Construction of wave-free potentials and multipoles in a two-layer fluid having free-surface boundary condition with higher-order derivatives

There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems associated with the Laplace’s equation (or Helmholtz equation) arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the multipole expansion method. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace’s equation or two-dimensional Helmholz equation. Construction of these wave-free potentials and multipoles are presented here in a systematic manner for a number of situations such as two-dimensional non-oblique and oblique waves, three dimensional waves in two-layer fluid with free surface condition with higher order partial derivative are considered. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.     

常水深下斜浪入射无波势研究（英文）

In this paper, a method to construct oblique wave-free potentials in the linearised theory of water waves for water with uniform finite depth is presented in a systematic manner. The water has either a free surface or an ice-cover modelled as a thin elastic plate. For the case of free surface, the effect of surface tension may be neglected or taken into account. Here, the wave-free potentials are singular solutions of the modified Helmholtz equation, having singularity at a point in the fluid region and they satisfy the conditions at the upper surface and the bottom of water region and decay rapidly away from the point of singularity. These are useful in obtaining solutions to oblique water wave problems involving bodies with circular cross-sections such as long horizontal cylinders submerged or half-immersed in water of uniform finite depth with a free surface or an ice-cover modelled as a floating elastic plate. Finally, the forms of the upper surface related to the wave-free potentials constructed here are depicted graphically in a number of figures to visualize the wave motion. The results for non-oblique wave-free potentials and the upper surface wave-free potentials are obtained. The wave-free potentials constructed here will be useful in the mathematical study of water wave problems involving infinitely long horizontal cylinders, either half-immersed or completely immersed in water.     

带有变化率的同轴圆柱体受到的波浪力研究(英文)

Wave diffraction of two concentric porous cylinders with varying porosity was studied by using an analytical method based on eigenfunction matching.The fluid domain around the cylinders is divided into three sub-domains and in each sub-domain an eigenfunction expansion of the velocity potential is obtained by satisfying the Laplace equation,the boundary conditions on the free surface and on the sea bed.The unknown coefficients of eigenfunction expansions are determined by boundary conditions on the porous hulls.In the paper,the boundary conditions are based upon the assumption that the flow in the porous medium is governed by Darcy’s law.Two porous-effect parameters applied on two porous cylinders are functions of the vertical coordinate instead of the constant.Wave loading on the outer and inner cylinder is presented in the numerical results.     

具有高阶导数自由表面边界条件的两层流体无波势和多级的构建（英文）

There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems associated with the Laplace's equation(or Helmholtz equation) arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the multipole expansion method. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace's equation or two-dimensional Helmholz equation. Construction of these wave-free potentials and multipoles are presented here in a systematic manner for a number of situations such as two-dimensional non-oblique and oblique waves, three dimensional waves in two-layer fluid with free surface condition with higher order partial derivative are considered. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.     

A numerical study of nonlinear wave interaction in regular and irregular seas: irrotational Green–Naghdi model

The irrotational Green–Naghdi model for nonlinear wave propagation in deep water is developed to simulate the irregular sea surface of a given directional wave spectrum. The model is derived from Hamilton's principle with a depthwise approximation to the flow field. The nonlinear boundary conditions are exactly satisfied on the actual free surface, and the continuity equation is satisfied exactly within the fluid domain. The ‘level’ of approximation in the depthwise direction is optimally chosen to simulate a given wave spectrum accurately with minimum computational effort. Several numerical techniques also are introduced to cut the computational cost further. Numerical results for two-dimensional nonlinear waves are presented.     

Exciting forces for a wave energy device consisting of a pair of coaxial cylinders in water of finite depth

Two coaxial vertical cylinders-one is a riding hollow cylinder and the other a solid cylinder of greater radius at some distance above an impermeable horizontal bottom,were considered.This problem of diffraction by these two cylinders,which were considered as idealization of a buoy and a circular plate,can be considered as a wave energy device.The wave energy that is created and transferred by this device can be appropriately used in many applications in lieu of conventional energy.Method of separation of variables was used to obtain the analytical expressions for the diffracted potentials in four clearly identified regions.By applying the appropriate matching conditions along the three virtual boundaries between the regions,a system of linear equations was obtained,which was solved for the unknown coefficients.The potentials allowed us to obtain the exciting forces acting on both cylinders.Sets of exciting forces were obtained for different radii of the cylinders and for different gaps between the cylinders.It was observed that changes in radius and the gap had significant effect on the forces.It was found that mostly the exciting forces were significant only at lower frequencies.The exciting forces almost vanished at higher frequencies.The problem was also investigated for the base case of no plate arrangement,i.e.,the case having only the floating cylinder tethered to the sea-bed.Comparison of forces for both arrangements was carried out.In order to take care of the radiation of the cylinders due to surge motion,the corresponding added mass and the damping coefficients for both cylinders were also computed.All the results were depicted graphically and compared with available results.     

