Nonlinear free surface action with an array of vertical cylinders

Nonlinear diffraction of regular waves by an array of bottom-seated circular cylinders is investigated in frequency domain, based on a Stokes expansion approach. A complete semi-analytical solution is developed which allows an efficient evaluation of the second-order potentials in the entire fluid domain, and the wave forces on the structure. Expressions are derived for the second-order potential in the vicinity of individual cylinders. These expressions have a simple form, thus providing an effective means for investigating the wave enhancement due to nonlinear interactions with multiple cylinders. Based on the present method, the wave run-up and free-surface elevations around an array of two, three and four cylinders are investigated numerically.     

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。     

Numerical simulation of a solitary wave interaction with submerged multi-bodies

The problems of a solitary wave passing over rectangular cylinders have been analysed. The numerical simulation is based on the full nonlinear two-dimensional Navier-Stokes equations which are solved by the finite difference method. The free surface is dealt with by the Volume of Fluid method (VOF). Results for a solitary wave passing over a single cylinder are compared with the experimental data of Seabra-Santos, Penouard and Temperville^[2] and better agreement is obtained than those obtained from the long wave equation based on the potential flow theory. Results are also given for two cylinders with different gaps. The project supported by the National Natural Science Foundation of China and the Development Foundation of Science and Technology of Shanghai Education Committee and the Royal Society.     

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.     

Water surface wave radiation generated by multiple cylinders oscillating with identical frequency

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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.     

WAVE PROPAGATION IN A TRANSVERSELY ISOTROPIC THERMOELASTIC SOLID CYLINDER OF ARBITRARY CROSS-SECTION

The wave propagation in an infinite, homogeneous, transversely isotropic solid cylinder of arbitrary cross-section is studied using Fourier expansion collocation method, within the frame work of linearized, three-dimensional theory of thermoelasticity. Three displacement potential functions are introduced, to uncouple the equations of motion and the heat conduction. The frequency equations are obtained for longitudinal and flexural (symmetric and antisymmetric) modes of vibration and are studied numerically for elliptic and parabolic cross-sectional zinc cylinders. The computed non-dimensional wave numbers are presented in the form of dispersion curves.     

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.     

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.     

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.     

Assessment of assembly shape and geometric effects on amplification of shear waves using discrete element method

The discrete element method (DEM) is a capable tool used to simulate shear wave propagation in granular assemblies for many years. Researchers have studied assembly shapes such as rectangles (in 2D simulations) or cylinders and cubes (in 3D simulations). This paper aimed to qualify the effect of assembly shape on the shear wave propagation and maximum amplification in the vertical plane (horizontal and vertical directions) caused by this propagation. To this end, shear wave propagations in different assembly shapes such as rectangle, trapezium, and triangle with rigid boundary conditions were simulated. A sine wave pulse was applied with a point source by moving a particle as the transmitter particle. To evaluate the shear wave velocity of the assemblies, the transmitter and receiver particles were simulated. All the simulations were performed with 2D DEM which is a useful tool to determine the amount and location of the maximum amplification factor of the assembly in both horizontal and vertical directions. An advantage of this study was assessing the effect of parameters such as input wave frequency, assembly height, shape, and aspect ratios on the amplification of the input waves.     

Hydroelastic analysis of surface wave interaction with concentric porous and flexible cylinder systems

The present study deals with the hydroelastic analysis of gravity wave interaction with concentric porous and flexible cylinder systems, in which the inner cylinder is rigid and the outer cylinder is porous and flexible. The problems are analyzed in finite water depth under the assumption of small amplitude water wave theory and structural response. The cylinder configurations in the present study are namely (a) surface-piercing truncated cylinders, (b) bottom-touching truncated cylinders and (c) complete submerged cylinders extended from free surface to bottom. As special cases of the concentric cylinder system, wave diffraction by (i) porous flexible cylinder and (ii) flexible floating cage with rigid bottom are analyzed. The scattering potentials are evaluated using Fourier–Bessel series expansion method and the least square approximation method. The convergence of the double series is tested numerically to determine the number of terms in the Fourier–Bessel series expansion. The effects of porosity and flexibility of the outer cylinder, in attenuating the hydrodynamic forces and dynamic overturning moments, are analyzed for various cylinder configurations and wave characteristics. A parametric study with respect to wave frequency, ratios of inner-to-outer cylinder radii, annular spacing between the two cylinders and porosities is done. In order to understand the flow distribution around the cylinders, contour plots are provided. The findings of the present study are likely to be of immense help in the design of various types of marine structures which can withstand the wave loads of varied nature in the marine environment. The theory can be easily extended to deal with a large class of problems associated with acoustic wave interaction with flexible porous structures.     

New analytical solutions to water wave radiation by vertical truncated cylinders through multi-term Galerkin method

New analytical solutions to water wave radiation by vertical truncated circular cylinders are developed based on linear potential flow theory. Two typical cylinder configurations of a surface-piercing cylinder and a submerged floating cylinder are considered. The multi-term Galerkin method is employed in the solution procedure, in which the fluid velocity on the interface between different regions is expanded into a set of basis function involving the Gegenbauer polynomials, and the cube-root singularity of fluid velocity at the side edges of the truncated cylinders is correctly modeled. The present solutions have the merits of very rapid convergence. The results with six-figure accuracy for added mass and radiation damping can be obtained using a few truncated numbers in the basis function for three motions (surge, heave and roll). The calculated results of the present solutions agree well with that by a higher-order boundary element method solution. Calculation examples are presented to investigate the influence of the motion frequency on the added mass and the radiation damping of the truncated cylinders with different geometric parameters. The present solutions can be used as a reliable benchmark for numerical solutions to water wave radiation by complicated structures.

