Reflection of multiple incident shock waves and shock interaction train

The reflection of multiple incident shock waves that converge to a single point on the reflecting surface is studied in this paper. The number of the incident shocks, denoted $K$ , is arbitrary. The interaction between the reflected shock of one incident shock and the other incident shocks may produce various possible configurations, such as type-I, type-II and type-IV shock interferences. The number of possible reflection configurations is shown to be an exponential function of ( $K-1$ ) with base 2. The possibility of pre-, middle- and post-Mach reflections, which means Mach reflection occurs for the first, middle and last incident shock, is revealed through numerical simulation for $K=3$ . For the particular case where the incident shocks are produced by equal variation of wedge surface deflection, the conventional von Neumann condition and detachment condition for the $k\mathrm{th}$ incident shock to have Mach reflection are derived. It is shown that the von Neumann condition for regular reflection is lowered and the detachment condition for Mach reflection is elevated as $k$ increases. The shock reflection patterns for $ K=1,2,\ldots ,10$ are obtained by numerical simulations. We observe a shock interaction train structure, where we have pre-Mach reflection followed by ( $K-1$ ) type-I or type-II shock interferences. We also observe that the Mach stem height decreases with $K$ well above the von Neumann condition and becomes non-monotonic near the von Neumann condition.