Hydrodynamic coefficients of a submerged pulsating sphere in finite depth

By extending the work of Linton (Linton, C.M., 1991. Radiation and diffraction of waver waves by a submerged sphere in finite depth. Ocean Engineering 18 (1/2), 61–74), the problem of radiation of water waves by a submerged pulsating sphere in finite depth is formulated using the multipole method. As in Linton (1991), this leads to an infinite system of linear equations, which are easily solved numerically. Simple expressions are derived for the hydrodynamic characteristics of such a body. Results showing the effect of varying both the immersion depth and the water depth on the hydrodynamic coefficients of the pulsating sphere are given. The paper resumes the work presented in Lopes (Lopes, D.B.S., 1999. On the study of the Archimedes wave swing device for wave energy utilization (in Portuguese). MSc on the Management and Modelling of the Marine Environment, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa.).     

A new approximation for ocean waves propagating over a beach with variable depth

In this paper, the phenomenon of ocean waves propagating over a beach with variable water depth is re-examined. Based on the assumption of shallow water, a linearised shallow water equation is solved with an arbitrary beach profile. These irregular beach profiles form a set of partial differential equation with variable coefficient as the governing equation, which is the main obstacle in obtaining analytical solutions. In this paper, two families of beach profile are used as examples. A parametric study is conducted to investigate the influence of the beach profiles on the water surface elevation (η) and velocities (u).     

On the growth of ocean waves

The availability of 10 h of continuous, uninterrupted field measurements of wind waves recorded in the western Pacific and containing a complete wave growth episode, has provided a distinct opportunity for us to make a novel, unprecedented examination of detailed wave growth processes. We found that the significance of the size of data used in the measurement, which can only be addressed with continuous and uninterrupted measurements, reflected the ineptness of the conventional approach toward further detailed understanding of realistic wave growth processes, as the conventional 20 min data size essentially stamped out any dynamics with time scale below 20 min. While our conventional understanding and modeling were generally operative and useful, they left no real vestige on time localized mechanisms such as wave grouping or wave breaking processes all with time scales much less than 20 min.     

Experimental investigation of the harmonic generation by waves over a submerged plate

Experiments in a wave flume have been performed to analyse the nonlinear interaction between regular gravity waves and a submerged horizontal plate used as breakwater. A new method, based on the Doppler shift generated by a moving probes, has been used to discriminate the incident fundamental mode and the reflected fundamental mode. The relationships of the reflection and transmission coefficients to the wave number at different submergence depth ratios are presented. The accurate discrimination, by this method, of the phase-locked and free modes allows the quantification of the higher harmonics generated by the breakwater and the analysis of the nonlinear interaction between the waves and the submerged plate. The transfer of energy from the fundamental mode to higher harmonics is very large in the cases of small submergence depth ratios. The vortices produced at the edges take part in the production of higher harmonics by interaction with the free surface but involve, at the same time, a dissipation process that increases the efficiency of the breakwater.     

Modelling ocean waves in ice-covered seas

Over the past decade there has been a rapid growth of interest in wave propagation through ice covers. This paper summarizes the author’s observation of the modeling efforts on this topic. Models can be theory-based, data-driven, or a combination of the two. A pure data-driven model relies on a large amount of observations and is only becoming available recently. Theory-based models on the other hand have a long history. They are always a simplified version of the reality. As our knowledge grows, theories become more complicated. A theory for waves-in-ice that captures all possible processes does not exist. However, when integrated with observation through calibration, these combined theory + data-based models may be used with some confidence. In this paper, different models, their basic concepts, their calibration and validation are discussed. The present theory-based models do not have the correct spectral attenuation trend as observed from field or laboratory experiments. Hence, through calibration they may fit different parts of the wave spectra but not all. Pure data-driven models can reproduce the correct trend, but its dependability outside the situation where the data are collected is uncertain. In addition to offering tools to forecast waves-in-ice, these model building and validating efforts point to missing mechanisms that should be carefully studied. Despite the many challenges towards building a satisfactory general waves-in-ice model, significant progress has been made for models that work reasonably well in the marginal ice zone. We anticipate much more data will become available in the coming years to help us improve the existing models.     

