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.     

Wave forces acting on a submerged horizontal circular cylinder in oblique waves and on a vertical cylinder in deep waves

This paper presents a method of estimating wave forces acting on a submerged horizontal circular cylinder fixed in oblique waves.The experiments show that drag and inertia coefficients in beam sea are available for calculating the wave forces in oblique waves.Wave forces exerted on a vertical circular cylinder in deep waves are also investigated.The experimental results show that wave forces acting on the vertical cylinder coincide approximately with hydrodynamic forces acting on a submerged circular cylinder in an oscillating fluid.     

Wave forces acting on a vertical circular cylinder with a constant forward velocity

Wave forces acting on submerged circular cylinders moving forward with a constant velocity in regular waves are investigated experimentally. Hydrodynamic forces acting on the cylinder forced to surge in a steady are also measured and hydrodynamic coefficients were obtained. Wave force coefficients obtained from wave force measurements are compared with the hydrodynamic coefficients from surging tests, and the similarity and difference between them are discussed. Experiments show that these coefficients are quite different from those of the cylinder without a forward velocity.     

Viscous Effects on Wave Forces on A Submerged Horizontal Circular Cylinder

Numerical simulations are carried out for wave action on a submerged horizontal circular cylinder by means of a viscous fluid model, and it is focused on the examination of the discrepancies between the viscous fluid results and the potential flow solutions. It is found that the lift force resulted from rotational flow on the circular cylinder is always in anti-phase with the inertia force and induces the discrepancies between the results. The influence factors on the magnitude of the lift force, especially the correlation between the stagnation-point position and the wave amplitude, and the effect of the vortex shedding are investigated by further examination on the flow fields around the cylinder. The viscous numerical calculations at different wave frequencies showed that the wave frequency has also significant influence on the wave forces. Under higher frequency and larger amplitude wave action, vortex shedding from the circular cylinder will appear and influence the wave forces on the cylinder substantially.     

Computation of the inertia-dominated forces on horizontal circular cylinders placed in oblique waves and near to a plane bed

Wave-force coefficients of horizontal circular cylinders inclined with respect to the incoming waves, are studied numerically under conditions when the effects of flow separation are insignificant. The mathematical model is set in terms of a boundary-value problem for the velocity potential of the wave, which is formulated under the assumption of the linear diffraction theory, and solved numerically by the boundary element method. The numerical calculations are performed in the vertical plane, assuming uniform water depths in the direction along the axis of the cylinder. A first-order correction to the pressures is introduced to take account of the asymmetry of the velocity field around the cylinder when it is close to the plane bed. The correction procedure is found to be highly effective in computing the transverse forces for small gap ratios. The numerical results show that irrespective of the values of the gap ratio, the in-line forces are always sensitive to the wave directionality. The transverse forces, however, show sensitivity only for the smaller gap ratios. It is also shown that by accounting for the wave directionality effects in the wave kinematics only, the forces could be estimated to a certain extent by using the hydrodynamic force coefficients of inertia and lift corresponding to the normal waves.     

A Finite Element Solution of Wave Forces on Submerged Horizontal Circular Cylinders

In this paper, a numerical model is established for estimating the wave forces on a submerged horizontal circular cylinder. For predicting the wave motion, a set of two-dimensional Navier-Stokes equations is solved numerically with a finite element method. In order to track the moving non-linear wave surface boundary, the Navier-Stokes equations are discretized in a moving mesh system. After each computational time step, the mesh is modified according to the changed wave surface boundary. In order to stabilize the numerical procedure, a three-step finite element method is applied in the time integration. The water sloshing in a tank and wave propagation over a submerged bar are simulated for the first time to validate the present model. The computational results agree well with the analytical solution and the experimental data.Finally, the model is applied to the simulation of interaction between waves and a submerged horizontal circular cylinder.The effects of the KC number and the cylinder depth on the wave forces are studied.     

