Statistical equilibrium of bubble oscillations in dilute bubbly flows

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 change͒ 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 initial͒ 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 ͑nonimpulsive͒ forcing and strong two-way coupled bubbly flows are briefly discussed.

Shock propagation through a bubbly liquid in a deformable tube

“…If bubble fission occurs after passage of the shock, the fission damping needs to be included in bubbledynamic modelling (Brennen 2002). Moreover, because polydispersity results in different frequency responses for different-sized bubbles, phase cancellations cause an additional apparent damping of the wave propagation (Smereka 2002;Colonius et al 2008;Ando, Colonius & Brennen 2009;Ando 2010).…”

Improvement of acoustic theory of ultrasonic waves in dilute bubbly liquids

“…The phase cancellations can be regarded as apparent damping of the wave propagation in the polydisperse mixture, and the damping mechanism becomes more effective as the bubble size distribution broadens. 13,14 As a result, the size distribution increases the attenuation, as seen in Fig. 2.…”

“…(3) and (4) or Eqs. (13) and (14) and are plotted in Fig. 2 from the phase velocity plot that the present modifications to ␦ and N have negligible impact on V. It is also found that the size distribution tends to smooth the transition in V at the resonant frequency.…”

“…For future use, we develop the asymptotic limits of Eqs. (13) and (14). In the quasistatic limit (i.e., → 0), it follows from Prosperetti 7,11 that…”

Shock Propagation in Polydisperse Bubbly Liquids

“…1) with the probable size R ref and standard deviation σ: delta function that models monodisperse mixtures. The measured distribution of gas bubble nuclei in water tunnels or seawater is reasonably fit by the lognormal distribution with σ ≈ 0.7 [3,24]. For example, the void fraction is computed as…”