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Colonius T, Hagmeijer R, Ando K, et al. Statistical equilibrium of bubble oscillations in dilute bubbly flows. Phys Fluids (1994). 2008;20(4):40902doi: 10.1063/1.2912517.
Colonius, T., Hagmeijer, R., Ando, K., & Brennen, C. E. (2008). Statistical equilibrium of bubble oscillations in dilute bubbly flows. Physics of fluids (Woodbury, N.Y. : 1994), 20(4), 40902. https://doi.org/10.1063/1.2912517
Colonius, Tim, et al. "Statistical equilibrium of bubble oscillations in dilute bubbly flows." Physics of fluids (Woodbury, N.Y. : 1994) vol. 20,4 (2008): 40902. doi: https://doi.org/10.1063/1.2912517
Colonius T, Hagmeijer R, Ando K, Brennen CE. Statistical equilibrium of bubble oscillations in dilute bubbly flows. Phys Fluids (1994). 2008 Apr;20(4):40902. doi: 10.1063/1.2912517. Epub 2008 Apr 30. PMID: 19547725; PMCID: PMC2698283.
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Phys Fluids (1994). 2008 Apr;20(4):40902. doi: 10.1063/1.2912517. Epub 2008 Apr 30.

Statistical equilibrium of bubble oscillations in dilute bubbly flows.

Physics of fluids (Woodbury, N.Y. : 1994)

Tim Colonius, Rob Hagmeijer, Keita Ando, Christopher E Brennen

PMID: 19547725 PMCID: PMC2698283 DOI: 10.1063/1.2912517

Abstract

The problem of predicting the moments of the distribution of bubble radius in bubbly flows is considered. The particular case where bubble oscillations occur due to a rapid (impulsive or step change) change in pressure is analyzed, and it is mathematically shown that in this case, inviscid bubble oscillations reach a stationary statistical equilibrium, whereby phase cancellations among bubbles with different sizes lead to time-invariant values of the statistics. It is also shown that at statistical equilibrium, moments of the bubble radius may be computed using the period-averaged bubble radius in place of the instantaneous one. For sufficiently broad distributions of bubble equilibrium (or initial) radius, it is demonstrated that bubble statistics reach equilibrium on a time scale that is fast compared to physical damping of bubble oscillations due to viscosity, heat transfer, and liquid compressibility. The period-averaged bubble radius may then be used to predict the slow changes in the moments caused by the damping. A benefit is that period averaging gives a much smoother integrand, and accurate statistics can be obtained by tracking as few as five bubbles from the broad distribution. The period-averaged formula may therefore prove useful in reducing computational effort in models of dilute bubbly flow wherein bubbles are forced by shock waves or other rapid pressure changes, for which, at present, the strong effects caused by a distribution in bubble size can only be accurately predicted by tracking thousands of bubbles. Some challenges associated with extending the results to more general (nonimpulsive) forcing and strong two-way coupled bubbly flows are briefly discussed.

References

  1. Phys Rev Lett. 2000 Oct 9;85(15):3165-8 - PubMed

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