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.
References
S. Bianchini and A. Bressan. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. (2nd series), 161(1):223–342, 2005. MR2150387 10.4007/annals.2005.161.223 S. Bianchini and A. Bressan. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. (2nd series), 161(1):223–342, 2005. MR2150387 10.4007/annals.2005.161.223
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. 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.
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. MR1857880 10.1002/cpa.10013 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. MR1857880 10.1002/cpa.10013
Gui-Qiang Chen and M. Feldman. Global solutions to shock reflection by large-angle wedges for potential flow. Annals of Math. To appear. MR2630061 10.4007/annals.2010.171.1067 Gui-Qiang Chen and M. Feldman. Global solutions to shock reflection by large-angle wedges for potential flow. Annals of Math. To appear. MR2630061 10.4007/annals.2010.171.1067
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. MR2667260 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. MR2667260
V. Elling. A possible counterexample to well-posedness of entropy solution and to Godunov scheme convergence. Math. Comp., 75:1721–1733, 2006. See also arxiv:math.NA/0509331. MR2240632 10.1090/S0025-5718-06-01863-1 V. Elling. A possible counterexample to well-posedness of entropy solution and to Godunov scheme convergence. Math. Comp., 75:1721–1733, 2006. See also arxiv:math.NA/0509331. MR2240632 10.1090/S0025-5718-06-01863-1
V. Elling and Tai-Ping Liu. The ellipticity principle for selfsimilar potential flow. J. Hyper. Diff. Eqns., 2(4):909–917, 2005. Preprint arxiv:math.AP-0509332. MR2195986 10.1142/S0219891605000646 V. Elling and Tai-Ping Liu. The ellipticity principle for selfsimilar potential flow. J. Hyper. Diff. Eqns., 2(4):909–917, 2005. Preprint arxiv:math.AP-0509332. MR2195986 10.1142/S0219891605000646
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. MR2549143 10.1007/978-3-540-75712-2_8 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. MR2549143 10.1007/978-3-540-75712-2_8
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. http://www.umich.edu/~velling/rims05.ps 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. http://www.umich.edu/~velling/rims05.ps
V. Elling and Tai-Ping Liu. Exact solutions for supersonic flow onto a solid wedge. Technical report, 2007. arxiv:0707.2108. 0707.2108 MR2549143 10.1007/978-3-540-75712-2_8 V. Elling and Tai-Ping Liu. Exact solutions for supersonic flow onto a solid wedge. Technical report, 2007. arxiv:0707.2108. 0707.2108 MR2549143 10.1007/978-3-540-75712-2_8
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. MR1676757 10.1002/(SICI)1097-0312(199906)52:6<763::AID-CPA4>3.0.CO;2-3 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. MR1676757 10.1002/(SICI)1097-0312(199906)52:6<763::AID-CPA4>3.0.CO;2-3
J. Glimm. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math., 18:697–715, 1965. MR194770 10.1002/cpa.3160180408 J. Glimm. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math., 18:697–715, 1965. MR194770 10.1002/cpa.3160180408
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. 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.
J. Hunter and M. Brio. Weak shock reflection. J. Fluid Mech., 410:235–261, 2000. MR1761754 10.1017/S0022112099008010 J. Hunter and M. Brio. Weak shock reflection. J. Fluid Mech., 410:235–261, 2000. MR1761754 10.1017/S0022112099008010
J. Hunter and A. Tesdall. Self-similar solutions for weak shock reflection. SIAM J. Appl. Math., 63(1):42–61, 2002. MR1952886 10.1137/S0036139901383826 J. Hunter and A. Tesdall. Self-similar solutions for weak shock reflection. SIAM J. Appl. Math., 63(1):42–61, 2002. MR1952886 10.1137/S0036139901383826
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. 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.
Tai-Ping Liu and Tong Yang. Well-posedness theory for hyperbolic conservation laws. Comm. Pure Appl. Math., 52:1553–1586, 1999. MR1711037 10.1002/(SICI)1097-0312(199912)52:12<1553::AID-CPA3>3.0.CO;2-S Tai-Ping Liu and Tong Yang. Well-posedness theory for hyperbolic conservation laws. Comm. Pure Appl. Math., 52:1553–1586, 1999. MR1711037 10.1002/(SICI)1097-0312(199912)52:12<1553::AID-CPA3>3.0.CO;2-S
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. 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.
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. 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.
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. MR1270863 10.1063/1.868246 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. MR1270863 10.1063/1.868246
P. Woodward and P. Colella. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comp. Phys., 54:115–173, 1984. MR748569 10.1016/0021-9991(84)90142-6 P. Woodward and P. Colella. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comp. Phys., 54:115–173, 1984. MR748569 10.1016/0021-9991(84)90142-6
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. MR2215512 10.1007/s10255-006-0296-5 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. MR2215512 10.1007/s10255-006-0296-5
