Analysis of Incompressible Flows over Thin Airfoils Using Integral Momentum Equation for Boundary Layer

Consider thin symmetric airfoil (at zero angle of attack). The perturbation velocity of the inviscid flow can be approximated by:

U fc y'(Q

u(x, 0) = di (9.37)

n 0 x – i

where y(x) is the equation of the upper surface of the profile.

To account for the viscous effects, the shape of the airfoil is augmented by the displacement thickness as y(x) + 6*(x).

The integral momentum equation can be solved coupled with the above integral where ui = U + u.

To account for the wake, an integral from the trailing edge to the far field is added to the above formula, namely

uw(x, 0) = – di (9.38)

n 0 x – І

The velocity profile in the wake must satisfy a different boundary condition at y = 0, x > c, namely du/dy = 0, rather than u = 0.

The extension to linearized subsonic flows is straightforward. The Prandtl/Glauert transformation can be used to modify the inviscid velocity distribution together with Oswatitsch’s modification of Pohlhausen method for compressible boundary layer as discussed before.

The finite plate at zero angle of attack is a special case, where y(x) = 0.

For thin airfoils at angle of attack, the viscous layers on the top and the bottom are different. Moreover, the inviscid flow analysis requires solution of an integral equation, not just an evaluation of an integral as in the symmetric case.

9.1.1.2 Analysis of Supersonic Flows over Thin Airfoils Using Integral Momentum Equation for Boundary Layer

Again, the body is augmented with the displacement thickness and the Oswatitsch method may be used to solve the integral momentum equation for the compressible boundary layer. The inviscid flow solution, using d’Alembert formula (according to Ackeret thin airfoil theory), is given by:

, U,, U,

u(x, 0+) = –(f + (x) – a), u(x, О-) = -(f ~(x) – a) (9.39)

P P

where в = MO – 1 and a is the incidence.

For attached supersonic flows, the solution is obtained with a marching scheme in the x-direction, in one sweep. Iteration are still needed at each step. The case of angle of attack can be easily treated by splitting the top and bottom regions, since there is no interaction between the two.