Water wave interaction with surface-piercing porous cylinders using the null-field integral equations

Following the successful experiences of solving water wave scattering problems for multiple impermeable cylinders by the authors' group, we extend the null-field integral formulation in conjunction with the addition theorem and the Fourier series to deal with the problems of surface-piercing porous cylinders in this paper. In the implementation, the null-field point can be exactly located on the real boundary free of calculating the Cauchy and Hadamard principal values, thanks to the introduction of degenerate kernels (or separable kernels) for fundamental solutions. This method is a semi-analytical approach, since errors attribute from the truncation of the Fourier series. Not only a systematic approach is proposed but also the effect on the near-trapped modes due to porous cylinders and disorder of layout is examined. Several advantages such as mesh-free generation, well-posed model, principal value free, elimination of boundary-layer effect and exponential convergence, over the conventional boundary element method (BEM) are achieved. It is found that the disorder has more influence to suppress the occurrence of near-trapped modes than the porosity. The free-surface elevation is consistent with the results of William and Li and those using the conventional BEM. Besides, the numerical results of the force on the surface of cylinders agree well with those of William and Li. Besides, the present method is a semi-analytical approach for problems containing circular and elliptical shapes at the same time.     

Numerical calculation of forces induced by short-crested waves on a vertical cylinder of arbitrary cross-section

Most off-shore oil platforms are supported by vertical cylinders extending to the ocean floor. An important problem in off-shore engineering is the calculation of the wave loading exerted on these vertical cylinders. Analytical solutions have been found for the case of plane incident waves incident on a circular cylinder by MacCamy and Fuchs [(1954), Wave forces on piles: a diffraction theory. U.S. Army Corps of Engineering, Beach Erosion Board, Technical Memorandum No. 69] and also for short-crested waves incident on a circular cylinder by Zhu [(1993), Diffraction of short-crested waves around a circular cylinder. Ocean Engng 20, 389–407]. However, for a cylinder of arbitrary cross-section, no analytic solutions currently exist. Au and Brebbia [(1983), Diffraction of water waves for vertical cylinders using boundary elements. Appl. Math. Modelling 7, 106–114] proposed an efficient numerical approach to calculate the wave loads induced by plane waves on vertical cylinders by using the boundary element method. However, wind-generated waves are better modelled by short-crested waves. Whether or not these short-crested waves can induce larger wave forces on a structure is of great concern to ocean engineers. In this paper wave loads, induced by short-crested incident waves, on a vertical cylinder of arbitrary cross-section are discussed. For a cylinder of certain cross-section, the wave loads induced by short-crested waves can be larger than those induced by plane waves with the same total wave number.     

大尺度海工圆柱的非线性波浪载荷

引入Stokes二阶水波理论,建立了对海洋工程中大尺度综合型圆柱式结构波浪载荷的二阶修正统一积分算式。通过对波势特征解法的推广及引入干扰波叠加法、最小二乘法,使原仅适合于轴对称单柱的二阶散射波浪载荷快速积分算法可完全适用于类型广泛的任意型实用圆柱的计算。几则实例(圆台,单、双柱)的计算检验了该方法的有效性。     

Numerical simulation of solitary wave scattering by a circular cylinder array

The interaction of a solitary wave with an array of surface-piercing vertical circular cylinders is investigated numerically. The wave motion is modeled by a set of generalized Boussinesq equations. The governing equations are discretized using a finite element method. The numerical model is validated against the experimental data of solitary wave reflection from a vertical wall and solitary wave scattering by a vertical circular cylinder respectively. The predicted wave surface elevation and the wave forces on the cylinder agree well with the experimental data. The numerical model is then employed to study solitary wave scattering by arrays of two circular cylinders and four circular cylinders respectively. The effect of wave direction on the wave forces and the wave runup on the cylinders is quantified.     

三阶Stokes型随机波浪载荷谱研究

应用非线性谱分析理论，对三阶Stokes型随机波浪载荷谱进行了分析，将波面方程及海水质点的水平速度用一阶波面的非线性组表示，导出了随机波浪谱的表达式。为了便于求解随机波浪的载荷谱，将阻力项展开为幂级数式，并应用非线性谱分析理论，确定了幂级数的系数，进而将波浪载荷表示为一阶波面及其导数的非线性组合，最后得出波浪载荷谱密度的表达式，并应用数值分析方法，得出单位桩柱波浪力及总波浪力谱密度。     

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.     

