The interaction of surface waves with fixed semi immersed cylinders having symmetric cross sections

The reflection of water waves by a semi immersed cylinder having a symmetric cross section is studied for both Dirichlet and Neumann boundary conditions on the cylinder. The method of conformal transformations as utilized by Ursell and by Tasai for the radiation problem is adapted to the present diffraction problem. The problem is solved by expansions of the reflected wave potential using nonorthogonal functions (wave free potentials). These functions are not complete, and an additional source and a dipole are required. Infinite systems of linear equations are obtained for the unknown expansion coefficients and the unknown strengths of the source and the dipole terms. Numerical results are obtained for the reflection coefficient, transmission coefficient, horizontal force on cylinder, vertical force on cylinder. In the long wave region analytical approximations are obtained for these functions when the cross section is circular. The reflection and transmission coefficients are very different for the two boundary conditions in the long wave region, the Dirichlet reflection coefficient being much larger than the corresponding Neumann coefficient. This behavior is similar to acoustic and electromagnetic diffraction problems in two dimensions. On leave of absence from Itek Corporation, Lexington (Mass.), U.S.A.     

Wave Motions in a Two-Layer Fluid Driven by Small Oscillations of a Cylinder Intersecting the Interface

The plane problem of steady-state small oscillations of a horizontal cylinder located at the interface between two fluids of different densities and indefinite depth is considered in the linear formulation. Boundary integral equations for the surface source distribution are derived. The behavior of the distributed singularities at points of intersection of the body contour and the interface is investigated. The problem of oscillations of a circular cylinder is solved by the multipole expansion method. The apparent mass and damping coefficients of the radiation problem and the reflection coefficient of the problem of scattering of an impinging wave by a floating body are calculated.     

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.

     

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.     

直角平面角点衬砌对稳态平面SH波的散射

利用波函数展开法求解了二维直角平面角点处圆弧形衬砌对稳态入射平面SH波的散射问题,得到问题的解析解．方法是首先构造出衬砌介质内外的总波场,它们能够预先满足直角平面两直角边界应力自由条件;再利用衬砌边界处的应力和位移连续条件写出确定散射波解中未知系数的方程组并求解．通过算例具体讨论了衬砌内边界处的环向应力集中系数和位移幅度比随无量纲波数、入射角的变化情况,结果表明它们存在不同程度的放大现象．     

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.     

Diffraction of solitary water waves around three-dimensional floating bodies

By using the matched asymptotic expansion method and the idea of edge layer, a mathematic model for describing the interaction between weakly nonlinear shallow-water waves and three-dimensional floating bodies is formed in the paper. As a numerical example, the diffraction of a solitary wave around a vertically floating circular cylinder has been investigated and the results are presented. The present method can further be extended to the study of wave diffraction around floating bodies of general shape. The project is supported by the National Natural Science Foundation of China.     

Convective heat loss of water in a circular pipe cooled at constant rate

A numerical study is made of the heat loss by natural convection of water within a horizontal circular cylinder with wall temperature decreasing at a constant rate. The particular situation of water with maximum density at 4 °C is formulated in dimensionless relations based on a linear relationship between the water thermal expansion coefficient and the temperature. Such an approach leads to an exhaustive solution in terms of a non linear Rayleigh number. The link is also established with the standard situation where the hypothesis of a linear relationship between density and temperature is applicable. In particular it is shown that the quasi steady state results obtained for a standard situation become equilibrium curves to which the system tends with increasing difference between the temperature of the boundary and 4 °C. A complete numerical solution is obtained for non linear Rayleigh numbers ranging between 0 and 10⁷. Previous numerical and experimental results on the horizontal circular cylinder are also discussed and recast in terms of the present dimensional approach.     

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.     

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.     

