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

Abstract

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 ͑3ST͒ were classified as "standard-3ST solutions" and "nonstandard-3ST solutions." It was shown that there are two situations in the nonstandard case: A situation whereby the 3ST provides solutions of which at least one is physical and a situation when the 3ST 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 3ST does not provide any solution. In these situations, a four-wave theory, which is also presented in this study, replaces the 3ST. 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 ͑GR͒. Recall that the wave configuration that was hypothesized by Guderley ͓"Considerations of the structure of mixed subsonic-supersonic flow patterns," Air Materiel Command

Patterns of steady axisymmetric supersonic compression flows with a Mach disk

“…The secondary shock polar 5 corresponding to δ i = 10 • intersects the initial shock polar 1 at a point with δ t = δ i + δ r > δ i on the right of the point corresponding to δ t = δ i . It is also known that the interaction of shocks in the latter case has to correspond to an irregular triple-shock configuration of the same type that occurs under the conditions of the von Neumann paradox (see [12,13]). The case δ t = δ i is transitional between the Mach and von Neumann reflections.…”

“…There are a considerable number of publications on experimental and theoretical studies of the von Neumann paradox in the 2D flows; a detailed review is presented in [13]. Numerical simulations were made to refine theoretical views of this subject in the framework of the ideal gas dynamic model.…”

Shock focusing in a planar convergent geometry: experiment and simulation

“…Either the leading-reflected wave is significantly weaker than that or there is a more complicated structure at the triple point that significantly reduces the pressure rise; notably, for the measured trajectory angle the triple-point configuration would be, according to Vasilev et al (2008), a Guderley reflection. The N2.5 case results are shown in table 5.…”

“…Additionally, Vasilev, Elperin & Ben-Dor (2008) have recently provided a detailed analysis of three-and four-wave solutions at the triple point, given the triple-point trajectory and the incident shock Mach number. The fourwave solution structures typically consist of a leading reflected shock followed by an attached expansion fan; this might be expected to result in a lower pressure rise (and a Mach stem/an incident shock kink angle) than a simple reflected shock of the same Mach number.…”

Non-uniqueness of solutions in asymptotically self-similar shock reflections

“…1. The Mach reflection type can be further subdivided into different regimes of reflections which have recently been classified by Semenov et al [22,23] and Vasilev et al [24]. The isentropic exponent γ, the geometry (angle of the wall), and the Mach number of the incident shock M i = D/c 0 uniquely define the problem; the type of reflection is subject to change with any of those variables.…”