Fundamental time scales of bubble fragmentation in homogeneous isotropic turbulence

Abstract

We investigate the fundamental time scales that characterise the statistics of fragmentation under homogeneous isotropic turbulence for air–water bubbly flows at moderate to large bubble Weber numbers, $We$ . We elucidate three time scales: $\tau _r$ , the characteristic age of bubbles when their subsequent statistics become stationary; $\tau _\ell$ , the expected lifetime of a bubble before further fragmentation; and $\tau _c$ , the expected time for the air within a bubble to reach the Hinze scale, radius $a_H$ , through the fragmentation cascade. The time scale $\tau _\ell$ is important to the population balance equation (PBE), $\tau _r$ is critical to evaluating the applicability of the PBE no-hysteresis assumption, and $\tau _c$ provides the characteristic time for fragmentation cascades to equilibrate. By identifying a non-dimensionalised average speed $\bar {s}$ at which air moves through the cascade, we derive $\tau _c=C_\tau \varepsilon ^{-1/3} a^{2/3} (1-(a_{max}/a_H)^{-2/3})$ , where $C_\tau =1/\bar {s}$ and $a_{max}$ is the largest bubble radius in the cascade. While $\bar {s}$ is a function of PBE fragmentation statistics, which depend on the measurement interval $T$ , $\bar {s}$ itself is independent of $T$ for $\tau _r \ll T \ll \tau _c$ . We verify the $T$ -independence of $\bar {s}$ and its direct relationship to $\tau _c$ using Monte Carlo simulations. We perform direct numerical simulations (DNS) at moderate to large bubble Weber numbers, $We$ , to measure fragmentation statistics over a range of $T$ . We establish that non-stationary effects decay exponentially with $T$ , independent of $We$ , and provide $\tau _r=C_{r} \varepsilon ^{-1/3} a^{2/3}$ with $C_{r}\approx 0.11$ . This gives $\tau _r\ll \tau _\ell$ , validating the PBE no-hysteresis assumption. From DNS, we measure $\bar {s}$ and find that for large Weber numbers ( $We>30$ ), $C_{\tau }\approx 9$ . In addition to providing $\tau _c$ , this obtains a new constraint on fragmentation models for PBE.

Breaking bubbles across multiple time scales in turbulence

“…However, this hypothesis has been challenged recently (Vela-Martín & Avila 2021;Qi et al 2022), and it is found that sub-bubble-scale eddies may also contribute to the breakup process, resulting in a shorter time scale. In addition to the eddy time scale, more time scales relevant to the breakup process have been proposed (Ni 2024), including the bubble natural oscillation time scale (Lamb 1879;Risso & Fabre 1998), the capillary time scale (Villermaux 2020; Rivière et al 2022;Ruth et al 2022), the bubble lifetime (Martínez-Bazán, Montanes & Lasheras 1999;Liao & Lucas 2009;Qi, Masuk & Ni 2020;Vela-Martín & Avila 2022), the large-scale shear time scale (Zhong & Ni 2023), and the convergent time scale (Qi et al 2020;Gaylo, Hendrickson & Yue 2023). However, the relationship among those time scales remains unclear, particularly how this relationship is connected to the Weber number We = ρ 2/3 D 5/3 /σ , where ρ, , D and σ are the density, turbulence dissipation rate, bubble diameter and surface tension coefficient, respectively.…”

Morphology of clean and surfactant-laden droplets in homogeneous isotropic turbulence

“…The rate of these exchanges is determined by the product of the interfacial flux and the interfacial area, emphasizing the pivotal role of interfacial characteristics. There has been considerable effort to estimate these exchanges via empirical correlations (Akita & Yoshida 1974;Kelly & Kazimi 1982;Delhaye & Bricard 1994) or via population balance equations relying on the droplet size distribution and on droplet breakage and coalescence models (Luo & Svendsen 1996;Babinsky & Sojka 2002;Andersson & Andersson 2006a,b;Martínez-Bazán et al 2010;Chan et al 2018;Chan, Johnson & Moin 2021;Gaylo, Hendrickson & Yue 2023). Hence, in this study we aim to answer the question, 'what is the interfacial area of surfactant-laden droplets in turbulence?'…”

“…It was shown also that, in the absence of coalescence, the breakage rate depends on the Weber number alone. Gaylo et al (2023) investigated the fragmentation of bubbles in statistically stationary homogeneous isotropic turbulence and characterized several fundamental time scales of the bubbles: the relaxation time, the expected lifetime and the time needed for the largest bubble to break down to the Kolmogorov-Hinze scale.…”