Wave Elevation Research Articles

Application of Stokes’ higher order theory and JONSWAP spectrum in modelling extreme wave events validation with historical data

Accurately modelling wave elevations is critical for the design and safety of offshore structures. This study explores the application of Stokes’ 5th Order Wave Theory and JONSWAP (Joint North Sea Wave Project) spectrum to simulate wave conditions similar to those experienced during the Draugen Ringing Event. Hindcasting on the dataset collected form Norwegian Meteorological Institute sensor system Draugen is performed. Through rigorous analysis performed by integrating these models, we aim to validate their accuracy and reliability against empirical data. The methodology includes generating the JONSWAP spectrum based on significant wave heights and peak periods, followed by computing wave surface elevations using Stokes’ 5th Order Wave Theory and spectral analysis. The outcome of the analysis shows a strong correlation between the modelled irregular wave elevations and actual wave data, demonstrating the effectiveness of these models.

Machine learning for predicting flow and pressure in an oscillating water column wave energy converter

The use of wave energy presents itself as a renewable alternative, diversifying the energy matrix and reducing the consumption of fossil fuels. One of the prominent devices for ocean wave harvesting is the Oscillating Water Column (OWC), which consists of a hydropneumatic chamber where the oscillation of the water surface compresses the confined air, driving a turbine. This work combines numerical simulation and machine learning algorithms, aiming to develop predictive models for mass flow rate and pressure through the OWC chamber. The database for the models was created using Design of Experiments (DOE) techniques, where different combinations of wave height and wavelength were simulated via Computational Fluid Dynamics (CFD) using the Finite Volume Method (FVM). Subsequently, the wave elevation height data is fed into a fully connected deep neural network, which uses the simulated data for learning and returns predictions based on the provided information. The model's performance is evaluated in terms of a cost function (Mean Squared Error). The employed model allows for the identification of flow rate and pressure peaks for the wave period provided at the input, and the results show differences of up to 5% in RMS (Root Mean Square) between the predictions and the CFD data. The main contribution of this work lies in the use of machine learning in predicting traditional operating conditions of an OWC device, introducing a concept that can be applied in the early stages of the design of such devices.

Modified free-surface synthetic Schlieren method to adjust measurement sensitivity in high-strain waves

This research revisits the free-surface synthetic Schlieren technique (FS-SS) (Moisy et al. in Exp Fluids 46(6):1021–1036, 2009) for the topographical measurement of the free surface of a liquid. In contrast to the aforementioned method, which utilizes the image refraction of a random dot pattern through a free surface to determine the surface gradients, this present study mathematically derives a method where the pattern may take an arbitrary three-dimensional shape. This is shown to be theoretically valid under a small pattern slope approximation. Our method is then verified against the previously mentioned, flat-pattern, free-surface synthetic Schlieren technique by resolving the free-surface elevation of plane waves across a channel, showing similar results in constant strain conditions, with improved results in variable strain conditions, particularly in cases where there is a very large difference in strain across the channel. The validation test cases investigated include a rectangular channel containing a transparent liquid with a random dot pattern placed below at a constant angle and a pattern placed on top of a cosine-shaped profile. Both of these setups are validated against the classical FS-SS technique involving a flat pattern. The new method involving an arbitrarily shaped pattern proposed here may increase the resolution in low-amplitude regions by increasing the surface–pattern distance below these regions and correspondingly reducing the sensitivity in high-strain regions by decreasing the surface–pattern distance. Geometries shown to produce advantageous results in waves that include both regions of very high strains and regions of very low amplitudes are explored, resolving the wave in both regions simultaneously. This shows promise in resolving multi-scale surface waves in highly viscous liquids, which may include very high-amplitude regions quickly followed by very low-amplitude regions due to damping effects.

Hydrodynamic Performance Investigations of OWC and Hybrid System: Geometry of OWC and Rectangular Submerged Breakwater

Due to its simplicity and high reliability, the Oscillating Water Column (OWC) has been a prominent focus of research. This study investigates the impact of OWC geometry on hydrodynamic performance by analyzing the effects of draft, chamber width, and Power Take Off (PTO) damping on energy conversion efficiency, reflection coefficient, transmission coefficient, and dissipation coefficient. The motion of the water column under different PTO damping conditions is analyzed. In addition, the hydrodynamic performance of the submerged breakwater-OWC hybrid system is studied and compared with that of a standalone OWC. The results show that a higher opening height improves the energy conversion efficiency of the OWC under medium-to-long waves. There is an optimal draft for the OWC to maximize the energy conversion efficiency for different wave periods. Meanwhile, the optimal draft increases with the wave period. Increasing PTO damping does not change the resonance period of the OWC, while the free surface elevation inside the chamber reduces. Within a range of periods around the maximum inclination, multi-point measurements of the wave elevation in the chamber are necessary to reduce measurement errors caused by the spatial non-uniformity of the water column. The submerged breakwater-OWC hybrid system exhibits a higher energy conversion efficiency under medium-to-long waves compared to a standalone OWC, and there is an optimal breakwater length.

