Skip to main content
SpringerLink
Log in

Transition boundary between regular and Mach reflections for a moving shock interacting with a wedge in inviscid and polytropic air

  • Original Article
  • Published:
Shock Waves Aims and scope Submit manuscript

Abstract

The transition boundary separating the region of regular reflection from the regions of single-, transitional-, and double-Mach reflections for a planar shock wave moving in air and interacting with an inclined wedge in a shock tube is studied by both analytical methods and computational-fluid-dynamic simulations. The analytical solution for regular reflection and the corresponding solutions from the extreme-angle (detachment), sonic, and mechanical-equilibrium transition criteria by von Neumann (Oblique reflection of shocks, Explosive Research Report No. 12, Navy Department, Bureau of Ordnance, U.S. Dept. Comm. Tech. Serv. No. PB37079 (1943). Also, John von Neumann, Collected Works, Pergamon Press 6, 238–299, 1963) are first revisited and revised. The boundary between regular and Mach reflection is then determined numerically using an advanced computational-fluid-dynamics algorithm to solve Euler’s inviscid equations for unsteady motion in two spatial dimensions. This numerical transition boundary is determined by post-processing many closely stationed flow-field simulations, to determine the transition point when the Mach stem of the Mach-reflection pattern just disappears and this pattern then transcends into that of regular reflection. The new numerical transition boundary is shown to agree well with von Neumann’s closely spaced sonic and extreme-angle boundaries for weak incident shock Mach numbers from 1.0 to 1.6, but this new boundary trends upward and above von Neumann’s sonic and extreme-angle boundaries by a couple of degrees at larger shock Mach numbers from 1.6 to 4.0. Furthermore, the new numerically determined transition boundary is shown to agree well with very few available experimental data obtained from previous experiments designed to reflect two symmetrical moving oblique shock waves along a plane without a shear or boundary layer.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27

Similar content being viewed by others

References

  1. Mach, E.: Über den verlauf der funkenwellen in der ebene und im raume. Sitzungsber. Akad. Wiss. Wien. (II, Abth), 77, 819–838 (1878)

  2. Smith, L.G.: Photographic investigation of the reflection of plane shocks in air, Division 2, National Defence Research Committee of the Office of Scientific Research and Development, OSRD Rep. No. 6271 (1945)

  3. White, D.R.: An experimental survey of the Mach reflection of shock waves, Technical Report II-10, Department of Physics, Princeton University, Aug. 1951. Also, Proceedings of the Second Midwestern Conference on Fluid Mechanics, March 17–19, 1952, Ohio State University, Engineering Experiment Station Bulletin 149, 21(3), 253–259 (1952)

  4. von Neumann, J.: Theory of shock waves, Progress Report, Division 8, National Defence Research Committee of the Office of Scientific Research and Development, U.S. Dept. Comm. Tech. Serv. No. PB32719 (1943). Also, John von Neumann, Collected Works, Pergamon Press, 6, 178–202 (1963)

  5. von Neumann, J.: Oblique reflection of shocks, Explosive Research Report No. 12, Navy Department, Bureau of Ordnance, U.S. Dept. Comm. Tech. Serv. No. PB37079 (1943). Also, John von Neumann, Collected Works, Pergamon Press 6, 238–299 (1963)

  6. von Neumann, J.: Refraction, intersection and reflection of shock waves, paper delivered at a conference on Shock Waves and Supersonic Flow on 19 March 1945 at Princeton University (1945). Also, John von Neumann, Collected Works, Pergamon Press 6, 300–308 (1963)

  7. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Wiley Interscience Publishers, New York (1948)

    MATH  Google Scholar 

  8. Bleakney, W., Taub, A.H.: Interaction of shock waves. Rev. Mod. Phys. 21(4), 584–605 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cabannes, H.: Lois de la réflexion des ondes de choc dans les écoulements plans non stationnaires, O.N.E.R.A. 80, 1–29 (1955)

  10. Kawamura, R., Saito, H.: Reflection of shock waves: 1. Pseudostationary case. J. Phys. Soc. Jpn. 11(5), 584–592 (1956)

    Article  Google Scholar 

  11. Ben-Dor, G., Glass, I.I.: Domains and boundaries of non-stationary oblique shock-wave reflexions. 1: Diatomic gas. J. Fluid Mech. 92(3), 459–496 (1979)

