Nonlinear Analysis of Stability of Plane Shock Waves in Media with Arbitrary Thermodynamic Properties

Abstract

S. D’yakov was the first who considered the problem of the shock wave (SW) stability in media with arbitrary thermodynamic properties [1]. He developed the linear theory of the plane SW stability on a basis of the normalmode analysis and obtained simple quantitative criteria for different types of the shock behavior: -1 1+2M (instability). The notations are the same as in [1], correct expression for the lower boundary of the neutral stability region L0 was given by V. Kontorovich in [2]. The same results were obtained in works [3-5] differing from [1] in the features of the linearized formulation and/or used mathematical technique. More recently (see, e.g., review [6]), it was found that the Hugoniot fragments meeting to the instability conditions lie within the region of the SW ambiguous representation. The solution choice in such regions is of great interest, the more so the feasibility of the unstable SW with an unlimited growth of the amplitude of the front perturbations raises justified doubts [6]. It is also problematic the spontaneous sound emission by the front of the neutrally stable SW predicted by the linear theory. The attempts of the theoretical analysis of the neutrally stable SW behavior in certain nonlinear formulations [6-7] gave sufficiently presumable and discrepant results. Besides, the neutral stable or unstable SW (at least, in the form predicted by the linear theory) are not yet observed in experiments. It is clear that the problem considered goes beyond the linear theory and must be solved in the framework of as far as possible complete nonlinear formulation. In the paper presented we briefly review results of our systematic nonlinear analysis of the plane SW stability problem. Due to space restrictions some details of problem formulations and solutions we are obliged to omit, but they may be found in our works [8-10] The emphasis is placed on the origin of the cellular shock front structure arising if the SW has ambiguous representation caused by the fulfilment of the instability condition L > 1+2M (this phenomenon has been first found in [10]).