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

Abstract

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 ${\sim}k^{-5/3}$ 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.

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

“…For SFST, the characteristic large surface deformations (and large Fr and We) weaken the free-surface boundary conditions, resulting in essentially isotropic near-surface turbulence (Yu et al 2019). Thus, the Kolmogorov spectrum E(r −1 ) ∝ 2/3 r 5/3 provides the near-surface turbulence energy spectrum (this is also confirmed in the DNS, see figure 4).…”

“…We note that for constant g, σ /ρ and ν, DNS-2 has a larger U and the same L as DNS-1. The DNS results capture the evolution of air-entraining SFST corresponding to large Fr and We (Yu et al 2019). Briefly, starting from a quiescent free surface, the FST grows rapidly due to production by the bulk flow mean shear, followed by a quasi-steady active entrainment SFST period, after which the FST eventually decays (and entrainment ceases) due to turbulence diffusion and dissipation.…”

“…We solve the three-dimensional two-phase incompressible Navier-Stokes equations with a fully nonlinear interface resolved by the cVOF method (Weymouth & Yue 2010), which conserves volume to machine accuracy. Yu et al (2019) provides the details of numerical implementations and verifications. For the data presented here, the mean shear-flow depth L and velocity deficit U are the characteristic length and velocity scales and normalize all the variables unless otherwise stated, with Fr 2 = U 2 /gL, We = ρU 2 L/σ and Re = UL/ν.…”

“…We report results within the quasi-steady active entrainment SFST period. In this period, turbulence forms vortical structures of different length scales near the free surface and entrains bubbles of different sizes (Yu et al 2019). Figure 4 shows the one-dimensional energy spectra of near-surface turbulence during the SFST period for DNS-1 (DNS-2 is similar).…”

“…For a relatively short time after the onset of entrainment, before subsurface mechanisms come into play, N a (r) closely resembles N E a (r) in terms of the power-law regimes and slopes. With increasing time, N a (r) evolves (primarily) as a result of turbulent bubble fragmentation and degassing, and can become qualitatively different from N E a (r) (Yu et al 2019). We perform DNS over a range of Froude (Fr) and Weber (We) numbers for the FST and obtain N a (r) at a relatively early time.…”