High-order Spectral Research Articles

1D -FNO - Residual convolutional neural network autoregressive prediction model for deep water wave train evolution based on high-order spectral (HOS) data

Accurate prediction of deep-water wave dynamics is essential for ocean engineering, meteorology, and maritime safety. Traditional physical models, though effective, face significant computational and time challenges with complex nonlinear dynamics. This study introduces an autoregressive prediction model using a one-dimensional adaptive Fourier Neural Operator-Residual Convolutional Neural Network (1D-FNO-ResNet) to enhance prediction accuracy. We developed a high-precision single-step prediction model capable of capturing wave characteristics such as height and fluid particle velocity. A fine-tuning training strategy based on multi-step prediction errors was proposed for autoregressive evolution. Performance comparisons between the ResNet and FNO-ResNet models were made under various fine-tuning steps. A correction model was also introduced to control error accumulation, improving autoregressive prediction steps. The model's generalization performance was validated with modulated and random wave trains under different conditions, optimizing predictive stability and accuracy. This approach demonstrates efficient 1D deep-water wave evolution prediction, offering a novel deep learning solution for wave dynamics prediction in ocean engineering, supporting wave energy conversion device design and optimization.

Development of a numerical wave tank of deep ocean engineering basin via high order spectral method

ABSTRACT In this study, we developed a numerical model for simulating wave environments of the deep ocean engineering basin (DOEB) by utilising the high order spectral (HOS) method. The approach employed the pseudospectral basis of the cosine function to automatically account for the tank wall boundary conditions. Additionally, the fast Fourier transform (FFT) algorithm was implemented to maximise the numerical efficiency. For selecting an appropriate wave generation technique for the DOEB, which distinguishes itself as a large-scale facility with controllable depth, in contrast to conventional wave basins, two-dimensional wave generation simulations were executed. These simulations entailed the application and subsequent evaluation of the embedded wave generation technique and the additional potential technique. Of these, the embedded wave generation technique, characterised by its suitability for numerical efficiency and adaptability to variable water depths, was chosen for implementation in the DOEB numerical tanks. The Stroke–source converting procedure was established to transform the stroke time history of the wave maker into the input wave source time history for the embedded wave generation technique, with due consideration to phase correction. In order to validate the developed numerical model, a wave generation experiment was conducted, and its results were compared with the numerical simulations. This comparative analysis revealed a high degree of correspondence between the nonlinear simulations and the experimental wave time history, demonstrating the model’s ability to accurately replicate higher-order wave components, including wave–wave interactions. The resulting numerical model serves as a valuable computer-aided tool, enhancing the efficiency of experiments within the DOEB.

Sum-frequency triad interactions among surface waves propagating through an ice sheet

We study nonlinear resonant wave–wave interactions which occur when ocean waves propagate into a thin floating ice sheet. Using multiple-scale perturbation analysis, we obtain theoretical predictions of the wave amplitude evolution as a function of distance travelled past the ice edge for a semi-infinite ice sheet. The theoretical predictions are supported by a high-order spectral (HOS) method capable of simulating nonlinear interactions in both open water and the ice sheet. Using the HOS method, the amplitude evolution predictions are extended to multiple (coupled) triad interactions and a single ice sheet of finite length. We relate the amplitude evolution to mechanisms with strong frequency dependence – ice bending strain, related to ice breakup, as well as wave reflection and transmission. We show that, due to sum-frequency interactions, the maximum strain in the ice sheet can be more than twice that predicted by linearised theory. For an ice sheet of finite length, we show that nonlinear wave reflection and transmission coefficients depend on a parameter in terms of wave steepness and ice length, and can have values significantly different than those from linear theory. In particular, we show that nonlinear sum-frequency interactions can appreciably decrease the total wave energy transmitted past the ice sheet. This work has implications for understanding the occurrence of ice breakup, wave attenuation due to scattering in the marginal ice zone and the resulting ice floe size distribution.

