Application of extended state space to nonlinear ship rolling

It is well known in the field of marine hydrodynamics that the added mass, damping and wave exciting forces are functions of frequency (Newman, 1977. Marine Hydrodynamics. MIT Press, Cambridge). Although most previous studies of nonlinear ship rolling motion have assumed that these forces do not vary with frequency, in this study the frequency dependent added mass and damping coefficients are approximated in the time domain with extended state space variables. Using numerical time simulation (integration), the extended state space model is compared to the constant coefficient model with a constant frequency forcing and the results for two constant value approximations of the added mass and damping are compared to the extended state space model with a multiple component pseudo random forcing.     

部分反射直墙前二维物体的绕射和辐射问题

基于映像理论将部分反射直墙前物体的散射问题,等效于开敞水域中原物体的散射和关于直墙映像体散射的线性叠加进行求解。采用高阶边界元方法建立了部分反射直墙前二维任意形状物体波浪绕射和辐射问题的数值分析模型,通过与已发表的海底方箱和淹没圆柱结果的对比验证了数值模型的准确性。应用该模型研究了直墙反射系数幅值及相位、方箱与墙间距离等参数对水面方箱上波浪激振力、附加质量和辐射阻尼的影响。结果表明：直墙反射系数幅值越大,波浪激振力、附加质量和辐射阻尼的波动越大,附加质量在一些频率下出现负值;相位角的变化会改变波浪激振力、附加质量和辐射阻尼曲线的偏移,在低频区对升沉附加质量有显著影响;方箱距离直墙越远,方箱上的波浪激振力、附加质量和辐射阻尼随波数振荡的频率越快,峰值频率向低频侧移动。     

Hull component interaction and scaling for TLP hydrodynamic coefficients

The purpose of this paper is to validate a new method that can be used by offshore platform designers to estimate the added mass and hydrodynamic damping coefficients of potential Tension Leg Platform hull configurations. These coefficients are critical to the determination of the platform response particularly to high frequency motions in heave caused by sum-frequency wave forcing i.e. “springing”. Previous research has developed the means by which offshore platform designers can extrapolate anticipated full-scale hydrodynamic coefficients based on the response of individual model scale component shapes. The work presented here further evaluates the component scaling laws for a single vertical cylinder and quantifies the effects due to hydrodynamic interaction. Hydrodynamic interaction effects are established through a direct comparison between the superposition of individual hull component coefficients and those evaluated directly from complete hull configuration models. The basis of this comparison is established by the experimental evaluation of the hydrodynamic coefficients for individual hull components as well as partial and complete platform models. The results indicate that hydrodynamic interaction effects between components are small in heave, and validate component scaling and superposition as an effective means for added mass and damping coefficient estimation of prototype platforms. It is found that the dependency of damping ratio with KC for a TLP is almost identical to that of a single column, thus offering a scaling methodology for prototype damping ratio values.     

Numerical modeling of perforated plates in oscillating flow

Perforated plates, relevant for several marine applications, are experimentally and numerically investigated. The numerical investigations are performed using a presently developed Navier–Stokes solver. Several comparison and sensitivity studies are presented, in order to validate and verify the solver. Forced heave experiments are performed on two perforated plates with perforation ratios 19% and 28%. Amplitude-dependent added mass and damping coefficients are presented. Good agreement is obtained between the solver and the present experiments. Consistent with existing data, the results show that the hydrodynamic coefficients of perforated plates are highly amplitude dependent. The damping force is found to dominate over added mass force. The damping force dominance increases with increasing perforation ratio. It is highlighted that plate-end flow separation has an important effect on the damping coefficient. The developed numerical solver is two-dimensional, but is found to yield reasonable estimates of hydrodynamic force coefficients when compared with a previous three-dimensional experimental investigation. This could indicate that three-dimensional effects are not dominant for the hydrodynamic forces of perforated plates, and that a two-dimensional viscous flow solver could have relevance as a tool for estimating hydrodynamic forces on three-dimensional perforated structures.     

