Random Wave Simulation Research Articles

Development of a Wind–Wave Coherence Function Based on Numerical Studies

The synchronization and intensity of fluctuating wind speeds and wave surfaces in wind–wave joint propagation processes are affected by the coherence of the marine ambient factors of fluctuating wind and random waves. This coherence further affects the precise calculations of wind–wave joint actions on marine structures. Therefore, a wind–wave joint propagation numerical flume was established based on the numerical simulation of random waves and fluctuating wind fields. A series of numerical simulations of wind–wave joint propagations were carried out. Based on the numerical results, the influences and influence laws of factors such as wind speed position height, significant wave height and wave spectrum peak frequency on the wind–wave coherence values were studied. According to the influence characteristics of these factors, a function of wind–wave coherence values for random wind–wave joint propagation was calculated. The coherence function takes frequency as the variable, while parameters include significant wave height, wind speed position height and wave spectrum peak frequency. Through a series of numerical simulation results, data fitting was used to calculate the parameter coefficients of the coherence function. The established random wind–wave coherence function can be described using the wind–wave joint fields of marine structures and the computational analyses of structural wind–wave joint actions.

An efficient method of defining the tail of a crest height distribution

The exceedance probability of wave crest elevation is a critical environmental input for the design/re-assessment of marine structures. With attention often focused on structural reliability, and in some cases survivability, the largest wave crests arising at the smallest exceedance probabilities, said to be located in the tail of a distribution, are of primary interest. This paper explains why present design practice may be non-conservative in the most extreme seas and outlines a new method by which the tail of the distribution can be defined using a relatively small number of deterministic wave events. This avoids the need to explore the entire distribution using very long (and expensive) random wave simulations. The new approach allows both an extension of the distribution to smaller exceedance probabilities and a concentration on the largest most design relevant crest heights. Having demonstrated the success of the proposed method by comparisons to laboratory data, the analysis is extended to include the effective prediction of the associated confidence intervals (CIs). With the highest waves subject to the largest statistical uncertainty, the paper explores the nonlinear changes in CI, demonstrates that these can also be accurately and efficiently defined, and explains how CI may be reduced. The focus of the paper lies in improved design calculations, based upon the nonlinear dynamics of extreme waves in realistic seas.

Monte Carlo Simulation of Random Waves at Finite Depth Based on the Joint Distribution of Wave Amplitude and its Associated Period in Nonlinear Random Waves of Finite Bandwidth

This study aims to develop a new methodology to generate random waves from a joint distribution of wave amplitude and its associated period. In doing so, the wave spectrum was derived from the joint distribution of wave amplitude and associated period and compared with the Wallops spectrum, which has been most frequently referred to in the literature. Numerical simulation shows that the nonlinearity of random waves makes a significant difference in the wave spectrum. These differences are summarized by a somewhat excessive peak period of the semi-empirical wave spectrum and underestimating wave energy over the high-frequency band. These behavioral characteristics of wave spectrum derived from the joint distribution of wave amplitude and its associated period are conceived to be due to the intrinsic property of the nonlinear random waves. The rationale of this reasoning can be found in the fact that non-negligible amount of wave energy shifts to the relatively short and long period band due to the resonant wave-wave interaction as nonlinearity is getting more pronounced as widely known in the coastal engineering community after studies of Hasselmann(1967) and Phillips(1980) on the developments of wind waves. Among the differences mentioned above, an excessively high peak period can be fully predicted by recalling that the wave period is getting shortened due to the phase difference between the free mode waves among the higher-order harmonic components appearing in random waves due to resonant wave-wave interaction and the resulting destructive interaction.

Nonlinear Wave Evolution in Interaction with Currents and Viscoleastic Muds

A numerical model is extended to investigate the nonlinear dynamics of surface wave propagation over mud in the presence of currents. A phase-resolving frequency-domain model for wave-current interaction is improved to account for wave modulations due to viscoelastic mud of arbitrary thickness. The model compares well with published laboratory data and performs slightly better than the model with viscous mud-induced wave damping mechanism. Monochromatic and random wave simulations are conducted to examine the combined effect of currents, mud-induced wave dissipation and modulation, and nonlinear wave-wave interactions on surface wave spectra. Results indicate that current effects on wave damping over viscoelastic mud is not as straightforward as that over viscous mud. For example, while opposing currents consistently increase damping of random waves over viscous mud, they can decrease damping over viscoelastic mud due to high variations in frequency-dependent damping stemming from mud’s elasticity. It is shown that a model that assumes the mud layer to be thin for simplification can overestimate wave damping over thick mud layers.

