Unsteady Thin Airfoil Theory for Transonic Flows with Embedded Shocks

Abstract

A simple modification of classical unsteady thin airfoil theory is presented which accounts for the presence and induced motion of the embedded partial chord shock waves which appear at supercritical transonic Mach numbers. The basic model is found to be divergent at zero frequency because of an unbounded growth of the shock excursion amplitude. This behavior is eliminated by introducing a term associated with the mean flow Mach number gradient near the shock. Numerical results are given for the loads induced by an oscillating flap. LASSICAL unsteady thin airfoil theory fails for low frequencies at the subsonic freestream Mach number at which local supersonic flow first occurs. The main cause of this sudden failure is the formation of a shock wave, which shields the forward region of the airfoil from aft generated disturbances. In the present study,l a simple modification of classical thin airfoil theory is given which accounts for the presence and induced motion of such shocks. Predicted airloads are shown to be in favorable agreement with both experimental observations and finite difference calculations. We assume that the unsteady flow is generated by in- finitesimal harmonic oscillations of an airfoil which is suf- ficiently thin and at sufficiently small mean angles of attack that the transonic small disturbance approximation is valid. The unsteady problem can then be linearized about the steady flow, taking due account of the (small amplitude) displacement of any embedded shocks. The linearized un- steady problem has been formulated for an arbitrary three dimensional planar lifting surface in an earlier paper.2 The linearized equations of motion depend through various coefficients on the steady local Mach number M0 (jc), which in general varies in some complicated way throughout the flowfield. Unless the functional form of M0 is very simple, the boundary value problem for the unsteady flow field must be solved by purely numerical techniques. This has been attempted using finite difference methods by Weatherill et al.3 and Traci et al. 4 However, neither of these studies properly accounted for the between the embedded shocks and the unsteady disturbance field. (Of course, this issue does not arise in nonlinear formulations of the unsteady problem, in which the interaction is implicit; flow with oscillating shocks has been successfully computed from the nonlinear small disturbance equation by, for example, Ballhaus and Goor jian.5) An alternative to direct numerical methods is to ap- proximate the steady Mach number distribution M0(x) in such a way that analytical or semianalytica l methods can be applied. Classical theory, where M0 is set equal to the freestream Mach number M^, is, of course, one example. The