Second-order Wave Theory Research Articles

Generation Of Highly Nonlinear Waves In A Short Wave Flume

The typical approach for generating nonlinear waves in physical models involves employing first- or second-order wave theory, requiring a large water depth at the wavemaker. When the prototype bathymetry shows a gentle slope, a large facility is required. However, practical constraints often make this unfeasible, leading to the use of steep transition slopes to obtain sufficient water depth at the generator. Incorporating a transition slope may generate unwanted free waves beyond the transition point, significantly impacting the wave parameters. The presence of these free waves causes the response of the tested structure to deviate from that found in the prototype. This paper offers guidelines for using transition slopes effectively while avoiding the generation of undesired free waves after the transition point.

Spatial energy evolution of focused waves generated in numerical wave tank

This paper numerically investigates the energy evolution of the focused wave and its correlation with input parameters for the wave generation. The focused wave is numerically generated based on the fully nonlinear potential wave theory which is solved by the Quasi Arbitrary Lagrangian Eulerian-Finite Element Method. The investigation is performed by dividing the generated wave energy into three categories according to their frequency intervals, i.e., the initially assigned frequency interval [f1fN], the frequencies lower (f < f1) and higher (f > fN) than the initial interval. The amount of the generated energy falling into three energy categories and the energy distribution in the initial frequency interval are analysed against amplitude parameters and frequency bands under three amplitude spectra. Four indicators are proposed to indicate the actual energy variations comparing to the initial design. It is found that for all the three frequency intervals the energies increase when the amplitude parameters increases or the given frequency shifts towards the lower frequency domain. The choice of wave amplitude spectrum plays a significant role to maintain the distribution of the energy in the design frequency range and to minimise the energies out of the design frequency range. The results from the second-order wave group theory are used to assist analyses of the mechanism of the energy evolution alongside the focused wave generation. It is found that the lower-frequency energy in the vicinity of the focusing point is dominated by the second-order difference waves, whereas the higher-frequency energy is mainly produced by the wave flap. The waves which are higher than the second-order and the complex interactions not considered by the second-order wave theory are found significantly disturbing the energy distribution in the originally assigned wave-frequency interval.

Toward the response of coastal bridges with updated RC shear key properties under the multi-hazard action of earthquakes and tsunamis

Coastal bridges are significantly endangered by the combined effects of earthquakes and tsunamis. The accurate prediction of bridge responses in such scenarios is vital to ensure their security. Reinforced concrete (RC) shear keys, which serve as sacrificial bridge components, can effectively mitigate inertial loads from superstructures, and thus safeguard pier columns. Previous research has shown that their mechanical properties can differ considerably under cyclic and persistent impacts. Therefore, this study focused on the seismic-tsunami responses of a simple-support coastal bridge equipped with RC shear keys. A horizontal failure shear key was incorporated because of its distinct performance observed previously. The OpenSees platform was employed to conduct the numerical work, with an updated material framework developed to modify the shear key behavior before the tsunami analysis. Natural ground motion records were selected to demonstrate the dynamic analysis, and the records were scaled using spectral accelerations (Sa) to account for the various damage states of shear keys resulting from superstructure collisions. The tsunami wave impact was modeled using the second-order solitary wave theory, with different wave conditions employed to consider the immersed conditions. The calculation results were compared for the bridges with and without updated shear key properties, showing distinctions in terms of time-history and maximum responses at certain seismic levels. Based on the analysis results, this study is expected to provide a reliable estimation of the seismic-tsunami responses of coastal bridges.