Faster than real-time, phase-resolving, data-driven model of wave propagation and wave–structure interaction

A machine learning time-series prediction approach is proposed for wave propagation and wave load prediction. Under unidirectional wave conditions and variable bathymetry, given a wave gauge upstream, a model is shown to reproduce wave elevation or wave forces downstream under irregular steep and either non-breaking or breaking conditions. Attempts to perform the opposite calculation, predicting upstream conditions from downstream measurements, results in higher error, likely due to information loss under breaking conditions. For choice of machine learning approach, comparisons show that the Time-series Dense Encoder (TiDE) approach results in a good balance between model complexity, stability, computational time, and error. Over a flat bottom, time-series of wave elevation can be predicted up to 10 wavelengths away, though with a degraded accuracy compared to shorter distances. Similar results are shown for time-series of forces on a vertical cylinder, showing better results than a simple Morison approach, as used in engineering tools such as OpenFAST, but with a similarly fast computational time. Generalizations show that training on irregular wave data permit extrapolations to periodic wave cases. Finally, the same method also is also demonstrated at field-scale, comparing results between two offshore buoys.

Velocity field measurements under Very Large Floating Structures interacting with surface waves

Increasing utilization of ocean space and a global push for renewable energy solutions has spurred interest in wave behavior around Very Large Floating Structures, like floating photovoltaic (PV) systems. Flexible PV modules may be more suitable for the varying wave conditions found in offshore environments. However, while viscoelastic models are commonly used for wave prediction, they show notable discrepancies with experiments, likely due to untested assumptions of inviscid flow. This experimental study aims to fill that gap by investigating both the wave characteristics and velocity fields underneath flexible and rigid structures using simultaneous Particle Image Velocimetry (PIV) and wave elevation measurements. Wave attenuation is observed for short wavelengths over the flexible structure length. The 2nd order Stokes wave theory provides a good approximation of the wave-induced horizontal velocity profiles under the flexible structure but underestimates the velocities under the rigid one which further lacks the typical exponential decay with water depth. The presence of a wave boundary layer is showcased and compared to an adaptation of the Stokes 2nd problem.

Multi-condition hawser tension prediction of offshore offloading system based on long and short-term memory network and transfer learning

Real-time prediction of hawser tension of the side-by-side oil offloading system of Floating Production Storage and Offloading (FPSO) can provide early warnings for hawser breakages and ship collision risk, thus improving structural, property, and environmental security. Although offline numerical models and Long Short-Term Memory (LSTM) could offer substantial precision, they confront intensive time costs in recalibrating or retraining models. This paper proposes an integration method of LSTM networks and transfer learning for real-time tension prediction considering inputs of actual remote wave elevations. We use short-term environmental data and numerical simulation data of correlated hawser tensions during offloading operations to train a pre-trained benchmark model for transfer learning. Then, a highly generalized and efficient transferred model is constructed by using a small sample to realize short-term tension predictions in time-varying environments. The results show the occurrence time and value of extreme tensions predicted by transfer learning nearly match the reference data, and their maximum errors are 3 s and 0.11, respectively, superior to LSTM direct training. Therefore, it provides sufficient demand for real-time prediction and early collision risk prevention in dynamically changing ocean environment. The research results could provide an alternative framework for intelligent monitoring of large-scale marine structures.

Real-time ocean wave prediction in time domain with autoregression and echo state networks

This study evaluates the potential of applying echo state networks (ESN) and autoregression (AR) for dynamic time series prediction of free surface elevation for use in wave energy converters (WECs). The performance of these models is evaluated on time series data at different water depths and wave conditions, including both measured and simulated data with a focus on real-time prediction of ocean waves at a given location without resolving for the surrounding ocean surface, in other words, short-time single-point forecasting. The work presented includes training the models on historical wave data and testing their ability to predict phase-resolved future surface wave patterns for short-time forecasts. Additionally, this study discusses the feasibility of deploying these models for extended time intervals. It provides valuable insights into the trade-offs between accuracy and practicality in the real-time implementation of predictive models for wave elevation, which are needed in wave energy converters to optimise the control algorithm.

