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.     

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 surface wave radiation generated by multiple cylinders oscillating with identical frequency

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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.     

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.     

Hydrodynamic forces and moments due to interaction of linear water waves with truncated partial-porous cylinders in finite depth

An exact analytical method is employed for studying the diffraction problems in an ocean due to the presence of a specific type of cylinders. In this current work, two models are studied: (i) a floating surface-piercing truncated partial-porous cylinder, (ii) a surface-piercing truncated partial-porous cylinder placed at the bottom. In both cases, the configuration of the composite cylinder is such that it consists of an impermeable inner cylinder rising above the free surface and a coaxial truncated porous cylinder around the lower part of the inner cylinder with the top of the porous cylinder being impermeable. By using linear water wave theory, a three-dimensional representation of the problem is developed based on eigenfunction expansion method. The condition on the porous boundary is defined by applying Darcy’s law. Pressure and velocity satisfy continuity conditions across the linear interface between the adjacent fluid domains. Hydrodynamic force, moment and wave run-up are calculated by using the velocity potentials. Comparisons are carried out with results of wave diffraction by a floating and bottom-mounted compound cylinder, i.e., when the whole cylinder is non-porous. Handy agreements are observed from these comparisons. Through numerical tests, various experiments are carried out to investigate the impact of various parameters, such as porous coefficients, draft ratio, the ratio of inner and outer radii, the water depth etc., on hydrodynamic force, moment and wave run-up. The results clearly indicate that an appropriate optimal ratio for various parameters may be considered in designing practical ocean structures with minimum adverse hydrodynamic effect. The appearance of resonance in the results and role of porosity in mitigating resonance effect are explained. Proposal to select various appropriate parameters for the best possible effect is put forward.     

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.     

Fluid trapping by two partially shrouded rotating cylinders

In this paper we examine the prospect of using localized flow control for biomimetic fluid trapping. The problem is of interest for applications that call for guided transport of fluid volumes.The study shows that trapping can be achieved with the help of two partially shrouded rotating cylinders in a side-by-side arrangement. Secondary flows that manifest successful trapping resemble recirculation zones forming under the crests of peristaltic deformation waves, in particular with respect to their response to increasing incident flow velocity. Varying the rotation speed of the cylinders provides means to control the amount of trapped fluid.Numerical calculations to support these conclusions are presented in the paper for 0≤Re≤100 and h≈2, where Re and h are, respectively, the Reynolds number and the center-to-center distance between two cylinders divided by the cylinder diameter. Experimental validation of numerical results is performed for 0≤Re≤4.     

Diffraction of regular waves by an arbitrarily oriented system of vertical circular cylinders

The linear theory of small-amplitude waves is used to construct a solution to the problem of the diffraction of surface gravity waves by a system of arbitrarily oriented vertical circular cylinders. Analytic expressions are obtained for the wave forces and overturning moments acting on each cylinder of the system. A system of two rows of cylinders with three cylinders in each row is considered as an example. It is shown that for certain relationships between the diameter of the cylinders, the distance between them, the angle of approach of the wave, and its wavelength, the maximal values of the hydrodynamic forces acting on the cylinders of the system with allowance for the interaction of the diffracted fields may be appreciably greater than when no allowance is made for it.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 177–181, January–February, 1984.     

Rayleigh limits for effective wavenumbers of randomly distributed porous cylinders. Comparison of explicit and implicit methods

The effective wavenumber of the coherent wave propagating in a fluid containing parallel porous cylinders randomly distributed in space is derived at the Rayleigh limit for (i) explicit formulas: Independent Scattering Approximation (ISA), Waterman and Truell (WT) and Linton and Martin (LM) and (ii) for implicit formulas: Coherent Potential Approximation (CPA) and Generalized Self Consistent Method (GSCM) applied to WT and to LM. The effective mass density and bulk modulus are also derived. The validity of all the effective quantities is checked by recovering, when the porosity of the scatterers tends to zero, the case of an inhomogeneous medium of elastic cylinders.     

Second-order diffraction by a bottom-seated compound cylinder

The second-order diffraction potential around a bottom-seated compound cylinder is considered. The solution method is based on a semi-analytical formulation for the double-frequency diffraction potentials, which are properly decomposed into a rational number of components in order to satisfy all boundary conditions involved in the problem. The solution process results in two different Sturm–Liouville problems which are treated separately in the ring-shaped fluid regions defined by the geometry of the structure. The matching of the potentials on the boundaries of adjacent fluid regions is established using the ‘free’ wave components of the potentials. Different Green's functions are constructed for each of the fluid regions surrounding the body. The calculation of integrals of the pressure distribution on the free surface is carried out using an appropriate Gauss–Legendre numerical technique. The efficiency of the method described in the present work is validated through comparative calculations. Finally, extensive numerical predictions are presented concerning the complete second-order excitation and the nonlinear wave elevation for various configurations of vertical axisymmetric bodies.     

Buckling behaviour of elliptical cylindrical shells and tubes under compression

This paper presents several issues that characterize the buckling behaviour of elliptical cylindrical shells and tubes under compression. First, a formulation of Generalised Beam Theory (GBT) developed to analyse the elastic buckling behaviour of non-circular hollow section (NCHS) members is presented. Since the radius varies along the cross-section mid-line, the main concepts involved in the determination of the deformation modes are adapted to account for the specific aspects related to elliptical cross-section geometry. After that, two independent sets of fully orthogonal deformation modes are determined: (i) local-shell modes satisfying the null membrane shear strain but exhibiting transverse extension and (ii) shell-type modes satisfying both assumptions of null membrane shear strain and null transverse extension. In order to illustrate the application, capabilities and versatility of the formulation, the local and global buckling behaviour of elliptical hollow section (EHS) members subjected to compression is analysed. In particular, in-depth studies concerning the influence of member length on the variation of the critical load and corresponding buckling mode shape are presented. Moreover, the GBT results are compared with estimates obtained by means of shell finite element analyses and are thoroughly discussed. The results show that short to intermediate length cylinders buckle mostly in local-shell modes, exhibiting only transverse extension, while intermediate length to long cylinders buckle mostly in shell-type modes (distortional and global modes), which are characterized by transverse bending and primary warping displacements. It is also shown that the present formulation is very efficient from the computational point of view since only three deformation modes (one local-shell, one distortional and one global) are required to evaluate the buckling behaviour of EHS cylinders for a wide range of lengths.     

