Multi-vessel seakeeping computations with linear potential theory

The wave-induced motion of two vessels in close proximity is studied using traditional two-dimensional (2D) and three-dimensional (3D) boundary element methods. Added mass, damping, and the behavior of the free surface between the vessels are examined in some detail. As expected based on previous work, the response of the water between the vessels is found to have a profound effect on the hydrodynamic forces. At “critical frequencies” corresponding to standing waves between the hulls, the hydrodynamic forces undergo significant drastic changes. The 2D and 3D results are compared, and the effects of a skew angle between the vessels are examined. Some of the consequences of the behavior at the critical frequencies for simulations in the time domain are examined, the most significant of which is a very lightly damped impulse response.     

The statistics of second order wave forces

A starting point in the statistical analysis of the slow drift motions of moored offshore structures is the determination of the statistical distribution of the second order exciting forces. In the past^2,6 an exponential probability density function has been used, this being the limiting case as the incident seastate becomes infinitely narrow banded. In the present note the second order force distribution is found for a general seastate and a significant deviation from an exponential distribution is found for those seastates which are likely to occur in practice. The statistical theory is based on a heuristic approximation to the second order force, which is shown to be consistent with a frequently used frequency domain approximation which has been suggested by Newman.⁸ Although Roberts² has shown that for very lightly damped systems the response statistics are independent of those of the exciting force, the present results may have a significant effect on the response statistics for higher levels of damping.     

An investigation into parametric roll resonance in regular waves using a partly non-linear numerical model

A partly non-linear time-domain numerical model is used for the prediction of parametric roll resonance in regular waves. The ship is assumed to be a system with four degrees of freedom, namely, sway, heave, roll and pitch. The non-linear incident wave and hydrostatic restoring forces/moments are evaluated considering the instantaneous wetted surface whereas the hydrodynamic forces and moments, including diffraction, are expressed in terms of convolution integrals based on the mean wetted surface. The model also accounts for non-potential roll damping expressed in an equivalent linearised form. Finally, the coupled equations of motion are solved in the time-domain referenced to a body fixed axis system.This method is applied to a range of hull forms, a post-Panamax C11 class containership, a transom stern Trawler and the ITTC-A1 containership, all travelling in regular waves. Obtained results are validated by comparison with numerical/experimental data available in the literature. A thorough investigation into the influence of the inclusion of sway motion is conducted. In addition, for the ITTC-A1 containership, an investigation is carried out into the influence of tuning the numerical model by modifying the numerical roll added inertia to match that obtained from roll decay curves.     

Bounded and unbounded coupled transverse response of parametrically excited vertical marine risers and tensioned cable legs for marine applications

The paper deals with the non-linear dynamic response in the transverse direction of vertical marine risers or a tensioned cable legs subjected to parametric excitation at the top of the structure. The dynamic model contains both elastic and bending effects. The analytical approach reveals that the dynamic lateral response is governed by effects originated from the coupling of modes in transverse direction. The mathematical model is being treated numerically by retaining a sufficient number of transverse modes. Numerical results are given for specific case studies and refer both to the time histories of the lateral response for all modes of motion, and to the corresponding power spectral densities obtained through FFT. The numerical predictions are suitably plotted and discussed. The calculations concern both the undamped and the damped dynamic system. The damping in the system is a non-linear Morison type term, which describes the effect of the hydrodynamic drag. Both coupled and uncoupled equations are treated and points as well as regions of coupled and uncoupled stability and instability are defined. It is shown that the impacts originated from the coupling, evaluate new instabilities for the respective undamped system. The numerical results obtained through FFT of the time histories, provide qualitative conclusions for the features of the dynamic response for the modes of motions considered. Special attention has been paid to the effect of the hydrodynamic drag for the parametric excitation frequencies that guide the dynamic system to lie within a region of coupled instability.     

Analysis of Dynamic Characteristics of Submarine Free Spanning Pipelines by Complex Damping Method

Considering the effect of the internal flowing fluid and the external marine environmental condition, the differential equation for the vortex-induced vibration (V1V) of the free spanning pipeline is derived and is discretized by the Hermit interpolation function. The free vibration equation with the damping term is solved by the complex damping method for the natural frequency, and then the effect of fluid damping on the natural frequency of the free spanning pipeline is analyzed.The results show that fluid damping has a significant influence on the damped natural frequency of the free spanning pipeline in the lock-in state, while it has little influence when the pipeline is out of the lock-in state. In the meantime,the change of the free span length has the same effect on the damped natural frequency and the undamped natural frequency.     

Analysis of second-order resonance in wave interactions with floating bodies through a finite-element method

A time-domain method is employed to analyse the resonant oscillations of the liquid confined within the two floating bodies. The velocity potentials at each time step are obtained through a finite-element method (FEM) with quadratic shape functions. The matrix equation of the FEM is solved through an iteration. The radiation condition is satisfied through a combination of the damping zone method and the Sommerfeld–Orlanski equation. A detailed analysis is made for two rectangular floating cylinders undergoing forced oscillation. The first-order potential reveals the resonant behaviour of the wave motion at certain frequencies ω_i, which is similar to sloshing in a tank. More interestingly, the second-order theory further reveals that when the oscillation frequency is at ω_i/2 or half of the resonant frequency, no first-order resonance is observed as expected, but the second-order resonant motion becomes evident, which does not seem to have been extensively investigated so far. Detailed results for two rectangular cylinders are provided to show some insights into the resonant effect due to the interaction between the bodies. The first- and second-order resonant phenomena have been observed and the result has shown that the second-order components have significant influence on the wave and force in some cases, especially at the second-order resonance.     

Experimental investigation of hull girder vibrations of a flexible backbone model in bending and torsion

Although the coupled horizontal–torsional vibrations of open ships have been investigated numerically for decades, the available experimental data in oblique seas seem rare. Model tests, considering natural frequencies of bending and torsional modes, have been conducted by the Centre for Ships and Ocean Structures (CeSOS) in the towing tank and ocean basin. A flexible backbone model was designed with five cut-outs on the top side of the aluminum beam to realize approximately torsional stiffness as well as vertical and horizontal bending stiffness. This paper mainly deals with measured bending and torsional vibrations in regular and irregular waves. The damping ratios, mode shapes, and modal moments were derived from experimental data, and a numerical model based on modal superimposition is established according to the measured hydroelastic properties. Some time-domain simulations are carried out considering the structural characteristics, and compared with measured results. The test data in regular waves are presented to show the possible factors of influence on the vibrations. The effect of bending and torsional vibrations on the extreme response values in irregular waves is estimated. The uncertainties in the experiments are discussed and conclusions are presented at the end of this paper.     

Investigation of motions of catamarans in regular waves—II

By extending the linear frequency domain theory, a quasi-non-linear time-domain technique has been developed to investigate the large amplitude motions of catamarans in regular waves. The non-linearity of hydrodynamic forces included in this practical method comes from variations of a ship's submerged portion. These forces are obtained from a database generated by the linear frequency domain method at each time step. The coupled equations, heave and pitch, are solved in the time domain by using the Runge-Kutta method with proper initial values. In order to investigate the non-linear effects of large amplitude motions of the V-1 catamaran in the head-sea condition, numerical results obtained from the linear and non-linear strip methods have been compared with those obtained from a series of experiments carried out in the towing tank of the Hydrodynamics Laboratory at the University of Glasgow. Based on the comparative studies, the numerical results obtained from the time-domain program can provide better predictions for the large amplitude motions of catamarans than the linear frequency domain method. It is concluded that the non-linear effects are significant when the model speeds and wave amplitudes increase. The peak values of large amplitude motions around the resonance frequencies, as obtained from the non-linear time-domain predictions as well as from measurements, are smaller than those obtained from the linear theory.     

Motion response simulation of a twin-hulled semi-submersible

The purpose of the study was to develop a prediction technique to simulate the motion response of a damaged platform under wave, wind and current forces. The equations of motion were obtained using Newton's second law and the numerical solution technique of non-linear equations of motion is explained for intact and damaged cases. The analysis technique employs large displacement non-linear equations of motion. Solutions were obtained in the time-domain to predict the motion characteristics. In this study, analysis procedures were developed to calculate: (a) wave loading on asymmetrical structural configurations; (b) hydrodynamic reaction forces (inertia or moment of inertia, damping and restoring forces) on asymmetrical shapes. During the damage simulation, change in the mass of the structure as well as wave and hydrodynamic reaction forces, were taken into account. The computer program developed for the time-domain simulation is introduced. In order to avoid slowly decaying transient motions of the structure due to wave excitation forces, an exponential ramp function is used. The application of a ramp function enables a quick convergence in the time-domain solution of equations of motion. Results of a numerical motion simulation program and the experimental studies are also presented in order to make comparisons. Comparison of the test results with the numerical simulations shows good agreement for heave, roll and pitch motions. The formulations and the computational procedures given in this paper provide useful tools for the investigation of the non-linear dynamic stability characteristics of floating structures in waves for intact, damaged and post-flooding conditions in six-degrees of freedom.     

