Effect of wave-directionality on second-order loads induced by the set-down

At first-order of approximation a sea-state may be considered as an infinite sum of Airy components with angular frequencies ω_i and wave-number vectors k_i. A second-order analysis shows the co-existence of long waves appearing at the difference frequencies ω_i—ω_j with wave-number vectors k_i—k_j. In shallow water they become appreciable in amplitudes and may induce slow-drift motion of moored structures.For small values of ω_i—ω_j,k_i—k_j may take all kinds of directions for an angular-spread wave system. Then it may be questioned how the in-line and transverse second-order accelerations compare to those obtained for a mono-directional wave-system.This analysis is carried out here by relating the spectra of the second-order horizontal accelerations to the directional wave-spectrum. Numerical applications are first performed for deep water. They show that at low frequencies, even for very narrowly spread wave systems, the transverse component is larger than the in-line component. In shallow water both components are dratically reduced as compared to the mono-directional case. As a consequence one may question the validity of model-testings or numerical models which take no account of the directionality of the wave-system.     

Fully nonlinear simulations of wave resonance by an array of cylinders in vertical motions

The finite element method(FEM) is employed to analyze the resonant oscillations of the liquid confined within multiple or an array of floating bodies with fully nonlinear boundary conditions on the free surface and the body surface in two dimensions.The velocity potentials at each time step are obtained through the FEM with 8-node quadratic shape functions.The finite element linear system is solved by the conjugate gradient(CG) method with a symmetric successive overelaxlation(SSOR) preconditioner.The waves at the open boundary are absorbed by the combination of the damping zone method and the Sommerfeld-Orlanski equation.Numerical examples are given by an array of floating wedgeshaped cylinders and rectangular cylinders.Results are provided for heave motions including wave elevations,profiles and hydrodynamic forces.Comparisons are made in several cases with the results obtained from the second order solution in the time domain.It is found that the wave amplitude in the middle region of the array is larger than those in other places,and the hydrodynamic force on a cylinder increases with the cylinder closing to the middle of the array.     

Fully Nonlinear Simulations of Wave Resonance by An Array of Cylinders in Vertical Motions

The finite element method (FEM) is employed to analyze the resonant oscillations of the liquid confined within multiple or an array of floating bodies with fully nonlinear boundary conditions on the free surface and the body surface in two dimensions. The velocity potentials at each time step are obtained through the FEM with 8-node quadratic shape functions. The finite element linear system is solved by the conjugate gradient (CG) method with a symmetric successive overelaxlation (SSOR) preconditioner. The waves at the open boundary are absorbed by the combination of the damping zone method and the Sommerfeld-Orlanski equation. Numerical examples are given by an array of floating wedge- shaped cylinders and rectangular cylinders. Results are provided for heave motions including wave elevations, profiles and hydrodynamic forces. Comparisons are made in several cases with the results obtained from the second order solution in the time domain. It is found that the wave amplitude in the middle region of the array is larger than those in other places, and the hydrodynamic force on a cylinder increases with the cylinder closing to the middle of the array.     

Time domain simulation of side-by-side floating bodies using a 3D numerical wave tank approach

The coupled system of two side-by-side fixed and/or floating bodies interacting with a large amplitude nonlinear wave is studied using a direct time domain solution method. The numerical model is based on a three-dimensional mixed Eulerian–Lagrangian (MEL) method under certain simplifying approximations permitting Rankine panel scheme to be implemented over a time-invariant boundary surface to solve the boundary value problem for the unknown velocity potentials. A 4th order Adams–Bashforth–Moulton scheme is used for time marching of rigid-body motion histories of the individual bodies and evolution of the free-surface including the gap region in which large resonant fluid motions occur. A systematic study has been carried out to evaluate the performance of the developed time domain method in simulating the forces and motions as well as the fluid motion in the gap region for the two body system under various arrangements and in different wave-headings. At first, the computed numerical results have been validated and verified with computational and experimental results available in literature for standard geometries such as vertical truncated cylinders and rectangular boxes. Secondly, effectiveness of the damping lid model which is introduced to suppress wave resonance in the gap region is investigated including its influence on maximum sway forces on fixed and floating rectangular barges in side-by-side configurations. Thirdly, comparative studies on absolute and relative motion response for two cases (two rectangular barges, and a FLNG-FPSO + shuttle tanker) in side-by-side arrangement are detailed to bring out the importance of nonlinearities arising due to steep nonlinear incident waves. Finally, coupled motions of the two-body system of an FPSO and a shuttle tanker floating in side-by-side configuration in a steep nonlinear wave field are studied in which the two bodies are connected through hawsers, and also the FPSO is moored to the ground. Additionally there is a fender between the two bodies.     