Wave behavior in a numerical wave tank at intermediate depth of water using stokes wave theory

The paper analyzes a second-order Stokes wave theory by CFD code and uses the volume of fluid (VOF) method in a 2D numerical wave tank (NWT) in the intermediate depth of water under different wave steepness conditions. The CFD simulation in an NWT generates a wave, validated with the already published experimental and numerical data to check the accuracy. It shows that current results are in good agreement with the published result and the difference in the results is within 2.1027%. Six tests under different wave heights for the time period [Formula: see text][Formula: see text]s have been conducted in an NWT to verify the analysis of the wave theory. Horizontal and vertical velocity, surface elevation, and the velocity vector of the particle motion are the important objectives in this analysis. The study shows that the free surface of the water is a high energy density zone. The time discretization has been achieved with the first-order implicit scheme and the second-order upwind scheme. This analysis shows that the ocean characteristic of second-order Stokes wave theory on the free surface is important to the engineering sciences and further investigation.

Optimization of the design of an oscillating water column for the Pacific Ocean conditions using the response surface methodology

Wave energy possesses a significant potential as a sustainable and renewable source of power due to its abundant presence in the Earth’s oceans. This research focused on the assessment of the hydrodynamic efficiency (ɛ) of a resonant chamber within an OWC (oscillating water column) system. Four geometric design parameters of the resonant chamber (h1/h, b2/h, d/b2 and α) were selected to optimize ɛ, being h the depth of the water; h1 and α correspond to the vertical length and angle of inclination of the front wall, respectively; b2 is the width of the resonant chamber, and d represents the diameter of the air outlet hole. To analyze the system, a combined approach utilizing CFD with the VoF (volume of fluid) method and the response surface methodology (RSM) was adopted. The RSM employed a FCCD (face-centered central composite design) to optimize the factors, while the VoF method enabled to simulate in a detailed way the fluid behavior. The ANSYS-Fluent software was employed to perform a transient simulation. A second-order regression model was constructed and validated based on the simulation results. The quadratic regression model exhibited a high Radj2 value (82.81%) and had a p-value associated ¡0.05 (4.66 × 10-8). Therefore, the model was highly significant in accurately representing the ɛ of the OWC resonant chamber. The highest ɛ value (67.01%) was obtained for h1/h, b2/h , d/b2 and α equal to 0.7786, 0.4455, 0.0863, and 89.38°, respectively. The numerical results were validated with the Stokes second-order wave theory and a good correlation was observed with the numerical findings. This study demonstrates that the designed and optimized OWC can be effectively utilized in the Colombian Pacific Ocean to generate electrical energy, promoting economic development in the area.

Study on p-y curve of large diameter pile under long-term cyclic loading

The pile foundation used in offshore wind turbine system is located in a complex marine environment and is subjected to various long-term cyclic loads such as wind, wave and current. However, the attenuation effect of long-term cyclic load on seabed soil has not been considered by most of the existing p-y curve calculation models. In order to solve this problem, the finite element numerical model of pile-soil interaction of large diameter single pile foundation of offshore wind turbine is established in this paper. Based on the Stokes second-order wave theory, the time history curve of wave and current loads on piles with different diameters are derived and calculated by programming. Then the soil stiffness attenuati on model subjected to marine cyclic loading is embedded into the finite element model by subroutine. Considering the influence of the number of ocean load cycles on the ultimate soil resistance of the soil around the pile and the initial foundation reaction modulus, the calculation results of different pile diameters and soil depths are analyzed and regressed. Finally, the p-y curve calculation model of large diameter single pile foundation of offshore wind turbine under long-term ocean cyclic load is obtained.

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.

Effect of relative opening on the moving thin plate under the action of ocean wave in the surging direction: A numerical simulation approach

