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) /A₅ₒ, 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) /A₅ₒ = CI \, g^-3 ^7/3 (2 a) ^-14/3, where g is gravity, and 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 = /uₑ₌ₒ 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.
Gaylo et al. (2025) studied this question.