Three-dimensional linear solution for wave propagation with slopingbottom

This paper presents a three-dimensional analytic linear wave solution for surface gravity wave propagation over a sloping bottom that is valid for small, but realistic, slopes. The sloping-bottom linear model is compared to published laboratory data and to predictions of two-dimensional, constant-bottom nonlinear theories. The model is shown to describe the measured wave-height growth in the wave transformation region up to a limiting local Ursell number U_r of 0.35-1.0, depending on the wave type, although, as a linear model, it does not predict the harmonics observed in that range. For U_r<0.35, the harmonics can generally be neglected and the sloping-bottom linear theory agrees closely with both the published wave-height data and third-order Stokes nonlinear theory. As a three-dimensional linear model, superposition can be invoked to synthesize and relate wave structure in the transformation region to complex incident ocean spectra with both wind wave and swell components that arrive with a range of incidence angles. As such, the sloping-bottom linear model presented here should be a convenient useful tool for ocean modeling through a significant portion of the wave transformation region     

Comparison of linear and nonlinear extreme wave statistics

An extremely large (“freak”) wave is a typical though rare phenomenon observed in the sea. Special theories (for example, the modulation instability theory) were developed to explain mechanics and appearance of freak waves as a result of nonlinear wave-wave interactions. In this paper, it is demonstrated that the freak wave appearance can be also explained by superposition of linear modes with the realistic spectrum. The integral probability of trough-to-crest waves is calculated by two methods: the first one is based on the results of the numerical simulation of a wave field evolution performed with one-dimensional and two-dimensional nonlinear models. The second method is based on calculation of the same probability over the ensembles of wave fields constructed as a superposition of linear waves with random phases and the spectrum similar to that used in the nonlinear simulations. It is shown that the integral probabilities for nonlinear and linear cases are of the same order of values     

A study of radiation stress and set-up in the nearshore region

In this paper the variations of radiation stress and mean water level are studied for the two-dimensional shoaling and breaking of progressive, periodic waves on a plane, gently sloping laboratory beach. Observations of radiation stress and mean water level were made for two initial conditions. The observed variations are compared to the variations as calculated by a linear theory, James' nonlinear theory, and a new nonlinear theory based on Cokelet's work. It is shown that the nonlinear theories are qualitatively and quantitatively superior to the linear theory, thus indicating that the effects of finite amplitude are important for the present case.     

Direct time domain analysis of floating structures with linear and nonlinear mooring stiffness in a 3D numerical wave tank

In this paper, motion response of a moored floating structure interacting with a large amplitude and steep incident wave field is studied using a coupled time domain solution scheme. Solution of the hydrodynamic boundary value problem is achieved using a three-dimensional numerical wave tank (3D NWT) approach based upon a form of Mixed-Eulerian–Lagrangian (MEL) scheme. In the developed method, nonlinearity arising due to incident wave as well as nonlinear hydrostatics is completely captured while the hydrodynamic interactions of radiation and diffraction are determined at every time step based on certain simplifying approximations. Mooring lines are modelled as linear as well as nonlinear springs. The horizontal tension for each individual mooring line is obtained from the nonlinear load-excursion plot of the lines computed using catenary theory, from which the linear and nonlinear line stiffness are determined. Motions of three realistic floating structures with different mooring systems are analyzed considering various combinations of linear and approximate nonlinear hydrodynamic load computations and linear/nonlinear mooring line stiffness. Results are discussed to bring out the influence and need for consideration of nonlinearities in the hydrodynamics and hydrostatics as well as the nonlinear modelling of the line stiffness.     

Consideration of fine-scale coastal oceanography and 3-D acoustics effects for the ESME sound exposure model

Results and recommendations for evaluating the effects of fine-scale oceanographic scattering and three-dimensional (3-D) acoustic propagation variability on the Effects of Sound on the Marine Environment (ESME) acoustic exposure model are presented. Pertinent acoustic scattering theory is briefly reviewed and ocean sound-speed fluctuation models are discussed. Particular attention is given to the nonlinear and linear components of the ocean internal wave field as a source of sound-speed inhomogeneities. Sound scattering through the mainly isotropic linear internal wave field is presented and new results relating to acoustic scattering by the nonlinear internal wave field in both along and across internal wave wavefront orientations are examined. In many cases, there are noteworthy fine-scale induced intensity biases and fluctuations of order 5-20 dB.     

Finite element analysis of two-dimensional nonlinear sloshing problems in random excitations

A two-dimensional nonlinear random sloshing problem is analyzed by the fully nonlinear wave velocity potential theory based on the finite element method. A rectangular container filled with liquid subjected to specified horizontal random oscillations is studied. Both wave elevation and hydrodynamic force are obtained. The spectra of random waves and forces have also been investigated, and the effects of the peak frequencies and spectral width of the specified spectrum used for the generation of the random oscillations are discussed. It is found that the energy mainly concentrates at the natural frequencies of the container and is dominant at the ith order natural frequency when the peak frequency is close to the ith order natural frequency. Some results are compared between the fully nonlinear solutions, the linear solutions and the linear plus second-order solutions.     

