Two-stage algorithm for the simulation of thin shocks in multirelaxing fluids

Abstract

An efficient two-stage algorithm is presented for simulating a weak shock in an arbitrary waveform propagating in a fluid with multiple relaxation mechanisms. The first stage of the algorithm is based on an evolution equation expressed in intrinsic coordinates for nonlinear propagation in a relaxing fluid [Hammerton and Crighton, JFM (1993)], and it includes effects of spherical spreading. Simulation using intrinsic coordinates allows the pressure waveform to become multivalued, avoiding the need to discretize thin shocks. At selected distances, a shock is inserted into the multivalued pressure waveform according to the equal-area rule, rendering the waveform single-valued. Waveforms calculated in this way agree with corresponding time-domain solutions of a Burgers equation augmented to include relaxation, while simulation with intrinsic coordinates requires orders of magnitude less computation time. An analytical solution for shock evolution [Crighton and Scott, Philos. Trans. R. Soc. A (1979)] is used to estimate shock thickness as a function of propagation distance for determining the appropriate distance at which the second stage, the time-domain solution of the augmented Burgers equation, can be used to continue propagation into the far field. [WAW is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]