Skip to main content Accessibility help
Hostname: page-component-59b7f5684b-569ts Total loading time: 0.728 Render date: 2022-09-30T21:58:47.198Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Multi-oscillations of a bubble in a compressible liquid near a rigid boundary

Published online by Cambridge University Press:  24 March 2014

Qianxi Wang*
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
Email address for correspondence:


Bubble dynamics near a rigid boundary are associated with wide and important applications in cavitation erosion in many industrial systems and medical ultrasonics. This classical problem is revisited with the following two developments. Firstly, computational studies on the problem have commonly been based on an incompressible fluid model, but the compressible effects are essential in this phenomenon. Consequently, a bubble usually undergoes significantly damped oscillation in practice. In this paper this phenomenon will be modelled using weakly compressible theory and a modified boundary integral method for an axisymmetric configuration, which predicts the damped oscillation. Secondly, the computational studies so far have largely been concerned with the first cycle of oscillation. However, a bubble usually oscillates for a few cycles before it breaks into much smaller ones. Cavitation erosion may be associated with the recollapse phase when the bubble is initiated more than the maximum bubble radius away from the boundary. Both the first and second cycles of oscillation will be modelled. The toroidal bubble formed towards the end of the collapse phase is modelled using a vortex ring model. The repeated topological changes of the bubble are traced from a singly connected to a doubly connected form, and vice versa. This model considers the energy loss due to shock waves emitted at minimum bubble volumes during the beginning of the expansion phase and around the end of the collapse phase. It predicts damped oscillations, where both the maximum bubble radius and the oscillation period reduce significantly from the first to second cycles of oscillation. The damping of the bubble oscillation is alleviated by the existence of the rigid boundary and reduces with the standoff distance between them. Our computations correlate well with the experimental data (Philipp & Lauterborn, J. Fluid Mech., vol. 361, 1998, pp. 75–116) for both the first and second cycles of oscillation. We have successively reproduced the bubble ring in direct contact with the rigid boundary at the end of the second collapse phase, a phenomenon that was suggested to be one of the major causes of cavitation erosion by experimental studies.

© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Adoua, R., Legendre, D. & Magnaudet, J. 2009 Reversal of the lift force on an oblate bubble in a weakly viscous linear shear flow. J. Fluid Mech. 628, 2341.CrossRefGoogle Scholar
Benjamin, T. B. & Ellis, A. T. 1966 The collapse of cavitation bubbles and the pressure thereby produced against solid boundaries. Phil. Trans. R. Soc. Lond. A 260, 221240.CrossRefGoogle Scholar
Best, J. P. 1993 The formation of toroidal bubbles upon collapse of transient cavities. J. Fluid Mech. 251, 79107.CrossRefGoogle Scholar
Best, J. P. 1994 The rebound of toroidal bubbles. In Bubble Dynamics and Interface Phenomena (ed. Blake, J. R., Boulton-Stone, J. M. & Thomas, N. H.), pp. 405412. Kluwer.CrossRefGoogle Scholar
Blake, J. R. & Gibson, D. C. 1987 Cavitation bubbles near boundaries. Annu. Rev. Fluid Mech. 19, 99123.CrossRefGoogle Scholar
Blake, J. R., Hooton, M. C., Robinson, P. B. & Tong, P. R. 1997 Collapsing cavities, toroidal bubbles and jet impact. Phil. Trans. R. Soc. Lond. A 355, 537550.CrossRefGoogle Scholar
Blake, J. R., Taib, B. B. & Doherty, G. 1986 Transient cavities near boundaries. Part 1. Rigid boundary. J. Fluid Mech. 170, 479497.CrossRefGoogle Scholar
