Theoretical and Computational Fluid Dynamics

, Volume 19, Issue 5, pp 303–317

Vortex ring modelling of toroidal bubbles

Original Article


During the collapse of a bubble near a surface, a high-speed liquid jet often forms and subsequently impacts upon the opposite bubble surface. The jet impact transforms the originally singly-connected bubble to a toroidal bubble, and generates circulation in the flow around it. A toroidal bubble simulation is presented by introducing a vortex ring seeded inside the bubble torus to account for the circulation. The velocity potential is then decomposed into the potential of the vortex ring and a remnant potential. Because the remnant potential is continuous and satisfies the Laplace equation, it can be modelled by the boundary-integral method, and this circumvents an explicit domain cut and associated numerical treatment. The method is applied to study the collapse of gas bubbles in the vicinity of a rigid wall. Good agreement is found with the results of Best (J. Fluid Mech. 251 79–107, 1993), obtained by a domain cut method. Examination of the pressure impulse on the wall during jet impact indicates that the high-speed liquid jet has a significant potential for causing damage to a surface. There appears to be an optimal initial distance where the liquid jet is most damaging.


Toroidal bubbles Boundary-integral method Potential flow theory 


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  1. 1.
    Benjamin, T.B., Ellis, A.T.: The collapse of cavitation bubbles and pressure thereby produced against solid boundary. Philos. Trans. R. Soc. Lond. A. 260, 221–240 (1966)ADSCrossRefGoogle Scholar
  2. 2.
    Guerri, L., Lucca, G., Prosperetti, A.: A numerical method for the dynamics of non-spherical cavitation bubbles, Proc. 2nd Int. Colloq. on Drops and Bubbles, p. 175. CA (1981)Google Scholar
  3. 3.
    Blake, J.R., Taib, B.B., Doherty, G.: Transient cavities near boundaries. Part 1. Rigid boundary. J. Fluid Mech. 170, 479 (1986)MATHGoogle Scholar
  4. 4.
    Baker, G.R., Moore, D.W.A.: The rise and distortion of a two-dimensional gas bubble in an inviscid liquid. Phys. of Fluids. 1(9), 1451 (1989)MathSciNetADSMATHGoogle Scholar
  5. 5.
    Best, J.P., Kucera, A.: A numerical investigation of non-spherical rebounding bubbles. J. Fluid Mech. 245, 137 (1992)ADSMATHGoogle Scholar
  6. 6.
    Brujan, E.A., Keen, G.S., Vogel, A., Blake, J.R.: The final stage of the collapse of a cavitation bubble close to a rigid boundary. Phys. of Fluids. 14(1), 85 (2002)CrossRefADSGoogle Scholar
  7. 7.
    Blake, J.R., Gibson, D.C.: Growth and collapse of a vapour cavity near a free surface. J. Fluid Mech. 111, 123 (1981)ADSGoogle Scholar
  8. 8.
    Blake, J.R., Taib, B.B., Doherty, G.: Transient cavities near boundaries. Part 2. Free surface. J. Fluid Mech. 181, 197 (1987)Google Scholar
  9. 9.
    Blake, J.R., Hooton, M.C., Robinson, P.B., Tong, P.R.: Collapsing cavities, toroidal bubbles and jet impact. Philos. Trans. R. Soc. Lond. A 355, 537–550 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Wang, Q.X., Yeo, K.S., Khoo, B.C., Lam, K.Y.: Strong interaction between buoyancy bubble and free surface. Theoret. and Comput. Fluid Dynamics 8, 73–88 (1996a)MATHGoogle Scholar
  11. 11.
    Wang, Q.X., Yeo, K.S., Khoo, B.C., Lam, K.Y.: Nonlinear interaction between gas bubble and free surface. Computers & Fluids 25, No. 7, 607 (1996b)CrossRefMATHGoogle Scholar
  12. 12.
    Chahine, G.L.: Numerical modeling of the dynamic behavior of bubble in nonuniform flow field. In: ASME Symp. on Numerical Methods for Multiphase Flows, Toronto (1990)Google Scholar
  13. 13.
    Chahine, G.L., Perdue, T.O.: Simulation of the three dimensional behaviour of an unsteady large bubble near a structure. In: Wang, T.G. (ed.) Drops and Bubbles, 3rd Int. Colloq., Monterey, CA, pp. 188–199. Amer. Inst. of Phys. (1988)Google Scholar
  14. 14.
    Wilkerson, S.A.: A boundary integral approach to three dimensional underwater explosion bubble dynamics. Ph.D. Dissertation, Johns Hopkins University, Baltimore, MD (1990)Google Scholar
