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.