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 argon

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

Abstract

The transition boundary separating the regions of regular and Mach reflections for a planar shock moving in argon and interacting with an inclined wedge in a shock tube is investigated using flow-field simulations produced by high-resolution computational fluid dynamics (CFD). The transition boundary is determined numerically using a modern and reliable CFD algorithm to solve Euler’s inviscid equations of unsteady motion in two spatial dimensions with argon treated as a polytropic gas. This numerically computed transition boundary for inviscid flow, without a combined thermal and viscous boundary layer on the wedge surface, is determined by post-processing many closely stationed flow-field simulations to accurately determine the transition-boundary 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 for argon 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 1.55, but it deviates upward and above the closely spaced sonic and extreme-angle boundaries by almost \(2^\circ \) at larger shock Mach numbers from 1.55 to 4.0. This upward trend of the numerical transition boundary for this sequel case with monatomic gases like argon (\(\gamma =5/3\)) and no boundary layer on the wedge surface (inviscid flow) is similar to the previous finding for the case of diatomic gases and air (\(\gamma =7/5\)). An alternative method used to determine one point on the transition boundary between regular and Mach reflections, from a collection of Mach-reflection patterns with a constant-strength shock and different far-field wedge angles, by linear and higher-order polynomial extrapolations to zero for triple-point trajectories versus wedge angle, is compared to the present method of using near-field data that are close to and surround the new transition boundary. Such extrapolation methods are shown to yield a different transition-boundary estimate that corresponds to the mechanical-equilibrium boundary of von Neumann. Finally, the significance of the computed inviscid transition boundary between regular and Mach reflections for monatomic and diatomic gases is explained relative to the case of viscous flow with a combined thermal and viscous boundary layer on the wedge surface.

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

Similar content being viewed by others

References

  1. von Neumann, J.: Oblique reflection of shocks. Explosive Research Report No. 12, Navy Department, Bureau of Ordnance, U.S. Department of Communication Technology Services No. PB37079 (1943) (John von Neumann, Collected Works). Pergamon Press, Oxford, vol. 6, pp. 238–299 (1963)

  2. 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). https://doi.org/10.1017/S0022112079000732

    Article  Google Scholar 

  3. 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). https://doi.org/10.1017/S0022112080002339

    Article  Google Scholar 

  4. Ben-Dor, G.: Shock Wave Reflection Phenomena, 1e, 2e. Springer, Berlin (1991, 2007). ISBN 978-3-540-71381-4. https://doi.org/10.1007/978-1-4757-4279-4

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

    Google Scholar 

  6. Colella, P., Henderson, L.F.: The von Neumann paradox for the diffraction of weak shock waves. J. Fluid Mech. 213, 71–94 (1990). https://doi.org/10.1017/S0022112090002221

    Article  MathSciNet  Google Scholar 

  7. Semenov, A.N., Berezkina, M.K., Krassovskaya, I.V.: Classification of pseudo-steady shock wave reflection types. Shock Waves 22(4), 307–316 (2012). https://doi.org/10.1007/s00193-012-0373-z

    Article  Google Scholar 

  8. Hornung, H.: Regular and Mach reflection of shock waves. Annu. Rev. Fluid Mech. 18, 33–58 (1986). https://doi.org/10.1146/annurev.fl.18.010186.000341

    Article  MathSciNet  MATH  Google Scholar 

  9. Bleakney, W., Taub, A.H.: Interaction of shock waves. Rev. Mod. Phys. 21(4), 584–605 (1949). https://doi.org/10.1103/RevModPhys.21.584

    Article  MathSciNet  MATH  Google Scholar 

  10. Kawamura, R., Saito, H.: Reflection of shock waves—1. Pseudostationary case. J. Phys. Soc. Jpn. 11(5), 584–592 (1956). https://doi.org/10.1143/JPSJ.11.584

    Article  Google Scholar 

  11. 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). https://doi.org/10.1017/S0022112000003165

    Article  MATH  Google Scholar 

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

    Google Scholar 

  13. 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 Report No. 6271 (1945)

  14. Henderson, L.F., Lozzi, A.: Experiments on transition of Mach reflexion. J. Fluid Mech. 68(1), 139–155 (1975). https://doi.org/10.1017/S0022112075000730

    Article  Google Scholar 

  15. 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). https://doi.org/10.1098/rspa.1980.0015

    Article  Google Scholar 

  16. 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). https://doi.org/10.1063/1.329871

    Article  Google Scholar 

  17. 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). https://doi.org/10.1007/BF00198446

    Article  Google Scholar 

  18. Smith, W.R.: Mutual reflection of two shock waves of arbitrary strengths. Phys. Fluids 2(5), 533–541 (1959). https://doi.org/10.1063/1.1705945

    Article  MathSciNet  MATH  Google Scholar 

  19. Henderson, L.F., Lozzi, A.: Further experiments on transition to Mach reflexion. J. Fluid Mech. 94(3), 541–559 (1979). https://doi.org/10.1017/S0022112079001178

    Article  Google Scholar 

  20. Barbosa, F.J., Skews, B.W.: Experimental confirmation of the von Neumann theory of shock wave reflection transition. J. Fluid Mech. 472, 263–282 (2002). https://doi.org/10.1017/S0022112002002343

    Article  MathSciNet  MATH  Google Scholar 

  21. Herron, T., Skews, B.: On the persistence of regular reflection. Shock Waves 21(6), 573–578 (2011). https://doi.org/10.1007/s00193-011-0341-z

    Article  Google Scholar 

  22. 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(4), 523–550 (2017). https://doi.org/10.1007/s00193-016-0697-1

