Shock Waves

, Volume 28, Issue 2, pp 253–266 | Cite as

Well-posed Euler model of shock-induced two-phase flow in bubbly liquid

Original Article


A well-posed mathematical model of non-isothermal two-phase two-velocity flow of bubbly liquid is proposed. The model is based on the two-phase Euler equations with the introduction of an additional pressure at the gas bubble surface, which ensures the well-posedness of the Cauchy problem for a system of governing equations with homogeneous initial conditions, and the Rayleigh–Plesset equation for radial pulsations of gas bubbles. The applicability conditions of the model are formulated. The model is validated by comparing one-dimensional calculations of shock wave propagation in liquids with gas bubbles with a gas volume fraction of 0.005–0.3 with experimental data. The model is shown to provide satisfactory results for the shock propagation velocity, pressure profiles, and the shock-induced motion of the bubbly liquid column.


Bubbly liquid Shock wave Two-phase Euler equations Bubble pulsations Well-posedness 1D calculations Experiments 



The present authors would like to thank their colleagues V.S. Aksenov, K.A. Avdeev, F.S. Frolov, I.O. Shamshin, and I. A. Sadykov for their valuable contribution to the experimental results used in this article. This work was supported by the Russian Foundation for Basic Research (project ofi-m 16-29-01065).


