Discrete Self Similarity in Filled Type I Strong Explosions

TL;DR

This paper introduces new solutions for strong explosion problems with non power law density profiles, revealing discrete self similarity in perturbations and verifying these solutions through numerical hydrodynamics.

Contribution

It extends previous work on type I solutions by demonstrating discrete self similarity in perturbations for non power law density profiles and verifying with numerical simulations.

Findings

Discrete self similarity in perturbations identified

Solutions verified through numerical hydrodynamic simulations

Clarifies boundary conditions for type I solutions

Abstract

We present new solutions to the strong explosion problem in a non power law density profi{}le. The unperturbed self similar solutions developed by Sedov, Taylor and Von Neumann describe strong Newtonian shocks propagating into a cold gas with a density profile falling off as $r^{-\omega}$, where $\omega\le\frac{7-\gamma}{\gamma+1}$ (filled type I solutions), and $\gamma$ is the adiabatic index of the gas. The perturbations we consider are spherically symmetric and log periodic with respect to the radius. While the unperturbed solutions are continuously self similar, the log periodicity of the density perturbations leads to a discrete self similarity of the perturbations, i.e., the solution repeats itself up to a scaling at discrete time intervals. We discuss these solutions and verify them against numerical integrations of the time dependent hydrodynamic equations. This is an extension of a previous investigation on type II solutions and helps clarifying boundary conditions for perturbations to type I self similar solutions.