2 Aerodynamics
99
Stokes integral formula is as follows:
=
C
V • d s =
¨
A
2(ω u − ω d )dσ
The velocity circulation of the flow around the airfoil is positive clockwise, and the vorticity in the boundary layer near the wall of the upper
wing is 2ω u . The vorticity 2ω d in the boundary layer near the wall of
the lower wing rotates counterclockwise, which is a negative contribution.
Therefore, it can be further written as
=
C
V • d s =
b
0
⎡
⎣
δ u (x)
0
2ω u dy −
δ d (x)
0
2ω d dy
⎤
⎦ dx =
b
0
γ (x)dx
γ (x) =
δ u (x)
0
2ω u dy −
δ d (x)
0
2ω d dy = γ u − γ d
γ u =
δ u (x)
0
2ω u dy, γ d =
δ d (x)
0
2ω d dy
where γ (x) is the surface vortex intensity along the chord, γ u is the value
of the upper wing (positive contribution), and γ d is the value of the lower
wing (negative contribution).
At the trailing edge of the airfoil, according to the Kutta and Joukowsky
conditions, if the airflow is to leave the trailing edge smoothly, there is γ (b)
= 0.0, so we can get
γ u = γ d ,
δ u (b)
0
2ω u dy =
δ d (b)
0
2ω d dy
As an approximation, as shown in Fig. 2.31, by definition of
2ω u ≈
V u
δ u (b)
, 2ω d ≈
V d
δ d (b)