Effects of bracings and motion coupling on resonance features of semi-submersible platform under irregular wave conditions

Resonance is a critical consideration in the design of offshore floating structures. This paper aims at analysing the nonlinear effects of bracings and motion coupling on the resonance features of a semi-submersible platform. An improved mathematical model based on potential theory is proposed to simulate the motion response of a semi-submersible platform under irregular wave conditions, considering both the variations of hydrostatic and hydrodynamic forces induced by the bracings entering and exiting the water and the nonlinear coupling induced by the platform motions. For comparison purposes, numerical simulations are also performed using a mathematical model without considering the aforementioned effects. Validated by results of wave basin tests and numerical simulations, the proposed model performs much better in capturing the characteristic resonance features of pitch motion in low-frequency region. The nonlinear hydrostatic effect of bracings leads to the increase of resonance frequency as the motion amplitude increases, while the hydrodynamic force on the bracings and the nonlinear motion coupling only influence the amplitude of resonance spectral peak. In addition, factors influencing the nonlinear effects such as the vertical position and diameter of bracings and the pitch restoring coefficient are further investigated. It is revealed that the deviation of pitch resonance frequency has evident dependence on the ratio between nonlinear and linear volumetric variations, and an empirical formula estimating the resonance frequency is proposed using the observed dependence. Theoretically, both smaller bracing radius and larger pitch restoring coefficient are beneficial for suppressing the resonance induced by the nonlinear effects. The proposed model can be an effective tool for predicting the motion response, and the understanding of the resonance features is helpful for the design of semi-submersibles.     

半潜式海上浮式风机气动阻尼特性研究

由于海上漂浮式风机具有较大的支撑平台运动,气动阻尼效应对海上漂浮式风机的运动响应带来了重要的影响, 日渐受到相关国内外学者的关注. 为了研究海上浮式风机的气动阻尼特性,本文推导了海上浮式风机气动阻尼力的数学模型,并借助模型实验和数值计算的方法,研究了半潜式海上浮式风机的气动阻尼特性及其作用规律. 结果表明,浮式风机的风轮旋转时的气动阻尼比风轮非旋转状态时更加明显;在作业工况下,气动阻尼对半潜式浮式风机平台的纵荡、纵摇、机舱的运动有明显的抑制作用,且主要体现为对半潜式浮式风机的平台运动固有频率响应的抑制作用,对波频范围的平台运动作用甚微. 其变化规律与风速大小、波浪载荷等有关,在风机的额定工况之前,气动阻尼通常与风速呈正相关关系,但是增长率有逐渐减小的趋势;在控制系统作用下,当入流风速接近或超过风机额定风速时,容易出现气动负阻尼现象,反而进一步强化浮式风机的运动响应,此时通过降低变桨距控制器的比例系数,即降低变桨距控制器的灵敏度,有助于增加海上浮式风机的气动阻尼效果,并且在一定程度上减缓负的气动阻尼的发生,改善海上浮式风机的运动响应.      

螺旋侧板和水深对浮式风机动态响应的影响

为研究漂浮式风力机平台动态响应的优化措施,提出平台附加螺旋侧板的方式。建立基于Spar平台的5MW风力机整机模型,利用有限元软件进行水动力计算,得到不同水深条件下,风力机平台在风、浪、流载荷联合作用下的频域特性,通过与不附加螺旋侧板情况下的动态特性参数对比,探讨螺旋侧板是否对结构的频响特性起到提升作用。结果表明,附加螺旋侧板后,结构的垂荡和纵摇的运动幅值和所受波浪力均得到了显著抑制;与纵荡和纵摇相比,垂荡运动的幅值和所受波浪力所受影响更显著;水深变化对结构响应的幅值有着显著影响。     

Tomographic reconstruction of internal wave patterns in a paraboloid

Using tomographic synthetic schlieren, we are able to reconstruct the three-dimensional density field of internal waves. In this study, the waves are radiating from an oscillating sphere positioned eccentrically at the surface of a paraboloidal domain filled with a uniformly stratified fluid. We find that the prediction by ray tracing corresponds well with the observed intensities of the wave field. Remarkably, for a specific value of the forcing frequency, we observe convergence of internal wave energy to an internal wave attractor. The attractor is found to dominate fluid motion in the plane perpendicular to the plane spanned by the symmetry axis and the oscillator position.     

