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Modelling bubble clusters in compressible liquids

Published online by Cambridge University Press:  21 October 2011

D. Fuster  and
T. Colonius
D. Fuster*
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
T. Colonius
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: [email protected]
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Abstract

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.

JFM classification

Drops and Bubbles: Bubble dynamics Drops and Bubbles: Cavitation Multiphase and Particle-laden Flows: Multiphase flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 688 , 10 December 2011 , pp. 352 - 389
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

‡

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

References

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, 3747–3750.Google Scholar
2. Akhatov, I., Parlitz, U. & Lauterborn, W. 1996 Towards a theory of self-organization phenomena in bubble-liquid mixtures. Phys. Rev. E 54 (5), 4990–5003.Google Scholar
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, EL69–EL74.Google Scholar
4. Ando, K., Colonius, T. & Brennen, C. E. 2011 Numerical simulation of shock propagation in a polydisperse bubbly liquid. Intl J. Multiphase Flow 37, 596–608.Google Scholar
5. Arora, M., Ohl, C. D. & Lohse, D. 2007 Effect of nuclei concentration on cavitation cluster dynamics. J. Acoust. Soc. Am. 121 (6), 3432–3436.CrossRefGoogle ScholarPubMed
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), 3–47.Google Scholar
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), 154505–154509.Google Scholar
8. Best, J. P. & Kucera, A. 1992 A numerical investigation of non-spherical rebounding bubbles. J. Fluid Mech. 245 (-1), 137–154.Google Scholar
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), 251–267.Google Scholar
10. Bremond, N., Arora, M., Dammer, S. M. & Lohse, D. 2006a Interaction of cavitation bubbles on a wall. Phys. Fluids 18, 121505–121515.Google Scholar
11. Bremond, N., Arora, M., Ohl, C. D. & Lohse, D. 2006b Controlled multibubble surface cavitation. Phys. Rev. Lett. 96 (22), 224501–224505.Google Scholar
12. Brennen, C. 1995 Cavitation and Bubble Dynamics, p. 254. Oxford University Press, ISBN 0195094093.Google Scholar
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, 259–273.Google Scholar
14. Chahine, G. L. 1993 The final stage of the collapse of a cavitation bubble near a rigid wall. J. Fluid Mech. 257, 147–181.Google Scholar
15. Chapman, R. B. & Plesset, M. S. 1970 Thermal Effects in the Free Oscillations of Gas Bubbles. Ft Belvoir Defense Technical information Center.Google Scholar
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, 103304–103316.Google Scholar
17. Commander, K. W. & Prosperetti, A. 1989 Linear pressure waves in bubbly liquids: comparison between theory and experiments. J. Acoust. Soc. Am. 85, 732–746.CrossRefGoogle Scholar
18. Delale, C. F. & Tryggvason, G. 2008 Shock structure in bubbly liquids: comparison of direct numerical simulations and model equations. Shock Waves 17 (6), 433–440.CrossRefGoogle Scholar
19. Devin, C. Jr 1959 Survey of thermal, radiation, and viscous damping of pulsating air bubbles in water. Tech. Rep. DTIC Document.CrossRefGoogle Scholar
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), 107–119.Google Scholar
21. Franck, J. A. & Colonius, T. 2010 Compressible large-eddy simulation of separation control on a wall-mounted hump. AIAA J. 48, 1098–1107.Google Scholar
22. Fuster, D., Hauke, G. & Dopazo, C. 2010 Influence of the accommodation coefficient on nonlinear bubble oscillations. J. Acoust. Soc. Am. 128, 5–10.Google Scholar
23. Gilmore, F. R. 1952 The Growth or Collapse of a Spherical Bubble in a Viscous Compressible Liquid. http://authors.library.caltech.edu/561/.Google Scholar
24. Hauke, G., Fuster, D. & Dopazo, C. 2007 Dynamics of a single cavitating and reacting bubble. Phys. Rev. E 75 (066310), 1–14.Google Scholar
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.Google Scholar
26. Honein, A. E. & Moin, P. 2004 Higher entropy conservation and numerical stability of compressible turbulence simulations. J. Comput. Phys. 201 (2), 531–545.Google Scholar
27. Ilinskii, Y. A., Hamilton, M. F. & Zabolotskaya, E. A. 2007 Bubble interaction dynamics in Lagrangian and Hamiltonian mechanics. J. Acoust. Soc. Am. 121, 786–795.CrossRefGoogle ScholarPubMed
28. Iordanskii, S. V. 1960 On the equation of motion for a liquid containing gas bubbles. Zh. Prikl. Mekh. Tekh. Fiz. 3, 102–110.Google Scholar
29. Johnsen, E. & Colonius, T. 2009 Numerical simulations of non-spherical bubble collapse. J. Fluid Mech. 629, 231–262.CrossRefGoogle ScholarPubMed
30. Keller, J. & Miksis, M. 1980 Bubble oscillations of large amplitude. J. Acoust. Soc. Am. 68 (2), 628–633.Google Scholar
31. Kogarko, B. S. 1964 One-dimensional unsteady motion of a liquid with an initiation and progression of cavitation. Dokl. Akad. Nauk SSSR 155, 779–782.Google Scholar
32. Lauterborn, W. & Bolle, H. 1975 Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary. J. Fluid Mech. 72 (02), 391–399.CrossRefGoogle Scholar
33. LeVeque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.Google Scholar
34. Magnaudet, J. & Eames, I. 2000 The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32 (1), 659–708.Google Scholar
35. Marchioro, M., Tanksley, M. & Prosperetti, A. 2000 Flow of spatially non-uniform suspensions. Part I: phenomenology. Intl J. Multiphase Flow 26 (5), 783–831.Google Scholar
36. Mattsson, K. & Nordström, J. 2004 Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199 (2), 503–540.Google Scholar
37. Mettin, R. & Lauterborn, W. 2003 Secondary acoustic waves in a polydisperse bubbly medium. J. Appl. Mech. Tech. Phys. 44 (1), 17–26.Google Scholar
38. Monaghan, J. J. 1982 Why particle methods work. SIAM J. Sci. Stat. Comput. 3, 422.Google Scholar
39. Oguz, H. N. & Prosperetti, A. 1990 Bubble entrainment by the impact of drops on liquid surfaces. J. Fluid Mech. 219, 143–179.Google Scholar
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), 313–334.Google Scholar
41. Peskin, C. S. 2003 The immersed boundary method. Acta Numerica 11, 479–517.CrossRefGoogle Scholar
42. Press, W. H., Teulkolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1992 Numerical Recipes in Fortran 77. Cambridge University Press.Google Scholar
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, 1–19. 123302.CrossRefGoogle Scholar
44. Prosperetti, A. 1997 A brief summary of L. van Wijngaarden’s work up till his retirement. Appl. Sci. Res. 58 (1), 13–32.CrossRefGoogle Scholar
45. Prosperetti, A., Crum, L. A. & Commander, K. W. 1988 Nonlinear bubble dynamics. J. Acoust. Soc. Am. 83, 502–514.Google Scholar
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).Google Scholar
47. Rayleigh, Lord 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 94.Google Scholar
48. Reisman, G. E, Wang, Y. C. & Brennen, C. E. 1998 Observations of shock waves in cloud cavitation. J. Fluid Mech. 344, 255.Google Scholar
49. Schenk, O., Bollhöfer, M. & Römer, R. A. 2008 On large-scale diagonalization techniques for the Anderson model of localization. SIAM Rev. 91–112.Google Scholar
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), 321–341.Google Scholar
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.Google Scholar
52. Storey, B. D. & Szeri, A. J. 2000 Water vapour, sonoluminiscence and sonochemistry. Proc. R. Soc. Lond. A 456, 1685–1709.Google Scholar
53. Tanguay, M. 2003 Computation of bubbly cavitating flow in shock wave lithotripsy. PhD thesis, California Institute of Technology, see also URL http://resolver.caltech.edu/CaltechETD:etd-05282004-130028.Google Scholar
54. Tomar, G., Fuster, D., Zaleski, S. & Popinet, S. 2010 Multiscale simulations of primary atomization using gerris. Comput. Fluids 39 (4), 1864–1874.Google Scholar
55. Toro, E. F. 1997 Riemann Solvers and Numerical Methods for Fluid Dynamics – A Practical Introduction. Springer.Google Scholar
56. Wang, Q. X. & Blake, R. 2010 Non-spherical bubble dynamics in a compressible liquid. Part 1. Travelling acoustic wave. J. Fluid Mech. 659, 191–224.Google Scholar
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), 872–880.Google Scholar
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.Google Scholar
59. van Wijngaarden, L. 1968 On the equations of motion for mixtures of liquid and gas bubbles. J. Fluid Mech. 33 (3), 465–474.Google Scholar
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), 1–7.Google Scholar
61. Yasui, K. 1997 Alternative model of single sonoluminiscence. Phys. Rev. E 56 (6), 6750–6760.Google Scholar
62. Zhang, D. Z. & Prosperetti, A. 1994a Averaged equations for inviscid disperse two-phase flow. J. Fluid Mech. 267, 185–219.Google Scholar
63. Zhang, D. Z. & Prosperetti, A. 1994b Ensemble phase-averaged equations for bubbly flows. Phys. Fluids 6, 2956–2970.Google Scholar
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, 425–453.Google Scholar