Spectral Analysis of Nonlinear Drag Forces

The spectral properties of nonlinear drag forces of random waves on vertical circular cylinders are analyzed in this paper by means of nonlinear spectral analysis. The analysis provides basic parameters for estimation of the characteristic drag forces. Numerical computation is also performed for the investigation of the effects of nonlinearity of the drag forces.The results indicate that the wave drag forces calculated by linear wave theory are larger than those calculated by the third order Stokes wave theory for given waves. The difference between them increases with wave height. The wave drag forces calculated by use of hnear approximation are about 5% smaller than their actual values when measured in the peak values of spectral densities. This will result in a safety problem for the design of offshore structures. Therefore, the nonlinear effect of wave drag forces should be taken into comidemtion in design and application of important offshore structures.     

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.     

Triangular Configuration Tension Leg Platform behaviour under random sea wave loads

This study investigates the dynamic response of a Triangular Configuration Tension Leg Platform (TLP) under random sea wave loads. The random wave has been generated synthetically using the Monte-Carlo simulation with the Peirson–Moskowitz (P–M) spectrum. Diffraction effects and second-order wave forces have not been considered. The evaluation of hydrodynamic forces is carried out using the modified Morison equation with water particle kinematics evaluated using Airy's linear wave theory. Wave forces are taken to be acting in the surge degree-of-freedom. The effect of coupling of various structural degrees-of-freedom (surge, sway, heave, roll, pitch and yaw) on the dynamic response of the TLP under random wave loads is studied. Parametric studies for random waves with different H_s and T_z under the presence of current have also been carried out. For the orientation of the TLP, surge, heave and pitch degrees-of-freedom responses are influenced significantly. The surge power spectral density function (PSDF) indicates that the mean square response is affected by the amplification at the natural frequency of the surge degree-of-freedom and also at the peak frequency of the wave loading. The PSDF of the heave response shows higher peak values near the surge frequency and near the peak frequency of the wave loading. Surge response, therefore, influences heave response to the maximum. Variable submergence seems to be a major source of nonlinearity and significantly enhances the responses in surge, heave and pitch degrees-of-freedom. In the presence of current, the response behaviour of the TLP is altered significantly introducing a non-zero mean response in all degrees-of-freedom.     

Interaction of linear waves with an array of infinitely long horizontal circular cylinders in a two-layer fluid of infinite depth

The linear water wave scattering and radiation by an array of infinitely long horizontal circular cylinders in a two-layer fluid of infinite depth is investigated by use of the multipole expansion method. The diffracted and radiated potentials are expressed as a linear combination of infinite multipoles placed at the centre of each cylinder with unknown coefficients to be determined by the cylinder boundary conditions. Analytical expressions for wave forces, hydrodynamic coefficients, reflection and transmission coefficients and energies are derived. Comparisons are made between the present analytical results and those obtained by the boundary element method, and some examples are presented to illustrate the hydrodynamic behavior of multiple horizontal circular cylinders in a two-layer fluid. It is found that for two submerged circular cylinders the influence of the fluid density ratio on internal-mode wave forces is more appreciable than surface-mode wave forces, and the periodic oscillations of hydrodynamic results occur with the increase of the distance between two cylinders; for four submerged circular cylinders the influence of adding two cylinders on the wave forces of the former cylinders is small in low and high wave frequencies, but the influence is appreciable in intermediate wave frequencies.     

On the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas

The radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas of finite depth is investigated. The analytical expressions for the radiated and diffracted potentials are derived as infinite series by use of the method of separation of variables. The unknown coefficients in the series are determined by the eigenfunction expansion matching method. The expressions for wave forces, hydrodynamic coefficients and reflection and transmission coefficients are given and verified by the boundary element method. Using the present analytical solution, the hydrodynamic influences of the angle of incidence, the submergence, the width and the thickness of the structure on the wave forces, hydrodynamic coefficients, and reflection and transmission coefficients are discussed in detail.     

