Revisiting the Strong Shock Problem: Converging and Diverging Shocks in Different Geometries

TL;DR

This paper provides a unified analysis of self-similar solutions for converging and diverging shocks across different geometries, revealing conditions for their validity and deriving analytical approximations for key parameters.

Contribution

It offers a comprehensive framework connecting various shock geometries and characterizes the self-similar solutions across all parameter regimes.

Findings

Only type II solutions are valid for converging shocks.

Some converging shocks do not produce reflected shocks.

Analytical approximations for the similarity exponent are derived.

Abstract

Self-similar solutions to converging (implosions) and diverging (explosions) shocks have been studied before, in planar, cylindrical or spherical symmetry. Here we offer a unified treatment of these apparently disconnected problems . We study the flow of an ideal gas with adiabatic index $\gamma$ with initial density $\rho\sim r^{-\omega}$, containing a strong shock wave. We characterize the self-similar solutions in the entirety of the parameter space $\gamma,\omega$, and draw the connections between the different geometries. We find that only type II self-similar solutions are valid in converging shocks, and that in some cases, a converging shock might not create a reflected shock after its convergence. Finally, we derive analytical approximations for the similarity exponent in the entirety of parameter space.