Blake, J. R., Taib, B. B. & Doherty, G. 1987 Transient cavities near boundaries. Part 2. Free surface. J. Fluid Mech. 181, 197212.CrossRefGoogle Scholar
Bonhomme, R., Magnaudet, J., Duval, F. & Piar, B. 2012 Inertial dynamics of air bubbles crossing a horizontal fluid–fluid interface. J. Fluid Mech. 707, 405443.CrossRefGoogle Scholar
Bonometti, T. & Magnaudet, J. 2007 An interface-capturing method for incompressible two-phase flows. Validation and application to bubble dynamics. Intl J. Multiphase Flow 33 (2), 109133.CrossRefGoogle Scholar
Brennen, C. E. 1995 Cavitation and Bubble Dynamics. Oxford University Press.Google Scholar
Brujan, E. A., Keen, G. S., Vogel, A. & Blake, J. R. 2002 The final stage of the collapse of a cavitation bubble close to a rigid boundary. Phys. Fluids 14 (1), 8592.CrossRefGoogle Scholar
Brujan, E. A. & Matsumoto, Y. 2012 Collapse of micrometre-sized cavitation bubbles near a rigid boundary. Microfluid. Nanofluid. 13, 957966.CrossRefGoogle Scholar
Brujan, E. A., Nahen, K., Schmidt, P. & Vogel, A. 2001 Dynamics of laser-induced cavitation bubbles near elastic boundaries: influence of the elastic modulus. J. Fluid Mech. 433, 283314.CrossRefGoogle Scholar
Calvisi, M. L., Iloreta, J. I. & Szeri, A. J. 2008 Dynamics of bubbles near a rigid surface subjected to a lithotripter shock wave: II. Reflected shock intensifies non-spherical cavitation collapse. J. Fluid Mech. 616, 6397.CrossRefGoogle Scholar
Chahine, G. L. & Bovis, A. 1980 Oscillation and collapse of a cavitation bubble in the vicinity of a two-liquid interface. In Cavitation and Inhomogeneities in Underwater Acoustics pp. 2329. Springer.CrossRefGoogle Scholar
Chahine, G. L. & Harris, G.1988a Multi-cycle underwater explosion bubble model. Part I: Theory and validation examples for free-field bubble problems. Report IHCR 98-64. US Naval Surface Warfare Center, Indian Head Division.Google Scholar
Chahine, G. L. & Harris, G.1988b Multi-cycle underwater explosion bubble model. Part II: Validation examples for hull girder whipping problems. Report IHCR 98-65. US Naval Surface Warfare Center, Indian Head Division.Google Scholar
Chahine, G. L. & Perdue, T. O.1988 Simulation of the three-dimensional behaviour of an unsteady large bubble near a structure. In Proc. 3rd Intl Colloq. on Drops and Bubbles, Monterey, CA.Google Scholar
Cole, R. H. 1948 Underwater Explosions. Princeton University Press.CrossRefGoogle Scholar
Coussios, C. C. & Roy, R. A. 2007 Applications of acoustics and cavitation to non-invasive therapy and drug delivery. Annu. Rev. Fluid Mech. 40, 395420.CrossRefGoogle Scholar
Curtiss, G. A., Leppinen, D. M., Wang, Q. X. & Blake, J. R. 2013 Ultrasonic cavitation near a tissue layer. J. Fluid Mech. 730, 245272.CrossRefGoogle Scholar
Delius, M. 1990 Effect of lithotripter shock waves on tissues and materials. In Proc. 12th ISNA: Frontiers of Nonlinear Acoustics (ed. Hamilton, M. F. & Blackstock, D. T.), pp. 3146. Elsevier.Google Scholar
Duncan, J. H., Milligan, C. D. & Zhang, S. G. 1996 On the interaction between a bubble and a submerged compliant structure. J. Sound Vib. 197 (1), 1744.CrossRefGoogle Scholar
Duncan, J. H. & Zhang, S. G. 1993 On the interaction of a collapsing cavity and a compliant wall. J. Fluid Mech. 226, 401423.CrossRefGoogle Scholar
Feng, Z. C. & Leal, L. G. 1997 Nonlinear bubble dynamics. Annu. Rev. Fluid Mech. 29, 201243.CrossRefGoogle Scholar
Geers, T. L. & Hunter, K. S. 2002 An integrated wave-effects model for an underwater explosion bubble. J. Acoust. Soc. Am. 111, 15841601.CrossRefGoogle ScholarPubMed
Geers, T. L., Lagumbay, R. S. & Vasilyev, O. V. 2012 Acoustic-wave effects in violent bubble collapse. J. Appl. Phys. 112, 054910.CrossRefGoogle Scholar
