The depth-integrated horizontal momentum equations and continuity equation are employed to develop a new model. The vertical velocity and pressure can be expressed exactly in terms of horizontal velocities and free-surface elevation, which are the only unknowns in the model. Dividing the water column into elements and approximating horizontal velocities using linear shape function in each element, a set of model equations for horizontal velocities at element nodes is derived by adopting the weighted residual method. These model equations can be applied for transient or steady free-surface flows by prescribing appropriate lateral boundary conditions and initial conditions. Here, only the wave–current–bathymetry interaction problems are investigated. Theoretical analyses are conducted to examine various linear wave properties of the new models, which outperform the Green–Naghdi-type models for the range of water depth to wavelength ratios and the Boussinesq-type models as they are capable of simulating vertically sheared currents. One-dimensional horizontal numerical models, using a finite-difference method, are applied to a wide range of wave–current–bathymetry problems. Numerical validations are performed for nonlinear Stokes wave and bichromatic wave group propagation in deep water, sideband instability, regular wave transformation over a submerged shoal and focusing wave group interacting with linearly sheared currents in deep water. Very good agreements are observed between numerical results and laboratory data. Lastly, numerical experiments of wave shoaling from deep to shallow water are conducted to further demonstrate the capability of the new model.

Oscillatory boundary layers over flat and rippled seabeds are well described in the literature. However, the presence of protruding vegetation stems has received no theoretical or experimental attention. The present work establishes an analytical constant viscosity model akin to the Stokes oscillatory boundary layer solution and a nonlinear varying-viscosity numerical model with a turbulence closure. The two models are used to describe the importance of vegetation and free stream velocity characteristics on spatially averaged oscillatory boundary layers: their friction factors, thickness and phase leads over the free-stream velocity. The models are periodic in time and resolve boundary and shear layers over the vertical, contrary to past efforts applying two-layer models. The models are extended to investigate the importance of finite wavelengths with steady streaming stresses and their associated mean velocity profile. Steady streaming is quantified both for the near-bottom streaming within the canopy and for the streaming in the shear layer above the canopy. Finally, akin to theoretical and experimental works on mean flows over unvegetated and flat seabeds due to oscillatory and nonlinear free-stream velocities, the numerical model investigates varying degree of nonlinearity for velocity- and acceleration-skewed velocity signals, and it is identified that the presence of vegetation stems gives rise to an additional contribution to the horizontal momentum balance which is not present for unvegetated conditions. Finally, it is discussed how the presence of a free surface, contrary to purely oscillatory conditions, alters the horizontal momentum balance within and above the canopy.

The impact of shoaling on linear water waves is well known, but it has only been recently found to significantly amplify both the intensity and frequency of rogue waves in nonlinear irregular wave trains atop coastal shoals. At least qualitatively, this effect has been partially attributed to the ‘rapid’ nature of the shoaling process, i.e. shoaling occurs over a distance far shorter than that required for waves to modulate themselves and adapt to the reduced water depth. Through a theoretical model and highly accurate nonlinear simulations, we disentangle the respective effects of the length and angle of a shoal's slope. We investigate the effects of the shoaling process rapidness on the evolution of key statistical and spectral sea-state parameters. We let the wave field evolve over a slope with constant angle in all cases while we vary the slope length. Our results indicate that the non-equilibrium dynamics is not affected by the slope length, because further extending the slope length does not influence the magnitude of the statistical and spectral measures as long as the non-equilibrium dynamics dominates the wave evolution. Thus, the shoaling effect on rogue waves is deduced to be mainly driven by the slope magnitude rather than the slope length.