Extreme motion response analysis of moored semi-submersibles

The motion response prediction of offshore structures may be carried out using time domain or frequency domain models or model tests. The frequency domain analysis uses the simplified, linearised form of the motion equations and it is very economical. The time domain analysis, unlike frequency domain models, is adequate to deal with non-linearities such as viscous damping and mooring forces, but it requires sophisticated solution techniques and it is expensive to employ. For moored semisubmersibles time domain techniques must be employed since there are strong nonlinearities in the system due to mooring line stiffness and damping and viscous drag forces. In the first part of this paper a time domain model to predict the dynamic response of a semi-submersibles are developed and the effect of thrusters and mooring line damping are incorporated into the time domain model. In the second part time domain simulations are carried out to find the total extreme motions and mooring forces.     

On a near-optimal control approach for a wave energy converter in irregular waves

Real-time smooth reactive control and optimal damping of wave energy converters in irregular waves is difficult in part because the radiation impulse response function is real and causal, which constrains the frequency-dependent added mass and radiation damping according to the Kramers–Kronig relations. Optimal control for maximum energy conversion requires independent synthesis of the impulse response functions corresponding to these two quantities. Since both are non-causal (one being odd and other even), full cancellation of reactive forces and matching of radiation damping requires knowledge or estimation of device velocity into the future. To address this difficulty and the non-causality of the exciting force impulse response function, this paper investigates the use of propagating-wave surface elevation up-wave of the device to synthesize the necessary forces. Long-crested waves are assumed, and the approach is based on the formulations of Naito and Nakamura [2] and Falnes [22]. A predominantly heaving submerged device comprised of three vertically stacked discs driving a linear power take-off is studied. The overall formulation leads to smooth control that is near-optimal, given the approximations involved in the time-shifting of the non-causal impulse response functions and the consequent up-wave distances at which wave surface elevation is required. Absorbed power performance with the near-optimal approach is compared with two other cases, (i) when single-frequency tuning is used based on non-real time adjustment of the reactive and resistive loads to maximize conversion at the spectral peak frequency, and (ii) when no control is applied with damping set to a constant value. Simulation results for wave spectra over a range of energy periods and significant wave heights are compared for the three situations studied. While practical implementation presents engineering challenges, in terms of time-averaged absorbed power, unconstrained near-optimal control is found to perform significantly better than single-frequency tuning in the spectra with longer energy periods (>10 s for the present device), and somewhat better in the spectra with shorter energy periods (here ≤10 s).     

Linear dynamics of hydrostatic adjustment to horizontally homogeneous heating

The process of hydrostatic adjustment to horizontally homogeneous heating in a stably stratified atmosphere of arbitrary thermal structure is investigated in the limit of small perturbations. A linear differential equation is derived for the vertical pressure distribution in the final balanced state. Solutions of this equation are compared with the time dependent solution which is found by numerically integrating the equations in time. During the process of hydrostatic adjustment acoustic‐buoyancy oscillations are generated. The amplitudes of these oscillations become so great that static instability is generated at heights above 100 km, depending on where and how abruptly the heat is added. As a crude representation of the unstable breakdown and damping of these waves, Rayleigh damping is introduced. If the associated damping coefficient in the upper atmosphere is sufficiently large (greater than the Brunt Väisälä frequency), the oscillations vanish. Below a height of about 50 km the steady state predicted by the above mentioned differential equation is reached approximately in 10 min.     

Second-order resonant heave,roll and pitch motions of a deep-draft semi-submersible: Theoretical and experimental results

Large volume semi-submersible units may present significant wave induced resonant motions in heave, roll and pitch. Evaluating the slow motions of such systems is important from the initial stages of their designs and therefore requires a model that is both accurate and expedite enough. In the present article, different options for modeling the second-order hydrodynamic forces and induced motions are discussed using as a case-study the PETROBRAS 52 unit—P-52. Computations of the low frequency forces are performed in the frequency domain by means of a commercial Boundary Element Method (BEM) code. Different hydrodynamic approximations are tested and evaluated by directly comparing the predicted responses with those measured in small-scale tests performed in a wave-basin. From the results obtained in theses comparisons, a methodology based on a white-noise approach of the force spectrum is proposed. The validity of such approximation is attributable to the typically low damping levels in heave, roll and pitch motions. Furthermore, results also indicate that the second order forces may be calculated disregarding the free-surface forcing components, an option that helps to reduce the computational burden even more, rendering the procedure suitable for preliminary design calculations.     

