Non-self-similar behavior of the von Neumann reflection

Abstract

The purpose of the present paper is to consider the von Neumann reflection (vNR) which takes place for a small incident shock Mach number and a small wedge angle. A series of experiments has been performed with ordinary smooth straight wedges and step-like wedges that simulate the former. The reflection configuration over the step-like wedge has suggested unsteady characteristics of the vNR. Contrary to the established notion of shock reflection phenomena over straight wedges, while the triple point trajectory was approximately a straight line through the apex of the wedge, the vNR showed unsteadiness in the relation between angles of incidence and reflection (ωi,ωr), and thus the flow-field proved to be non-self-similar near the triple point. Based on the measured values, the incident angle for the reflected wave and the Mach number of the flow ahead of the reflected wave were estimated. These values show that the reflected wave is not a Mach wave, but it moves on the (ωi,ωr)-plane almost along a trivial solution. In particular, the flow Mach number ahead of the reflected wave approaches unity, which leads to analytical formulas for angles of incidence and reflection as functions of the incident shock Mach number only. The reflected wave degenerates to a normal Mach wave as the incident shock proceeds. In the case of the vNR, the von Neumann paradox takes place only at an early stage of reflection, and the paradox is resolved later, since the flow properties such as angles of incidence and reflection are given by the trivial solution which is a particular solution for the three-shock theory.