D'yakov-Kontorovich instability of expanding shock waves

TL;DR

This paper investigates the D'yakov-Kontorovich instability in expanding shock waves, revealing a power-law growth of shock-front perturbations and identifying conditions under which the instability occurs.

Contribution

It demonstrates that expanding shock waves exhibit a true instability with power-law growth, contrasting with the classic oscillatory instability in steady shocks.

Findings

Instability manifests as power-law growth of shock-front perturbations.

Instability occurs for high angular mode numbers.

Shock divergence acts as a stabilizing factor.

Abstract

In the range of $h_c<h<1+2 M_2$, where $h$ is the D'yakov-Kontorovich parameter, $h_c$ is its critical value corresponding to the onset of the spontaneous acoustic emission, and $M_2$ is the downstream Mach number, the classic analysis predicts a special form of the instability of isolated steady planar shock waves: non-decaying oscillations of shock-front ripples. For spherically and cylindrically expanding steady shock waves, we demonstrate instead an instability in a literal sense, a power-law growth of shock-front perturbations with time. As the parameter $h$ increases from $h_c$ to $1+2 M_2$, the instability power index grows from zero to infinity. Shock divergence is a stabilizing factor, and instability is found for high angular mode numbers.