Chapter 2 Shock reflection in gas dynamics

Abstract

Abstract This paper is about multi-dimensional shocks and their interactions. The latter take place either between two shocks or between a shock and a boundary. Our ultimate goal is the analysis of the reflection of a shock wave along a ramp, and then at a wedge. Various models may be considered, from the full Euler equations of a compressible fluid, to the Unsteady Transonic Small Disturbance (UTSD) equation. The reflection at a wedge displays a self-similar pattern that may be viewed as a two-dimensional Riemann problem. Most of mathematical problems remain open. Regular Reflection is the simplest situation and is well-understood along an infinite ramp. More complicated reflections occur when the strength of the incident shock increases and/or the angle between the material boundary and the shock front becomes large. This is the realm of Mach Reflection. Mach Reflection involves a so-called triple shock pattern, where typically the reflection of the incident shock detaches from the boundary, and a secondary shock, the Mach stem, ties the interaction point to the wall. The triple shock pattern is pure if it is made only of the incident, reflected and secondary shocks, but of no other wave. As predicted by J. von Neumann, pure triple shock structures are impossible. A common belief was that this impossibility is of thermodynamical nature. We prove here that the obstruction is of kinematical nature, thus is independent either of an equation of state or of an admissibility condition. This holds true for all situations: Euler models, irrotational flows and UTSD, the latter case having been known for a decade. Because the Regular Reflection problem along a wedge gathers several major technical difficulties (a free boundary, a domain singularity, a solution singularity, a mixed-type system of PDEs, a type degeneracy across the sonic line), its solvability is still far from our knowledge, except in the simplest context of potential flows with small incidence, a problem solved recently by G.-Q. Chen and M. Feldman. Good though partial results have been obtained by S. Canic et al. for the UTSD model and by Y. Zheng for the Euler system. As far as the Euler equations are concerned, we improve and derive with higher mathematical rigour our pointwise estimates of 1994. Our improvements concern most of the estimates: • We give a now rigorous proof of the minimum principle for the pressure, whenever the flow is piecewise smooth, • Our new bound of the size of the subsonic domain applies now to data of arbitrary strength and incidence, • This together with the observation that the entropy increases, yields much better pointwise estimates of field variables, • We prove that there must exist a vortical singularity, at least in the barotropic case: the vorticity of the flow may not be square integrable, • Last but not least, we give a rigorous justification that the flow is uniform between the ramp, the pseudo-sonic line and the reflected shock, the latter being straight.