Maximum theoretical power absorption of connected floating bodies under motion constraints

In this paper, a mathematical model is presented to evaluate the maximum mean power that can be absorbed by a three-dimensional system of connected floating bodies in waves under a weighted global constraint. The constraint imposed on the motion amplitudes of the system can be used to limit the motions of the bodies in order to make sure that the assumption of linear theory for wave–structure interaction remains valid. The absorbed power of a structure can be considered as the difference between excitation and radiated power without consideration of practical power take off. The solutions for the maximum relative capture widths of rigidly connected two rafts have a good agreement with the calculated ones for the same rafts being seen as a single whole structure. Meanwhile, the results of maximum power absorption of two hinged slender rafts in cuboid shape give a good agreement as well with those by using slender-body approximations. Then dynamics of a hinged two rafts are studied and results show how the maximum efficiency is affected by raft width and restricting the motions.     

Phase control through load control of oscillating-body wave energy converters with hydraulic PTO system

Oscillating bodies constitute an important class of wave energy converters, especially for offshore deployment. Phase control by latching has been proposed in the 1970s to enhance the wave energy absorption by oscillating bodies (especially the so-called point absorbers). Although this has been shown to be potentially capable of substantially increasing the amount of absorbed energy, the practical implementation in real irregular waves of optimum phase control has met with theoretical and practical difficulties that have not been satisfactorily overcome. The present paper addresses the case of oscillating-body converters equipped with a high-pressure hydraulic power take-off mechanism (PTO) that provides a natural way of achieving latching: the body remains stationary for as long as the hydrodynamic forces on its wetted surface are unable to overcome the resisting force (gas pressure difference times cross-sectional area of the ram) introduced by the hydraulic PTO system. A method of achieving sub-optimal phase-control is developed, based on the theoretical time-domain modelling of a single-degree of freedom oscillating body in regular and irregular waves, by adequately delaying the release of the body in order to approximately bring into phase the body velocity and the diffraction (or excitation) force on the body, and in this way get closer to the well-known optimal condition derived from frequency-domain analysis for an oscillating body in regular waves.     

Maximum wave-power absorption under motion constraints

An expression is derived for the maximum mean power that can be absorbed by a system of oscillating bodies in waves under a global constraint on their motions. The particular case of a single half-immersed sphere is used to show how the ‘point absorber’ result predicting capture widths in excess of unity must be modified. The theory is also applied to the submerged cylinder wave-energy device and curves are presented which show how the maximum efficiency is affected by restricting the motion of the device.     

Nonlinear Capillary-Gravity Waves Produced by a Vertically Oscillating Plate

The nonlinear capillary-gravity wave produced by a vertically oscillating plate,in which thecontact-angle model is considered,is studied by use of the Boundary Integral Equation Method(BIEM).The present numerical experiment shows that the code is robust and efficient for modeling the generationand propagation of capillary-gravity waves.It is found that the wave heights of stationary periodicnonlinear waves radiated away from the plate are dependent on the parameters involved in the contact-an-gle model.The effect of the contact-angle hysteresis and the nonlinearity of capillary-gravity waves on thewave profile is discussed in the paper.     

Fully nonlinear simulations of wave resonance by an array of cylinders in vertical motions

The finite element method(FEM) is employed to analyze the resonant oscillations of the liquid confined within multiple or an array of floating bodies with fully nonlinear boundary conditions on the free surface and the body surface in two dimensions.The velocity potentials at each time step are obtained through the FEM with 8-node quadratic shape functions.The finite element linear system is solved by the conjugate gradient(CG) method with a symmetric successive overelaxlation(SSOR) preconditioner.The waves at the open boundary are absorbed by the combination of the damping zone method and the Sommerfeld-Orlanski equation.Numerical examples are given by an array of floating wedgeshaped cylinders and rectangular cylinders.Results are provided for heave motions including wave elevations,profiles and hydrodynamic forces.Comparisons are made in several cases with the results obtained from the second order solution in the time domain.It is found that the wave amplitude in the middle region of the array is larger than those in other places,and the hydrodynamic force on a cylinder increases with the cylinder closing to the middle of the array.     

The effect of sub-optimal control and the spectral wave climate on the performance of wave energy converter arrays

A linear hydrodynamic model is used to assess the sensitivity of the performance of a wave energy converter (WEC) array to control parameters. It is found that WEC arrays have a much smaller tolerance to imprecision of the control parameters than isolated WECs and that the increase in power capture of WEC arrays is only achieved with larger amplitudes of motion of the individual WECs. The WEC array radiation pattern is found to provide useful insight into the array hydrodynamics. The linear hydrodynamic model is used, together with the wave climate at the European Marine Energy Centre (EMEC), to assess the maximum annual average power capture of a WEC array. It is found that the maximum annual average power capture is significantly reduced compared to the maximum power capture for regular waves and that the optimum array configuration is also significantly modified. It is concluded that the optimum configuration of a WEC array will be as much influenced by factors such as mooring layout, device access and power smoothing as it is by the theoretical optimum hydrodynamic configuration.     

