Evolution of Weak Shocks in One Dimensional Planar and Non-planar Gasdynamic Flows

TL;DR

This paper derives asymptotic decay laws for weak shock waves and their associated discontinuities in one-dimensional gasdynamic flows, using multiple approximation methods and analyzing their behavior over time.

Contribution

It introduces a unified analysis of shock decay laws using four different approximation methods, including singular surface theory and nonlinear geometrical optics.

Findings

Decay of first order discontinuity is faster than shock decay.

Precursor disturbances evolve like acceleration waves.

Asymptotic laws are consistent across multiple approximation theories.

Abstract

Asymptotic decay laws for planar and nonplanar shock waves and the first order associated discontinuities that catch up with the shock from behind are obtained using four different approximation methods. The singular surface theory is used to derive a pair of transport equations for the shock strength and the associated first order discontinuity, which represents the effect of precursor disturbances that overtake the shock from behind. The asymptotic behaviour of both the discontinuities is completely analysed. It is noticed that the decay of a first order discontinuity is much faster than the decay of the shock; indeed, if the amplitude of the accompanying discontinuity is small then the shock decays faster as compared to the case when the amplitude of the first order discontinuity is finite (not necessarily small). It is shown that for a weak shock, the precursor disturbance evolves like an acceleration wave at the leading order. We show that the asymptotic decay laws for weak shocks and the accompanying first order discontinuity are exactly the ones obtained by using the theory of nonlinear geometrical optics, the theory of simple waves using Riemann invariants, and the theory of relatively undistorted waves. It follows that the relatively undistorted wave approximation is a consequence of the simple wave formalism using Riemann invariants.