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

Abstract

We consider the size spectrum of entrained bubbles under strong free-surface turbulence (SFST). We investigate the entrainment bubble-size spectrum per unit (mean) interface area, , with dimension length , and develop a physical/mechanistic model for through energy arguments. The model obtains two distinct regimes of , separated by bubble-size scale . For bubble radius , the effects of gravity dominate those of the surface tension force , and , where is the turbulence dissipation rate. For , surface tension is more important and . From the model, we show that , the capillary length scale, and not the generally assumed Hinze scale . For an air–water interface and Earth gravity, 1.5 mm. The model provides an – entrainment regime map that identifies a critical dissipation rate (constant for given and ) above which there is appreciable air entrainment, thus separating SFST and weak FST. We confirm the theoretical model and its predictions using two-phase, high-fidelity direct numerical simulations of a canonical FST flow using the conservative volume-of-fluid method: the respective power laws of and for and ; the value ; the scaling ; and the predictions of the – entrainment regime map.