Self-Similar Solutions for Converging Shocks and Collapsing Cavities

Abstract

A complete analysis is attempted of the self-similar solutions for the converging shock and collapsing cavity problems, in spherical and cylindrical geometry, for a perfect gas with arbitrary adiabatic exponent $\gamma > 1$. Emphasis is given to the rich variety of previously neglected nonanalytic soulutions, and to a full exploration of the relevant parameter space. Distinctions are made between what must be determined numerically and what can be derived algebraically. New solutions are described which contain additional converging shocks, arriving at the origin concurrently with the initial shock or free surface. Some of these new solutions are entirely analytic, except at the shocks themselves, and some are not; in some cases, only one secondary shock is possible, in other cases an arbitrary number. The physical significance of previously rejected partial solutions is discussed. The stability of solutions is discussed in a narrow (one-dimensional) sense. Finally, a study is urged of the asymptotic approach (or nonapproach) to self-similarity of direct numerical integrations of the original partial differential equations; it is argued that the evidence for approach to a unique self-similar solution is not convincing.