Numerical Investigation on Dynamics of the Tendon System of A TLP by Applying Absolute Nodal Coordinate Formulation

In the present study, the dynamics of the tendon system of a tension-leg platform(TLP) is investigated through the absolute nodal coordinate formulation(ANCF). Based on the energy conversion principle, the stiffness, generalized elastic force, external load and mass matrices of the element are deduced to perform the element assembling by using the finite element method. Then the motion equation of the tendon/riser is established. In this study, the TLP in the International Ship Structures Committee(ISSC) model under the first and second wave forces is considered as the case study. The simulation is performed in the MATLAB environment. Moreover, the accuracy and reliability of the programs are verified for cases of beam model with theoretical solutions. It is found that the motion response of tendons is affected by the TLP movement and environmental load, simultaneously. Then, the motion response is calculated using the SESAM software and exported as the boundary of ANCF tendons. Finally, the static and dynamic characteristics of the four tendons of ISSC TLP are analyzed systematically by the ANCF method.Performed analysis proves the effectiveness and feasibility of the ANCF method. It is concluded that the proposed method is a powerful scheme for calculating the dynamics of tendon/riser in the field of ocean engineering.     

Numerical investigation of secondary load cycle and ringing response of a vertical cylinder

A numerical wave tank is established based on two-phase FVM model and VOF method and verified with the physical experiment in Grue et al. (1994). Focusing waves with different wave steepness passing a vertical cylinder are investigated by Numerical simulations. The phenomenon called ‘secondary load cycle’ which may lead to ringing response, is observed and discussed. The presence of secondary load cycle could be related to Froude Number (Fr). The possible transition region of the present and absent secondary load cycle is Fr = 0.4. Sub and super harmonic wave components appear in the propagation of waves, second-order wave theory could give a good prediction. Morison equation with linear wave theory could predict well the wave forces of vertical cylinder with small steepness without the secondary load cycle, but cannot capture the crests/troughs of the wave forces with the secondary load cycle. Crest improvements are achieved by second-order wave theory. A spectral analysis based on wavelet transform is applied to wave loads. The frequency of the secondary load may be up to 13 times the wave frequency, which may cause the ringing response expanding to a higher frequency range. Strong ringing response occurs in steep wave, it could be extended up to 15 times wave trough-to-trough frequency due to the secondary load cycle. The damping has slightly influence on the peak of resonance response, but it will lead to faster decay of subsequent response, if the damping ratio is large.     

三阶Stokes型随机波浪载荷谱研究

应用非线性谱分析理论，对三阶Stokes型随机波浪载荷谱进行了分析，将波面方程及海水质点的水平速度用一阶波面的非线性组表示，导出了随机波浪谱的表达式。为了便于求解随机波浪的载荷谱，将阻力项展开为幂级数式，并应用非线性谱分析理论，确定了幂级数的系数，进而将波浪载荷表示为一阶波面及其导数的非线性组合，最后得出波浪载荷谱密度的表达式，并应用数值分析方法，得出单位桩柱波浪力及总波浪力谱密度。     

Numerical analysis on random wave-induced porous seabed response

Research on the response of random wave on offshore structures has received great deal of attention of many researchers and engineers in the design of marine structures. Most previous investigations have been limited to the regular waves. In this paper, based on Longuet–Higgins random wave theory and finite element method, a numerical model for random wave-induced seabed response is established. The seabed is treated as poroelastic medium and characterized by Biot’s partly dynamic equations (u–p model). The JONSWAP spectrum is adopted in Longuet–Higgins model, which is based on the cumulative superposition of linear diffraction solution. Based on the numerical results, the effects of random wave on seabed response are investigated by comparing with the corresponding Stokes wave and cnoidal wave. Then, a parametric study is conducted to examine the effect of wave and soil characteristic on the seabed.     

随机波作用下海床动态响应分析

真实的海洋波浪是随机的,而前人对海床的动态响应分析大都是选用线性波或者Stokes波理论,对海床的模拟大都采用Biot拟静力模型,忽略了流体速度及土体位移加速度的影响。联合使用Longuet-Higgins随机波模型(采用Jonswap谱)以及动力u-p形式的海床响应计算模型,使用COMSOL Multiphysics多场耦合软件的PDE模块输入方程进行有限元计算,得到随机波作用下整体海床动态响应结果。将随机波结果与一阶Stokes波和椭圆余弦波结果进行对比,并对渗透系数和饱和度进行参数分析,研究表明渗透系数和饱和度对于随机波作用下海床动态响应影响显著。     

Radiation and diffraction of linear water waves by an infinitely long submerged rectangular structure parallel to a vertical wall

The radiation and the diffraction of linear water waves by an infinitely long floating rectangular structure submerged in water of finite depth with leeward boundary being a vertical wall are analyzed in this paper by using the method of separation of variables. Analytical expressions for the radiated and diffracted potentials are derived as infinite series with unknown coefficients determined by the eigenfunction expansion matching method. The expressions for wave forces and hydrodynamic coefficients are given. A comparison is made between the results obtained by the present analytical solution and those obtained by the boundary element method. By using the present analytical solution, the hydrodynamic influences of the submergence, the width, the thickness of the structure, and the distance between the structure and the wall on the wave forces and hydrodynamic coefficients are discussed in detail.     