Geers, T. L. & Zhang, P. 1994 Doubly asymptotic approximations for submerged structures with internal fluid volumes. Trans. ASME: J. Appl. Mech. 61, 893906.CrossRefGoogle Scholar
Guerri, L., Lucca, G. & Prosperetti, A. 1981 A numerical method for the dynamics of non-spherical cavitation bubbles. In Proc. 2nd Int. Colloq. on Drops and Bubbles, California NASA JPL Publications 82-7, p. 175California Institute of Technology.Google Scholar
Herring, C.1941 The theory of the pulsations of the gas bubbles produced by an underwater explosion. Report. US National Defense Research Committee.Google Scholar
Hua, J. & Lou, J. 2007 Numerical simulation of bubble rising in viscous liquid. J. Comput. Phys. 222 (2), 769795.CrossRefGoogle Scholar
Hung, C. F. & Hwangfu, J. J. 2010 Experimental study of the behaviour of mini-charge underwater explosion bubbles near different boundaries. J. Fluid Mech. 651, 5580.CrossRefGoogle Scholar
Iloreta, J. I., Fung, N. M. & Szeri, A. J. 2008 Dynamics of bubbles near a rigid surface subjected to a lithotripter shock wave: I. Consequences of interference between incident and reflected waves. J. Fluid Mech. 616, 4361.CrossRefGoogle Scholar
Jayaprakash, A., Chao-Tsung, H. & Chahine, G. 2010 Numerical and experimental study of the interaction of a spark-generated bubble and a vertical wall. J. Fluids Engng 134 (3), 031301.Google Scholar
Jayaprakash, A., Singh, S. & Chahine, G. 2011 Experimental and numerical investigation of single bubble dynamics in a two-phase bubbly medium. J. Fluids Engng 133, 121305.CrossRefGoogle Scholar
Johnsen, E. & Colonius, T. 2008 Shock-induced collapse of a gas bubble in shockwave lithotripsy. J. Acoust. Soc. Am. 124, 20112020.CrossRefGoogle ScholarPubMed
Johnsen, E. & Colonius, T. 2009 Numerical simulations of non-spherical bubble collapse. J. Fluid Mech. 629, 231262.CrossRefGoogle ScholarPubMed
Keller, J. B. & Kolodner, I. I. 1956 Damping of underwater explosion bubble oscillations. J. Appl. Phys. 27 (10), 11521161.CrossRefGoogle Scholar
Kornfeld, M. & Suvorov, L. 1944 On the destructive action of cavitation. J. Appl. Phys. 15, 495506.CrossRefGoogle Scholar
Klaseboer, E., Fong, S. W., Turangan, C. K., Khoo, B. C., Szeri, A. J., Calvisi, M. L., Sankin, G. N. & Zhong, P. 2007 Interaction of lithotripter shockwaves with single inertial cavitation bubbles. J. Fluid Mech. 593, 3356.CrossRefGoogle ScholarPubMed
Klaseboer, E., Hung, K. C., Wang, C., Wang, C. W., Khoo, B. C., Boyce, P., Debono, S. & Charlier, H. 2005 Experimental and numerical investigation of the dynamics of an underwater explosion bubble near a resilient/rigid structure. J. Fluid Mech. 537, 387413.CrossRefGoogle Scholar
Lauterborn, W. & Bolle, H. 1975 Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary. J. Fluid Mech. 72, 391399.CrossRefGoogle Scholar
Lauterborn, W. & Kurz, T. 2010 Physics of bubble oscillations. Rep. Prog. Phys. 73, 106501.CrossRefGoogle Scholar
Lauterborn, W. & Ohl, C. D. 1997 Cavitation bubble dynamics. Ultrason. Sonochem. 4, 6575.CrossRefGoogle ScholarPubMed
Lauterborn, W. & Vogel, A. 2013 Shock wave emission by laser generated bubbles. In Bubble Dynamics and Shock Waves (ed. Delale, C. F.), pp. 67103. Springer.CrossRefGoogle Scholar
Lee, M., Klaseboer, E. & Khoo, B. C. 2007 On the boundary integral method for the rebounding bubble. J. Fluid Mech. 570, 407429.CrossRefGoogle Scholar
Leighton, T. 1994 The Acoustic Bubble. Academic Press.Google Scholar
Lenoir, M. 1979 A calculation of the parameters of the high-speed jet formed in the collapse of a bubble. J. Appl. Mech. Tech. Phys. 20 (3), 333337.Google Scholar
Leslie, T. A. & Kennedy, J. E. 2006 High-intensity focused ultrasound principles, current uses, and potential for the future. Ultrasound Q. 22, 263272.CrossRefGoogle ScholarPubMed
Lezzi, A. & Prosperetti, A. 1987 Bubble dynamics in a compressible liquid. Part. 2. Second-order theory. J. Fluid Mech. 185, 289321.CrossRefGoogle Scholar