Flume confinement effect on wave-induced dynamic pressures on twin-tandem cylinders

The estimates of wave-induced dynamic pressures around surface piercing twin-tandem rigid vertical circular cylinders fixed at the bottom of a wave flume, obtained using the source distribution method, are compared with the measured values. The influence of flume confinement on dynamic pressures is estimated by the method of images. The experiments are carried out on cylinders of 40 cm diameter mostly in deep water conditions in a wave flume 4 m wide, for different values of scattering parameters and spacings between the cylinders.     

Fully nonlinear analysis of near-trapping phenomenon around an array of cylinders

The wave diffraction around an array of fixed vertical circular cylinders is simulated in a numerical wave tank by using a fully nonlinear model in the time domain. The emphasis of the paper lies in the insightful investigation of the nonlinear properties of the near-trapping phenomenon associated with the multiple cylinders. The numerical model is validated by analytical solutions as well as experimental data for waves propagating past two and four vertical cylinders in certain arrangements. An array of four identical circular cylinders at the corners of a square with an incident wave along the diagonal of the square is the main focus here for investigating the near-trapping phenomenon. When near-trapping occurs, the present study shows that an extremely high wave elevation near the cylinders can be observed. At the same time, the hydrodynamic forces on different cylinders are found to be either in phase or out of phase, leading to some characteristic force patterns acting on the whole structure. Due to the nature of the numerical model adopted, nonlinearity at different orders can be captured using a harmonic analysis. In addition to first- and second-order near-trapping, the third-order (triple-frequency) nonlinear component is presented for the first time. For the configuration selected, it is found that at one specific incident wave frequency and direction one trapped mode is excited by second-order effects, while a different trapped mode (having similar symmetries) is excited by the third harmonic of the incident wave frequency.     

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.     

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].     

张力腿平台水动力参数计算

张力腿平台作为一种深水平台 ,其半顺应半固定的运动特征 ,成为在复杂的深海海洋环境中进行海洋石油勘探、开采一种重要结构型式。在张力腿平台波浪载荷的计算中根据产生波动流场的因素不同将波浪载荷分为绕射效应和辐射效应。本文对辐射效应深入研究并以附加质量、附加阻尼等水动力参数的形式计算了辐射效应对张力腿平台的载荷影响。在计算水动力参数时将平台简化为浮式直立柱群 ,采用势流理论 ,引进改进平面波法计入柱体间水动力相互作用得到辐射波速度势的半解析解 ,引入大间距假设、通过非平面波修正 ,精确地求解了柱群的辐射波的载荷作用     

Modelling linear wave transformation induced by dissipative structures—Random waves

The relevant theory is presented and numerical results are compared with the analytical solution for the interaction of non-breaking waves with an array of vertical porous circular cylinders on a horizontal bed. The extension to the cases of unidirectional and multidirectional waves is obtained by means of a transfer function. The influence of the mechanical properties of porous structures and wave irregularity on wave transformation is analysed. Results for unidirectional and multidirectional wave spectra are compared to those obtained for regular waves. The model presented reproduces well the analytical results and provides a tool for analysing several engineering problems.     

Interaction of regular waves with a group of dual porous circular cylinders

Diffraction of linear waves around a group of dual porous cylinders consisting of a thin and porous outer cylinder with an impermeable inner cylinder is investigated analytically based on the eigenfunction expansion method proposed by Spring and Monkmeyer [Spring BH, Monkmeyer PL. Interaction of plane waves with vertical cylinders. In: Proceedings 14th international coastal engineering conference. 1974. p. 1828–47] and further modified by Linton and Evans [Linton CM, Evans DV. The interaction of waves with arrays of vertical circular cylinders. Journal of Fluid Mechanics 1990;215:549–69]. The present formulation is an extension of the work of Wang and Ren [Wang KH, Ren X. Wave interaction with a concentric porous cylinder system. Ocean Engineering 1994;21(4):343–60], wherein; the interaction of linear waves with a single concentric porous cylinder system was studied. This paper aims at investigating the influence of multiple interactions between the cylinders in the group on the hydrodynamic wave forces, wave run-up and free-surface elevation in their vicinity. Further, the study focuses on the variation of the forces and run-up on the individual cylinders within the group compared to that on isolated cylinders.     

