Shock Waves

, Volume 29, Issue 1, pp 221–234 | Cite as

Simulation of shock-induced bubble collapse using a four-equation model

  • E. GoncalvesEmail author
  • Y. Hoarau
  • D. Zeidan
Original Article


This paper presents a numerical study of the interaction between a planar incident shock wave with a cylindrical gas bubble. Simulations are performed using an inviscid compressible one-fluid solver based upon three conservation laws for the mixture variables, namely mass, momentum, and total energy along with a supplementary transport equation for the volume fraction of the gas phase. The study focuses on the maximum pressure generated by the bubble collapse. The influence of the strength of the incident shock is investigated. A law for the maximum pressure function of the Mach number of the incident shock is proposed.


Bubble collapse Shock waves Pressure peak Jet formation 



The authors would like to thank the reviewers for the constructive suggestions leading to substantial improvement of the text.


  1. 1.
    Plesset, M., Chapman, R.: Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary. J. Fluids Mech. 47, 283–290 (1971). CrossRefGoogle Scholar
  2. 2.
    Jamaluddin, A., Ball, G., Turangan, C., Leighton, T.: The collapse of single bubbles and approximation of the far-field acoustic emissions for cavitation induced by shock wave lithotripsy. J. Fluids Mech. 677, 305–341 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bourne, N.: On the collapse of cavities. Shock Waves 11, 447–455 (2002). CrossRefGoogle Scholar
  4. 4.
    Haas, J., Sturtevant, B.: Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluids Mech. 181, 41–76 (1987). CrossRefGoogle Scholar
  5. 5.
    Terashima, H., Tryggvason, G.: A front-tracking/ghost-fluid method for fluid interfaces in compressible flows. J. Comput. Phys. 228(11), 4012–4037 (2009). CrossRefzbMATHGoogle Scholar
  6. 6.
    Ball, G., Howell, B., Leighton, T., Schofield, M.: Shock-induced collapse of a cylindrical air cavity in water: a free-Lagrange simulation. Shock Waves 10, 265–276 (2000). CrossRefzbMATHGoogle Scholar
  7. 7.
    Hu, X., Khoo, B., Adams, N., Huang, F.: A conservative interface method for compressible flow. J. Comput. Phys. 219, 553–578 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lauer, E., Hu, X., Hickel, S., Adams, N.: Numerical investigation of collapsing cavity arrays. Phys. Fluids 24, 052104 (2012). CrossRefGoogle Scholar
  9. 9.
    Liu, T., Khoo, B., Yeo, K.: Ghost fluid method for strong shock impacting on material interface. J. Comput. Phys. 190(2), 651–681 (2003). CrossRefzbMATHGoogle Scholar
  10. 10.
    Nourgaliev, R., Dinh, T., Theofanous, T.: Adaptive characteristics-based matching for compressible multifluid dynamics. J. Comput. Phys. 213, 500–529 (2006). CrossRefzbMATHGoogle Scholar
  11. 11.
    Abgrall, R.: How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. J. Comput. Phys. 125(1), 150–160 (1996). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Allaire, G., Clerc, S., Kokh, S.: A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181(2), 577–616 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Apazidis, N.: Numerical investigation of shock induced bubble collapse in water. Phys. Fluids 28, 046101 (2016). CrossRefGoogle Scholar
  14. 14.
    Coralic, V., Colonius, T.: Shock-induced collapse of a bubble inside a deformable vessel. Eur. J. Mech. B/Fluids 40, 64–74 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Johnsen, E., Colonius, T.: Numerical simulations of non-spherical bubble collapse. J. Fluids Mech. 629, 231–262 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Massoni, J., Saurel, R., Nkonga, B., Abgrall, R.: Some models and Eulerian methods for interface problems between compressible fluids with heat transfer. Int. J. Heat Mass Transf. 45, 1287–1307 (2002). CrossRefzbMATHGoogle Scholar
