Abstract

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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.