The vorticity jump across a shock in a non-uniform flow

Abstract

The vorticity jump across an unsteady curved shock propagating into a two-dimensional non-uniform flow is considered in detail. The exact general expression for the vorticity jump across a shock is derived from the gasdynamics equations. This general expression is then simplified by writing it entirely in terms of the Mach number of the shock M_S and the local Mach number of the flow ahead of the shock M_U.

The vorticity jump is very large at places where the curvature of the shock is very large, even in the case of weak shocks. Vortex sheets form behind shock-shocks (associated with kinks in the shock front).

The ratio of vorticity production by shock curvature to vorticity production by baroclinic effects is O(½(γ−1)M²_U), where γ is ratio of specific heats, which is very small if the flow ahead of the shock is only weakly compressible. If, however, the tangential gradient along the shock of M²_U is large then baroclinic production is significant; this is the case in turbulent flows with large gradients of turbulent kinetic energy ½M²_U. The vorticity jump across a weak shock decreases in proportion to shock intensity if the flow ahead of the shock is rotational, rather than in proportion to the cube of shock intensity as is often assumed, and thus is not negligible. It is also shown that vorticity may be generated across a straight shock even if the flow ahead of the shock is irrotational. The importance of the contribution to the vorticity jump by non-uniformities in the flow ahead of the shock has not been recognized in the past.

Examples are given of the vorticity jump across strong and weak shocks in a variety of flows exhibiting some properties of turbulence.