全非线性方法对于沉浸双圆柱定常前行运动引起自由表面流的研究（英文）

The free surface flow generated by twin-cylinders in forced motion submerged beneath the free surface is studied based on the boundary element method. Two relative locations, namely, horizontal and vertical, are examined for the twin cylinders. In both cases, the twin cylinders are starting from rest and ultimately moving with the same constant speed through an accelerating process. Assuming that the fluid is inviscid and incompressible and the flow to be irrotational, the integral Laplace equation can be discretized based on the boundary element method. Fully-nonlinear boundary conditions are satisfied on the unknown free surface and the moving body surface. The free surface is traced by a Lagrangian technique. Regriding and remeshing are applied, which is crucial to quality of the numerical results. Single circular cylinder and elliptical cylinder are calculated by linear method and fully nonlinear method for accuracy checking and then fully nonlinear method is conducted on the twin cylinder cases, respectively. The generated wave elevation and the resultant force are analysed to discuss the influence of the gap between the two cylinders as well as the water depth. It is found that no matter the kind of distribution, when the moving cylinders are close to each other, they suffer hydrodynamic force with large absolute value in the direction of motion. The trend of force varying with the increase of gap can be clearly seen from numerical analysis. The vertically distributed twin cylinders seem to attract with each other while the horizontally distributed twin cylinders are opposite when they are close to each other.     

箱式超大型浮体在不平底部海域中的水弹性响应

采用作者们(2007)提出的函数展开法描述箱式超大型浮体漂浮在不平底部海域中的水弹性响应问题.将速度势沿静水面位置展开,其系数是水平位置的函数.根据速度势满足拉普拉斯方程就可将这些系数简化到只有两个未知函数.海底条件和平板动力学方程或联合的自由面条件可以分别在Ⅰ区和Ⅱ区建立两个方程用于确定这两个未知函数.不同区域的解要在公共界面上进行匹配.在求解水弹性问题时,展开法必须要截断,从原理上来说,保留的项数越多,其解可以适用于更高阶的kh,其中k是波数,h是水深.作为一个例子,文中给出了只保留一项的结果,而这正好是人们所熟悉的浅水方程.     

球体波能转换装置捕获能量的理论研究

球体波能转换装置在波浪的激励下可以同时在水平和竖直两个模式振荡。基于线性微幅波浪和势流理论,将波浪与球体相互作用简化为球体固定不动,波浪绕过球体和球体在静止流体中运动。线性叠加以后得到总的速度势,计算出球体波能转换装置的平均功率和能量俘获宽度因子的解析表达式。理论研究表明:装置与波浪运动的相位差是影响波能的最重要的参数,最优相位差为-π/2,并且给出了最优相位差调节的理论表达式。另外,对波长、波幅、球体半径、水深和球心距静水面的距离等因素也进行了分析。     

有限水深中垂直下潜弹性薄板的水波散射(英文)

The problem of water wave scattering by a thin vertical elastic plate submerged in uniform finite depth water is investigated here.The boundary condition on the elastic plate is derived from the Bernoulli-Euler equation of motion satisfied by the plate.Using the Green’s function technique,from this boundary condition,the normal velocity of the plate is expressed in terms of the difference between the velocity potentials(unknown)across the plate.The two ends of the plate are either clamped or free.The reflection and transmission coefficients are obtained in terms of the integrals’involving combinations of the unknown velocity potential on the two sides of the plate,which satisfy three simultaneous integral equations and are solved numerically.These coefficients are computed numerically for various values of different parameters and depicted graphically against the wave number in a number of figures.     

