Size distribution of large air bubbles entrained by strong free-surface turbulence

In air-entraining flows, there is often strong turbulence beneath the free surface. We consider the entrainment of bubbles at the free surface by this strong free-surface turbulence (FST). Our interest is the entrainment size distribution (per unit free surface area) I(a)/AFS , for bubbles with radius a greater than the capillary scale ( ≈ 1.3\ mm for air–water on Earth), where gravity dominates surface tension. We develop a mechanistic model based on entrained bubble size being proportional to the minimum radius of curvature of the initial surface deformation. Using direct numerical simulation of a flow that isolates entrainment by FST, we show that, consistent with our mechanism, I(a)/AFS = CI \, g⁻³ ε 7/3 (2 a)-14/3 , where g is gravity, and ε is the turbulence dissipation rate. In the limit of negligible surface tension, CI≈ 3.62 , and we describe how CI decreases with increasing surface tension. This scaling holds for sufficiently strong FST such that near-surface turbulence is nearly isotropic, which we show is true for turbulent Froude number Fr²T = ε /uᵣₘₛ g > 0.1 . While we study FST entrainment in isolation, our model corroborates previous numerical results from shear-driven flow, and experimental results from open-channel flow, showing that the FST entrainment mechanism that we elucidate can be important in broad classes of air-entraining flows.