二维直角平面内固定圆形夹杂对稳态入射反平面剪切波的散射

利用复变函数法、多极坐标及傅立叶级数展开技术求解了二维直角平面内固定圆形夹杂对稳态入射反平面剪切（shearing horizontal, SH）波的散射问题。首先构造出介质内不存在夹杂时的入射波场和反射波场,然后建立介质内存在夹杂时由夹杂边界产生的能够自动满足直角边应力自由条件的散射波解，从而利用叠加原理写出介质内的总波场。利用夹杂边界处位移条件和傅立叶级数展开方法列出求解散射波中未知系数的无穷代数方程组，在满足计算精度的前提下通过有限项截断，得到相应有限代数方程组的解，最后通过算例具体讨论了二维直角平面水平边界点的位移幅度比和相位随量纲一波数、入射波入射角及夹杂位置的不同而变化的情况，结果表明了算法的有效实用性。     

Acoustic radiation force on a rigid cylinder in an off-axis Gaussian beam near an impedance boundary

The exact equations of the axial and transverse acoustic radiation force functions of a Gaussian beam arbitrarily incident on an infinite rigid cylinder close to an impedance boundary and immersed in an ideal fluid are deduced by expressing the incident wave, the scattering wave and the boundary reflected wave in terms of the cylindrical wave function. The effects of the beam waist, the sound reflection coefficient, the cylinder position and the distance from the impedance boundary on the acoustic radiation force are studied using numerical simulations. The simulation results show that the amplitude of the acoustic radiation force function increases with beam width. Moreover, the values of the acoustic radiation force in both the axial and transverse directions reach those of a plane wave when the beam width is considerably larger than the wavelength of the Gaussian beam. The properties of the impedance boundary and the position of the cylinder in the Gaussian beam have a considerable effect on the magnitude and direction of the force. The simulation results, particularly in the case of a transverse force, indicate the presence of a negative acoustic radiation force that is related to the nondimensional frequency and position of the cylinder in the Gaussian beam.     

A lattice Boltzmann model for two‐dimensional sound wave in the small perturbation compressible flows

A multi‐entropy‐level lattice Boltzmann model for two‐dimensional sound wave equation in the small perturbation flows is presented. In this model, we used higher‐order moment method, multi‐scale technique and the Chapman–Enskog expansion, and multi‐entropy‐level to obtain sound wave equation with isentropic equation. As numerical examples, the Doppler effects in the sound wave propagation, the sound scattering from circular cylinder are simulated. The numerical results show that this model can be used to simulate sound wave propagation. Copyright © 2010 John Wiley & Sons, Ltd.     

Real-time nonlinear cylinder wave force reconstruction in stochastic wave field considering second-order wave effects

This study provides a novel method for reconstructing real-time nonlinear wave forces on a large-scale circular cylinder by considering second-order wave effects. Potential theory is utilized for deriving the expression of wave forces with the measured data of wave elevation. Approximate expressions of quadratic transfer functions are built with undetermined coefficients, which are resolved by using the historical data of measured wave elevation. Two different algorithms, including fast Fourier transform (FFT) and recursive least squares (RLS), are adopted for real-time reconstruction. Hydrodynamic tests are conducted in the wave flume on a circular cylinder to examine the effectiveness of the nonlinear reconstruction method. Comparative results demonstrate that the accuracy of real-time reconstructed wave forces is significantly enhanced by the present method. The over-prediction errors at force crests and the under-prediction errors at force troughs have been reduced. Furthermore, comparative results show that the nonlinear method implemented by the FFT algorithm provides more accurate results, whereas the RLS algorithm is more time cost efficient.     

An efficient curvilinear non‐hydrostatic model for simulating surface water waves

An efficient curvilinear non‐hydrostatic free surface model is developed to simulate surface water waves in horizontally curved boundaries. The generalized curvilinear governing equations are solved by a fractional step method on a rectangular transformed domain. Of importance is to employ a higher order (either quadratic or cubic spline function) integral method for the top‐layer non‐hydrostatic pressure under a staggered grid framework. Model accuracy and efficiency, in terms of required vertical layers, are critically examined on a linear progressive wave case. The model is then applied to simulate waves propagating in a canal with variable widths, cnoidal wave runup around a circular cylinder, and three‐dimensional wave transformation in a circular channel. Overall the results show that the curvilinear non‐hydrostatic model using a few, e.g. 2–4, vertical layers is capable of simulating wave dispersion, diffraction, and reflection due to curved sidewalls. Copyright © 2010 John Wiley & Sons, Ltd.     