Uncertainty Analysis of Wind-Wave Predictions in Lake Michigan

With all the improvement in wave and hydrodynamics numerical models, the question rises in our mind that how the accuracy of the forcing functions and their input can affect the results. In this paper, a commonly used numerical third-generation wave model, SWAN is applied to predict waves in Lake Michigan. Wind data are analyzed to determine wind variation frequency over Lake Michigan. Wave predictions uncertainty due to wind local effects are compared during a period where wind has a fairly constant speed and direction over the northern and southern basins. The study shows that despite model calibration in Lake Michigan area, the model deficiency arises from ignoring wind effects in small scales. Wave prediction also emphasizes that small scale turbulence in meteorological forces can increase prediction errors by 38%. Wave frequency and coherence analysis show that both models can predict the wave variation time scale with the same accuracy. Insufficient number of meteorological stations can result in neglecting local wind effects and discrepancies in current predictions. The uncertainty of wave numerical models due to input uncertainties and model principals should be taken into account for design risk factors.     

Simulation of wave power devices

Classical frequency and time domain models of a single degree of freedom wave power device are presented. In the time domain, a convolution integral is conventionally used to represent the fluid dynamic radiation force, characterised by added mass and damping in the frequency domain. This integral is replaced by an approximate ordinary differential equation (ODE) model which is faster and more convenient in simulations. A time domain model of the fluid dynamics of an oscillating water column (OWC) device is derived to illustrate the technique. Digital simulations of the OWC are used to compare the accuracy of the classical and ODE models. The simulation of the ODE model runs about six times as fast as the classical model based on convolution, yet characterises the fluid dynamics accurately.     

Hull/mooring/riser coupled dynamic analysis and sensitivity study of a tanker-based FPSO

A computer program is developed for hull/mooring/riser coupled dynamic analysis of a tanker-based turret-moored FPSO (Floating Production Storage and Offloading) in waves, winds, and currents. In this computer program, the floating body is modeled as a rigid body with six degrees of freedom. The first- and second-order wave forces, added mass, and radiation damping at various yaw angles are calculated from the second-order diffraction/radiation panel program WAMIT. The wind and current forces for various yaw angles of FPSO are modeled following the empirical method suggested by OCIMF (Oil Company International Marine Forum).

The mooring/riser dynamics are modeled using a rod theory and finite element method (FEM), with the governing equations described in a generalized coordinate system. The dynamics of hull, mooring lines, and risers are solved simultaneously at each time step in a combined matrix for the specified connection condition. For illustration, semi-taut chain-steel wire-chain mooring lines and steel catenary risers are employed and their effects on global FPSO hull motions are investigated. To better understand the physics related to the motion characteristics of a turret-moored FPSO, the role of various hydrodynamic contributions is analyzed and assessed including the effects of hull and mooring/riser viscous damping, second-order difference-frequency wave-force quadratic transfer functions, and yaw-angle dependent wave forces and hydrodynamic coefficients. To see the effects of hull and mooring/riser coupling and mooring/riser damping more clearly, the case with no drag forces on those slender members is also investigated. The numerical results are compared with MARIN's wave basin experiments.     