二维水波辐射问题的二阶时域解

用二阶理论在时域范围内研究二维二阶非线性水波的辐射问题,一阶、二阶问题分别满足各自的自由表面条件和物面条件,采用人工阻尼来吸收反射波.计算采用有限元法,对水平半圆柱体在水面的振荡问题进行了计算,计算了波高时间历程和水动力,计算结果与有关文献中的数据相比符合较好.     

Sensitivity of extreme hydroelastic load effects to changes in ship hull stiffness and structural damping

A new hybrid method for the time-domain nonlinear simulation of the hydroelastic load effects and the peak over threshold (POT) method for the calculation of the short-term extreme responses are briefly described and applied to a flexible SL-7 class containership and a flexible liquefied natural gas (LNG) ship. Three stiffness levels, three stiffness distributions and three modal damping ratios are used to study the influence of the hull flexibility and structural damping on the short-term prediction of extreme vertical hydroelastic load effects. The results give justification for some simplified treatment of the first vertical flexible mode in early design stage when structural details are not available.     

Ocean waves propagating over a porous seabed of finite thickness

In the past few decades, considerable efforts have been devoted to the phenomenon of wave-seabed interaction. However, conventional investigations for determining wave characteristics have been focused on the wave nonlinearity. On the other hand, most previous works have been only concerned with the seabed response under the wave pressure, which was obtained from the assumption of a rigid seabed. In this paper, the inertia forces and employing a complex wave number are considered in the whole problem. Based on Biot’s poro-elastic theory, the problem of wave-seabed interaction is first treated analytically for a homogeneous bed of finite thickness and a new wave dispersion relationship is also obtained, in which the soil characteristics are included. The numerical results indicate that the effects of soil parameters significantly affect the wave characteristics (such as the damping of water wave, wave length and wave pressure). Furthermore, the effects of inertia forces on the wave-induced seabed response cannot always be ignored under certain combination of wave and soil conditions.     

Limiting heave response of a wave energy device by draft adjustment with upper surface immersion

Many studies have been published concerning the influence of the immersed shape (in still water) of a floating body on its response and power capture from ocean waves. With a few notable exceptions, much of this analysis has assumed small amplitude motion and linear models have been employed to predict response. The form of the upper surface of such a body has received little attention. Here, we show how the shape of the upper (top) surface of a floating body can be designed to ensure that the response amplitude of the body is within a specified value. This is of considerable importance to the survivability of wave energy devices. The approach used is to achieve a large increase of both natural period and hydrodynamic damping for only a small change of float mass. These two factors impose a hydrodynamic limit on the displacement which may be exploited to avoid the ‘end-stop’ problem often encountered in wave device design. To demonstrate the change of response, experimental measurements are presented of the response of an axisymmetric float with rounded base and conical upper surface with rounded perimeter due to a range of regular, irregular and focused wave conditions. Power extraction is not considered since the mechanically undamped response represents the worst case. In contrast to a simple, straight-sided axisymmetric float, a smaller change of mass is required to satisfy a particular response amplitude limit. Although a significant reduction is not expected, hydrodynamic damping may reduce with increasing physical scale, and this remains to be quantified.     

On the response of a Coulomb‐damped poroelastic bed to water waves

Abstract

The problem of forced vibration of a slightly inelastic porous bed by water waves is treated analytically on the basis of a linearized expression of the nonlinear damping term for the grain‐to‐grain friction in bed soils and the linear theory by Biot (1962a [Jour. Appl. Physics, 33:1482–1498]) on the elastic wave propagation in porous media. A dispersion relation of water waves is obtained as a function of wave frequency, water depth, permeability, Poisson's ratio, rigidity, and specific loss of bed soil. Three types of elastic waves are induced in a bed by water waves: a shear wave and a compressional wave in the skeletal frame of soil, and a compressional wave in the pore fluid. The compressional wave, due to the motion of the pore fluid relative to the skeletal frame of soil, is highly damped by the viscosity of pore fluid and only a short range effect near the boundaries of discontinuity, such as a sea‐seabed interface. The seabed response to water waves is characterized by the two Mach numbers, i.e., the ratio of water‐wave speed to shear‐wave speed in soil and the ratio of water‐wave speed to compressional‐wave speed in soil. Most of the water‐wave propagation problems fall into the subsonic flow condition, where elastic waves in the bed travel faster than water waves.

For sandy beds, generally the speeds of compressional and shear waves are much higher than the phase velocity of the water wave. For this case, the solution of the Coulomb‐damped poroelastic bed response presented in this paper approaches the solution of the massless poroelastic bed response in Yamamoto et al. (1978 [Jour. Fluid Mech., 87(1): 193–206]). The damping of water waves due to internal grain‐to‐grain friction is equally or more significant than the damping due to percolation in sand beds.

