Time-Domain Higher-Order Boundary Element Method for Simulating High Forward-Speed Ship Motions in Waves

The hydrodynamic performance of a high forward-speed ship in obliquely propagating waves is numerically examined to assess both free motions and wave field in comparison with a low forward-speed ship.This numerical model is based on the time-domain potential flow theory and higher-order boundary element method,where an analytical expression is completely expanded to determine the base-unsteady coupling flow imposed on the moving condition of the ship.The ship in the numerical model may possess different advancing speeds,i.e.stationary,low speed,and high speed.The role of the water depth,wave height,wave period,and incident wave angle is analyzed by means of the accurate numerical model.It is found that the resonant motions of the high forward-speed ship are triggered by comparison with the stationary one.More specifically,a higher forward speed generates a V-shaped wave region with a larger elevation,which induces stronger resonant motions corresponding to larger wave periods.The shoaling effect is adverse to the motion of the low-speed ship,but is beneficial to the resonant motion of the high-speed ship.When waves obliquely propagate toward the ship,the V-shaped wave region would be broken due to the coupling effect between roll and pitch motions.It is also demonstrated that the maximum heave motion occurs in beam seas for stationary cases but occurs in head waves for high speeds.However,the variation of the pitch motion with period is hardly affected by wave incident angles.

Using a Time-domain Higher-order Boundary Element Method to Simulate Wave and Current Diffraction from a 3-D Body

To study wave-current actions on 3-D bodies a time-domain numerical model was established using a higher-order boundary element method(HOBEM).By assuming small flow velocities,the velocity potential could be expressed for linear and higher order components by perturbation expansion.A 4th-order Runge-Kutta method was applied for time marching.An artificial damping layer was adopted at the outer zone of the free surface mesh to dissipate scattering waves.Validation of the numerical method was carried out on run-up,wave exciting forces,and mean drift forces for wave-currents acting on a bottom-mounted vertical cylinder.The results were in close agreement with the results of a frequency-domain method and a published time-domain method.The model was then applied to compute wave-current forces and run-up on a Seastar mini tension-leg platform.

Error Analysis of A New Higher Order Boundary Element Method for A Uniform Flow Passing Cylinders

A higher order boundary element method(HOBEM)is presented for inviscid flow passing cylinders in bounded or unbounded domain.The traditional boundary integral equation is established with respect to the velocity potential and its normal derivative.In present work,a new integral equation is derived for the tangential velocity.The boundary is discretized into higher order elements to ensure the continuity of slope at the element nodes.The velocity potential is also expanded with higher order shape functions,in which the unknown coefficients involve the tangential velocity.The expansion then ensures the continuities of the velocity and the slope of the boundary at element nodes.Through extensive comparison of the results for the analytical solution of cylinders,it is shown that the present HOBEM is much more accurate than the conventional BEM.

A Method for Calculation of Low-Frequency Slow Drift Motions Based on NURBS for Floating Bodies

Through a higher-order boundary element method based on NURBS （Non-uniform Rational B-splines）, the calculation of second-order low-frequency forces and slow drift motions is conducted for floating bodies. In the floating body＇s inner domain, an auxiliary equation is obtained by applying a Green function which satisfies the solid surface condition. Then, the auxiliary equation and the velocity potential equation are combined in the fluid domain to remove the solid angle coefficient and the singularity of the double layer potentials in the integral equation. Thus, a new velocity potential integral equation is obtained. The new equation is extended to the inner domain to reheve the irregular frequency effects; on the basis of the order analysis, the comparison is made about the contribution of all integral terms with the result in the second-order tow-frequency problem; the higher-order boundary element method based on NURBS is apphed to calculate the geometric position and velocity potentials; the slow drift motions are calculated by the spectrum analysis method. Removing the solid angle coefficient can apply NURBS technology to the hydrodynamic calculation of floating bodies with complex surfaces, and the extended boundary integral method can reduce the irregular frequency effects. Order analysis shows that free surface integral can be neglected, and the numerical results can also prove the correctness of order analysis. The results of second-order low-frequency forces and slow drift motions and the comparison with the results from references show that the application of the NURBS technology to the second-order low-frequency problem is of high efficiency and credible results.