Hydrodynamics of A Flexible Riser Undergoing the Vortex-Induced Vibration at High Reynolds Number

This study proposed a method to obtain hydrodynamic forces and coefficients for a flexible riser undergoing the vortex-induced vibration (VIV), based on the measured strains collected from the scale-model testing with the Reynolds numbers ranging from 1.34E5 to 2.35E5. The riser is approximated as a tensioned spatial beam, and an inverse method based on the FEM of spatial beam is adopted for the calculation of hydrodynamic forces in the cross flow (CF) and inline (IL) directions. The drag coefficients and vortex-induced force coefficients are obtained through the Fourier Series Theory. Finally, the hydrodynamic characteristics of a flexible riser model undergoing the VIV in a uniform flow are carefully investigated. The results indicate that the VIV amplifies the drag coefficient, and the drag coefficient does not change with time when the CF VIV is stable. Only when the VIVs in the CF and IL directions are all steady vibrations, the vortex-induced force coefficients keep as a constant with time, and under “lock-in” condition, whether the added-mass coefficient changes with time or not, the oscillation frequency of the VIV keeps unchanged. It further shows that the CF excitation coefficients at high frequency are much smaller than those at the dominant frequency, while, the IL excitation coefficients are in the same range. The axial distributions of the excitation and damping region at the dominant frequency and high frequency are approximately consistent in the CF direction, while, in the IL direction, there exists a great difference.     

海洋石油平台TMD振动控制及参数优化

研究了随机波浪载荷作用下调谐质量阻尼器（ＴＭＤ）对桩基钢结构海洋平台的减振效果,采用谱分析法对ＴＭＤ参数进行优化,优化ＴＭＤ 使平台的位移响应标准偏差比无ＴＭＤ下降１２．４％ 。并研究了ＴＭＤ参数在优化域内的失调对响应的影响,ＴＭＤ刚度失调比阻尼失调要敏感,欠阻尼失调比过阻尼失调要敏感。从振害累积概念出发,对谐激励下ＳＤＯＦ－ ＴＭＤ的Ｒａｎｄａｌｌ参数优化方法提出了改进。     

Identification of non-linear effects in predicting the motion response of an offshore platform

This paper presents the results of an extensive parametric study to investigate various non-linear aspects of the prediction of the large-amplitude motion responses of a semi-submersible. The main objectives of the parametric studies were to investigate the following aspects, which can non-linearly influence the motion responses and which cannot be studied by linear frequency-domain prediction techniques. These aspects are the effects on the motion responses of flooding time and mass; non-linear wave-exciting and rigid-body induced motion (i.e. added mass and damping) forces; non-linear restoring forces; steady wind and current; variation of GM (transverse metacentric height); and the initial position of the semi-submersible. The investigations were carried out for a particular semi-submersible geometry using a numerical simulation technique in the time domain. The simulations were performed for the model during intact, progressive and post-flooding conditions under the combined loading of regular waves, steady wind and current for two different heading angles. This paper is thus intended to provide some insight into the physical effects of the non-linear terms in the equations of motion which are associated with the wave-excitation forces, rigid-body induced motion forces and restoring forces. Since the resulting motion responses could have a steady component as well as the oscillatory one, the force and motion phenomena were also highlighted through the computation of these components.     

Added mass and damping of a vertical cylinder in finite-depth waters

A comprehensive set of theoretical added masses and wave damping data for a floating circular cylinder in finite-depth water is presented. The hydrodynamic problem is solved by matching eigen functions of the interior and exterior problems. The resulting infinite system is solved directly and found to have excellent truncation characteristics. Added mass and damping are given for heave, sway, and roll motion, as well as coupling coefficients for sway and roll. It is shown that the heave added mass is logarithmic singular and the damping approaches a constant in the low-frequency limit. Transition of the behaviour in finite-depth water to deep water is also discussed.     

Optimal identification of potential-radiation hydrodynamics for moored floating structures—a new general approach in state space

In this paper, a new state-space model of the potential-radiation hydrodynamics in moored ocean engineering floating structures and its parameter identification are presented. The raw data for this goal are the added mass and potential-damping matrices in frequency-discrete domain.In a preliminary study of existing approaches in the literature, two mathematical models and their estimation are comparatively analysed in detail. These served in the development of the new approach that shares certain main advantages of the previous approaches.The model is identified in a least-squares sense using a weighted norm and a free control parameter to accomplish a trade-off between quality and stability. This reduces numerical instability problems and also keeps the analytical and computational benefits of a parametric state-space model. The model can be conveniently expressed in any usual canonical form in state space.The application of the model acceptably accurate reproduces the behaviour of the potential-radiation hydrodynamic forces in time. Case studies involving a semisubmersible and buoys are shown to demonstrate the features of the proposed approach.     