层化海洋中的二阶波散射作用问题

引入Stokes有限振幅水波的多色波作用机制 ,采用二层层化海洋模型 ,分析了层化海洋中非线性波对结构的散射作用问题。提出了二层海洋中二阶水波散射条件的数学表达式 ,进而推导了二层海洋中二阶波对一般型圆柱散射作用的格林积分解式 ,并给出了对解式中自由面及分层界面无限积分的递推算式。     

正交直墙前直立圆柱的波浪绕射的解析研究

应用镜像原理,将正交直墙前单个圆柱的波浪绕射问题,转化为4个对称布置圆柱的4向入射波浪的绕射问题。然后应用速度势的特征展开方法和Graf加法定理,建立了正交直墙前垂直圆柱的波浪绕射解析解。通过数值计算研究了圆柱与正交直墙间距离大小、波浪入射角等因素与圆柱上总波浪作用力的解析关系。     

Approximate hydrodynamic analysis of multi-column ocean structures

An approximate, semi-analytical technique is developed to calculate the wave-induced loads and added-mass and radiation damping coefficients for idealized ocean structures consisting of arrays of large-diameter, truncated, vertical circular cylinders. The approach involves replacing the complete wave field scattered/radiated by one cylinder in the vicinity of another cylinder by a plane wave together with a non-planar correction term and so is essentially a large-spacing approximation. Numerical results are presented which illustrate the accuracy of the approximate approach and provide estimates of the hydrodynamic characteristics of several practical array configurations.     

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.     

Water wave interaction with an array of bottom-mounted surface-piercing porous cylinders

The interaction of water waves with arrays of bottom-mounted, surface-piercing circular cylinders is investigated theoretically. The sidewall of each cylinder is porous and thin. Under the assumptions of potential flow and linear wave theory, a semi-analytical solution is obtained by an eigenfunction expansion approach first proposed for impermeable cylinders by Spring and Monkmeyer (1974), and later simplified by Linton and Evans (1990). Analytical expressions are developed for the wave motion in the exterior and all interior fluid regions. Numerical results are presented which illustrate the effects of various wave and structural parameters on the hydrodynamic loads and the diffracted wave field. It is found that the porosity of the structures may result in a significant reduction in both the hydrodynamic loads experienced by the cylinders and the associated wave runup.     

On the estimation of Morison force coefficients and their predictive accuracy for very rough circular cylinders

This paper makes an assessment of the various method that may be used to analyse experiment data on the force experienced by a circular cylinder in waves and combined wave and current flows to estimate drag and inertia coefficients for use in Morison's equation. Most of the widely used techniques are considered together with a weighted least squares approach for time domain analysis. A set of data obtained from experiments on heavily roughened circular cylinders of diameters 0.513 and 0.216 m in the Delta wave flume at De Voorst in Holland in waves and simulated current has been analysed in turn by all these techniques. The experiment data was split into two halves. The first was used for the analyses and the second was used to assess the predictive accuracy of Morison's equation. Using the force coefficients obtained from the different analysis techniques corresponding predicted force time series were constructed using the particle kinematics measured in the second parts of the data sets. These predicted time series were then compared with the corresponding measured force time histories. The root mean square error and the bias in the estimation of maximum force in each wave cycle are used as measures of predictive accuracy and as a basis for comparing the efficiency of the different analysis techniques. It was found that the weighted least square method generally gave the best predictive accuracy, but only by a small margin.     

Effects of Second Order Scattered Wave in Stratified Ocean

By using the theories on Stokes multicolored water waves and taking the two- layer ocean as a basic model of stratified ocean, the paper analyzes the problems related to the effects of the nonlinear water wave on offshore structures. A mathematical expression is presented to describe second order wave radiation conditions. Using integral principle, the analytical integral solutions are given to evaluate second order scattered wave loads on general vertical circular cylinders in the two-layer ocean, and the special recurrence formulas for infinite integrals over free and stratified surfaces are derived.