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Analytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge
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Image of FIG. 1.
FIG. 1.

Definition of parameters across an oblique shock wave.

Image of FIG. 2.
FIG. 2.

Schematic illustration of the wave configuration of a MR.

Image of FIG. 3.
FIG. 3.

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

Image of FIG. 4.
FIG. 4.

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

Image of FIG. 5.
FIG. 5.

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

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

Schematic illustration of a GR.

Image of FIG. 8.
FIG. 8.

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.

Image of FIG. 9.
FIG. 9.

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

Image of FIG. 10.
FIG. 10.

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).


Generic image for table
Table I.

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


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Analytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge