Skip to main content

Modelling bubble clusters in compressible liquids

  • D. Fuster (a1) and T. Colonius (a1)

We present a new model for bubbly cavitating flows. Based on volume-averaged equations, a subgrid model is added to account for a bubble, or multiple bubbles, within each computational cell. The model converges to the solution of ensemble-averaged bubbly flow equations for weak oscillations and monodisperse systems. In the other extreme, it also converges to the theoretical solution for a single oscillating bubble, and captures the bubble radius evolution and the pressure disturbance induced in the liquid. A substantial saving of computational time is achieved compared to ensemble-averaged models for polydisperse mixtures.

Corresponding author
Email address for correspondence:
Hide All

Present address: CNRS (UMR 7190), Université Pierre et Marie Curie, Institut Jean le Rond d’Alembert, France

Hide All
1. Akhatov, I., Mettin, R., Ohl, C. D., Parlitz, U. & Lauterborn, W. 1997 Bjerknes force threshold for stable single bubble sonoluminiscence. Phys. Rev. E 55, 37473750.
2. Akhatov, I., Parlitz, U. & Lauterborn, W. 1996 Towards a theory of self-organization phenomena in bubble-liquid mixtures. Phys. Rev. E 54 (5), 49905003.
3. Ando, K., Colonius, T. & Brennen, C. E. 2009 Improvement of acoustic theory of ultrasonic waves in dilute bubbly liquids. J. Acoust. Soc. Am. EL 126, EL69EL74.
4. Ando, K., Colonius, T. & Brennen, C. E. 2011 Numerical simulation of shock propagation in a polydisperse bubbly liquid. Intl J. Multiphase Flow 37, 596608.
5. Arora, M., Ohl, C. D. & Lohse, D. 2007 Effect of nuclei concentration on cavitation cluster dynamics. J. Acoust. Soc. Am. 121 (6), 34323436.
6. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M. & Krysl, P. 1996 Meshless methods: an overview and recent developments. Comput. Meth. Appl. Mech. Engng 139 (1–4), 347.
7. Bergmann, R., van der Meer, D., Stijnman, M., Sandtke, M., Prosperetti, A. & Lohse, D. 2006 Giant bubble pinch-off. Phys. Rev. Lett. 96 (15), 154505154509.
8. Best, J. P. & Kucera, A. 1992 A numerical investigation of non-spherical rebounding bubbles. J. Fluid Mech. 245 (-1), 137154.
9. Blake, J. R., Keen, G. S., Tong, R. P. & Wilson, M. 1999 Acoustic cavitation: the fluid dynamics of non–spherical bubbles. Phil. Trans. R. Soc. Lond. A 357 (1751), 251267.
10. Bremond, N., Arora, M., Dammer, S. M. & Lohse, D. 2006a Interaction of cavitation bubbles on a wall. Phys. Fluids 18, 121505121515.
11. Bremond, N., Arora, M., Ohl, C. D. & Lohse, D. 2006b Controlled multibubble surface cavitation. Phys. Rev. Lett. 96 (22), 224501224505.
12. Brennen, C. 1995 Cavitation and Bubble Dynamics, p. 254. Oxford University Press, ISBN 0195094093.
13. Caflisch, R. E., Miksis, M. J., Papanicolaou, G. C. & Ting, L. 1985 Effective equations for wave propagation in bubbly liquids. J. Fluid Mech. 153, 259273.
14. Chahine, G. L. 1993 The final stage of the collapse of a cavitation bubble near a rigid wall. J. Fluid Mech. 257, 147181.
15. Chapman, R. B. & Plesset, M. S. 1970 Thermal Effects in the Free Oscillations of Gas Bubbles. Ft Belvoir Defense Technical information Center.
16. Climent, E. & Magnaudet, J. 2006 Dynamics of a two-dimensional upflowing mixing layer seeded with bubbles: bubble dispersion and effect of two-way coupling. Phys. Fluids 18, 103304103316.
17. Commander, K. W. & Prosperetti, A. 1989 Linear pressure waves in bubbly liquids: comparison between theory and experiments. J. Acoust. Soc. Am. 85, 732746.
18. Delale, C. F. & Tryggvason, G. 2008 Shock structure in bubbly liquids: comparison of direct numerical simulations and model equations. Shock Waves 17 (6), 433440.
19. Devin, C. Jr 1959 Survey of thermal, radiation, and viscous damping of pulsating air bubbles in water. Tech. Rep. DTIC Document.
20. Foldy, L. L. 1945 The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers. Phys. Rev. 67 (3–4), 107119.
21. Franck, J. A. & Colonius, T. 2010 Compressible large-eddy simulation of separation control on a wall-mounted hump. AIAA J. 48, 10981107.
22. Fuster, D., Hauke, G. & Dopazo, C. 2010 Influence of the accommodation coefficient on nonlinear bubble oscillations. J. Acoust. Soc. Am. 128, 510.
23. Gilmore, F. R. 1952 The Growth or Collapse of a Spherical Bubble in a Viscous Compressible Liquid.
24. Hauke, G., Fuster, D. & Dopazo, C. 2007 Dynamics of a single cavitating and reacting bubble. Phys. Rev. E 75 (066310), 114.
25. Hinsch, K. 1976 The dynamics of bubble fields in acoustic cavitation. In Proc. 6th Intl Symp. on Nonlinear Acoustics, Moscow 1975, pp. 26–34.
26. Honein, A. E. & Moin, P. 2004 Higher entropy conservation and numerical stability of compressible turbulence simulations. J. Comput. Phys. 201 (2), 531545.
27. Ilinskii, Y. A., Hamilton, M. F. & Zabolotskaya, E. A. 2007 Bubble interaction dynamics in Lagrangian and Hamiltonian mechanics. J. Acoust. Soc. Am. 121, 786795.
28. Iordanskii, S. V. 1960 On the equation of motion for a liquid containing gas bubbles. Zh. Prikl. Mekh. Tekh. Fiz. 3, 102110.
29. Johnsen, E. & Colonius, T. 2009 Numerical simulations of non-spherical bubble collapse. J. Fluid Mech. 629, 231262.
30. Keller, J. & Miksis, M. 1980 Bubble oscillations of large amplitude. J. Acoust. Soc. Am. 68 (2), 628633.
31. Kogarko, B. S. 1964 One-dimensional unsteady motion of a liquid with an initiation and progression of cavitation. Dokl. Akad. Nauk SSSR 155, 779782.
32. Lauterborn, W. & Bolle, H. 1975 Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary. J. Fluid Mech. 72 (02), 391399.
33. LeVeque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.
34. Magnaudet, J. & Eames, I. 2000 The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32 (1), 659708.
35. Marchioro, M., Tanksley, M. & Prosperetti, A. 2000 Flow of spatially non-uniform suspensions. Part I: phenomenology. Intl J. Multiphase Flow 26 (5), 783831.
36. Mattsson, K. & Nordström, J. 2004 Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199 (2), 503540.
37. Mettin, R. & Lauterborn, W. 2003 Secondary acoustic waves in a polydisperse bubbly medium. J. Appl. Mech. Tech. Phys. 44 (1), 1726.
38. Monaghan, J. J. 1982 Why particle methods work. SIAM J. Sci. Stat. Comput. 3, 422.
39. Oguz, H. N. & Prosperetti, A. 1990 Bubble entrainment by the impact of drops on liquid surfaces. J. Fluid Mech. 219, 143179.
40. Parlitz, U., Mettin, R., Luther, S., Akhatov, I., Voss, M. & Lauterborn, W. 1999 Spatio-temporal dynamics of acoustic cavitation bubble clouds. Phil. Trans. R. Soc. Lond. A 357 (1751), 313334.
41. Peskin, C. S. 2003 The immersed boundary method. Acta Numerica 11, 479517.
42. Press, W. H., Teulkolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1992 Numerical Recipes in Fortran 77. Cambridge University Press.
43. Preston, A. T., Colonius, T. & Brennen, C. E. 2007 A reduced order model of diffusive effects on the dynamics of bubbles. Phys. Fluids 19, 119. 123302.
44. Prosperetti, A. 1997 A brief summary of L. van Wijngaarden’s work up till his retirement. Appl. Sci. Res. 58 (1), 1332.
45. Prosperetti, A., Crum, L. A. & Commander, K. W. 1988 Nonlinear bubble dynamics. J. Acoust. Soc. Am. 83, 502514.
46. Puente, G. F. & Bonetto, F. J. 2005 Proposed method to estimate the liquid-vapour accommodation coefficient based on experimental sonoluminescence data. Phys. Rev. E 71 (5).
47. Rayleigh, Lord 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 94.
48. Reisman, G. E, Wang, Y. C. & Brennen, C. E. 1998 Observations of shock waves in cloud cavitation. J. Fluid Mech. 344, 255.
49. Schenk, O., Bollhöfer, M. & Römer, R. A. 2008 On large-scale diagonalization techniques for the Anderson model of localization. SIAM Rev. 91112.
50. Schenk, O., Waechter, A. & Hagemann, M. 2007 Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization. J. Comput. Opt. Appl. 36 (2–3), 321341.
51. Seo, J. H., Lele, S. K. & Tryggvason, G. 2010 Investigation and modelling of bubble–bubble interaction effect in homogeneous bubbly flows. Phys. Fluids 22, 063302.
52. Storey, B. D. & Szeri, A. J. 2000 Water vapour, sonoluminiscence and sonochemistry. Proc. R. Soc. Lond. A 456, 16851709.
53. Tanguay, M. 2003 Computation of bubbly cavitating flow in shock wave lithotripsy. PhD thesis, California Institute of Technology, see also URL
54. Tomar, G., Fuster, D., Zaleski, S. & Popinet, S. 2010 Multiscale simulations of primary atomization using gerris. Comput. Fluids 39 (4), 18641874.
55. Toro, E. F. 1997 Riemann Solvers and Numerical Methods for Fluid Dynamics – A Practical Introduction. Springer.
56. Wang, Q. X. & Blake, R. 2010 Non-spherical bubble dynamics in a compressible liquid. Part 1. Travelling acoustic wave. J. Fluid Mech. 659, 191224.
57. Wang, Y. C. & Brennen, C. E. 1999 Numerical computation of shock waves in a spherical cloud of cavitation bubbles. Trans. ASME: J. Fluids Engng 121 (4), 872880.
58. van Wijngaarden, L. 1964 On the collective collapse of a large number of gas bubbles in water. In Proceedings 11th Int. Cong. Appl. Mech., pp. 854–861.
59. van Wijngaarden, L. 1968 On the equations of motion for mixtures of liquid and gas bubbles. J. Fluid Mech. 33 (3), 465474.
60. Xu, N., Apfel, R. E., Khong, A., Hu, X. & Wang, L. 2003 Water vapour diffusion effects on gas dynamics in a sonoluminescing bubble. Phys. Rev. Lett. E 68 (016309), 17.
61. Yasui, K. 1997 Alternative model of single sonoluminiscence. Phys. Rev. E 56 (6), 67506760.
62. Zhang, D. Z. & Prosperetti, A. 1994a Averaged equations for inviscid disperse two-phase flow. J. Fluid Mech. 267, 185219.
63. Zhang, D. Z. & Prosperetti, A. 1994b Ensemble phase-averaged equations for bubbly flows. Phys. Fluids 6, 29562970.
64. Zhang, D. Z. & Prosperetti, A. 1997 Momentum and energy equations for disperse two-phase flows and their closure for dilute suspensions. Intl J. Multiphase Flow 23, 425453.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed