Time domain analysis of second-order wave diffraction by an array of vertical cylinders

A time domain method is employed to analyze interactions of water waves and a group or an array of cylinders. The nonlinear free-surface boundary conditions are satisfied based on the perturbation method up to the second order. The first- and second-order velocity potential problems at each time step are solved through a finite element method (FEM). The mesh required is generated based on a 2-D unstructured grid on a horizontal plane and its extension in the vertical direction. The matrix equation of the FEM is solved through an iteration and the initial solution is obtained from the result at the previous time step. The radiation condition is imposed through a combination of the damping zone method and the Sommerfeld–Orlanski equation. Results for various configurations, including two cylinders, four cylinders, an array of 10 cylinders and two arrays of eight cylinders are provided to show the effect of the interaction and their behaviour near the trapped mode.     

RESONANCES IN WAVE DIFFRACTION/RADIATION FOR ARRAYS OF ELASTICALLY CONNECTED CYLINDERS

As shown by Maniar & Newman in 1997, for a long array of bottom-mounted cylinders in the open sea, resonant modes occur as “near-trapping” and large diffraction forces are excited on the cylinders. The mechanism of such a resonant phenomenon was subsequently explained by the present authors in connection with the Dirichlet trapped modes for an array of cylinders aligned perpendicular to the walls in a wave channel. This paper examines similar resonant phenomena for radiation problems. Considered is an array of elastically connected cylinders in a wave channel. The cylinders are surface-piercing and extend to the sea-bottom. They constitute an array in a line, and each cylinder is allowed to oscillate only in the direction parallel to the line. Nonradiating wave modes, which cause only added mass force and no hydrodynamic damping are demonstrated to exist for an array of cylinders across the wave channel. Each mode corresponds to a “dry-mode” for the periodic array of elastically connected cylinders. This result leads to the existence of pure-resonant modes for a periodic array of elastically connected cylinders across the channel. Trapped modes for the corresponding diffraction problem are obtained as the limiting case when the stiffness of the springs has an infinite value.     

Water wave trapping in a long array of bottomless circular cylinders

This study tries to identify wave trapping situations by engaging and properly combining two well established phenomena: (i) the trapped modes induced by arrays of cylinders and (ii) the pumping trapped modes which are known to occur in moonpools. To this end, the fundamental hydrodynamic boundary value problem for arrays of bottomless cylinders was solved using standard domain decomposition. The method employed expansions of the solutions for the velocity potentials in polar harmonics combined with the eigenfunction expansions technique. The solution sought for the velocity potentials is achieved using the “direct” method of approach which accordingly requires the employment of a sophisticated matrix manipulation process.The elaboration of the concerned concept was motivated by three basic tasks: (i) to identify whether arrays of truncated and bottomless cylinders indeed preserve the occurrence of Neumann, Dirichlet and near trapped modes, extensively investigated for bottom-seated cylinders; (ii) to examine whether the expected pumping modes in moonpools modify the characteristics of the hydrodynamic resonance regimes (trapped modes) in the open liquid space between the cylinders and vice versa and (iii) to explore the possibility to suggest relevant configurations as parts of integrated mechanisms for practical applications, focusing a fortiori to clusters of hydrodynamically interacting Oscillating Water Columns (OWCs).The method developed is generic and can be employed for arbitrary configurations of multi-body arrays accommodating bottomless cylinders with uneven geometrical characteristics. Trapped modes are identified numerically as peaks in loading and this fact has been explicitly demonstrated in rows of cylinders. Therefore, the numerical results shown and discussed in the present are based on a specific in-line array that has been investigated in the past for bottom-seated cylinders. The investigated subject, i.e. whether the combined wave trapping induced by the examined configuration could be conceived as an efficient water wave power extraction mechanism is approached and discussed through dedicated computations of the free-surface displacements in the moonpools.     

Hydrodynamic interaction between two vertical cylinders in water waves

I.IntroductionWiththeconstructionoflargeoffshorestructures-wavediffractionandradiationproblemscausedbyseveralbodiesbecomeincreasinglyimportant.LargeoffShoreplatforms,wave-powerextractiondevices,Iargestoragefacilitiesandoffshorefloatingairportsl']havebeenp…     

Oscillations of elastically mounted cylinders in regular waves

Under the assumption of potential flow and linear wave theory, a semi-analytic method based on eigenfunciton expansion is proposed to predict the hydrody-namic forces on an array of three bottom-mounted, surface-piercing circular cylinders. The responses of the cylinders induced by wave excitation are determined by the equa-tions of motion coupled with the solutions of the wave radiation and diffraction problems. Experiments for three-cylinder cases are then designed and performed in a wave flume to determine the accuracy of this method for regular waves.     

多个椭圆柱波浪力的一种解析解

波浪在大尺寸结构表面产生不可忽略的散射波, 该散射波在多柱体体系中继续传播, 并在同体系中的其他柱体上产生高次散射波. 本文基于椭圆坐标系和绕射波理论首先推导了波浪作用下椭圆单柱体产生的散射波压力公式, 随后考虑该散射波在多柱体系中的传播, 将其视为第二次入射波, 推导出柱体上第二次散射波压力公式, 同理可以推导出高次散射波压力公式, 最后得到椭圆多柱体波浪力解析解, 并用数值解验证了本文解析方法的正确性. 本文以双柱体和四柱体体系为例, 分析了不同参数(波数、净距、波浪入射角度等)下, 高次散射波对柱体上波浪作用的影响. 结果表明: 波数较大的情况下, 高次散射波引起柱体上的波浪力不能忽略; 结构间距较大的情况下, 虽然高次波的作用有减小的趋势但仍然明显; 高次散射波来自多个柱体对入射波的散射, 柱体数目的增加后, 高次波的影响会增加, 结构所受的高次波作用因参数变化而起的波动会变剧烈; 高次波对上游柱体波浪力的贡献较对下游柱体的贡献大.      

The application of finite element analysis to the solution of Stokes wave diffraction problems

A new finite-element based method of calculating non-linear wave loads on offshore structures in extreme seas is presented in this paper. The diffraction wave field is modelled using Stokes wave theory developed to second order. Wave loads and free surface elevations are obtained for fixed surface-piercing structures by solving a boundary value problem for the second-order velocity potential. Special attention has been given to the radiation condition for the second-order diffraction field. Results are presented for three test examples, the vertical cylinders of Kim and Yue and of Chakrabarti, and an elliptic cylinder. These results demonstrate that early problems with the application of second-order theory arising from inadequate radiation conditions have been overcome.     

On solitary wave diffraction by multiple,in-line vertical cylinders

The interaction of solitary waves with multiple, in-line vertical cylinders is investigated. The fixed cylinders are of constant circular cross section and extend from the seafloor to the free surface. In general, there are N of them lined in a row parallel to the incoming wave direction. Both the nonlinear, generalized Boussinesq and the Green–Naghdi shallow-water wave equations are used. A boundary-fitted curvilinear coordinate system is employed to facilitate the use of the finite-difference method on curved boundaries. The governing equations and boundary conditions are transformed from the physical plane onto the computational plane. These equations are then solved in time on the computational plane that contains a uniform grid and by use of the successive over-relaxation method and a second-order finite-difference method to determine the horizontal force and overturning moment on the cylinders. Resulting solitary wave forces from the nonlinear Green–Naghdi and the Boussinesq equations are presented, and the forces are compared with the experimental data when available.     

Numerical predictions for unsteady viscous flow past an array of cylinders

The unsteady two-dimensional flow around an array of circular cylinders submerged in a uniform onset flow is analysed. The fluid is taken to be viscous and incompressible. The array of cylinders consists of two horizontal rows extending to infinity in the upstream and downstream directions. The centre-to-centre distance between adjacent cylinders is fixed at three diameters, and the rows are staggered. Advantage is taken of spatially periodic boundary conditions in the flow direction. This reduces the computational domain to a rectangular region surrounding a single circular cylinder. Two cases, for Reynolds numbers of 1000 and 10,000, are presented.     

短峰波与双层开孔圆筒柱相互作用的数值分析

应用比例边界有限元法（SBFEM）研究了短峰波与双层开孔圆筒柱相互作用。该方法将整个计算域划分成两个有限子域和1个无限子域,利用SBFEM使空间维数降低一阶,并在降维方向保持解析的特点,只需对求解域外圆柱边界进行离散。通过比例坐标与直角坐标之间的转换,并利用变分原理推导了各个子域的SBFEM方程。然后,SBFEM针对有限域和无限域分别采用贝塞尔函数和汉克尔函数作为基函数来求解对应域的解,并将计算结果与解析解进行比较,验证了该方法是一种用很少单元便能得到精确结果的高效方法。进一步研究了诸如相对波数ka、内外柱半径比a/b和内外壁孔隙影响系数G等对双层开孔圆筒柱结构所受波浪荷载、结构内外侧波浪爬升及绕射波轮廓的影响。为双层开孔圆筒柱水动力分析和结构设计提供了有价值的参考。     

THE DIFFRACTION OF SECOND-ORDER BICHROMATIC WAVES BY A SEMI-IMMERSED HORIZONTAL RECTANGULAR CYLINDER

The diffraction of second-order bichromatic Stokes waves by a semi-immersed horizontal rectangular cylinder (prism) is investigated theoretically. The problem is assumed two-dimensional and the fluid domain is divided into three regions: upwave, beneath and downwave of the structure. Analytical expressions for the velocity potentials in each region at both first- and second-order are obtained by an eigenfunction expansion approach. The solutions in each fluid region are linked through matching conditions on the imaginary fluid interfaces between them. Semi-analytical expressions are derived for the sum-and difference-frequency hydrodynamic loads and the free-surface elevations upwave and downwave of the structure to second-order. Numerical results are presented which illustrate the influence of the different wave and structural parameters on these quantities at both first- and second-order.     

Hysteresis phenomenon at interaction of shock waves generated by a cylinder array

This paper describes investigations of the interaction between bow shock waves generated by cylindrical bodies in a supersonic flow. Numerical simulations are performed using the inviscid Euler equations for cylinders whose axes are parallel to each other and normal to the flow direction. Mostly an infinite periodical cylinder array is considered, but the case of two cylinders is also briefly discussed. Three different regimes of the shock wave interaction, a regular interaction, a Mach interaction, and a choked flow, have been observed for the flow through the periodical cylinder array. In the case of the flow around two bodies, the choked flow is replaced by a regime with a collective bow shock. The transition between different flow regimes is studied by varying the inflow Mach number or the distance between the cylinders. A hysteretic behavior at the transition between the regular and Mach interactions has been observed. The transition is governed by the theoretical detachment and von Neumann criteria based on the local shock wave inclination at the interaction point.     

Propagation and scattering of waves by dense arrays of impenetrable cylinders in a waveguide

A coupled numerical scheme, based on modal expansions and boundary integral representations, is developed for treating propagation and scattering by dense arrays of impenetrable cylinders inside a waveguide. Numerical results are presented and discussed concerning reflection and transmission, as well as the wave details both inside and outside the array. The method is applied to water waves propagating over an array of vertical cylinders in constant depth extended all over the water column, operating as a porous breakwater unit in a periodic arrangement (segmented breakwater). Focusing on the reflection and transmission properties, a simplified model is also derived, based on Foldy–Lax theory. The latter provides an equivalent index of refraction of the medium representing the porous structure, modeled as an inclusion in the waveguide. Results obtained by the present fully coupled and approximate models are compared against experimental measurements, collected in wave tank, showing good agreement. The present analysis permits an efficient calculation of the properties of the examined structure, reducing the computational cost and supporting design and optimization studies.     