极浅水多体波浪能发电装置锚泊系统设计

采用三维势流理论对多体波浪能发电装置进行频域水动力计算,分析极浅水效应和多体间相互作用对单体发电装置的水动力性能的影响。利用Orcaflex软件建立多体波浪能发电装置数值模型,浮体间采用铰接进行连接,并利用弹簧单元模拟发电液压缸的弹性阻尼作用。为极浅水多体波浪能发电装置设计合理的锚泊方案,采用时域动态耦合方法进行分析,同时考虑极浅水效应、多体影响和二阶低频力,根据API规范对自存工况下的锚泊缆强度进行校核,并对确定选用的锚泊方案进行敏感性分析。通过对计算结果进行分析,发现在进行浅水多浮体锚泊设计时,低频波浪载荷、多体相互作用和海底摩擦不可忽视;对于极浅水的锚泊系统设计,Lazy wave 型锚泊形式优于传统悬链线式锚泊。上述研究工作为近海极浅水海域新型海洋能发电装置锚泊系统的设计提供了新的参考和思路。     

上游静止柱体对圆柱体结构尾激振动的影响

文章运用半隐式特征线分裂算子有限元算法对串列布置的静止上游柱体和下游圆柱的尾激振动问题进行了数值模拟研究。数值结果表明:上游柱体结构的形状、尺寸比(d/D)和折减速度(Ur)三个参数对下游圆柱体结构的动力响应、运动轨迹与涡脱落模态有着显著的影响,且与单圆柱工况相比存在明显区别;随着尺寸比的增大,上/下游柱体结构之间的互扰作用会由流致效应逐渐转变为尾流效应,使得频率特性发生变化,并会导致下游圆柱体结构的振动响应增强;当d/D=0.5和1.0时,下游圆柱体结构的运动轨迹主要为“8”字形;当d/D=1.5时,除了“8”字形外,运动轨迹还会呈现双弯刀形、“双8”字形和不规则形状。通过对流体力系数与位移时程曲线及位移PSD曲线特性进行分析,揭示了其相互作用的内在机理。另外,下游圆柱体结构的尾流场特性也会随参数的变化而变化,其涡脱落模态主要为2S、P+S和2P三种。     

经过海峡的双层流体由底部波动产生的斜向波散射问题研究(英文)

The problem of oblique wave(internal wave) propagation over a small deformation in a channel flow consisting of two layers was considered.The upper fluid was assumed to be bounded above by a rigid lid,which is an approximation for the free surface,and the lower one was bounded below by an impermeable bottom surface having a small deformation;the channel was unbounded in the horizontal directions.Assuming irrotational motion,the perturbation technique was employed to calculate the first-order corrections of the velocity potential in the two fluids by using Green’s integral theorem suitably with the introduction of appropriate Green’s functions.Those functions help in calculating the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom deformation.Three-dimensional linear water wave theory was utilized for formulating the relevant boundary value problem.Two special examples of bottom deformation were considered to validate the results.Consideration of a patch of sinusoidal ripples(having the same wave number) shows that the reflection coefficient is an oscillatory function of the ratio of twice the x-component of the wave number to the ripple wave number.When this ratio approaches one,the theory predicts a resonant interaction between the bed and the interface,and the reflection coefficient becomes a multiple of the number of ripples.High reflection of incident wave energy occurs if this number is large.Similar results were observed for a patch of sinusoidal ripples having different wave numbers.It was also observed that for small angles of incidence,the reflected energy is greater compared to other angles of incidence up to.These theoretical observations are supported by graphical results.     

分舱形式对圆柱壳结构声学性能影响研究

为研究舱壁布置形式对圆柱壳结构声学性能的影规律,文章应用有限元方法对等间距与不等间距分舱圆柱壳模型的振动响应进行了计算,以此为输入可计算获得壳体的均方法向速度级声学传递函数,并在典型频率处应用波数谱分析方法将壳体空间域振动场转换到波数域上,对壳体的振动进行波形分离与量化,分析获得了舱壁等间距布置与不等间距布置圆柱壳在典型激励下的结构声学性能,并解释其机理.研究表明:对垂向激励,圆柱壳不等间距分舱相比等间距分舱具有较好的结构声学性能,但对轴向激励,两者区别不明显.若将不等间距分舱与局部结构补强措施相结合,则可使得圆柱壳在两典型工况下均能获得更优的结构声学性能.     

