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

Eugene I. Vasilev, Tov Elperin, Gabi Ben-Dor

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30 Scopus citations


The reflection of weak shock waves has been reconsidered analytically using shock polars. Based on the boundary condition across the slipstream, the solutions of the three-shock theory (3 ST) were classified as "standard-3 ST solutions" and "nonstandard-3 ST solutions." It was shown that there are two situations in the nonstandard case: A situation whereby the 3 ST provides solutions of which at least one is physical and a situation when the 3 ST provides a solution which is not physical, and hence a reflection having a three-shock confluence is not possible. In addition, it is shown that there are initial conditions for which the 3 ST does not provide any solution. In these situations, a four-wave theory, which is also presented in this study, replaces the 3 ST. It is shown that four different wave configurations can exist in the weak shock wave reflection domain, a Mach reflection, a von Neumann reflection, a ?R (this reflection is not named yet!), and a modified Guderley reflection (GR). Recall that the wave configuration that was hypothesized by Guderley ["Considerations of the structure of mixed subsonic-supersonic flow patterns," Air Materiel Command Technical Report No. F-TR-2168-ND, ATI No. 22780, GS-AAF-Wright Field No. 39, U.S. Wright-Patterson Air Force Base, Dayton, OH (October 1947); Theorie Schallnaher Strömungen (Springer-Verlag, Berlin, 1957)] and later termed Guderley reflection did not include a slipstream (see Fig. 7). Our numerical study revealed that the wave structure proposed by Guderley must be complemented by a slipstream (see Fig. 4) in order to be relevant for explaining the von Neumann paradox. Hereafter, for simplicity, this modified GR wave configuration will be also termed Guderley reflection. The domains and transition boundaries between these four types of reflection are elucidated.

Original languageEnglish
Article number046101
JournalPhysics of Fluids
Issue number4
StatePublished - 1 Jan 2008
EventSymposium of the International Union of Theoretical and Applied Mechanics on Recent Advances in Multiphase Flows: Numerical and Experimental, IUTAM - Istanbul, Turkey
Duration: 11 Jun 200714 Jun 2007

ASJC Scopus subject areas

  • Computational Mechanics
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Fluid Flow and Transfer Processes


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