On one-dimensional planar and nonplanar shock waves in a relaxing gas

Abstract

The paper examines the evolutionary behavior of shock waves of arbitrary strength propagating through a relaxing gas in a duct with spatially varying cross section. An infinite system of transport equations, governing the strength of a shock wave and the induced discontinuities behind it, are derived in order to study the kinematics of the shock front. The infinite system of transport equations, when subjected to a truncation approximation, provides an efficient system of only finite number of ordinary differential equations describing the shock propagation problem. The analysis, which accounts for the dynamical coupling between the shock fronts and the flow behind them, describes correctly the nonlinear steepening effects of the flow behind the shocks. Effects of relaxation on the evolutionary behavior of shocks are discussed. The first-order truncation approximation accurately describes the decay behavior of weak shocks; the usual decay laws for weak shocks in a nonrelaxing gas are exactly recovered. The results concerning shocks of arbitrary strength are compared with the characteristic rule. In the limit of vanishing shock strength, the transport equation for the first-order induced discontinuity leads to an exact description of an acceleration wave. In the strong shock limit, the second-order truncation criterion leads to a propagation law for imploding shocks which is in agreement (within 5% error) with the Guderley’s exact similarity solution.