Nonlinear deterministic reconstruction and prediction of remotely measured ocean surface waves

Algorithms for reconstructing and predicting nonlinear ocean wave fields from remote measurements are presented. Three types of synthetic observations are used to quantify the influence of remote measurement modulation mechanisms on the algorithms’ performance. First, the observations correspond to randomly distributed surface elevations. Then, they are related to a marine radar model – the second type takes the wave shadowing modulation into account whereas the third one also includes the tilt modulation. The observations are numerically generated based on unidirectional waves of various steepness values. Linear and weakly nonlinear prediction algorithms based on analytical models are considered, as well as a highly nonlinear algorithm relying on the high-order spectral (HOS) method. Reconstructing surfaces from shadowed observations is found to have an impact limited to the non-visible regions, while tilt modulation affects the reconstruction more generally due to the indirect, more complex extraction of wave information. It is shown that the accuracy of the surface reconstruction mainly depends on the correct modelling of the wave shape nonlinearities. Modelling the nonlinear correction of the dispersion relation, in particular the frequency-dependent wave phase effects in the case of irregular waves, substantially improves the prediction. The suitability of the algorithms for severe wave conditions in finite depth and using non-perfect observations is assessed through wave tank experiments. It shows that only the third-order HOS solution predicts the right amplitude and phase of an emerging extreme wave, emphasizing the relevance of the corresponding physical modelling.

A Graphic Processing Unit–High-Order Spectral (GPU-HOS) Numerical Wave Tank for the Simulation of Directional Wave Field Evolution over a Long Time

Existing numerical models for direct simulations of stochastically directional waves are limited by their computational burden, and only a few of them can be applied to modeling directional waves on account of nonlinear evolutions on large time scales. In response to these drawbacks, this paper proposes a modified rectangular numerical wave tank based on the high-order spectral (HOS) method. Three main arrangements are responsible for the model improvement. Firstly, the relaxation technique is applied on the four sides of the tank for generating and attenuating waves, with an enhancement in the total computational efforts required of less than 5% being achieved. Secondly, for this paper, the HOS method for solving the nonlinear evolution of waves was modified to a GPU-speedup version for the first time, and the speedup increases dramatically (up to 16-fold) when the grid number is O(106). Thirdly, the higher-order Runge–Kutta method with order 8 is adopted for the purpose of suppressing cumulative temporal errors. Furthermore, several numerical results are presented for the validation of the model. With our modified numerical wave tank, the nonlinear evolutions of random directional waves can be efficiently studied over a large time and space scale.

An accurate and efficient HOS-meshfree CFD coupling method for simulating strong nonlinear wave–body interactions

The study on strong nonlinear wave–structure interaction (WSI) is crucial in ocean engineering. The Smoothed Particle Hydrodynamics (SPH) method is popular in this field because of its advantages in simulating free surface flow. However, to predict the details of WSI, a high resolution of particles is usually required, which results in a huge computational cost. In this paper, a potential-meshfree CFD weak coupling method based on the High Order Spectral (HOS) and SPH method is developed, which combines the advantages of the former in generating nonlinear waves quickly and accurately and the latter in adapting to strong nonlinear free surface problems. The accuracy of the coupling method was verified through the simulations of regular and irregular waves compared with experiment data. It shows the coupling method can effectively improve computational efficiency and extensively reduce the numerical dissipation. Furthermore, a simulation of the interaction between the focus wave and the floating body was conducted by using the coupling method, the original SPH method and the Constrained Interpolation Profile (CIP) method. Among all numerical methods, the coupling method is the best one for predicting the impact pressure of the floating body and is the only one that can capture the second impacting peak.

Establishment and validation of a viscous-potential coupled and graphics processing unit accelerated numerical tank based on smoothed particle hydrodynamics and high-order spectral methods