This paper aims to develop the simulated numerical methodology for sloping beach with nonlinear free-surface waves of sloshing water in a 2D rectangular tank. The current computational tool would be the first step in accurately modeling wave energy converters. The 2D tank model was developed by ANSYS FLUENT software, and Navier–Stokes equations are discretized on a regular structured grid based on the finite volume method (FVM). The model is validated for nonlinear sloshing cases. In addition, tank flexibility is analyzed for its impact on liquid sloshing. The test was conducted in five different wave steepness conditions. There were five conditions of wave steepness used in the test. This study looked at fluid–structure interaction around moving plates. The output power of the plate wave energy converter for different plate heights and streamline motion around the plate are important studies in the flow field. Variations of drag coefficient with the wave steepness are the important study in this paper. It shows that the drag coefficient increases with the increase in wave steepness and plate length. In particular, the new structure can give a new, highly accurate formula for determining the wave power of the ocean structures. Its application is seen in wave energy converter (WEC), where the interaction of solid objects, such as wind turbines with water waves, extracts renewable wave energy from the ocean. A highly efficient CFD approach method is introduced to capture the interaction of solid object with a two-phase flow. The effect of wave steepness using second-order stokes wave theory on a moving thin plate in intermediate water is a new discussion. This research is a highly relevant source of information in the development of many practical systems and has direct employment in the design of WEC in order to extract ocean energy.

Modeling wave attenuation by submerged flexible vegetation with XBeach phase-averaged model

Understanding and modeling wave attenuation by flexible vegetation have emerged as essential issues in coastal and ocean engineering. Given that the existing XBeach phase-averaged wave model can only simulate wave attenuation by rigid vegetation, and previous studies of wave energy dissipation by vegetation are mainly restricted to linear wave theory. This study aims to extend the XBeach phase-averaged wave model to simulate wave attenuation by both homogeneous and heterogeneous submerged flexible vegetation, and determine wave attenuation simulation deviation based on different wave theories. Hence, the existing flexible vegetation dynamic model is incorporated into the XBeach phase-averaged wave model. The reliability of the coupled model in simulating wave attenuation by both homogeneous and heterogeneous submerged flexible vegetation is revealed. Simulation results also demonstrate that the computed wave attenuation rate based on Stokes second-order wave theory is larger than that based on linear wave theory within the parameters used in this investigation. Meanwhile, wave attenuation simulation deviation between these two wave theories enlarges as the Ursell number increases within the parameters used in this investigation. This study can broaden the application of the XBeach model and lay the technical foundation for future engineering practice of nature-based coastal defence.

Correct generation of the bound set-down for surface gravity wave groups in laboratory experiments of intermediate to shallow depth

Using linear (first-order) wave generation theory in laboratory experiments, leads to significant contamination of the wave field by free non-linear (second-order) error waves, increasingly so at shallower depths. Second-order wave generation theory has previously been established, and so has correct generation of the bound set-down, made up from second-order bound waves in the sub-harmonic part of the spectrum, for bichromatic and irregular wave fields in shallow to intermediate depth. In the present work, different from previous studies, we validate second-order wave theory explicitly for isolated wave groups, which provide a demanding test on the correct generation of sub-harmonic bound waves and the stroke length of the wavemaker. We do so for shallow to intermediate water depth, where some previous attempts at full elimination of sub-harmonic error waves have been hampered by limited paddle stroke. We overcome these limitations by applying second-order wavemaker theory to a piston-type paddle with an extended paddle stroke that can thence generate the bound set-down correctly. We show that sub-harmonic error waves are eliminated by considering wave groups in relative depths k0d = 0.6–1.1, with important applications in coastal engineering experiments, such as run-up and overtopping.

Numerical investigation on the propagation and evolution of focused waves over a sloping bed