PORO-FSSI-FOAM model for seafloor liquefaction around a pipeline under combined random wave and current loading

In this study, unlike most previous investigations have limited to the regular wave conditions or combined wave and current conditions, a numerical model for seabed response around a pipeline in a trenched layer due to random waves is established. Three different wave spectra including the JONSWAP spectrum, Bretchneider–Mitsuyasu spectrum and Pierson–Moscowitz spectrum are considered in the present model for the simulation of random waves. Numerical examples indicate that the irregular wave-induced pore-water pressure can randomly amplify the upward gradient of the pore-water pressure in the porous seabed, making the backfill sand layer more likely to be liquefied at a deeper depth. It is observed that the upper width of the trench layer becomes larger once the seabed liquefaction occurs, which can effectively prevent the seabed liquefied surface from penetrating into the bottom of the pipeline by increasing the backfill thickness to twice of the pipe diameter. The superimposition of the irregular waves on the following current has more significant effect on the maximum-development of the seabed liquefaction depth, which is particularly obvious when the sand-bed is in the random wave system by using the JONSWAP spectrum.

A numerical study on nonlinear surface wave evolution over viscoelastic mud

Accurate prediction of wave energy dissipation over mud is essential for understanding nearshore circulation, sediment transport processes, and design of engineering projects along muddy coasts. It is desirable to be able to simulate wave propagation over a wide range of mud behaviors as wave evolution is highly dependent on mud rheological characteristics. In this study, we develop a numerical model to simulate surface wave evolution over viscoelastic muds. This new wave-mud interaction model is comprised of a frequency-domain phase-resolving model for wave propagation that explicitly solves nonlinear wave-wave interactions, and a model for mud-induced surface wave dissipation and modulation. Model results show satisfactory agreement with laboratory measurements of wave height and attenuation rates over mud, and shows improvement over the model with a viscous mud mechanism. The model is then used to investigate the combined effect of mud viscoelasticity and nonlinear wave-wave interactions on surface wave evolution. Cnoidal and random wave simulations are conducted. In general, qualitative measures such as shape of cnoidal waves or pattern of variation in Hrms of random waves are dictated by direct mud-induced damping which due to resonance effects, has a substantially different structure over viscoelastic mud compared to viscous mud. Nonlinear interactions affect spectral shape and distribution of energy loss across the spectrum. Subharmonic interactions cause indirect damping of high frequencies but ameliorate damping of harmonics around mud’s resonance frequency. Therefore, neglecting mud elasticity can result in significant errors in estimation of bulk wave characteristics and spectral shape.

Parameterization of variations for wave energy over bottom slopes

Wave energy transfer through a shallow water region, produced by nonlinearity, was investigated using random wave simulations based on JONSWAP spectra in a physical experiment. To confirm the role of topography, slopes of β = 1/15, 1/30, and 1/45 were installed in a flume. The shoaling effect was first examined by comparing the experimental data with the theoretically predicted values. A wavelet-based bispectrum was then used to investigate the role of the bottom slope in energy transfer. Next, the evolution of energy transfer in the total, primary, and higher harmonic frequency bands was investigated using the wavelet-based energy spectrum. Based on the data from each slope, empirical formulas expressing the dimensionless energy parameters as a function of the local water depth were derived. The bottom slope was confirmed to have an influence on the variation in the total energy and primary harmonic energy, but its effect was shown to be negligible for the higher harmonic energy. Novel formulas were derived to represent the combined slope effect.