The smoothed particle hydrodynamics (SPH) method is an entirely Lagrangian and truly meshless numerical tool that is inherently suitable for simulating violent viscous free-surface flows, whilst it usually renders considerable simulation time when solving large-scale engineering problems. The high-order spectral (HOS) method is an accurate and very efficient numerical tool for modeling a large-scale inviscid gravity wave field, whilst it is hard to tackle complex physical phenomena due to the dependence on the potential flow theory. This paper aims to propose a viscous-potential coupled and graphics processing unit accelerated numerical model for establishing a high-fidelity and high-efficiency numerical wave tank. In this coupling scheme, the SPH method is applied in the near-field where violent physical processes exist to accurately capture complex free-surface behaviors such as overturning and breaking, whilst the HOS method is applied in the far-field where the water undergoes gentle motion to efficiently generate a desired gravity wave. A dedicated SPH–HOS coupling interface is also presented to accurately transmit the flow information from the HOS side to the SPH side. Meanwhile, an SPH–HOS buffer region with several key techniques is also proposed to ensure a successful coupling between the two numerical methods. Different benchmarks demonstrate that the present SPH–HOS coupling method absorbs the respective advantages of the two numerical methods and shows great feasibility and reliability in solving hydrodynamic problems involving gravity wave generation, propagation, deformation, and its interaction with marine structures.

Hydrodynamic analysis and validation of the floating DeepCwind semi-submersible under 3-h irregular wave with the HOS and CFD coupling method

Recent studies, such as the Offshore Code Comparison, Collaboration, Continued, with Correlation (OC5) led by NREL, have identified significant under-prediction of low-frequency pitch and surge responses in the DeepCwind semi-submersible platform when comparing model test results with predictions from engineering tools like OpenFAST, which rely on potential flow theory assumptions for hydrodynamics. To address this mismatch, a newer experimental campaign was conducted as part of the OC6 Phase 1A study. In this work, a coupling numerical approach was employed to replicate various scenarios of the floating DeepCwind semi-submersible, including decay, regular, and 3-h irregular waves, which were explored during the OC6 Phase 1A model testing. The numerical technique involved utilizing far-field nonlinear waves simulated using the High-Order Spectral (HOS) method and applying them to a near-field Computational Fluid Dynamics (CFD) domain. This allowed for the solution of wave-structure interactions using the overset mesh capabilities of StarCCM+. The critical hydrodynamic responses of the freely floating DeepCwind semi-submersible were modeled and validated against the model test results. The results obtained from the coupling approach were then presented and compared with simulations using OpenFAST and results from other researchers. The coupling strategy demonstrated improved agreement, especially for low-frequency responses in surge and pitch motion, under the 3-h irregular wave conditions, compared to OpenFAST simulations. While the validation investigation showed overall success in capturing low-frequency and wave-frequency surge and heave motion, there was an underestimation of low-frequency pitch motion in the TCOMS coupling tool. This underprediction can be attributed, at least in part, to the effect of wave overtopping on the rear columns, which resulted in deviations from the experimental pitch motion, particularly during extreme events.

Machine learning simulation of one-dimensional deterministic water wave propagation

Deterministic phase-resolved prediction of the evolution of surface gravity waves in water is challenging due to their complex spatio-temporal dynamics. Physics-based methods of varying complexity are available, but the conflicting objectives of numerical efficiency and accuracy impede real-time wave prediction. Data-driven methods may be able to overcome this challenge by using training data generated by complex numerical methods. This work explores the potential of a machine learning (ML) approach based on a fully convolutional encoder–decoder architecture for the efficient and accurate prediction of water waves. The high-order spectral (HOS) method forms the foundation for the generation of the training data. The HOS method is applied for different, consecutive orders of nonlinearity starting from first order up to fourth order. The JONSWAP wave energy spectrum serves as the basis for modeling the one-dimensional irregular sea states. The overall objective of this work is to evaluate whether the complex non-linear physical processes can be identified and learned by the ML approach. The trained ML flow mapper is used to perform time integration of an initial sea state. The results indicate that the proposed ML approach is able to reproduce the distinctive physical processes of the different orders of nonlinearities. It is shown that the ML approach enables fast and accurate predictions of one-dimensional waves over a time horizon that spans multiple peak periods.