  17. 17.
    Shyue, K.: A volume-fraction based algorithm for hybrid barotropic and non-barotropic two-fluid flow problems. Shock Waves 15, 407–423 (2006). CrossRefzbMATHGoogle Scholar
  18. 18.
    Baer, M., Nunziato, J.: A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiph. Flow 12, 861–889 (1986). CrossRefzbMATHGoogle Scholar
  19. 19.
    Zeidan, D.: Assessment of mixture two-phase flow equations for volcanic flows using Godunov-type methods. Appl. Math. Comput. 272, 707–719 (2016). MathSciNetGoogle Scholar
  20. 20.
    Daude, F., Galon, P., Gao, Z., Blaud, E.: Numerical experiments using a HLLC-type scheme with ALE formulation for compressible two-phase flows five-equation models with phase transition. Comput. Fluids 94, 112–138 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kapila, A., Menikoff, R., Bdzil, J., Son, S., Stewart, D.: Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations. Phys. Fluids 13(10), 3002–3024 (2001). CrossRefzbMATHGoogle Scholar
  22. 22.
    Murrone, A., Guillard, H.: A five equation reduced model for compressible two phase flows problems. J. Comput. Phys. 202(2), 664–698 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Goncalves, E., Charriere, B.: Modelling for isothermal cavitation with a four-equation model. Int. J. Multiph. Flow 59, 54–72 (2014). CrossRefGoogle Scholar
  24. 24.
    Kunz, R., Boger, D., Stinebring, D., Chyczewski, T., Lindau, J., Gibeling, H., Venkateswaran, S., Govindan, T.: A preconditioned Navier–Stokes method for two-phase flows with application to cavitation prediction. Comput. Fluids 29(8), 849–875 (2000). CrossRefzbMATHGoogle Scholar
  25. 25.
    Goncalves, E., Patella, R.F.: Numerical simulation of cavitating flows with homogeneous models. Comput. Fluids 38(9), 1682–1696 (2009). CrossRefzbMATHGoogle Scholar
  26. 26.
    Zheng, J., Khoo, B., Hu, Z.: Simulation of wave–flow–cavitation interaction using a compressible homogenous flow method. Commun. Comput. Phys 14(2), 328–354 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Goncalves, E.: Numerical study of expansion tube problems: Toward the simulation of cavitation. Comput. Fluids 72, 1–19 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ishii, C., Hibiki, T.: Thermo-Fluid Dynamics of Two-Phase Flow. Springer, New York (2006).
  29. 29.
    Metayer, O.L., Massoni, J., Saurel, R.: Elaborating equations of state of a liquid and its vapor for two-phase flow models. Int. J. Therm. Sci. 43, 265–276 (2004). CrossRefGoogle Scholar
  30. 30.
    Goncalves, E., Zeidan, D.: Simulation of compressible two-phase flows using a void ratio transport equation. Commun. Comput. Phys. (in press)Google Scholar
  31. 31.
    Batten, P., Clarke, N., Lambert, C., Causon, D.: On the choice of wave speeds for the HLLC Riemann solver. SIAM J. Sci. Comput. 18(6), 1553–1570 (1997). MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Toro, E., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4, 25–34 (1994). CrossRefzbMATHGoogle Scholar
  33. 33.
    Hawker, N., Ventikos, Y.: Interaction of a strong shockwave with a gas bubble in a liquid medium: a numerical study. J. Fluids Mech. 701, 59–97 (2012). CrossRefzbMATHGoogle Scholar
  34. 34.
    Ozlem, M., Schwendeman, D., Kapila, A., Henshaw, W.: A numerical study of shock-induced cavity collapse. Shock Waves 22, 89–117 (2012).

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut Pprime, UPR 3346 CNRSISAE-ENSMAFuturoscope ChasseneuilFrance
  2. 2.ICUBE Laboratory, UMR 7357 CNRSUniversity of StrasbourgStrasbourgFrance
  3. 3.School of Basic Sciences and HumanitiesGerman Jordanian UniversityAmmanJordan

Personalised recommendations