Momentum and gravity effects during the constant velocity water entry of wedge-shaped sections

Computational fluid dynamics analysis was used to investigate the added mass momentum, flow momentum and gravity effects during the constant velocity water entry of wedge-shaped sections with deadrise angles from 5° to 45°. It is shown that the added mass continues to increase for a time after chine immersion and that added mass can be estimated in terms of a constant added mass coefficient and an effective wetted width. A momentum theory is presented in which the water entry force is explained as the sum of the rate of change of added mass momentum, which becomes zero at immersion to chine depth ratios greater than about three, and the rate of change of flow momentum, which continues at deep immersions. The effect of gravity on the water entry force is given as the hydrostatic force together with the force necessary to create the potential energy in the water pile up. Hydrodynamic forces are not significantly changed by the effect of gravity on the flow fields.     

Hydrodynamic performance of solid and porous heave plates

Heave plates have been widely utilized in floating offshore structures as they can provide additional damping and added mass to improve the hydrodynamic response of the system. This study investigates the hydrodynamic characteristics (added mass and damping) of oscillatory solid or porous disks using model scale experiments. All experiments were conducted via forced oscillation model tests using a planar motion mechanism (PMM). The hydrodynamic coefficients of the solid or porous disk obtained from the force measurements are analysed and presented. The sensitivities of the damping and added mass coefficients to both motion amplitude and the disk porosity are examined.     

The effects of yawing motion with different frequencies on the hydrodynamic performance of floating vertical-axis tidal current turbines

Under real sea conditions, the hydrodynamic performance of floating vertical-axis tidal current turbines is affected by waves and currents. The wave circular frequency is a significant factor in determining the frequencies of the wave-induced motion responses of turbines. In this study, the ANSYS-CFX software (manufacturer: ANSYS Inc., Pittsburgh, Pennsylvania, United States) is used to analyse the hydrodynamic performance of a vertical-axis turbine for different yawing frequencies and to study how the yawing frequencies affect the main hydrodynamic coefficients of the turbine, including the power coefficient, thrust coefficient, lateral force coefficient, and yawing moment coefficient. The time-varying curves obtained from the CFX software are fitted using the least-squares method; the damping and added mass coefficients are then calculated to analyse the influence of different yawing frequencies. The simulation results demonstrate that when analysing non-yawing turbines rotating under constant inflow, the main hydrodynamic coefficient time-varying curves of yawing turbines exhibit an additional fluctuation. Furthermore, the amplitude is positively correlated with the yawing frequency, and the oscillation amplitudes also increase with increasing yawing frequency; however, the average values of the hydrodynamic coefficients (except the power coefficient) are only weakly influenced by yawing motion. The power coefficient under yawing motion is lower than that under non-yawing motion, which means that yawing motion will cause the annual energy production of a turbine to decrease. The fitting results show that the damping term and the added mass term exert effects of the same level on the loads and moments of vertical-axis turbines under yawing motion. The results of this study can facilitate the study of the motion response of floating vertical-axis tidal current turbine systems in waves.     

The hydrodynamic coefficients for an oscillating rectangular structure on a free surface with sidewall

Based on a two dimensional linear water wave theory, the boundary element method (BEM) is developed and applied to study the heave and the sway problem of a floating rectangular structure in water to finite depth with one side of the boundary is a vertical sidewall and the other boundary is an open boundary. Numerical results for the added mass and radiation damping coefficients are presented. These coefficients are not only depend on the submergence and the width of the structure, but also depend on the clearance between structure and sidewall. Negative added mass and sharp peaks in the damping and added mass coefficients have been found when the clearance with a value close to integral times of half wave length of wave generated by oscillation structure. The important effect of the clearance on the added mass and radiation damping coefficients are discussed in detail. An analytical solution method is also presented. The BEM solution is compared with the analytical solution, and the comparison shows good agreement.