A Study on the High-Order Spectral Model Capability to Simulate a Fully Developed Nonlinear Sea States

Modeling a nonlinear ocean wave is one of the primary concerns in ocean engineering and naval architecture to perform an accurate numerical study of wave-structure interactions. The high-order spectral (HOS) method, which can simulate nonlinear waves accurately and efficiently, was investigated to see its capability for nonlinear wave generation. An open-source (distributed under the terms of GPLv3) project named "HOS-ocean" was used in the present study. A parametric study on the "HOS-ocean" was performed with three-hour simulations of long-crested ocean waves. The considered sea conditions ranged from sea state 3 to sea state 7. One hundred simulations with fixed computational parameters but different random seeds were conducted to obtain representative results. The influences of HOS computational parameters were investigated using spectral analysis and the distribution of wave crests. The probability distributions of the wave crest were compared with the Rayleigh (first-order), Forristall (second-order), and Huang (empirical formula) distributions. The results verified that the HOS method could simulate the nonlinearity of ocean waves. A set of HOS computational parameters was suggested for the long-crested irregular wave simulation in sea states 3 to 7.

Numerical investigation on nonlinear evolution behavior and water particle velocity of wave crests for narrow-band wave field with Gaussian spectrum

The nonlinear evolution of initial narrow-band Gaussian spectrum wave field, as well as the water particle velocity of wave crests for Freak waves are investigated by phase-resolved numerical model of high-order spectral (HOS) method. The long-time evolution behavior (up to 2000 wave periods) are considered under the effects of initialized parameters, including initial spectral width, wave steepness and BFI. The effects of initial BFI on occurrence probability of Freak waves indicate various distinctive stages. Moreover, the evolution characteristics of spectral width, local wave steepness, BFI and kurtosis are analyzed. And near-periodic oscillating pattern of spectral width after its long-time evolution is obtained and effected by initial spectral width and wave steepness with different features. Besides, the positive (or negative) linear correlations between wave height extreme value and local wave steepness & kurtosis (or BFI evolution value) are effected by initial spectral width primarily. At last, the behavior of water particle velocity (including the extreme value) of Freak wave crest is discussed. And the nonlinear correlation between wave crest height and water particle velocity of wave crest is obtained quantitatively.

An efficient methodology for the simulation of nonlinear irregular waves in computational fluid dynamics solvers based on the high order spectral method with an application with OpenFOAM

An efficient methodology for simulating nonlinear irregular waves in a Computational Fluid Dynamics (CFD) solver is proposed. The High Order Spectral (HOS) method is used to generate nonlinear irregular waves in an open ocean and a numerical wave tank. The inverse Fast Fourier Transforms (FFTs) and multi-dimensional interpolation from the HOS simulation results are used for the efficient reconstruction of nonlinear waves in the CFD solver. The proposed procedure is published as an open-source project called Grid2Grid, which is developed to interface with a generic CFD solver. It provides the function Application Programming Interface (API), which can communicate with different programming languages. Extreme wave events were used to validate the proposed procedure. The predicted wave breaking events are reported both in OpenFOAM CFD and HOS simulations, and the wave elevations of CFD during simulations show good agreement with experiments and with HOS simulation.

Space-time statistics of extreme ocean waves in crossing sea states

The study of extreme ocean waves has gained considerable interest in recent years, due to their importance for offshore design and navigation safety, and several theoretical approaches have been developed for their statistical description. However, in the case of crossing seas, where two or more wave systems of different characteristics are present, a full understanding of the main physical mechanisms responsible for the occurrence of very high individual waves is still lacking. As a consequence, the prediction of extremes in such conditions currently relies on integrated parameters of the total sea state, such as the spectral wave steepness. In this study, to gain further insight into the role of the crossing wind sea and swell wave systems in producing extreme individual waves, we investigate realistic sea states during typhoon Kong-rey (2018) using an ensemble of numerical simulations obtained from a phase-resolving wave model based on the High-Order Spectral (HOS) method. The reliability of the numerical fields is assessed for the first time with stereo wave measurements of the sea surface elevation field collected from an offshore platform in the area of interest. We show that, in specific conditions, space-time extreme crest heights in crossing seas can be larger than in unimodal seas due to second-order bound wave interactions between the wind sea and the swell. To improve existing prediction capabilities, we propose a novel formulation for the wave steepness in crossing seas, which includes nonlinear effects up to the second order and accounts for the spectral parameters of the interacting wave systems.

