- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Simulation of the high Mach number motion for bubble collapse in a compressible Euler fluid using Basilisk

Daniels Krimans*

Steven J. Ruuth

Seth Putterman

  • *Contact author: krimans@physik.uni-kiel.de
PDF

Phys. Rev. E 112, 065107 – Published 22 December, 2025

DOI: https://doi.org/10.1103/19vc-d3vd

Abstract

Cavitation is a process where bubbles form and collapse within a fluid with dynamic, spatially varying pressure. This phenomenon can concentrate energy density by 12 orders of magnitude, creating light-emitting plasma or damaging nearby surfaces. A key question in cavitation theory and experiments is: What are the upper limits of energy density achievable through this spontaneous multiscale process? Among the many physical processes at play, we focus on fluid compressibility, modeled using the Tait-Murnaghan equation of state for a homentropic Euler fluid. Sonoluminescence represents the extreme limit of cavitation. Here, we examine an extreme case of experimentally realizable sonoluminescence, where spherical cavities have an initial radius that is 10 to 20 times their ambient radius and change their radius by a factor of over 100 during the collapse. To capture such extreme motion, with Mach numbers relative to ambient sound speed greater than one during the final stages of implosion, require methods beyond the classic approaches of Rayleigh and Gilmore. In this direction, we applied an all-Mach solver developed in the Basilisk framework, actively used to model bubble dynamics. Some of the reasons for its popularity include the fact that it is also expected to work for high Mach numbers, transport processes, nonspherical motion, shock wave capture, and involves low computational cost. Additionally, it has been tested for compressible motion at lower Mach numbers. Given its recent and extensive application, we endeavored to determine the specific solver's capability in resolving rapid bubble collapses. This solver was designed to handle the hydrodynamics of both the gas inside the bubble and the compressible fluid outside. However, we discovered that even at early times, when the Mach number is small and the fluid is approximately incompressible, the solver cannot accurately resolve the hydrodynamic motion within practical computational limits. To capture high Mach number motion and resolve dynamics in the sonoluminescence regime, we employed the well-established uniform bubble approximation for the ideal gas inside the bubble. Within this approximation, the all-Mach solver achieved numerically converging results describing the evolution of the bubble wall. Although compressibility slows down the collapse, these bubbles reach velocities exceeding the ambient speed of sound of the surrounding fluid. Our method works for various fluids and is applied to liquid lithium as well as water. Our results reproduce the equation-of-state-dependent asymptotic power-law region predicted by analytic calculations for water and liquid lithium in the case of an empty cavity. This confirms our method's ability to capture high Mach number motion and suggests that the asymptotic regime could be experimentally observed. When the cavity is filled with an ideal gas, the transition to Mach number greater than one in liquid lithium occurs later in the collapse than for water, making liquid lithium a possible candidate for achieving greater concentration of energy density. Furthermore, an outgoing shock wave, which can diagnose cavitation in opaque fluids such as liquid lithium, is captured without implementing an ad hoc construction algorithm for solutions based on characteristics or other approaches.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (45)