浮体二阶运动下钢悬链式立管触地区动态响应分析

浮体运动是引起钢悬链式立管(steel catenary riser,简称SCR)动态响应和疲劳损伤的关键因素,目前研究SCR问题时,为简化计算往往仅考虑平台一阶运动,忽略二阶运动影响。而实际上不同浮体结构的二阶运动响应特征明显,拟以SCR服役张力腿平台(tension leg platform,简称TLP)为例,探讨浮体二阶运动对SCR触地区动态响应的影响。建立考虑海床刚度退化的管土作用模型以改进现有的CABLE3D RSI程序,通过编写程序接口,将有限元分析得到的平台实际运动响应导入,研究平台不同运动作用下SCR触地区的位移、动力响应及疲劳分布情况。根据波流作用方向将TLP二阶慢漂运动分为近端和远端漂移两种工况,发现二阶运动下立管与海床的作用范围会增大,且触地区不仅发生高频小幅振荡运动,同时伴随低频大幅运动响应;平台远端漂移时,管内张力敏感程度高,而近端漂移时触地区的弯矩显著增大,都会不同程度提高触地区的疲劳损伤率。研究可为服役不同浮体的SCR响应预测与疲劳分析提供参考和借鉴。     

Minimization of drift force on a floating cylinder by optimizing the flexural rigidity of a concentric annular plate

The deployment of suitable configurations of mutually interacting floating bodies for efficiently controlling their hydrodynamic interactions towards the reduction of the wave drift forces and, thus, of the mooring lines’ loads, has, nowadays, gained a great scientific interest. In this paper, the hydrodynamic behaviour of a floating cylinder and a concentric annular flexible plate is analysed in the frequency domain aiming at the minimization of the drift forces acting on the cylinder by optimizing the flexural rigidity of the plate. The diffraction/radiation problem is solved using a higher-order boundary element method. The analysis is implemented assuming that both floating bodies oscillate freely in heave, while for the plate, flexible modes are, additionally, considered for describing its structural deformations. The required modes shapes are determined in vacuum (“dry” mode superposition approach) through analytical expressions. The flexural rigidity of the plate, D, is optimized at a specific wave number using a real-coded genetic algorithm. Initially, results are compared with numerical results of other investigators for the case of two rigid concentric floating cylinders. Next, extended results are presented, focusing on the effect of D, including its optimum value, on various physical quantities describing the behaviour of both the cylinder and the plate. Contrary to the isolated cylinder, the presence of the plate introduces sharp peaks in the variation pattern of the drift force of the cylinder, bounded at specific wave numbers, where resonance of the seiche mode of water motion in the annular cavity or of specific flexible modes of the plate occurs. However, by reducing D to its optimum value, the cylinder’s drift force obtains practically zero values at the target wave number, due to an efficient improvement of the wave field in the annular cavity around the cylinder. Moreover, a great reduction of the drift force compared to the isolated cylinder is achieved in the subsequent high frequency range.     

顺应式海洋平台慢漂运动物理模型试验研究

为研究顺应式海洋平台慢漂运动的影响因素,以截断圆柱和漂浮方箱为例进行了不规则波作用下的慢漂运动模型试验。测量了不同系泊刚度条件下的漂浮方箱以及相同系泊刚度条件下的截断圆柱和漂浮方箱在静水中自由衰减运动和在不规则波中的运动响应,并将运动响应分解成一阶波频运动响应和二阶低频运动响应,分析了系泊刚度和浮体形状对浮体运动的影响。通过物理模型试验发现了系泊刚度及浮体形状对顺应式系泊浮体一阶运动标准差和二阶低频运动平均漂移值和标准差的关系。结果表明由于顺应式浮体的固有周期远离波浪谱峰周期时,系泊刚度以及浮体形状对慢漂运动的一阶运动响应影响不大;二阶低频运动相对偏离平衡位置的平均值和标准差均随系泊刚度增大而减小,浮体形状同样对慢漂运动的二阶低频纵荡运动响应影响较大。试验结果为实际海洋工程的外形选择和系泊刚度选择提供数据支持。     

