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: 𝜏𝑟 , the characteristic age of bubbles when their subsequent statistics become stationary; 𝜏ℓ , the expected lifetime of a bubble before further fragmentation; and 𝜏𝑐 , the expected time for the air within a bubble to reach the Hinze scale, radius 𝑎𝐻 , through the fragmentation cascade. The time scale 𝜏ℓ is important to the population balance equation (PBE), 𝜏𝑟 is critical to evaluating the applicability of the PBE no-hysteresis assumption, and 𝜏𝑐 provides the characteristic time for fragmentation cascades to equilibrate. By identifying a non-dimensionalised average speed 𝑠¯ at which air moves through the cascade, we derive 𝜏𝑐=𝐶𝜏𝜀−1/3𝑎2/3(1−(𝑎𝑚𝑎𝑥/𝑎𝐻)−2/3) , where 𝐶𝜏=1/𝑠¯ and 𝑎𝑚𝑎𝑥 is the largest bubble radius in the cascade. While 𝑠¯ is a function of PBE fragmentation statistics, which depend on the measurement interval 𝑇 , 𝑠¯ itself is independent of 𝑇 for 𝜏𝑟≪𝑇≪𝜏𝑐 . We verify the 𝑇 -independence of 𝑠¯ and its direct relationship to 𝜏𝑐 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 𝜏𝑟=𝐶𝑟𝜀−1/3𝑎2/3 with 𝐶𝑟≈0.11 . This gives 𝜏𝑟≪𝜏ℓ , validating the PBE no-hysteresis assumption. From DNS, we measure 𝑠¯ and find that for large Weber numbers ( 𝑊𝑒>30 ), 𝐶𝜏≈9 . In addition to providing 𝜏𝑐 , this obtains a new constraint on fragmentation models for PBE.