The Mach Reflection of Weak Shocks

We present numerical solutions of weak shock Mach reflections that contain a remarkably complex sequence of supersonic patches, triple points, and expansion fans immediately behind the leading triple point. This structure resolves the von Neumann triple point paradox of weak shock Mach reflection. During the second world war, von Neumann carried out an extensive study of shock reflection [5]. He suggested a number of criteria for the transition from regular to Mach reflection, and compared his theoretical results with observations. There was generally excellent agreement for strong shocks, but for weak shocks serious discrepancies were found. In particular, a pattern closely resembling simple Mach reflection is observed for weak shocks, but no standard triple point configuration is compatible with the jump relations across shocks and contact discontinuities. This discrepancy was called the ‘von Neumann triple point paradox’ by Birkhoff [1]. Many different resolutions of this paradox have been suggested over the years [3]. In this paper, we present numerical solutions of initial and boundary value problems for the unsteady and steady transonic small disturbance equations that provide an asymptotic description of weak shock Mach reflection. These solutions contain a sequence of supersonic patches, shocks, expansion fans, and triple points in a tiny region behind the leading triple point. We conjecture that this sequence is infinite for an inviscid weak shock Mach reflection. At each triple point, there is an additional expansion fan, thus resolving the apparent conflict with von Neumann’s theoretical arguments. Furthermore, an infinite sequence of shrinking supersonic patches resolves theoretical difficulties connected with the transition from supersonic to subsonic flow at the rear of the supersonic region. The existence of a supersonic patch and an expansion fan at the triple point of a weak shock Mach reflection was proposed by Guderley [2], although he did not give evidence that this is what actually occurs, nor did he suggest that there is, in fact, a sequence of supersonic patches and triple points. Previous numerical solutions of weak shock Mach reflections with supersonic patches behind the triple point were obtained in [4, 7, 8], but none of these solutions were sufficiently resolved to show the true structure of the solution. An asymptotic shock reflection problem for the unsteady transonic small disturbance equations, which have the normalized form