Phase-resolved ocean wave forecast with simultaneous current estimation through data assimilation

In Wang & Pan (J. Fluid Mech., vol. 918, 2021, A19), the authors developed the first ensemble-based data assimilation (DA) capability for the reconstruction and forecast of ocean surface waves, namely the EnKF-HOS method coupling an ensemble Kalman filter (EnKF) and the high-order spectral (HOS) method. In this work, we continue to enrich the method by allowing it to simultaneously estimate the ocean current field, which is in general not known a priori and can (slowly) vary in both space and time. To achieve this goal, we incorporate the effect of ocean current (as unknown parameters) on waves to build the HOS-C method as the forward prediction model, and obtain a simultaneous estimation of (current) parameters and (wave) states via an iterative EnKF (IEnKF) method that is necessary to handle the complexity in this DA problem. The new algorithm, named the IEnKF-HOS-C method, is first tested in synthetic problems with various forms (steady/unsteady, uniform/non-uniform) of current. It is shown that the IEnKF-HOS-C method is able to not only estimate the current field accurately, but also boost the prediction accuracy of the wave field (even) relative to the state-of-the-art EnKF-HOS method. Finally, using real data from a shipborne radar, we show that the IEnKF-HOS-C method successfully recovers the current speed that matches the in situ measurement by a floating buoy.

Numerical Investigation on Temporal Evolution Behavior for Triad Resonant Interaction Induced by Steady Free-Surface Flow over Rippled Bottoms

Investigating the wave hydrodynamics of free-surface flow over rippled bottoms is a continuing concern due to the existence of submarine multiple sandbars and ambient flow in coastal and estuarial areas. More attention to free-surface wave stimulation has been received from the perspective of resonant wave-wave interaction, which is an intensive way for wave energy transfer and a potential way for wave component generation. However, the basic behavior of the triad resonant interaction of this problem is still limited and unclear. In this study, the triad resonant interaction induced by steady free-surface flow over rippled bottoms is numerically investigated by means of the High-Order Spectral (HOS) method. By considering the interactions among free-surface waves, ambient current, and rippled bottoms, the numerical model is applied for this situation based on Zakharov equation with ambient flow term. The temporal evolution of the triad resonant wave amplitude has been numerically investigated and compared well with multiple-scale expansion perturbation theory. Specifically, the temporal evolution behaviors of all six triad resonant wave components are confirmed by both numerical simulation and nonlinear perturbation analysis.

A GPU-accelerated domain decomposition method for numerical analysis of nonlinear waves-current-structure interactions

A fully nonlinear near-far fields coupling solver is proposed in this work. The target is to provide an efficient numerical tool for wave-current-structure interaction simulations. A GPU-accelerated high-order spectral (HOS) solver is developed to simulate the wave propagation and interaction with current in large far field and an incompressible flow solver with free surface capture scheme from the OpenFOAM library is adopted in the near field to solve the nonlinear wave loads and seakeeping response of the ship hull. The newly developed HOS solver is firstly validated by two wave-current interaction test cases, including a regular wave train travelling on collinear following and opposing current, and regular waves blocked by strong opposing current. Then, the near and far field coupling solver is applied to a simulation of the seakeeping response of a cruising ship hull in head wave condition. Lastly, The proposed method is applied to a more practical application, i.e. the self-propulsion of a hull in short crested spreading irregular waves. The numerical results are compared to the lab measurement data or numerical results, good agreements have been obtained.

Wave amplification caused by Bragg resonance on parabolic-type topography

Inspired by the focusing principle of a parabolic-type mirror and the wave Bragg resonance, a parabolic-type spatial layout with a Bragg bottom is proposed, which can significantly amplify wave height. The wave propagation characteristics on the parabolic-type topography were numerically investigated using a developed numerical model based on the High-Order Spectral (HOS) method. The amplification properties of the wave height on the proposed parabolic topography with a Bragg bottom were systematically analyzed considering different ripple heights, ripple numbers, topographic widths and focal lengths. The results show that the focusing effect of wave height increases with the increase of the ripple height, ripple number and topographic width. However, when the ripple number and topographic width are bigger than certain constants, their effect on wave focusing becomes insignificant. Compared with the traditional parallel-type layout, the ripple number of the proposed parabolic-type layout can be reduced to one-fifth to achieve the same amplified effect of the parallel layout with the same ripple height. Moreover, the focus length also influences the wave focusing by changing the wave collection angle. From the presented studies, the optimal collection angle of the parabolic-type layout is approximately in the range of 82°–115°.

