Mathematical modelling of response amplitude operator for roll motion of a floating body: Analysis in frequency domain with numerical validation

This paper investigates mathematical modelling of response amplitude operator （RAO） or transfer function using the frequency-based analysis for uncoupled roll motion of a floating body under the influence of small amplitude regular waves. The hydrodynamic coefficients are computed using strip theory formulation by integrating over the length of the floating body. Considering sinusoidal wave with frequency （ω ） varying between 0.3 rad/s and 1.2 rad/s acts on beam to the floating body for zero forward speed, analytical expressions of RAO in frequency domain is obtained. Using the normalization procedure and frequency based analysis, group based classifications are obtained and accordingly governing equations are formulated for each case. After applying the fourth order Runge-Kutta method numerical solutions are obtained and relative importance of the hydrodynamic coefficients is analyzed. To illustrate the roll amplitude effects numerical experiments have been carried out for a Panamax container ship under the action of sinusoidal wave with a fixed wave height. The effect of viscous damping on RAO is evaluated and the model is validated using convergence, consistency and stability analysis. This modelling approach could be useful to model floating body dynamics for higher degrees of freedom and to validate the result.     

采用解析和数值方法分析浮动式矩形浮箱的波浪诱导载荷研究（英文）

A three-dimensional mathematical hydrodynamic model associated with surface wave radiation by a floating rectangular box-type structure due to heave, sway, and roll motions in finite water depth is investigated based on small amplitude water wave theory and linear structural response. The analytical expressions for the radiation potentials, wave forces, and hydrodynamic coefficients are presented based on matched eigenfunction expansion method(MEFEM). The correctness of the analytical results of...     

深水起重铺管船横摇运动性能分析(英文)

For a large floating vessel in waves,radiation damping is not an accurate prediction of the degree of roll unlike other degrees of freedom motion.Therefore,to get the knowledge of roll motion performance of deepwater pipelay crane vessels and to keep the vessel working safety,the paper presents the relationship between a series of dimensionless roll damping coefficients and the roll response amplitude operator(RAO).By using two kinds of empirical data,the roll damping is estimated in the calculation flow.After getting the roll damping coefficient from the model test,a prediction of roll motion in regular waves is evaluated.According to the wave condition in the working region,short term statistics of roll motion are presented under different wave parameters.Moreover,the relationship between the maximal roll response level to peak spectral wave period and the roll damping coefficient is investigated.Results may provide some reference to design and improve this kind of vessel.     

Dynamic interaction of parallel moving ships in close proximity

Nowadays,there are many studies conducted in the field of marine hydrodynamics which focus on two vessels traveling and floating in sufficiently close proximity to experience significant interactions.The hydrodynamic behavior of parallel moving ships in waves is an interesting and important topic of late.A numerical investigation has been carried out for the prediction of wave exciting forces and motion responses of parallel moving ships in regular waves.The numerical solution was based on 3D distribution technique and using the linear wave theory to determine the exciting forces and ship＇s motion.The speed effects have been considered in the Green function for more realistic results.The numerical computations of wave exciting forces and motion responses were carried out for a Mariner and Series 60 for the purpose of discovering different Froude numbers and different separation distances in head sea conditions.Based on the numerical computations,it was revealed that the sway,roll and yaw have a significant effect due to hydrodynamic interaction.     

斜浪下FLNG与LNGC旁靠时相对运动的时域分析（英文）

Strong hydrodynamic interactions during the side-by-side offloading operation between floating liquefied natural gas(FLNG) and liquefied natural gas carrier(LNGC) can induce high risks of collision. The weather vane effect of a single-point mooring system normally results in the satisfactory hydrodynamic performance of the side-by-side configuration in head seas. Nevertheless, the changes in wave directions in real sea conditions can significantly influence the relative motions. This article studies the relative motions of the side-by-side system by using the theoretical analysis method and the numerical calculation method. Based on the three-dimensional potential theory modified by artificial damping-lid method, the frequency-domain hydrodynamic coefficients can be improved to calculate the retardation functions for the multi-body problem. An in-house code is then developed to perform the time-domain simulation of two vessels, through which the relative motions are subsequently obtained. A range of oblique waves are chosen for the extensive calculation of relative motions between the two vessels, which are further analyzed in terms of the phase shift of motion responses induced by specific resonant wave patterns. Investigation results show that wave directions have a significant influence on the relative sway, roll, and yaw motions. Under the circumstance that the absolute phase shift between the roll motions of two vessels approaches 180°, stronger relative motions are induced when LNGC is on the weather side.Moreover, the gap water resonances at high frequencies tend to cause the dangerous opposed oscillation of two vessels in the sway and yaw modes, whereas FLNG reduces the gap water resonances and relative motions when located on the weather side.     