Dynamics of arrays of floating point-absorber wave energy converters with inter-body and bottom slack-mooring connections

If point absorbers are employed in the extensive exploitation of the offshore wave energy resource, they should be deployed in arrays, the distance between the elements in the array being possibly tens of meters. In such cases, it may be more convenient that the array is spread moored to the sea bottom through only some of its elements, located in the periphery, while the other array elements are prevented from drifting and colliding with each other by connections to adjacent elements. An array of identical floating point absorbers located at the grid points of an equilateral triangular grid is considered in the paper. A spread set of slack-mooring lines connect the peripheric floaters to the bottom. A weight is located at the centre of each triangle whose function is to pull the three floaters towards each other and keep the inter-body mooring lines under tension. The power take-off system (PTO) is a linear damper activated by the buoy heaving motion. The whole system–buoys, moorings and power take-off systems–is assumed linear, so that a frequency domain analysis may be employed. Hydrodynamic interference between the oscillating buoys in array is accounted for. Equations are presented for a set of three identical point absorbers. This is then extended to more complex equilateral triangular grid arrays. Results from numerical simulations, with regular and irregular waves, are presented for the motions and power absorption of hemispherical converters in arrays of three and seven elements and for different mooring parameters and wave incidence angles. Comparisons are given with the unmoored and independently-moored buoy situations.     

Long waves dissipation and harmonic generation by coastal vegetation

In this paper, we study the harmonic generation and energy dissipation as water waves propagating through coastal vegetation. Applying the homogenization theory, linear wave models have been developed for a heterogeneous coastal forest in previous works (e.g. [17], [10], [11]). In this study, the weakly nonlinear effects are investigated. The coastal forest is modeled by an array of rigid and vertically surface-piercing cylinders. Assuming monochromatic waves with weak nonlinearity incident upon the forest, higher harmonic waves are expected to be generated and radiated into open water. Using the multi-scale perturbation theory, micro-scale flows in the vicinity of cylinders and macro-scale wave dynamics are separated. Expressing the unknown variables (e.g. velocity, free surface elevation) as a superposition of different harmonic components, the governing equations for each mode are derived while different harmonics are interacting with each other because of nonlinearity in the cell problem. Different from the linear models, the leading-order cell problem for micro-scale flow motion, driven by the macro-scale pressure gradient, is now a nonlinear boundary-value-problem, while the wavelength-scale problem for wave dynamics remains linear. A modified pressure correction method is employed to solve the nonlinear cell problem. An iterative scheme is introduced to connect the micro-scale and macro-scale problems. To demonstrate the theoretical results, we consider incident waves scattered by a homogeneous forest belt in a constant shallow depth. Higher harmonic waves are generated within the cylinder array and radiated out to the open water region. The comparisons of numerical results obtained by linear and nonlinear models are presented and the behavior of different harmonic components is discussed. The effects of different physical parameters on wave solutions are discussed as well.     

Fully Nonlinear Simulations of Wave Resonance by An Array of Cylinders in Vertical Motions

The finite element method (FEM) is employed to analyze the resonant oscillations of the liquid confined within multiple or an array of floating bodies with fully nonlinear boundary conditions on the free surface and the body surface in two dimensions. The velocity potentials at each time step are obtained through the FEM with 8-node quadratic shape functions. The finite element linear system is solved by the conjugate gradient (CG) method with a symmetric successive overelaxlation (SSOR) preconditioner. The waves at the open boundary are absorbed by the combination of the damping zone method and the Sommerfeld-Orlanski equation. Numerical examples are given by an array of floating wedge- shaped cylinders and rectangular cylinders. Results are provided for heave motions including wave elevations, profiles and hydrodynamic forces. Comparisons are made in several cases with the results obtained from the second order solution in the time domain. It is found that the wave amplitude in the middle region of the array is larger than those in other places, and the hydrodynamic force on a cylinder increases with the cylinder closing to the middle of the array.     