Lind, S. J. & Phillips, T. N. 2012 The influence of viscoelasticity on the collapse of cavitation bubbles near a rigid boundary. Theor. Comput. Fluid Dyn. 26 (1–4), 245277.CrossRefGoogle Scholar
Lindau, O. & Lauterborn, W. 2003 Cinematographic observation of the collapse and rebound of a laser-produced cavitation bubble near a wall. J. Fluid Mech. 479, 327348.CrossRefGoogle Scholar
Lundgren, T. S. & Mansour, N. N. 1991 Vortex ring bubbles. J. Fluid Mech. 72, 391399.Google Scholar
Minsier, V., De Wilde, J. & Proost, J. 2009 Simulation of the effect of viscosity on jet penetration into a single cavitating bubble. J. Appl. Phys. 106, 084906.CrossRefGoogle Scholar
Naudé, C. F. & Ellis, A. T. 1961 On the mechanism of cavitation damage by non hemispherical cavities collapsing in contact with a solid boundary. Trans. ASME D: J. Basic Engng 83, 648656.CrossRefGoogle Scholar
Pearson, A., Blake, J. R. & Otto, S. R. 2004 Jets in bubbles. J. Engng Maths 48 (3–4), 391412.CrossRefGoogle Scholar
Pedley, T. J. 1968 The toroidal bubble. J. Fluid Mech. 32, 97112.CrossRefGoogle Scholar
Philipp, A. & Lauterborn, W. 1998 Cavitation erosion by single laser-produced bubbles. J. Fluid Mech. 361, 75116.CrossRefGoogle Scholar
Plesset, M. S. & Chapman, R. B. 1971 Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary. J. Fluid Mech. 47, 283290.CrossRefGoogle Scholar
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145185.CrossRefGoogle Scholar
Popinet, S. & Zaleski, S. 2002 Bubble collapse near a solid boundary: a numerical study of the influence of viscosity. J. Fluid Mech. 464, 137163.CrossRefGoogle Scholar
Prosperetti, A. & Lezzi, A. 1986 Bubble dynamics in a compressible liquid. Part. 1. First-order theory. J. Fluid Mech. 168, 457478.CrossRefGoogle Scholar
Rayleigh, Lord 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 9498.CrossRefGoogle Scholar
Roberts, W. W., Hall, T. L., Ives, K., Wolf, J. S., Fowlkes, J. B. & Cain, C. A. 2006 Pulsed cavitational ultrasound: a noninvasive technology for controlled tissue ablation (histotripsy) in the rabbit kidney. J. Urol. 175, 734738.CrossRefGoogle ScholarPubMed
Shima, A., Takayama, K., Tomita, Y. & Miura, N. 1981 An experimental study on effects of a solid wall on the motion of bubbles and shock waves in bubble collapse. Acustica 48, 293301.Google Scholar
Song, W. D., Hong, M. H., Luk’yanchuk, B. & Chong, T. C. 2004 Laser-induced cavitation bubbles for cleaning of solid surfaces. J. Appl. Phys. 95 (6), 2952.CrossRefGoogle Scholar
Szeri, A. J., Storey, B. D., Pearson, A. & Blake, J. R. 2003 Heat and mass transfer during the violent collapse of nonspherical bubbles. Phys. Fluids 15, 25762586.CrossRefGoogle Scholar
Taib, B. B.1985 Boundary integral method applied to cavitation bubble dynamics. PhD thesis, University of Wollonggong.Google Scholar
Taylor, G. I. 1942 Vertical motion of a spherical bubble and the pressure surrounding it. In Underwater Explosion Research vol. 2, pp. 131144. Office of Naval Research.Google Scholar
Tomita, Y. & Shima, A. 1986 Mechanisms of impulsive pressure generation and damage pit formation by bubble collapse. J. Fluid Mech. 169, 535564.CrossRefGoogle Scholar
Turangan, C. K., Jamaluddin, A. R., Ball, G. J. & Leighton, T. G. 2008 Free-Lagrange simulations of the expansion and jetting collapse of air bubbles in water. J. Fluid Mech. 598, 125.CrossRefGoogle Scholar
Van Dyke, M. D. 1975 Perturbation Methods in Fluid Mechanics. 2nd edn. Parabolic Press.Google Scholar
Vogel, A., Lauterborn, W. & Timm, R. 1989 Optical and acoustic investigations of the dynamics of laser-produced cavitation bubbles near a solid boundary. J. Fluid Mech. 206, 299338.CrossRefGoogle Scholar
