Analytical and Numerical Studies of the Non-uniformity induced Type II Asymmetric Cap Shock Mach Reflection in Over-expanded Supersonic Jets

Abstract

A combined analytical and numerical study is conducted to investigate the asymmetric cap-shock non-uniform Mach Reflection (csMR) phenomenon outside of an over-expanded supersonic jet. Prior analytical works have only considered the wedge-induced steady symmetric and asymmetric Mach reflection (MR) configurations, as well as symmetric MR in an open jet. However, there is another structure occurring in nozzle flow fields known as a non-uniformity induced cap-shock pattern. We derive a new analytical model to predict the wave structure of the asymmetric csMR in the absence of internal shocks by extending on a prior symmetric Mach reflection model. Different from the wedge flow case, flow non-uniformity is incorporated by assuming different upstream Mach numbers in both upper and lower domains, where the asymmetry is predicted through averaged flowfields and slipstream inclination angles. The numerical approach utilises an Euler solver for comparisons to the developed theory. It is found that the model adequately predicts the shock structure obtained from numerical simulations, and can be utilised for various sets of parameters to capture the overall direct Mach reflection o[DiMR+DiMR] configuration. The von Neumann criterion is also well captured by the new analytic model, along with the Mach stem profile and shock curvatures. Based on the analytical and numerical observations, a hypothesis is also made regarding the stability of MR structures within an over-expanded jet.