Regular reflection in self-similar potential flow and the sonic criterion. (English) Zbl 1328.76038

Elling, Volker

Commun. Math. Anal. 8, No. 2, 22-69 (2010).

Summary: Reflection of a shock from a solid wedge is a classical problem in gas dynamics. Depending on the parameters either a regular or a irregular (Mach-type) reflection results. We construct regular reflection as an exact self-similar solution for potential flow. For some upstream Mach numbers \(M_I\) and isentropic coefficients \(\gamma\), a solution exists for all wedge angles \(\theta\) allowed by the sonic criterion. This demonstrates that, at least for potential flow, weaker criteria are false.

MSC:

76H05	Transonic flows
76M10	Finite element methods applied to problems in fluid mechanics

Keywords:

shock; regular reflection; sonic criterion; potential flow

References:

[1]	G. Ben-Dor. Shock Wave Reflection Phenomena . Springer, 1992. · Zbl 1142.76300
[2]	S. Bianchini and A. Bressan. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. (2nd series) , 161(1):223-342, 2005. · Zbl 1082.35095
[3]	A. Bressan, G. Crasta, and B. Piccoli. Well-Posedness of the Cauchy Problem for \(n\times n\) Systems of Conservation Laws. Number 694 in Memoirs of the AMS. American Mathematical Society, July 2000. · Zbl 0958.35001
[4]	S. Čanić, B.L. Keyfitz, and Eun Heui Kim. A free boundary problem for a quasi-linear degenerate elliptic equation: regular reflection of weak shocks. Comm. Pure Appl. Math. , 55(1):71-92, 2002. · Zbl 1124.35338
[5]	Gui-Qiang Chen and M. Feldman. Global solutions to shock reflection by large-angle wedges for potential flow. Annals of Math. · Zbl 1277.35252
[6]	V. Elling. Nonuniqueness of entropy solutions and the carbuncle phenomenon. In Proceedings of the 10th Conference on Hyperbolic Problems (HYP2004) , volume I, pages 375-382. Yokohama Publishers, 2005. · Zbl 1102.35350
[7]	V. Elling. A possible counterexample to well-posedness of entropy solution and to Godunov scheme convergence. Math. Comp. , 75:1721-1733, 2006. See also · Zbl 1114.35124
[8]	V. Elling and Tai-Ping Liu. The ellipticity principle for selfsimilar potential flow. J. Hyper. Diff. Eqns. , 2(4):909-917, 2005. · Zbl 1089.35049
[9]	V. Elling and Tai-Ping Liu. Exact solutions to supersonic flow onto a solid wedge. In Proceedings of the 11th Conference on Hyperbolic Problems (HYP2006) , pages 101-112. Springer, 2006. · Zbl 1354.76084
[10]	V. Elling and Tai-Ping Liu. Physicality of weak Prandtl-Meyer reflection. In RIMS Kokyuroku , volume 1495, pages 112-117. Kyoto University, Research Institute for Mathematical Sciences, May 2006. http://www.umich.edu/ velling/rims05.ps. URL:
[11]	V. Elling and Tai-Ping Liu. Exact solutions for supersonic flow onto a solid wedge. Technical report, 2007. arXiv: · Zbl 1143.76030
[12]	L.C. Evans. Partial Differential Equations . American Mathematical Society, 1998. · Zbl 0902.35002
[13]	I. Gamba, R. Rosales, and E. Tabak. Constraints on possible singularities for the unsteady transonic small disturbance (UTSD) equations. Comm. Pure Appl. Math. , 52(6):763-799, 1999. · Zbl 0928.35033
[14]	J. Glimm. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. , 18:697-715, 1965. · Zbl 0141.28902
[15]	K.G. Guderley. The Theory of Transonic Flow . Pergamon Press, Oxford, 1962. · Zbl 0112.41205
[16]	H.G. Hornung, H. Oertel, and R.J. Sandeman. Transition to mach reflexion of shock waves in steady and pseudosteady flow with and without relaxation. J. Fluid Mech. , 90:541-560, 1979.
[17]	J. Hunter and M. Brio. Weak shock reflection. J. Fluid Mech. , 410:235-261, 2000. · Zbl 0959.76036
[18]	J. Hunter and A. Tesdall. Self-similar solutions for weak shock reflection. SIAM J. Appl. Math., 63(1):42-61, 2002. · Zbl 1060.76070
[19]	M.S. Ivanov, D. Vandromme, V.M. Fomin, A.N. Kudryavtsev, A. Hadjadj, and D.V. Khotyanovsky. Transition between regular and mach reflection of shock waves: new numerical and experimental results. Shock Waves , 11:199-207, 2001. · Zbl 0978.76514
[20]	P. Krehl and M. van der Geest. The discovery of the Mach reflection effect and its demonstration in an auditorium. Shock Waves , 1:3-15, 1991.
[21]	Jiequan Li, Tong Zhang, and Shuli Yang. The Two-Dimensional Riemann Problem in Gas Dynamics. Addison Wesley Longman, 1998. · Zbl 0935.76002
[22]	G. Lieberman. Oblique derivative problems in Lipschitz domains II. Discontinuous boundary data. J. reine angew. Math. , 389:1-21, 1988. · Zbl 0648.35033
[23]	Tai-Ping Liu and Tong Yang. Well-posedness theory for hyperbolic conservation laws. Comm. Pure Appl. Math. , 52:1553-1586, 1999. · Zbl 1034.35073
[24]	G.D. Lock and J.M. Dewey. An experimental investigation of the sonic criterion for transition from regular to mach reflection of weak shock waves. Experiments in Fluids , 7:289-292, 1989.
[25]	E. Mach and J. Wosyka. Über die Fortpflanzungsgeschwindigkeit von Explosionsschallwellen. Sitzungsber. Akad. Wiss. Wien (II. Abth.) , 72:44-52, 1875.
[26]	J. von Neumann. Oblique reflection of shocks. Technical Report 12, Navy Dep., Bureau of Ordnance, Washington, D.C., 1943. In: Collected works, v. 6, p. 238-299.
[27]	E. Tabak and R. Rosales. Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection. Phys. Fluids , 6:1874-1892, 1994. · Zbl 0831.76039
[28]	P. Woodward and P. Colella. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comp. Phys. , 54:115-173, 1984. · Zbl 0573.76057
[29]	Yuxi Zheng. Systems of Conservation Laws . Birkhäuser, 2001. · Zbl 0971.35002
[30]	Yuxi Zheng. Two-dimensional regular shock reflection for the pressure gradient system of conservation laws. Acta Math. Appl. Sin. Engl. Ser. , 22(2):177-210, 2006. · Zbl 1106.35034