内孤立波作用下竖直圆柱体的横向力研究

本文对内孤立波作用下圆柱体的水动力特性进行了数值研究。对不同直径、不同水深比、不同浸没深度的圆柱进行了水平方向受力分析,从多个因素出发,研究了内孤立波对水下竖直圆柱的水平作用情况。通过分析水平作用力和力矩的时程曲线,得到内孤立波行进过程中对于圆柱体的横向作用力和力矩与时间的关系。通过对比不同工况下内孤立波橫向作用力幅值,得到内孤立波作用力幅值与圆柱直径、水深比以及浸没深度的关系。结果表明,内孤立波横向作用力(力矩)幅值与圆柱直径呈线性相关关系,但与水深比相关关系不明显,随着圆柱浸没深度的增加,内孤立波横向作用力幅值呈现出先增大后减小的趋势。     

Interaction of Linear Waves with Infinitely Long Horizontal Cylinders Studied by Boundary Element Method

The two-dimensional problems concerning the interaction of linear water waves with cylinders of arbitrary shape in two-layer deep water are investigated by use of the Boundary Integral Equation Method (BIEM). Simpler new expressions for the Green functions are derived, and verified by comparison of results obtained by BIEM with those by an analytical method. Examined are the radiation and scattering of linear waves by two typical configurations of cylinders in two-layer deep water. Hydrodynamic behaviors including hydrodynamic coefficients, wave forces, reflection and transmission coefficients and energies are analyzed in detail, and some interesting physical phenomena are observed.     

Hydrodynamic forces on a partially submerged cylinder at high Reynolds number in a steady flow

The hydrodynamic forces on the stationary partially submerged cylinder are investigated through towing test with Reynolds number ranging from 5 × 10⁴ to 9 × 10⁵. Three test groups of partially submerged cylinders with submerged depths of 0.25 D, 0.50 D, and 0.75 D and one validation group of fully submerged cylinders are conducted. During the experiments, the hydrodynamic forces on the cylinders are measured using force sensors. The test results show a considerable difference in the hydrodynamic coefficients for the partially submerged cylinders versus the fully submerged cylinders. A significant mean downward lift force is first observed for the partially submerged cylinders in a steady flow. The maximum of the mean lift coefficients can reach 1.5. Two distinct features are observed due to the effects of overtopping: random distributions in the mean drag coefficients and a clear quadratic relationship between the mean lift coefficients and the Froude number appear in the non-overtopping region. However, the novel phenomenon of a good linear relationship with the Froude number for the mean hydrodynamic coefficients is clearly shown in the overtopping region. In addition, fluctuating hydrodynamic coefficients are further proposed and investigated. These results are helpful to have a better understanding of the problem and to improve related structural designs.     

两层流体中振荡水平圆柱潜体水动力特性

研究了有限深两层流体中水平圆柱潜体的振荡辐射问题。在线性势流理论框架内,建立求解辐射势的多极展开理论,提出附加质量和阻尼系数的计算方法,进行了数值计算分析,并且与均匀流中的情况进行比较。结果表明,在圆柱潜体的某个振荡频率范围内,流体的分层效应对其附加质量和阻尼系数是有重要影响的。同时,对水平圆柱潜体振荡产生的自由面和内界面波动特性进行了数值分析。     

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.     

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。     

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.     

Diffraction of Water Waves by A Vertically Floating Cylinder in A Two-Layer Fluid

In this paper, the diffraction of water waves by a vertically floating cylinder in a two-layer fluid of a finite depth is studied. Analytical expressions for the hydrodynamic loads on the vertically floating cylinder are obtained by use of the method of eigenfunction expansions. The hydrodynamic loads on the vertically floating cylinder in a two-layer fluid inelude not only the surge, heave and pitch exciting forces due to the incident wave of the surface-wave mode, but also those due to the incident wave of the internal-wave mode. This is different from the case of a homogenous fluid. Some given examples show that, for a two-layer fluid system with a small density difference, the hydrodynamic loads for the surface-wave mode do not differ significantly from those due to surface waves in a single-layer fluid, but the hydrodynamic loads for the internal-wave mode are important over a wide range of frequencies. Moreover, also considered are the free surface and interface elevations generated by the diffraction wave due to the incident wave of the surface-wave and interhal-wave modes, and transfer of energy between modes.     