Motion responses of barge carrying liquid tank

The interaction between the liquid sloshing in a rectangular tank equipped inside the barge and the barge responses has been investigated through a comprehensive experimental program. The barge was subjected to both regular and random wave excitations under beam sea condition. Three relative fill levels (h_s/l) with liquid fill depth (h_s) to length of tank (l) ratio of 0.163, 0.325 and 0.488 were considered. In addition, the barge responses of equivalent dry weight condition corresponding to each fill level were measured to understand the influence of sloshing. While the excitation wave frequency equals to first mode natural sloshing frequency, a noticeable decrease in the sway response has been observed. However, the effect of sloshing oscillation on the heave response is insignificant. A split up of roll resonance was observed for the aspect ratio of 0.163 due to the coupling effect of roll motion and sloshing.     

海洋中波浪下迎浪驳船运动对两矩形驳船间窄缝内水体共振的影响

A three-dimensional time-domain potential flow model is developed and applied to simulate the wave resonance in a gap between two side-by-side rectangular barges. A fourth-order predict-correct method is implemented to update free surface boundary conditions. The response of an up-wave barge is predicted by solving the motion equation with the Newmark-β method. Following the validation of the developed numerical model for wave radiation and diffraction around two side-by-side barges, the influence of up-wave barge motion on the gap surfaceresonance is investigated in two different locations of the up-wave barge relative to the back-wave barge at various frequencies. The results reveal that the freely floating up-wave barge significantly influences the resonance frequency and the resonance wave amplitude. Simultaneously, the up-wave barge located in the middle of the back-wave barge leads to a reduction in the resonance wave amplitude and motion response when compared with other configurations.     

An experimental study of the wave excitation in the gap between two closely spaced bodies,with implications for LNG offloading

The side-by-side offloading of liquid natural gas (LNG) at offshore terminals involves a fixed and a floating body in close proximity; the offshore terminal being the fixed body and the LNG tanker the floating body. The closeness of the two bodies leads to the formation of a long and relatively narrow gap, within which there is the potential for large amplifications of the water surface elevation. The present paper uses experimental results to characterise both the size and nature of the excitation within the gap. It also illustrates the effect of the vessel motion on this amplification by considering a 1:100 scaled model of an LNG tanker as well as its fixed approximation. It is found that the body's ability to move acts to increase the frequency at which resonant amplification within the gap occurs (the resonance frequency). The incident wave conditions considered include regular and irregular waves in both beam- and head-sea orientations; the latter leading to very different gap end conditions. The nature of the resonant amplification for the floating LNG tanker is shown to be similar for the two orientations, suggesting that the gap end conditions do not drive the resonant amplification. Consideration of the nonlinearity within the gap illustrates that resonant amplification occurs at the resonance frequency, irrespective of whether the fluid motion is first or second harmonic. The present paper provides data relevant to the safe offloading operations of an LNG tanker and demonstrates the importance of incorporating the vessel motion in numerical modelling procedures.     

Second-order resonance among an array of two rows of vertical circular cylinders

Experimental and theoretical works are conducted on the first-order and second-order wave interactions among an array of two rows of vertical circular cylinders.We show that both the first-order and the second-order resonance of dynamic free-surface displacement could in some cases be quite relevant among an array of two rows of vertical cylinders not only in theory but also in real phenomena. In some cases, on the other hand, we show that large free-surface displacement predicted theoretically is quite attenuated in reality probably due to viscous dissipation.     