     

阶梯压电层合梁的波动动力学特性

采用行波理论系统地研究了压电阶梯梁的自由振动分析以及强迫响应的分析方法. 基于分布 参数理论研究了压电阶梯梁的波传播特性，忽略柔性梁横向剪切和转动惯量的影响，给出了 梁的轴向和横向的简谐波解. 将压电阶梯梁离散化为单元，考虑压电片的刚度和质量的影响， 建立了节点散射模型. 应用位移连续和力平衡条件，推导了节点的波反射和波传递矩阵，在 此基础上，引入波循环矩阵的概念，给出波循环矩阵、波传递系数矩阵的确定方法. 应用波 循环矩阵可以有效地计算结构的固有频率. 另外，应用波传递系数研究了压电陶瓷作动器位 置对其驱动能力的影响. 得出两个主要结论：1)作动器靠近悬臂梁固定端将有较强的驱动 能力，悬臂梁边界反射行波产生弯曲消失波有利于增大压电波的模态传递系数；2)模态传 递系数与固有频率的灵敏度密切相关，波传递系数越大, 对应该处固有频率变化灵敏度越大. 另外，数值算例表明了行波方法比有限元方法具有更高的计算精度.     

孤立波与多个潜水障碍物相互作用的数值研究

本文以完整二维Navier-Stokes方程为控制方程,采用VOF界面跟踪技术和差分方法,数值计算孤立波与多个淹没水下物体相互作用的问题。本文对潜水物体的高度接近水深时,孤立波通过水下孤立直立方柱、两个间距较大的水下直立方柱和两个间距较小的水下直立方柱等三种情况分别进行了计算,给出了波形随时间的演化图,可以看到反射波、前传波和跟随振荡型小波列的生成。对孤立波通过水下孤立直立方柱情形的计算结果,与实验结果和势流理论结果进行了比较。     

压电空心圆柱中的波特性

基于三维线性压电弹性理论,采用Legendre多项式方法研究了电开路时压电空心圆柱中轴向波的传播特性.给出了结构的频散曲线及其相应的非压电情况,展示了压电的影响.比较了压电对轴对称模态和弯曲模态影响的区别.从电势分布的角度分析了压电对弯曲扭转波的影响.最后讨论了径厚比和极化方向对波传播压电效应的影响.     

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     

Calculation of autooscillations of the type of azimuthal waves occurring at loss of stability of flow of a viscous fluid between concentric cylinders rotating in opposite directions

Starting with the Navier-Stokes equation we use the Lyapunov-Schmidt method to investigate the nature of the loss of stability of Couette flow between cylinders as the Reynolds number passes through its critical value. We consider the rotation of the cylinders in opposite directions with the ratio of the angular velocities such that the role of the most dangerous disturbances passes over from rotationally symmetric to nonrotationally symmetric disturbances. Branching nonstationary secondary flows (autooscillations) are found in the form of azimuthal waves; the longitudinal wave number and the azimuthal wave number m are assumed given. The amplitude of autooscillations and the wave velocity are calculated for m = 1, and it is shown that depending on the value of both weak excitation of stable and strong excitation of unstable autooscillations are possible and the wave number for which the critical Reynolds number is a minimum corresponds to a stable wave regime in the supercritical region. The linear problem of the stability of the circular flow of a viscous fluid with respect to nonrotationally symmetric disturbances is discussed in [1–3]. Di Prima [1] solved the problem numerically by the Galerkin method when the gap is small and the cylinders rotate in the same direction. Di Prima's analysis is extended in [2] to cylinders rotating in opposite directions, and in [3] it is extended to gaps which are not small. The nonlinear stability problem is treated in [4], where for fixed = 3 and cylinders rotating in opposite directions the axisymmetric stationary secondary flow the Taylor vortex is calculated. The formation of azimuthal waves in the fluid between the cylinders was studied experimentally in detail by Coles [5].Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 2, pp. 68–75, March–April, 1976.     

Hydrodynamic interactions of water waves with a group of independently oscillating truncated circular cylinders

In this study, we examine the water wave radiation by arrays of truncated circular cylinders. Each cylinder can oscillate independently in any rigid oscillation mode with a prescribed amplitude, including translational and rotational modes such as surge, sway, heave, pitch, roll, and their com-binations. Based on the eigenfunction expansion and Graf ’s addition theorem for Bessel functions, we developed an ana-lytical method that includes the effects of evanescent modes in order to analyze such arrays of cylinders. To investi-gate the effects of several influential factors on convergence, our objective is to dramatically reduce the number of tests required and determine the influencing relationships between truncation number and convergence behavior for different factor combinations. We use the orthogonal test method to fulfill the objective. Lastly, we present our results regarding the effects of evanescent modes on hydrodynamic coeffi-cients.