Reconsideration of the So-Called von Neumann Paradox in the Reflection of a Shock Wave over a Wedge

Abstract

Numerous experimental investigations on the reflection of plane shock waves over straight wedges indicated that there is a domain, frequently referred to as the weak shock wave domain, inside which the resulted wave configurations resemble the wave configuration of a Mach reflection although the classical three-shock theory does not provide an analytical solution. This paradox is known in the literature as the von Neumann paradox. While numerically investigating this paradox Colella & Henderson [1] suggested that the observed reflections were not Mach reflections but another reflection, in which the reflected wave at the triple point was not a shock wave but a compression wave. They termed them it von Neumann reflection. Consequently, based on their study there was no paradox since the three-shock theory never aimed at predicting this wave configuration. Vasilev & Kraiko [2] who numerically investigated the same phenomenon a decade later concluded that the wave configuration, inside the questionable domain, includes in addition to the three shock waves a very tiny Prandtl-Meyer expansion fan centered at the triple point. This wave configuration, which was first predicted by Guderley [3], was recently observed experimentally by Skews & Ashworth [4] who named it Guderley reflection. The entire phenomenon was re-investigated by us analytically. It has been found that there are in fact three different reflection configurations inside the weak reflection domain: • A von Neumann reflection – vNR, • A yet not named reflection – ?R, • A Guderley reflection – GR. The transition boundaries between MR, vNR, ?R and GR and their domains have been determined analytically. The reported study presents for the first time a full solution of the weak shock wave domain, which has been puzzling the scientific community for a few decades. Although the present study has been conducted in a perfect gas, it is believed that the reported various wave configurations, namely, vNR, ?R and GR, exist also in the reflection of shock waves in condensed matter.