Interaction between waves and an array of floating porous circular cylinders

The present study theoretically as well as experimentally investigates the interaction between waves and an array of porous circular cylinders with or without an inner porous plate based on the linear wave theory.To design more effective floating breakwaters,the transmission rate of waves propagating through the array is evaluated.Each cylinder in the array is partly made of porous materials.Specifically,it possesses a porous sidewall and an impermeable bottom.In addition,an inner porous plate is horizontally fixed inside the cylinders.It dissipates the wave more effectively and eliminates the sloshing phenomenon.The approach suggested by Kagemoto and Yue(1986) is adopted to solve the multiple-scatter problem,while a hierarchical interaction theory is adopted to deal with hydrodynamic interactions among a great number of bodies,which efficiently saves computation time.Meanwhile,a series of model tests with an array of porous cylinders is performed in a wave basin to validate the theoretical work and the calculated results.The draft of the cylinders,the location of the inner porous plate,and the spacing between adjacent cylinders are also adjusted to investigate their effects on wave dissipation.     

Investigations on 3D effects and correlation between wave height and lip submergence of an offshore stationary OWC wave energy converter

Understanding the hydrodynamic interactions between ocean waves and the oscillating water column (OWC) wave energy converter is crucial for improving the device performance. Most previous relevant studies have focused on testing onshore and offshore OWCs using 2D models and wave flumes. Conversely, this paper provides experimental results for a 3D offshore stationary OWC device subjected to regular waves of different heights and periods under a constant power take–off (PTO) damping simulated by an orifice plate of fixed diameter. In addition, a 3D computational fluid dynamics (CFD) model based on the RANS equations and volume of fluid (VOF) surface capturing scheme was developed and validated against the experimental data. Following the validation stage, an extensive campaign of computational tests was performed to (1) discover the impact of testing such an offshore OWC in a 2D domain or a wave flume on device efficiency and (2) investigate the correlation between the incoming wave height and the OWC front wall draught for a maximum efficiency via testing several front lip draughts for two different rear lip draughts under two wave heights and a constant PTO damping. It is found that the 2D and wave flume modelling of an offshore OWC significantly overestimate the overall power extraction efficiency, especially for wave frequencies higher than the chamber resonant frequency. Furthermore, a front lip submergence equal to the wave amplitude affords maximum efficiency whilst preventing air leakage, hence it is recommended that the front lip draught is minimized.     

Dynamic Properties and Energy Conversion Efficiency of A Floating Multi-Body Wave Energy Converter

The present study proposed a floating multi-body wave energy converter composed of a floating central platform, multiple oscillating bodies and multiple actuating arms. The relative motions between the oscillating bodies and the floating central platform capture multi-point wave energy simultaneously. The converter was simplified as a forced vibration system with three degrees of freedom, namely two heave motions and one rotational motion. The expressions of the amplitude-frequency response and the wave energy capture width were deduced from the motion equations of the converter. Based on the built mathematical model, the effects of the PTO damping coefficient, the PTO elastic coefficient, the connection length between the oscillating body and central platform, and the total number of oscillating bodies on the performance of the wave energy converter were investigated. Numerical results indicate that the dynamical properties and the energy conversion efficiency are related not only to the incident wave circle frequency but also to the converter's physical parameters and interior PTO coefficients. By adjusting the connection length, higher wave energy absorption efficiencies can be obtained. More oscillating bodies installed result in more stable floating central platform and higher wave energy conversion efficiency.     

An iterative procedure for the diffraction of water waves by multiple cylinders

Diffraction of water waves by an array of vertical cylinders of circular cross section is studied. In order to account for first order interaction among the cylinders, the body boundary condition is satisfied for each cylinder considering the scattered wave field from other cylinders in an iterative way. After each iteration, coefficients in the partial wave decomposition of the wave potential are modified. Convergence is fast for the whole range of frequencies and for a large number of bodies, compared with exact algebraic methods of Linton and Evans [J. Fluid Mech. 46 (1990) 549] and Kagemoto and Yue [J. Fluid Mech. 166 (1) (1986) 189].     

Climate Variations of Certain Wave Parameters at the Inlet of Novorossiysk Bay

The main purpose of the study is an analysis of the wave climate of Novorossiysk Bay (Black Sea) for 1979–2019. The analysis is based on mathematical modeling results obtained with the modern DHI MIKE 21 SW spectral wave model. The wave model was verified using numerous instrumental data of the wind-wave parameters in the Black Sea and Sea of Azov. As follows from the research, the average power of storm waves in Novorossiysk Bay area is about 8 kW/m. The maximum power ranges from 50 to 100 kW/m but can reach 200 kW/m or more. The total duration of storm waves for one year is 40–60 days. The maximum duration of one storm can be up to three days but does not exceed two days on average. The mean storm duration is usually 10–11 h. The largest number of storms (above 130) was recorded in 2004; on average, there are about 100 storms annually. In recent decades, in the area of Novorossiysk Bay, the maximum wind-wave heights and the total number of storms has increased with statistical control. It is also highly probable that the average annual power of combined waves and maximum storm duration have increased; i.e., the time of continuous wave impact has increased.