Generation of internal waves by a turbulent jet in a stratified fluid

The process of generation of internal waves by an initially cylindrical, turbulent jet with a Gaussian profile of the average horizontal velocity component in a fluid with stable linear density stratification is investigated by direct numerical simulation. It is shown that on time intervals Nt < 30, where N is the buoyancy frequency, the vertical velocity pulsations collapse, which is accompanied by the generation of internal waves whose spatial period is close to the wavelength of the spiral mode of jet instability in a homogeneous fluid. The wave dynamics and kinematics can be satisfactorily described by the linear theory for a pulsed source and their parameters are in good agreement with the parameters of the “coherent” internal waves generated by a stratified wake in a laboratory experiment. At large times the wave generation ceases and the variations of the fluid density are localized in the neighborhood of the centers of large-scale vortices formed in the horizontal plane in the neighborhood of the jet.     

Surface characters of internal waves generated by Rankine ovoid

A linear theory on the internal waves generated in the stratified fluid with a pycnocline is presented in this paper. The internal wave fields such as the velocity fields in the stratified fluid and velocity gradient fields at the free surface are also investigated by means of the theoretical and numerical method. From the numerical results, it is shown that the internal wave generated by horizontally moving Rankine ovoid is a sort of trapped wave which propagates in a wave guide, and its waveform is a kind of Mach front-type internal wave in the pycnocline. Influence of the internal wave on the flow fields at the free surface is represented by the velocity gradient fields resulted from the internal waves generated by motion of the Rankine ovoid. At the same time, it is also shown that under the hypothesis of inviscid fluid, the synchronism between the surface velocity gradient fields at the free surface and the internal wave fields in the fluid is retained. This theory opens a possibility to study further the modulated spectrum of the Bragg waves at the free surface.The project supported by the National Natural Science Foundation of China (40576010). The English text was polished by Keren Wang.     

The phase configuration of the waves around an accelerating disturbance in a rotating stratified fluid

Ray theory is extended to consider the case of an accelerating disturbance which is producing waves in a rotating stratified fluid. Starting from the equations of motion, dispersion relations are derived for surface gravity waves, capillary waves, Rossby waves and internal-inertial waves. The wave system is studied in each case for the problem of a body starting impulsively from rest and for a body starting from rest and moving with constant acceleration.     

分层流体中运动源生成的内波研究进展

针对两类密度分布模型------连续分层流体和间断分层流体, 综述了在运动潜体生成的Kelvin型和非Kelvin型内尾迹研究方面的现状, 内容侧重于运动源生成内波的解析理论和分层拖曳水槽中内尾迹实验方面的研究成果. 介绍了在连续分层流体中运动源生成的Kelvin型非线性内波的一般方程和在间断分层流体中Kelvin型内波的势流分析的一般方法; 概述了运动源诱生的先锋内孤立子、代数孤立子和平孤立波3类特殊非线性内波的研究进展, 其中运动潜体生成的平孤立内波被作者实验证实是一类极限孤立波, 并首次建立了共轭流动模型予以描述; 综合分析了在密度线性分布流体中潜体运动生成内波的动力学过程多样性特征, 其中包括内尾迹近场和远场的时空结构、不稳定结构、涡旋与湍流耦合结构以及湍流与内波相互作用结构等.      

Turbulent Dynamics of a Critically Reflecting Internal Gravity Wave

Results from computational fluid dynamics experiments of internal wave reflection from sloping boundaries are presented. In these experiments the incident wave lies in a plane normal to the slope. When the angle of wave energy propagation is close to the bottom slope the reflection causes wave breakdown into a quasi-periodic, turbulent boundary layer. Boundary layer energetics and vorticity dynamics are examined and indicate the importance of the three-dimensional turbulence. The boundary layer exhibits intermittent turbulence: approximately every 1.2 wave periods the boundary layer mixes energetically for a duration of about one-third of a wave period, and then it restratifies until the next mixing event. Throughout the wave cycle a strong thermal front is observed to move upslope at the phase speed of the incident waves. Simulations demonstrate that the net effects of turbulent mixing are not confined to the boundary layer, but are communicated to the interior stratified fluid by motions induced by buoyancy effects and by the wave field, resulting in progressive weakening of the background density gradient. Transition to turbulence is determined to occur at Reynolds numbers of approximately 1500, based upon the wavelength and maximum current velocity of the oncoming wave train. The boundary layer thickness depends on the Reynolds number for low Richardson numbers, with a characteristic depth of approximately one-half of the vertical wavelength of the oncoming wave. Received 21 May 1997 and accepted 14 October 1997     