Reflection of surface waves by submerged cylinders

Reflection from submerged cylinders are studied by means of integral equations. By expressing the solution as a distribution of vortices, the integral equations become non-singular for closed contours. It is shown that the method gives a short and easy proof for the classical result that no reflection occurs for the circular cylinder. The reflection power for the elliptic contour and the flat plate are studied when the bodies are situated deeply below the surface.     

Transient dynamic response of submerged sphere shell with an opening subjected to underwater explosion

This investigation applies the time domain FEM/DAA coupling procedure to predict the transient dynamic response of submerged sphere shell with an opening subjected to underwater explosions. The elastic–plastic material behavior of the transient fluid–structure interaction relate to structural response equation also presented herein. This analysis also examines the transient responses of structures to different charge distances. The effects of standoff distance on pressure time history of the shell to underwater explosion (3001b TNT) are presented. Additionally, the transient dynamic responses to underwater explosion shockwaves in the sea and the air are compared.     

Experimental study of strong reflection of regular water waves over submerged breakwaters in tandem

A series of laboratory experiments was carried out to investigate the strong reflection of regular water waves over a train of submerged breakwaters. Rectangular and trapezoidal shapes of submerged breakwaters are employed and compared for reflecting capability of incident waves. Measured reflection coefficients of regular waves over impermeable submerged breakwaters are verified by comparing with those of the eigenfunction expansion method. A very good agreement is observed. Reflection coefficients of permeable submerged breakwaters are less than those of impermeable breakwaters. The trapezoidal shape is recommended for a submerged breakwater in terms of reflecting capability and practical application.     

Radiation and diffraction of linear water waves by an infinitely long submerged rectangular structure parallel to a vertical wall

The radiation and the diffraction of linear water waves by an infinitely long floating rectangular structure submerged in water of finite depth with leeward boundary being a vertical wall are analyzed in this paper by using the method of separation of variables. Analytical expressions for the radiated and diffracted potentials are derived as infinite series with unknown coefficients determined by the eigenfunction expansion matching method. The expressions for wave forces and hydrodynamic coefficients are given. A comparison is made between the results obtained by the present analytical solution and those obtained by the boundary element method. By using the present analytical solution, the hydrodynamic influences of the submergence, the width, the thickness of the structure, and the distance between the structure and the wall on the wave forces and hydrodynamic coefficients are discussed in detail.     

Analysis of wave induced drift forces acting on a submerged sphere in finite water depth

A solution is presented for the wave induced drift forces acting on a submerged sphere in a finite water depth based on linearised velocity potential theory. In order to obtain the velocity potential, use has been made of multipole expansions in terms of an infinite series of Legendre functions with unknown coefficients. The series expression for the second order mean forces (drift forces) is provided by integrating the fluid pressure over the body surface. The horizontal drift force is also expressed by a series solution obtained using the far-field method.     

Interaction mechanisms among waves,currents and a submerged plate

Wave-induced loads on a submerged plate, representative of submerged breakwater, coastal-bridge deck and a certain type of wave energy converter, in a uniform current are investigated in this study using fully nonlinear numerical wave tanks (NWTs) based on potential flow theory. The coupling effect of wave and current is explored, and the underlying interaction mechanisms of the hydrodynamic forces are described. The presence of a background current modifies the frequency dispersion. It produces changes of the water-surface elevation, and also has an effect on wave-induced loads. Depending on the nonlinearity, higher harmonic wave components are generated above the submerged plate. These contribute to the wave forces. It is found that the horizontal and the vertical force, hence the moment, are affected in the opposite way by the currents. The Doppler shifted effect dominates the vertical force and the moment on the plate. Whereas, the Doppler shifted effect and the generation of higher wave harmonics play opposite roles on the horizontal forces. The contribution of 2^nd order harmonics is found to be up to 30% of the linear component. The current-induced drag force, represented by the advection term ρU∂φ/∂x in the pressure equation, is found to lead to a decrease in the moment for the most range of wavelengths considered, and an increase in the moment for a small range of longer waves.     

A finite element model of interaction between viscous free surface waves and submerged cylinders

This paper describes the simulation of the flow of a viscous incompressible Newtonian liquid with a free surface. The Navier–Stokes equations are formulated using a streamline upwind Petrov–Galerkin scheme, and solved on a Q-tree-based finite element mesh that adapts to the moving free surface of the liquid. Special attention is given to fitting the mesh correctly to the free surface and solid wall boundaries. Fully non-linear free surface boundary conditions are implemented. Test cases include sloshing free surface motions in a rectangular tank and progressive waves over submerged cylinders.     

