Current engineering standards lack the ability to predict the peak impact forces of breaking waves impinging directly upon coastal structures. In this study solitary waves impacting vertical and tapered walls are investigated. To capture the detailed physics of the wave profile that impacts the wall, two-dimensional direct numerical simulations are applied to model the wave traveling over a simplified bathymetry consisting of an initially uniform depth, followed by a uniform beach ramp and then terminating in a uniform depth inshore region and vertical wall. Such an approach can simulate wave runup on land and then the impact with the vertical or tapered walls. The wall location in the bathymetry was varied to simulate different types of wave impacts, including non-breaking, plunging, and bores. The resulting wave characteristics and wall impact pressures were compared across these varying regimes. The associated wave impact force was extracted and compared to various standards used in coastal engineering, and severe underestimation has been found for plunging and weak plunging type impacts. To address this, in this study, a dimensionless distance parameter has been proposed to provide a unifying trend in regards to the peak impact forcing across the various impact types.

This paper presents a computationally fast implementation of the third-order wave interaction theory of Madsen and Fuhrman (2012) for multidirectional irregular waves. The evaluation stage is reformulated from direct summation of sinusoidal waves in the physical domain to spectral accumulation, followed by two-dimensional inverse fast Fourier transform, reducing the cost of full-grid evaluation from 𝒪(N_gN_terms) per time step to 𝒪(N_terms)+𝒪(N_glog N_g). Numerical verification against direct physical-space evaluation shows machine-precision agreement, with typical errors of order 10^-13 – 10^-12 in tests using a finite-depth, high-steepness, multidirectional crossing-sea case. Representative benchmark cases show speedup of roughly three orders of magnitude. The method provides a practical route for higher-order nonlinear surface-field evaluation in larger multidirectional bound wave calculations and related time series where repeated time step evaluations are required, and more broadly suggests spectral-domain accumulation as a scalable strategy for problems involving large-scale superposition of sinusoidal wave components.

This study investigates the performance of a vertically-Lagrangian multi-layer model on numerically simulating shoaling and breaking two-dimensional solitary waves during both the breaking and post-breaking processes. The energy dissipation of the breaking event for the multi-layer waves is analyzed and compared to prior direct numerical simulation work with the same bathymetric and wave cases. It shows very similar data collapse to shallow-water inertial theory. For post-breaking behavior, bore characteristics are compared to an experimental study of bores formed from breaking solitary waves and similar results are found. While the multi-layer method was not found to behave sufficiently well for direct force measurement at a vertical wall, the resulting bore characteristic behavior is found to be sufficient for use in theoretical estimations of the impact force on the wall. These findings in this study suggest that vertically-Lagrangian multilayer models resolve propagating bores sufficiently well when trying to estimate dynamic loads on vertical seawalls with minimal model tuning.

We present numerical simulations investigating the evolution of liquid droplets into baglike structures in turbulent airflows. The droplet bag breakup problem is of significance for many multiphase processes in scientific and engineering applications. Turbulent fluctuations are introduced synthetically into a mean flow, and the droplet is inserted when the air-phase turbulence reaches a statistically stationary state. The morphological evolution of the droplet under different turbulence configurations is retrieved and analyzed in comparison with laminar aerobreakup results. While the detailed evolution history of individual droplets varies widely between different realizations of the turbulent flow, common dynamic and morphological evolution patterns are observed. The presence of turbulence is found to enhance the drag coefficient of the droplet as it flattens. At late times, the droplet becomes tilted and increasingly corrugated under strong turbulence intensity. We quantify these phenomena and discuss their possible governing mechanisms associated with turbulence intermittency. Lastly, the influences of liquid-gas viscosity ratio are examined and the implications of air-phase turbulence on the later bag film breakup process are discussed.

We develop a phase-based manipulation technique, which can be adopted for many physical problems that involve separating the slave harmonics from the free waves. This technique extends the existing literature by developing a general solution to the problem while enabling the incorporation of arbitrary numbers and combinations of phases for computation. We further extend our matrix formulation to phase-based reconstruction applications, where a missing phase can be predicted from other phase-shifted signals with all slave harmonics included. This allows a clean separation between weakly nonlinear effects from the strong nonlinear process such as the breaking of water waves. We examine the possible sources for the phase-based reconstruction error, including the bandwidth of the input signal, the nonlinear effects, different phase combinations and phase misalignment. We aim to provide a reference for the optimal choice of the number and combination of phases to be selected. We use surface gravity-driven water waves as a case study.