On the nonlinear hydrodynamic forces for a ship advancing in waves

In this paper, using a second-order steady-state approach and a three-dimensional (3D) pulsating source distribution method derives the nonlinear hydrodynamic forces on a ship advancing in waves. The nonlinear hydrodynamic forces considered here consist of the mean lateral drifting force and the added resistance, which can be expressed as products of the ship-motion responses, the radiation potential, diffraction potential and the incident-wave potential. All related velocity potentials applied in the calculations are in 3D form. The Series 60 and Marine ship hulls are used for numerical calculations and the results are compared with existing experimental data and two-dimensional (2D) solutions. The comparisons show that the results obtained in the paper generally agree with experimental data well. It is also found that the nonlinear hydrodynamic forces obtained based on the present 3D source distribution methods are indeed improved in some calculations compared with the 2D method, especially for the mean lateral drifting force.     

Monthly variation of some parameters about internal solitary waves in the South China sea

In this paper, by non-dimensional analysis, it is found that finite-depth theory is more appropriate to the study of internal solitary waves (ISWs) in the South China Sea (SCS) than shallow-water theory. The 1-degree grid data of monthly mean temperature and salinity data at standard levels in the SCS are used to solve the linearized vertical eigenvalue problem. The nonlinear parameter and the wave phase speed are computed, then the nonlinear phase speed and the characteristic half-width of ISWs are calculated respectively by two different theories to investigate the difference between these two parameters in the SCS. The nonlinearity is the strongest near the continental slope of the SCS or islands where the bottom topography changes sharply, it is stronger in summer than that in winter; it increases (decreases) as pycnocline depth deepens (shallows), stratification strengthens (weakens) and pycnocline thickness thins (thickens). The nonlinear wave phase speed and the characteristic half-width are the largest in deep sea area, they then reduce peripherally in shallower water. The nonlinear wave phase speed in the SCS changes slightly with time, but the characteristic half-width changes somewhat larger with time. In most of the SCS basin, the nonlinear wave phase speed derived from shallow-water theory is very close to that derived from finite-depth theory, but the characteristic half-width derived from shallow-water theory is about 0.2–0.6 times larger than that derived from finite-depth theory. The ISW induced horizontal current velocity derived from shallow-water theory is larger than that derived from finite-depth theory. Some observed and numerical modeled ISW characteristic half-widths are compared with those derived from shallow-water and finite-depth theories, respectively. It is shown that, the characteristic half-widths derived from finite-depth theory agree better with observational and numerical modeled results than those derived from shallow-water theory in most cases, finite-depth theory is more applicable to the estimation of ISW characteristic half-widths in the northern SCS. It is also suggested that, to derive the precise ISW parameters in further study, the physical non-dimensional ratios which are related with ISW characteristic half-width, amplitude, thermocline and water depths should be calculated, so that an appropriate theory can be chosen for estimation.     

Finite Element Analysis to Two-Dimensional Nonlinear Sloshing Problems

A two-dimensional nonlinear sloshing problem is analyzed by means of the fully nonlinear theory and time domain second order theory of water waves. Liquid sloshing in a rectangular container subjected to a horizontal excitation is sim-ulated by the finite element method. Comparisons between the two theories are made based on their numerical results, It is found that good agreement is obtained for the case of small amplitude oscillation and obvious differences occur for large amplitude excitation. Even though, the second order solution can still exhibit typical nonlinear features of nonlinear wave and can be used instead of the fully nonlinear theory.     

Probability distribution of surface slope for nonlinear random sea waves and calculation of whitecap coverage

Based on the nonlinear model of two-dimensional random sea waves, a statistical distribution of wave surface slope exact to the third order is derived by using the expansion of the characteristic function and direct calculations of each order moment. Based on the distribution of wave surface slope derived in this paper, a whitecap coverage is proposed by using the limit surface slope as a criterion of wave breaking. The whitecap coverage expressed by the model depends on three parameters which can be determined in principle by the linear wave spectrum and three kinds of wave-wave interaction.     

Nonlinear wave interaction with a vertical circular cylinder: wave forces

Second-order wave forces on a large diameter vertical circular cylinder, computed according to a semi-analytic nonlinear diffraction theory, are compared to results of 22 laboratory experiments with regular waves. In general, predicted forces agree quite well with measured forces. In most tests, both measured and predicted maximum forces exceeded linear theory by 5 to 15%. In a few cases, however, the measured forces were less than those predicted by linear theory, in contrast to the second-order predictions. It is shown that these results are related to the phasing of various linear and nonlinear wave force components, and are consistent with those obtained by other investigators.     

