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We investigate air entrainment and bubble statistics in three-dimensional breaking waves through novel direct numerical simulations of the two-phase air–water flow, resolving the length scales relevant for the bubble formation problem, the capillary length and the Hinze scale. The dissipation due to breaking is found to be in good agreement with previous experimental observations and inertial scaling arguments. The air entrainment properties and bubble size statistics are investigated for various initial characteristic wave slopes. For radii larger than the Hinze scale, the bubble size distribution, can be described by $N(r,t)=B(V_{0}/2{\\rm\\pi})({\\it\\varepsilon}(t-{\\rm\\Delta}{\\it\\tau})/Wg)r^{-10/3}r_{m}^{-2/3}$ during the active breaking stages, where ${\\it\\varepsilon}(t-{\\rm\\Delta}{\\it\\tau})$ is the time-dependent turbulent dissipation rate, with ${\\rm\\Delta}{\\it\\tau}$ the collapse time of the initial air pocket entrained by the breaking wave, $W$ a weighted vertical velocity of the bubble plume, $r_{m}$ the maximum bubble radius, $g$ gravity, $V_{0}$ the initial volume of air entrained, $r$ the bubble radius and $B$ a dimensionless constant. The active breaking time-averaged bubble size distribution is described by $\\bar{N}(r)=B(1/2{\\rm\\pi})({\\it\\epsilon}_{l}L_{c}/Wg{\\it\\rho})r^{-10/3}r_{m}^{-2/3}$ , where ${\\it\\epsilon}_{l}$ is the wave dissipation rate per unit length of breaking crest, ${\\it\\rho}$ the water density and $L_{c}$ the length of breaking crest. Finally, the averaged total volume of entrained air, $\\bar{V}$ , per breaking event can be simply related to ${\\it\\epsilon}_{l}$ by $\\bar{V}=B({\\it\\epsilon}_{l}L_{c}/Wg{\\it\\rho})$ , which leads to a relationship for a characteristic slope, $S$ , of $\\bar{V}\\propto S^{5/2}$ . We propose a phenomenological turbulent bubble break-up model based on earlier models and the balance between mechanical dissipation and work done against buoyancy forces. The model is consistent with the numerical results and existing experimental results.

We report on an investigation of bubble-induced turbulence. Bubbles of a size larger than the dissipative scale cannot be treated as pointwise inclusions, and generate important hydrodynamic fields in the carrier fluid when in motion. Furthermore, bubble motions may induce a collective agitation due to hydrodynamic interactions which display some turbulent-like features. We tackle this complex phenomenon numerically, performing direct numerical simulations with a volume-of-fluid method. In the first part of the work, we perform both two-dimensional and three-dimensional tests in order to determine appropriate numerical and physical parameters. We then carry out a highly resolved simulation of a three-dimensional bubble column, with a set-up and physical parameters similar to those used in laboratory experiments. This is the largest simulation attempted for such a configuration and is only possible thanks to adaptive grid refinement. Results are compared both with experiments and previous coarse-mesh numerical simulations. In particular, the one-point probability density function of the velocity fluctuations is in good agreement with experiments. The spectra of the kinetic energy show a clear $k^{-3}$ scaling. The mechanisms underlying the energy transfer and notably the possible presence of a cascade are unveiled by a local scale-by-scale analysis in physical space. The comparison with previous simulations indicates to what extent simulations not fully resolved may yet give correct results, from a statistical point of view.

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.

The effect of viscosity on jet formation for bubbles collapsing near solid boundaries is studied numerically. A numerical technique is presented which allows the Navier–Stokes equations with free-surface boundary conditions to be solved accurately and efficiently. Good agreement is obtained between experimental data and numerical simulations for the collapse of large bubbles. However, the bubble rebound in our simulation is larger than that observed in laboratory experiments. This leads us to conclude that compressible and thermal effects should be taken into account to obtain a correct model of the rebound. A parametric study of the effect of viscosity on jet impact velocity is undertaken. The jet impact velocity is found to decrease as viscosity increases and above a certain threshold jet impact is impossible. We study how this critical Reynolds number depends on the initial radius and the initial distance from the wall. A simple scaling law is found to link this critical Reynolds number to the other non-dimensional parameters of the problem.

We investigate the influence of capillary effects on wave breaking through direct numerical simulations of the Navier–Stokes equations for a two-phase air–water flow. A parametric study in terms of the Bond number, $\\mathit{Bo}$ , and the initial wave steepness, ${\\it\\epsilon}$ , is performed at a relatively high Reynolds number. The onset of wave breaking as a function of these two parameters is determined and a phase diagram in terms of $({\\it\\epsilon},\\mathit{Bo})$ is presented that distinguishes between non-breaking gravity waves, parasitic capillaries on a gravity wave, spilling breakers and plunging breakers. At high Bond number, a critical steepness ${\\it\\epsilon}_{c}$ defines the onset of wave breaking. At low Bond number, the influence of surface tension is quantified through two boundaries separating, first gravity–capillary waves and breakers, and second spilling and plunging breakers; both boundaries scaling as ${\\it\\epsilon}\\sim (1+\\mathit{Bo})^{-1/3}$ . Finally the wave energy dissipation is estimated for each wave regime and the influence of steepness and surface tension effects on the total wave dissipation is discussed. The breaking parameter $b$ is estimated and is found to be in good agreement with experimental results for breaking waves. Moreover, the enhanced dissipation by parasitic capillaries is consistent with the dissipation due to breaking waves.

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 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).

Mesoscale eddies, although being on scales of O (20–100) km, have a disproportionate role in shaping the mean stratification, which varies on the scale of O (1000) km. With the increase in computational power, we are now able to partially resolve the eddies in basin-scale and global ocean simulations, a model resolution often referred to as mesoscale permitting. It is well known, however, that due to gridscale numerical viscosity, mesoscale-permitting simulations have less energetic eddies and consequently weaker eddy feedback onto the mean flow. In this study, we run a quasigeostrophic model at mesoscale-resolving resolution in a double gyre configuration and formulate a deterministic closure for the eddy rectification term of potential vorticity (PV), namely, the eddy PV flux divergence. Our closure successfully reproduces the spatial patterns and magnitude of eddy kinetic and potential energy diagnosed from the mesoscale-resolving model. One novel point about our approach is that we account for nonlocal eddy feedbacks onto the mean flow by solving the “subgrid” eddy PV equation prognostically in addition to the mean PV.

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.

We investigate droplet impact on a solid substrate in order to understand the influence of the gas in the splashing dynamics. We use numerical simulations where both the liquid and the gas phases are considered incompressible in order to focus on the gas inertial and viscous contributions. We first confirm that the dominant gas effect on the dynamics is due to its viscosity through the cushioning of the gas layer beneath the droplet. We then describe an additional inertial effect that is directly related to the gas density. The two different splashing mechanisms initially suggested theoretically are observed numerically, depending on whether a jet is created before or after the impacting droplet wets the substrate. Finally, we provide a phase diagram of the drop impact outputs as the gas viscosity and density vary, emphasizing the dominant effect of the gas viscosity with a small correction due to the gas density. Our results also suggest that gas inertia influences the splashing formation through a Kelvin–Helmholtz-like instability of the surface of the impacting droplet, in agreement with former theoretical works.