Numerical investigation of shear-flow free-surface turbulence and air entrainment at large Froude and Weber numbers

We investigate two-phase free-surface turbulence (FST) associated with an underlying shear flow under the condition of strong turbulence (SFST) characterized by large Froude (Fr) and Weber (We) numbers. We perform direct numerical simulations of three-dimensional viscous flows with air and water phases. In contrast to weak FST (WFST) with small free-surface distortions and anisotropic underlying turbulence with distinct inner/outer surface layers, we find SFST to be characterized by large surface deformation and breaking accompanied by substantial air entrainment. The interface inner/outer surface layers disappear under SFST, resulting in nearly isotropic turbulence with similar to k(-5/3) scaling of turbulence kinetic energy near the interface (where k is wavenumber). The SFST air entrainment is observed to occur over a range of scales following a power law of slope -10/3. We derive this using a simple energy argument. The bubble size spectrum in the volume follows this power law (and slope) initially, but deviates from this in time due to a combination of ongoing broad-scale entrainment and bubble fragmentation by turbulence. For varying Fr and We, we find that air entrainment is suppressed below critical values Fr-cr and We(cr). When Fr-2 > Fr-cr(2) and We > We(cr), the entrainment rate scales as Fr-2 when gravity dominates surface tension in the bubble formation process, while the entrainment rate scales linearly with We when surface tension dominates.