浅水区波浪非线性效应对砂质海床动力响应的影响

以广义Biot动力固结理论为基础,运用一阶椭圆余弦波和二阶Stokes波等非线性波浪理论考虑浅水区波浪荷载的非线性效应,在时域上采用有限元方法对非线性波浪力作用下饱和砂质海床的动力响应进行了数值求解,并与线性波浪作用下海床动力响应特性进行了对比分析。结果表明,随着波长与水深之比L／d及无量纲参数T(g/d)^1/2的增大,非线性波浪对海床动力响应的影响增大。与线性波浪理论相比,孔隙水压力与有效应力幅值的增大效应非常显著。因此在近海海洋建筑物设计与工程场地评价中,波浪力的非线性特性必须引起注意。     

Efficient computation of cross-spectral densities in the stochastic modelling of waves and wave loads

A new method is presented for efficient calculation of auto- and cross-spectral densities in the stochastic modelling of ocean waves and wave loads. As part of the short-term response analyses, the method may contribute to more efficient long-term response prediction. Specifically the cross-spectral densities of the first order wave excitation forces are considered, but the method is straightforwardly generalized to other cross-spectral densities, e.g. for wave elevation, wave kinematics or second order loads. The method can be used with any choice of directional spreading function, but special attention is given to the commonly used cos-2s type directional distribution. In addition to the development of the new method, the traditional method using the trapezoidal rule for numerical quadrature is improved by developing an adaptive way of choosing the number of integration points. The accuracy of the adaptive method and the new method is investigated, revealing rapid convergence for both methods. However, the new method appears more robust as it avoids so-called spurious hat errors. When applied to two different pontoon type floating bridges the adaptive method and the new method both achieve a great improvement in computational effort compared to the traditional trapezoidal rule method. When the dimensions of the floating bridge increase, i.e. the number of pontoons and their relative distances increase, the new method is superior with respect to computation time.     

A general method for calculating hydrodynamic forces

This paper presents the derivation of a general method for calculating wave forces on the cylindrical members of offshore structures. By means of the proposed method one can calculate the wave loading on cylindrical members of fixed or floating offshore structures orientated randomly in waves. This method of calculating wave forces is based on the linear Airy wave theory. Calculation procedure of wave force components is presented in great detail on the basis of wave particle kinematic properties obtained from the linear Airy wave theory. In the procedure of calculating wave forces presented, definitions of the wave reference system for propagating wave, the structure reference system for the platform and the member reference system for the tubular members of the structure are first established, and then the calculation of wave forces is given in terms of its components, which are pressure, acceleration and velocity forces, including current forces. At the end of the paper, expressions of total heave, sway and surge forces and total roll, pitch and yaw moments acting on the platform are given as a sum of these forces acting on each member of the platform. The calculation procedure derived in this paper provides a very efficient means of calculating wave forces and moments during the time-domain simulations of a floating platform experiencing large amplitude motion in intact, progressive flooding and damaged conditions. Comparisons of the predictions with the measurements which will be presented elsewhere reveal that the calculation procedure developed can predict large amplitude oscillatory and steady motion characteristics of an intact and damaged platform in waves with an acceptable degree of accuracy.     

Mean and low frequency wave drifting forces on floating structures

A recently developed method, based on three-dimensional potential theory, to compute the mean wave drifting forces on a free floating structure in regular waves, is extended to include low frequency oscillatory components which arise when the structure is floating in regular wave groups consisting of two regular waves with small difference frequency. This completes the information necessary for the determination of the wave drifting forces under arbitrary irregular wave conditions.In regular wave groups the drifting forces not only depend on the first order velocity potential and the first body motions, but also on the wave exciting forces due to the low frequency part of the second order potential. For the general three-dimensional case the latter contribution can only be determined numerically and at the expense of long computation times. Since this contribution is generally not large compared to components which may be determined using linear potential theory it is included using a simple approximation. Results of the method of approximation are compared with some two-dimensional cases for which exact solutions are known.Results of computations of the total mean and low frequency surge forces on a rectangular barge and a column stabilized semi-submersible platform are presented. For both structures, the computed mean surge drifting forces in regular head waves are compared with results of model tests.The computed components of the total mean drifting forces are presented. It appears that for both the barge and the semi-submersible the same components are of importance.For the semi-submersible, the computed low frequency second order surge forces in head waves are compared with results obtained from a test in irregular head waves using cross-bispectral analysis methods.     