On quantitative errors of two simplified unsteady models for simulating unidirectional nonlinear random waves on large scale in deep sea

To investigate nonlinear random wave dynamics or statistics, direct phase-resolved numerical simulation of nonlinear random waves in deep sea on large-spatial and long-temporal scales are often performed by using simplified numerical models, such as those based on the Nonlinear Schrödinger Equation (NLSE). They are efficient and can give sufficiently acceptable results in many cases but they are derived by assuming narrow bandwidth and small steepness. So far, there has been no formula to precisely predict the quantitative errors of such simplified models. This paper will present such formulas for estimating the errors of enhanced NLSE based on the Fourier transform and quasi spectral boundary integral method when they are applied to simulate ocean waves on large-spatial and long-temporal scales (about 128 peak wavelengths and 1000 peak periods). These formulas are derived by fitting the errors of the simplified models, which are estimated by comparing their wave elevations with these obtained by using a fully nonlinear model for simulating the cases with initial conditions defined by two commonly used ocean wave spectra with a wide range of parameters. Based on them, the suitable regions for the simplified models to be used are shown.

Modification of endurance wave analysis based on New-wave theory

The main objective of this investigation is to modify endurance wave analysis (EWA) based on New-wave theory for convenient assessment of offshore structures under extreme waves. EWA is a novel time historybased approach to evaluate structural performance in various excitation levels of wave loading by intensifying wave records. In this article, modifications are proposed to determine appropriate time duration and intensifying trend for reliable and practical assessment of offshore structures. In this way, a simplified model of a real jacket platform is utilised to investigate the application of this approach under extreme wave conditions in the Persian Gulf and results are compared with typical three hour random wave simulations. It is shown that in extreme waves, except resonance states, proposed method can estimate the structural response with acceptable accuracy and very low computational costs.

Boussinesq modeling of longshore currents in the SandyDuck experiment under directional random wave conditions

To investigate surf zone hydrodynamics in a directional random wave environment, a numerical simulation of the SandyDuck experiment was performed for 2 October 1997 using the Boussinesq model FUNWAVE. This simulation utilized the field topography and the offshore spectrum constructed from observed datasets. The simulation results agreed well with the experimental data to reveal a wave height distribution of the random waves as well as the well-developed longshore current and its energetic fluctuation. The simulation also revealed that longshore variations were induced by not only the field topography but also the phase interaction of multidirectional random waves in the surf zone wavefield, and subsequently complicated the mixing process with fluctuations of longshore currents and rip currents in the nearshore current field. Based on the Boussinesq modeling, further simulations were performed using incident wave spectra that varied in the frequency and directional spreading. The simulation results confirmed that random waves with broader frequency-directional spectra tended to produce weaker mean longshore currents, which were associated with the early dissipation of relatively short wave components and the small moments of broad wave spectra. In addition, it was found that the energetic nearshore fluctuations, which involved complicated evolutions of vortices, did not vary significantly according to the frequency and directional spreading. However, the results of the monochromatic unidirectional wave revealed relatively weak and organized nearshore fluctuations (i.e., instabilities), even though this simulation also revealed a stronger longshore current with stronger wave breaking than did the random wave simulations.

Wave spectral modeling of multidirectional random waves in a harbor through combination of boundary integral of Helmholtz equation with Chebyshev point discretization

A mathematical model is presented to predict the wave spectrum in a complex geometry harbor domain due to the diffraction of multidirectional random waves propagating through entrance of the harbor from the open sea. An accurate description of the harbor geometry, bathymetry and an incident wave spectrum are required. The solution of Helmholtz equation in the bounded and unbounded region is obtained by 2-D boundary integral method with the consideration of partial reflection and corner point approximation. Moreover, the Chebyshev point discretization is applied on boundary of the harbor to improve the accuracy of the numerical scheme. The principle of superposition is adopted to simulate the directional random waves based on the solution of monochromatic incident waves. The form of Mitsuyasu’s frequency spectrum was utilized for the random wave simulation. The wave spectra for multidirectional random waves are analyzed at various recorder stations in the Pohang New Harbor (PNH). Comparison of simulation results with in situ measurement data recorded at various recorder observatories in the PNH demonstrates the feasibility and accuracy of our proposed method. After this study, we analyzed the wave height distribution in the whole PNH domain. Based on the simulation results, some tactic has been introduced to reduce the wave amplitude in the PNH. Our primary results confirm that present numerical model is accurate and flexible to implement on various real harbors to predict the multidirectional random wave diffraction.

