Water Of Arbitrary Depth Research Articles

Modulation effect of uniform flow on three-dimensional freak wave generation in arbitrary water depth

In arbitrary water depths, the influence of uniform flow, which includes transverse and longitudinal flows, on the generation of three-dimensional (3D) freak waves is examined. A modified Davey–Stewartson equation is derived using potential flow theory and the multiscale method. This equation describes the evolution of 3D freak wave amplitude under the influence of uniform flow. The relationship between two-dimensional (2D) modulational instability (MI) and the generation of 3D freak waves, as represented by the modified 3D Peregrine Breather solution, is explored. The characteristics of 2D MI depend on the orientation of the longitudinal and transverse perturbations. In shallow waters, the generation of freak waves by MI is challenging due to the minimal orientation difference, and longitudinal flows hardly affect the occurrence of MI. Variations in relative water depth can contribute to forming shallow-water freak waves. In finite-depth waters, oblique modulation leads to MI, whereas in deep and infinite-depth waters, longitudinal modulation gains significance. In environments of finite-depth, deep, and infinite-depth waters, the divergence (convergence) effect of longitudinal favorable (adverse) currents reduces (increases) the MI growth rate and suppresses (facilitates) freak wave generation.

Hydro- and aero-elastic response of floating offshore wind turbines to combined waves and wind in frequency domain

An analytical approach and numerical solution to determine coupled aeroelastic and hydroelastic response of floating offshore wind turbines of arbitrary shape to combined wind and wave loads is presented. The model considers simultaneously the aerodynamic and hydrodynamic loads on an FOWT and integrates these with finite element method for structural analysis due to the combined loads. The hydrodynamic and aerodynamic loads are determined based on the linear wave diffraction theory and steady blade element momentum method, respectively, and the solution is obtained in frequency domain. The structure may be fixed or floating, located in arbitrary water depth, and may host single or multiple wind towers. The model captures the complete translational and rotational motions of the body in three dimensions, and the elasticity of the blades, tower and the floating platform. To assess the performance of the model, rigid and elastic responses of a FOWT to combined wave and wind loads are computed and compared with available laboratory measurements and other theoretical approaches where possible, and overall very good agreement is observed. The model developed in this study addresses directly three shortcomings of existing approaches used for the analysis of FOWTs, namely (i) determination of the elastic responses of the entire structure including the floating platform, (ii) analysis of the motion and elastic response of FOWTs in frequency domain, and (iii) assessment of responses of FOWTs with single or multiple wind towers.

Investigating the sidewall's effects on [formula omitted]-shaped floating breakwaters interacting with water waves by the scaled boundary FEM

The effect of the existing vertical sidewall on the hydrodynamic characteristics of a π-shaped floating breakwater interacting with water waves in a fluid domain of finite, arbitrary water depth is analyzed. The Scaled Boundary Finite Element Method (SBFEM) is adopted to solve the Laplace governing equation based on the potential flow assumption in a two-dimensional vertical plane. The accuracy and efficiency of the SBFEM-based model are evaluated by comparing the calculated results of the current model with the literature. The presence of a sidewall generally produces more significant responses due to the creation of standing waves and the appearance of the resonance phenomenon, which synchronizes some wavelengths with the distance between the sidewall and the π-shape floating structure. Furthermore, the present model accurately predicts the hydrodynamic behavior, either with sharp corners or a few elements in the solution domain.

Wave inhomogeneities in a wave basin of constant water depth

Experimental testing of large marine structures in hydrodynamic laboratories requires an accurate reproduction of waves over a large area of the basin. However, with a uniform motion of the wavemaker, wave interactions with the basin’s boundaries can result in an inhomogeneous wave field, even before beach reflection takes place. In this paper, the main geometrical parameters causing inhomogeneities are investigated. For that purpose, a fast linear model is implemented and applied to describe wave fields in basins of arbitrary geometry but constant water depth. Comparisons with experiments in SINTEF’s Ocean Basin demonstrate the validity of the model for regular waves. Finally, an efficient method to reduce wave field inhomogeneities by optimizing the motion of the paddles of a multiflap wavemaker is presented and validated experimentally.

Gravity-capillary waves modulated by linear shear flow in arbitrary water depth* *Project supported by the National Natural Science Foundation of China (Grant No. 41830533) and the National Key Research and Development Program of China (Grant Nos. 2016YFC1401404 and 2017YFA0604102).

Considering that the fluid is inviscid and incompressible and the flow is irrotational in a fixed frame of reference and using the multiple scale analysis method, we derive a nonlinear Schrödinger equation (NLSE) describing the evolution dynamics of gravity-capillary wavetrains in arbitrary constant depth. The gravity-capillary waves (GCWs) are influenced by a linear shear flow (LSF) which consists of a uniform flow and a shear flow with constant vorticity. The modulational instability (MI) of GCWs with the LSF is analyzed using the NLSE. The MI is effectively modified by the LSF. In infinite depth, there are four asymptotes which are the boundaries between MI and modulational stability (MS) in the instability diagram. In addition, the dimensionless free surface elevation as a function of time for different dimensionless water depth, surface tension, uniform flow and vorticity is exhibited. It is found that the decay of free surface elevation and the steepness of free surface amplitude change over time, which are greatly affected by the water depth, surface tension, uniform flow and vorticity.

A Unified Breaking Onset Criterion for Surface Gravity Water Waves in Arbitrary Depth

AbstractWe investigate the validity and robustness of the Barthelemy et al. (2018, https://doi.org/10.1017/jfm.2018.93) wave‐breaking onset prediction framework for surface gravity water waves in arbitrary water depth, including shallow water breaking over varying bathymetry. We show that the Barthelemy et al. (2018) breaking onset criterion, which they validated for deep and intermediate water depths, also segregates breaking crests from nonbreaking crests in shallow water, with subsequent breaking always following the exceedance of their proposed generic breaking threshold. We consider a number of representative wave types, including regular, irregular, solitary, and focused waves, shoaling over idealized bed topographies including an idealized bar geometry and a mildly to steeply sloping planar beach. Our results show that the new breaking onset criterion is capable of detecting single and multiple breaking events in time and space in arbitrary water depth. Further, we show that the new generic criterion provides improved skill for signaling imminent breaking onset, relative to the available kinematic or geometric breaking onset criteria in the literature. In particular, the new criterion is suitable for use in wave‐resolving models that cannot intrinsically detect the onset of wave breaking.

SPH evaluation of the hydrodynamic consequences induced by reef degradation

An apparent increase in the frequency and intensity of natural disturbances and anthropogenic activities accelerates the global degradation of coral reefs. The emergency planning for coastal erosion and flooding along the low-lying coasts fronted by coral reefs underscores the need to predict and evaluate reasonably the hydrodynamic consequences of reef degradation. Here the ability of the well validated Smoothed Particle Hydrodynamics (SPH) porous model is enhanced by integrating a momentum source wave maker for wave groups generation in water of arbitrary depth. Validated results show the SPH model can predict properly the propagation and transformation of wave groups over the complex coral reef. The model is finally adopted to predict the hydrodynamic consequences induced by reef degradation both under the reef resonance and off-resonance conditions. This work provides a general framework for understanding the effects of reef degradation on the spatial distributions of wave setup, wave-induced current and low frequency wave energy over the reef-flat under the action of bichromatic wave groups with different frequency differences.

Wave packet focusing in shallow water

ABSTRACT In coastal engineering, as shoaling waves break in shallow water, most wave-breaking research has been focused on depth-induced wave breaking. However, violent wave breaking under storm conditions may be caused by superposition of many shoaling wave components. In order for simulating extreme waves in open ocean, theories to focus wave-packets, assuming that wave speed of each wave component is unchanged during propagation (i.e. deep water or shallower water without bathymetry changes), have been derived, while shoaling effects of the focusing process in shallower waters have not been considered in the previous studies. The extension to shoaling coastal wave-packets may aid in simulating extreme coastal wave breaking under violent storm conditions via laboratory experiments. Here, we derive the general wave-packet focusing conditions for arbitrary water depth. The proposed conditions were used to simulate shoaling wave-packet focusing on flat and sloped bottoms in shallow water experiments for validation purposes. The effects of depth-induced wave breaking in the focusing wave groups are discussed in this paper. The current focusing technique can fully control the focus location and time in wave experiments with an arbitrary bottom structure, which may be useful for further extreme wave research in coastal engineering.

Observations of Air‐Sea Momentum Flux Variability Across the Inner Shelf

AbstractOver the open ocean, the aerodynamic drag coefficient is typically well predicted; however, the impact depth‐limited processes have on the drag remains underexplored. A case study is presented here where winds, waves, and currents were simultaneously observed from a mobile platform that repeatedly transected the inner shelf of Monterey Bay, CA. Eddy covariance‐derived drag coefficients were compared to several bulk parameterizations, including all of the roughness variations of COARE 3.5 and two explicitly depth‐limited models. The analysis demonstrated that the drag was underestimated by O(2–4) times and the variability with wind speed or cross‐shore distance was not well predicted. The drag based on a recent depth‐limited roughness length model performed substantially better than the rest of the bulk estimates, which were all within 15% of each other and effectively equivalent given typical operational uncertainties. The measured friction velocity was compared to a wave‐dependent parameterization and generalizing the model to arbitrary water depth significantly improved the mean observation‐model difference to within 30%. Latent variability in the observation‐model comparison was associated with stability, wind direction, and wave steepness. The wind stress angle variability was also analyzed. Stress veering was correlated with the alongshore surface current within 2 km from shore (r2= 0.7–0.95, p < 0.05); offshore of this margin, consistent wind stress veering was observed and may be attributable to a secondary, low‐frequency swell system. These results demonstrate that it remains a persistent challenge to accurately predict wind stress variability in the nearshore, especially at locations with complex wave and current fields.

On the limiting Stokes wave of extreme height in arbitrary water depth

Both Schwartz (J. Fluid Mech., vol. 62 (3), 1974, pp. 553–578) and Cokelet (Phil. Trans. R. Soc. Lond., vol. 286 (1335), 1977, pp. 183–230) failed to gain convergent results for limiting Stokes waves in extremely shallow water by means of perturbation methods, even with the aid of extrapolation techniques such as the Padé approximant. In particular, it is extremely difficult for traditional analytic/numerical approaches to present the wave profile of limiting waves with a sharp crest of$120^{\circ }$included angle first mentioned by Stokes in the 1880s. Thus, traditionally, different wave models are used for waves in different water depths. In this paper, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear equations, we successfully gain convergent results (and especially the wave profiles) of the limiting Stokes waves with this kind of sharp crest in arbitrary water depth, even including solitary waves of extreme form in extremely shallow water, without using any extrapolation techniques. Therefore, in the frame of the HAM, the Stokes wave can be used as a unified theory for all kinds of waves, including periodic waves in deep and intermediate depths, cnoidal waves in shallow water and solitary waves in extremely shallow water.

Numerical computations of two-dimensional flexural-gravity solitary waves on water of arbitrary depth

This work is concerned with flexural-gravity solitary waves on water of finite depth. The deformation of the elastic sheet is modelled based on the Cosserat theory of hyperelastic shells satisfying Kirchhoff’s hypotheses. Both steady and unsteady waves are computed numerically for the full Euler equations by using a conformal mapping technique. Complete bifurcation diagrams of solitary waves are presented, and various dynamical experiments, including the evolution of unstable solitary waves and the generation of stable ones, are carried out via direct time-dependent simulations. In particular, we show that generalized solitary waves can also be excited by moving loads on the elastic cover.

Pressure induced by the interaction of water waves with nearly equal frequencies and nearly opposite directions

We present second-order expressions for the free-surface elevation, velocity potential and pressure resulting from the interaction of surface waves in water of arbitrary depth. When the surface waves have nearly equal frequencies and nearly opposite directions, a second-order pressure can be felt all the way to the sea bottom. There are at least two areas of applications: reflective structures and microseisms. Microseisms generated by water waves in the ocean are small vibrations of the ground resulting from pressure oscillations associated with the coupling of ocean surface gravity waves and the sea floor. They are recorded on land-based seismic stations throughout the world and they are divided into primary and secondary types, as a function of spectral content. Secondary microseisms are generated by the interaction of surface waves with nearly equal frequencies and nearly opposite directions. The efficiency of microseism generation thus depends in part on ocean wave frequency and direction. Based on the second-order expressions for the dynamic pressure, a simple theoretical analysis that quantifies the degree of nearness in amplitude, frequency, and incidence angle, which must be reached to observe the phenomenon, is presented.

Automatic Optimization for Large-Scale Real-Time Coastal Water Simulation

We introduce an automatic optimization approach for the simulation of large-scale coastal water. To solve the singular problem of water waves obtained with the traditional model, a hybrid deep-shallow-water model is estimated by using an automatic coupling algorithm. It can handle arbitrary water depth and different underwater terrain. As a certain feature of coastal terrain, coastline is detected with the collision detection technology. Then, unnecessary water grid cells are simplified by the automatic simplification algorithm according to the depth. Finally, the model is calculated on Central Processing Unit (CPU) and the simulation is implemented on Graphics Processing Unit (GPU). We show the effectiveness of our method with various results which achieve real-time rendering on consumer-level computer.

A Generalized Multiple Discrete Interaction Approximation for resonant four-wave interactions in wind wave models

For several decades, the Discrete Interaction Approximation (DIA) for nonlinear resonant four-wave interactions has been the engine of third-generation wind-wave models. The present study presents a Generalized Multiple DIA (GMD) which expands upon the DIA by (i) expanding the definition of the representative quadruplet, (ii) formulating the DIA for arbitrary water depths, (iii) providing complimentary deep and shallow water scaling terms and (iv) allowing for multiple representative quadruplets. The GMD is rigorously derived to be an extension of the DIA, and is backward compatible with it. The free parameters of the GMD are optimized holistically, by optimizing full model behavior in the WAVEWATCH III® wave model as reported in a companion paper. Here, a cascade of GMD configurations with increasing complexity, accuracy and cost is presented. First, the performance of these configurations is discussed using idealized test cases used to optimize the GMD. It is shown that in deep water, GMD configurations can be found which remove most of the errors of the DIA. The GMD is also capable of representing four-wave interactions in extremely shallow water, although some remaining spurious behavior makes applications of this part of the GMD less suitable for operational wave models. Finally, several GMD configurations are applied to an idealized hurricane case, showing that results from idealized test cases indeed are representative for real-world applications, and confirming that such GMD configurations are economically feasible in operational wind wave models. Finally, the DIA results in surprisingly large model errors in hurricane conditions.

Frequency Domain Response at Pacific Coast Harbors to Major Tsunamis of 2005–2011

Tsunamis waves caused by submarine earthquake or landslide might contain large wave energy, which could cause significant human loss and property damage locally as well as in distant region. The response of three harbors located at the Pacific coast (i.e. Crescent City Harbor, Los Angeles/Long Beach Port, and San Diego Harbor) to six well-known tsunamis events generated (both near-field and far-field) between 2005 and 2011 are examined and simulated using a hybrid finite element numerical model in frequency domain. The model incorporated the effects of wave refraction, wave diffraction, partial wave reflection from boundaries, entrance and bottom energy dissipation. It can be applied to harbor regions with arbitrary shapes and variable water depth. The computed resonant periods or modes of oscillation for three harbors are in good agreement with the energy spectral analysis of the time series of water surface elevations recorded at tide gauge stations inside three harbors during the six tsunamis events. The computed wave induced currents based on the present model are also in qualitative agreement with some of the reported eye-witness accounts absence of reliable current data. The simulated results show that each harbor responded differently and significantly amplified certain wave period(s) of incident wave trains according to the shape, topography, characteristic dimensions and water depth of the harbor basins.

Effects of Finite Water Depth on Natural Frequencies of Suspended Water Tanks

Linear sloshing problems (inviscid irrotational flows) in suspended tanks are revisited, with an intention to address some issues in the previous study based on shallow water wave theory. Time-periodic solutions are considered, which describe the synchronized oscillation of the water and tank, reached after the initial transient dies out. The solutions are developed for arbitrary water depths, and separate explicitly the propagating and evanescent wave components of the fluid motion, illuminating clearly the physics and converging rapidly. At the limit of infinite string length, these solutions describe the sloshing motions in tanks that are free to oscillate horizontally on a frictionless plane. Various effects on the lowest sloshing mode are discussed, emphasizing the physical interpretations and examining the limitations of the shallow water approximations. Comparisons with existing laboratory experiments are made, showing agreements with the analysis.

