Second-Order Potentials and Forces for Two-Dimensional Diffraction Problem of Finite Water Depth

-In this paper, an analytical solution in the outer region of finite water depth is derived for the second-order diffraction potential, which gives a clear physical meaning of the wave transmission and reflection characteristics in the far field. A numerical method-simple Green's function technique-for calculating the second-order diffraction potential in the inner region is also described. Numerical results are provided for the second-order wave forces on a semi-submerged cylinder. It is found that the contribution of second-order diffraction potential to second-order wave forces is important. The effect of water depth and submerged depth on the wave force is also discussed.

Effects of Second-Order Sum-and Difference-Frequency Wave Forces on the Motion Response of a Tension-Leg Platform Considering the Set-down Motion

This paper presents a study on the motion response of a tension-leg platform(TLP) under first-and second-order wave forces, including the mean-drift force, difference and sum-frequency forces. The second-order wave force is calculated using the full-field quadratic transfer function(QTF). The coupled effect of the horizontal motions, such as surge, sway and yaw motions, and the set-down motion are taken into consideration by the nonlinear restoring matrix. The time-domain analysis with 50-yr random sea state is performed. A comparison of the results of different case studies is made to assess the influence of second-order wave force on the motions of the platform. The analysis shows that the second-order wave force has a major impact on motions of the TLP. The second-order difference-frequency wave force has an obvious influence on the low-frequency motions of surge and sway, and also will induce a large set-down motion which is an important part of heave motion. Besides, the second-order sum-frequency force will induce a set of high-frequency motions of roll and pitch. However, little influence of second-order wave force is found on the yaw motion.

SECOND-ORDER INTERACTION OF IRREGULAR WAVES WITH A TRUNCATED COLUMN

A complete semi-analytical solution is obtained for second-order diffraction of plane bichromatic waves by a fixed truncated circular column.The fluid domain is divided into interior and exterior regions.In the exterior region,the second-order velocity potential is expressed in terms of‘locked-wave’and‘free-wave’ components,both are solved using Fourier and eigenfunction expansions.The re- sulting‘locked wave’potential is expressed by one-dimensional Green's integrals with oscillating integrands.In order to increase computational efficiency,the far-field part of the integrals are carried out analytically.Solutions in both regions are matched on the interface by the potential and its normal derivative continuity conditions.Based on the present approach,the sum-and difference-frequency potentials are efficiently evaluated and are used to generate the quadratic transfer functions which correlates the incident wave spectrum with second-order forcing spectrum on the column.The sum-frequency QTFs for a TLP column are present,which are compared for some frequency pairs with those from a fully numerical procedure.Satisfactory agreement has been obtained.QTF spectra for a case study TLP column,generated using the semi-analytical solution are presented.Also given are the results for nonlinear wave field around the column.

Comparisons of Wave Force Model Effects on the Structural Responses and Fatigue Loads of a Semi-Submersible Floating Wind Turbine

The selection of wave force models will significantly impact the structural responses of floating wind turbines.In this study,comparisons of wave force model effects on the structural responses and fatigue loads of a semi-submersible floating wind turbine(SFWT)were conducted.Simulations were performed by employing the Morison equation(ME)with linear or second-order wave kinematics and potential flow theory(PFT)with first-or second-order wave forces.A comparison of regular waves,irregular waves,and coupled wind/waves analyses with the experimental data showed that many of the simulation results and experimental data are relatively consistent.However,notable discrepancies are found in the response amplitude operators for platform heave,tower base bending moment,and tension in mooring lines.PFT models give more satisfactory results of heave but more significant discrepan-cies in tower base bending moment than the ME models.In irregular wave analyses,low-frequency resonances were captured by PFT models with second-order difference-frequency terms,and high-frequency resonances were captured by the ME models or PFT models with second-order sum-frequency terms.These force models capture the response frequencies but do not reasonably predict the response amplitudes.The coupled wind/waves analyses showed more satisfactory results than the wave-only analyses.However,an important detail to note is that this satisfactory result is based on the overprediction of wind-induced responses.

ON VELOCITY POTENTIALS DUE TO PULSATING PRESSURE DISTRIBUTIONS ON THE FREE SURFACE OF INFINITE-DEPTH WATERS AND THE RADIATION CONDITIONS FOR SECOND-ORDER DIFFRACTION PROBLEMS

In the present paper,two-and three-dimensional velocity potentials generated by pulsating pressure distributions of infinite extent on the free surface of infinite-depth waters are strictly derived based on special cases of concentrated pulsating pressure.The far-field asymptotic behaviour of the potentials and the radiation conditions to be satisfied by them are discussed. It is proved in a general sense that the potentials should be composed of a forced wave component,a free wave component and a local disturbance component.The radiation condition of the forced wave component should correspond to the far-field asymptotic behaviour of the pressure distribution,Hence,the formulation of radiation conditions for the second-order diffraction potentials has theoretically become clear,The radiation conditions for two-and three-dimensional problems are explicitly given in the paper.

AN ANALYTICAL SOLUTION OF SECOND－ORDER WAVE FORCE ON A VERTICAL CIRCULAR CYLINDER

So far many investigations have been made on nonlinear wave diffraction problemfor a large-diameter vertical circular cylinder. However,there are still some problemsworthy to be further discussed. It includes that the second order radiation condition isnot very clear and the inhomogeneous term of second order free surface boundarycondition makes the calculation of second order wave force either not easier to beperformed accurately due to its slowly decaying with radial distance or toocomplicated for practical application. In this paper, the second order radiationcondition is posed of the circumferential Fourier components of second orderpotential, instead of the second order potential. It is found that the circumferenatialFourier cormponents of second order potential have to satisfy Sommerfeld radiationcondition. By means of the mathematical formulae derived in this paper, theinhomogeneous term of second order free surface boundary condition were simplifiedand then an exact expression of second order wave force was obtained, which issimpler in form and easier to be used in practical calculation. The calculation resultsagree well with some experimental data.