Vogel, A., Schweiger, P., Frieser, A., Asiyo, M. & Birngruber, R. 1990 Intraocular Nd:YAG laser surgery: damage mechanism, damage range and reduction of collateral effects. IEEE J. Quantum Electron. 26, 22402260.CrossRefGoogle Scholar
Wang, Q. X. 1998 The numerical analyses of the evolution of a gas bubble near an inclined wall. Theor. Comput. Fluid Dyn. 12, 2951.CrossRefGoogle Scholar
Wang, Q. X. 2004 Numerical modelling of violent bubble motion. Phys. Fluids 16 (5), 16101619.Google Scholar
Wang, Q. X. 2013 Underwater explosion bubble dynamics in a compressible liquid. Phys. Fluids 25, 072104.CrossRefGoogle Scholar
Wang, Q. X. & Blake, J. R. 2010 Non-spherical bubble dynamics in a compressible liquid. Part 1. Travelling acoustic wave. J. Fluid Mech. 659, 191224.CrossRefGoogle Scholar
Wang, Q. X. & Blake, J. R. 2011 Non-spherical bubble dynamics in a compressible liquid. Part 2. Acoustic standing wave. J. Fluid Mech. 679, 559581.CrossRefGoogle Scholar
Wang, Q. X., Yeo, K. S., Khoo, B. C. & Lam, K. Y. 1996a Nonlinear interaction between gas bubble and free surface. Comput. Fluids 25 (7), 607628.CrossRefGoogle Scholar
Wang, Q. X., Yeo, K. S., Khoo, B. C. & Lam, K. Y. 1996b Strong interaction between buoyancy bubble and free surface. Theor. Comput. Fluid Dyn. 8, 7388.CrossRefGoogle Scholar
Wang, Q. X., Yeo, K. S., Khoo, B. C. & Lam, K. Y. 2005 Vortex ring modelling for toroidal bubbles. Theor. Comput. Fluid Dyn. 19 (5), 303317.CrossRefGoogle Scholar
Wang, S. P., Zhang, A., Liu, Y. L. & Zeng, D. R. 2013 Numerical simulation of bubble dynamics in an elastic vessel. Eur. Phys. J. E 36, 119.CrossRefGoogle Scholar
Wardlaw, A. Jr. & Luton, J. A. 2000 Fluid–structure interaction for close-in explosions. Shock Vib. J. 7, 265275.CrossRefGoogle Scholar
Wardlaw, A. Jr., Luton, J. A., Renzi, J. J. & Kiddy, K. 2003a Fluid–structure coupling methodology for undersea weapons. In Fluid Structure Interaction II pp. 251263. WIT Press.Google Scholar
Wardlaw, A. B., Luton, J. A., Renzi, J. R., Kiddy, K. C. & McKeown, R. M.2003b The Gemini Euler solver for the coupled simulation of underwater explosions. NSWCIHD/IHTR-2500.Google Scholar
Yang, B. & Prosperetti, A. 2008 Vapour bubble collapse in isothermal and non-isothermal liquids. J. Fluid Mech. 601, 253279.CrossRefGoogle Scholar
Yang, Y. X., Wang, Q. X. & Keat, T. S. 2013 Dynamic features of a laser-induced cavitation bubble near a solid boundary. Ultrason. Sonochem. 20, 10981103.CrossRefGoogle Scholar
Yoon, S. S. & Heister, S. D. 2004 Analytical formulas for the velocity field induced by an infinitely thin vortex ring. Int. J. Numer. Methods Fluids 44, 665672.CrossRefGoogle Scholar
Yu, P. W., Ceccio, S. L. & Tryggvason, G. 1995 The collapse of a cavitation bubble in shear flows – a numerical study. Phys. Fluids 7 (11), 26082616.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Bertelo, C. A. & Hu, H. H. 2007 An arbitrary Lagrangian–Eulerian method for simulating bubble growth in polymer foaming. J. Comput. Phys. 226 (2), 22292249.CrossRefGoogle Scholar
Young, F. R. 1989 Cavitation. McGraw-Hill.Google Scholar
Zhang, S. G. & Duncan, J. H. 1994 On the non-spherical collapse and rebound of a cavitation bubble. Phys. Fluids 6 (7), 23522362.CrossRefGoogle Scholar
Zhang, S. G., Duncan, J. H. & Chahine, G. L. 1993 The final stage of the collapse of a cavitation bubble near a rigid wall. J. Fluid Mech. 257, 147181.CrossRefGoogle Scholar
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Multi-oscillations of a bubble in a compressible liquid near a rigid boundary
Available formats

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Multi-oscillations of a bubble in a compressible liquid near a rigid boundary
Available formats

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Multi-oscillations of a bubble in a compressible liquid near a rigid boundary
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *