Fundamental time scales of bubble fragmentation in homogeneous isotropic turbulence

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(l), 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(l) 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 (s) over bar at which air moves through the cascade, we derive tau(c) = C(tau epsilon)(-1/3)a(2/3)(1-(a(max)/a(H))(-2/3)), where C-tau = 1/(s) over bar and a(max) is the largest bubble radius in the cascade. While (s) over bar is a function of PBE fragmentation statistics, which depend on the measurement interval T, (s) over bar itself is independent of T for tau(r) << T << tau(c). We verify the T-independence of (s) over bar and its direct relationship to tau(c) using Monte Carlo simulations. We perform direct numerical simulations (DNS) at moderate to large bubbleWeber 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 epsilon(-1/3)a(2/3) with C-r approximate to 0.11. This gives tau(r) << tau(l), validating the PBE no-hysteresis assumption. From DNS, we measure (s) over bar and find that for large Weber numbers (We > 30), C-tau approximate to 9. In addition to providing tau(c), this obtains a new constraint on fragmentation models for PBE.