The propagation of weak shocks in non-uniform flows

A new theory of the propagation of weak shocks into non-uniform, two-dimensional flows is introduced. The theory is based on a description of shock propagation in terms of a manifold equation together with compatibility conditions for shock strength and its normal derivatives behind the shock. This approach was developed by Ravindran & Prasad (1993) for shocks of arbitrary strength propagating into a medium at rest and is extended here to non-uniform media and restricted to moderately weak shocks. The theory is tested against known analytical solutions for cylindrical and plane shocks, and against a full direct numerical simulation (DNS) of a shock propagating into a sinusoidal shear flow. The test against DNS shows that the present theory accurately predicts the evolution of a moderately weak shock front, including the formation of shock-shocks due to shock focusing. The theory is then applied to the focusing of an initially parabolic shock, and to the propagation of an initially straight shock into a variety of simple flows (sinusoidal shear, vortex array, point-vortex array) exhibiting some fundamental properties of turbulent flows. A number of relations are deduced for the variation of shock quantities with initial shock strength M S0 and the Mach number of the flow ahead of the shock M U (e.g. separation of shock-shocks and maximum shock strength at a focus). It is found that shock-shocks are likely to form in turbulent flows with M t / M 1N > 0.14–0.25, where M t is the average Mach number of the turbulence and M 1N is the Mach number of the shock in a flow at rest. The shock moves up to 1.5% faster in a two-dimensional vortex array than in uniform flow.