Hydroelastic response of a circular plate in waves on a two-layer fluid of finite depth

The hydroelastic response of a circular, very large floating structure （VLFS）, idealized as a floating circular elastic thin plate, is investigated for the case of time-harmonic incident waves of the surface and interfacial wave modes, of a given wave frequency, on a two-layer fluid of finite and constant depth. In linear potential-flow theory, with the aid of angular eigenfunction expansions, the diffraction potentials can be expressed by the Bessel functions. A system of simultaneous equations is derived by matching the velocity and the pressure between the open-water and the plate-covered regions, while incorporating the edge conditions of the plate. Then the complex nested series are simplified by utilizing the orthogonality of the vertical eigenfunctions in the open-water region. Numerical computations are presentedto investigate the effects of different physical quantities, such as the thickness of the plate, Young＇s modulus, the ratios ofthe densities and of the layer depths, on the dispersion relations of the flexural-gravity waves for the two-layer fluid.Rapid convergence of the method is observed, but is slower at higher wave frequency. At high frequency, it is found that there is some energy transferred from the interfacial mode to the surface mode.     

Nonlinear progressive waves in water of finite depth — an analytic approximation

An analytical solution using homotopy analysis method is developed to describe the nonlinear progressive waves in water of finite depth. The velocity potential of the wave is expressed by Fourier series and the nonlinear free surface boundary conditions are satisfied by continuous mapping. Unlike the perturbation method, the present approach is not dependent on small parameters. Thus solutions are possible for steep waves. Furthermore, a significant improvement of the convergence rate and region is achieved by applying Homotopy-Padé Approximants. The calculated wave characteristics of the present solution agree well with previous numerical and experimental results.     

Effect of a submerged plate on the near-bed dynamics underincoming waves in deep water conditions

It is well known that wave induced bottom oscillations become more and more negligible when the water depth exceeds half the wavelength of the surface gravity wave. However, it was experimentally demonstrated for regular waves that the bottom pressure oscillations at both first and second wave harmonic frequencies could be significant even for incoming waves propagating in deep water condition in the presence of a submerged plate [16]. For a water depth h of about the wavelength of the wave, measurements under the plate (depth immersion of top of plate h/6, length h/2) have shown bottom pressure variations at the wave frequency, up to thirty times larger than the pressure expected in the absence of the plate. In this paper, not only regular but also irregular wave are studied together with wave following current conditions. This behavior is numerically verified by use of a classical linear theory of waves. The wave bottom effect is explained through the role of evanescent modes and horizontally oscillating water column under the plate which still exist whatever the water depth. Such a model, which allows the calculation of the velocity fields, has shown that not only the bottom pressure but also the near bed fluid velocity are enhanced. Two maxima are observed on both sides of the location of the plate, at a distance of the plate increasing with the water depth. The possible impact of such near bed dynamics is then discussed for field conditions thanks to a scaling based on a Froude similarity. It is demonstrated that these structures may have a significant impact at the sea bed even in very deep water conditions, possibly enhanced in the presence of current.     

Experimental study of the breaking limit of multi-directional random waves passing over an impermeable submerged breakwater

This study investigates experimentally the breaking wave height of multi-directional random waves passing over an impermeable submerged breakwater. Experiments have been conducted in a three-dimensional wave basin equipped with a multi-directional random wave generator. A special type of wave gauge has been newly devised to record the water surface elevations in the breaker zone as accurately as possible. The records are analyzed to estimate the location and limit of wave breaking. Comparisons have also been made with the results of regular waves. The influence of the incident wave conditions on the breaking wave height normalized by the breakwater dimensions has been investigated. Empirical formulae have been presented to estimate the breaking limit of multi-directional random waves based on the experimental records. The formulae have been tested and found to work well not only for multi-directional random waves, but for regular waves as well.     

