Self-Similar Solutions for Weak Shock Reflection

Abstract

We present numerical solutions of a two-dimensional Riemann problem for the unsteady transonic small disturbance equations that provides an asymptotic description of the Mach reflection of weak shock waves. We develop a new numerical scheme to solve the equations in selfsimilar coordinates and use local grid refinement to resolve the solution in the reflection region. The solutions contain a remarkably complex structure: there is a sequence of triple points and tiny supersonic patches immediately behind the leading triple point that is formed by the reflection of weak shocks and expansion waves between the sonic line and the Mach shock. An expansion fan originates at each triple point, thus resolving the von Neumann paradox of weak shock reflection. These numerical solutions raise the question of whether there is an infinite sequence of triple points in an inviscid weak shock Mach reflection.

Guderley reflection for higher Mach numbers in a standard shock tube

“…When analysing the contours in Fig. 2b, indication is given of the possible existence of at least two supersonic patches along the Mach stem further satisfying Tesdall and Hunter's [9] observations, and the formation of a GMR.…”

“…This resulted in the region behind the triple point being better resolved and the supersonic patch to be identified. For incident shock Mach numbers ranging from 1.05 to 1.1 on a 10 • ramp, their results clearly detected the fourth wave, namely the expansion fan, and an indication of a shock terminating the expansion wave as predicted in [9]. The authors advised that the observed four-wave structure be referred to as Guderley reflection.…”

“…They contemplated that the supersonic patch could be terminated by a shock, and that there could be a series of such patches along the Mach stem. With the use of a highly refined mesh in the vicinity of the triple point, Tesdall and Hunter [9] resolved a remarkably complex flow structure subsequently named Guderley Mach reflection (GMR) [8], shown in Fig. 1c.…”

“…1c. The flow pattern consists of a sequence of triple points and tiny decreasing supersonic patches along the Mach stem, formed by the reflection of weak shocks and expansion fans between the sonic line and the Mach stem [9].…”

Two-dimensional Riemann problems for the compressible Euler system

“…This brings up Euler system's von Neumann triple point paradox, for which John Hunter and collaborators have produced numerical evidence of Guderley reflection [10,35,36]. See Skews and Ashworth [30] for physical experimental evidence.…”