  1. J. R. Blake and D. C. Gibson, Cavitation bubbles near boundaries, Annu. Rev. Fluid Mech. 19, 99 (1987).
  2. P. Kumar and R. Saini, Study of cavitation in hydro turbines—A review, Renew. Sustain. Energy Rev. 14, 374 (2010).
  3. A. J. Walton and G. T. Reynolds, Sonoluminescence, Adv. Phys. 33, 595 (1984).
  4. B. P. Barber, R. A. Hiller, R. Löfstedt, S. J. Putterman, and K. R. Weninger, Defining the unknowns of sonoluminescence, Phys. Rep. 281, 65 (1997).
  5. L. Rayleigh, On the pressure developed in a liquid during the collapse of a spherical cavity, London, Edinburgh, Dublin Philos. Mag. J. Sci. 34, 94 (1917).
  6. B. P. Barber and S. Putterman, Observation of synchronous picosecond sonoluminescence, Nature (London) 352, 318 (1991).
  7. G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, UK, 2000).
  8. R. H. Cole, Underwater Explosions (Princeton University Press, Princeton, NJ, 1948).
  9. C. Hunter, On the collapse of an empty cavity in water, J. Fluid Mech. 8, 241 (1960).
  10. W. Wagner and A. Pruß, The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use, J. Phys. Chem. Ref. Data 31, 387 (2002).
  11. G. E. Vazquez and S. J. Putterman, Temperature and pressure dependence of sonoluminescence, Phys. Rev. Lett. 85, 3037 (2000).
  12. A. Bass, S. J. Ruuth, C. Camara, B. Merriman, and S. Putterman, Molecular dynamics of extreme mass segregation in a rapidly collapsing bubble, Phys. Rev. Lett. 101, 234301 (2008).
  13. M. C. Ramsey and R. W. Pitz, Energetic cavitation collapse generates 3.2 Mbar plasma with a 1.4 J driver, Phys. Rev. Lett. 110, 154301 (2013).
  14. D. Krimans and S. Putterman, Power law singularity for cavity collapse in a compressible Euler fluid with Tait–Murnaghan equation of state, Phys. Fluids 35, 086114 (2023).
  15. R. Garcia and F. G. Hammitt, Cavitation damage and correlations with material and fluid properties, J. Basic Eng. 89, 753 (1967).
  16. Y. Toriyabe, E. Yoshida, J. Kasagi, and M. Fukuhara, Acceleration of the d+d reaction in metal lithium acoustic cavitation with deuteron bombardment from 30 to 70 keV, Phys. Rev. C 85, 054620 (2012).
  17. H. W. Davison, Compilation of thermophysical properties of liquid lithium, NASA Technical Note No. D-4650 (NASA, Washington, DC, 1968).
  18. S. Blairs, Review of data for velocity of sound in pure liquid metals and metalloids, Int. Mater. Rev. 52, 321 (2007).
  19. K. V. Khishchenko, Equation of state for lithium in shock waves, Math. Montisnigri 41, 91 (2018).
  20. Y. Shu, Y. Kono, I. Ohira, R. Hrubiak, C. Kenney-Benson, M. Somayazulu, N. Velisavljevic, and G. Shen, Structural changes in liquid lithium under high pressure, J. Phys. Chem. B 124, 7258 (2020).
  21. R. Löfstedt, K. Weninger, S. Putterman, and B. P. Barber, Sonoluminescing bubbles and mass diffusion, Phys. Rev. E 51, 4400 (1995).
  22. V. Q. Vuong and A. J. Szeri, Sonoluminescence and diffusive transport, Phys. Fluids 8, 2354 (1996).
  23. F. R. Gilmore, The growth or collapse of a spherical bubble in a viscous compressible liquid, California Institute of Technology Technical Report No. 26-4 (California Institute of Technology, Pasadena, CA, 1952).
  24. R. Hickling and M. S. Plesset, Collapse and rebound of a spherical bubble in water, Phys. Fluids 7, 7 (1964).
  25. R. Pecha and B. Gompf, Microimplosions: Cavitation collapse and shock wave emission on a nanosecond time scale, Phys. Rev. Lett. 84, 1328 (2000).
  26. K. R. Weninger, P. G. Evans, and S. J. Putterman, Time correlated single photon Mie scattering from a sonoluminescing bubble, Phys. Rev. E 61, R1020 (2000).
  27. F. Denner and S. Schenke, Modeling acoustic emissions and shock formation of cavitation bubbles, Phys. Fluids 35, 012114 (2023).
  28. M. Koch, C. Lechner, F. Reuter, K. Köhler, R. Mettin, and W. Lauterborn, Numerical modeling of laser generated cavitation bubbles with the finite volume and volume of fluid method, using openfoam, Comput. Fluids 126, 71 (2016).
  29. W. Lauterborn and T. Kurz, Physics of bubble oscillations, Rep. Prog. Phys. 73, 106501 (2010).
  30. X.-X. Liang, N. Linz, S. Freidank, G. Paltauf, and A. Vogel, Comprehensive analysis of spherical bubble oscillations and shock wave emission in laser-induced cavitation, J. Fluid Mech. 940, A5 (2022).
  31. C. C. Wu and P. H. Roberts, Shock-wave propagation in a sonoluminescing gas bubble, Phys. Rev. Lett. 70, 3424 (1993).
  32. D. Fuster, C. Dopazo, and G. Hauke, Liquid compressibility effects during the collapse of a single cavitating bubble, J. Acoust. Soc. Am 129, 122 (2011).
  33. S. Miller, H. Jasak, D. Boger, E. Paterson, and A. Nedungadi, A pressure-based, compressible, two-phase flow finite volume method for underwater explosions, Comput. Fluids 87, 132 (2013).
  34. V.-T. Nguyen, T.-H. Phan, and W.-G. Park, Modeling of shock wave produced by collapse of cavitation bubble using a fully conservative multiphase model, Phys. Fluids 35, 116102 (2023).
  35. D. Fuster and S. Popinet, An all-Mach method for the simulation of bubble dynamics problems in the presence of surface tension, J. Comput. Phys. 374, 752 (2018).
  36. Y. Saade, D. Lohse, and D. Fuster, A multigrid solver for the coupled pressure-temperature equations in an all-Mach solver with VoF, J. Comput. Phys. 476, 111865 (2023).
  37. R. Löfstedt, B. P. Barber, and S. J. Putterman, Toward a hydrodynamic theory of sonoluminescence, Phys. Fluids 5, 2911 (1993).
  38. S. Hilgenfeldt, M. P. Brenner, S. Grossmann, and D. Lohse, Analysis of Rayleigh–Plesset dynamics for sonoluminescing bubbles, J. Fluid Mech. 365, 171 (1998).
  39. S. J. Ruuth, S. Putterman, and B. Merriman, Molecular dynamics simulation of the response of a gas to a spherical piston: Implications for sonoluminescence, Phys. Rev. E 66, 036310 (2002).
  40. A. Bass, S. Putterman, B. Merriman, and S. J. Ruuth, Symmetry reduction for molecular dynamics simulation of an imploding gas bubble, J. Comput. Phys. 227, 2118 (2008).
  41. D. Schanz, B. Metten, T. Kurz, and W. Lauterborn, Molecular dynamics simulations of cavitation bubble collapse and sonoluminescence, New J. Phys. 14, 113019 (2012).
  42. L. D. Landau and E. M. Lifshitz, Fluid Mechanics: Volume 6 (Course of Theoretical Physics), 2nd ed. (Butterworth-Heinemann, Oxford, UK, 1987).
  43. L. D. Landau and E. M. Lifshitz, Statistical Physics: Volume 5 (Course of Theoretical Physics), 3rd ed. (Butterworth-Heinemann, Oxford, UK, 1980).
  44. S. Popinet, A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equations, J. Comput. Phys. 302, 336 (2015).
  45. J. Anderson, Computational Fluid Dynamics: The Basics with Applications, 8th ed. (McGraw-Hill, Columbus, OH, 1995).

Outline

Information

Phys. Rev. E 112, 065107– Published 22 December, 2025
  • Received 1 July 2025
  • Accepted 26 November 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation