Wave Steepness Research Articles

Numerical Modelling of Beach Profile Evolution with and without an Artificial Reef

With the recent development from grey infrastructures to green infrastructures, artificial reefs become more popular in coastal protection projects. To investigate the responses of beach profile evolution to the presence of an artificial reef, a non-hydrostatic model is established. Both hydrodynamic and morphodynamic evolution for the beach with and without an artificial reef are compared under regular wave conditions. In addition, the protected beach profile evolution by an artificial reef is discussed under irregular wave conditions. Three key parameters in non-hydrostatic simulation are considered for sensitivity analysis, including maximum wave steepness criterium (maxbrsteep), water depth factor (depthscale), and equilibrium sediment concentration factor (sedcal). The numerical results under regular wave conditions indicate that the artificial reef enhances wave attenuation by inducing wave breaking. In addition, the artificial reef reduces local flow velocity and offshore sediment transport by 51%, therefore decrease the total erosion by 53%. Over the artificial reef, wave skewness and asymmetry go through a drastic change. Under irregular wave conditions, short waves contribute to the wave energy mainly and reflection-induced standing wave effects decline considerably. It demonstrates that the artificial reef can protect the beach from regular and irregular waves by reducing erosion and offshore transport of suspended sediments. Moreover, in the wave breaking area, the increase of maximum wave steepness criterium may give arise to the wave height. The morphological evolution is more sensitive to water depth factor than equilibrium sediment concentration factor, because the former is a controlling factor for beach profile characteristics while the latter forms the sandbar varying irregularly in shape.

Numerical Investigation of Optimal Hydrodynamic Performance by Changing the Orifice Ratio and Relative Opening of a Land-Fixed Rectangular-based OWC

The energy that can be extracted from the ocean is inexhaustible. An oscillating water column (OWC) is a wave energy converter that extracts this energy. A numerical investigation has been conducted by altering relative opening (α) and orifice ratio (τ) to assess the maximal energy of a land-fixed rectangular-based OWC model in a nonlinear wave field. The power of OWC has also been evaluated by the wave steepness (H/L) alteration. The numeric analysis has been imposed to obtain the optimal power using Fluent software in a three-dimensional tank. Validation of the present numeric model’s result correlates with the printed empirical data. The Finite Volume Method (FVM) solves RANS equations, and the relevant waves are generated at the inlet of the numerical tank by the inlet velocity approach. The efficiency (η) increases with relative openings (α) increase. The efficiency (η) decreases with wave steepness (H/L) increase. The η reaches the optimum shown in the study at H/L = 0.02 and τ = 1.03% for entire values of α. The excellent energy of around 71.3% is attained at α =75% and H/L = 0.02. This study is a highly relevant source of information that finds the optimal efficiency of a land-fixed rectangular base OWC and gives prior knowledge of the performance of OWC before the real-life experiment.

Numerical study on static drift tests for ship maneuvering in regular waves

This study is driven by the coupled seakeeping–maneuvering analysis, which predicts ship maneuverability in waves by combining seakeeping analysis with a mathematical maneuvering model. The traditional maneuvering model and its corresponding coefficients are developed based on calm sea conditions, where wave effects are typically disregarded as a temporary compromise. It is an inevitable need to revisit the mathematical maneuvering model or coefficients by considering wave effects. To this end, in this study, a series of static drift tests are employed in waves to investigate the influence of wave effects on the sway damping coefficients of ship maneuvering motion. These tests are conducted using a numerical tank, which overcomes spatial constraints and maintains a reliable wave field for ship–wave interactions. Three wave parameters, namely wavelength, wave steepness, and wave direction, are considered to assess their impacts on the wave-induced motions and forces. Furthermore, the wave effects on the maneuvering coefficients are summarized and discussed using direct observations of ship-turning motions in waves. Additionally, the benefits of computational fluid dynamics analysis are utilized to showcase the strong nonlinear phenomena of ship–wave interactions.

Modelling the interaction of Alfvénic fluctuations with coronal mass ejections in the low solar corona

Context.Alfvénic fluctuations of various scales are ubiquitous in the corona; their non-linear interactions and eventual turbulent cascade result in an important heating mechanism that accelerates the solar wind. These fluctuations may be processed by large-scale, transient, and coherent heliospheric structures such as coronal mass ejections (CMEs). In this study we investigate the interactions between Alfvénic solar wind fluctuations and CMEs using magnetohydrodynamic (MHD) simulations.Aims.We study the transmission of upstream solar wind fluctuations into the CME leading to the formation of CME sheath fluctuations. Additionally, we investigate the influence of the fluctuation frequencies on the extent of the CME sheath.Methods.We used an ideal MHD model with an adiabatic equation of state. An Alfvén pump wave is injected into the quiet solar wind by perturbing the transverse magnetic field and velocity components, and a CME is injected by inserting a flux-rope modelled as a magnetic island into the quasi-steady solar wind.Results.The upstream Alfvén waves experience a decrease in wavelength and change in the wave vector direction due to the non-radial topology of the CME shock front. The CME sheath inhibits the transmission of long-wavelength fluctuations due to the presence of non-radial flows in this region. The frequency of the solar wind fluctuations also affects the steepening of MHD fast waves causing the CME shock propagation speed to vary with the solar wind fluctuation frequencies.

On the probability of down-crossing and up-crossing rogue waves

By means of the direct numerical simulation of directional waves on the surface of deep water, it is shown that extreme waves can exhibit such asymmetry that the occurrence of deeper troughs is several times more likely on the wave rear slopes. This effect becomes most pronounced in the case of steep short-crested waves. It is not related to the Benjamin–Feir instability but is a result of complex contribution from nonlinear combination harmonics, mainly cubic in nonlinearity. The discovered asymmetry can lead to remarkably different estimates of the rogue wave probability based on either down- or up-zero-crossing methods for individual wave selection, commonly used in the oceanography.

A numerical study on wave overwash on viscoelastic floating covers using smoothed particle hydrodynamics

Large floating covers on the water surface, such as very large floating platforms or ice covers in the marginal ice zone, can exhibit viscoelastic response during their interactions with progressive gravity waves. When the wave magnitude is exceedingly large, overtopping can also occur with wave overwash appearing on top of the cover. In the present study, we investigate the numerical simulations of the transient wave overwash on the viscoelastic floating cover using the smoothed particle hydrodynamics (SPH) approach. In particular, a new numerical scheme is derived whereby the floating cover is represented through discretization into elementary rigid sections interconnected via rotational hinges with damping, with the hinge characteristics matching the viscoelastic properties of the cover. Both laminar viscosity and sub-particle-scale turbulence model for the water motion are considered in the SPH simulation. For validation, experiments of wave interaction with a viscoelastic floating cover with and without wave overtopping were also performed in the present study. The results show that SPH can predict well the wave reflection and transmission by the floating cover with transient overwash on top, as well as the threshold of incident wave steepness that initiates the wave overtopping. Overall, the current study demonstrates for the first time that SPH approach can be an effective tool to solve hydro-viscoelasticity problems by simulating the highly complex wave interactions with viscoelastic floating covers including overtopping.

The influence of spectral bandwidth and shape on deep-water wave breaking onset

Deep-water surface wave breaking affects the transfer of mass, momentum, energy and heat between the air and sea. Understanding when and how the onset of wave breaking will occur remains a challenge. The mechanisms that form unforced steep waves, i.e. nonlinearity or dispersion, are thought to have a strong influence on the onset of wave breaking. In two dimensions and in deep water, spectral bandwidth is the main factor that affects the roles these mechanism play. Existing studies, in which the relationship between spectral bandwidth and wave breaking onset is investigated, present varied and sometimes conflicting results. We perform potential-flow simulations of two-dimensional focused wave groups on deep water to better understand this relationship, with the aim of reconciling existing studies. We show that the way in which steepness is defined may be the main source of confusion in the literature. Locally defined steepness at breaking onset reduces as a function of bandwidth, and globally defined (spectral) steepness increases. The relationship between global breaking onset steepness and spectral shape (using the parameters bandwidth and spectral skewness) is too complex to parameterise in a general way. However, we find that the local surface slope of maximally steep non-breaking waves, of all spectral bandwidths and shapes that we simulate, approaches a limit of$1/\tan ({\rm \pi} /3)\approx 0.5774$. This slope-based threshold is simple to measure and may be used as an alternative to existing kinematic breaking onset thresholds. There is a potential link between slope-based and kinematic breaking onset thresholds, which future work should seek to better understand.

Numerical simulation study of wave dissipation performance of step‐type breakwaters

AbstractTo improve the force endurance capacity of plate breakwaters, a step‐type plate breakwater was proposed. The performance of the step‐type breakwater was investigated by numerical simulation method. First, a two‐dimensional numerical regular wave tank was constructed using FLUENT software, and the effectiveness of the tank application was verified by comparing with theoretical value and test data. Then, the numerical tank was used to simulate various working conditions, and the variation rules of the breakwater transmission coefficient with wave height, period, perforation rate, relative plate width, wave steepness, and number of layers were obtained. Finally, the difference in the wave dissipation mechanism between the step‐type breakwater and the “”‐type breakwater was discussed. The results indicated that the transmission coefficient of the step‐type breakwater was small, which also proved to be good at dissipating long‐period waves. In general, the wave dissipation performance of the step‐type breakwater was better than that of the “”‐type breakwater. The research can provide some clues for the research and design of corresponding breakwaters.

The dependence of wave height probability distributions on physical parameters from measurements near Sakhalin Island

The data of long-term surface waves measurements with bottom sensors near Sakhalin Island were used to build instrumental probability distributions for exceedance of wave heights. Waves with heights exceeding the significant wave height by more than three times were recorded. Specific features of the observations conducted during the periods of open and ice-covered sea surfaces are discussed. The subsets of statistically homogeneous data are arranged through selection considering the natural physical dimensionless parameters of the task, that control the effects of finite depth and nonlinearity (namely, the wave steepness, the wave amplitude to water depth ratio, the Ursell parameter). The manifestation of these and composite parameters in theoretical probability distributions of wave heights is discussed. The effects of nonlinearity and the measurement point depth on probability distributions are estimated with a focus on abnormally high waves. In particular, it is shown that for the place of registration an increase in the wave amplitude to water depth ratio parameter leads to a decrease in the probability of abnormally high waves. This behavior is consistent with the Glukhovsky theoretical distribution. The waves characterized by relatively large dimensionless depth parameter exhibit a higher probability of substantial exceedance of the significant height and are better described by the Rayleigh distribution.

Numerical validations and investigation of a semi-submersible floating offshore wind turbine platform interacting with ocean waves using an SPH framework

In this work, we propose numerical validations of the DeepCwind semi-submersible floating platform configuration for a single horizontal axis wind turbine using data from two experimental testing investigations. A Smoothed Particle Hydrodynamics solver is employed to estimate fluid induced loads, whereas the mooring connections are handled via an external library. The first validation setup is based on the DeepCwind offshore wind semi-submersible concept moored with a system of taut-lines and tested for free-decay surge and heave motion (OC6-Phase Ia). The damping evaluation yields a fair estimation of the heave damping behavior, whereas much more dissipation is experienced for the surge. The second validation features a full hydrodynamic characterization of the frequency-related load patterns induced by three different sea-state representations (mono-, bi-chromatic, and irregular waves) (OC6-Phase Ib). The model accurately matches the hydrodynamic load estimation for the whole spectrum of investigated wave components, perfectly capturing the non-linear behavior shown by the considered wave patterns. This work concludes with a systematic study on the motion response, mooring tension, pressure and vorticity, suggesting that: the wave steepness criterion alone cannot identify the most restrictive load case; waves with spectral characteristics close to the heave resonance period lead to higher tensions in the mooring systems, whereas the maximum fluid-induced loads on the hull are decoupled from displacement peaks, showing an average reduction of 30% with respect to the maxima; very steep waves maximize the likelihood of wave overtopping and slamming loads, resulting in locally induced overpressure on the free-board of up to 100% higher than expected for similar wave heights with milder profiles. The input data for these last tests is released for the sake of reproduction.

The superharmonic instability and wave breaking in Whitham equations

The Whitham equation is a model for the evolution of surface waves on shallow water that combines the unidirectional linear dispersion relation of the Euler equations with a weakly nonlinear approximation based on the Korteweg–De Vries equation. We show that large-amplitude, periodic, traveling-wave solutions to the Whitham equation and its higher-order generalization, the cubic Whitham equation, are unstable with respect to the superharmonic instability (i.e., a perturbation with the same period as the solution). The threshold between superharmonic stability and instability occurs at the maxima of the Hamiltonian and L2-norm. We examine the onset of wave breaking in traveling-wave solutions subject to the modulational and superharmonic instabilities. We present new instability results for the Euler equations in finite depth and compare them with the Whitham results. We show that the Whitham equation more accurately approximates the wave steepness threshold for the superharmonic instability of the Euler equations than does the cubic Whitham equation. However, the cubic Whitham equation more accurately approximates the wave steepness threshold for the modulational instability of the Euler equations than does the Whitham equation.

Wave Reflection of Concrete Armour Unit: A 2D Physical Model Study

A breakwater is a coastal structure that breaks the wave energy coming towards the beach. When a wave hits an object, it will be reflected entirely or partially. Initially, the primary layer units on the coastal protection structure were only composed of large piles of natural stone. It is becoming increasingly difficult to find a natural stone of the right size, and to prevent its scarcity; natural stones are commonly replaced with concrete blocks. This study conducted the reflection coefficient analysis on the BPPT Lock concrete armor unit. Physical model testing was carried out on the wave channel of the Coastal and Port Infrastructure Laboratory, Department of Ocean Engineering ITS. From this study, it was found that the reflection coefficient on BPPT Lock ranges from 0.512 to 0.828 at a wave steepness of 0.002088 to 0.0117 with a structural slope angle of 1: 2. The reflection coefficient value is influenced by wave height, wave period, and the slope angle of the structure.

Sea surface reconstruction from marine radar images using deep convolutional neural networks

The sea surface reconstructed from radar images provides valuable information for marine operations and maritime transport. The standard reconstruction method relies on the three-dimensional fast Fourier transform (3D-FFT), which introduces empirical parameters and modulation transfer function (MTF) to correct the modulation effects that may cause errors. In light of the convolutional neural networks’ (CNN) success in computer vision tasks, this paper proposes a novel sea surface reconstruction method from marine radar images based on an end-to-end CNN model with the U-Net architecture. Synthetic radar images and sea surface elevation maps were used for training and testing. Compared to the standard reconstruction method, the CNN-based model achieved higher accuracy on the same data set, with an improved correlation coefficient between reconstructed and actual wave fields of up to 0.96-0.97, and a decreased non-dimensional root mean square error (NDRMSE) of around 0.06. The influence of training data on the deep learning model was also studied. Additionally, the impact of the significant wave height and peak period on the CNN model’s accuracy was investigated. It has been demonstrated that the accuracy will fluctuate as the wave steepness increases, but the correlation coefficient remains above 0.90, and the NDRMSE remains less than 0.11.

Experimental and CFD Assessment of Harmonic Characteristics of Point-Absorber Wave-Energy Converters with Nonlinear Power Take-Off System

The wave-energy excitation of point absorbers is highly associated with their resonant movement, and harmonic characteristics are of increasing concern in affecting resonance. However, the commonly used linearized power take-off (PTO) systems underestimate the impact of harmonics. The purpose of this study is to address the knowledge gap in assessing the contribution of hydraulic PTO systems to higher harmonic wave loads and velocities. In the present work, higher harmonics in point-absorber wave-energy converters (PA-WECs) with hydraulic power take-off (PTO) systems are investigated through both experimental and computational fluid dynamics (CFD) methods. The fast Fourier transform is used to decompose the high-order harmonics. To account for the influence of nonlinear wave–wave interaction on harmonics, the isolated PA-WEC is used as a basis for comparison with the paired PA-WECs. The influence of wave steepness is also estimated at two resonance periods. Results indicate that the additional resonance of the paired PA-WECs may be attributed to the harmonic wave loads at longer wave periods. Harmonic wave loads of paired PA-WECs typically have a more substantial impact and increase more rapidly with increasing wave steepness compared to isolated PA-WECs. Furthermore, as the wave steepness increases, there are significant enhancements in both the harmonic wave loads and heaving velocity, which strongly correlate with the instantaneous maximum hydraulic power. Consequently, our study will contribute to enhancing the maximum power output in the design of future point absorber arrays.

Hydrodynamic analysis of the combined structure of offshore monopile wind turbine foundation and aquaculture cage

Reliable prediction in hydrodynamics is crucial for the design of marine structures, especially for columnar structures such as offshore wind turbine foundations. A combined structure of fixed monopile wind turbine foundation and bottom-seated circular aquaculture cage is proposed for efficiently utilizing the offshore space and reducing the wave forces on the monopile wind turbine foundation. A further developed volume-averaged numerical model is adopted to simulate the pressure drop due to clean and fouled cages by introducing the porous media theory, in which the coefficients of source terms are determined by the least square method based on hydrodynamic force on cage from existing experimental data. The Stokes fifth-order wave theory is considered reliable in simulating the nonlinear waves on slopes and corresponding inline wave force by comparing with multiple existing experimental data. Then the effects of incident wave steepness, seabed bottom slopes and cage diameters on simulation results are analyzed. Results reveal the fouled cage can effectively reduce inline wave forces relative to the clean cage of the combined structure, accompanying a significant resonance effect of wave motion inside the cage.