It is well recognized that wave focusing is one of main mechanisms leading to nonlinear extreme/rouge waves in the real sea. The evolution of this nonlinear focused wave group (i.e., wave generated by wave focusing), characterized by the wave shape and kinematics, is significantly affected by the bottom topography. Since the terrain of the nearshore area is generally in the form of a mild slope, it is of both academic and practical values to investigate its propagation and evolution over a sloping bed. In this paper, a two-dimensional fully nonlinear numerical model is established based on the potential flow theory and the time-domain high-order boundary element method. The hybrid Euler-Lagrangian method is used to track the instantaneous water surface, and the fourth-order Runge-Kutta method is used to update the water surface and the velocity potential at the next time step. The focused wave is generated by prescribing velocities from second-order Stokes wave theory at the immerged sources boundary, and is forced to focus at a pre-defined location and time by the principle of wave focusing. By changing the bed sloping and the input wave amplitude, it was found that the focused wave crest decreases with the increase of the bed sloping, and the actual focal location is somehow shifted behind the theoretical value. When the initial focusing position is pre-defined at the beginning of the slope, the free surface elevations and the wave energy of higher-order harmonics above the slope are both smaller than the corresponding values on the flat bottom. However, when the focusing position is pre-set at the back part of the slope, the conclusions are opposite. The presence of the slope facilitates the energy conversion from low frequencies to high frequencies.

Experimental simulation study on dynamic response of offshore wind power pile foundation under complex marine loadings

Offshore wind power structures are subjected to complex dynamic loads in marine environment during their service. At present, the real sea conditions can be simulated to the maximum extent by the mixed test technology of boundary layer wind tunnels, wave pools and shaking tables. However, due to the high cost of test and the lack of a complete and mature combined simulation laboratory which can be operated for wind, wave and seismic loads, it is difficult to carry out such joint tests. It is advantageous to use self-developed mechanical devices to simulate marine environmental loads within the controllable errors and which is in simple operation. A series of dynamic model tests were carried out on saturated sand foundation in order to explore the feasibility of studying dynamic response of offshore wind power pile foundations through simulating marine environmental loads by simplified methods. Based on Stokes second-order wave theory and harmonic superposition method, the time history of wind, wave and current load is derived and programmed, then the wind, wave and current load on offshore fan model structure is simulated by using self-developed complex dynamic loading system. Finally, the dynamic response of single pile foundation in saturated sand under complex load is analyzed according to the test results. The test results show that the load displacement curve of single pile foundation in saturated sand is of work hardening type. The maximum pile bending moment at the peak of dynamic load increases with the increase of loading times, but the cumulative effect of bending moment caused by the change of load amplitude should be considered. The dynamic load in marine environment will cause the increase of pore pressure around the pile, but the increase of pore pressure on loading side is not obvious in comparison to that on the opposite side. The displacement change of pile body is basically in accordance with the fluctuation trend of test load and increases with load. The cumulative rate of displacement of single pile decreases with the increase of number of cycles under the long-term cyclic wind wave load. In this study, the method combining theoretical calculation of wind, wave and current load with model test of complex loading system can consider the characteristics of dynamic loads of marine environment more accurately, compared with the empirically determined cyclic load waveforms in conventional tests, the obtained response law is more in accord with actual sea conditions.

Application of Arbitrary Lagrangian–Eulerian strips with fully nonlinear wave kinematics for force estimation

The initial design of offshore constructions generally uses a combination of model tests and second-order wave theory to reconstruct the wave kinematics to determine wave forces using the Morison equation. This routine becomes difficult to apply to steep waves without model test data where the second-order theory is not valid. In this article, a novel approach combining a fully nonlinear numerical wave tank (NWT) using the σ-grid with strip theory and the Morison equation in an Arbitrary Lagrangian–Eulerian (ALE) framework is presented. This provides improved wave kinematics from the nonlinear numerical wave tank, an improved representation of the instantaneous velocity field associated with the instantaneous location of the free surface, making an accurate estimation of the wave hydrodynamics of highly nonlinear and breaking waves possible at a reasonable computational cost.The method is verified by comparing the calculated wave kinematics for a fifth-order Stokes wave with the analytical solution. The procedure for the estimation of the hydrodynamic loads on a cylinder is validated using comparisons with model test data for wave forces on a cylinder due to regular, irregular and focussed waves. A good agreement is seen for the estimated wave forces, including for the breaking waves in the irregular wave train. Some overestimation of the wave forces due to irregular waves is seen due to the frequency-independent force coefficient used in the Morison equation. The results demonstrate the potential of the ALE method combined with a nonlinear NWT to evaluate a three-hour irregular sea state simulation and generate wave loading statistics for the initial stochastic design of offshore structures without model test data.

