Approximate unsteady thin-airfoil theory for subsonic flow

Abstract

An analytical solution of the Possio integral equation is obtained for low reduced-frequency to, correct to or- der wM/(l-M2). The resulting loading, lift, and moment differ from that derived from GASP theory by Osborne, in agreement with recent work of Amiet, who has shown GASP theory to be inapplicable to two- dimensional flow with shed vorticity. The solution is applied to a generalized gust, a power law upwash, plunging motion, pitching motion, and a sinusoidal gust. Comparisons with numerical solutions are given for lift in the latter three cases. They show that one of the forms of the solution, the Osborne lift times a phase correction, is remarkably accurate up to coM/(l —M2) = ir/4. This approximation should prove convenient and useful in applications. HE nonsteady load distribution on two-dimension al thin airfoils oscillating in subsonic flow is governed by the Possio integral equation, which has no known exact analytical solution. There have been many numerical approaches to its solution over the years, and also a number of approximate analytical approaches. For a given airfoil motion, the forces depend on two parameters, the stream Mach number M and the reduced frequency of oscillation co. The task of any ap- proximate analytical solution is to give a satisfactory ap- proximation in as large a region of the M, co plane (M< 1) as possible. Some years ago, Miles1 briefly discussed the com- pressibility correction to the incompressible problem (which has an exact solution). He transformed the governing dif- ferential equation and boundary conditions by a combined Galilean and Lorentz transformation, expanded to first order in coM//3 2, where /32 = 1 — M2, and derived a compressibilit y correction rule. In another paper, Miles2 derived a quasisteady theory by expanding the Possio integral equation to first order in co. Recently, Amiet and Sears3 applied the method of matched asymptotic expansions to small- perturbation subsonic flows, and, again, obtained Miles' correction rule1 in the course of their work, which has been called GASP (Glauert-Amiet-Sears-Prandtl) theory. Osborne4 applied this correction rule to obtain analytical formulas for the lift and moment of oscillating, thin two- dimensional airfoils flying in a generalized gust, a sinusoidal gust, and in plunging motion, although his results were in finite form only for the sinusoidal gust. Kemp5 showed that the results for the generalized gust and the plunging motion also could be put in finite form, and added the other in- teresting case of pitching motion. (He also showed, sub- sequently6, that the lift and moment for any integer power law upwash distribution could be derived by purely algebraic