Interfacial deformation and energy exchange in free-surface turbulence

This study investigates the dynamics of free-surface turbulence (FST) using direct numerical simulations (DNS). We focus on the energy exchange between the deformed free-surface and underlying turbulence, examining the influence of high Reynolds ( ${\textit{Re}}$ ) and Weber ( ${\textit{We}}$ ) numbers at low to moderate Froude ( ${\textit{Fr}}$ ) numbers. The two-fluid DNS of FST at the simulated conditions is able to incorporate air entrainment effects in a statistical steady state. Results reveal that a high ${\textit{We}}$ number primarily affects entrained bubble shapes (sphericity), while ${\textit{Fr}}$ significantly alters free-surface deformation, two-dimensional compressibility and turbulent kinetic energy (TKE) modulation. Vortical structures are mainly oriented parallel to the interface. At lower ${\textit{Fr}}$ , kinetic energy is redistributed between horizontal and vertical components, aligning with rapid distortion theory, whereas higher ${\textit{Fr}}$ preserves isotropy near the surface. Evidence of a reverse or dual energy cascade is verified through third-order structure functions, with upscale transfer near the integral length scale, and enhanced vertical kinetic energy in upwelling eddies. Phase-based discrete wavelet transforms of TKE show weaker decay at the 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 subgrid modelling.