The reflection of a weak shock wave is considered using a shock polar. We present a sufficient condition under which the von Neumann paradox appears for the Euler equations. In an attempt to resolve the von Neumann paradox for the Euler equations, two new types of reflection configuration, one called the von Neumann reflection (vNR) and the other called the Guderley reflection (GR), are observed in numerical calculations. Finally, we obtain that GR is a reasonable configuration and vNR is an unreasonable configuration to resolve the von Neumann paradox.
Li WANG
. THE VON NEUMANN PARADOX FOR THE EULER EQUATIONS[J]. Acta mathematica scientia, Series B, 2021
, 41(3)
: 753
-763
.
DOI: 10.1007/s10473-021-0308-9
[1] Mach E. Uber den verlauf von funkenwellenin der ebene und im raume. Akademie der Wissenschaften Wien, 1878, 77:819-838
[2] von Neumann J. Oblique reflection of shock. Washington, DC, USA:Explos Res Rept 12, Navy Dept Bureau of Ordinance, 1943
[3] Colella P, Henderson L F. The von Neumann paradox for the diffraction of a weak shock waves. J Fluid Mech, 1990, 213:71-94
[4] Skews B W, Ashworth J T. The physical nature of weak shock wave reflection. J Fluid Mech, 2005, 542:105-114
[5] Hunter J K, Brio M. Weak shock reflection. J Fluid Mech, 2000, 410:235-261
[6] Tesdall A M, Hunter J K. Self-similar solution for weak shock reflection. SIAM J Appl Math, 2002, 63:42-61
[7] Tesdall A M, Sanders R, Keyfitz B L. The triple point paradox for the nonlinear wave system. SIAM J Appl Math, 2006, 67:321-336
[8] Guderley K G. Considerations on the structure of mixed subsonic supersonic flow patterns//Tech Report F-TR-2168-ND. Wright Field, Dayton, Ohio:Headquarters Air Materiel Command, 1947
[9] Chen G Q, Rigby M. Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. Acta Mathematica Scientia, 2018, 38B(5):1485-1514
[10] Olim M, Dewey J M. A revised three shock solution for the Mach reflection of weak shocks. Shock Waves, 1992, 2(3):167-176
[11] Lai G, Sheng W C. Nonexistence of the Von Neumann reflection configuration for the triple point paradox. SIAM J Appl Math, 2011, 71:2072-2092
[12] Vasilev E I. High-resolution simulation for the Mach reflection of weak shock waves. In Proceedings of the Fourth European Computational Fluid Dynamics Conference, 1998, 1:520-527
[13] Tesdall A M, Hunter J K. Self-similar solutions for weak shock reflection. SIAM J Appl Math, 2002, 63:42-61
[14] Tesdall A M, Sanders R, Keyfitz B L. The triple point paradox for the nonlinear wave system. SIAM J Appl Math, 2006, 67:321-336
[15] Skews B W, Ashworth J T. The physical nature of weak shock wave reflection. J Fluid Mech, 2005, 542:105-114
[16] Skews B W, Li G, Paton R. Experiments on Guderley Mach reflection. Shock Waves, 2009, 19:95-102
[17] Tesdall A M, Sanders R, Keyfitz B L. Self-similar solutions for the triple point paradox in gas dynamics. SIAM J Appl Math, 2008, 68:1360-1377
[18] Vasilev E, Kraiko A. Numerical simulation of weak shock diffraction over a wedge under the von Neumann paradox conditions. Comput Math Phys, 1999, 39:1335-1345
[19] Zakharian A, Brio M, Hunter J K, Webb G. The von Neumann paradox in weak shock reflection. J Fluid Mech, 2000, 422:193-205
[20] Hunter J K, Brio M. Weak shock reflection. J Fluid Mech, 2000, 410:235-261
[21] Lai G, Sheng W C. Two-dimensional centered wave flow patches to the Guderley Mach reflection configuratins for steady flows in gas dynamics. Journal of Hyperbolic Differential Equations, 2016, 13(1):107-128
[22] Ben-Dor G. Shock wave reflection phenomena. New York:Springer-Verlag, 1992
[23] Vasilev E I, Elperin T, Ben-Dor G. Analytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge. Physics of Fluids, 2008, 20(4):046101
[24] Courant R, Friedrichs K O. Supersonic Flow and Shock Wave. New York:Interscience Publishers Inc, 1948