Large sea spray drops - of up to 2mm in diameter - constitute one of the most uncertain factors controlling the intensification of hurricanes and severe storms because their generation mechanisms are not understood. Wave splashing produces among the largest spray drops, but observational data regarding these drops is difficult to obtain and hence cannot inform current modelling efforts. In this study, we instead propose a sea spray generation function (SSGF) for ocean wave splashing by assembling a model from first principles. First, we introduce the transverse collision of two cylindrical liquid rims as the basic mechanism for drop production. We characterize the resulting drop production in terms of three competing processes: ligament production and merging, drop generation by end-pinching, and gravity which arrests the mechanism. Second, we formulate a theoretical model which explains the drop size distributions produced by the colliding rims and test it against existing experimental and numerical data. Finally, the model can be developed into a full SSGF by incorporating sea state information with relatively few tuning parameters. The model is flexible and can be extended by including related effects such as finite droplet lifetime and secondary breakup. Altogether, our model suggests that wave splashing can efficiently produce numerous secondary droplets, challenging prior assumptions that it is an inefficient generation mechanism for sea spray.

Rainfall has been observed to damp water waves. However, the long observed effects of rainfall on water waves have not been much investigated numerically. In this paper, numerical simulations are conducted to explore the effect of rainfall on two-dimensional water waves. The Basilisk software, used herein, solves the two-phase, incompressible, Navier-Stokes equations on adaptive Cartesian meshes. In the present simulations, a monochromatic wave is generated within the domain boundary and periodically moves from left to right through the domain. Rainfall with representative drop diameter distributions, as well as accurate terminal velocities, is generated to fall on the monochromatic wave. The energy of the receiving body of water is tracked for evidence of dissipation of the wave induced by the rainfall. The simulation is run for different values of the wave steepness, ranging from non-breaking waves to plunging breakers. It is found that, in no wind conditions, the presence of rainfall can reduce wave energy, particularly in the non-breaking case. In the future, a more realistic configuration with three-dimensional waves and wind will be considered.

Depth-limited overturning wave shape affects water turbulence and sediment suspension. Experiments have shown that wind affects shoaling and overturning wave shape, with uncertain mechanism. Here, we study wind effects (given by the wind Reynolds number) on solitary wave shoaling and overturning with the two-phase direct numerical simulations model Basilisk run in two dimensions on steep bathymetry for fixed wave Reynolds number and Bond number. For all wind, the propagating solitary wave sheds a two-dimensional turbulent air wake and has nearly uniform speed with minimal wave energy changes over the rapidly varying bathymetry. Wave-face slope is influenced by wind, and shoaling wave shape changes are consistent with previous studies. As overturning jet impacts, wind-dependent differences in overturn shape are quantified. The non-dimensional breakpoint location and overturn area have similar wind dependence as previous experience, whereas the overturn aspect ratio has opposite wind dependence. During shoaling, the surface viscous stresses are negligible relative to pressure. Surface tension effects are also small but grow rapidly near overturning. In a wave frame of reference, surface pressure is low in the lee and contributes 2-5 % to the velocity potential rate of change in the surface dynamic boundary condition, which, integrated over time, changes the wave shape. Reasons why the overturn aspect ratio is different than in experiment and why a stronger simulated wind is required are explored. The dramatic wind effects on overturning jet area, and thus to the available overturn potential energy, make concrete the implications of wind-induced changes to wave shape.

We present direct numerical simulations of the splashing process between two cylindrical liquid rims. This belongs to a class of impact and collision problems with a wide range of applications in science and engineering, and motivated here by splashing processes in breaking ocean waves. Interfacial perturbations with a truncated white noise frequency profile are introduced to the rims before their collision, whose subsequent morphological development is simulated by solving the two-phase incompressible Navier-Stokes equation with the adaptive mesh refinement (AMR) technique, within the Basilisk software environment. We first derive analytical solutions predicting the unsteady interfacial and velocity profiles of the expanding sheet forming between the two rims, and develop scaling laws for the evolution of the lamella rim under capillary deceleration. We then analyse the formation and growth of transverse ligaments ejected from the lamella rims, which we find to originate from the initial corrugated geometry of the perturbed rim surface. Novel scaling models are proposed for predicting the decay of the ligament number density due to the ongoing ligament merging phenomenon, and found to agree well with the numerical results presented here. The role of the mechanism in breaking waves is discussed further and necessary next steps in the problem identified.

