Subsonic Flow Past an Airfoil

Before studying three dimensional subsonic flow, we are going to start our analysis with two dimensional flows. There are two options to do so. We either rederive the equations for two dimensional flows or we take the integral of the three dimen­sional acceleration potential in y to make the equations independent of spanwise direction. Here, the latter approach is preferred since we have already obtained the necessary expression to be integrated only. Integrating 5.36 from —1 to 1in spanwise direction we obtain

(5.38)

Let us now write the amplitude of the downwash in terms the lifting pressure using the Kernel function for two dimensional flows (Bisplinghoff et al. 1996).

b

w(x) =———– 2 Ap(-)K M, – d-, —b < x < b (5.39-a)

Pi u2 b

—b

Here, if we take k = kx—- and u = k^j—-, the kernel function K, in terms of Mach number and the reduced frequency reads as

Equation 5.39-a is called the Possio integral equation. Possio tried to solve his equation in terms of the Fourier series, however, his solution technique confronted with the convergence problem. A different approach from Possio to remedy the convergence problem is to employ Fourier like series for the lifting pressure expressed in – = b cosh coordinates. A new way of approximating the pressure discontinuity is

1

Ap(9)= A0 cot – + An——— , 0 < в < n (5.40)

2 n

n=1

The cotangent term in 5.40 gives an integrable singularity for the lifting pressure at the leading edge while satisfying the Kutta condition with zero lifting pressure at the trailing edge. The integral of the second term of Kernel function, 5.39-b, can be evaluated numerically. In doing so, keeping the number of control points on the chord equal to the number of terms in Eq. 5.40 enables us to have a number of algebraic equations equal to the number unknowns with complex ele­ments. The right hand sides of the equation, fi, are the known values of the prescribed airfoil motion to result in following set of linear equations.

Here, x = b cos u denotes the coordinates for the control points.