Solutions of the converging and diverging shock problem in a medium with varying density

TL;DR

This paper analyzes self-similar solutions of converging and diverging shock waves in a medium with varying density, comparing analytical results with hydrodynamic simulations for validation.

Contribution

It provides detailed analysis of the Guderley problem with variable density profiles and validates analytical solutions against numerical simulations.

Findings

Analytical solutions match well with hydrodynamic simulations.

Reflected shock coefficient is characterized for various parameters.

Self-similar solutions serve as effective code verification tests.

Abstract

We consider the solutions of the Guderley problem, consisting of a converging and diverging hydrodynamic shock wave in an ideal gas with a power law initial density profile. The self-similar solutions, and specifically the reflected shock coefficient, which determines the path of the reflected shock, are studied in detail, for cylindrical and spherical symmetries and for a wide range of values of the adiabatic index and the spatial density exponent. Finally, we perform a comprehensive comparison between the analytic solutions and Lagrangian hydrodynamic simulations, by setting proper initial and boundary conditions. A very good agreement between the analytical solutions and the numerical simulations is obtained. This demonstrates the usefulness of the analytic solutions as a code verification test problem.