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.     

Stokes flow of micro-polar fluids by peristaltic pumping through tube with slip boundary condition

This paper studies the Stokes flow of micro-polar fluids by peristaltic pumping through the cylindrical tube under the effect of the slip boundary condition. The motion of the wall is governed by the sinusoidal wave equation. The analytical and numerical solutions for the axial velocity, the micro-polar vector, the stream function, the pressure gradient, the friction force, and the mechanical efficiency are obtained by using the lubrication theory under the low Reynolds number and long wavelength approximations. The impacts of the emerging parameters, such as the coupling number, the micro-polar parameter, the slip parameter on pumping characteristics, the friction force, the velocity profile, the mechanical efficiency, and the trapping phenomenon are depicted graphically. The numerical results infer that large pressure is required for peristaltic pumping when the coupling number is large, while opposite behaviors are found for the micro-polar parameter and the slip parameter. The size of the trapped bolus reduces with the increase in the coupling number and the micro-polar parameter, whereas it blows up with the increase in the slip parameter.     

流体动力干扰对单排圆柱桩列波浪力的影响

多物体之间的流体动力干扰特性对超大型海洋结构物的设计和研究十分重要用波动源在截面周线上分布的方法，就垂直桩柱间三维流体动力干扰对波浪力的影响进行了系统的研究，桩柱的数目可达１００余根得到了柱间流体动力干扰力学机理的若干新的特性尤应指出的是，当桩柱根数超过某一数量后，桩柱上的受力表现出有规律的连续依赖性当柱数很大时，无论柱数是奇数还是偶数，中间大部分的桩柱都将表现出均匀的受力特征这些特性的发现对其他形式多体结构物流体动力干扰的研究也有重要的借鉴和指导意义     

Numerical Modeling of the Performance of Lubricated Journal Bearings Using Navier-Stokes Equations

Two-dimensional incompressible Navier-Stokes equations are solved numerically to model the thermohydrodynamic performance of a dynamically loaded journal bearing which is modeled as eccentrically rotating cylinders. The region between those cylinders are occupied by Newtonian lubricants, whose physical properties such as viscosity and thermal conductivity are assumed to be functions of local temperature. A single domain pseudospectral method which combines Fourier expansions and Chebyshev polynomials for spatial discretization is introduced in conjunction with appropriate time marching scheme for the unsteady incompressible Navier-Stokes equations. The selection of these polynomial functions is favorable since both FFT algorithms for Fourier and Chebyshev expansions are easily available. In this numerical model, the journal is dynamically loaded by an external force and set free, so that its center moves in such a way to strike a balance between the applied load and the hydrodynamic forces. The pseudo-spectral scheme is then applied to a few classical problems, such as concentric rotating cylinders and journal bearings with lubricants of constant and varying (temperature dependent) viscosity to establish the validity of the numerical scheme in simulating these problems realistically as well as to gauge the convergence characteristics and relevant numerical issues. The numerical modeling has been found to be reasonably accurate and robust enough to serve as a tool for the study the flow in the region between the journal and the bearing.     

Trapped modes in bent elastic rods

We investigate the existence of trapped modes in elastic rods of constant circular cross-section that possess bends of arbitrary curvature and straighten out at infinity; such trapped modes consist of finite energy localized in regions of maximal curvature. An asymptotic model assuming smallness of dimensionless curvature is developed to describe the trapping. Existence conditions depending on Poisson’s ratio are offered, and the equations from which they derive are numerically validated. A physical explanation of why trapped modes should be expected is also given.     

Branching of secondary modes in the flow between rotating cylinders

The steady secondary flows (Taylor vortices) of a viscous incompressible fluid between concentric rotating cylinders are studied. The range of wave numbers a and Reynolds numbers R in which there are several secondary flow modes is determined. The branching of these modes is studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp.47–53, January–February, 1977.     

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.     

Experiments on the dynamic behavior of fluid-coupled concentric cylinders

This paper describes an experimental vibration study of fluid-coupled coaxial cylinders that simulates the vibration of a reactor vessel with a thermal liner. The model cylinders are made of acrylic. Thickness and gap-size parameter studies are performed by a series of different compinations of three outside cylinders and nine inside cylinders that have variable thicknesses and diameters. Damping ratios are measured on a mode-by-mode basis for several combinations of cylinders. The vibrated cylinders are mounted to a rigid stand, with the cuter cylinder supported at both ends and the inner cylinder supported at either one end (pendulum mode) or both ends, as the case may be. The natural frequencies are obtained first in air and then with coaxial cylinders coupled by water. The mode shapes are obtained by circumferential (shell modes) and axial (shell/beam modes) mapping of the response with two diametrically opposite ‘roving’ Dymac eddy probes. In general, the natural vibration of the system has two distinct responses in-phase and/or out-of-phase modes, i.e., the radial displacement phase relationship between inner and outer cylinders. In the out-of-phase modes the frequency is shown to decrease to either zero or a very low limiting value as the gap size cecreases. The opposite occurs for in-phase modes. Damping ratios are found to be much higher for out-of-phase modes and for relatively rigid cylinders than for in-phase modes and flexible cylinders, respectively.