Numerical Study of Pseudospectral Methods in Shock Wave Applications

In this paper we use shock capturing spectral methods to simulate compressible flows in the presence of shock waves. Three applications are considered. The first is an interaction of a one-dimensional shock with an entropy wave. The second case deals with interactions between shock wave and an entropy wave in two space dimensions and the third case is two-dimensional shock-vortex interactions. The first two applications are found in the study of turbulence in high speed flow. The last application is a key element in understanding the acoustic dynamics in aeroacoustic and in design of supersonic jet in aerodynamics. The purpose of this study is to show the feasibility of simulating shocked flow with spectral methods. The numerical methods involved are the Chebyshev and Fourier collocation methods. The Euler equations of gas dynamics is discretized by pseudospectral (collocation) methods in space and a nonlinearly stable third-order Runge-Kutta methods in time. The fluxes are evaluated pointwise directly and not by the cell-averaging technique. A significant reduction in CPU time and storage usage are achieved by incorporating several well-established numerical techniques, such as grid transformation and filtering, into the spectral algorithm. The results of this study indicate that spectral method is well suited not only for smooth problems but also for those with discontinuity.