Stability of Shock Waves Attached to Wedges and Cones

Abstract

A numerical study of a reduced set of the Euler equations shows that both weak and strong attached shock waves are stable configurations depending on the boundary conditions imposed. The results obtained are in good agreement with experimental observations and with the minimum entropy principle. A heuristic ex- planation of the behavior observed is given in terms of quasistatic considerations. OR flow over a wedge or cone, whose deflection angle is less than the angle associated with shock detachment for the incoming supersonic flow, it is a well-known fact that the steady-state Euler equations admit two different solutions. Each solution is distinguished by the strength of the attached oblique shock. Thus a solution is labeled strong or weak if the shock-wave inclination is, respectively, greater than or less than the shock-wave inclination at detachment. For the strong shock solution, the flow is always subsonic downstream of the shock; for the weak shock solution, the flow is supersonic downstream of the shock, except for a small range of deflection angles in the neighborhood of the detachment angle where the weak shock is followed by subsonic flow. This small region will be ignored in the present work. The existence of multiple solutions has attracted considerable theoretical scrutiny, which has mainly succeeded in obscuring the problem. Indeed, our understanding of this problem has changed little since 1948, when Courant and Friedrichs1 wrote .. .The question arises which of the two actually occurs. It has frequently been stated that the strong one is unstable and that, therefore, only the weak one could occur. A convincing proof of this instability has apparently never been given. Quite aside from the question of stability, the problem of determining which of the possible shocks occurs cannot be formulated and an- swered without taking the boundary conditions at infinity into account. The confusion has come about because the question posed by Courant and Friedrichs and every succeeding investigator has not been the proper question to ask. The question should not be which solution actually occurs, since we have ex- perimental evidence2 that both the weak and the strong solutions do occur, but under what conditions do one or the other solutions occur. The answer to this latter question obviously depends on the boundary conditions, as already indicated by Courant and Friedrichs. Moreover, since the nature of steady-state solution is elliptic for the strong shock and hyperbolic for the weak shock, we should suspect that each problem is governed by a different set of boundary conditions. The purpose of this paper is, therefore, to study the stability of the weak and strong solutions with boundary conditions which are appropriate to each case under in- vestigation.