非线性海洋内波的理论、模型与计算

海水因盐度与温度的垂向差异造成密度层结现象, 进而由于海洋系统的内部扰动(如海潮流过局部隆起的海底地形)与外部扰动(如死水现象)造成等密面的波动, 这一现象称为“内波”. 内波在全球范围内大量存在, 尤其是在海峡入海口等密度层结现象较为明显和稳定的区域会有内波频繁活动. 海洋通常呈现“三明治”状的结构: 密度相对稳定的混合层与深水层, 以及位于中间密度连续过渡的密跃层. 密跃层的整体脉动对于海洋工程和海洋生态环境有重大的影响; 而密跃层内部的波动对于潜艇的非声探测(反过来说, 对于潜艇的隐身作战)具有潜在的应用价值. 而造成这些重大影响的根源在于内波在水平和垂直方向都具备传播能力, 这是有别于海洋表面波浪的关键之处.本文针对两类海洋密度模型-连续分层模型与间断分层模型, 从理论研究、数值模拟、实验室机理实验等方面论述了研究海洋内波的各类非线性模型(包括弱非线性的Korteweg-de Vries方程、Benjamin-Ono方程, Kadomtsev-Petviashvili方程等著名模型以及强非线性Miyata-Choi-Camassa方程、非线性势流理论、带密度变化的不可压缩Navier-Stokes方程等), 讨论各自的适用范围, 并借此探讨内波在海洋质量动量能量输运中所起的至关重要的作用.      

Internal waves arising in the presence of the unsteady motion of a source in a continuously stratified fluid

A general approach to the investigation of linear internal and surface waves in a stably stratified fluid, arising from different types of perturbations, is presented in [1]. The methods of calculation of the internal waves generated by the motion of a mass source are developed for particular cases of steady horizontal motion in a continuously stratified fluid in [2] and arbitrary motion in an unbounded exponentially stratified fluid in [3]. Internal waves generated by other types of perturbations have also been investigated but only for particular cases of motion (see, for example, [4]). This paper presents a method for the calculation of unsteady, linear, internal gravity waves arising in an inviscid incompressible fluid with continuous stable stratification.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 122–130, July–August, 1985.     

Structure of internal waves in a channel

One method of describing wave motion in a fluid with continuous stratification is to use normal waves (modes). The propagation of internal gravity waves in closed rectangular regions whose boundaries coincide with planes through which there is no normal motion is essentially different from wave motion in an unbounded medium [1, 2]. This paper describes a theoretical and experimental investigation of the propagation of internal waves in an exponentially stratified fluid in a horizontal channel of finite height.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–110, January–February, 1935.In conclusion the authors wish to express their gratitude to A. T. Onufriev for his interest in their work.     

Quasi-periodic waves and quasi-solitary waves in stratified fluid of slowly varying depth

The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper. The model considered here consists of a two-layer incompressible constantdensity inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall. The Korteweg-de Vries (KdV) equation with varying coefficients is derived with the aid of the reductive perturbation method. By using the method of multiple scales, the approximate solutions of this equation are obtained. It is found that the unevenness of bottom may lead to the generation of so-called quasi-periodic waves and quasisolitary waves, whose periods, propagation velocities and wave profiles vary slowly. The relations of the period of quasiperiodic waves and of the amplitude, propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented. The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.Project Supported by National Natural Science Foundation of China.     

Low-frequency wave propagation in an elastic plate floating on a two-layered fluid

For some technical applications related to the ice–sea interaction, it is necessary to predict waveguide properties of elastic plates floating on a relatively thin layer of water, which has a non-uniform density distribution across its depth. The issue of particular concern is propagation of low-frequency waves in such a coupled waveguide. In the present paper, a stratified fluid is modelled as two homogeneous, inviscid and incompressible layers with slightly different densities. The lighter layer of fresh water lies on top of the heavier layer of salty water. The former one produces fluid loading at the pre-stressed plate, whereas the latter one is bounded by the sea bottom. The classical asymptotic methods are employed to identify significant regimes of wave motion in such a three-component waveguide. Dispersion diagrams obtained from approximate dispersion relations are compared with their exact counterparts. The phenomena of veering and generation of waves with zero group velocity induced by pre-stress are identified and quantified.     