关于Boussinesq型水波方程理论和应用研究的综述

Boussinesq型方程是研究水波传播与演化问题的重要工具之一,本文就1967-2018年常用的Boussinesq型水波方程从理论推导和数值应用两个方面进行了回顾,以期推动该类方程在海岸(海洋)工程波浪水动力方向的深入研究和应用。此类方程推导主要从欧拉方程或Laplace方程出发。在一定的非线性和缓坡假设等条件下,国内外学者建立了多个Boussinesq型水波方程,并以Stokes波的相关理论为依据,考察了这些方程在相速度、群速度、线性变浅梯度、二阶非线性、三阶非线性、波幅离散、速度沿水深分布以及和(差)频等多方面性能的精度。将Boussinesq型水波方程分为水平二维和三维两大类,并对主要Boussinesq型水波方程的特性进行了评述。进而又对适合渗透地形和存在流体分层情况下的Boussinesq型水波方程进行了简述与评论。最后对这些方程的应用进行了总结与分析。     

Incorporating irregular nonlinear waves in coupled simulation and reliability studies of offshore wind turbines

Design of an offshore wind turbine requires estimation of loads on its rotor, tower and supporting structure. These loads are obtained by time-domain simulations of the coupled aero-servo-hydro-elastic model of the wind turbine. Accuracy of predicted loads depends on assumptions made in the simulation models employed, both for the turbine and for the input wind and wave conditions. Currently, waves are simulated using a linear irregular wave theory that is not appropriate for nonlinear waves, which are even more pronounced in shallow water depths where wind farms are typically sited. The present study investigates the use of irregular nonlinear (second-order) waves for estimating loads on the support structure (monopile) of an offshore wind turbine. We present the theory for the irregular nonlinear model and incorporate it in the commonly used wind turbine simulation software, FAST, which had been developed by National Renewable Energy Laboratory (NREL), but which had the modeling capability only for irregular linear waves. We use an efficient algorithm for computation of nonlinear wave elevation and kinematics, so that a large number of time-domain simulations, which are required for prediction of long-term loads using statistical extrapolation, can easily be performed. To illustrate the influence of the alternative wave models, we compute loads at the base of the monopile of the NREL 5MW baseline wind turbine model using linear and nonlinear irregular wave models. We show that for a given environmental condition (i.e., the mean wind speed and the significant wave height), extreme loads are larger when computed using the nonlinear wave model. We finally compute long-term loads, which are required for a design load case according to the International Electrotechnical Commission guidelines, using the inverse first-order reliability method. We discuss a convergence criteria that may be used to predict accurate 20-year loads and discuss wind versus wave dominance in the load prediction. We show that 20-year long-term loads can be significantly higher when the nonlinear wave model is used.     

Nonlinear effect on inertia component of wave forces on a cylinder

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Numerical Simulation of Waves Generated by Seafloor Movements

Waves generated by vertical seafloor movements are simulated by use of a fully nonlinear two-dimensional numerical wave tank. In the souree region, the seafloor lifts to a designated height by a generation function. The numerical tests show that the linear theory is only valid for estimating the wave behaviors induced by the seafloor movements with a small amplitude, and the fully nonlinear numerical model should be adopted in the simulation of the wave generation by the large amplitude seafloor movements. Without the background surface waves, many numerical tests on the stable maximum elevations η0^max are carried out by beth the linear theory and the fully nonlinear model. The results of two models are compared and analyzed. For the fully nonlinear model, the influences of the amplitudes and the horizontal lengths on η^max are stronger than that of the characteristic duration times. Furthermore, results reveal that there are significant differences be- tween the linear theory and the fully nonlinear model. When the influences of the background surface waves are considered, the corresponding numerical analyses reveal that with the fully nonlinear model the η0^max near-linearly varies with the wave amplitudes of the surface waves, and the η0^max has significant dependences on the wave lengths and the wave phases of the surface waves. In addition, the differences between the linear theory and the fully nonlinear model are still obvious, and these differences are significantly affected by the wave parameters of the background surface waves, such as the wave amplitude, the wave length and the wave phase.     

Vertical structure of the quasi-two-dimensional velocity field of a viscous incompressible flow and the problem of nonlinear friction

An approximate theory is constructed to describe quasi-two-dimensional viscous incompressible flows. This theory takes into account a weak circulation in the vertical plane and the related divergence of the two-dimensional velocity field. The role of the nonlinear terms that are due to the interaction between the vortex and potential components of velocity and the possibility of taking into account the corresponding effects in the context of the concept of bottom friction are analyzed. It is shown that the nonlinear character of friction is a consequence of the three-dimensional character of flow, which results in the effective interaction of vortices with vertical and horizontal axes. An approximation of the effect of this interaction in quasi-two-dimensional equations is obtained with the use of the coefficient of nonlinear friction. The results based on this approximation are compared to the data of laboratory experiments on the excitation of a spatially periodic fluid flow.     

