### 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.

Original language | English |
---|---|

Article number | 040902 |

Journal | Physics of Fluids |

Volume | 20 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2008 Apr |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mechanics of Materials
- Computational Mechanics
- Physics and Astronomy(all)
- Fluid Flow and Transfer Processes
- Condensed Matter Physics

### Cite this

*Physics of Fluids*,

*20*(4), [040902]. https://doi.org/10.1063/1.2912517

**Statistical equilibrium of bubble oscillations in dilute bubbly flows.** / Colonius, Tim; Hagmeijer, Rob; Ando, Keita; Brennen, Christopher E.

Research output: Contribution to journal › Article

*Physics of Fluids*, vol. 20, no. 4, 040902. https://doi.org/10.1063/1.2912517

}

TY - JOUR

T1 - Statistical equilibrium of bubble oscillations in dilute bubbly flows

AU - Colonius, Tim

AU - Hagmeijer, Rob

AU - Ando, Keita

AU - Brennen, Christopher E.

PY - 2008/4

Y1 - 2008/4

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=43149111059&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43149111059&partnerID=8YFLogxK

U2 - 10.1063/1.2912517

DO - 10.1063/1.2912517

M3 - Article

AN - SCOPUS:43149111059

VL - 20

JO - Physics of Fluids

JF - Physics of Fluids

SN - 1070-6631

IS - 4

M1 - 040902

ER -