On the von Neumann paradox of weak Mach reflection

Abstract

Recent experimental and numerical studies of weak Mach reflections are examined. It is shown that the fundamental reason for the von Neumann paradox is that his theory of Mach reflection is based on the assumption that the flow downstream of the reflected wave and the Mach shock near the wave triple point is uniform. The assumption is shown to be valid for strong Mach reflection which agrees with experiment, but invalid for weak Mach reflection whicb does not agree with experiment. It is also shown that viscous effects are dominant when the incident shock is within about 100 mean free path lengths of the corner, but not otherwise. The analytical theory of the entire subsonic region supports these conclusions.* Results based on conclusions made at the" Workshop on weak Mach reflection", March 31 ~April 1, '88 at Tokyo Denki University.

An elliptic problem arising from the unsteady transonic small disturbance equation

“….." bivariate form B~satisiies'the "bounds ' --' '~~~~~-In order to show t@ we construct sub-and super-solutions. We shalI show that for every J < c and M > I/e, the functions J@ -g and M@ -g are sub-and super-solutions respectively to (23), where~satisfies (15). This is equivalent to showing that Jand M@ are sub-and super-solutions to…”

“…In particular, we may now suppose f is given by (15), then the solution of (21), which we have found, is the solution we want, save for the possibility that Bg does not coincide with B for our weak solution v. We rule this out in the next section.…”

“…The first integral in (33) can be estimated using the fact that ii is bounded from below by e & Therefore, there exists a Cl >0 such that where -the bivariate forna B is defined by (16) and f by (15 The theory in this paper also maybe extended to unbounded domains 0:…”

“…We handle this by supposing~to be an arbitrary function, in (21), (not related to g by (15)), and then letting~be the function defined by (15) to obtain the existence of a weak solution for problem (14).…”

“….. gas dynamics equations iq the form of a particularly simple nonlinearity, and may give a description of the transition between regular and Mach reflection for weak" shocks. This phenomenon has been extensively studied numerically and experimentally (see, for example, [5], [15]); however, there are only a few related aimIytical results. We mention Morawetz's study of a full-pot entialequation model, [10], which is in the same spirit as the TSD model.…”