Mooring Stability Study for Novel Wave Energy Converter Based on Regular Wave

The mooring system not only plays a vital role in keeping wave energy generators floating stably, but also affects the success of engineering design. Combining wave force theory and the hydrological data obtained from the field measurements of a certain sea area in the Bohai Sea, the Stokes second-order wave theory was adopted to design the mooring system of a new type of power-generating device. At the same time, the study uses the Aqwa software to gather the dynamic data of a power-generating device in a real test, and then makes models and carries out regular wave tests so as to verify the viability of the mooring system and the stability of the whole power-generating device. All of this work will provide a theoretical basis for the manufacture of an engineering prototype and its reliable supply of power.

A data-driven model for nonlinear marine dynamics

The design and engineering of ships and platforms that operate in the ocean environment requires understanding of a nonlinear dynamical system that responds according to complex interaction with a wide range of sea and wind conditions. Time domain observation of nonlinear marine dynamics with either experiments or high-fidelity numerical simulation tools is costly due to the random nature of the ocean and the full range of environmental and loading conditions that are experienced in the lifetime of a ship or platform. In this paper, a data-driven method is presented to predict the complex nonlinear input–output relationship typical of marine systems. A Long Short-Term Memory neural net is used to learn nonlinear wave propagation and the nonlinear roll of a ship section in beam seas. Training data are generated with second-order wave theory or a volume-of-fluid computational fluid dynamics, although the method is directly applicable to data that is generated by other means such as nonlinear potential flow or experimental measurements. The cost and the amount of data to apply the method are estimated and measured. The data-driven results are compared with unseen data to demonstrate the accuracy and feasibility.

Research on wave and tidal current law prediction and coupling mechanism applied to ocean energy utilization

The energy efficiency of integrated wave and current turbine is closely related to the velocity and direction of wave force and tidal current force.To study the blade motion of the wave-tidal current integrated variable-pitch turbine, raise the capacity efficiency of the turbine, take Fishing grounds of Changjiang estuary Port as an example,based on the tidal current and wave features data in recent two years,time series prediction model was established to predict the changes of wave and tidal current in the next 24 hours respectively, and the prediction accuracy is up to 93%. The wave theory applicable to Fishing grounds of Changjiang estuary Port was confirmed as the second-order strokes wave theory, the wave-tidal current coupling equivalent velocity vector was deduced, which is related to the tidal current velocity, flow direction, average wave direction, average wave period and significant wave height. The predicted wave-current coupling velocity vector is obtained after the predicted 24 hours wave-current data was substituted into it, which can provide a method and basis for the design of marine energy turbine’s variable angle law in the future, and can be applied to the development of new energy and the prediction of Marine environment.

Flow structure and bottom friction of the nonlinear turbulent boundary layer under stormy waves

Wave environment equivalent to the full-scale nearshore storm was generated in a large wave flume, and the flow within wave boundary layer was measured using an acoustic Vectrino profiler. Analysis were made on various factors including the velocity profile, phase lead, boundary layer thickness, steady streaming, turbulence as well as bottom shear stress, and comparisons were made with existing studies using oscillatory flow tunnels. Results show that the phase lead in the boundary layer is less significant compared to previous tunnel experiments, generally smaller than 20°. Free surface and vertical velocity lead to prominent wave-induced Reynolds stress at the top of the boundary layer, resulting in a thinner boundary layer. Both wave-induced Reynolds stress and wave shape asymmetry contribute to the steady streaming, making its direction either onshore or offshore. Relationship of friction factor versus relative roughness generally agrees with the exponential expression in the literatures for rough turbulent boundary layer, however the value of friction factor is larger. It is stressed that the friction factor would be several times higher if the free stream velocity is calculated using linear wave theory, which is questionable as the measured waves in the large wave flume were shown to be highly nonlinear. To compensate the nonlinear effects which significantly contribute to the maximum bed shear stress, a velocity skewness factor based on second-order Stokes wave theory was introduced. For predicting the intra-wave bed shear stress, a time-varying friction factor was also constructed, which was proved to have acceptable precision compared with experimental data. These formulations have simple, explicit expressions that can be used in practical engineering applications, especially for the nonlinear waves under nearshore storm conditions.