Frequency/time domain modeling of a direct drive point absorber wave energy converter

Two different methods to model a point absorber wave energy converter (WEC) with direct drive linear power take-off (PTO) are proposed in the present study: the frequency domain (FD) method and the time domain (TD) method. In the FD analysis, the frequency response function (FRF) of the WEC device is obtained via the equation of motion, and the expressions of power capture width in regular and random waves are derived as well. In the TD modeling, based on a state space approximation of the convolution term in the motion equation, both regular wave and random wave simulations are carried out. The regular wave simulation results indicate that the state space approximation is sufficiently accurate and the capture width reaches the maximum in the vicinity of the natural frequency. In the random wave simulations, the effects of buoy size, the PTO damping and wave climate on the power capture width are discussed in detail, which leads to the conclusion that the capture widths are influenced by the natural frequency of the WEC device, peak frequency of the wave spectrum, the amplitude of FRF and PTO damping. Furthermore, the increase of the capture width is at the cost of a relatively large buoy size and PTO damping when control is not included.

Iteration Coupling Simulation of Random Waves and Wave‐Induced Currents

A two‐way coupling algorithm for wave‐current interaction is developed and implemented into a nearshore circulation model to investigate the effects of fully wave‐current interaction on irregular wave transformation over an elliptic shoal. The wave field is simulated by a spectral wave model WABED, and the wave‐induced current is solved by a quasi‐three‐dimensional model WINCM. The surface roller effects are represented in the formulation of surface stress, and the roller characteristics are solved by a roller evolution model. The proposed two‐way coupling algorithm can describe both the generation of wave‐induced current and the current‐induced wave transformation, which is more physically reasonable than the one‐way approaches. The model test with a laboratory experiment shows that wave‐induced currents have an important influence on the wave transformation, for example, the wave energy defocusing due to the strong jet‐like current along the centerline of the shoal. It is revealed that the accuracy of simulated wave field can be significantly improved by taking into account the two‐way wave‐current interaction.

Non-Gaussian Random Wave Simulation by Two-Dimensional Fourier Transform and Linear Oscillator Response to Morison Force

The one-dimensional fast Fourier transform (FFT) has been applied extensively to simulate Gaussian random wave elevations and water particle kinematics. The actual sea elevations/kinematics exhibit non-Gaussian characteristics that can be represented mathematically by a second-order random wave theory. The elevations/kinematics formulations contain frequency sum and difference terms that usually lead to expensive time-domain dynamic analyses of offshore structural responses. This study aims at a direct and efficient two-dimensional FFT algorithm for simulating the frequency sum terms. For the frequency-difference terms, inverse FFT and forward FFT are implemented, respectively, across the two dimensions of the wave interaction matrix. Given specified wave conditions, the statistics of simulated elevations/kinematics compare well with not only the empirical fits but also the analytical solutions based on a modified eigenvalue/eigenvector approach, while the computational effort of simulation is very economical. In addition, the stochastic analyses in both time domain and frequency domain show that, attributable to the second-order nonlinear wave effects, the near-surface Morison force and induced linear oscillator response are more non-Gaussian than those subjected to Gaussian random waves.

A three-dimensional modeling of multidirectional random wave diffraction by rectangular submarine pits

A three-dimensional modeling of multidirectional random-wave diffraction by a group of rectangular submarine pits is presented in this paper. The fluid domain is divided into N interior regions representing the pit area and an overall exterior region separated by the imaginary pit boundaries. In the interior region, the analytical expressions of the Fourier series expansion for velocity potentials in the pit regions have been derived with the unknown coefficients determined from a series of Green's function based boundary integral equations. The boundary integral approach has also been applied to obtain the velocity potential and free-surface elevation in the exterior region. The Pierson–Moskowitz (P–M) frequency spectrum was selected for the random wave simulation using the superposition of solutions of a finite number of decomposed wave components. Numerical results for the cases of regular waves and random waves are presented to examine the influences of the pit geometry and incident wave condition on the overall wave field. The general diffraction pattern of alternate bands of increase and decrease of relative wave height in front of the pit system can be observed. It is found that, in the shadow region, the relative wave height is reduced. As the number of pit increases, the effectiveness of reducing the relative wave height behind the multiple-pit system increases. However, the relative wave height within the pit area and in front of the leading pit shows increasing trend. It is noticed that under the random-wave condition, the level of increase and decrease of the relative wave height due to the existence of submarine pits is less pronounced than that observed from results in regular-wave condition.