Interactions between fully nonlinear water waves and cylinder arrays in a wave tank

Fully nonlinear interactions between water waves and vertical cylinder arrays in a numerical tank are studied based on a finite element method (FEM). The three-dimensional (3D) mesh is constructed through an extension of a 2D Delaunay surface grid along the vertical line. The velocity potential is obtained by solving a linear matrix system of FEM, and a difference scheme is then used to calculate the velocity on the free surface to track its movement. Waves and hydrodynamic forces are obtained for both bottom mounted and truncated cylinders. The simulations have provided many results to show the nature of mutual interference between cylinders in arrays and its effects on waves and forces at the nearly trapped mode frequency. The effect of the tank wall on waves and forces has been investigated, and the nonlinear features of waves and forces have also been discussed.     

Generation and propagation of transient nonlinear waves in a wave flume

A semi-analytical nonlinear wavemaker model is derived to predict the generation and propagation of transient nonlinear waves in a wave flume. The solution is very efficient and is achieved by applying eigenfunction expansions and FFT. The model is applied to study the effect of the wavemaker and its motion on the generation and propagation of nonlinear waves. The results indicate that the linear wavemaker theory may be applied to predict only the generation of waves of low steepness for which the nonlinear terms in the kinematic wavemaker boundary condition and free-surface boundary conditions are of secondary importance. For waves of moderate steepness and steep waves these nonlinear terms have substantial effects on wave profile and wave spectrum just after the wavemaker. A wave spectrum corresponding to a sinusoidally moving wavemaker possesses a multi-peak form with substantial nonlinear components, which disturbs or may even exclude physical modeling in wave flumes. The analysis shows that the widely recognized weakly nonlinear wavemaker theory may only be applied to describe the generation and propagation of waves of low steepness. This is subject to further restrictions in shallow and deep waters because the kinematic wavemaker boundary condition as well as the nonlinear interaction of wave components and the evolution of wave energy spectrum is not properly described by weakly nonlinear wavemaker theory. Laboratory experiments were conducted in a wave flume to verify the nonlinear wavemaker model. The comparisons show a reasonable agreement between predicted and measured free-surface elevation and the corresponding amplitudes of Fourier series. A reasonable agreement between theoretical results and experimental data is observed even for fairly steep waves.     

Oceanic thermohaline intrusions: theory

This is a review of theories governing growth and evolution of thermohaline intrusive motions. We discuss theories based on eddy coefficients and salt finger flux ratios and also on molecular Fickian diffusion, drawing relationships and parallels where possible. We discuss linear theories of various physical configurations, effects of rotation and shear, and nonlinear theories. A key requirement for such theories to become quantitatively correct is the development and field testing of relationships between double-diffusive fluxes and average vertical gradients of temperature and salinity. While we have some ideas about the functional dependencies and rough observational constraints on the magnitudes of such flux/gradient relationships, many questions will not be answered until usable ‘flux laws’ exist. Furthermore, numerical experiments on double-diffusive intrusions are currently feasible, but will have more quantitative meaning when fluxes are parameterised with such laws. We conclude that more work needs to be done in at least two areas. Firstly, tests of linear theory against observations should continue, particularly to discover the extent to which linear theories actually explain the genesis of intrusions. Secondly, theoretical studies are needed on the nonlinear effects that control the evolution and finite amplitude state of intrusions, since these determine the lateral fluxes of salt, heat, and momentum.     

Tsunami simulation with Green–Naghdi theory

Green–Naghdi (GN) theory is a fully nonlinear wave theory which has been used with success to simulate nonlinear water waves. In previous applications of GN theory to water wave problems the ocean bottom was assumed to be time invariant. In this work no such restriction is made and GN theory is used to simulate tsunami caused by bottom fluctuation. As first test cases we simulate two-dimensional nonlinear surface waves generated by positive bottom movements. The results in the generation region for three different seabed movements compare well against earlier experimental data. The results in the downstream region for impulsive seabed movements show some discrepancies in wave phase and amplitude compared with earlier experimental values. It is suspected that the viscous effects may have played a role. The GN theory is then used to study three-dimensional near-field tsunami amplitudes caused by submarine landslides and slumps spreading in two orthogonal directions. The GN results agree with previous linear solution very well when the ratio of the velocities is v₁/v₂=1.0. But GN theory give more believable results for the case of v_T/v=0.1 and v₁/v₂=0.1.