Analytical and numerical solutions of the shock tube problem in a channel with a pseudo-perforated wall

Abstract

An approximate analytical solution is obtained for the shock tube problem in a rectangular channel with an array of rectangular grooves in the lower wall. The analytical solution is based on the approximate quasi-1D shock adiabat for a shock wave that propagates in a channel with periodically located barriers. This problem is also studied numerically. It is found that the approximate analytical solution correctly predicts the propagation velocity of the leading discontinuity and the flow parameters at this discontinuity. The leading gas-dynamic discontinuity is followed by a relaxation zone in which there are long-wave and short-wave flow oscillations. These oscillations are caused by waves arising from the interaction of the flow behind the leading shock wave with the side and bottom walls of the grooves. An approximate analytical solution allows one to obtain the values of the flow parameters at the end of the relaxation zone. These values are in good agreement with the numerical calculation.