Scattering of surface water waves involving semi-infinite floating elastic plates on water of finite depth

Two problems of scattering of surface water waves involving a semi-infinite elastic plate and a pair of semi-infinite elastic plates,separated by a gap of finite width,floating horizontally on water of finite depth,are investigated in the present work for a two-dimensional time-harmonic case.Within the frame of linear water wave theory,the solutions of the two boundary value problems under consideration have been represented in the forms of eigenfunction expansions.Approximate values of the reflection and transmission coefficients are obtained by solving an over-determined system of linear algebraic equations in each problem.In both the problems,the method of least squares as well as the singular value decomposition have been employed and tables of numerical values of the reflection and transmission coefficients are presented for specific choices of the parameters for modelling the elastic plates.Our main aim is to check the energy balance relation in each problem which plays a very important role in the present approach of solutions of mixed boundary value problems involving Laplace equations.The main advantage of the present approach of solutions is that the results for the values of reflection and transmission coefficients obtained by using both the methods are found to satisfy the energy-balance relations associated with the respective scattering problems under consideration.The absolute values of the reflection and transmission coefficients are presented graphically against different values of the wave numbers.     

Dynamic coupled analysis of the floating platform using the asynchronous coupling algorithm

This paper discusses the numerical modeling of the dynamic coupled analysis of the floating platform and mooring/risers using the asynchronous coupling algorithm with the purpose to improve the computational efficiency when multiple lines are connected to the platform. The numerical model of the platform motion simulation in wave is presented. Additionally, how the asynchronous coupling algorithm is implemented during the dynamic coupling analysis is introduced. Through a comparison of the numerical results of our developed model with commercial software for a SPAR platform, the developed numerical model is checked and validated.     

Coupled motions of two ships in irregular waves in time domain

A three-dimensional time domain approach is used to study the coupled motions of two ships with forward speed in waves. In this approach, the boundary condition is satisfied on the mean wetted hull surface of the moving bodies and the free surface condition is linearized. The problem is solved by using a transient free-surface Green function source distribution on the submerged hulls. After solving the response amplitude operator, the method of spectral analysis is employed to clearly express the motion energy spectrum and significant amplitude of two ships. For verifying the code, two same circular cylinders at beam wave are selected to calculate coupled motions by comparison with the results obtained by 3D frequency method which has been proved to be efficient for solving such problems.Two Wigley ships of different sizes with the same forward speed are chosen for numerical calculation of the interaction effect, and some useful suggestions are obtained for underway replenishment at sea.     

Wave Scattering by Porous Bottom Undulation in a Two Layered Channel

The scattering of plane surface waves by bottom undulations in channel flow consisting of two layers is investigated by assuming that the bed of the channel is composed of porous material. The upper surface of the fluid is bounded by a rigid lid and the channel is unbounded in the horizontal directions. There exists only one wave mode corresponding to an internal wave. For small undulations, a simplified perturbation analysis is used to obtain first order reflection and transmission coefficients in terms of integrals involving the shape function describing the bottom. For sinusoidal bottom undulations and exponentially decaying bottom topography, the first order coefficients are computed. In the case of sinusoidal bottom the first order transmission coefficient is found to vanish identically. The numerical results are depicted graphically in a number of figures.     

可渗透海床上具有小底部变形的斜弯曲重力波的相互作用:格林函数法（英文）

The interaction of oblique incident water waves with a small bottom deformation on a porous ocean-bed is examined analytically here within the framework of linear water wave theory. The upper surface of the ocean is assumed to be covered by an infinitely extended thin uniform elastic plate, while the lower surface is bounded by a porous bottom surface having a small deformation. By employing a simplified perturbation analysis, involving a small parameter δ(1), which measures the smallness of the deformation, the governing Boundary Value Problem(BVP) is reduced to a simpler BVP for the first-order correction of the potential function. This BVP is solved using a method based on Green's integral theorem with the introduction of suitable Green's function to obtain the first-order potential, and this potential function is then utilized to calculate the first-order reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom deformation. Consideration of a patch of sinusoidal ripples shows that when the quotient of twice the component of the incident field wave number propagating just below the elastic plate and the ripple wave number approaches one, the theory predicts a resonant interaction between the bed and the surface below the elastic plate. Again, for small angles of incidence, the reflected wave energy is more as compared to the other angles of incidence. It is also observed that the reflected wave energy is somewhat sensitive to the changes in the flexural rigidity of the elastic plate, the porosity of the bed and the ripple wave numbers. The main advantage of the present study is that the results for the values of reflection and transmission coefficients obtained are found to satisfy the energy-balance relation almost accurately.     