直立浮式柱群波浪荷载计算

为了研究类似张力腿平台（ＴＬＰ）等海洋工程结构物的运动,浮式直立柱群的波浪荷载计算是十分重要的。应用匹配渐进和特征函数展开法得到单柱流场速度势的半解析解,再引入大间距假设,将柱群中某一柱体对其它柱体的扰动作用用平面入射波来代替。同时对该问题进行了非平面波修正,计入柱间水动力相互作用,解决了柱群的绕射问题。应用该方法计算了不同条件下的双柱波浪荷载的幅频特征。     

Three-dimensional analysis of submerged, moored, horizontal, rigid cylinders used as breakwaters

Wave attenuation by moored cylinders is considered. The cylinders are submerged with their axes horizontal. Linear potential theory is applied. Three-dimensional motions of the cylinders subjected to normal and oblique monochromatic waves are determined using potential theory and a boundary integral method. Each cylinder has length 9.1 m (30 ft) and radius 1.5 m (5 ft), with its top 1.5 m (5 ft) below the still water line and its bottom 3.0 m (10 ft) above the seabed. Free-surface elevations are obtained for a single cylinder and for two cylinders in series. These configurations are effective wave barriers for a range of wave frequencies and incident angles.     

Wave drift enhancement effects in multi column structures

A theoretical assessment is made of mean wave drift forces on groups of vertical circular cylinders, such as the columns of a floating offshore platform. A complete analytical solution is obtained for two cylinders extending from seabed to free surface, and a long wave approximation is found to provide reliable predictions of the drift force in line with the waves at low frequencies. For moderate separation between the two cylinders, this force is found to tend at low frequencies to a value four times the force on an isolated cylinder.A numerical method is employed to study two surface piercing cylinders truncated below the free surface, and an arrangement of four vertical cylinders characteristic of a floating offshore platform. The mean vertical drift force is found to be reasonably well approximated, over the frequency range of practical interest, by the force on an individual cylinder considered in isolation multiplied by the number of cylinders in the group. Interaction effects, however, have a profound influence on the total horizontal drift force. At low frequencies this force is found to tend to the force on an isolated cylinder multiplied by the squate of the number of cylinders in the group.     

Added mass and damping of horizontal circular cylinder sections

This note brings together and extends previously known results for the added masses and dampings of surface-piercing or completely submerged horizontal circular cylinders. For vertical (heave) motions, but not horizontal (sway) motions, a striking qualitative change occurs in these hydrodynamic coefficients when the cylinder breaks the surface. This is simply explained in terms of low-frequency asymptotics and the equivalent wavemaker method.     

Radiation and scattering of surface waves by a group of submerged,horizontal, circular cylinders

One of the major methods available for investigating the interaction of water waves with arbitrarily shaped structures is based on the classical theory of Green's functions. For multiple bodies, however, this technique can become expensive in terms of both computer storage and execution time and it is desirable for special geometrics where possible, to seek simpler methods. In this paper, the radiation and scattering of surface waves by a group of parallel, horizontal, circular cylinders, submerged in deep water is studied using a method involving multipole potentials. The method is developed for any number of submerged, parallel, horizontal cylinders with arbitrary positions and radii. In particular, hydrodynamic coefficients are determined for various configurations of two and three cylinders and a comparison is made with results obtained for a cylinder in isolation.     

Wave interaction with multiple horizontal cylinders

The interaction of waves with submerged two-dimensional circular cylinder groups is investigated. Linear wave theory is used in the analysis and the viscous effects are neglected. The boundary value problem for the wave potential is based on Green's theorem, and the resulting integral equation is solved numerically. The added mass and damping coefficients in sway, heave and roll of oscillating twin-cylinders and the total wave forces on a fixed cylinder in a group are examined. The effects of the free surface, and particularly, the spacing of the cylinders are shown. The numerical results are tested against known solutions. Results for a two-cylinder configuration at different orientations are presented. It is found that the interaction between closely spaced cylinders is large. The spacings at which the interaction effect is important are shown graphically.