Modelling of multi-bodies in close proximity under water waves—Fluid forces on floating bodies

This work presents two-dimensional numerical results of the dependence of wave forces of multiple floating bodies in close proximity on the incident wave frequency, gap width, body draft, body breadth and body number based on both viscous fluid and potential flow models. The numerical models were validated by the available experimental data of fluid oscillation in narrow gaps. Numerical investigations show that the large amplitude responses of horizontal and vertical wave forces appear around the fluid resonant frequencies. The convectional potential flow model is observed to un-physically overestimate the magnitudes of wave forces as the fluid resonance takes place. By introducing artificial damping term with appropriate damping coefficients μ∈[0.4, 0.5], the potential flow model may work as well as the viscous fluid model, which agree with the damping coefficients used in our previous work for the predication of wave height under gap resonance. In addition, the numerical results of viscous fluid model suggest that the horizontal wave force is highly dependent on the water level difference between the opposite sides of an individual body and the overall horizontal wave force on the floating system is generally smaller than the summation of wave force on each body.     

波浪方向对两矩形驳船间窄缝内流体共振影响的数值研究

应用二阶非线性三维势流时域模型模拟两个并靠矩形驳船间的窄缝内流体共振现象。模型中,速度势分解为入射势和未知的散射势,其中散射势通过求解边界积分方程计算得到。采用四阶预报校正的方法对自由水面边界条件在时间上进行数值积分。本文研究了波浪方向对窄缝内一阶、二阶波高的影响。计算结果表明,入射波浪的角度不会影响窄缝内流体的共振频率,同时窄缝内流体发生共振时,最大的共振波高总是发生在窄缝中点处。然而,共振时对应的最大波高随着入射角度的变化而变化。同时,入射波浪的频率与窄缝内流体共振频率的相对大小导致窄缝内最大波高位置向上游或者下游移动。     

Hydrodynamic analysis of three–unit arrays of floating annular oscillating–water–column wave energy converters

This paper deals with the hydrodynamic analysis of an array of Oscillating Water Column (OWC) devices, made up of coaxial cylinders, which are floating either independently or as a unit forming a floating platform. The platform is considered either free – floating or as TLP configuration connected to the sea bottom. Numerical results concerning the three boundary value problems, namely, the diffraction, the motion – and the pressure – dependent radiation ones are given. They have been obtained through an analytical solution method using matched axisymmetric eigenfunction expansion formulations. In all cases the interaction phenomena with neighbouring bodies have been taken properly into account using the physical idea of multiple scattering. Numerical results for the first – and the mean second – order wave forces, the hydrodynamic interaction coefficients along with pressure hydrodynamic parameters, inner air pressure and free–surface oscillation amplitude inside and outside of each device are parametrically evaluated and supplemented by experimental data.     

Approximation of near-resonant wave motion using a damped harmonic oscillator model

In this work, transient resonant motions excited in linearised scattering interactions are approximated using a simple damped harmonic oscillator model. The scattering interactions considered involve the diffraction of an incident wave-train with a dominant regular time-harmonic component by structures which enclose a portion of the free surface. Provided a single resonant mode only is excited significantly, the fluid oscillation in the vicinity of the structure will primarily be composed of the resonant mode and incident wave mode contributions. The forced damped harmonic oscillator equation is used to predict the fluid motion and, in particular, the elevation of the internal free surface. The predictions are compared to the results from a numerical time-domain solver based on the linearised water-wave equations. It is shown that, given a good estimate of the location of the resonance in the complex frequency plane and a priori knowledge of the form of the incident wave, the model can successfully predict the time-dependent behaviour of the motion. Both two- and three-dimensional scattering problems are considered with a variety of scatterers in each case.     

Finite element analysis of two-dimensional nonlinear sloshing problems in random excitations

A two-dimensional nonlinear random sloshing problem is analyzed by the fully nonlinear wave velocity potential theory based on the finite element method. A rectangular container filled with liquid subjected to specified horizontal random oscillations is studied. Both wave elevation and hydrodynamic force are obtained. The spectra of random waves and forces have also been investigated, and the effects of the peak frequencies and spectral width of the specified spectrum used for the generation of the random oscillations are discussed. It is found that the energy mainly concentrates at the natural frequencies of the container and is dominant at the ith order natural frequency when the peak frequency is close to the ith order natural frequency. Some results are compared between the fully nonlinear solutions, the linear solutions and the linear plus second-order solutions.     