A modified approach to wavemaker modeling for high-order spectral numerical wave tanks

In this study, an enhanced novel numerical wave tank (NWT), based on the high-order spectral (HOS) method, is established to simulate the propagation of fully nonlinear water waves. A fundamental issue of the HOS-NWT approach is the wave generation problem, which is accounted for using an additional potential concept. We propose a modified wavemaker model by feeding wave properties from the tank bottom to the instantaneous wave surface, according to a chosen wave theory at the tank entrance. Gravity water waves at intermediate and shallow water depths are adopted in this study to verify the stability, efficiency, and accuracy of the modified and conventional methods for wave generation, by comparison with stream function solutions. The results of both methods for in-silico waves are almost identical to the stream function solutions for intermediate water waves. However, the modified model performs better than the conventional one for long shallow water wave simulations. Through Fourier analysis, evident nonlinear characteristics of the time series of the wave elevations are confirmed. The modified HOS-NWT is also extended to investigate nonlinear wave–wall interactions, and satisfactory agreement is achieved with the numerical Fourier solutions.

Reduced order model for nonlinear multi-directional ocean wave propagation

In this paper, we introduce a reduced order model (ROM) for the propagation of nonlinear multi-directional ocean wave-fields. The ROM relies on Galerkin projection of Zakharov equations embedded in the high-order spectral (HOS) method, which describes the evolution of nonlinear waves. The dominant flow features of wave evolution are computed from proper orthogonal decomposition (POD) and these modes are used for the projection. The HOS scheme to compute the vertical velocity is treated in a novel way for an efficient implementation of POD-based ROM. We refer to this alternative formalism of HOS as HOS-simple. The final reduced order model (ROM) is derived from the Galerkin projection of HOS-simple. For the case of irregular waves, where the number of modes required are in the range of 200, the ROM has no significant advantage since both HOS and HOS-simple are much faster than real-time. The real advantage is demonstrated in multi-directional (or short-crested) irregular waves, where the ROM is the only model capable of achieving real-time computation, a major improvement to the standard HOS method. The potential use of the ROM in propagating short-crested waves from far-field to near-field for real-world applications involving wave probes in a wave tank/controlled environment as well as X-band radar in open ocean is also demonstrated.

The Focusing Waves Induced by Bragg Resonance with V-Shaped Undulating Bottom

Intensive wave reflection occurs when the wavelengths of the incident waves and bottom undulations are in a 2:1 ratio. Existing studies have included the Bragg resonance phenomenon of waves passing over a continuous undulating bottom parallel to and oblique to the shoreline. More generally, the Bragg resonance mechanism is used as a means of coastal protection, rather than wave power generation. To focus the wave energy in a specific area, here, we propose sinusoidal sandbars of a horizontal V-shaped pattern, which is formed by two continuous undulating bottoms inclined at an angle to each other and the center axis of the angle is perpendicular to the shoreline. Based on the high-order spectral (HOS) numerical model, both the characteristics of Bragg resonance induced by the regular waves and random waves are investigated. In the scenario of regular waves, it shows that the wave-focusing effect is related to the angle of the V-shaped undulating bottom, and the optimal angle of inclination for the V-shaped undulating bottom is 162.24°. On that basis, considering the interactions between the random waves and the V-shaped undulating bottom of 162.24°, the Bragg resonance characteristics of random waves are studied. The BFI factor combining wave steepness and spectrum width can evaluate the focusing intensity of the Bragg resonance of the random waves. For BFI, in the range of 0.15–1.0, the values of Hsmax/Hs0 linearly increase with the increase of BFI.