^{1}, Tov Elperin

^{2}and Gabi Ben-Dor

^{2}

### 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 (3ST) 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 (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 Year: 1947); Theorie Schallnaher Strömungen (Springer-Verlag, Berlin, Year: 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.

This study was conducted under the auspices of the Dr. Morton and Toby Mower Chair of Shock Wave Studies.

INTRODUCTION

THEORETICAL BACKGROUND

The three-shock theory

The four-wave theory

SHOCK-POLAR PRESENTATION

Shock-polar presentation of the flow field in the vicinity of the triple point

SUMMARY

### Key Topics

- Shock waves
- 89.0
- Mach numbers
- 24.0
- Subsonic flows
- 14.0
- Supersonic flows
- 11.0
- Boundary value problems
- 9.0

## Figures

Definition of parameters across an oblique shock wave.

Definition of parameters across an oblique shock wave.

Schematic illustration of the wave configuration of a MR.

Schematic illustration of the wave configuration of a MR.

Schematic illustration of the wave configurations of a vNR, the newly presented reflection ?R, and the GR. Gray denotes subsonic flow.

Schematic illustration of the wave configurations of a vNR, the newly presented reflection ?R, and the GR. Gray denotes subsonic flow.

Schematic illustration of a four-wave configuration and definition of parameters.

Schematic illustration of a four-wave configuration and definition of parameters.

-polar presentation of a possible MR solution for which (a) ( , , and ); (b) ( , , and ); and (c) ( , , and ).

-polar presentation of a possible MR solution for which (a) ( , , and ); (b) ( , , and ); and (c) ( , , and ).

The wave configurations of the three possible solutions of the three-shock theory whose graphical solutions are shown in Figs. 5(a)–5(c) , respectively.

The wave configurations of the three possible solutions of the three-shock theory whose graphical solutions are shown in Figs. 5(a)–5(c) , respectively.

Schematic illustration of a GR.

Schematic illustration of a GR.

The -polar combinations for and : (a) (MR); (b) 38.2° ; (c) 34.5° (vNR); (d) 33.9° ; (e) 32.5° (?R); (f) 31.8° ; and (g) 30° (GR). Recall that . The sonic point is marked on all the shock polars.

The -polar combinations for and : (a) (MR); (b) 38.2° ; (c) 34.5° (vNR); (d) 33.9° ; (e) 32.5° (?R); (f) 31.8° ; and (g) 30° (GR). Recall that . The sonic point is marked on all the shock polars.

Evolution-tree-type presentation of the transition criteria between the various reflections.

Evolution-tree-type presentation of the transition criteria between the various reflections.

Domains of and the transition boundaries between the various shock wave reflection configurations in the plane for (a) and (b) . Curve 1: The transition curve, i.e., . Curve 2: The transition curve, i.e., . This curve also separates the domains in which the 3ST does or does not have a physical solution. Curve 3: The transition curve, i.e., . Curve 4: The curve on which . Below this curve, the flow behind the incident shock wave is subsonic and a reflection cannot take place! Curve 5: Above this curve, the 3ST has at least one mathematical solution (not necessarily physical) and below it, the 3ST does not have any mathematical solution ( , , is the incident shock wave Mach number, is the reflecting wedge angle, and is the triple-point trajectory angle).

Domains of and the transition boundaries between the various shock wave reflection configurations in the plane for (a) and (b) . Curve 1: The transition curve, i.e., . Curve 2: The transition curve, i.e., . This curve also separates the domains in which the 3ST does or does not have a physical solution. Curve 3: The transition curve, i.e., . Curve 4: The curve on which . Below this curve, the flow behind the incident shock wave is subsonic and a reflection cannot take place! Curve 5: Above this curve, the 3ST has at least one mathematical solution (not necessarily physical) and below it, the 3ST does not have any mathematical solution ( , , is the incident shock wave Mach number, is the reflecting wedge angle, and is the triple-point trajectory angle).

## Tables

Summary of the reflection types that occur when is decreased for and .

Summary of the reflection types that occur when is decreased for and .

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