Accurate computation of hydroelastic waves in shallow water is critical because many hydroelastic wave applications are nearshores, such as sea-ice and floating infrastructures. In this paper, Boussinesq assumptions for shallow water are employed to derive nonlinear Boussinesq-type equations of hydroelastic waves, in which non-uniform distribution of structural stiffness and varying water depth are considered rigorously. Application of Boussinesq assumptions enables complicated three-dimensional problems to be reduced and formulated on the two-dimensional horizontal plane, therefore the proposed Boussinesq-type models are straightforward and versatile for a wide range of hydroelastic wave applications. Two configurations, a floating plate and a submerged plate, are studied. The first-order linear governing equations are solved analytically with periodic conditions assuming constant depth and uniform stiffness, and the linear dispersion relations are subsequently derived for both configurations. For flexural-gravity waves of a floating plate, unique behaviours of flexural-gravity waves different from shallow-water waves are discussed, and a generalized solitary wave solution is investigated. A nonlinear numerical solver is developed, and nonlinear flexural-gravity waves are found to have smaller wavelength and celerity than their linear counterparts. For hydroelastic waves of a submerged plate, dual-mode analytical solutions are discovered for the first time. Numerical computation has demonstrated that a plate with decreasing submerged depth is able to transfer wave energy from the deeper water to the surface layer.

The probability density function (PDF) for the free surface elevation in an irregular sea has an integral formulation when based on the cumulant generating function. To leading order, the result is Gaussian, whereas nonlinear extensions have long been limited to Gram–Charlier series approximations. As shown recently by Fuhrman et al. (J. Fluid Mech., vol. 970, 2023, A38), however, the second-order integral can be represented exactly in closed form. The present work extends this further, enabling determination of this PDF to even higher orders. Towards this end, a new ordinary differential equation (ODE) governing the PDF is first derived. Asymptotic solutions in the limit of large surface elevation are then found, utilizing the method of dominant balance. These provide new analytical forms for the positive tail of the PDF beyond second order. These likewise clarify how high-order cumulants (involving statistical moments such as the kurtosis) govern the tail, which is shown to get heavier with each successive order. The asymptotic solutions are finally utilized to generate boundary conditions, such that the governing ODE may be solved numerically, enabling novel determination of the PDF at third and higher order. Successful comparisons with challenging data sets confirm accuracy. The methodology thus enables the PDF of the surface elevation to be determined numerically, and the asymptotic tail analytically, to any desired order. Results are worked out explicitly up to fifth order. The theoretical probability of extreme surface elevations (typical of rogue waves) may thus be assessed quantitatively for highly nonlinear irregular seas, requiring only relevant statistical quantities as input.

Equations relating the pressure at a horizontal seabed, the free-surface profile and the surface pressure are derived for two-dimensional irrotational steady water waves with arbitrary pressure at the free surface. Special cases include gravity, capillary, flexural and wind waves. Without approximations, we show that the free-surface recovery from the bottom pressure requires the resolution of only one first-order ordinary differential equation independent of the surface pressure, thus providing a new general recovery method valid for a broad class of water waves. Another equation provides an explicit expression for the surface pressure as a function of the bottom pressure and of the free surface. Thus, if unknown, the surface pressure can also be recovered if one extra measurement is available. This new recovery procedure is illustrated analytically for the linear approximation of a flexural–capillary–gravity wave, and numerically for fully nonlinear capillary–gravity waves.

Flow within submerged canopies influences the transport of nutrients, sediment, pollutants, plant seeds and the settlement of larvae. To improve our understanding of mass transport within canopies, a simple model is proposed to predict the total time-varying velocity within submerged rigid canopies (representing coral reefs) and flexible canopies (representing seagrasses and saltmarshes). The model divides the momentum equations into a canopy layer and free-stream layer. The difference in the time derivative of the velocity between the two layers is balanced by the sum of the shear stress and canopy drag, both of which depend on the in-canopy total velocity. The present model extended the shear stress model developed for steady current to combined current and wave conditions without additional calibrating coefficients. The model agreed well with the in-canopy velocity measured in the present and several previous studies. Importantly, the proposed model significantly improved the accuracy of canopy time-mean velocity prediction, which reduced the root mean square error by more than 50 %, compared with previous models. The model revealed that the addition of waves can significantly decrease the in-canopy time-mean velocity.

Based on the multiple-scale expansion technique, a new set of extended nonlinear Schrödinger (ENLS) equations up to the third order is derived to account for the additional high-order bottom and dispersion effects as well as the nonlinear wave interaction on wave transformation over periodic sandbars of sinusoidal geometry. By employing the small-amplitude wave assumption, a closed-form analytical solution for Bragg scattering is obtained from the linearised ENLS equations, which demonstrates that a downshift of wave frequency of the maximum reflection is mainly due to the inclusion of the high-order bottom effect. The factors that affect the downshift of the resonant frequency are identified and a theoretical expression in parabolic form is derived to quantify the downshift magnitude. The fully ENLS equations are further analysed to reveal the additional wave nonlinear effects on Bragg scattering characteristics. Under the condition of infinitesimal sandbar amplitude, the ENLS equations render a theoretical expression of the critical value of $kh$ when the nonlinear wave self-modulation effect and the nonlinear wave cross-modulation effect are equal, whereas the former effect is responsible for wavenumber upshifting and the latter downshifting. When $kh$ is larger than the critical value, the increase of wave nonlinearity will enhance the downshift magnitude of the Bragg resonance, and vice versa. For finite amplitude of the bottom sandbar, the ENLS equations are solved numerically to examine the influence of both wave nonlinearity and sandbar amplitude on the characteristics of Bragg resonance. The results reveal that as the increase of sandbar amplitude, the critical $kh$ increases monotonically.

A novel boundary integral approach for the recovery of overhanging (or not) rotational water waves (with constant vorticity) from pressure measurements at the bottom is presented. The method is based on the Cauchy integral formula and on an Eulerian–Lagrangian formalism to accommodate overturning free surfaces. This approach eliminates the need to introduce a priori a special basis of functions, thus providing a general means of fitting the pressure data and, consequently, recovering the free surface. The effectiveness and accuracy of the method are demonstrated through numerical examples.

For turbulent bubbly flows, multi-phase simulations resolving both the liquid and bubbles are prohibitively expensive in the context of different natural phenomena. One example is breaking waves, where bubbles strongly influence wave impact loads, acoustic emissions and atmospheric-ocean transfer, but detailed simulations in all but the simplest settings are infeasible. An alternative approach is to resolve only large scales, and model small-scale bubbles adopting sub-resolution closures. Here, we introduce a large eddy simulation smoothed particle hydrodynamics (SPH) scheme for simulations of bubbly flows. The continuous liquid phase is resolved with a semi-implicit isothermally compressible SPH framework. This is coupled with a discrete Lagrangian bubble model. Bubbles and liquid interact via exchanges of volume and momentum, through turbulent closures, bubble breakup and entrainment, and free-surface interaction models. By representing bubbles as individual particles, they can be tracked over their lifetimes, allowing closure models for sub-resolution fluctuations, bubble deformation, breakup and free-surface interaction in integral form, accounting for the finite time scales over which these events occur. We investigate two flows: bubble plumes and breaking waves, and find close quantitative agreement with published experimental and numerical data. In particular, for plunging breaking waves, our framework accurately predicts the Hinze scale, bubble size distribution, and growth rate of the entrained bubble population. This is the first coupling of an SPH framework with a discrete bubble model, with potential for cost-effective simulations of wave–structure interactions and more accurate predictions of wave impact loads.

To date, the predominant means for computing the probability density function (p.d.f.) for the free surface elevation of a nonlinear, irregular water wave field, free of assumptions involving narrow-bandedness and small directionality, is the approximate Gram–Charlier series solution of Longuet-Higgins (J. Fluid Mech., vol. 17, 1963, pp. 459–480, hereafter LH63). In this paper we re-visit the derivation of this p.d.f. to second order in the wave steepness, utilizing both moment and cumulant generating functions. We show that LH63's approximate solution based on the cumulant generating function, in fact, matches that derived from the moment generating function. Moreover, through a change of variables coupled with complex analysis, it is shown that the approximation employed by LH63 is unnecessary, and the second-order p.d.f. stemming from the cumulant generating function can be represented exactly in terms of the Airy function. The new second-order p.d.f. predicts increased probability of extreme positive surface elevations typical of e.g. rogue waves, relative to both second- and third-order solutions of LH63. This heavy positive tail is inherent, and is explained through comparison of the asymptotic limits of the p.d.f.s for large surface elevations. A semi-theoretical method is also proposed for remedying non-physical spurious oscillations that arise in the negative tail, based on the envelope of the Airy function with negative arguments. This modified negative tail is valid for irregular wave fields having skewness less than or equal to 0.2. The new p.d.f.s are compared against those based on data sets generated from second-order irregular wave theory as well as a fully nonlinear, spectrally accurate numerical wave model. Good accuracy is collectively demonstrated for directionally spread irregular seas in both finite and infinite water depths for a range of directional spreading.