Deterministic Sea Wave Reconstruction and Prediction Based on Coherent S-Band Radar Using Condition Number Regularized Least Squares

Coherent S-band radar is a remote sensing observation device with high spatial-temporal resolution and can be used to achieve deterministic sea wave reconstruction and prediction (DSWRP) technology. However, coherent S-band radar can observe nonlinear details of the sea surface due to its high resolution, which makes the propagation operator matrix an ill-conditioned overdetermined matrix. To solve this problem, this paper proposes a DSWRP scheme using condition number regularized least squares (CN-RLS) for coherent S-band radar. First, the space-time velocity information was obtained from the radar echo. Second, the CN-RLS method solved the phase-resolved model coefficients. Finally, the deterministic wave field was predicted according to the solved model coefficients. The proposed scheme was verified by simulation data and the real radar dataset observed by the coherent S-band wave-measuring radar onboard the ship XIANGYANGHONG-18 in the East China Sea in April 2024. The predicted wave elevation of the proposed method was compared with the wave elevation observed based on the X-band wave-measuring radar, and the root mean square error (RMSE) and correlation coefficient (CC) were 0.22 m and 0.76, respectively, which show that the proposed method could effectively implement the DSWRP technology.

Mixed neural operator learning on the solitary wave propagation over slope topography and inverse problem

This study proposes the mixed neural operator (MNO) learning framework, which further combines with the particle swarm optimization (PSO) to address challenges of solitary wave propagation over topography. The forward problem is defined as the evolution prediction of the solitary wave propagating over topography, while the inverse problem is defined as an optimization to identify the topography parameter based on the solitary wave elevation. Both the forward and inverse problems can be considered within a single framework and the dataset are provided by the classical Korteweg–de Vries (KdV) equation. The MNO framework is shown to simulate the evolution of solitary waves over topography, accurately capturing the wave elevation under different topographical conditions. By comparing with different neural operators, it is found that the U-shape neural operator is the most suitable for the KdV equation simulation. The coefficient of determination for the inverse problem based on the combination of MNO and PSO can reach 0.992, showing great potential of the approach in topography recognition. Finally, the proposed learning framework is preliminary applied to the prediction of the tsunami runup onto a complex beach, and a good agreement is also achieved between the direct simulation and the learning framework prediction.

Effects of gap entrance configuration on gap resonances between two free-heaving barges: Vortex shedding behaviors

This paper mainly concentrates on the influences of the vortex shedding pattern and the behaviors of the generated vortex bubbles on the hydrodynamic performances of two identical side-by-side barges when encountering the linear and nonlinear gap resonances. The vortex shedding pattern around the entrance of the gap is highly influenced by the gap entrance configuration, and thus, two kinds of gap entrance configurations, i.e., round edge and square edge, are taken into account. The vortex shedding patterns are identified during the linear and nonlinear gap resonances for all conditions. Meanwhile, the effects of incident-wave height and barges' heave responses are considered when interacting with different edge shapes. Numerical results demonstrated that both the gap resonances and barges' heave responses are influenced by the vortex generation pattern. Significantly, the magnitudes of the wave elevation at gap during the linear and nonlinear gap resonances are influenced by the coupling effect between the vortex bubbles' pre-behaviors, i.e., vortex shedding pattern, and their post-behaviors, i.e., vortex bubbles' behaviors post-shedding from shear layers, underscoring the importance of understanding vortex shedding patterns and the behaviors of vortex bubble in investigating the gap resonances.

An Experimental Study of Wave Impact Loads on an FPSO Bow in 2D Wave-Tank

In harsh environments, an floating production storage and offloading (FPSO) is occasionally damaged by impact loads, such as bow flare slamming and green water. This study conducted an impact load measurement experiment on a model of an FPSO bow in a 2D wave tank. Three types of frequency-focused waves (steep, spilling, and plunging) were generated, and the speed and slope of the waves were measured. Seven wave probes were placed in a row, and the wave elevation was measured to determine the speed and slope of the waves. In addition, the side of the 2D wave tank was photographed with a high-speed camera. The speed and slope of the waves obtained from the wave probe array agreed well with those obtained from the photographs taken using a high-speed camera. In the case of a steep wave, wave runup occurred at the bow before the wave reached the bow of the FPSO, so no impact load was generated, and only hydrostatic pressure was measured. Impact loads were generated in the spilling and plunging waves, and the magnitude of impact loads using the Von Karman’s estimation formula and the impact loads measured in model tests showed similar values.

Impact of rogue waves on semi-submersible platforms: An experimental investigation of wave run-up and air-gap responses

The wave run-up and air-gap responses of semi-submersible platforms under extreme sea conditions are crucial design considerations for ensuring the structural integrity and safety of marine engineering operations. This study investigated the impact of rogue waves on the wave run-up and air-gap responses of semi-submersible platforms using a transient focusing wave group to accurately simulate irregular seas. Comparative model tests were conducted on a semi-submersible platform under both rogue and non-rogue-wave conditions. The platform motions and air-gap responses were analysed using spectral analysis, extreme-value statistics, and wavelet methods. Rogue waves were found to significantly affect the air-gap response spectrum, increasing the nonlinearity of both the wave run-up and air-gap responses. Different locations on the platform exhibited varying statistical and spectral response characteristics. Rogue waves modified the instantaneous properties of wave elevation, thereby increasing the energy levels in the system. This study confirmed that rogue waves significantly influence the air-gap responses and wave run-up of semi-submersible platforms, suggesting that they should be considered when designing and planning the operations of these platforms. This study offers a comprehensive understanding of the complex interactions between rogue waves and large maritime structures and provides insights for improving safety and performance under extreme sea conditions.

Numerical analysis of coupling effect between sloshing and motion responses of a KCS in uni- and bi-directional waves

The seakeeping behaviour can be severely affected by the coupling between sloshing and ship motions in waves. The sloshing behaviour of liquid-loaded ships in bi-directional waves differs significantly from that in uni-directional waves. Given that ships operate in short-crested or multi-directional waves during their whole service period, it is significant to gain a deeper understanding of the effect of wave directionality on the coupling between sloshing and ship motions. In this study, a KRISO container ship (KCS) model with type-B liquefied natural gas (LNG) cargo tanks is adopted for coupling analysis. The 2D long-crested regular wave is adopted as an uni-directional wave, and three dimensional (3-D) cross wave is adopted as a bi-directional wave. The computational fluid dynamics (CFD) method is applied for the sloshing and seakeeping simulation of the ship model in different wave fields. The differences in wave elevation, sloshing forces, and coupling effects between ship motions and sloshing are studied in these two wave fields by adjusting control parameters such as ship heading angle and filling ratio. The comparative results indicate that sloshing behaviour and coupled motion responses in cross waves should be paid enough attention during ship design and evaluation.

Nonlinear model predictive dynamic positioning of a remotely operated vehicle with wave disturbance preview

Exploiting preview information of disturbances within robot control is an effective method of handling state perturbations. For underwater vehicles operating in wave-dominated environments, disturbances significantly influence vehicle response and pose a threat to operational safety. In consideration of this, a complete end-to-end control architecture is developed in this work for disturbance rejection during station keeping tasks under wave perturbations, encompassing a nonlinear model predictive controller (NMPC) combined with a deterministic sea wave predictor (DSWP). The wave predictor exploits a continuous measurement of the wave elevation at a location upstream of the vehicle to form a forecast of the temporal evolution of the wave elevation at the vehicle location. The predicted wave parameters are then used to estimate the impending wave-induced hydrodynamic loading, enabling explicit consideration of accurate short-term disturbance forecasts within the controller, ensuring this preview is incorporated in the optimised control sequence. Experimental testing confirms the validity of the wave predictor, producing RMSE as low as 0.017 m. The proposed strategy is then simulated under various wave conditions, optimising control actions inclusive of the preview information. The NMPC outperforms a similar disturbance mitigating feed-forward controller, with average improvements of up to 52% and is found to perform best even with noisy, lower accuracy wave predictions, as well as with communication time delays, providing a high degree of robustness. This demonstrates the potential of the proposed control framework to effectively tackle disturbance mitigation of underwater vehicles subject to large magnitude wave disturbances, expanding the operational envelop of autonomous systems towards forbidding ocean environments.

Effects of gap entrance configuration on gap resonances between two free-heaving barges: Higher-order harmonics

The hydrodynamic characteristics of linear and nonlinear gap resonances between two identical side-by-side free-heaving barges are investigated in a numerical wave tank based on the constrained interpolation profile method. This study focuses on the influence of the gap entrance configuration on key hydrodynamic parameters during gap resonances, comparing conditions of round and square edges. Additionally, the effects of incident wave height and the barge's heave responses are examined. The distributions of the first four harmonic components of the key parameters are illustrated, including the wave elevation at the gap, wave run-up on each barge, and wave forces. Numerical results reveal that the gap entrance configuration influences more on the linear gap resonance rather than the nonlinear gap resonances. The higher-order components of the wave elevation at the gap are more sensitive to the incident wave height rather than the edge shape. The influences of the edge shape on the wave forces are mainly manifested in the magnitude of the wave forces rather than in their tendencies. Furthermore, the response time during the development stage of gap resonance is analyzed. The findings indicate that gap resonance develops more quickly with square edges or when the incident wave height increases.

Data Assimilation and Parameter Identification for Water Waves Using the Nonlinear Schrödinger Equation and Physics-Informed Neural Networks

The measurement of deep water gravity wave elevations using in situ devices, such as wave gauges, typically yields spatially sparse data due to the deployment of a limited number of costly devices. This sparsity complicates the reconstruction of the spatio-temporal extent of surface elevation and presents an ill-posed data assimilation problem, which is challenging to solve with conventional numerical techniques. To address this issue, we propose the application of a physics-informed neural network (PINN) to reconstruct physically consistent wave fields between two elevation time series measured at distinct locations within a numerical wave tank. Our method ensures this physical consistency by integrating residuals of the hydrodynamic nonlinear Schrödinger equation (NLSE) into the PINN’s loss function. We first showcase a data assimilation task by employing constant NLSE coefficients predetermined from spectral wave properties. However, due to the relatively short duration of these measurements and their possible deviation from the narrow-band assumptions inherent in the NLSE, using constant coefficients occasionally leads to poor reconstructions. To enhance this reconstruction quality, we introduce the base variables of frequency and wavenumber, from which the NLSE coefficients are determined, as additional neural network parameters that are fine tuned during PINN training. Overall, the results demonstrate the potential for real-world applications of the PINN method and represent a step toward improving the initialization of deterministic wave prediction methods.

Experimental study on plunging breaking waves impacting a vertical cylinder with three different model scales

Scale effects, which can influence the reliability and effectiveness of experimental results, are often unavoidable but seldom explored, particularly in the context of plunging waves. To investigate the scale effects of plunging wave impacts on a vertical cylinder, experiments were conducted in a wave flume at model scales of 1:20, 1:30 and 1:40. The wavemaker movements were scaled based on the Froude-similarity law and the pressures on the front surface of the cylinder were analyzed. The results indicated that plunging waves were generated across these three scales, with fine agreement in overall wave elevations and a local waveform similarity of 80–90%. The analysis of pressures showed that the distribution of peak pressures on the cylinders remained consistent across different scales; however, scale effects were closely related to the intensity of wave impacts. Specifically, in cases of relatively weak impact, such as when the wavefront struck the cylinder before breaking or even after an intense wave breaking with intense gas-liquid mixtures, the pressure discrepancies were less than 20% among these different scales. This discrepancy can exceed 50% near the wave peaks during the vertical wave front or early jet formation stages with strong impacts.

Uncertainty of Wave Spectral Shape and Parameters Associated with the Spectral Estimation

The uncertainty in estimating the wave spectrum from the records of wave elevation by heave–pitch–roll buoys is studied, considering the effects of the estimation method and the spectral resolution adopted in the process. This investigation utilizes measurements from a wave buoy moored in deep water in the South Atlantic Ocean. First, the spectra are computed using the autocorrelation function and the direct Fourier method. Second, the spectral resolution is tested in terms of degrees of freedom. The degrees of freedom are varied, and the resulting spectra and integrated parameters are computed, showing significant variability. A simple and robust methodology for determining the wave spectrum is suggested, which involves calculating the average energy density in each frequency band. The results of this methodology reduce the variability of the estimated parameters, improving overall accuracy while preserving frequency resolution, which is crucial in complex sea states. Additionally, to demonstrate the feasibility of the implemented approach, the final spectrum is fitted using an empirical model ideal for that type of spectrum. Finally, the performance and the goodness of the fit process for the final averaged curve are checked by widely used statistical metrics, such as R2 = 0.97 and root mean square error = 0.49.

A Deep Learning Strategy for the Retrieval of Sea Wave Spectra from Marine Radar Data

In the context of sea state monitoring, reconstructing the wave field and estimating the sea state parameters from radar data is a challenging problem. To reach this goal, this paper proposes a fully data-driven, deep learning approach based on a convolutional neural network. The network takes as input the radar image spectrum and outputs the sea wave directional spectrum. After a 2D fast Fourier transform, the wave elevation field is reconstructed, and accordingly, the sea state parameters are estimated. The reconstruction strategy, herein presented, is tested using numerical data generated from a synthetic sea wave simulator, considering the spectral proprieties of the Joint North Sea Wave Observation Project model. A performance analysis of the proposed deep-learning estimation strategy is carried out, along with a comparison to the classical modulation transfer function approach. The results demonstrate that the proposed approach is effective in reconstructing the directional wave spectrum across different sea states.