    Article  Google Scholar 

  12. Ben-Dor, G., Glass, I.I.: Domains and boundaries of non-stationary oblique shock-wave reflexions. 2: Monatomic gas. J. Fluid Mech. 96(4), 735–756 (1980)

    Article  Google Scholar 

  13. Ben-Dor, G.: Shock Wave Reflection Phenomena, 1e, 2e. Springer-Verlag Books, Berlin (1991, 2007)

  14. Glass, I.I., Sislian, J.P.: Nonstationary Flows and Shock Waves. Clarendon Press, Oxford (1994)

    Google Scholar 

  15. Semenov, A.N., Berezkina, M.K., Krassovskaya, I.V.: Classification of pseudo-steady shock wave reflection types. Shock Waves 22(4), 307–316 (2012)

    Article  Google Scholar 

  16. Henderson, L.F., Lozzi, A.: Experiments on transition of Mach reflexion. J. Fluid Mech. 68, 139–155 (1975)

    Article  Google Scholar 

  17. Henderson, L.F., Siegenthaler, A.: Experiments on the diffraction of weak blast waves: the von Neumann paradox. Proc. R. Soc. Lond. A 369(1739), 537–555 (1980)

    Article  Google Scholar 

  18. Walker, D.K., Dewey, J.M., Scotten, L.N.: Observation of density discontinuities behind reflected shocks close to the transition from regular to Mach reflection. J. Appl. Phys. 53(3), 1398–1400 (1982)

    Article  Google Scholar 

  19. Lock, G.D., Dewey, J.M.: An experimental investigation of the sonic criterion for transition from regular to Mach reflection of weak shock waves. Exp. Fluids 7(5), 289–292 (1989)

    Article  Google Scholar 

  20. Kobayashi, S., Adachi, T., Suzuki, T.: On the unsteady transition phenomenon of weak shock waves. Theor. Appl. Mech. 49, 271–278 (2000)

    Google Scholar 

  21. Hornung, H.: Regular and Mach reflection of shock waves. Ann. Rev. Fluid Mech. 18, 33–58 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. Henderson, L.F., Takayama, K., Crutchfield, W.Y., Itabashi, S.: The persistence of regular reflection during strong shock diffraction over rigid ramps. J. Fluid Mech. 431, 273–296 (2001)

    Article  MATH  Google Scholar 

  23. Adachi, T., Sakurai, A., Kobayashi, S.: Effect of boundary layer on Mach reflection over a wedge surface. Shock Waves 11(4), 271–278 (2002)

    Article  MATH  Google Scholar 

  24. Thompson, P.A.: Compressible-Fluid Dynamics. Rensselaer Polytechnic Institute Press, New York (1988)

    Google Scholar 

  25. Polachek, H., Seeger, R.J.: Shockwave interactions. In: Emmons, H.W. (ed.) Fundamental of Gas Dynamics. High Speed Aerodynamics and Jet Propulsion, Vol III, pp. 482–525 (1958)

  26. Henderson, L.F.: Regions and boundaries of diffracting shock wave systems. ZAMM 67(2), 73–86 (1987)

    Article  Google Scholar 

  27. Henderson, L.F., Lozzi, A.: Further experiments on transition to Mach reflexion. J. Fluid Mech. 94, 541–559 (1979)

    Article  Google Scholar 

  28. Hryniewicki, M.K., Groth, C.P.T., Gottlieb, J.J.: Parallel implicit anisotropic block-based adaptive mesh refinement finite-volume scheme for the study of fully resolved oblique shock wave reflections. Shock Waves 25(4), 371–386 (2015)

    Article  Google Scholar 

  29. McDonald, J.G., Sachdev, J.S., Groth, C.P.T.: Application of Gaussian moment closure to microscale flows with moving embedded boundaries. AIAA J. 52(9), 1839–1857 (2014)

    Article  Google Scholar 

  30. Northrup, S.A., Groth, C.P.T.: Parallel implicit adaptive mesh refinement scheme for unsteady fully-compressible reactive flows. AIAA Paper 2013-2433 (2013)

  31. Williamschen, M.J., Groth, C.P.T.: Parallel anisotropic block-based adaptive mesh algorithm for three-dimensional flows. AIAA Paper 2013-2442 (2013)

