On the radiation and diffraction of water waves by a rectangular buoy

An analytical method is presented to analyze the radiation and diffraction of water waves by a rectangular buoy in an infinite fluid domain of finite water depth. Analytical expressions for the radiated potentials and the diffracted potentials are obtained by use of the method of separation of variables. The unknown coefficients in the expressions are determined by use of the eigenfunction expansion matching method. The added masses and damping coefficients for the buoy heaving, swaying and rolling in calm water are obtained by use of the corresponding radiated potentials. Wave excitation forces are calculated by two different approaches, one is by use of the radiated potentials through Haskind’s theorem and the other is by the diffracted potential. It can be seen that the latter approach for wave forces on a rectangular buoy is much simpler than the former. To verify the correctness of the method, two specific examples in the past references are recomputed and the obtained results are in good agreement with those by use of other methods, which shows that the present method is correct.     

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.     

On the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas

The radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas of finite depth is investigated. The analytical expressions for the radiated and diffracted potentials are derived as infinite series by use of the method of separation of variables. The unknown coefficients in the series are determined by the eigenfunction expansion matching method. The expressions for wave forces, hydrodynamic coefficients and reflection and transmission coefficients are given and verified by the boundary element method. Using the present analytical solution, the hydrodynamic influences of the angle of incidence, the submergence, the width and the thickness of the structure on the wave forces, hydrodynamic coefficients, and reflection and transmission coefficients are discussed in detail.     

Interaction of linear waves with an array of infinitely long horizontal circular cylinders in a two-layer fluid of infinite depth

The linear water wave scattering and radiation by an array of infinitely long horizontal circular cylinders in a two-layer fluid of infinite depth is investigated by use of the multipole expansion method. The diffracted and radiated potentials are expressed as a linear combination of infinite multipoles placed at the centre of each cylinder with unknown coefficients to be determined by the cylinder boundary conditions. Analytical expressions for wave forces, hydrodynamic coefficients, reflection and transmission coefficients and energies are derived. Comparisons are made between the present analytical results and those obtained by the boundary element method, and some examples are presented to illustrate the hydrodynamic behavior of multiple horizontal circular cylinders in a two-layer fluid. It is found that for two submerged circular cylinders the influence of the fluid density ratio on internal-mode wave forces is more appreciable than surface-mode wave forces, and the periodic oscillations of hydrodynamic results occur with the increase of the distance between two cylinders; for four submerged circular cylinders the influence of adding two cylinders on the wave forces of the former cylinders is small in low and high wave frequencies, but the influence is appreciable in intermediate wave frequencies.     

The hydrodynamic coefficients for an oscillating rectangular structure on a free surface with sidewall

Based on a two dimensional linear water wave theory, the boundary element method (BEM) is developed and applied to study the heave and the sway problem of a floating rectangular structure in water to finite depth with one side of the boundary is a vertical sidewall and the other boundary is an open boundary. Numerical results for the added mass and radiation damping coefficients are presented. These coefficients are not only depend on the submergence and the width of the structure, but also depend on the clearance between structure and sidewall. Negative added mass and sharp peaks in the damping and added mass coefficients have been found when the clearance with a value close to integral times of half wave length of wave generated by oscillation structure. The important effect of the clearance on the added mass and radiation damping coefficients are discussed in detail. An analytical solution method is also presented. The BEM solution is compared with the analytical solution, and the comparison shows good agreement.     

Wave radiation by a floating rectangular structure in oblique seas

The linear wave radiation by a long floating rectangular structure in oblique seas of finite depth is investigated by use of the method of separation of variables and the eigenfunction expansion matching method. Analytical expressions for the radiated potentials, wave forces and hydrodynamic coefficients are given. The correctness of these expressions is verified through two specific examples investigated previously by other numerical methods. Using the present analytical solution, the hydrodynamic effects of the angle of incidence, the draft and the width of the structure on the wave forces and hydrodynamic coefficients are discussed in detail which may provide some useful information for the design of rectangular structures in oblique seas.     

两层流体中多个振荡水平圆柱潜体水动力特性

采用解析方法研究了无限水深两层流体中多个振荡水平圆柱潜体的水动力特性。在线性势流理论框架内,基于多极子方法建立了辐射势的解析表达式,在此基础上导出了附加质量和阻尼系数的计算公式,采用边界元方法对这种解析方法进行验证,同时研究了两种不同工况下多个振荡水平圆柱潜体水动力的特性,结果表明,两层流体的密度比、圆柱的淹没深度以及圆柱的排列方式和间距等参数的变化对水平圆柱群附加质量和阻尼系数有很大的影响。     

On diffraction and radiation problem for two cylinders in water of finite depth

The hydrodynamic problem arising form the interaction of linear water waves with a wave energy device consisting of two coaxial vertical cylinders of different radii is investigated. One cylinder is riding in waves, while another is submerged in fluid. By use of the method of separation of variables and the method of matched eigenfunction expansion, analytical expressions for the potentials are obtained. Using the expressions for the potentials, analytical expressions for the hydrodynamic coefficients and exciting forces/moments on the device are obtained. Numerical results of the hydrodynamic coefficients and exciting forces/moments are presented for some ratios of the radius of the submerged cylinder to that of the riding one. It is found that the radius of the submerged cylinder has a significant influence on the hydrodynamic coefficients and exciting forces/moments for relatively bigger radius of the submerged cylinder at low frequencies.     