A new probability density function for the surface elevation in irregular seas

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.

A method of retrieving significant wave height based on shadowing from X-band marine radar images

ABSTRACT In this paper, the shadow statistical method, which has the merit of without calibration, is further investigated for retrieving significant wave height (SWH) based on shadowing from X-band marine radar images. For the shadow statistical method, the Smith fitting function (SFF) is utilized to fit the illumination probability obtained from the radar image and calculate the SWH. By deeply investigating the estimation of wave steepness from the shadow image, it is found that the analytic solution of wave steepness is not obtained, due to the complementary error function and exponential term in the SFF. Meanwhile, several screening processes of gradually discarding illumination probability are required to obtain accurate root mean square (RMS) wave steepness, which makes the calculation time-consuming. To solve these problems, both the complementary error function and exponential term in the SFF are approximated under the working conditions of marine radar in practice, and the analytic solution of the wave steepness is described with the illumination probability. The shore-based X-band radar images at Pingtan station from November 9 to 17, 2014 are collected to validate the validity of the proposed method. By using 254 sets of radar data with the SWH range of 0.5 ~ 3.5 m, the experimental results demonstrate that the correlation coefficient (CC) of the estimated SWH between the proposed method and the traditional method is 0.996, and the root mean square error (RMSE) is 0.034. The average running time is significantly reduced from 2.76 s to 0.44 s.

3D NUMERICAL STUDY OF HYDROYNAMICS AROUND A MONOPILE FOUNDATION UNDER COMBINED WAVE CURRENT EFFECT

A numerical investigation of hydrodynamics around a monopile foundation is presented in the paper. An open-source CFD program REEF3D is used for the present study. The REEF3D model solves the Reynolds-averaged Naiver-Stokes equation with k-w turbulence closure to obtain the flow hydrodynamics. The code utilizes a staggered cartesian grid system. Also, Message Passing Interface (MPI) based parallelization is used for domain decomposition. In order to ensure the precision of the flow field, the model is first validated for four different combined wave-current scenarios, where the currents and waves are following each other. The results obtained from the validation studies exhibit good agreement with the already reported experimental results. Initially, a grid convergence study has been performed to ensure that the Numerical wave tank is independent of the mesh size. Finally, numerical experiments corresponding to four different combined wave-current cases for different wave steepness are simulated using the code.

AVERAGE OVERTOPPING DISCHARGE PREDICTION FOR BERM BREAKWATERS

The present paper deals with overtopping prediction for berm breakwaters in line with the EurOtop methodology. The basis for the paper is the recent advances proposed for EurOtop for conventional breakwaters with respect to the influence of the wave steepness and the crest width. New model tests have been performed to investigate the applicability of these influence factors to berm breakwaters. To cover a white spot in existing data for berm breakwaters, the model tests included wave conditions with very low wave steepness. The results show that the recently developed influence factors for conventional breakwaters also improve predictions for berm breakwaters. Based heron an additional influence factor for the dimensionless berm width is established. The berm width was in previous studies made dimensionless by the wave height, but the present study indicates that the wavelength is more appropriate.

On the Consistent Parametric Description of the Wave Age Dependence of the Sea Surface Roughness

Abstract The wind drag on the sea surface is characterized by the aerodynamic roughness of the sea surface, z0, which is regulated by surface wind waves. Many studies have related the dimensionless form of z0 to the wave age parameter estimated from spectral peak information. These parametric relationships have been well developed for the wind-driven sea but not for mixed seas. Based on an analysis using observations from a fixed platform in the northern South China Sea, the deficiency of the spectral peak information in the parametric description z0 when swells dominate is indicated. Instead, a consistent parametric description of z0 can be obtained by using the wave age estimated from the mean wave period, and normalizing z0 by the mean wavelength. Normalizing z0 by the significant wave height introduces a spurious residual dependence of z0 on the wave steepness. A parametric relationship is developed between the dimensionless z0 (normalized by the mean wavelength) and the wave age from the mean wave period. A comparison of this new relationship to the wind-speed-only formulation in COARE 3.5 is provided. Significance Statement In this paper, a consistent parametric description of the wave age dependence of the surface aerodynamic roughness is presented, with a wide range of sea states from dominant wind-driven seas to mixed seas in which the swells are dominant.