Perturbation of a Discontinuous Transonic Flow

Abstract

The main difficulty in perturbing a discontinuous transonic How is in the representation of the shift in the location of the discontinuity (shock wave). Herein presented is a method of overcoming this difficulty by using a distorted airfoil as the initial case rather than the real physical airfoil; the distortion is chosen such that the location is unchanged by the perturbation. The distorted airfoil is obtained by the use of a strained coordinate system. A direct consequence of the theory is the derivation of an algebraic similarity relation between related airfoils with waves at different locations. Results for simple examples are shown. N important problem in aerodynamics is the accurate prediction of the pressures on an airfoil that is oscillating with small amplitude in a transonic flow; this prediction is necessary for the satisfactory estimation of flutter parameters. An important physical feature of such flows can be the presence of an oscillating wave, which should be accurately represented in any solution procedure because of the relatively large pressure fluctuations in the region bounded by the extremities of the motion. It is therefore desirable to represent the wave by the correct discontinuity rather than by the rapid compression exhibited by the commonly used shock capture finite-difference methods.' Such a discontinuous representation of the can be obtained in steady flow by using either a finite- difference method with fitting,2 or by the integral equation method.3 The fitting technique has recently been applied4 to unsteady flows. The present work is ultimately directed toward the development of a method of treating oscillating waves mainly through the integral equation approach. In the limit of zero frequency, the problem of computing the flow around an oscillating airfoil reduces to a steady perturbation problem with the airfoil geometry perturbed by an amount proportional to the amplitude of the oscillation. The feature of a increment is retained in this problem since it is unlikely that the perturbed airfoil will have a wave in the same location as the initial airfoil. The steady perturbation case is therefore a good starting point for deriving a fundamental approach for computing oscillatory flows, and it is this problem that is considered in this paper. In addition, a perturbation solution is useful in other ways; for example, when the flow over a given airfoil for a range of freestream Mach numbers is required, since once one nonlinear result is obtained, the other required results can be . obtained from the linear perturbation solution. As suggested earlier, the main difficulty in perturbing a discontinuous transonic flow is in the representation of the shift in the position of the discontinuity (shock wave), because for most small perturbations the physical aspects of the flow require that there be the same number of waves in both the initial and perturbed states but at different locations. An examination of the usual form for a perturbation solution indicates that this physical feature cannot be correctly represented except in the trivial case where no movement occurs. In the method herein presented, the