Six consecutive solitary waves with identical wave height and separation time are generated to study the flow structures during the uprush–downwash interactions in the swash zone. Using particle image velocimetry, the cross-shore velocity fields are captured. Two different wave conditions are examined with different wave-height-to-water-depth ratios, i.e. $H_o/h=0.11$ and 0.22. The uprush–downwash interaction reaches quasi-steady state from the third solitary wave for both cases. For the former case, a weak non-stationary hydraulic jump appears during the downwash flow for all the six consecutive waves. The weak hydraulic jump evolves into a momentarily ‘stationary’ broken bore when the next wave arrives. For the latter case, the larger wave height generates stronger wave breaking. No non-stationary hydraulic jump is observed as the duration of downwash flow is relatively short. The flow reverses to the onshore direction before the downwash Froude number reaches the hydraulic jump condition. The temporal and spatial evolution of turbulence structure at the quasi-steady state is quantified using the spatial spectral analysis, the integral length scale and turbulence eddy viscosity. The results suggest that the large-scale energy generated during the uprush–downwash interaction modified the slope of the turbulence energy spatial spectrum in the inertial subrange from $-$5/3 to $-$1 in the larger length scale region, indicating the energy cascade depends not only on the dissipation rate, but also on the turbulent kinetic energy from the large-scale turbulence structure because of the large-scale energy injection in the inertial subrange.

Bathymetric changes have been experimentally shown to affect the occurrence of rogue waves. We recently derived a non-homogeneous correction to the spectral analysis, allowing us to describe the evolution of the rogue wave probability over a shoal. Here, we extend this work to the evolution of the excess kurtosis of the surface elevation, that plays a central role in estimating rare event probabilities. Furthermore, we provide an upper bound to the excess kurtosis. In intermediate and deep water regimes, a shoal does not affect wave steepness nor bandwidth significantly, so that the vertical asymmetry between crests and troughs, the excess kurtosis and the exceedance probability of wave height stay rather constant. In contrast, in shallower water, a sharp increase in wave steepness increases the vertical asymmetry, resulting in a growth of both the tail of the exceedance probability and the excess kurtosis.

