Summary
We describe the problem of weak shock reflection off a wedge and discuss the triple point paradox that arises. When the shock is sufficiently weak and the wedge is thin, Mach reflection appears to be observed but is impossible according to what von Neumann originally showed in 1943. We summarize some recent numerical results for weak shock reflection problems for the unsteady transonic small disturbance equations, the nonlinear wave system, and the Euler equations. Rather than finding a standard but mathematically inadmissible Mach reflection with a shock triple point, the solutions contain a complex structure: there is a sequence of triple points and supersonic patches in a tiny region behind the leading triple point, with an expansion fan originating at each triple point. The sequence of patches may be infinite, and we refer to this structure as Guderley Mach reflection. The presence of the expansion fans at the triple points resolves the paradox. We describe some recent experimental evidence which is consistent with these numerical findings.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Brio and J. K. Hunter. Mach reflection for the two-dimensional Burgers equation. Phys. D, 60:194–207, 1992.
W. Bleakney and A. H. Taub. Interaction of shock waves. Rev. Modern Physics, 21:584–605, 1949.
R. Courant and K. O. Friedrichs. Supersonic Flow and Shock Waves. Springer, 1976.
P. Colella and L. F. Henderson. The von Neumann paradox for the diffraction of weak shock waves. J. Fluid Mech., 213:71–94, 1990.
S. Čanić and B. L. Keyfitz. Quasi-one-dimensional Riemann problems and their role in self-similar two-dimensional problems. Arch. Rational Mech. Anal., 144:233–258, 1998.
S. Čanić, B. L. Keyfitz, and E. H. Kim. Mixed hyperbolic-elliptic systems in self-similar flows. Bol. Soc. Bras. Mat., 32:1–23, 2001.
S. Čanić, B. L. Keyfitz, and E. H. Kim. Free boundary problems for nonlinear wave systems: Mach stems for interacting shocks. SIAM J. Math. Anal., 37:1947–1977, 2005.
K. G. Guderley. Considerations of the structure of mixed subsonic-supersonic flow patterns. Air Material Command Tech. Report, F-TR-2168-ND, ATI No. 22780, GS-AAF-Wright Field 39, U.S. Wright-Patterson Air Force Base, Dayton, Ohio, October 1947.
K. G. Guderley. The Theory of Transonic Flow. Pergamon Press, Oxford, 1962.
J. K. Hunter and M. Brio. Weak shock reflection. J. Fluid Mech., 410:235–261, 2000.
L. F. Henderson. On a class of multi-shock intersections in a perfect gas. Aero. Q., 17:1–20, 1966.
L. F. Henderson. Regions and boundaries for diffracting shock wave systems. Z. Angew. Math. Mech., 67:73–86, 1987.
J. K. Hunter and A. M. Tesdall. Weak shock reflection. In D. Givoli, M. Grote, and G. Papanicolaou, editors, A Celebration of Mathematical Modeling. Kluwer Academic Press, New York, 2004.
B. L. Keyfitz and M. C. Lopes Filho. A geometric study of shocks in equations that change type. J. Dynam. Differential Equations, 6:351–393, 1994.
J. von Neumann. Oblique reflection of shocks. Explosives Research Report 12, Bureau of Ordinance, 1943.
J. von Neumann. Collected Works, Vol. 6. Pergamon Press, New York, 1963.
R. D. Richtmeyer. Principles of Mathematical Physics, Vol. 1. Springer, 1981.
B. Skews and J. Ashworth. The physical nature of weak shock wave reflection. J. Fluid Mech., 542:105–114, 2005.
J. Sternberg. Triple-shock-wave intersections. Phys. Fluids, 2:179–206, 1959.
A. Sasoh, K. Takayama, and T. Saito. A weak shock wave reflection over wedges. Shock Waves, 2:277–281, 1992.
A. M. Tesdall and J. K. Hunter. Self-similar solutions for weak shock reflection. SIAM J. Appl. Math., 63:42–61, 2002.
E. G. Tabak and R. R. Rosales. Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection. Phys. Fluids, 6:1874–1892, 1994.
A. M. Tesdall, R. Sanders, and B. L. Keyfitz. The triple point paradox for the nonlinear wave system. SIAM J. Appl. Math., 67:321–336, 2006.
E. Vasil’ev and A. Kraiko. Numerical simulation of weak shock diffraction over a wedge under the von Neumann paradox conditions. Comput. Math. Math. Phys., 39:1335–1345, 1999.
A. Zakharian, M. Brio, J. K. Hunter, and G. Webb. The von Neumann paradox in weak shock reflection. J. Fluid Mech., 422:193–205, 2000.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer Science + Business Media B.V.
About this chapter
Cite this chapter
Sanders, R., Tesdall, A.M. (2008). The von Neumann Triple Point Paradox. In: Glowinski, R., Neittaanmäki, P. (eds) Partial Differential Equations. Computational Methods in Applied Sciences, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8758-5_6
Download citation
DOI: https://doi.org/10.1007/978-1-4020-8758-5_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-8757-8
Online ISBN: 978-1-4020-8758-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)