Lattice Boltzmann simulation of non-Newtonian flows past confined cylinders

A second-order lattice Boltzmann algorithm is used for Power-Law non-Newtonian flow simulation. The shear dependent behavior of the fluid is implemented through calculating the shear locally from the lattice distribution functions. A step by step verification procedure is taken to ensure the accuracy and the physical correctness of the numerical simulation. The flow past a series of tandem arrangement of two cylinders is computed in a confined domain. The effects of Reynolds number, the Power-Law index, and the distance between two cylinders on both the flow field and the drag coefficients of the cylinders are examined in detail.     

Axisymmetric longitudinal wave propagation in a finite prestretched compound circular cylinder made of incompressible materials

The propagation of an axisymmetric longitudinal wave in a finite prestrained compound (composite) cylinder is investigated using a piecewise-homogeneous body model and the three-dimensional linearized theory of wave propagation in prestressed body [13–15]. The inner and outer cylinders are assumed to be made of incompressible neo-Hookean materials. Numerical results on the influence of the prestrains in the inner and outer cylinders on wave dispersion are presented and discussed. These results are obtained for the case where the inner solid cylinder is stiffer than the outer hollow cylinder. In particular, it is established that the pretension of the cylinders increases the wave velocity     

利用 Laplace 变换确定双层厚壁长圆筒的轴对称界面碰撞压力

利用Ｌａｐｌａｃｅ变换，考虑轴对称弹性波的影响，采用特征函数展开法求解双层厚壁长圆筒受爆炸载荷作用下的轴对称弹性碰撞冲击问题，着重研究前几次碰撞冲击引起的轴对称界面碰撞压力。并对轴对称界面碰撞压力与间隙量、爆炸载荷幅值、爆炸载荷衰减系数之间的关系以及相关的动力响应作了初步的分析。     

波浪与外壁开孔双筒柱群的相互作用

应用速度势的特征函数展开和透空壁内两壁间压力差和流体速度成正比的线性模型,建立 了波浪与外壁开孔同轴双筒桩柱群相互作用的线性解析解. 应用这一模型进行了数值计算, 用以检验孔隙系数对双筒柱上的波浪力和波面高度的影响. 结果表明,外壁孔隙系数的增加 对减小波浪力和柱外波面高度有很大影响.     

Diffraction by a grating of cylinders with an arbitrary cross-section

The diffraction of a monochromatic plane electromagnetic wave by a grating consisting of parallel, electrically perfectly conducting cylinders is investigated. Two separate two-dimensional scalar problems are dealt with viz. the case ofE-polarization and the case ofH-polarization. A new Green's function formulation of the problem is employed leading to integral equations for the unknown field distributions on an elementary scatterer of the grating. Numerical results are presented for a grating of cylinders with either square or circular cross-section.     

Diffraction of cnoidal waves by vertical cylinders in shallow water

Diffraction of nonlinear waves by single or multiple in-line vertical cylinders in shallow water is studied by use of different nonlinear, shallow-water wave theories. The fixed, in-line, vertical circular cylinders extend from the free surface to the seafloor and are located in a row parallel to the incident wave direction. The wave–structure interaction problem is studied by use of the nonlinear generalized Boussinesq equations, the Green–Naghdi shallow-water wave equations, and the linearized version of the shallow-water wave equations. The wave-induced force and moment of the Green–Naghdi and the Boussinesq equations are presented when the incoming waves are cnoidal, and the forces are compared with the experimental data when available. Results of the linearized equations are compared with the nonlinear results. It is observed that nonlinearity is very important in the calculation of the wave loads on circular cylinders in shallow water. The variation of wave loads with wave height, wavelength and the spacing between cylinders is studied. Effect of the neighboring cylinders, and the shielding effect of upwave cylinders on the wave-induced loads on downwave cylinders are discussed.     

Second-order torsion problem of a homogeneous isotropic compressible multiply-connected elastic cylinder

For simply-connected regions, some solutions are available for the second-order torsion problem of homogeneous isotropic compressible elastic cylinders based on the theory given by Green and others. In the present paper, these theories are extended to cover the second-order torsion problem for multiply-connected regions. As an example, results for torsion of a confocal elliptical ring are given.