     

Hydrodynamic interactions in floating cylinder arrays—I. Wave scattering

The hydrodynamic interactions due to wave scattering between the numbers of an array of stationary, truncated circular cylinders simulating the columns of an idealized tension-leg platform (TLP) are investigated. The method of solution for the fluid velocity potential involves replacing scattered waves by equivalent plane waves together with non-planar correction terms. This technique is, therefore, essentially a large spacing approximation. Use of this approach makes it possible to determine the hydrodynamic interactions between the array members utilizing only the diffraction characteristics of an isolated cylinder.Numerical results are presented for six array configurations consisting of 2–6 cylinders representing the legs of idealized TLPs. Calculations of the wave loads on these cylinders have been performed for a range of wave and structural parameters. It is found that, for certain parameter combinations, the influence of neighbouring bodies on the total wave field leads to hydrodynamic loading on individual columns which is significantly greater than the loading they would experience in isolation. The presented results demonstrate the significance of hydrodynamic interactions between TLP columns and clearly indicate that these effects should be considered by the designers and researchers associated with TLPs.     

Wave parameter gradients along the wave ray

Computer methods have been used to track selected deep-water waves along their orthogonals to the surf zone. Three bottom orbital parameters were computed at each step: diameter, maximum velocity and acceleration. These three parameters were then plotted along the several wave rays, thereby providing shoreward gradients.The three gradients have essentially the same geometry, so that any one can be taken as an indicator of the other two. Bottom slope changes are responsible for departure of the gradients from linearity. The bottom orbital gradients are quite different from gradients of wave height and wave length. These results from coasting waves cannot be applied directly to the case of forced, or wind-driven, waves.The non-linear nature of the bottom orbital gradients indicates that the near-shore bottom should exhibit a crude banding, more-or-less parallel with the coast, with different sediment transport and ripple-mark geometry from band to band.     

Experimental study of interactions between multi-directional focused wave and vertical circular cylinder,part II: Wave force

We describe experiments with multi-directional focused waves interacted with a vertical circular cylinder in a 3D wave basin. The focus of this study is on the run-up of multi-directional focused waves, wave forces, and wave pressures on the cylinder. Part I, the study on wave run-up, has already been presented by Li et al. (2012). In this paper, the analysis of the wave force on the vertical cylinder is presented.In this experiment, a cylinder with 0.25 m in diameter was adopted and different wave parameters, such as focused wave amplitude, peak frequency, frequency bandwidth and directional spreading index, are considered. The model scale k_pa (k_p is the wave number corresponding to peak frequency, a is the radium of the cylinder) varies from 0.32 to 0.65. The maximum forces of multi-directional focused wave on cylinder were measured and investigated. The results showed that the wave parameters have a significant influence on the wave force, and that the spatial profile of the surface of multi-directional focused wave can also affect its force on the cylinder, which is different from two-dimensional wave. In addition, the ‘secondary loading cycle’ phenomenon was also observed and discussed. In our experiments, the ‘secondary loading cycles’ occur when kA > 0.36 for all cases. While in some referred small scale experiments, the secondary load cycles are observed even for kA = 0.2, when the waves are longer enough. To larger model scale, the pronounced secondary load cycle occurs with larger wave steepness waves.     

The low-frequency limit for wave-power systems

Relations are derived for the maximum power absorbed by a system of devices in the low-frequency or long-wavelength limit. A Fourier series is used to represent the far-field radiation potential, and the number of terms in the Fourier series determines the maximum output of the system subject to the restriction of optimum motions. For an array with N devices, or a single device with N separate degrees of freedom, the maximum output is a function of N and the angle of the incident waves.     

Hydrodynamics identities and wave-drift force of a porous body

In the present study, hydrodynamic interactions between water waves and porous bodies are investigated. Various hydrodynamic identities, such as the Haskind relation, Bessho-Newman relation etc., are systematically re-examined. Some of these identities, such as the symmetry of added mass and damping and the Haskind-Hanaoka relation, are still valid for porous bodies even without modification to the identities. However, when energy dissipation due to porosity is involved, appropriate supplementation is required to properly consider porous effects. In addition, the calculation of wave drift forces acting on a porous body is formulated either by pressure integration or using momentum conservation as basis. We conclude that porosity dissipation makes a more substantial contribution to wave drift forces than does conventional dissipation created by scattered and radiated waves.