Short crested wave–current forces around a large vertical circular cylinder

An analytical solution for the diffraction of short crested incident wave along positive x-axis direction on a large circular cylinder with uniform current is derived. The important influences of currents on wave frequency, water run-up, wave force, inertia and drag coefficients on the cylinder profiles are investigated for short-crested incident wave. Based on the numerical results, we find wave frequency of short crested wave system is affected by incident angle and the strength of the currents. The wave frequency increases or decreases with increasing current speed following or opposing wave propagating direction. It shows that the effects of current speeds, current directions on water run-up on the circular cylinder with different radius for different wave numbers are very conspicuous when the incident wave changes from long crested plane waves to short-crested waves. With the increase of current speed, the water run-up on the cylinder becomes more and more high, and will exceed that of long crested plane wave and short crested wave case without currents even though the current speed is small. The total wave loads, inertia coefficient and drag coefficient exerted on a cylinder with currents would be larger compared to the wave loads exerted pure short-crested waves. Therefore, ocean engineers should consider the short crested wave–current load on marine constructs carefully.     

A semi-analytical approach to the study of an elastic circular cylinder confined in a cylindrical fluid domain subjected to small-amplitude transient motions

This paper deals with the transient motions experienced by an elastic circular cylinder in a cylindrical fluid domain initially at rest and subjected to small-amplitude imposed displacements. Three fluid models are considered, namely potential, viscous and acoustic, to cover different fluid–structure interaction regimes. They are derived here from the general compressible Navier–Stokes equations by a formal perturbation method so as to underline their links and ranges of validity a priori. The resulting fluid models are linear owing to the small-amplitude-displacement hypothesis. For simplicity, the elastic flexure beam model is chosen for the circular cylinder dynamics. The semi-analytical approach used here is based on the methods of Laplace transform in time, in vacuo eigenvector expansion with time-dependent coefficients for the transverse beam displacement and separation of variables for the fluid. Moreover, the viscous case is handled with a matched asymptotic expansion performed at first order. The projection of the fluid forces on the in vacuo eigenvectors leads to a fully coupled system involving the modal time-dependent displacement coefficients. These coefficients are then obtained by matrix inversion in the Laplace domain and fast numerical inversion of the Laplace transform. The three models, written in the form of convolution products, are described through the analysis of their kernels, involving both the wave propagation phenomena in the fluid domain and the beam elasticity. Last, the three models are illustrated for a specific imposed motion mimicking shock loading. It is shown that their combination permits coverage of a broad range of motions.     

波浪与外圆弧开孔壁双圆筒柱的相互作用

应用比例边界有限元法(SBFEM)研究了短峰波与圆筒外接圆弧开孔结构物的相互作用. 求解时将外接圆弧延伸构建一个虚拟圆, 该圆的孔隙影响系数可由矩阵G_0统一进行表达. 整个流场可划分成一个有限域和一个无限域. SBFEM只需对虚拟圆边界进行离散, 使空间维数降低一阶, 在圆的半径方向保持解析, 并且无限域处的辐射边界条件能自动满足. 利用变分原理推导SBFEM方程, 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得对应域的解. 将计算结果与解析解和其他数值方法进行了比较, 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算法. 进一步研究了诸如短峰波波向、结构的几何、材料参数等因素对结构所受波浪载荷及绕射波轮廓的影响, 并进行了分析.      

Interaction between a submerged evacuated cylindrical shell and a shock wave—Part II: Numerical aspects of the solution

The interaction between a submerged elastic circular cylindrical shell and an external shock wave is addressed. A linear, two-dimensional formulation of the problem is considered. A semi-analytical solution is obtained using a combination of the classical analytical approach based on the use of the Laplace transform and separation of variables, and finite difference methodology. The study consists of two parts. Part I focuses on the simulation and analysis of the acoustic fields induced during the interaction. Both the diffraction (absolutely rigid cylinder) and complete diffraction–radiation (elastic shell) are considered. Special attention is paid to the lower-magnitude shell-induced waves representing radiation by the elastic waves circumnavigating the shell. The focus of Part II is on the numerical analysis of the solution. The convergence of the series solution and finite-difference scheme is analysed. The computation of the response functions of the problem is discussed as well, as is the effect of the bending stiffness on the acoustic field. The membrane model of the shell is considered to analyse such effect, which, in combination with the models addressed in Part I, allows for the analysis of the evolution of the acoustic field around the structure as its elastic properties change from an absolutely rigid cylinder to a membrane. The results of the numerical simulations are compared to available experimental data, and a good agreement is observed.