Many supervised machine learning methods have revolutionised the empirical modelling of complex systems. These empirical models, however, are usually "black boxes" and provide only limited physical explanations about the underlying systems. Instead, so-called "knowledge discovery" methods can be used to explore the governing equations that describe observed phenomena. This paper focuses on how we can use such methods to explore underlying physics and also model a commonly observed yet not fully understood phenomenon - the breaking of ocean waves. In our work, we use symbolic regression to explore the equation that describes wave-breaking evolution from a dataset of in silico waves generated using expensive numerical methods. Our work discovers a new boundary equation that provides a reduced-order description of how the surface elevation (i.e., the water-air interface) evolves forward in time, including the instances when the wave breaks - a problem that has defied traditional approaches. Compared to the existing empirical models, the unique equation-based nature of our model allows further mathematical interpretation, which provides an opportunity to explore the fundamentals of breaking waves. Further expert-AI collaborative research reveals the physical meaning of each term of the discovered equation, which suggests a new characteristic of breaking waves in deep water - a decoupling between the water-air interface and the fluid velocities. This novel reduced-order model also hints at computationally efficient ways to simulate breaking waves for engineering applications.

We present novel numerical simulations investigating the bag breakup of liquid droplets. We first examine the viscous effect on the early-time drop deformation, comparing with theory and experiment. Next, a bag film forms at late time and is susceptible to spurious mesh-induced breakup in numerical simulations, which has prevented previous studies from reaching grid convergence of fragment statistics. We therefore adopt the manifold death (MD) algorithm which artificially perforates thin films once they reach a prescribed critical thickness independent of the grid size, controlled by a numerical parameter $L_{sig}$ . We show grid convergence of fragment statistics when utilising the MD algorithm, and analyse the fragment behaviour and bag film disintegration mechanisms including ligament breakup, node detachment and rim destabilisation. Our choice of the critical thickness parameter $L_{sig}$ is limited by numerical constraints and thus has not been matched to experiment or theory; consequently, the current simulations yield critical bag film perforation thicknesses larger than experimentally observed. The influence of the MD algorithm configuration on the bag breakup phenomena and statistics will be investigated in future work. We also study the effects of moderate liquid Ohnesorge number ( $0.005 \leqslant Oh \leqslant 0.05$ ) on the bag breakup process and fragment statistics, where a non-monotonic dependency of the average diameter of bag film fragments on $Oh$ is found. These results highlight the utility of the MD algorithm in multiphase simulations involving topological changes, and pave the way for physics-based numerical investigations into spume generation at the air–sea interface.

The effects of surfactants on a mechanically generated plunging breaker are studied experimentally in a laboratory wave tank. Waves are generated using a dispersively focused wave packet with a characteristic wavelength of $\lambda _0 = 1.18$ m. Experiments are performed with two sets of surfactant solutions. In the first set, increasing amounts of the soluble surfactant Triton X-100 are mixed into the tank water, while in the second set filtered tap water is left undisturbed in the tank for wait times ranging from 15 min to 21 h. Increasing Triton X-100 concentrations and longer wait times lead to surfactant-induced changes in the dynamic properties of the free surface in the tank. It is found that low surface concentrations of surfactants can dramatically change the wave breaking process by changing the shape of the jet and breaking up the entrained air cavity at the time of jet impact. Direct numerical simulations (DNS) of plunging breakers with constant surface tension are used to show that there is significant compression of the free surface near the plunging jet tip and dilatation elsewhere. To explore the effect of this compression/dilatation, the surface tension isotherm is measured in all experimental cases. The effects of surfactants on the plunging jet are shown to be primarily controlled by the surface tension gradient ( $\Delta \mathcal {E}$ ) while the ambient surface tension of the undisturbed wave tank ( $\sigma _0$ ) plays a secondary role.

While understanding breaking waves is crucial for the development of parametrizations used in ocean wave modeling for both deep and shallow water, the complete process of wave breaking is not well understood. Here we present direct numerical simulations of twodimensional solitary waves that shoal and break on a uniform beach in shallow water, with the presence of storm surge represented by an inshore region. The storm surge depth, beach slope, and wave amplitude are varied to study the dependence of energy dissipation in the breaker on wave and bathymetric parameters. We classify wave breaker types and find a separation between plunging and spilling breakers when scaled by breaking amplitude and depth. We compare energy dissipation during the breaking process with results from the literature without storm surge. A representation of energy dissipation in this solitary wave breaker data is also compared with prior experiments and simulations of breakers in deep water, and possibilities of a unifying model are explored. We conclude that a previously developed shallow-water inertial dissipation model for wave breaking on a uniform slope can be extended to this storm surge environment with good data collapse, and we further discuss possibilities for a general parametrization of wave breaking valid across different depth regimes.

