Weakly disturbed steady transonic flows of a vibrationally relaxing gas

Abstract

The formation of shock waves in transonic flows is a problem which is far from being resolved. Conventional gas dynamics contains numerous examples of the construction of transonic flows free of discontinuities and flows in which discontinuities occur in the transition through the speed of sound (see [I], for example). Experiments have confirmed the existence of both continuous [2] transonic flows and transonic flows containing discontinuities [3]. The exceptional nature of nonshock flow in a local supersonic region was shown in [4], which suggested to one author that continuous stagnation of a transonic flow is not possible [5]. In the opinion Kuo and Sirs [6], such a flow is either unstable or contains shock waves or both. One possible approach to solving this problem is the study of transonic flows for their stability in relation to steady and nonsteady disturbances that might develop in the flow due to irregularities on the walls bounding the flow or the arrival of weak nonsteady waves at the sonic line. Such an investigation is complex in the general case, with approximate methods usually being used to obtain a solution. The stability of a transonic flow relative to small nonsteady disturbances in a small neighborhood of the sonic line was examined in [7, 8] on the basis of a quasi-unidimensional approximation. An attempt to perform an analysis in general form was made in [9], where results agreeing with the data in [7] were obtained after certain simplifying assumptions were made. Kuz'min [i0] analyzed the stability of a transonic flow on the basis of linearization of the Linn-Reissner-Tsien equation. It was found that transonic flow is stable in relation to a small change in the shape of the walls of the nozzle and the conditions at the inlet if the acceleration of the specified flow is everywhere positive. The examples of steady continuous transonic flows that have been constructed exist only near certain types of solid boundaries, thus giving rise to the view that such flows are the exception. A study of the behavior of steady perturbations of such flows due to irregularities in the walls bounding the flow will make it possible to study the development of the features of the flow. Aspects of the formation of shock waves and the conditions for nonshock flow in transonic flows were examined in [II]. It was shown that shock waves form if the slope of the walls of the channel near the mouth of the nozzle is too gentle. Discontinuities may form at a point on the sonic line or downstream, in the supersonic region. Numerical calculations of plane transonic flow with a local supersonic zone have shown that a shock wave is formed inside the region of supersonic flow [12]. Below, we examine the behavior of steady and nonsteady perturbations of steady transonic flows of a vibrationally relaxing gas. Use of the methods in [8] makes it possible to reduce the problem of determining the stability of a transonic flow against small nonsteady distortions to the study of a certain nonlinear partial differential equation. Analysis of the solutions of this equation for specific states of the medium shows that flow with a transition from the supersoni c regime to the subsonic regime is stable relative to small distortions if the gas is in the equilibrium state or if it is nonequilibrium with unexcited vibrational degrees of freedom. In this case, transonic flow will also be stable in the course of the transition from the subsonic to the supersonic regime. In our study of the behavior of steady distortions, we examined the supersonic region of transonic flow. We derive the equations of the transonic approximation as in traditional