    Article  Google Scholar 

  23. Alzamora Previtali, F., Timofeev, E., Kleine, H.: On unsteady shock wave reflections from wedges with straight and concave tips. 45th AIAA Fluid Dynamics Conference, Dallas, TX, AIAA Paper 2015-2642 (2015). https://doi.org/10.2514/6.2015-2642

  24. Henderson, L.F., Crutchfield, W.Y., Virgona, R.J.: The effects of thermal conductivity and viscosity of argon on shock waves diffracting over rigid ramps. J. Fluid Mech. 331, 1–36 (1997). https://doi.org/10.1017/S0022112096003850

    Article  Google Scholar 

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

    Google Scholar 

  26. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience Publisher, New York (1948)

    MATH  Google Scholar 

  27. 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). https://doi.org/10.1007/s00193-015-0572-5

    Article  Google Scholar 

  28. Freret L., Groth, C.P.T.: Anisotropic non-uniform block-based adaptive mesh refinement for three-dimensional inviscid and viscous flows. 22nd AIAA Computational Fluid Dynamics Conference, Dallas, TX, AIAA Paper 2015-2613 (2015). https://doi.org/10.2514/6.2015-2613

  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). https://doi.org/10.2514/1.J052576

    Article  Google Scholar 

  30. Northrup, S.A., Groth, C.P.T.: Parallel implicit adaptive mesh refinement scheme for unsteady fully-compressible reactive flows. 21st AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences, San Diego, CA, AIAA Paper 2013-2433 (2013). https://doi.org/10.2514/6.2013-2433

  31. Williamschen, M.J., Groth, C.P.T.: Parallel anisotropic block-based adaptive mesh refinement algorithm for three-dimensional flows. 21st AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences, San Diego, CA, AIAA Paper 2013-2442 (2013). https://doi.org/10.2514/6.2013-2442

  32. Zhang, Z.J., Groth, C.P.T.: Parallel high-order anisotropic block-based adaptive mesh refinement finite-volume scheme. 20th AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences, Honolulu, HI, AIAA Paper 2011-3695 (2011). https://doi.org/10.2514/6.2011-3695

  33. Gao, X., Northrup, S.A., Groth, C.P.T.: Parallel solution-adaptive method for two-dimensional non-premixed combusting flows. Prog. Comput. Fluid Dyn. 11(2), 76–95 (2011). https://doi.org/10.1504/PCFD.2011.038834

    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). https://doi.org/10.1016/j.jcp.2010.01.001

    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). https://doi.org/10.1080/10618560600917583

    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). https://doi.org/10.1080/10618560410001729135

    Article  MATH  Google Scholar 

  37. Hryniewicki, M.K.: On the transition boundary between regular and Mach reflections from a wedge in inviscid and polytropic gases. PhD Thesis, UTIAS, University of Toronto (2016)

  38. von Becker, E.: Instationäire grenzschichten hinter verdichtungsstößen und expansionswellen. Prog. Aerosp. Sci. 1, 104–173 (1961). https://doi.org/10.1016/0376-0421(61)90005-7

    Article  MATH  Google Scholar 

  39. Hornung, H.G., Taylor, J.R.: Transition from regular to Mach reflection of shock waves. Part 1. The effect of viscosity in the pseudosteady case. J. Fluid Mech. 123, 143–153 (1982). https://doi.org/10.1017/S0022112082002997

    Article  Google Scholar 

  40. Hornung, H.: The effect of viscosity on the Mach stem length in unsteady strong shock reflection. In: Meier, G.E.A., Obermeier, F. (eds.) Flow of Real Fluids. Lecture Notes in Physics, vol. 235, pp. 82–91. Springer, Berlin (1985). https://doi.org/10.1007/3-540-15989-4_73

    Chapter  Google Scholar 

  41. Reichenbach, H.: Roughness and heated-layer effects on shock-wave propagation and reflection—experimental results. Ernst Mach Institute, EMI Report E24/85. West Germany, Freiburg (1985)

  42. Lee, J.-H., Glass, I.I.: Pseudo-stationary oblique-shock-wave reflections in frozen and equilibrium air. Prog. Aerosp. Sci. 21(1), 33–80 (1984). https://doi.org/10.1016/0376-0421(84)90003-4

    Article  Google Scholar 

  43. Shirouzu, M., Glass, I.I.: Evaluation of assumptions and criteria in pseudostationary oblique shock-wave reflections. Proc. R. Soc. Lond. A 406(1830), 75–92 (1986). https://doi.org/10.1098/rspa.1986.0065

    Article  Google Scholar 

  44. Wheeler, J.: An interferometric investigation of the regular to Mach reflection transition boundary in pseudostationary flow in air. UTIAS Technical Note No. 256, University of Toronto Institute for Aerospace Studies (1986)

  45. Kleine, H., Timofeev, E., Hakkaki-Fard, A., Skews, B.: The influence of Reynolds number on the triple point trajectories at shock reflection off cylindrical surfaces. J. Fluid Mech. 740, 47–60 (2014). https://doi.org/10.1017/jfm.2013.634

    Article  Google Scholar 

Download references

Acknowledgements

The contributions of Lucie Freret in making the anisotropic algorithm for adaptive mesh refinement more effective and computationally efficient are greatly appreciated. Important papers on shock-induced boundary layers sent by Hans G. Hornung to the first author are gratefully acknowledged. Mach-reflection discussions with Evgeny Timofeev by the first and third authors were very helpful and much 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, via 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

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

Authors
  1. J. J. Gottlieb
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. K. Hryniewicki
    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 E. Timofeev.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gottlieb, J.J., Hryniewicki, M.K. & Groth, C.P.T. Transition boundary between regular and Mach reflections for a moving shock interacting with a wedge in inviscid and polytropic argon. Shock Waves 29, 795–816 (2019). https://doi.org/10.1007/s00193-018-0873-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00193-018-0873-6

Keywords

Search

Navigation