フーチングを有するケーソン式防波堤に作用する波圧の算定式の検証

The present design procedures for the breakwater with footing ignore a wave pressure acting on the footing under the assumption that the compressive force on the upper face of the footing can be cancelled by the uplift force on the bottom face of the footing. This assumption is acceptable, but the experiments have confirmed that the compressive force becomes much lager for the case of long footing than the uplift force and works to stabilize the breakwater. We have proposed two empirical formulas for the distributions of uplift and compressive pressures. However, the validity of the formulas was not confirmed for random wave experiments. The present paper describes the confirmation of their validity thorough the experiments in random wave and numerical simulations. The empirical formulas are slightly improved.

Simulation of nonlinear random finite depth waves coupled with an elastic structure

Time accurate computation of random nonlinear shallow-water waves is required for the design of offshore wind turbines, and also in a multitude of other areas. Assuming the fluid to be inviscid and irrotational, nonlinear finite amplitude waves in a liquid body of finite depth are computed based on potential theory, i.e. the velocity potential must satisfy the Laplace equation at every instant in time. The free surface boundary conditions give a partial differential equation for the velocity potential and the free surface elevation on the free boundary. Similar coupling conditions are obtained at the boundary in contact with the deformable body. During the computation, the repeated solution of Laplace’s equation can be obtained by solving a boundary integral equation with the BEM and a fast multipole solver. The incident wave field is given by a stochastic wave process far from the structure, where the scattered (diffracted and radiated) wave field is comparatively small. This is finally coupled to another fluid description at the structure, which has a fully nonlinear description of both incident and scattered wave. The effect of the interacting flexible structure is included via a coupling computation. The chosen software architecture allows a modular coupling concept reflected in the algorithmically strong coupling of the various fluid domains, the structure, and the soil. As a numerical example, an offshore wind turbine is simulated, which has an additional fluid interaction with the stochastic wind field.

Fully dispersive nonlinear water wave model in curvilinear coordinates

A vertically integrated fully dispersive nonlinear wave model is expressed in curvilinear coordinates with non-orthogonal grids for the simulation of broad-banded nonlinear random water waves in regions of arbitrary geometry. The transformation is performed for both dependent and independent variables, hence an irregular physical domain is converted into a rectangular computational domain with contravariant velocities. Use of contravariant velocity components as dependent variables ensures easy and accurate satisfaction of the wall condition for lateral enclosures surrounding a physical domain, such as a coastal area, channel, or harbor. The numerical scheme is based on finite-difference approximations with staggered grids which results in implicit formulations for the momentum equations and a semi-explicit formulation for the continuity equation. Linear long wave propagation in a channel of varying cross-section and linear random wave propagation in a circular channel are presented as test cases for comparisons with the corresponding analytical solutions. Cnoidal and Stokes waves in a circular channel are also simulated as examples to nonlinear wave propagation within curved walls.

A Spectral Response Surface Method for Calculating Crest Elevation Statistics

The statistics of wave crest elevation in a random, directionally spread sea are calculated using a novel approach. The nonlinearity of steep waves is modelled to second order using Sharma and Dean kinematics and a spectral response surface method is used to deduce the crest elevation corresponding to a given probability of exceedance. The spectral response surface method works in the probability domain, making it several times faster than conventional time domain simulation of random waves. However, the results from the two methods show good agreement. As expected, nonlinearity makes extreme crests higher than the corresponding linear ones.

The noise spectral parameters and the energy of breaking waves experimental study

The process of wave breaking and whitecap creation is one of the most important and least understood phenomena associated with the evolution of the surface gravity waves in the open sea. This process is the main way of energy and momentum transfer between ocean and atmosphere. However, it is very difficult to estimate, under real sea conditions, the frequency of breaking wave events or the fraction of sea surface covered by whitecaps and the amount of dissipated energy produced by wave breaking. A controlled experiment was carried out in the Ocean Basin Laboratory at MARINTEK, Trondheim (Norway). The simulation of random waves of the prescribed spectra provided a very realistic pattern of the sea surface. The number of breaking waves was estimated using photography method and wave staff recording. Acoustic measurements during the experiments were conducted in order to examine the relationship between the noise spectral parameters and both the whitecap coverage and dissipation energy of breaking waves for different types of waves. A comparison of simultaneous video observations, wave staff records of the surface wave above the hydrophones, with the spectral parameters of acoustical signals made it possible to find physical links between processes.