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The statistics of breaking wave fields are characterised within a novel multi-layer framework, which generalises the single-layer Saint-Venant system into a multi-layer and non-hydrostatic formulation of the Navier–Stokes equations. We simulate an ensemble of phase-resolved surface wave fields in physical space, where strong nonlinearities, including directional wave breaking and the subsequent highly rotational flow motion, are modelled, without surface overturning. We extract the kinematics of wave breaking by identifying breaking fronts and their speed, for freely evolving wave fields initialised with typical wind wave spectra. The $$\varLambda (c)$$ distribution, defined as the length of breaking fronts (per unit area) moving with speed $$c$$ to $$c+{\rm d}c$$ following Phillips ( J. Fluid Mech. , vol. 156, 1985, pp. 505–531), is reported for a broad range of conditions. We recover the $$\varLambda (c) \propto c^{-6}$$ scaling without wind forcing for sufficiently steep wave fields. A scaling of $$\varLambda (c)$$ based solely on the root-mean-square slope and peak wave phase speed is shown to describe the modelled breaking distributions well. The modelled breaking distributions are in good agreement with field measurements and the proposed scaling can be applied successfully to the observational data sets. The present work paves the way for simulations of the turbulent upper ocean directly coupled to a realistic breaking wave dynamics, including Langmuir turbulence, and other sub-mesoscale processes. 

We investigate wind wave growth by direct numerical simulations solving for the two-phase Navier–Stokes equations. We consider the ratio of the wave speed $$c$$ to the wind friction velocity $$u_*$$ from $$c/u_*= 2$$ to 8, i.e. in the slow to intermediate wave regime; and initial wave steepness $ak$ from 0.1 to 0.3; the two being varied independently. The turbulent wind and the travelling, nearly monochromatic waves are fully coupled without any subgrid-scale models. The wall friction Reynolds number is 720. The novel fully coupled approach captures the simultaneous evolution of the wave amplitude and shape, together with the underwater boundary layer (drift current), up to wave breaking. The wave energy growth computed from the time-dependent surface elevation is in quantitative agreement with that computed from the surface pressure distribution, which confirms the leading role of the pressure forcing for finite amplitude gravity waves. The phase shift and the amplitude of the principal mode of surface pressure distribution are systematically reported, to provide direct evidence for possible wind wave growth theories. Intermittent and localised airflow separation is observed for steep waves with small wave age, but its effect on setting the phase-averaged pressure distribution is not drastically different from that of non-separated sheltering. We find that the wave form drag force is not a strong function of wave age but closely related to wave steepness. In addition, the history of wind wave coupling can affect the wave form drag, due to the wave crest shape and other complex coupling effects. The normalised wave growth rate we obtain agrees with previous studies. We make an effort to clarify various commonly adopted underlying assumptions, and to reconcile the scattering of the data between different previous theoretical, numerical and experimental results, as we revisit this longstanding problem with new numerical evidence. 

We perform direct numerical simulations of a gas bubble dissolving in a surrounding liquid. The bubble volume is reduced due to dissolution of the gas, with the numerical implementation of an immersed boundary method, coupling the gas diffusion and the Navier–Stokes equations. The methods are validated against planar and spherical geometries’ analytical moving boundary problems, including the classic Epstein–Plesset problem. Considering a bubble rising in a quiescent liquid, we show that the mass transfer coefficient $$k_L$$ can be described by the classic Levich formula $$k_L = (2/\sqrt {{\rm \pi} })\sqrt {\mathscr {D}_l\,U(t)/d(t)}$$ , with $d(t)$ and $U(t)$ the time-varying bubble size and rise velocity, and $$\mathscr {D}_l$$ the gas diffusivity in the liquid. Next, we investigate the dissolution and gas transfer of a bubble in homogeneous and isotropic turbulence flow, extending Farsoiya et al. ( J. Fluid Mech. , vol. 920, 2021, A34). We show that with a bubble size initially within the turbulent inertial subrange, the mass transfer coefficient in turbulence $$k_L$$ is controlled by the smallest scales of the flow, the Kolmogorov $$\eta$$ and Batchelor $$\eta _B$$ microscales, and is independent of the bubble size. This leads to the non-dimensional transfer rate $${Sh}=k_L L^\star /\mathscr {D}_l$$ scaling as $${Sh}/{Sc}^{1/2} \propto {Re}^{3/4}$$ , where $${Re}$$ is the macroscale Reynolds number $${Re} = u_{rms}L^\star /\nu _l$$ , with $$u_{rms}$$ the velocity fluctuations, $L^*$ the integral length scale, $$\nu _l$$ the liquid viscosity, and $${Sc}=\nu _l/\mathscr {D}_l$$ the Schmidt number. This scaling can be expressed in terms of the turbulence dissipation rate $$\epsilon$$ as $${k_L}\propto {Sc}^{-1/2} (\epsilon \nu _l)^{1/4}$$ , in agreement with the model proposed by Lamont & Scott ( AIChE J. , vol. 16, issue 4, 1970, pp. 513–519) and corresponding to the high $Re$ regime from Theofanous et al. ( Intl J. Heat Mass Transfer , vol. 19, issue 6, 1976, pp. 613–624). 

Bubble-mediated gas exchange in turbulent flow is critical in bubble column chemical reactors as well as for ocean–atmosphere gas exchange related to air entrained by breaking waves. Understanding the transfer rate from a single bubble in turbulence at large Péclet numbers (defined as the ratio between the rate of advection and diffusion of gas) is important as it can be used for improving models on a larger scale. We characterize the mass transfer of dilute gases from a single bubble in a homogeneous isotropic turbulent flow in the limit of negligible bubble volume variations. We show that the mass transfer occurs within a thin diffusive boundary layer at the bubble–liquid interface, whose thickness decreases with an increase in turbulent Péclet number, $$\widetilde {{Pe}}$$ . We propose a suitable time scale $$\theta$$ for Higbie ( Trans. AIChE , vol. 31, 1935, pp. 365–389) penetration theory, $$\theta = d_0/\tilde {u}$$ , based on $$d_0$$ the bubble diameter and $$\tilde {u}$$ a characteristic turbulent velocity, here $$\tilde {u}=\sqrt {3}\,u_{{rms}}$$ , where $$u_{{rms}}$$ is the large-scale turbulence fluctuations. This leads to a non-dimensional transfer rate $${Sh} = 2(3)^{1/4}\sqrt {\widetilde {{Pe}}/{\rm \pi} }$$ from the bubble in the isotropic turbulent flow. The theoretical prediction is verified by direct numerical simulations of mass transfer of dilute gas from a bubble in homogeneous and isotropic turbulence, and very good agreement is observed as long as the thin boundary layer is properly resolved.