Shock Waves

, Volume 23, Issue 6, pp 583–594 | Cite as

Numerical investigations of the porosity effect on the shock focusing process

Original Article


The effect of cylindrical obstacles and the porosity in between them along the path of a converging cylindrical shock is studied through numerical simulations. An initially cylindrical converging shock wave is perturbed by cylindrical obstacles placed radially in its path. High pressures and temperatures are achieved as the shock wave is focused. Results show that the shape of the shock wave close to the point of convergence as well as the porosity and type of shock wave reflection the converging shock undergoes influence the peak values. Various configurations of the obstacle size and number are considered. The Guderley constant for each case is compared with previous reported experimental values.


Shock focusing Regular reflection  Irregular reflection Overlapping structured grids 



The authors would sincerely like to thank the anonymous reviewers for their insightful and generous comments.


  1. 1.
    Apazidis, N., Lesser, M.: On generation and convergence of polygonal-shaped shock waves. J. Fluid Mech. 309, 301–319 (1996)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Apazidis, N., Lesser, M., Tillmark, N., Johansson, B.: An experimental and theoretical study of converging polygonal shock waves. Shock Waves 12, 39–58 (2002)CrossRefMATHGoogle Scholar
  3. 3.
    Berger, S., Sadot, O., Ben-Dor, G.: Experimental investigation on the shock-wave load attenuation by geometrical means. Shock Waves 20, 29–40 (2010)CrossRefGoogle Scholar
  4. 4.
    Betelu, S., Aronson, D.: Focusing of noncircular self-similar shock waves. Phys. Rev. Lett. 87, 074501 (2001)CrossRefGoogle Scholar
  5. 5.
    Britan, A., Igra, O., Ben-Dor, G., Shapiro, H.: Shock wave attenuation by grids and orifice plates. Shock Waves 16, 1–15 (2006)CrossRefGoogle Scholar
  6. 6.
    Britan, A., Karpov, A.V., Vasilev, E.I., Igra, O., Ben-Dor, G., Shapiro, E.: Experimental and numerical study of shock wave interaction with perforated plates. J. Fluids Eng. 126(3), 399–409 (2004)CrossRefGoogle Scholar
  7. 7.
    Butler, D.S.: Converging spherical and cylindrical shocks. Rep. No 54/54, Armament Research and Development Establishment, Ministry of Supply, Fort Halstead, Kent, GB (1954)Google Scholar
  8. 8.
    Chaudhuri, A., Hadjadj, A., Sadot, O., Ben-Dor, G.: Numerical study of shock-wave mitigation through matrices of solid obstacles. Shock Waves 23(1), 91–101 (2013)CrossRefGoogle Scholar
  9. 9.
    Chaudhuri, A., Hadjadj, A., Sadot, O., Glazer, E.: Computational study of shock-wave interaction with solid obstacles using immersed boundary methods. Int. J. Numer. Methods Eng. 89(8), 975–990 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chesshire, G., Henshaw, W.D.: Composite overlapping meshes for the solution of partial differential equations. J. Comput. Phys. 90(1), 1–64 (1990)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chester, W.: The quasi-cylindrical shock tube. Phil. Mag. 45(7), 1239–1301 (1954)MathSciNetGoogle Scholar
  12. 12.
    Chisnell, R.F.: The motion of a shock wave in a channel, with applications to cylindrical and spherical shocks. J. Fluid Mech. 2(3), 286–298 (1957)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chisnell, R.F.: An analytic description of converging shock waves. J. Fluid Mech. 354, 357–375 (1998)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dimotakis, P.E., Samtaney, R.: Planar shock cylindrical focusing by a perfect-gas lens. Phys. Fluids 18(3), 031705 (2006)CrossRefGoogle Scholar
  15. 15.
    Dosanjh, D.S.: Interaction of Grids with Traveling Shock Waves. In: NASA Technical Note TN 3680 (1956)Google Scholar
  16. 16.
    Eliasson, V., Apazidis, N., Tillmark, N.: Controlling the form of strong converging shocks by means of disturbances. Shock Waves 17, 29–42 (2007)CrossRefGoogle Scholar
  17. 17.
    Eliasson, V., Henshaw, W., Appelö, D.: On cylindrically converging shock waves shaped by obstacles. Physica D Nonlinear Phenomena 237, 2203–2209 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Eliasson, V., Tillmark, N., Szeri, A.J., Apazidis, N.: Light emission during shock wave focusing in air and argon. Phys. Fluids 19, 106106 (2007)CrossRefGoogle Scholar
  19. 19.
    Fong, K., Ahlborn, B.: Stability of converging shock waves. Phys. Fluids 22(3), 416–421 (1979)CrossRefGoogle Scholar
  20. 20.
    Gardner, J.H., Book, D.L., Bernstein, I.B.: Stability of imploding shocks in the CCW approximation. J. Fluid Mech. 114, 41–58 (1982)CrossRefMATHGoogle Scholar
  21. 21.
    Guderley, G.: Starke kugelige und zylindrische Verdichtungsstöße in der Nähe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrt Forsch. 19, 302–312 (1942)Google Scholar
