POSSIO’S INTEGRAL EQUATION

The results of the preceding sections can be combined to give an integral equation expressing the boundary-value problem of an oscillating airfoil. The total vertical component of velocity due to a lift distribution L(£, t) is, according to Eq. 16 of § 14.3, given by the Cauchy integral:

v{x, 0, 0 = Г Щ, t) K(M, x~£)d£ (1)

Pou J-1

where

v(x, 0, t) = v(x)eiu>t, Щ, 0,-t) = Щ)еіиЛ

The function v(x, 0, t) is given by the boundary condition (Eq. 7 of § 13.2) for a given airfoil in a given oscillation mode. The problem is then to determine Щ, t) from Eq. 1 under the side condition that L 0 at the trailing edge f = 1 (the Kutta-Joukowski condition).[39] Equation 1 is called the Possio’s integral equation. f The kernel K(M, x — f) depends on the Mach number M, the reduced frequency к = 1 w/U, and the dis­tance x — f. It has a singular point at x = f.

Possio’s equation is a special case of a general type of singular integral equation studied earlier by Carleman. J It is a Cauchy-type integral equation of the first kind. With the imposition of the Kutta condition at the trailing edge, a unique solution is known to exist. Several methods of reducing this equation into an ordinary (nonsingular) integral equation are known. But so far no solution of Possio’s equation has been obtained in closed form. Numerical or graphical methods must be employed.

For this purpose the singularities of K(M, z) may be isolated in the following manner according to Schwarz:1416

F(M) , ,

K(M, z) = – Д—’ + і G(M) log |z| + Д(М, z)

where

z = kx (k = reduced frequency)

F(M) Vl – M G{M) =—————— Д=—

2w 2ттл/і – Мг

In particular, at zero Mach number,

where Ci(z) and Si(z) are the cosine and sine integrals, respectively.* Д(Л/, z), defined by Eq. 2, is a continuous function of z. Numerical tables of the functions К and Д are published by Possio, Dietze, Schwarz, and Schade. f

A different separation is given by Dietze:14,2

K(M, z) = AKyiM, z) + AK. JM, z) + ДО, z) (5)

where

ДД(М, 2) = ~ + kn – I – k12 logz + z(kla + к, л log |z|) (6)

The real and imaginary parts of the coefficients k2n are tabulated in Dietze’s paper.14,2