uV

γ

∞

= −

⇒ ∆ = − = −

( )( ) 2pxC xV

γ

∞

⇒ =

Symmetric Airfoil Solution

For a symmetric airfoil (i.e. 0dzdx

= ), the vortex strength is:

θθαθγ

sincos12)( +

= ∞V

But, recall:

(1 cos )2cx θ= −

Thin Airfoil Theory Summary

16.100 2002 3

2

2

cos 1 2

sin 1 cos

1 1 2

sin 2 (1 )

xc

xc

x xc c

θ

θ θ

θ

⇒ = −

= −

= − −

= −

1( ) 2

(1 )

xcx V

x xc c

γ α ∞

−⇒ =

−

1( ) 2

xcx V xc

γ α ∞

−=

Thus, 1

4p

xcC xc

α−

∆ = .

Some things to notice: • At trailing edge 0=∆ pC .

⇒ Kutta condition is enforced which requires upper lowerp p= • At leading edge, ∞→∆ pC ! “Suction peak” required to turn flow around

leading edge which is infinitely thin. The instance of a suction peak exists on true airfoils (i.e. not infinitely thin) though pC∆ is finite (but large).

Suction peaks should be avoided as they can result in 1. leading edge separation 2. low (very low) pressure at leading edge which must rise towards trailing edge

⇒ adverse pressure gradients ⇒ boundary layer separation.

xc

∆Cp

Thin Airfoil Theory Summary

16.100 2002 4

Cambered Airfoil Solutions For a cambered airfoil, we can use a “Fourier series”–like approach for the vortex strength distribution:

1flat plate camberedcontributions contributions

1 cos( ) 2 sinsino n

nV A A nθγ θ θ

θ

∞

∞=

+ ⇒ = +

∑

Plugging this into the flow tangency condition for the camber line gives (after some work):

0 00

0 00

1

2 cosn

dzA ddx

dzA n ddx

π

π

α θπ

θ θπ

= −

=

∫

∫

After finding the nA ’s, the following relationships can be used to find ,

acmC C , etc.

4

2 1

0 0 0 10

2 ( )

( )4

1 1(cos 1)2

c ac

LO

m m

LO

C

C C A A

dz d A Adx

π

π α απ

α θ θ απ

= −

= = −

= − − = − −∫

Note: in thin airfoil theory, the aerodynamic center is always at the quarter-chord ( 4c ), regardless of the airfoil shape or angle of attack.