Flow and turbulent structures generated by the interaction of forced incoming oscillatory flow with a circular, vertical cylinder placed in an open channel with a horizontal bed are investigated using eddy-resolving simulations. Validation simulations performed with a Keulegan–Carpenter (KC) number of 20 and multiple-mode forcing corresponding to the laboratory experiment of Sumer et al. (J. Fluid Mech., vol. 332, 1997, pp. 41–70) show that detached eddy simulation (DES) predicts more accurately the amplification of the bed shear stress beneath the horseshoe vortex system (upstream side of the cylinder) and the maximum magnitude of the bed shear stress at the downstream (wake) side of the cylinder compared with unsteady Reynolds-averaged Navier–Stokes simulations. High-Reynolds-number DES simulations are then conducted with 1.5 ≤ KC ≤ 30.8 and one-mode sinusoidal forcing of the streamwise velocity in the approaching flow to investigate the changes in the wake vortex-flow regimes, the coherence of the horseshoe vortices and the generation of other near-bed coherent structures in the wake during the oscillatory cycle. The flow is periodic, no horseshoe vortices form and no vortices are shed in the wake for KC = 1.5. By contrast, for KC ≥ 8 horseshoe vortices are present over part of the oscillatory cycle and up to three wake vortices are shed over each half-cycle as KC is increased to 30.8. For an intermediate range of KC numbers, one (KC = 15.4) or two (KC = 8) of the vortices forming at the back of the cylinder during each half-cycle are washed around it when the flow reverses. The main horseshoe vortex and other horizontal near-bed vortices have a large capacity to amplify the bed shear stresses when the incoming velocity magnitude is significantly less than its peak value. Assuming the depth-averaged velocity in the incoming (undisturbed) oscillatory flow is the same in simulations conducted with different KC numbers, the peak values of the sediment entrainment potential measured by the mean (cycle-averaged) volumetric flux of sediment entrained from the bed over one oscillatory cycle occur for 8 ≤ KC ≤ 15.4. For all KC numbers, the in-line force variation over the oscillatory cycle is fairly well approximated by the Morison equation. For KC = 1.5, the in-line force is only due to inertia effects. For KC = 30.8, the maximum and minimum values of the phase-averaged in-line force are approximatively in phase with those of the incoming flow velocity. For KC ≥ 15, the phase-averaged in-line force coefficients vary between 0.8 and 1.1 during most of the oscillatory cycle (e.g. when the incoming flow velocity is not close to zero). This is different from cases with KC ≤ 8 where the in-line force coefficient is equal to zero twice during the oscillatory cycle as the in-line force becomes equal to zero for non-zero values of the incoming velocity. The largest cycle-to-cycle variations of the in-line force coefficient and in-line force are observed around KC = 8. For KC = 8 and 15.4, the cylinder is subject to relatively large phase-averaged spanwise drag forces that are comparable to the peak phase-averaged streamwise drag forces. As KC is increased to 30.8, the phase-averaged spanwise drag force becomes zero over the whole oscillatory cycle but the cylinder is still subject to large instantaneous spanwise forces over part of the oscillatory cycle.

The root systems of mangroves, a tree species found in intertidal tropical and subtropical coastal zones, provide a natural barrier that dissipates wave energy effectively and reduces sediment erosion. Here, we use a combination of experiments and numerical simulations to examine the wake and drag characteristics of porous arrays of cylinders, which serve as simplified models of mangrove root networks. Optimal arrangements of the arrays are obtained by coupling Navier–Stokes simulations with a multi-objective optimization algorithm, which seeks configurations that minimize wake enstrophy and maximize drag on the porous structure. These optimal configurations are investigated using particle image velocimetry, and the internal and external flows around the porous arrays are analysed using a combination of proper orthogonal decomposition and Lagrangian particle tracking. Large variations in drag and enstrophy are observed by varying the relative positions of the cylinders, which indicates that the geometrical arrangement of porous arrays plays a prominent role in determining wake and drag characteristics. A sensitivity analysis suggests that enstrophy is more sensitive than drag to specific cylinder placement, and depends on distinctive flow patterns that develop in the interior due to interactions among neighbouring cylinders. Arrays with higher drag involve a combination of larger projected frontal area and minimal flux through the interior, leading to increased wake enstrophy, which is unfavourable for particle deposition and erosion. Based on the analysis of characteristics associated with the optimal arrays, several manually designed arrays are tested, and they display the expected behaviour with regard to drag and enstrophy.

Subaqueous vortex ripples in equilibrium are characterized by their unique geometry and dimensions. Motivated by the recent direct numerical simulation study of oscillatory turbulent flow over a wavy bottom by Önder & Yuan (J. Fluid Mech., vol. 858, 2019, pp. 264–314), the objective of this study is to further investigate the fluid dynamical controls that determine the distinctive equilibrium dimensions of vortex ripples. We use direct numerical simulations to investigate the differences in flow kinetics between sinusoidal oscillatory flow over equilibrium and out-of-equilibrium vortex ripples. In comparison with the equilibrium case, the spanwise coherent vortices, the averaged bottom shear stress on overlying flow and the shear stress distribution on the ripple surface are identified as the key fluid dynamical controls on equilibrium dimensions. Based on these controls, we propose mechanisms in the selection of vortex ripple dimensions. We observe that the flow adjusts in such a way that the interaction between overlying flow and vortex ripples tends to generate the strongest coherent vortices while the ripple surface (or overlying flow) experiences the smallest shear stress averaged over ripple wavelength during the selection process. Through a triple decomposition of the flow, the component of the ripple-induced fluctuation is found to dictate these fluid dynamical controls, which implies that this component plays an important role in the evolution of vortex ripples.