Mooring forces and motion responses of pontoon-type floating breakwaters

The experimental and theoretical investigations on the behaviour of pontoon-type floating breakwaters are presented. A two-dimensional finite element model is adopted to study the behaviour of pontoon-type floating breakwaters in beam waves. The stiffness coefficients of the slack mooring lines are idealized as the linear stiffness coefficients, which can be derived from the basic catenary equations of the cable. The theoretical model is supported by an experimental programme conducted in a wave flume. The motion responses and mooring forces are measured for three different mooring configurations, and the results are reported and discussed in detail in this paper. The wave attenuation characteristics are presented for the configurations studied.     

Wave diffraction through submerged flexible breakwaters

This paper presents an analytical, computationally efficient method for the wave reflection and dynamic displacement of a submerged flexible breakwater. The solution of the two-dimensional linearized hydrodynamic problem introduced is based on the eigenfunction expansion technique. The breakwater is assumed to be thin, impermeable, flexible, moored to the bed through tethers and kept in tension by means of a floating buoy at its tip. The beam structure is considered to be either clamped or hinged at the sea bed, situated in an arbitrary water depth and subjected to normal linear waves. Numerical examples presented by this method are compared with those obtained by the Boundary Integral Equation Method, presented by Williams et al. Comparisons show an excellent agreement over a wide range of parameters for the wave reflection and the dynamic displacement. Numerical results are presented, mainly to show the effect of the breakwater rigidity and the method of fixation on the wave reflection and the structural displacement over a wide range of wave frequencies.     

Probability distribution of random wave forces in weakly nonlinear random waves

Based on the second-order random wave theory, the joint statistical distribution of the horizontal velocity and acceleration is derived using the characteristic function expansion method. From the joint distribution and the Morison equation, the theoretical distributions of drag forces, inertia forces and total random wave forces are determined. The distribution of inertia forces is Gaussian as that derived using the linear wave model, whereas the distributions of drag forces and total random forces deviate slightly from those derived utilizing the linear wave model. It is found that the distribution of wave forces depends solely on the frequency spectrum of sea waves associated with the first order approximation and the second order wave–wave interaction.     

Freely floating-body simulation by a 2D fully nonlinear numerical wave tank

Nonlinear interactions between large waves and freely floating bodies are investigated by a 2D fully nonlinear numerical wave tank (NWT). The fully nonlinear 2D NWT is developed based on the potential theory, MEL/material-node time-marching approach, and boundary element method (BEM). A robust and stable 4th-order Runge–Kutta fully updated time-integration scheme is used with regriding (every time step) and smoothing (every five steps). A special φ_n-η type numerical beach on the free surface is developed to minimize wave reflection from end-wall and wave maker. The acceleration-potential formulation and direct mode-decomposition method are used for calculating the time derivative of velocity potential. The indirect mode-decomposition method is also independently developed for cross-checking. The present fully nonlinear simulations for a 2D freely floating barge are compared with the corresponding linear results, Nojiri and Murayama’s (Trans. West-Jpn. Soc. Nav. Archit. 51 (1975)) experimental results, and Tanizawa and Minami’s (Abstract for the 6th Symposium on Nonlinear and Free-surface Flow, 1998) fully nonlinear simulation results. It is shown that the fully nonlinear results converge to the corresponding linear results as incident wave heights decrease. A noticeable discrepancy between linear and fully nonlinear simulations is observed near the resonance area, where the second and third harmonic sway forces are even bigger than the first harmonic component causing highly nonlinear features in sway time series. The surprisingly large second harmonic heave forces in short waves are also successfully reproduced. The fully updated time-marching scheme is found to be much more robust than the frozen-coefficient method in fully nonlinear simulations with floating bodies. To compare the role of free-surface and body-surface nonlinearities, the body-nonlinear-only case with linearized free-surface condition was separately developed and simulated.     

Linear analysis of air cushion vehicle seakeeping response

A linearized analysis of the response of an air cushion vehicle running in waves is described. The analysis uses the linear systems approach where the vehicle is considered to be a “black box”, i.e. the response characteristics are determined experimentally from input-output relationships. The wave forces and moments are expressed in a form that produces the proper limiting behavior for infinite wavelength. Predicted motion response is shown to compare well with experimental data.