波浪在半球上绕射的解析解

基于线性势流理论,推导了无限水深中波浪在半球上绕射问题的解析解.给出了球坐标系下无限水深入射波速度势的表达式;利用多极子展开法,以包含未知系数和连带勒让德函数的无限序列的形式给出了流场中绕射速度势的表达式;通过物面边界条件建立了方程组,进而求解了未知系数.计算了作用在半球上总的波浪激励力,并通过与利用Haskind关系得到的结果以及他人的数值结果进行比较,验证了文中结果的正确性.

Hydrodynamic Performance of Multiple-Row Slotted Breakwaters

This study examines the hydrodynamic performance of multiple-row vertical slotted breakwaters. We developed a mathematical model based on an eigenfunction expansion method and a least squares technique for Stokes second-order waves. The numerical results obtained for limiting cases of double-row and triple-row walls are in good agreement with results of previous studies and experimental results. Comparisons with experimental measurements of the reflection, transmission, and dissipation coefficients （CR, Cr, and CE） for double-row walls show that the proposed mathematical model adequately reproduces most of the important features. We found that for double-row walls, the CR increases with increasing wave number, kd, and with a decreasing permeable wall part, din. The Cr follows the opposite trend. The CE slowly increases with an increasing kd for lower kd values, reaches a maximum, and then decreases again. In addition, an increasing porosity of dm would significantly decrease the CR while increasing the Cr. At lower values of kd, a decreasing porosity increases the CE, but for high values of kd, a decreasing porosity reduces the Ce. The numerical results indicate that, for triple-row walls, the effect of the arrangement of the chamber widths on hydrodynamic characteristics is not significant, except when kd〈0.5 Double-row slotted breakwaters may exhibit a good wave-absorbing performance at kd〉0.5, where by the horizontal wave force may be smaller than that of a single wall. On the other hand, the difference between double-row and triple-row vertical slotted breakwaters is marginal.

有限水深二维绕射问题的二阶速度势和水动力

本文给出有限水深二维物体二阶绕射势在外域中的解析表达式，从而准确满足二阶绕射势的辐射条件。二阶绕射势在内域自由表面上的边界条件则由一阶势的数值微分求得。然后对内域用简单Ｇｒｅｅｎ函数法求得二阶绕射势。本文对二维浮体和潜体在不同水深和潜深情况下的绕射问题进行了计算，求得了二阶绕射势和二阶定常及倍频波浪力。

Coupled dynamic response analysis of multi-column floating offshore wind turbine with low center of gravity

To realize the application of the floating offshore wind turbine(FOWT)from deep to relatively shallow waters,a new concept of multi-column floating wind turbine platform with low center of gravity(CG)is designed and validated.The multi-column low CG platform is designed to support a 6MW wind turbine class and operated at a water depth of 50m in the South China Sea.The frequency domain software WADAM and time domain software NREL-FAST are used to simulate coupled dynamic responses of the floating wind turbine system with second-order wave loads considering.The dynamic behaviors of multi-column low CG FOWT system under normal operation and parked conditions are presented.The influence of second-order wave force on the motion responses of the multi-column platform,fore-aft force and moment of the tower base and mooring force are researched respectively.The results demonstrate that the coupled dynamic responses at rated operating condition and extreme condition meet the normal operating requirements and extreme survival requirements of FOWT system in the shallow water(50m)of South China Sea.In addition,it is found that,the wave frequency response gradually replaces the second-order low frequency response as the main influencing factor of the coupled dynamic response of the FOWT system with the increasing severity of the sea states.However,in general,the magnitude of second-order low frequency response increases with the increasing severity of the design load case.Thus,in the subsequent design of the shallow water FOWT system,the second-order effects should be paid enough attention.

不同二阶绕射波浪力理论公式的结果互比

该文研究了一种简单快速回转体的二阶波浪力计算方法,并给出了直立圆柱和截断圆柱的二阶波浪力的计算结果,将所得计算结果与其它3种不同的理论公式的结果进行了对比,验证了该文计算方法的正确性。因该文方法不涉及自由表面上无穷积分,表达式简单、计算速度快,所以更适合快速计算二阶速度势所引起的二阶波浪力。

消减波浪荷载的桥梁沉箱基础外形设计研究

基于波浪场中动水压强的空间分布特征,提出了消减跨海桥梁沉箱基础波浪荷载的方法:一是优化基础横截面外形,调节基础横截面外轮廓不同位置波浪压力的相位差,减小基础所受到波浪力和力矩;二是优化基础竖截面外形,减小水面附近波浪能量集中区域的截面尺寸,增大靠近海床面区域的截面尺寸,将基础下部的迎浪面设置为外伸斜面,利用斜面上波浪压力竖向分力产生的力矩抵消部分水平分力产生的力矩,从而消减桥梁基础上的总波浪力矩。对于大尺度矩形和圆端矩形截面的桥梁深水沉箱基础,基于势流绕射理论和边界积分方法进行分析,结果表明,相比于矩形截面,采用圆端矩形截面可有效减小基础的波浪力和力矩;相比于上下等截面的基础,采用下部迎浪面设置外伸斜面的基础可以大幅减小基础的波浪力矩。研究成果可为跨海桥梁深水基础设计提供参考。