<p>Energy dissipation due to the breaking of surface waves remains an important open topic in both the open ocean and in coastal waters. Here we will discuss similarities between the deep- and shallow-water regimes. To do this, we first present data from direct numerical simulations of shoaling and breaking solitary waves in bathymetric depth transition. In an abrupt depth transition, we investigate the influence of the severity of the depth transition on whether the incident wave will break, finding good agreement with experimental data of Losada et al. (1988). We next investigate the energy dissipation rate in a gradual, linear depth transition. The resulting dataset is compared with an array of existing physics-based scaling arguments, and finds especially good agreement with an inertial model of Mostert & Deike (2020). We then discuss possible scaling approaches for understanding breaker dissipation in shallow water and draw comparisons with deep-water data and models. We will conclude with some insights towards a potential universal breaking parametrisation.</p>

The bubble size distribution below a breaking wave is of paramount interest when quantifying mass exchanges between the atmosphere and oceans. Mass fluxes at the interface are driven by bubbles that are small compared to the Hinze scale $d_h$, the critical size below which bubbles are stable, even though individually these are negligible in volume. Combining experimental and numerical approaches, we report a power law scaling 3/2 of the diameter of small bubble size distribution, for sufficiently large separation of scales between the injection size and the Hinze scale. From an analysis of individual bubble break-up events, we show that break-ups generating small bubbles are driven by capillary effects, and that their break-up time scales with the diameter to the 3/2, which physically explains the sub-Hinze scaling observed.

[Abridged]We present high-resolution three-dimensional direct numerical simulations of breaking waves solving the two-phase Navier-Stokes equations. We investigate the role of the Reynolds and Bond numbers on the energy, bubble and droplet statistics of strong plunging breakers, and explore the asymptotic regimes at high Reynolds and Bond numbers to be compared with laboratory breaking waves. Energetically, the breaking wave transitions from laminar to three-dimensional turbulent flow on a timescale that depends on the turbulent Reynolds number up to a limiting value of $Re_\lambda \sim 100$, consistent with the mixing transition observed in other canonical turbulent flows. We characterize the role of capillary effects on the impacting jet and ingested main cavity shape and subsequent fragmentation process. We confirm two regimes in the bubble size distribution, separated by the Hinze scale $r_H$. We extend the buoyant-energetic scaling of Deike et al. (2016) to account for the cavity shape and its scale separation from the Hinze scale. We show resolved bubbles up to one order of magnitude below the Hinze scale and observe a good collapse of the numerical data compared to laboratory breaking waves (Deane and Stokes 2002). We resolve droplet statistics at high Bond number and our data show good agreement with recent experiments (Erinin et al., 2009) in various statistics. We discuss velocity distributions for the droplets, finding ejection velocities up to four times the phase speed of the wave, which are produced during the most intense splashing events of the breaking process.

We investigate the modes of deformation of an initially spherical bubble immersed in a homogeneous and isotropic turbulent background flow. We perform direct numerical simulations of the two-phase incompressible Navier-Stokes equations, considering a low-density bubble in the high-density turbulent flow at various Weber numbers (the ratio of turbulent and surface tension forces) using the air-water density ratio. We discuss a theoretical framework for the bubble deformation in a turbulent flow using a spherical harmonic decomposition. We propose, for each mode of bubble deformation, a forcing term given by the statistics of velocity and pressure fluctuations, evaluated on a sphere of the same radius. This approach formally relates the bubble deformation and the background turbulent velocity fluctuations, in the limit of small deformations. The growth of the total surface deformation and of each individual mode is computed from the direct numerical simulations using an appropriate Voronoi decomposition of the bubble surface. We show that two successive temporal regimes occur: the first regime corresponds to deformations driven only by inertial forces, with the interface deformation growing linearly in time, in agreement with the model predictions, whereas the second regime results from a balance between inertial forces and surface tension. The transition time between the two regimes is given by the period of the first Rayleigh mode of bubble oscillation. We discuss how our approach can be used to relate the bubble lifetime to the turbulence statistics and eventually show that at high Weber numbers, bubble lifetime can be deduced from the statistics of turbulent fluctuations at the bubble scale.