  1. 1.
    Frolov, S.M., Frolov, F.S., Aksenov,V.S., Avdeev, K.A., Petrov, A.D.: Pump-jet pulse detonation engine (variants) and method for creating hydro-jet thrust. Patent application PCT/RU2013/001148 on 23.12.2013. Accessed 2 July 2015
  2. 2.
    Avdeev, K.A., Aksenov, V.S., Borisov, A.A., Tukhvatullina, R.R., Frolov, S.M., Frolov, F.S.: Numerical simulation of momentum transfer from a shock wave to a bubbly medium. Russ. J. Phys. Chem. B 9(3), 363–374 (2015). doi: 10.1134/S1990793115030021 CrossRefGoogle Scholar
  3. 3.
    Wijngaarden, L.V.: On equations of motion for mixtures of liquid and gas bubbles. J. Fluid Mech. 33(3), 465–474 (1968). doi: 10.1017/S002211206800145X CrossRefMATHGoogle Scholar
  4. 4.
    Kutateladze, S.S., Nakoryakov, V.E.: Heat and Mass Transfer and Waves in Gas–Liquid Systems. Izdatel’stvo Nauka, Novosibirsk (1984)Google Scholar
  5. 5.
    Kameda, M., Matsumoto, Y.: Shock waves in a liquid containing small gas bubbles. Phys. Fluid 8(2), 322–335 (1996). doi: 10.1063/1.868788 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kameda, M., Shimaura, N., Higashino, F., Matsumoto, Y.: Shock waves in a uniform bubbly flow. Phys. Fluid 10(10), 2661–2668 (1998). doi: 10.1063/1.869779 CrossRefGoogle Scholar
  7. 7.
    Mori, J., Hijikata, K., Komine, A.: Propagation of pressure waves in two-phase flow. Int. J. Multiph. Flow 2(2), 139–152 (1975). doi: 10.1016/0301-9322(75)90004-X CrossRefGoogle Scholar
  8. 8.
    Sychev, A.I.: Intense shock waves in bubble media. Tech. Phys. 55(6), 783–788 (2010). doi: 10.1134/S1063784210060058 CrossRefGoogle Scholar
  9. 9.
    Nakoryakov, V.E., Pokusaev, B.G., Shreiber, I.R., Kuznetsov, V.V., Malykh, N.V.: Wave processes in two-phase systems. In: Kutateladze, S.S. (ed.) Inst. Therm. Phys. Sib. Br. USSR Academy of Sciences, p 54. (1975)Google Scholar
  10. 10.
    Borisov, A.A., Gelfand, B.E., Timofeev, E.I.: Shock waves in liquids containing gas bubbles. Int. J. Multiph. Flow 9(5), 531–543 (1983) doi: 10.1016/0301-9322(83)90016-2
  11. 11.
    Gelfand, B.E., Gubin, S.A., Kogarko, B.S., Kogarko, S.M.: Investigations of compression waves in a mixture of liquid with gas bubbles. Doklady USSR Acad. Sci., 213, No.5. (1973)Google Scholar
  12. 12.
    Avdeev, K.A., Aksenov, V.S., Borisov, A.A., Frolov, S.M., Frolov, F.S., Shamshin, I.O.: Momentum transfer from a shock wave to a bubbly liquid. Russ. J. Phys. Chem. B 9(6), 895–900 (2015). doi: 10.1134/S1990793115060032 CrossRefGoogle Scholar
  13. 13.
    Nigmatulin, R.I.: Dynamics of Multiphase Media, vol. 1. Hemisphere, New York (1990)Google Scholar
  14. 14.
    Preston, A.T., Colonius, T., Brennen, C.E.: A reduced-order model of diffusive effects on the dynamics of bubbles. Phys. Fluid 19(12), 123302 (2007). doi: 10.1063/1.2825018 CrossRefMATHGoogle Scholar
  15. 15.
    Gidaspow, D.: Modeling of two phase flow. Heat transfer 7, 163–168 (1974)Google Scholar
  16. 16.
    van Wijngaarden, L.: Some problems in the formulation of the equations for gas/liquid flows. In: Koiter, W.T. (ed.) Theoretical and Applied Mechanics. North-Holland Publishing Company, Amsterdam, pp. 249–260 (1976)Google Scholar
  17. 17.
    Stewart, H.B.: Stability of two-phase flow calculation using two-fluid models. J. Comput. Phys. 33(2), 259–270 (1979). doi: 10.1016/0021-9991(79)90020-2 CrossRefMATHGoogle Scholar
  18. 18.
    Klebanov, L.A., Kroshilin, A.E., Nigmatulin, B.I., Nigmatulin, R.I.: On the hyperbolicity, stability and correctness of the Cauchy problem for the system of equations of two-speed motion of two-phase media. J. Appl. Math. Mech. 46(1), 66–74 (1982). doi: 10.1016/0021-8928(82)90084-3 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ramshaw, J.D., Trapp, J.A.: Characteristics, stability, and short-wavelength phenomena in two-phase flow equation systems. Nucl. Sci. Eng. 66(1), 93–102 (1978)CrossRefGoogle Scholar
  20. 20.
    Radvogin, YuB, Posvyanskii, V.S., Frolov, S.M.: Stability of 2D two-phase reactive flows. J. Phys. IV 12, 437–444 (2002). doi: 10.1051/jp4:20020313 Google Scholar
  21. 21.
    Stuhmiller, J.H.: The influence of interfacial pressure forces on the character of two-phase flow model equations. Int. J. Multiph. Flow 3(6), 551–560 (1977). doi: 10.1016/0301-9322(77)90029-5 CrossRefMATHGoogle Scholar
  22. 22.
    Yeom, G.S., Chang, K.S., Baek, S.W.: Robust Waf-Hll scheme for compressible two-pressure two-velocity multiphase flow model. Eng. Appl. Comput. Fluid Mech. 6(1), 144–162 (2012). doi: 10.1080/19942060.2012.11015410 Google Scholar
  23. 23.
    Yeom, G.S., Chang, K.S.: Two-dimensional two-fluid two-phase flow simulation using an approximate Jacobian matrix for HLL scheme. Numer. Heat Transf. Part B Fundam. 56(5), 372–392 (2010). doi: 10.1080/10407790903507998 CrossRefGoogle Scholar
  24. 24.
    Gavrilyuk, S., Saurel, R.: Mathematical and numerical modeling of two-phase compressible flows with micro-inertia. J. Comput. Phys. 175(1), 326–360 (2002). doi: 10.1006/jcph.2001.6951 MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Burdukov, A.P., Kuznetsov, V.V., Kutateladze, S.S., Nakoryakov, V.E., Pokusaev, B.G., Shreiber, I.R.: Shock wave in a gas–liquid medium. J. Appl. Mech. Tech. Phys. 14(3), 349–352 (1973). doi: 10.1007/BF00850948 CrossRefGoogle Scholar
  26. 26.
    Noordzij, L. Shock waves in bubble–liquid mixtures. Phys. Communs 3 (1971)Google Scholar
  27. 27.
    Plesset, M.S., Prosperetti, A.: Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145–185 (1977). doi: 10.1146/annurev.fl.09.010177.001045
  28. 28.
    Schlichting, H.: Boundary-Layer Theory. McGraw-Hill, New-York (1968). doi: 10.1007/978-3-662-52919-5
  29. 29.
    Lamb, H.: Hydrodynamics. Cambridge University Press, Cambridge (1932)MATHGoogle Scholar
  30. 30.
    Clift, R., Grace, J.R., Weber, M.E.: Bubbles, Drops, and Particles. Courier Corporation, New York (2005)Google Scholar
  31. 31.
    Harlow, F.H., Amsden, A.A.: A numerical fluid dynamics calculation method for all flow speeds. J. Comput. Phys. 8(2), 197–213 (1971). doi: 10.1016/0021-9991(71)90002-7 CrossRefMATHGoogle Scholar
  32. 32.
    Zel’dovich, Y.B., Raizer, Y.P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Dover Publications, New York (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Semenov Institute of Chemical PhysicsMoscowRussian Federation
  2. 2.National Nuclear Research University “MEPhI”MoscowRussian Federation

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