Internal gravity waves in critical generation modes and in the vicinity of trajectories of motion of perturbation sources

The field of internal gravity waves in a layer of an arbitrary stratified fluid is studied for critical generation modes and in the vicinity of trajectories of motion of the perturbation sources. The exact solutions describing the structure of a separate mode of the wave field in the vicinity of the perturbation source in the critical generation modes are investigated, and expressions for the total field representing the sum of all wave modes are obtained. In the vicinity of the trajectories of the perturbation sources, asymptotic representations of the eigenfunctions and eigenvalues of the basic vertical spectral problem of internal waves are constructed in the approximation of large wave numbers and asymptotic expressions for a separate mode of the wave field are obtained that describe the spatial structure and features of the fields of internal gravity waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 70–79, September–October, 2008.     

Internal wave structure in a three-layer fluid with a stratified middle layer

The behavior of internal waves in a vertically bounded channel differs considerably from the wave motion in an infinite stratified fluid. In [1] the phase structure of the internal waves in an exponentially stratified layer of fluid between rigid horizontal planes was experimentally and theoretically investigated. A characteristic feature of such a channel is the boundedness of the phase and group velocities of each mode. Below, the case of an exponentially stratified channel between layers of homogeneous unbounded fluid is considered.In conclusion, the authors wish to thank A. T. Onufriev for his interest in their work.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–132, May–June, 1987.     

传统Spar平台垂荡主共振时非线性耦合响应的研究

考虑瞬时波面影响,建立了Spar平台垂荡-纵摇运动的参数激励耦合运动方程,应用多尺度法导出了波浪频率接近垂荡固有频率时响应方程的一阶摄动解并做数值验证。计算结果显示:当波浪高度达到一定值时,纵摇运动中出现大量的亚谐频率成分;随着波浪高度进一步增大,纵摇运动将出现明显的幅值跳跃现象。研究结果表明,能量渗透现象是纵摇运动失稳的主要原因,波浪激励力矩对于纵摇本身的贡献很小。由于垂荡模态存在能量饱和现象,多余的垂荡能量将会向纵摇模态转移,导致出现大幅值的纵摇运动。     

连续分层流体中垂直薄板的水动力特性

研究了在线性连续分层流体中水波与半潜式刚性垂直薄板相互作用的问题. 在 Boussinesq近似下，基于分离变量法，导出了具有自由面的平面前进波的色散关系，建立 了半潜式刚性垂直薄板的反射与透射能量、水平波浪力的计算方法. 对给定的频率，当它大 于浮力频率时，流场中只有一种模态的平面前进波，当它小于浮力频率时，流场 中有无数多个模态的平面前进波，并证明了对每一种模态的入射波，其它每个模态水波的反射与透射能量是 相等的. 对水面漂浮和座底半潜式薄板的反射与透射能量，以及作用在薄板上的水平波浪力 进行了数值计算分析，表明了流体的线性连续分层效应对这些水动力的影响是不可忽视 的. 特别地，在入射波频率小于浮力频率时，与第1模态入射波的能量转化量及其对薄板产 生的水平波浪力相比，其它模态入射波的能量转化量及其对薄板产生的水平波浪力都要大得 多.     

Distributed fault detection and isolation for leader–follower multi-agent systems with disturbances using observer techniques

This paper presents an adaptive optimal control method to suppress the oscillation of a floating platforms system in random seas. The system consists of multiple modules which are connected by flexible connectors with strong geometrical nonlinearity. To cope with the uncertain waves, a disturbance estimator is introduced to assess the actual wave excitation. This adaptive scheme of the estimator is integrated with an optimal control method subject to a limitation on the control outputs of actuators, where the optimal control process is carried out by the sequential quadratic program method. In numerical experiments, a floating platform with five semi-submersible modules is considered. Control process deals with 20 control variables to stabilize the surge, sway and yaw motions of a 30-DOF floating system, subject to unknown irregular waves and limitation of control output. Numerical results have verified the efficiency of the control strategy and show that the proposed control method performs very well.