Effects of Viscosity on Steady Reflection of Weak Shock Waves

Abstract

The viscosity effects on weak shock wave reflection are investigated with the Navier– Stokes and DSMC flow solvers. It is shown that the viscosity plays a crucial role in the vicinity of three-shock intersection. Instead of a singular triple point, in viscous flow there is a smooth shock transition zone, where one-dimensional shock jump relations cannot be applied. At the parameters corresponding to the von Neumann reflection, when no inviscid three-shock solution exists, the three-shock configuration is also observed. The existence of a viscous zone in the region of shock-wave interaction allows a continuous transition from the parameters behind the Mach stem to the parameters behind the reflected shock, which is impossible in the three-shock theory. I. Background and motivation Many interesting phenomena that occur in oblique shock wave reflection have been discovered in the past. The main feature herein is the existence of two possible configurations of shocks, regular and irregular. Regular reflection consists of the incident shock wave and the reflected shock wave with supersonic flow behind the reflected shock. Irregular reflection, which is in most cases called Mach reflection due to E. Mach who first discovered this phenomenon, is a complex shock wave pattern that combines the incident and reflected shock waves and the Mach stem. A contact discontinuity (slip surface) emanates from the triple point due to inequality of entropy in the flow passing through the incident and reflected shocks and the flow passing through the Mach stem. Classical theoretical methods such as shock polar analysis and the three-shock theory based on Rankine–Hugoniot jump conditions across the oblique shocks were developed by J. von Neumann to describe the shock wave configurations at various flow parameters and to predict transitions between different types of shock wave interaction. These theoretical methods predict well most of the features of shock wave interaction. Steady shock wave reflection is very important in aerodynamics and has been extensively studied in recent years with an emphasis on strong shock waves (for flow Mach numbers higher than 2.2 in air). For supersonic civil aviation, however, the lower Mach number range is of greatest interest. Regular and irregular interactions of different types are inherent in such critical phenomena as off-design inlet flows, inlet starting, and flow stalling. Interactions and reflections of weak shock waves are typical for supersonic inlet flows at low and moderate Mach numbers (M=1-2). There are many problems of irregular shock reflections in steady flows which are not yet investigated. One of the most exciting phenomena that occurs in irregular reflection of weak shock waves is a shock wave reflection in the range of flow parameters where the von Neumann’s three-shock theory does not produce any solution whereas the experiments reveal 1 a three-shock structure similar to the Mach reflection pattern. This inconsistency is referred to as the von Neumann paradox, and the observed reflection pattern is called the von Neumann reflection (vNR). Inviscid numerical simulations