It has been reported that, in submarine landslides down a slope, only a small part of the landslide momentum was transferred to the tsunami while most was lost in the further propagation over the flat bed. The high momentum of landslides off a slope meant a long run-out and a great impact on underwater infrastructure. Nevertheless, little attention has been paid to the variation of momentum of deformable submarine landslides off a slope, i.e. the loss of momentum when the slides flow away from the slope over a flat bottom. In this paper, the translational momentum of deformable submarine granular landslides running down a non-erodible inclined bed is investigated with a two-phase smoothed particle hydrodynamics model. After flowing down the slope, the transport rate and flux of the landslide translational momentum along the propagation over the flat bottom are examined. The effects of physical variables of the slide, particularly the grain size, the initial compaction and the front intrusion angle on the variation of the translational momentum, are explored. Accordingly, scaling relations of the spatial-temporal maximum transport rate and flux of the landslide translational momentum, as well as those of the slide final run-out and the generated leading wave height, are proposed. These scaling relations, although based on numerical data of small-scale granular landslides, demonstrate a preliminary attempt to develop practical expressions in the framework of momentum for estimating the run-out of real submarine landslides and the impact on underwater infrastructure.

In the propagation and evolution of sea waves, previous studies pointed out that the occurrence of the freak wave height is significantly related to the quasi-resonant four-wave interaction in the modulated waves. From numerical--experimental study over an uneven bottom, the nonlinear effect caused by the bathymetry change also contributes to the occurrence of extreme events in unidirectional waves. To comprehensively analyse the two-dimensional wavefield, this study develops an evolution model for a directional random wavefield based on the depth-modified nonlinear Schrödinger equation, which considers the nonlinear resonant interactions and the wave shoaling the shallow water. Through Monte Carlo simulation, we discuss the directional effect on the four-wave interaction in the wave train and the maximum wave height distribution from deep to shallow water with a slow varying slope. The numerical result indicates that the directional spreading has a dispersion effect on the freak wave height. In a shallow-water environment, this effect becomes weak, and the bottom topography change is the main influencing factor in the wave evolution.

Undulating and breaking bores are generated in the laboratory using a programmable long-stroke wavemaker. By changing the stroke length and the speed of the wavemaker, both non-decaying and decaying bores are generated and studied. Bore strength, height and duration are measured and compared with the solutions derived by using the method of characteristics, with excellent agreement. The measurements for inundation depth, runup height and flood duration are checked with the formulas presented in Barranco & Liu (J. Fluid Mech., vol. 915, 2021). The comparisons show that the formulas are also accurate for the non-decaying bores generated by the wavemaker. The maximum inundation depth predicted by the formula for zero bore length at the beach toe agrees with the laboratory observations for decaying bores. Using a high-speed particle image velocimetry system, the ensemble-averaged velocities and fluctuating velocities under undulating bores and breaking bores are measured in constant water depth and in the vicinity of the still water shoreline. Detailed analyses of the velocity fields are presented and discussed. For the undulating bore a long quiescent flood duration is observed, while for the breaking bore the up-rush flow changes into down-rush flow almost linearly.

A multiscale asymptotic theory is formulated for surface gravity waves and currents in finite-depth water with a vegetation canopy that provides a drag force on both flows with known drag coefficients. It assumes that the density is uniform and the depth is uniform pro tem and that the wave frequency is fast compared to the current advective rate. It is a quasi-linear theory in which the wave dynamics is independent of current and drag to leading order but provides perturbative corrections, and in which wave nonlinear interactions are neglected while quadratic wave-averaged wave fluxes and quadratic wave-drag effects are retained. The primary surface wave is modified by drag and current interactions, and the wave-averaged current momentum balance includes a wave-augmented drag force and several vortex forces due to Earth's rotation, current vorticity, Stokes drift and drag-induced wave vorticity. The wave-averaged current equations derived here are a suitable basis for future large-eddy simulation and submesoscale circulation computational models.