Scale dependence of entrainment bubble size distribution in free-surface turbulence

Air entrainment by free-surface turbulence plays important roles in both natural processes and engineering applications. We consider the size spectrum of surface entrained bubbles under strong free-surface turbulence (SFST) and develop a physical/mechanistic model for the entrainment bubble-size spectrum per unit interface area $\mathcal{N}_e(r)$. The model defines the spectrum dependence on gravity $g$, surface tension $\sigma/\rho$, and turbulence dissipation $\epsilon$, and obtains two distinct entrainment regimes separated by bubble-size scale $r_0$. From the model we show that $r_0=r_c=\frac{1}{2}\sqrt{\sigma/\rho g}$, the capillary length scale, and not the Hinze scale $r_H$ as is generally assumed. For an air-water interface and earth gravity, $r_c \approx 1.5$mm. We confirm the theoretical model by high-fidelity, two-phase, volume-conserving direct numerical simulations (DNS) of a canonical SFST flow. We will present: (1) the respective power-laws of the two regimes; (2) the value $r_0=r_c\ne r_H$; (3) the scaling of $\mathcal{N}_e$ with $g$,$\sigma/\rho$ and $\epsilon$; and (4) confirmation of the $\epsilon-r$ entrainment regime map predicted by the model.