波浪与带有窄缝结构作用的非线性数值模拟

针对波浪与带有窄缝结构作用产生的流体共振问题,采用基于域内源造波技术的时域高阶边界元方法并在窄缝内流体引入人工阻尼,建立了自由水面满足完全非线性边界条件的二维时域数值波浪水槽模型。求解中采用混合欧拉-拉格朗日方法追踪流体瞬时水面,运用四阶龙格库塔方法更新下一时间步的波面和速度势,利用加速度势的方法来求得作用结构上的瞬时波浪荷载。通过与已发表文献的数值与实验结果对比,验证了模型的准确性。同时通过大量的数值计算研究了共振条件入射波浪非线性对反射波高、透射波高和窄缝内波高、及结构波浪荷载和压力分布的影响。     

简单Green函数法求解三维水动力系数

本文提出用简单Green函数1/r求解无航速的三维物体在水面、水下或在无界域中作各向振荡时水动力系数的数值方法。对于无穷远处扩散条件转换成辐射面条件的问题作了专门的考虑,即用圆柱将物体包围在其内部,该圆柱将流场分成内、外两部分,外部域中速度势的特征函数展开式与内部域中用Green公式求得的解相匹配。 根据本计算方法编制的程序对一些典型情况作了试算,其中包括在水面振荡的半球、方盒和圆盘的水动力系数曲线;水面及水底对圆球水动力系数的影响等。为考核本方法,与其它计算方法得到的结果作了比较。对方盒的三维结果与切片法作了比较。     

带有月池的非对称结构物水波辐射和激励力（英文）

A highly efficient "hybrid integral-equation method" for computing hydrodynamic added-mass, wave-damping, and wave-exciting force of general body geometries with a vertical axis of symmetry is presented. The hybrid method utilizes a numerical inner domain and a semi-infinite analytical outer domain separated by a vertical cylindrical matching boundary.Eigenfunction representation of velocity potential is used in the outer domain; the three-dimensional potential in the inner domain is solved using a "two-dimensional" boundary element method with ring sources and ring dipoles to exploit the body symmetry for efficiency. With proper solution matching at the common boundary, both radiation and diffraction potentials can be solved efficiently while satisfying the far-field radiation condition exactly. This method is applied to compute the hydrodynamic properties of two different body geometries: a vertical-walled moonpool with a bottom plate that restricts the opening and a spar-like structure with a diverging bottom opening inspired by designs of floating Oscillating Water Columns. The effects of the size of the bottom opening on the hydrodynamic properties of the body are investigated for both geometries. The heave motion of the floater as well as the motion of the internal free surface under incident wave excitation are computed and studied for the spar-like structure.     

Second-order wave run-up on a vertical cylinder adjacent to a plane wall based on the application of quadratic transfer function in bi-directional waves

A numerical model was developed in this study to simulate the wave diffraction caused by an arbitrarily shaped structure in the presence of bi-directional incident waves based on a higher-order boundary element method (HOBEM). Based on the developed model, the wave elevation quadratic transfer function (QTF) in bi-directional waves, which is defined as the second-order wave elevation caused by two incident waves of unit amplitude from two directions, can be determined. The developed model was subsequently used to investigate the wave interaction with a cylinder situated near a vertical wall. The image principle was applied to transform the original problem into an equivalent one of wave diffraction caused by two symmetrical cylinders in open seas exposed to bi-directional incident waves. The second-order wave run-up on the cylinder can then be determined using the wave elevation QTF obtained from an analysis of the equivalent problem. A detailed numerical analysis was then conducted. Numerical results revealed that the presence of the vertical wall can apparently disturb the wave diffraction process from the cylinder, and lead to significantly amplified second-order wave elevation within the region between the wall and cylinder. In addition, the respective contribution from the first- and second-order components to the overall wave elevation around the cylinder was discussed.     

A time-domain higher-order boundary element method for 3D forward-speed radiation and diffraction problems

A time-domain higher-order boundary element method for seakeeping analyses in the framework of linear potential theory is newly developed. Ship waves generated by two modified Wigley models advancing at a constant forward speed in calm water or incident waves and the resultant radiation and diffraction forces are computed to validate this code. A rectangular computational domain moving with the same forward speed as the ship is introduced, in which an artificial damping beach is installed at an outer portion of the free surface except the downstream side for satisfying the radiation condition. The velocity potential on the ship hull and the normal velocity on the free surface are calculated directly by solving the boundary integral equation. An explicit time-marching scheme is employed for updating both kinematic and dynamic free-surface boundary conditions, with an embedding of a second-order upwind difference scheme for the derivative in the x-direction to stabilize the calculation. Extensive results including the exciting forces, added mass and damping coefficients, wave profiles, and wave patterns for blunt Wigley and slender Wigley hulls with forward speed are presented to validate the efficiency of the proposed 3D time-domain approach. The corresponding physical tests of the radiation and diffraction problems in a towing tank are also carried out. Computed numerical results show good agreement with the corresponding experimental data and other numerical solutions.