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.

We investigate mechanisms governing the initial growth and nonlinear evolution of interfacial waves in horizontal two-fluid plane Couette–Poiseuille flows. Nonlinear coupling of the Kelvin–Helmholtz interfacial instability with resonant wave interactions has been shown to be capable of rapidly generating long waves through the transfer of energy from linearly unstable short waves to stable long-wave components within the context of potential flow theory. The objective of this work is to determine whether that coupled mechanism persists in laminar and turbulent viscous flows. Utilizing both theoretical and computational methods, we analyse the initial Orr–Sommerfeld instability to quantify the frequencies and growth/decay rates of each wave mode for two-fluid laminar and turbulent channel flows. The obtained dispersion relation allows for the identification of resonant and/or near-resonant triads among (unstable and damped) wave components in an interfacial wave spectrum. We perform direct numerical simulations (DNS) of the two-phase Navier–Stokes equations with a fully nonlinear interface to formally establish the validity of our theoretical predictions for viscous flows. DNS results show the existence of a nonlinear energy cascade from unstable short- to damped long-wavelength waves due to resonant subharmonic and/or triadic interactions in both laminar Couette and turbulent Poiseuille flows. Spectral analysis of the interfacial evolution confirms that the combined instability–resonance mechanism persists in the presence of viscosity despite being derived under the assumption of potential flow theory. Finally, we perform a detailed examination of experimentally measured wave power spectra from Jurman et al. (J. Fluid Mech., vol. 238, 1992, pp. 187–219) and carry out a numerical sensitivity study of the flow conditions to demonstrate and verify the existence of the coupled instability–resonance mechanism in physical systems. Our analysis accurately predicts the initial instability and the resulting nonlinear energy cascade through subharmonic and triadic interfacial wave resonances.

We consider the problem of nonlinear resonant interactions of interfacial waves with the presence of a linear interfacial instability in an inviscid two-fluid stratified flow through a horizontal channel. The resonant triad consists of a (linearly) unstable wave and two stable waves, one of which has a wavelength that can be much longer than that of the unstable component. Of special interest is the development of the long wave by energy transfer from the base flow due to the coupled effect of nonlinear resonance and interfacial instability. By use of the method of multiple scales, we derive the interaction equations which govern the time evolution of the amplitudes of the interacting waves including the effect of interfacial instability. The solution of the evolution equations shows that depending on the flow conditions, the (stable) long wave can achieve a bi-exponential growth rate through the resonant interaction with the unstable wave. Moreover, the unstable wave can grow unboundedly even when the nonlinear self-interaction effect is included, as do the stable waves in the associated resonant triad. For the verification of the theoretical analysis and the practical application involving a broadbanded spectrum of waves, we develop an effective direct simulation method, based on a high-order pseudo-spectral approach, which accounts for nonlinear interactions of interfacial waves up to an arbitrary high order. The direct numerical simulations compare well with the theoretical analysis for all of the characteristic flows considered, and agree qualitatively with the experimental observation of slug development near the entrance of two-phase flow into a pipe.