Model for shock interaction with sharp area reduction

Abstract

The interaction of a planar shock wave, propagating in a conduit of constant cross-section with a sharp area change, is modelled analytically. The area ratio between the wide (incident) conduit and the narrow (continuing) conduit is taken to be very large. Previously published studies for shock interaction with an area change assumed a short smooth connecting nozzle, where, most often, steady flow evolves. It is assumed that the flow converges through the sharp area transition via a smoothly-formed quasi-one-dimensional streamtube, reaching sonic (choked) or subsonic (unchoked) velocity. In the choked case, a centred rarefaction wave is required to match pressures between the sonic flow and the transmitted shock state. The model leads to a pair of equations for velocity and pressure at a contact discontinuity, analogous to the wave interaction curves that resolve a Riemann problem. It is found that the overpressure amplification ratio of the transmitted shock is in the range of 1.5—2.0 for very strong to very weak incident shocks, respectively. A good agreement is obtained between the model predictions and two-dimensional generalized Riemann problem (GRP) simulations. The model is also applicable to a configuration where a smooth (and short) nozzle replaces the sharp area transition. The numerical simulations then produce an amplification factor A t, somewhat higher than the corresponding sharp area value. This is taken to mean that the virtual streamtube formed in the sharp area reduction case has a throat (sonic) cross-section area lower than the actual area of the narrow tube.