  15. 15.
    Harris, P.J.: A numerical model for determining the motion of a bubble close to a fixed rigid structure in a fluid. Int. J. Numer. Methods Eng. 33, 1813 (1992)CrossRefMATHGoogle Scholar
  16. 16.
    Harris, P.J.: A numerical method for predicting the motion of a bubble close to a moving rigid structure. Commun. Numer. Methods Eng. 9, 81 (1993)MATHGoogle Scholar
  17. 17.
    Wang, Q.X.: The numerical analyses of the evolution of a gas bubble near an inclined wall. Theoret. & Comput. Fluid Dynamics 12, 29–51 (1998)MATHGoogle Scholar
  18. 18.
    Wang, Q.X.: Numerical simulation of violent bubble motion. Phys. of Fluids 16(5), 1610–1619 (2004)Google Scholar
  19. 19.
    Pozrikidis, C.: Numerical simulation of three-dimensional bubble oscillations by a generalized vortex method theoret. and comput. Fluid Dynamics 16(2), 151–169 (2002)MATHMathSciNetGoogle Scholar
  20. 20.
    Wang, C., Khoo, B.C.: An indirect boundary element method for three-dimensional explosion bubbles. J. Comput. Phys. 194 (2), 451–480 (2004)ADSMATHGoogle Scholar
  21. 21.
    Plesset, M.S., Prosperetti, A.: Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145 (1977)CrossRefADSGoogle Scholar
  22. 22.
    Blake, J.R., Gibson, D.C.: Cavitation bubbles near boundaries. Annu. Rev. Fluid Mech. 19, 99–123 (1987)CrossRefADSGoogle Scholar
  23. 23.
    Best, J.P.: The formulation of toroidal bubbles upon collapse of transient cavities. J. Fluid Mech. 251, 79–107 (1993)MATHADSGoogle Scholar
  24. 24.
    Best, J.P.: The rebound of toroidal bubbles. In: Blake, J.R., Boulton-Stone, J.M., Thomas, N.H. (eds.) Bubble Dynamics and Interface Phenomena, Fluid Mechanics and its Applications, vol. 23, pp. 405–412. Kluwer Academic Publishers, Dordrecht (1994)Google Scholar
  25. 25.
    Pearson, A., Cox, E., Blake, J.R., Otto, S.R.: Bubble interactions near a free surface. Eng. Anal. Bound. Elem. 28(4), 295–313 (2004a)CrossRefMATHGoogle Scholar
  26. 26.
    Pearson, A., Blake, J.R., Otto, S.R.: Jets in bubbles. J Eng. Math. 48(3–4), 391–412 (2004b)MathSciNetMATHGoogle Scholar
  27. 27.
    Zhang, S., Duncan, J.H., Chahine, G.L.: The final stage of the collapse of a cavitation bubble near of a rigid wall. J. Fluid Mech. 257, 147–181 (1993)ADSMATHGoogle Scholar
  28. 28.
    Zhang, S., Duncan, J.H.: On the non-spherical collapse and rebound of cavitation bubble. Physics of Fluids 6(7), 2352–2362 (1994)CrossRefADSMATHGoogle Scholar
  29. 29.
    Szymczak, W.G., Roger, J.C.W., Solomon, J.M., Berger, A.E.: A numerical algorithm for hydrodynamic free boundary problems. J. Comput. Phys. 106, 319–336 (1993)MathSciNetADSMATHGoogle Scholar
  30. 30.
    Pedley, T.J.: The toroidal bubble. J. Fluid Mech. 32, 97–112 (1968)MATHADSGoogle Scholar
  31. 31.
    Lundgren, T.S., Mansour, N.N.: Vortex ring bubbles. J. Fluid Mech. 72, 391–399 (1991)Google Scholar
  32. 32.
    Taib, B.B.: Boundary integral methods applied to cavitation bubble dynamics. Ph.D. Thesis, Univ. of Wollongong, NSW, Australia (1985)Google Scholar
  33. 33.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, NY (1965)Google Scholar
  34. 34.
    Birkhoff, G., Zarantonello, E.H.: Jet, Cavities and Wakes. Academy Press, NY (1957)Google Scholar
  35. 35.
    Keller, J.B., Milewski P.A., Vanden-Broeck, J.M.: Breaking and merging of liquid sheets and filaments. J. Eng. Math. 42(3–4), 283–290 (2002)MathSciNetMATHGoogle Scholar
  36. 36.
    Zhang, Y.L., Yeo, K.S., Khoo, B.C., Wang, C.: 3D jet impact and toroidal bubbles. J. Comput. Phys. 166, 336–360 (2001)CrossRefADSMATHGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • Q. X. Wang
    • 1
  • K. S. Yeo
    • 2
  • B. C. Khoo
    • 2
  • K. Y. Lam
    • 2
  1. 1.Maritime Research Centre, Division of Environmental and Water Resources EngineeringSchool of Environmental and Civil Engineering, Nanyang Technological UniversitySingaporeThe Republic of Singapore
  2. 2.Department of Mechanical EngineeringNational University of SingaporeSingaporeThe Republic of Singapore

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