Solutions of the imploding shock problem in a medium with varying density

TL;DR

This paper analyzes self-similar solutions of imploding shock waves in gases with power-law density profiles, exploring their behavior across different geometries and parameters, and demonstrating their use in validating hydrodynamic codes.

Contribution

It provides detailed analysis of the similarity exponent for imploding shocks in varying density media and shows how these solutions can be used for code validation.

Findings

Derived similarity exponents for cylindrical and spherical geometries.

Validated hydrodynamic codes using analytic solutions.

Explored effects of adiabatic index and density exponent on shock behavior.

Abstract

We consider the solutions of the Guderley problem, consisting of an imploding strong shock wave in an ideal gas with a power law initial density profile. The self-similar solutions, and specifically the similarity exponent which determines the behavior of the accelerating shock, are studied in detail, for cylindrical and spherical symmetries and for a wide range of the adiabatic index and the spatial density exponent. We then demonstrate how the analytic solutions can be reproduced in Lagrangian hydrodynamic codes, thus demonstrating their usefulness as a code validation and verification test problem.