Wave force coefficients for horizontally submerged rectangular cylinders

The results of wave force measurements carried out on a section of horizontally submerged rectangular cylinders, which are used as pontoons in many offshore structures, are reported in this paper. Two rectangular cylinders with aspect (depth–breadth) ratios equal to $\frac{1}{2}$ and $\frac{3}{4}$ and a square section (aspect ratio=1.0) cylinder are chosen for this study. Experiments are carried out in a wave tank at a water depth of 2.2 m at low Keulegan–Carpenter (KC) numbers to measure the horizontal and vertical wave forces acting on a 100 mm section, located at mid-length of the cylinders. For each cylinder, tests are carried out for two relative depths of submergence of 2.68 and 4.68. Measured wave forces in regular and irregular waves are then used to derive drag (C_D) and inertia coefficients (C_M). The analysis show that at very low KC numbers the inertia coefficients for all cylinders approached the potential flow values for both horizontal and vertical forces. The drag coefficients at low KC numbers exhibited large values and they decreased sharply with increase in KC number. For the square cylinder, where relatively a large KC number is obtained compared to other cylinders, inertia coefficients reached minimum values in the range of KC of about 3–4 and increased thereafter. In this range, C_M values are about 50% or so, smaller than the same at KC close to zero. The results of the experiments reveal that aspect ratio has large influence on hydrodynamic coefficients.     

Nonlinear simulations of wave-induced motions of a freely floating body using WCSPH method

Nonlinear interactions between waves and floating bodies are investigated using the weakly compressible Smoothed Particle Hydrodynamic (WCSPH) method. An improved algorithm based on the dynamic boundary particles (DBPs) is proposed to treat the moving boundary of the floating body. The force exerted on the floating body boundary particle by the particles surrounding it is evaluated using the volume integration of the stress tensors obtained from the momentum equation in its compact support. The improved WCSPH model is validated by the experimental results. The numerical test cases of the vertical oscillation of a rectangular box, the damped rolling oscillation of a floating box and the wave forces on a fixed rectangular box are then carried out to demonstrate the performance of the proposed model. Finally the evolution in time of the dynamic response of the freely floating body under nonlinear waves are discussed and compared with experimental results.     

On two-dimensional moonpool resonance for twin bodies in a two-layer fluid

This paper studies the moonpool resonance of two heaving rectangular bodies in a two-layer fluid. A mathematical model is proposed based on an eigenfunction matching approach. The motion of the two-dimensional bodies is assumed to be vertical and harmonic. Heave added mass and damping coefficients are computed to examine the hydrodynamic behavior of the twin bodies. The free surface and internal wave elevations are obtained near the resonant frequencies. The presented results and analyses reveal that there exist both Helmholtz and higher-order resonances in the two-layer fluid system, which is similar to the single-layer fluid case. It is also found that the resonances are closely associated with the free surface elevation inside moonpool gap, not the wave elevation at the interfacial surface. In addition, parametric studies have been performed to identify the dependencies of hydrodynamic behavior on geometry and density stratification.     

Moored floating breakwater response to regular and irregular waves

The simple, yet versatile numerical technique particularly suitable for investigating the problem of the wave attenuation by moored floating breakwater was recently developed by the author. In order to verift the theory, nearly full scale model tests were conducted in a large wave tank ($\text{3.6}\text{m wide}\text{× 4.5}\text{m high}\text{× 106}\text{m long}$). Both random waves and monochromatic waves were used to compare the results. A breakwater with a rectangular cross-section and a hydrodynamically shaped «three-cycle cylinderå breakwater were tested. Incident wave spectra were successfully decomposed from the multi-reflected sea spectra. Frequency response functions of transmitted wave, sway, heave and roll motions of the breakwater as well as mooring forces were all experimentally determined and compared with the theory. Generally, excellent agreements between the theory, the random wave tests and the monochromatic wave tests were obtained for the hydrodynamically shaped breakwater. Except near the modal frequencies of body motion generally good agreement between theory and experiment was obtained for the rectangular breakwater. Near the modal frequencies, the body motion was damped by the flow separation at the sharp corners of the rectangular breakwater. Generation of higher harmonics in wave, body motion and mooring forces was observed and measured, but was generally small. The slow drift oscillation and its effects on the performance of the spring moored breakwaters were also small. From the comaprisons of the small scale test and the large scale tests, it was found that the scale effects were negligibly small on the performance of the spring-moored breakwaters.