Abstract

A numerical study was made of imploding shock waves. The objective of the present paper is to clarify the propagation and instability of the converging cylindrical and elliptical shock waves. In order to make the simulation, a second order explicit MacCormack type finite difference method was employed to solve the unsteady compressible Euler equations. At first, a one-dimensional simulation was preformed for converging cylindrical shock waves. The results were found to be in good agreement with the Chester, Chisnell and whitham relation. Also, the numerical results were in good agreement with other published 1-D numerical simulations (e.g. method of characteristic). A two-dimensional simulation was also developed for initially perturbed implosions. The governing fluid dynamics equations have been solved in polar coordinates. The 2-D scheme is an extension to the 1-D case, using the operator splitting, where the multi-dimensional solution is evolved from a factored sequence of one dimensional operators. From the two-dimensional study of converging cylindrical shocks, it has been concluded that although the shock has been perturbed initially in the azimuthal direction, it has a tendency to keep its continuous shock front curvature until it gets close to the centre. Eventually, at the centre a break down in the shock front is inevitable. Finally, for the elliptical shock the perturbation parameter ($\xi$) was found to vary with shock travel in a complex manner that cannot be represented by a simple power law. This finding is in good agreement with recent experimental results.