  22. 22.
    Henshaw, W.D., Schwendeman, D.W.: An adaptive numerical scheme for high-speed reactive flow on overlapping grids. J. Comput. Phys. 191(2), 420–447 (2003)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Henshaw, W.D., Schwendeman, D.W.: Moving overlapping grids with adaptive mesh refinement for high-speed reactive and non-reactive flow. J. Comput. Phys. 216(2), 744–779 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Hornung, H., Pullin, D., Ponchaut, N.: On the question of universality of imploding shock waves. Acta Mech. 201, 31–35 (2008)CrossRefMATHGoogle Scholar
  25. 25.
    Hosseini, S.H.R., Takayama, K.: Implosion of a spherical shock wave reflected from a spherical wall. J. Fluid Mech. 530, 223–239 (2005)CrossRefMATHGoogle Scholar
  26. 26.
    Kjellander, M., Tillmark, N., Apazidis, N.: Experimental determination of self-similarity constant for converging cylindrical shocks. Phys. Fluids 23(11), 116103 (2011)CrossRefGoogle Scholar
  27. 27.
    Kleine, H.: Time resolved shadowgraphs of focusing cylindrical shock waves. In: Study Treatise at the Stoßenwellenlabor, RWTH Achen, FRG (1985)Google Scholar
  28. 28.
    Mishkin, E.A., Fujimoto, Y.: Analysis of a cylindrical imploding shock wave. J. Fluid Mech. 89(1), 61–78 (1978)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Naiman, H., Knight, D.: The effect of porosity on shock interaction with a rigid, porous barrier. Shock Waves 16, 321–337 (2007)CrossRefMATHGoogle Scholar
  30. 30.
    Nakamura, Y.: Analysis of self-similar problems of imploding shock waves by the method of characteristics. Phys. Fluids 26(5), 1234–1239 (1983)CrossRefMATHGoogle Scholar
  31. 31.
    de Neef, T., Nechtman, C.: Numerical study of the flow due to a cylindrical implosion. Comput. Fluids 6(3), 185–202 (1978)CrossRefMATHGoogle Scholar
  32. 32.
    Perry, R.W., Kantrowitz, A.: The production and stability of converging shock waves. J. Appl. Phys. 22(7), 878–886 (1951)CrossRefGoogle Scholar
  33. 33.
    Ponchaut, N.F., Hornung, H.G., Pullin, D.I., Mouton, C.A.: On imploding cylindrical and spherical shock waves in a perfect gas. J. Fluid Mech. 560, 103–122 (2006)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Ramsey, S.D., Kammb, J.R., Bolstad, J.H.: The Guderley problem revisited. Int. J. Comput. Fluid Dyn. 26(2), 79–99 (2012)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Schwendeman, D.: On converging shock waves of spherical and polyhedral form. J. Fluid Mech. 454, 365–386 (2002)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Schwendeman, D., Whitham, D.: On converging shock waves. Proc. R. Soc. Lond. A413, 297–311 (1987)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Seeraj, S., Skews, B.W.: Dual-element directional shock wave attenuators. Exp. Therm. Fluid Sci. 33(3), 503–516 (2009)CrossRefGoogle Scholar
  38. 38.
    Shi, H., Yamamura, K.: The interaction between shock waves and solid spheres arrays in a shock tube. Acta Mech. Sin. 20(3), 219–227 (2004)CrossRefGoogle Scholar
  39. 39.
    Skews, B.W., Kleine, H.: Flow features resulting from shock wave impact on a cylindrical cavity. J. Fluid Mech. 580, 481–493 (2007)CrossRefMATHGoogle Scholar
  40. 40.
    Stanyukovich, K.P.: Unsteady motion of continuous media. Pergamon Press, Oxford (1960)Google Scholar
  41. 41.
    Taieb, D., Ribert, G., Hadjadj, A.: Numerical simulations of shock focusing over concave surfaces. AIAA J. 48(8), 1739–1747 (2010) Google Scholar
  42. 42.
    Takayama, K., Kleine, H., Grönig, H.: An experimental investigation of the stability of converging cylindrical shock waves in air. Exp. Fluids 5, 315–322 (1987)Google Scholar
  43. 43.
    Takayama, K., Onodera, O., Hoshizawa, Y.: Experiments on the stability of converging cylindrical shock waves. Theor. Appl. Mech. 32, 117–127 (1984)Google Scholar
  44. 44.
    Vandenboomgaerde, M., Aymard, C.: Analytical theory for planar shock focusing through perfect gas lens and shock tube experiment designs. Phys. Fluids 23, 016101 (2011)CrossRefGoogle Scholar
  45. 45.
    Watanabe, M., Takayama, K.: Stability of converging cylindrical shock waves. Shock Waves 5, 149–160 (1991)CrossRefGoogle Scholar
  46. 46.
    Welsh, R.L.: Imploding shocks and detonations. J. Fluid Mech. 29, 61–79 (1967)CrossRefMATHGoogle Scholar
  47. 47.
    Whitham, G.: Linear and nonlinear waves. Wiley, New York (1974)MATHGoogle Scholar
  48. 48.
    Zel’dovich, Y., Raizer, Y.: Physics of shock waves and high-temperature hydrodynamic phenomena. Dover Publications, New York (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.University of Southern CaliforniaLos AngelesUSA

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