Interfacial deformation and energy exchange in strong free-surface turbulence

Abstract: This study investigates the dynamics of strong free-surface turbulence (SFST) using direct numerical simulations (DNS). We focus on the energy exchange between the deformed free-surface and underlying turbulence, examining the influence of Reynolds ($Re$), Froude ($Fr$), and Weber ($We$) numbers. The two-fluid DNS of SFST at high $Fr$ and $We$ is able to incorporate air entrainment effects in a statistical steady-state. Results reveal that high $We$ primarily affects entrained bubble shapes (sphericity), while $Fr$ significantly alters free-surface deformation, two-dimensional compressibility, and turbulent kinetic energy (TKE) modulation. Vorticity flux across the interface occurs from viscous diffusion of surface-parallel structures. At lower $Fr$, kinetic energy is redistributed between horizontal and vertical components, aligning with rapid distortion theory (RDT), whereas higher $Fr$ preserves isotropy near the surface. Evidence of a reverse or dual energy cascade is verified through third-order structure functions, with a strong net reverse cascade near the integral length scale, and enhanced vertical kinetic energy in upwelling eddies. Discrete wavelet transforms (DWT) of TKE show weaker decay at smallest scales near the interface, suggesting contributions from gravitational energy conversion and reduced dissipation. The wavelet energy spectra also exhibits different scaling laws across the wavenumber range, with a $-3$ slope within the inertial subrange. These findings highlight scale- and proximity-dependent effects on two-phase TKE transport, with implications for sub-grid modeling. This work underscores the need for advanced analytical tools that allow for localization in both physical and spectral domains to further elucidate the complex energy cascade mechanisms in SFST.