海浪感应电磁场的理论计算

根据麦克斯韦电磁理论,在地磁场中运动的海水将产生感应电磁场。利用一个简单的数学物理模型对感应电磁场在海水内部的分布进行了计算,结果表明海浪产生的电磁场明显依赖于海浪波动的周期及浪高。在100 m的海水深度内,海浪产生的磁感应强度的大小为纳特数量级,而电场强度的大小为几个微伏每米。在同一海水深度处,磁感应强度随海水波动的周期呈现近线性变化,而电场强度的大小有一个极值,该极值随海水深度的增加向长周期方向移动。海浪产生的电磁场是影响海洋电磁探测数据精度的主要噪声之一。     

Reflection and diffraction of internal solitary waves by a circular island

We have investigated the reflection and diffraction of first-mode and second-mode solitary waves by an island, using a three-dimensional nonhydrostatic numerical model. The model domain consists of a circular island 15 km in diameter in an ocean 300 m deep. We use prescribed density anomalies in an initially motionless ocean to produce highly energetic internal solitary waves; their subsequent propagation is subject to island perturbations with and without the effect of earth’s rotation. In addition to reflected waves, two wave branches pass around the island and reconnect behind it. Island perturbations to the first-mode and second-mode waves are qualitatively similar, but the latter is more profound because of the longer contact time and, in the presence of earth’s rotation, the scale compatibility between Rossby radius of the second baroclinic mode and the island diameter. Without earth’s rotation, reflected and diffracted waves are symmetrical relative to the longitudinal axis passing through the island center. With earth’s rotation, the current following the wave front veers to the right due to Coriolis deflection. For a westward propagating incoming wave, the deflection favors northward wave propagation in the region between the crossover point and the island, shifting the wave reconnection point behind the island northward. It also displaces the most visible part of the reflected waves to the southeast. In the presence of earth’s rotation, a second-mode incoming wave produces island-trapped internal Kelvin waves, which are visible after the passage of the wave front.     

Evolution of waves and currents over a submerged laboratory shoal

The vertically-integrated effect of interaction between waves and wave-induced currents on wave transformation over a submerged elliptic shoal was investigated based on numerical simulations of the Vincent and Briggs experiment [Vincent, C.L., Briggs, M.J., 1989. Refraction- diffraction of irregular waves over a mound. Journal of Waterway, Port, Coastal and Ocean Engineering, 115(2), pp. 269–284.]. The numerical simulations were performed using two numerical wave-current model systems: one, a combination of the wave model SWAN and the current model SHORECIRC, and the other, a combination of the wave model REF/DIF and the same current model. A time-dependent, phase-resolving wave and current model, FUNWAVE, was also utilized to simulate the experiment. In the simulations, the developed wave-induced currents defocused waves behind the shoal and brought on a wave shadow zone that showed relatively low wave height distributions. For the breaking case of monochromatic waves, the wave heights computed using FUNWAVE showed good agreement with the measurements and the resulting wave-induced currents showed a jet-like velocity distribution in transverse direction. And the computed results of the two model combinations agreed better with the measurements than the computed results obtained by neglecting wave-current interaction. However, it was found that for the case in which transverse interference pattern caused by refracted waves was strong, REF/DIF-SHORECIRC did not correctly evaluate radiation stresses, the gradients of which generate wave-induced currents. SWAN-SHORECIRC, which cannot deal with the interference patterns, predicted a jet-like wave-induced current. For breaking random wave cases, the computed results of the two model combinations and FUNWAVE agreed well with the measurements. The agreements indicate that it is necessary to take into account the effect of wave-induced current on wave refraction when wave breaking occurs over a submerged shoal.     

Statistical distribution of wave-surface elevation for second-order random directional ocean waves in finite water depth

Based on the second-order random wave solutions of water wave equations in finite water depth, a statistical distribution of the wave-surface elevation is derived by using the characteristic function expansion method. It is found that the distribution, after normalization of the wave-surface elevation, depends only on two parameters. One parameter describes the small mean bias of the surface produced by the second-order wave-wave interactions. Another one is approximately proportional to the skewness of the distribution. Both of these two parameters can be determined by the water depth and the wave-number spectrum of ocean waves. As an illustrative example, we consider a fully developed wind-generated sea and the parameters are calculated for various wind speeds and water depths by using Donelan and Pierson spectrum. It is also found that, for deep water, the dimensionless distribution reduces to the third-order Gram–Charlier series obtained by Longuet-Higgins [J. Fluid Mech. 17 (1963) 459]. The newly proposed distribution is compared with the data of Bitner [Appl. Ocean Res. 2 (1980) 63], Gaussian distribution and the fourth-order Gram–Charlier series, and found our distribution gives a more reasonable fit to the data.