Shock Reflection in Axisymmetric Internal Flows

  • B. Shoesmith
  • S. Mölder
  • H. Ogawa
  • E. Timofeev
Conference paper


The flow downstream of an axisymmetric conical shock wave, with a downstream pointing apex, can be predicted by solving the Taylor-Maccoll equations. Previous research, however, has suggested that these theoretical flowfields are not fully realisable in practice, and that a Mach reflection forms towards the centreline of the flow. This phenomenon is investigated for the case where the freestream Mach number is 3.0 and the shock angle is 150\(^\circ \). A range of complementary prediction techniques that include the solution to the Taylor-Maccoll equations, the method of characteristics, curved shock theory and CFD, are used to gain insight into this flow. The case where a cylindrical centrebody is placed along the axis of symmetry is studied for several values of centrebody radius that are expected to produce regular reflection at the centrebody surface. An analysis of pressure gradients suggests that the flowfield downstream of the reflected shock does not contribute to the process of transition from regular to Mach reflection at these conditions.



The present research of B.S. and E.T. is supported by the Fonds de recherche du Quèbec - Nature et technologies (FRQNT) via the Team Research Project program and the National Science and Engineering Research Council (NSERC) via the Discovery Grant program. B.S. gratefully acknowledges the McGill Engineering Doctoral Award (MEDA) funded in part by the Faculty of Engineering, McGill University. Rabi Tahir’s support regarding Masterix code is greatly appreciated.


  1. 1.
    Courant, R., Friedrichs, K.O.: Supersonic flow and shock waves. Interscience (1948)Google Scholar
  2. 2.
    Grozdovskii, G.L.: Supersonic axisymmetric conical flows with conical shocks adjacent to uniform parallel flows. Prikl Mat. Mekh. 23(2), 379–383 (1959)MathSciNetGoogle Scholar
  3. 3.
    Isakova, N.P., Kraiko, A.N., P’yankov, K.S., Tillyayeva, N.I.: The amplification of weak shock waves in axisymmetric supersonic flow and their reflection from an axis of symmetry. J. Appl. Math. Mech. 76, 451–465 (2012)Google Scholar
  4. 4.
    Kraiko, A.N., Tillyayeva, N.I.: Axisymmetric-conical and locally conical flows without swirling. J. Appl. Mech. Tech. Phy. 55(2), 282–298 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Rylov, A.I.: On the impossibility of regular reflection of a steady-state shock wave from the axis of symmetry. Appl. Math. Mech. 54(2), 245–249 (1990)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Mölder, S., Gulamhussein, A., Timofeev, E., Voinovich, P.A.: Focusing of conical shocks at the centerline of symmetry. In: Proceedings of the 21th ISSW, Vol. 2, pp. 875–880, Panther Publishing (1997)Google Scholar
  7. 7.
    Timofeev, E., Mölder, S., Voinovich, P., Hosseini, S.H.R., Takayama, K.: Shock wave reflections in axisymmetric flow. In: Lu, F. (ed) Shock Waves, CD-ROM Proc of the 23th International Symp on Shock Waves, Fort Worth, USA, 22–27 July, 2001, University of Texas at Arlington, pp. 1486–1493 (2001)Google Scholar
  8. 8.
    Mölder, S.: Internal axisymmetric conical flow. AIAA J. 5(7), 1252–1255 (1967)CrossRefGoogle Scholar
  9. 9.
    Mölder, S.: Curved aerodynamic shock waves. PhD Thesis, McGill University (2012)Google Scholar
  10. 10.
    Masterix, ver. 3.40, RBT Consultants, Toronto, Ontario, Canada (2003–2015)Google Scholar
  11. 11.
    Savchuk, A., Timofeev, E.: On further development of an unstructured space-marching technique. AIAA Paper 2011–3690 (2011)Google Scholar
  12. 12.
    Délery, J.: Handbook of compressible aerodynamics. Wiley (2010)Google Scholar
  13. 13.
    Mölder, S.: Curved shock theory. Shock Waves 26(4), 337–353 (2015)CrossRefGoogle Scholar
  14. 14.
    Saito, T., Voinovich, P., Timofeev, E., Takayama, K.: Development and application of high-resolution adaptive numerical techniques in Shock Wave Research Center. In: Toro, E.F. (ed) Godunov Methods: Theory and Applications, pp. 763–784. Kluwer Academic/Plenum Publishers, New York, USA (2001)Google Scholar
  15. 15.
    Mölder, S., Timofeev, E., Emanuel, G.: Shock detachment from curved surfaces. In: Kontis, K. (ed) Proc of the 28th International Symposium on Shock Waves, Manchester, UK, 17-22 July, 2011, Springer, Vol. 2, pp. 593–598 (2011)Google Scholar
  16. 16.
    Mölder, S., Timofeev, E.: Regular-to-Mach reflection transition on curved surfaces. In: Book of Papers, XXIV ICTAM, 21–26 August 2016, Montreal, Canada (2016)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • B. Shoesmith
    • 1
  • S. Mölder
    • 2
  • H. Ogawa
    • 3
  • E. Timofeev
    • 1
  1. 1.Department of Mechanical EngineeringMcGill UniversityMontrealCanada
  2. 2.Department of Aerospace EngineeringRyerson UniversityTorontoCanada
  3. 3.School of EngineeringRMIT UniversityMelbourneAustralia

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