Identification of Swell in Nearshore Surface Wave Energy Spectra

An approach for routine identification of swell and sea in nominally fully developed, omnidirectional, surface water wave energy spectra measured in arbitrary water depth is developed, applied, and discussed. The methodology is an extension of earlier work with deepwater spectra and involves identifying the frequency at which wave steepness is maximized and relating this to the swell separation frequency. The TMA parameterized spectrum is employed to establish a relationship between the two frequencies so that the methodology can be used when wind data are unavailable. The methodology is developed for finite water depth and tested using a dataset that includes both acoustic Doppler current profiler and wave buoy data, recorded simultaneously at the same location. For cases where the sea and swell are clearly, visually distinguishable in the omnidirectional spectra, the new method accurately distinguishes between the two, but it can also be used to identify sea and swell in unimodal spectra.

Nonlinear Crest Height Distribution in Three-Dimensional Ocean Waves

The interest and studies on nonlinear waves are increased recently for their importance in the interaction with floating and fixed bodies. It is also well-known that nonlinearities influence wave crest and wave trough distributions, both deviating from the Rayleigh law. In this paper, a theoretical crest distribution is obtained, taking into account the extension of Boccotti’s quasideterminism theory (1982, “On Ocean Waves With High Crests,” Meccanica, 17, pp. 16–19), up to the second order for the case of three-dimensional waves in finite water depth. To this purpose, the Fedele and Arena (2005, “Weakly Nonlinear Statistics of High Random Waves,” Phys. Fluids, 17(026601), pp. 1–10) distribution is generalized to three-dimensional waves on an arbitrary water depth. The comparison with Forristall’s second order model (2000, “Wave Crest Distributions: Observations and Second-Order Theory,” J. Phys. Oceanogr., 30(8), pp. 1931–1943) shows the theoretical confirmation of his conclusion: The crest distribution in deep water for long-crested and short-crested waves are very close to each other; in shallow water the crest heights in three-dimensional waves are greater than values given by the long-crested model.

The effect of third-order nonlinearity on statistical properties of random directional waves in finite depth

Abstract. It is well established that third-order nonlinearity produces a strong deviation from Gaussian statistics in water of infinite depth, provided the wave field is long crested, narrow banded and sufficiently steep. A reduction of third-order effects is however expected when the wave energy is distributed on a wide range of directions. In water of arbitrary depth, on the other hand, third-order effects tend to be suppressed by finite depth effects if waves are long crested. Numerical simulations of the truncated potential Euler equations are here used to address the combined effect of directionality and finite depth on the statistical properties of surface gravity waves; only relative water depth kh greater than 0.8 are here considered. Results show that random directional wave fields in intermediate water depths, kh=O(1), weakly deviate from Gaussian statistics independently of the degree of directional spreading of the wave energy.

Energy dissipation in non-uniform mangrove forests of arbitrary depth

This study aims to develop a predictive model of wave propagation through non-uniform forest in water of arbitrary depth. The theoretical model solves a full boundary value problem for wave propagation with dissipation for non-uniform mangrove forest and arbitrary water depth. In particular, wave−trunk interaction and wave breaking were found to be the dominant dissipation mechanisms. A modified mild-slope equation including dissipation is applied for wave model over changing water depth within the mangrove forest. The non-linear governing equations for wave−trunk interactions are linearised using the concept of stochastic minimalisation. The numerical results of the model are verified and compared with the experimental wave data in Can Gio Mangrove Biosphere Reserve, Southern Vietnam. The results for wave height, wave spectrum and wave-induced velocities as well as for the coefficients of transmission and reflection suggest that most of the energy is dissipated within the mangrove forest even at relatively small distance. The effect of wave breaking plays a more important role on wave attenuation in sparse forest, however, it is smaller compared to the effect of wave–trunk interactions in the denser forest.