Probability density function for wave elevation based on Gaussian mixture models

In this article, Gaussian mixture models are used to estimate the probability density function of wave elevation in the context of the second-order random wave theory. Two approaches are used to construct the Gaussian mixture probability distribution. One is the moment estimate in which the unknown parameters are determined by matching the moments of Gaussian mixture model with those of the real wave process. The other is the maximum likelihood estimation in which the expectation–maximization (EM) algorithm is used to determine the parameters in statistical models. The proposed Gaussian mixture distribution is favorably validated by using Monte Carlo simulations in comparison with other theoretical distribution models. Numerical results reveal a clear dependence of the probability distribution structure on the wave steepness and the spectral shape. Finally, three sets of observation data are applied to further confirm the accuracy and efficiency of Gaussian mixture model.

Numerical investigation of secondary load cycle and ringing response of a vertical cylinder

A numerical wave tank is established based on two-phase FVM model and VOF method and verified with the physical experiment in Grue et al. (1994). Focusing waves with different wave steepness passing a vertical cylinder are investigated by Numerical simulations. The phenomenon called ‘secondary load cycle’ which may lead to ringing response, is observed and discussed. The presence of secondary load cycle could be related to Froude Number (Fr). The possible transition region of the present and absent secondary load cycle is Fr = 0.4. Sub and super harmonic wave components appear in the propagation of waves, second-order wave theory could give a good prediction. Morison equation with linear wave theory could predict well the wave forces of vertical cylinder with small steepness without the secondary load cycle, but cannot capture the crests/troughs of the wave forces with the secondary load cycle. Crest improvements are achieved by second-order wave theory. A spectral analysis based on wavelet transform is applied to wave loads. The frequency of the secondary load may be up to 13 times the wave frequency, which may cause the ringing response expanding to a higher frequency range. Strong ringing response occurs in steep wave, it could be extended up to 15 times wave trough-to-trough frequency due to the secondary load cycle. The damping has slightly influence on the peak of resonance response, but it will lead to faster decay of subsequent response, if the damping ratio is large.

Stochastic analysis of the non-Gaussian airgap response of a semi-submersible using frequency domain analysis

Airgap is a crucial design parameter for semi-submersibles. The analysis of the extreme airgap statistics is challenging due to nonlinearities in the wave elevation and platform motions, as well as their interactions and phase relationships. In this paper, an efficient frequency domain approach is developed for airgap analysis, using second-order diffraction analysis to model the wave loads. The wave amplification effect is predicted using second-order random wave theory together with linear diffraction/radiation analysis, and the drag forces are linearized. The linear and quadratic transfer functions (sum- and difference frequencies) of the airgap response are derived, preserving all the phase relationships. The crossing rate is a key quantity for characterizing the extreme of a stochastic process. Using the transfer functions, the crossing rate of the non-Gaussian airgap process is derived via a statistical method that preserves the statistical dependency between the linear, sum- and difference-frequency components. In the case studies, the spectral density of the airgap obtained from frequency domain analysis are compared to time domain simulations (which are performed as a benchmark), and the results are found to be in excellent agreement. Moreover, the proposed approach generally provides good estimation of the crossing rate. The results also reveal insights into the airgap behavior; for example, the airgap at the platform center is nearly Gaussian, whereas the off-center locations are significantly non-Gaussian. Finally, the influence of various nonlinearities on the airgap statistics are systematically investigated.