APPROXIMATE SOLUTIONS OF POSSIO’S EQUATION

It is a general feature of Carleman’s integral equation that the solution admits a singularity. For an airfoil, such a singularity is located at the leading edge where the acceleration potential should tend to infinity like V(l —x)j{ 1 + x) when — 1. This particular form of the singu­larity can be derived from Carleman’s general theory, or from analogy with the incompressible-flow case. The intensity of the singularity can

* Jahnke and Emde, Tables of Functions, p. 3. Dover Publications.

t See § 14.7 and Bibliography. Summaries of numerical tables are given in Refs. 14.1 and 14.8.

be determined by the Kutta-Joukowski condition that the lift be zero at the trailing edge.

When we write L(x, i) — L(x)ela, i, the function L(x) consists of a term const V(1 — x)j([ + x) and a nonsingular part. Possio14,12 and Frazer14-6 write the nonsingular part as a series involving a number of constant coefficients and determine these constants by the collocation method. Schade14,15 writes the nonsingular part of L(x) as a series in Legendre polynomials. The nonsingular part of the kernel Ky(M, z) as well as the upwash distribution v(x, 0) are also expressed as series in Legendre polynomials. The undetermined coefficients are then obtained on the basis of the orthogonality of Legendre polynomials in the interval – 1 < 1.

The best-known method is probably Dietz’s iteration procedure,14-2 which will be outlined below. Let us introduce the notation of the composition product

P K(M, х-І) Щ) dH = K-L (1)

p0U2J-i

— P m X-S) m dS =– Ka ■ L (2)

PqU* J-i

Then Possio’s equation can be written as

K-L = v (3)

where v{x) is the given vertical velocity component on the airfoil. If the fluid is incompressible, the corresponding lift distribution is given by L0, and Eq. 3 is reduced to

K0-L0 = v (4)

The solution of this equation is known explicitly. (See § 13.5.) Let us write L = L0+ AL0 which defines AL0; we have

K0- L0 = v — К ■ L — К ■ L0- K – AL0 (5)

or

К • AL„ = % (6)

if

v^{K0 ~K)-L, (7)

The function vy being known, Eq. 6 is completely analogous to Eq. 3. Hence, the same procedure can be applied. Let

AL0 i- – f~ A /.}

where Ly is the solution of the incompressible-fluid problem

Vy = KQ- Ly

SEC. 14.6 OSCILLATING AIRFOILS IN SUPERSONIC FLOW and ALj is governed by the equation

Continuing this process, we obtain the approximate solution L = L0 + + L2 + • ‘ ‘ + Ln + A Ln

where Lm is the solution of the incompressible-fluid problem K0 ‘ Lm = vm (m < ri)

while

Vm — (Ко К) ■ Lm_J

v0 = v being specified, and

K-ALn= (K0 – K)Ln

The convergence of this procedure has not been proved, but is indicated by Dietz’s numerical examples. The rapidity of convergence is found to deteriorate for increasing M and k.

Recently, Fettis14-4 introduced a method that avoids the iterative procedure and gives a relatively simple solution on the basis of an ap­proximate kernel, in which the nonsingular part of Possio’s kernel is replaced by a polynomial.

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