For clay beds, the speed of the shear wave in soil becomes low and comparable to the phase speed of the water wave. The bed motion for this case is considerably amplified due to the near‐resonance vibration of shear mode of bed vibration. The water wavelength on a clay bed is significantly shortened compared to the water wavelength over a rigid bed. The water wave damping due to internal grain‐to‐grain friction in soil becomes much larger compared to the water wave damping due to percolation in clay beds. Long water waves over a soft clayey bed attenuate within several wavelengths of travel distance.     

Viscosity and nonlinearity effects on the forces and waves generated by a floating twin hull under heave oscillation

Nonlinear hydrodynamics of a twin rectangular hull under heave oscillation is analyzed using numerical methods. Two-dimensional nonlinear time-domain solutions to both inviscid and viscous problems are obtained and the results are compared with linear, inviscid frequency-domain results obtained in [26] to quantify nonlinear and viscous effects. Finite-difference methods based on boundary-fitted coordinates are used for solving the governing equations in the time domain [2]. A primitive-variables based projection method [6] is used for the viscous analysis and a mixed Eulerian–Lagrangian formulation [11] for inviscid analysis. The algorithms are validated and the order of accuracy determined by comparing the results obtained from the present algorithm with the experimental results of Vugt [22] for a heaving rectangle in the free surface. The present study on the twin-hull hydrodynamics shows that at large and non-resonant regular frequencies, and small amplitude of body oscillation, the fluid viscosity does not significantly affect the wave motion and the radiation forces. At low frequencies however the viscosity effect is found to be significant even for small amplitude of body oscillation. In particular, the hydrodynamic force obtained from the nonlinear viscous analysis is found to be closer to the linear inviscid force than the nonlinear inviscid force to the linear inviscid force, the reason for which is attributed to the wave dampening effect of viscosity. Since the wave lengths generated at smaller frequencies of oscillation are longer and therefore the waves could have a more significant effect on the dynamic pressure on the bottom of the hulls which contribute to the heave force, the correlation between the heave force and the wave elevation is found to be larger at smaller frequencies. Because of nonlinearity, the wave radiation and wave damping force remained nonzero even at and around the resonant frequencies – with the resonant frequencies as determined in [26] using linear potential flow theory. As to be expected, the nonlinear effect on the wave force is found to be significant at all frequencies for large amplitude of oscillation compared to the hull draft. The effect of viscosity on the force, by flow separation, is also found to be significant for large amplitude of body oscillation.     

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.     

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.     

Effects of side keels on patrol vessel safety in astern waves

Planing hull vessel is well-known to have the problem of low damping and susceptible to large roll angles when encountering beam seas especially when stopped or operated at low speed. One approach to reduce this problem is to incorporate a pair of side keels. The effects of the side keels on both roll damping and resistance of a planing hull patrol vessel were experimentally evaluated by varying length, breadth and the position of the side keels. Tests data of the side keels have been presented on the roll damping coefficients and the resistance forces of the vessel. Next, the roll damping coefficients were utilised in time domain simulation program based on a six-degree-of-freedom mathematical model to predict the roll response and assess the vessel safety in astern waves for the case with and without the side keels. The results from the time domain simulation program have indicated that the effects of a pair of side keels can improve the vessel safety.     

CFD approach to roll damping of ship with bilge keel with experimental validation

The most common method of reducing roll motion of ship-shaped floating systems is the use of bilge keel which act as damping elements. The estimation of the damping introduced by bilge keel is still largely based on empirical methods. The present work adopts the CFD approach to the estimation of roll damping, both without and with bilge keel and validates the results with experiments conducted in a wave flume. Specifically, free oscillation tests are conducted at model scale to obtain roll damping, both by experiments and CFD simulation and reasonably good comparisons are obtained. The experiments also include PIV study of the flow field and attempt has been made to correlate the measured flow field with that obtained by CFD. The CFD methodology has the potential to determine rationally the size and orientation of bilge keels in design with reasonably accurate estimate of the additional roll damping that it provides to ship's roll motion.     

Effects of mooring line drag damping on response statistics of vessels excited by first- and second-order wave forces

The drag-induced damping in a mooring cable due to combined first- and second-order wave excited motion of a moored vessel has been determined by statistical linearisation. A dynamic stiffness approach developed elsewhere is used to deal with the dynamics of the mooring cables. The power spectral densities of low- and wave-frequency responses are obtained which clearly show the influence of mooring line damping. The non-Gaussian probability density functions (pdf) and expected crossing rates of vessel responses and dynamic cable tensions are determined using the Kac–Seigert technique, and the influence of drag damping is highlighted.