OSCILLATING AIRFOILS IN TWO-DIMENSIONAL. COMPRESSIBLE FLOW

An oscillating airfoil in a two-dimensional compressible flow will be treated in this chapter. All the hypotheses made in § 13.1 regarding the linearization are again assumed here.

The linearized equation for the acceleration potential <f>, referred to a frame of reference at rest relative to the fluid at infinity is given by Eq. 15 of § 12.5. If the coordinate system is moving with a speed U in the negative ж-axis direction relative to the fluid at infinity, the field equation of small disturbances can be obtained by transforming that equation according to the Galilean transformation

x — x’ — Ut’, у = у’, t = f

The following field equation is obtained in this new coordinate system for a two-dimensional flow in the (x, y) plane:

(2)

where the primes are omitted. For an observer fixed on the moving coordinates, the fluid at infinity has a velocity U in the positive a;-axis direction, a is the speed of sound in the undisturbed flow.

As in the last chapter, the principle of superposition is valid, and it suffices to treat oscillating airfoils of zero camber, zero thickness, and at zero mean angle of attack.

In a subsonic flow, (U < a), an elementary solution of Eq. 2 was obtained by Possio who derived an integral equation governing ф and obtained some numerical results by a method of collocation in 1937. These calculations were repeated and extended by Frazer (1941) and Frazer and Skan (1942). The kernel of Possio’s equation was tabulated by Schwarz (1943). Approximate solutions were proposed by Schade (1944), Dietze (1942-44), and Fettis (1952). An exact solution of Possio’s equation in closed form is yet unknown.

A different approach to the subsonic oscillating-airfoil theory was pursued independently by Reissner and Sherman (1944), Biot (1946),

Timman (1946), Haskind (1947), and Kiissner (1953). The boundary – value problem was directly attacked by the introduction of (confocal) elliptic coordinates. An explicit solution of the problem can then be obtained in terms of Mathieu functions. A great deal of mathematical work is required, however, to bring the solution into a form suitable for numerical calculations.

In contrast to the subsonic case, the linearized supersonic case is extremely simple. This is so because of the simple physical condition that (in the two-dimensional case) the flows above and below the airfoil are independent of each other, and that the flow over the airfoil is inde­pendent of the conditions in the wake. The corresponding differential equation, of the hyperbolic type, can be solved by a number of methods. The first solution was given by Possio in 1937 by a method of super­position of sources and sinks. Von Borbely solved the problem by the method of Laplace transformation (1942), and Temple and Jahn solved it by Riemann’s method (1945). Extensive numerical results were obtained by Schwarz, Temple and Jahn, Garrick and Rubinow, and others (see bibliography).

In §§ 14.1 through 14.4, Possio’s integral equation for the subsonic flow will be derived in a manner used by Biot, Karp, Shu, and Weil.14-1 Approximate methods of solution of Possio’s equation will be outlined in § 14.5.

The supersonic case is discussed in § 14.6, where, following Stewart – son,14-40 the Laplace transformation method is used. In § 14.7, the results known so far are tabulated for a quick reference.

The lift of an oscillating wing can also be computed from the indicial responses to suddenly started motions. It turns out that, if the main­stream Mach number is close to one, it is simpler to calculate the indicial responses. Heaslet, Lomax, and Spreiter14-24 have shown (1949) how the lift due to a harmonic vertical-translation oscillation of a flat plate in a flow with main-stream Mach number equal to one can be calculated according to the linearized theory.