Propagation of curved shock fronts using shock ray theory and comparison with other theories

Abstract

Shock ray theory (SRT) has been found to be useful and computationally efficient in finding successive positions of a curved weak shock front. In this paper, we solve some piston problems and show that the shock ray theory with two compatibility conditions gives shock positions, which are very close to those obtained by solving the same problems by the numerical solution of Euler's equations (Euler solutions). Comparison of the results obtained by shock ray theory and geometrical shock dynamics (GSD) of Whitham (J. Fluid Mech. vol. 2, 1957, p. 146) with the Euler solution shows that the shock ray theory gives more accurate results for any piston motion. The aim of the work is not just this comparison, but also to investigate the role of the nonlinearity in accelerating the process of the evolution of a shock, produced by an explosion of a non-circular finite charge, into a circular shock front. We find that the nonlinear waves propagating on the shock front appreciably accelerate this process. We also discuss a situation, for shock Mach number very close to 1, when GSD and shock ray theory may fail to give any result.