液舱晃荡与浮体非线性耦合运动分析(英文)

Nonlinear interactions among incident wave, tank-sloshing and floating body coupling motion are investigated. The fully nonlinear sloshing and body-surface nonlinear free surface hydrodynamics is simulated using a Non-Uniform Rational B-Spline (NURBS) higher-order panel method in time domain based on the potential theory. A robust and stable improved iterative procedure (Yan and Ma, 2007) for floating bodies is used for calculating the time derivative of velocity potential and floating body motion. An energy dissipation condition based on linear theory adopted by Huang (2011) is developed to consider flow viscosity effects of sloshing flow in nonlinear model. A two-dimensional tank model test was performed to identify its validity. The present nonlinear coupling sway motion results are subsequently compared with the corresponding Rognebakke and Faltinsen (2003)’s experimental results, showing fair agreement. Thus, the numerical approach presented in this paper is expected to be very efficient and realistic in evaluating the coupling effects of nonlinear sloshing and body motion.     

3-D computational method of wave loads on turret moored FPSO tankers

A three-dimensional method of calculating wave loads of turret moored FPSO (Floating Production Storage and Offloading) tankers is presented. The linearized restoring forces acting on the ship hull by the mooring system are calculated according to the catenary theory, which are expressed as the function of linear stiffness coefficients and the displacements of the upper ends of mooring chains. The hydrodynamic coefficients of the ship are calculated by the three-dimensional potential flow theory of the linear hydrodynamic problem for ships with a low forward speed. The equations of ship motions are established with the effect of the restoring forces from the mooring system included as linear stiffness coefficients. The equations of motions are solved in frequency domain, and the responses of wave-induced motions and loads on the ship can be obtained. A computer program based on this method has been developed, and some calculation examples are illustrated. Analysis results show that the method can give satisfying prediction of wave loads.     

复杂形体远程操控水下运载器水动力阻尼模型建模和检验(英文)

In this paper,numerical modeling and model testing of a complex-shaped remotely-operated vehicle(ROV) were shown.The paper emphasized the systematic modeling of hydrodynamic damping using the computational fluid dynamic software ANSYS-CFXTM on the complex-shaped ROV,a practice that is not commonly applied.For initial design and prototype testing during the developmental stage,small-scale testing using a free-decaying experiment was used to verify the theoretical models obtained from ANSYS-CFXTM.Simulation results are shown to coincide with the experimental tests.The proposed method could determine the hydrodynamic damping coefficients of the ROV.     

超大型浮体连接器载荷时域计算（英文）

Loads generated after an air crash, ship collision, and other accidents may destroy very large floating structures(VLFSs) and create additional connector loads. In this study, the combined effects of ship collision and wave loads are considered to establish motion differential equations for a multi-body VLFS. A time domain calculation method is proposed to calculate the connector load of the VLFS in waves. The Longuet–Higgins model is employed to simulate the stochastic wave load. Fluid force and hydrodynamic coefficient are obtained with DNV Sesam software. The motion differential equation is calculated by applying the time domain method when the frequency domain hydrodynamic coefficient is converted into the memory function of the motion differential equation of the time domain. As a result of the combined action of wave and impact loads, high-frequency oscillation is observed in the time history curve of the connector load. At wave directions of 0° and 75°, the regularities of the time history curves of the connector loads in different directions are similar and the connector loads of C1 and C2 in the X direction are the largest. The oscillation load is observed in the connector in the Y direction at a wave direction of 75° and not at 0°. This paper presents a time domain calculation method of connector load to provide a certain reference function for the future development of Chinese VLFS     

能量耗散效应的多域边界元法(英文)

