Advertisement

The von Neumann Triple Point Paradox

  • Richard Sanders
  • Allen M. Tesdall
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 16)

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.

Key words

self-similar solutions two-dimensional Riemann problems triple point paradox 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BH92.
    M. Brio and J. K. Hunter. Mach reflection for the two-dimensional Burgers equation. Phys. D, 60:194–207, 1992.MATHCrossRefMathSciNetGoogle Scholar
  2. BT49.
    W. Bleakney and A. H. Taub. Interaction of shock waves. Rev. Modern Physics, 21:584–605, 1949.MATHCrossRefADSMathSciNetGoogle Scholar
  3. CF76.
    R. Courant and K. O. Friedrichs. Supersonic Flow and Shock Waves. Springer, 1976.Google Scholar
  4. CH90.
    P. Colella and L. F. Henderson. The von Neumann paradox for the diffraction of weak shock waves. J. Fluid Mech., 213:71–94, 1990.CrossRefADSMathSciNetGoogle Scholar
  5. ČK98.
    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.MATHCrossRefADSMathSciNetGoogle Scholar
  6. ČKK01.
    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.CrossRefGoogle Scholar
  7. ČKK05.
    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.Google Scholar
  8. Gud47.
    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.Google Scholar
  9. Gud62.
    K. G. Guderley. The Theory of Transonic Flow. Pergamon Press, Oxford, 1962.MATHGoogle Scholar
  10. HB00.
    J. K. Hunter and M. Brio. Weak shock reflection. J. Fluid Mech., 410:235–261, 2000.MATHCrossRefADSMathSciNetGoogle Scholar
  11. Hen66.
    L. F. Henderson. On a class of multi-shock intersections in a perfect gas. Aero. Q., 17:1–20, 1966.Google Scholar
  12. Hen87.
    L. F. Henderson. Regions and boundaries for diffracting shock wave systems. Z. Angew. Math. Mech., 67:73–86, 1987.CrossRefGoogle Scholar
  13. HT04.
    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.Google Scholar
  14. KF94.
    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.MATHCrossRefADSMathSciNetGoogle Scholar
  15. Neu43.
    J. von Neumann. Oblique reflection of shocks. Explosives Research Report 12, Bureau of Ordinance, 1943.Google Scholar
  16. Neu63.
    J. von Neumann. Collected Works, Vol. 6. Pergamon Press, New York, 1963.Google Scholar
  17. Ric81.
    R. D. Richtmeyer. Principles of Mathematical Physics, Vol. 1. Springer, 1981.Google Scholar
  18. SA05.
    B. Skews and J. Ashworth. The physical nature of weak shock wave reflection. J. Fluid Mech., 542:105–114, 2005.MATHCrossRefADSMathSciNetGoogle Scholar
  19. Ste59.
    J. Sternberg. Triple-shock-wave intersections. Phys. Fluids, 2:179–206, 1959.MATHCrossRefADSGoogle Scholar
  20. STS92.
    A. Sasoh, K. Takayama, and T. Saito. A weak shock wave reflection over wedges. Shock Waves, 2:277–281, 1992.CrossRefADSGoogle Scholar
  21. TH02.
    A. M. Tesdall and J. K. Hunter. Self-similar solutions for weak shock reflection. SIAM J. Appl. Math., 63:42–61, 2002.MATHCrossRefMathSciNetGoogle Scholar
  22. TR94.
    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.MATHCrossRefADSMathSciNetGoogle Scholar
  23. TSK06.
    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.CrossRefMathSciNetGoogle Scholar
  24. VK99.
    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.MATHMathSciNetGoogle Scholar
  25. ZBHW00.
    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.MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  • Richard Sanders
    • 1
  • Allen M. Tesdall
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Fields InstituteTorontoCanada
  3. 3.Department of MathematicsUniversity of HoustonHoustonUSA

Personalised recommendations