  32. Zhang, J.Z., Groth, C.P.T.: Parallel high-order anisotropic block-based adaptive mesh refinement finite-volume scheme. AIAA Paper 2011-3695 (2011)

  33. Gao, X., Northrup, S., Groth, C.P.T.: Parallel solution-adaptive method for two-dimensional non-premixed combusting flows. Prog. Comput. Fluid Dyn. 11(2), 76–95 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gao, X., Groth, C.P.T.: A parallel solution-adaptive scheme for three-dimensional turbulent non-premixed combusting flows. J. Comput. Phys. 229(9), 3250–3275 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Gao, X., Groth, C.P.T.: A parallel adaptive mesh refinement algorithm for predicting turbulent non-premixed combusting flows. Int. J. Comput. Fluid Dyn. 20(5), 349–357 (2006)

    Article  MATH  Google Scholar 

  36. Sachdev, J.S., Groth, C.P.T., Gottlieb, J.J.: A parallel solution-adaptive scheme for multi-phase core flows in solid propellant rocket motors. Int. J. Comput. Fluid Dyn. 19(2), 159–177 (2005)

    Article  MATH  Google Scholar 

  37. LeVeque, R.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

  38. Ivan, L., Groth, C.P.T.: High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows. J. Comput. Phys. 257(A), 860–862 (2014)

  39. Barth, T.J.: Recent developments in high order k-exact reconstruction on unstructured meshes. AIAA Paper 93-0668 (1993)

  40. Venkatakrishnan, V.: On the accuracy of limiters and convergence to steady state solutions. AIAA Paper 93-0880 (1993)

  41. Harten, A., Lax, P.D., Van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  42. Einfeldt, B.: On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal. 25(2), 294–318 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rankine, W.J.M.: On the thermodynamic theory of waves of finite longitudinal disturbance. Philos. Trans. R. Soc. 160, 277–288 (1870)

    Article  Google Scholar 

  44. Taylor, G.I.: The conditions necessary for discontinuous motion in gases. Proc. R. Soc. Lond. A 84(571), 371–377 (1910)

    Article  MATH  Google Scholar 

  45. Becker, R., Stoßwelle und detonation. Z. Phys. 8, 321–362 (1922). Also, English translation, NACA TM No. 505 (1929)

  46. Smith, W.R.: Mutual reflection of two shock waves of arbitrary strengths. Phys. Fluids 2(5), 533–541 (1959)

  47. Barbosa, F.J., Skews, B.W.: Experimental confirmation of the von Neumann theory of shock wave reflection transition. J. Fluid Mech. 472(1), 263–282 (2002)

  48. Herron, T., Skews, B.W.: On the persistence of regular reflection. Shock Waves 21(6), 573–578 (2011)

  49. Previtali, F.A., Timofeev, E., Kleine, H.: On unsteady shock wave reflections from wedges with straight and concave tips. AIAA Paper 2015-2642 (2015)

  50. Ivanov, M.S., Gimelshein, S.F., Beylich, A.E.: Hysteresis effect in stationary reflection of shock waves. Phys. Fluids 7(4), 685–667 (1995)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The contributions of Lucie Freret in making the anisotropic algorithm of AMR more efficient are greatly appreciated. Computational resources for performing all of the calculations reported in this research were provided by the SciNet High-Performance Computing Consortium at the University of Toronto and Compute/Calcul Canada, through funding from the Canada Foundation for Innovation (CFI) and the Province of Ontario, Canada.

Author information

Authors and Affiliations

  1. Institute for Aerospace Studies, University of Toronto, 4925 Dufferin Street, Toronto, ON, M3H 5T6, Canada

    M. K. Hryniewicki, J. J. Gottlieb & C. P. T. Groth

Authors
  1. M. K. Hryniewicki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. J. Gottlieb
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. P. T. Groth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. J. Gottlieb.

Additional information

Communicated by B. Skews.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hryniewicki, M.K., Gottlieb, J.J. & Groth, C.P.T. Transition boundary between regular and Mach reflections for a moving shock interacting with a wedge in inviscid and polytropic air. Shock Waves 27, 523–550 (2017). https://doi.org/10.1007/s00193-016-0697-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00193-016-0697-1

Keywords

Search

Navigation