The wave diffraction and radiation around a floating body is considered within the framework of the linear potential theory in a fairly perfect fluid.The fluid domain extended infinitely in the horizontal directions but is limited by the sea bed,the body hull,and the part of the free surface excluding the body waterplane,and is subdivided into two subdomains according to the body geometry.The two subdomains are connected by a control surface in fluid.In each subdomain,the velocity potential is described by using the usual boundary integral representation involving Green functions.The boundary integral equations are then established by satisfying the boundary conditions and the continuous condition of the potential and the normal derivation across the control surface.This multi-domain boundary element method(MDBEM) is particularly interesting for bodies with a hull form including moonpools to which the usual BEM presents singularities and slow convergence of numerical results.The application of the MDBEM to study the resonant motion of a water column in moonpools shows that the MDBEM provides an efficient and reliable prediction method.     

Time Domain Calculation of Connector Loads of a Very Large Floating Structure

Loads generated after an air crash, ship collision, and other accidents may destroy very large floating structures(VLFSs) and create additional connector loads. In this study, the combined effects of ship collision and wave loads are considered to establish motion differential equations for a multi-body VLFS. A time domain calculation method is proposed to calculate the connector load of the VLFS in waves. The Longuet–Higgins model is employed to simulate the stochastic wave load. Fluid force and hydrodynamic coefficient are obtained with DNV Sesam software. The motion differential equation is calculated by applying the time domain method when the frequency domain hydrodynamic coefficient is converted into the memory function of the motion differential equation of the time domain. As a result of the combined action of wave and impact loads, high-frequency oscillation is observed in the time history curve of the connector load. At wave directions of 0° and 75°, the regularities of the time history curves of the connector loads in different directions are similar and the connector loads of C1 and C2 in the X direction are the largest. The oscillation load is observed in the connector in the Y direction at a wave direction of 75° and not at 0°. This paper presents a time domain calculation method of connector load to provide a certain reference function for the future development of Chinese VLFS     

并靠两浮体上的波浪漂移力(英文)

An innovative hydrodynamic theory and numerical model were developed to help improve the efficiency, accuracy, and convergence of the numerical prediction of wave drift forces on two side-by-side deepwater floating bodies. The wave drift forces were expressed by the double integration of source strength and the corresponding Green function on the body surface, which is consistent with the far field formula based on momentum conservation and sharing the advantage of near field calculations providing the drift force on each body. Numerical results were validated through comparing the general far field model and pressure integral model, as well as the middle field model developed using the software HydroStar.     

Scattering of Oblique Surface Waves by the Edge of Small Deformation on a Porous Ocean Bed

The scattering of oblique incident surface waves by the edge of a small cylindrical deformation on a porous bed in an ocean of finite depth, is investigated here within the framework of linearized water wave theory. Using perturbation analysis, the corresponding problem governed by modified Helmholtz equation is reduced to a boundary value problem for the first-order correction of the potential function. The first-order potential and, hence, the reflection and transmission coefficients are obtained by a method based on Green’s integral theorem with the introduction of appropriate Green’s function. Consideration of a patch of sinusoidal ripples shows that when the quotient of twice the component of the incident field wave number along x-direction and the ripple wave number approaches one, the theory predicts a resonant interaction between the bed and the free-surface, and the reflection coefficient becomes a multiple of the number of ripples. Again, for small angles of incidence, the reflected energy is more as compared to the other angles of incidence. It is also observed that the reflected energy is somewhat sensitive to the changes in the porosity of the ocean bed. From the derived results, the solutions for problems with impermeable ocean bed can be obtained as particular cases.     

Analytical study of magnetohydrodynamic propulsion stability

In this paper an analytical solution for the stability of the fully developed flow drive in a magneto-hydro-dynamic pump with pulsating transverse Eletro-magnetic fields is presented. To do this, a theoretical model of the flow is developed and the analytical results are obtained for both the cylindrical and Cartesian configurations that are proper to use in the propulsion of marine vessels. The governing parabolic momentum PDEs are transformed into an ordinary differential equation using approximate velocity distribution. The numerical results are obtained and asymptotic analyses are built to discover the mathematical behavior of the solutions. The maximum velocity in a magneto-hydro-dynamic pump versus time for various values of the Stuart number, electro-magnetic interaction number, Reynolds number, aspect ratio, as well as the magnetic and electrical angular frequency and the shift of the phase angle is presented. Results show that for a high Stuart number there is a frequency limit for stability of the fluid flow in a certain direction of the flow. This stability frequency is dependent on the geometric parameters of a channel.