Stability of subsonic gasdynamic flows

Abstract

A system of differential equations describing small perturbations of the steady flow of a non-viscous ideal gas in a channel of variable cross section is analyzed in this paper. The equations of nonsteady flow and the boundary conditions are linearized, and the solution of the linearized equations is sought in the form v(x)expt λt, where v(x) is an eigenfunction while λ is the natural frequency for the boundary problem being studied. With such an approach the problem is reduced to finding the solutions to ordinary differential equations with variable coefficients which depend on the parameter λ. Analytical solutions of this system are obtained for small values of λ and for values of ¦λ¦≫1. The results can be used to calculate the growth of high-frequency and low-frequency perturbations imposed on subsonic, supersonic, and mixed (i.e., with transitions through the velocity of sound) gasdynamic flows, to analyze the stability of subsonic sections, and to verify and supplement various numerical methods for calculating unsteady flows and numerical methods for studying stability in gasdynamics. The application of the solutions found for small and large λ is demonstrated on a study of flow stability behind a shock wave (a direct compression shock in the present formulation). Analytical expressions are obtained for the determination of λ from which it follows that the flow stability behind a shock essentially depends on the shape of the channel at the place where the shock is located in the steady flow, which was noted earlier in [1], and on the conditions of the reflection of small perturbations in the exit cross section of the channel, which was first pointed out in [2].