Analytical method of radiation by a wave energy device with dual rectangular floating bodies

The system with one floating rectangular body on the free surface and one submerged rectangular body has been applied to a wave energy conversion device in water of finite depth. The radiation problem by this device on a plane incident wave is solved by the use of an eigenfunction expansion method, and a new analytical expression for the radiation velocity potential is obtained. The wave excitation force is calculated via the known incident wave potential and the radiation potential with a theorem of Haskind employed. To verify the correctness of this method, an example is computed respectively through the bound element method and analytical method. Results show that two numerical methods. are in good agreement, which shows that the present method is applicable. In addition, the trends of hydrodynamic coefficients and wave force are analyzed under different conditions by use of the present analytical method.     

The surge motion of a circular cylinder containing a concentric cylindrical hole in finite depth

This paper deals with hydrodynamic forces of a single semisubmerged circular cylinder containing a concentric cylindrical hole constrained to move in a water domain of finite depth. The fluid domain is divided into inner and outer regions. The Laplace equations governing velocity potentials for the three regions are solved by separation of variables and expressed in terms of eigenfunctions of the resulting equations which satisfy appropriate boundary conditions. Continuity of pressure and velocity at the interface of the regions provides the necessary equations from which the velocity potentials, pressures and forces are obtained. Numerical results are plotted for added mass and damping coefficients for different draft-to-depth and radius-to-depth values and for various wave amplitudes.     

A modified scaled boundary finite-element method for problems with parallel side-faces. Part II. Application and evaluation

The modified scaled boundary finite-element method (SBFEM), keeping the advantages of the original SBFEM, eliminates the restriction of the scaling center location so that this approach can solve two-dimensional problems with parallel side-faces. In this paper, the modified SBFEM is applied to solutions of two types of problems—wave diffraction by a single and twin surface rectangular obstacles and wave radiation induced by an oscillating mono-hull and twin-hull structures in a finite depth of water. For wave diffraction problems, numerical results agree extremely well with the analytic solution for the single obstacle case and other numerical results of a different approach for the twin obstacle case. For wave radiation problems, the particular solutions to the scaled boundary finite-element equation are presented for cases of heave, sway and roll motions. The added mass and damping coefficients for heave, sway and roll motions of a two-dimensional rectangular container are computed and the numerical results are compared with those from independent analytical solution and numerical solution using the boundary element method (BEM). It is found that the SBFEM method achieves equivalent accuracy to the conventional BEM with only a few degrees of freedom. In the last example, wave radiation by a two-dimensional twin-hull structure is analyzed. Comparisons of the results with those obtained using conventional Green's function method (GFM) demonstrate that the method presented in this paper is free from the irregular frequency problems.     

两层流体中多个圆柱浮体波浪力的解析方法

采用解析方法研究了线性入射波作用下两层流体中多个圆柱形淹没浮体的渡浪力特性.首先基于多极子展开方法,建立了散射势函数的解析表达式,并进一步得到浮体散射渡浪力的计算公式,然后利用边界元方法验证了本文的解析解,最后分析了不同参数的变化对双圆柱形浮体结构波浪力的特有影响.     

直墙前矩形月池水波振荡研究分析

月池内流体存在活塞和晃荡两类振荡现象。基于线性势流理论,推导了波浪斜向入射下,直墙前矩形月池辐射和绕射问题的解析解。通过分离变量法和特征函数展开法求解了速度势函数,根据边界条件来确定速度势函数中的未知系数,由速度势函数计算斜向波与矩形月池相互作用的水动力系数和波浪激励力,对它们的变化规律进行了分析讨论,研究了底部开口大小、波浪入射角度对矩形月池水动力特性的影响以及直墙远近对波浪力的影响。结果表明,月池底部开口大小对流体水平作用的影响较小,而对流体垂直作用的影响较大;波浪入射角度的变化对矩形月池横荡和横摇运动时的水动力特性有一定的影响;在一定条件下,直墙的存在会使得月池在水平方向所受到的波浪力比开敞水域中的要大。     

Reflection and transmission properties of double submerged rectangular blocks

An analytic method is used to study the reflection and transmission coefficients of the double submerged rectangular blocks (DSRBs) in oblique waves. The scattering potentials are obtained by means of the eigenfunction expansion method, and expressions for the reflection and transmission coefficients are determined. The boundary element method is employed to verify the correctness of the present analytical method. The DSRBs have better performance than the single submerged rectangular block (SSRB) in certain cases. The reflection and transmission properties of the DSRBs are investigated for some specific cases, and the influences of the geometric parameters are also presented.     

