The Triple Point Paradox for the Nonlinear Wave System

Abstract

We present numerical solutions of a two-dimensional Riemann problem for the nonlinear wave system which is used to describe the Mach reflection of weak shock waves. Robust low order as well as high resolution finite volume schemes are employed to solve this equation formulated in self-similar variables. These, together with extreme local grid refinement, are used to resolve the solution in the neighborhood of an apparent but mathematically inadmissible shock triple point. Rather than observing three shocks meeting in a single standard triple point, we are able to detect a primary triple point containing an additional wave, a centered expansion fan, together with a sequence of secondary triple points and tiny supersonic patches embedded within the subsonic region directly behind the Mach stem. An expansion fan originates at each triple point. It is our opinion that the structure observed here resolves the von Neumann triple point paradox for the nonlinear wave system. These solutions closely resemble the solutions obtained in [A. M. Tesdall and J. K. Hunter, SIAM J. Appl. Math., 63 (2002), pp. 42-61] for the unsteady transonic small disturbance equation.

Analytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge

“…Recently, Tesdall et al 25 performed similar calculations using the model of the nonlinear wave system. They achieved a high numerical resolution that was sufficient for the analysis of the fine structure of the flow field near the triple point under the conditions of the von Neumann paradox.…”

Further Results on Guderley Mach Reflection and the Triple Point Paradox

“…It was conjectured in [11] that this sequence is infinite for an inviscid weak shock reflection. Since the detection of this multiple patch structure in [11], the same phenomenon has been found in solutions of the nonlinear wave system in [12], and, more recently, in the compressible Euler equations in [13].…”

“…However, there is a lack of agreement on the question of whether a shock terminates the supersonic patch (as occurs on a transonic airfoil), and on whether or not a sequence of such patches, each with a terminating shock, actually occurs. The multiple-patch structure has been found in solutions of the UTSD equations, the nonlinear wave system, and the full Euler equations, and it appears likely that a sequence of supersonic patches and triple points is a generic feature of some class of hyperbolic conservation laws, possibly characterized by "acoustic waves," as proposed in [12]. A natural question is whether the single-patch structure is also typical in solutions, possibly in a different range of parameter values.…”

Numerical solutions to the self-similar transonic two-dimensional nonlinear wave system

“…To simplify the notations, we let F = F − U and G = G− U, and the problem in conserved form (5) can be rewritten as [4,15] U +F +G +2U = 0.…”

“…These methods appear to capture the shock interaction of the nonlinear wave system accurately. Since our goal is to capture self-similar solutions, we rewrite the system using the self-similar variables [4] and implement the numerical schemes to the self-similar system, rather than solve the initial value problem for the unsteady nonlinear wave system in the original variables.…”