Reconsideration of the So-Called von Neumann Paradox in the Reflection of a Shock Wave over a Wedge
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  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  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 , was recently observed experimentally by Skews & Ashworth  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.
S. Itoh and K. Hokamoto
E. I. Vasilev et al., "Reconsideration of the So-Called von Neumann Paradox in the Reflection of a Shock Wave over a Wedge", Materials Science Forum, Vol. 566, pp. 1-8, 2008