Wave radiation due to oscillation of two parallel spaced cylinders

The radiation problem for two parallel-spaced cylinders is studied. The solution is expressed explicitly in terms of well-behaved convergent series with elementary functions, which are convenient for numerical computation and readily applicable for two-dimensional two-body potential problems. The added mass and damping coefficients together with the phase angles of radiated wave potentials for the forced heave and sway motions of two identical submerged cylinders are presented. The results are useful for determination of the hydrodynamic properties of multi-hull semi-submersibles. In view of the close relationship between a radiation and a scattering problem, the application of the results to the problem of energy extraction from water waves is also noted.     

Hydrodynamic analysis and shape optimization for vertical axisymmetric wave energy converters

The absorber is known to be vertical axisymmetric for a single-point wave energy converter (WEC). The shape of the wetted surface usually has a great influence on the absorber’s hydrodynamic characteristics which are closely linked with the wave power conversion ability. For complex wetted surface, the hydrodynamic coefficients have been predicted traditionally by hydrodynamic software based on the BEM. However, for a systematic study of various parameters and geometries, they are too multifarious to generate so many models and data grids. This paper examines a semi-analytical method of decomposing the complex axisymmetric boundary into several ring-shaped and stepped surfaces based on the boundary discretization method (BDM) which overcomes the previous difficulties. In such case, by using the linear wave theory based on eigenfunction expansion matching method, the expressions of velocity potential in each domain, the added mass, radiation damping and wave excitation forces of the oscillating absorbers are obtained. The good astringency of the hydrodynamic coefficients and wave forces are obtained for various geometries when the discrete number reaches a certain value. The captured wave power for a same given draught and displacement for various geometries are calculated and compared. Numerical results show that the geometrical shape has great effect on the wave conversion performance of the absorber. For absorbers with the same outer radius and draught or displacement, the cylindrical type shows fantastic wave energy conversion ability at some given frequencies, while in the random sea wave, the parabolic and conical ones have better stabilization and applicability in wave power conversion.     

Approximation to the hydrodynamics of floating pontoons under oblique waves

The hydrodynamic properties of long rigid floating pontoon interacting with linear oblique waves in water of finite arbitrary depth are examined theoretically. The flow is idealized as linearized, velocity potentials are expressed in the form of eigen-function expansions with unknown coefficients. The fluid domain is split into three regions, region (1) wave-ward of the structure, region (2) in the lee of the structure, and region (3) beneath the structure. The different hydrodynamic quantities of interest such as the exciting forces, added mass and damping coefficients, reflection and transmission coefficients were studied for an applicable range of wave/structure parameters. Assuming rigid body motions, dynamic responses of the moored structure is approximately calculated through three equations of motion. Floating pontoons proved to be a convenient alternative for protection from waves in shallow water. The present method of solution was found to be computationally efficient, and results are comparable to those obtained through other techniques.     

Wave-induced surge motion of a tension leg structure

An analytical solution for the coupling problem of a two-dimensional tension leg structure interacting with a monochromatic linear wave train in an inviscid and incompressible fluid is presented. The tension legs are considered to be linearly elastic. The flow is further assumed to be irrotational and single-valued velocity potentials can then be defined.The boundary value problem is incorporated into a scattering and a radiation problem. The boundary value problems are then solved separately and combined to resolve all unknowns. The complete solutions of the velocity potentials are represented by the series of eigen-functions, and the surge motion of the structure is described in terms of the incident wave properties.The analytical solution is compared with a computer-coded numerical solution utilizing the boundary element method. The solutions agree very well, and both predict a resonant frequency for a specific structure which is different from the natural frequency of the structure due to the presence of the evanescent waves caused by the structure.     

Wave radiation and diffraction by a two-dimensional floating body with an opening near a side wall

The radiation and diffraction problem of a two-dimensional rectangular body with an opening floating on a semi-infinite fluid domain of finite water depth is analysed based on the linearized velocity potential theory through an analytical solution procedure. The expressions for potentials are obtained by the method of variation separation, in which the unknown coefficients are determined by the boundary condition and matching requirement on the interface. The effects of the position of the hole and the gap between the body and side wall on hydrodynamic characteristics are investigated. Some resonance is observed like piston motion in a moon pool and sloshing in a closed tank because of the existence of restricted fluid domains.     

Numerical study of the motions and drift force of a floating OWC device

The motion and the drift force of a floating OWC (oscillating water column) wave energy device in regular waves are studied taking account of the oscillating surface-pressure due to the pressure drop across the duct of the air chamber. The potential problem inside the chamber is formulated by making use of the Green integral equation associated with the Rankine-type Green function while the outer problem with the Kelvin-type Green function. The added mass, wave damping and excitation coefficients as well as the motion and drift force of the OWC device are calculated for various values of parameter related to the pressure drop.