An iterative Rankine boundary element method for wave diffraction of a ship with forward speed

A 3-D time-domain seakeeping analysis tool has been newly developed by using a higher-order boundary element method with the Rankine source as the kernel function. An iterative time-marching scheme for updating both kinematic and dynamic free-surface boundary conditions is adopted for achieving numerical accuracy and stability. A rectangular computational domain moving with the mean speed of ship is introduced. A damping beach at the outer portion of the truncated free surface is installed for satisfying the radiation condition. After numerical convergence checked, the diffraction unsteady problem of a Wigley hull traveling with a constant forward speed in waves is studied. Extensive results including wave exciting forces, wave patterns and pressure distributions on the hull are presented to validate the efficiency and accuracy of the proposed 3-D time-domain iterative Rankine BEM approach. Computed results are compared to be in good agreement with the corresponding experimental data and other published numerical solutions.

Bankine源高阶边界元法求解势流问题

本文以简单Ｇｒｅｅｎ函数１／ｒ为基本解，采用高阶边界元求解势流问题。利用８结点等参元来逼近流域边界的几何形状和未知量，利用变形的边界积分方程和坐标变换技术有效地处理了在场点的立体角计算和数值积分中所谓的“奇异性”问题，得到足够精度的结果。对零航速的半球振荡问题的数值解表明，１／ｒ源高阶边界元法能给出满意的结果，且消除了“不规则频率”现象。

FPSO水动力特性的完全非线性数值模拟

[目的]为提高海洋结构物的安全性能,针对波浪与结构物相互作用的问题开展完全非线性数值模拟研究。[方法]基于三维完全非线性时域势流理论及高阶边界元法(HOBEM),建立波浪与结构物相互作用的开敞水域模型。采用速度势分离技术将整个问题分解为入射部分和散射部分,入射势由理论解给定。采用混合欧拉-拉格朗日(MEL)方法追踪瞬时自由水面的流体质点,并采用四阶龙格-库塔法对瞬时自由水面进行更新。引进虚拟函数计算波浪载荷,而非直接求解速度势的时间导数。在自由水面的外侧设置人工阻尼层,防止波浪从远场边界反射。自由水面网格仅在初始时刻生成一次,并采用弹簧近似法在不改变网格节点顺序的情况下对瞬时水面进行网格重构,以避免数值不稳定。[结果]在验证所提出数值模型有效性和精确性的基础上,针对某浮式生产储卸油轮(FPSO)模型的水动力特性进行数值模拟,发现考虑非线性影响时FPSO的运动响应在共振区段明显增大,证明了传统线性方法的预报结果趋于危险。[结论]研究成果既可为海洋浮式结构物的设计提供更可靠的预报工具,也可为其实际应用提供理论依据。

Numerical investigation of solitary wave action on tworectangular boxes with a narrow gap

Based on the time-domain higher-order boundary element method(HOBEM), a two-dimensional numerical wave flume is developed to investigate solitary wave interaction with two rectangular boxes with a narrow gap.In the numerical model, the fully nonlinear boundary conditions are satisfied on the free surface, the mixed Eulerian-Lagrangian method is adopted to track the transient water surface and the fourth-order Runga-Kutta method is used to predict the velocity potential and wave elevation on the free surface. The acceleration potential technique is used to compute the transient wave forces along the wetted object surface. A piston-type wavemaker is used to generate solitary waves. The proposed model is validated by comparing the simulated wave run-up and the wave loads with the published experimental and numerical results of the reflection of a solitary wave from a vertical wall. Then, numerical experiments are performed to study the effects of the narrow gap and the size of each box on the wave run-ups at the two sides of the two-box system and in the narrow gap between two boxes, and the wave loads on the two boxes. The interaction between double solitary waves with a time interval between them with a two-box system is also investigated.

Wave Slamming on An OWSC Wave Energy Converter in Coupled Wave-Current Conditions with Variable-Depth Seabed

Coastal wave energy resources have enormous exploitation potential due to shorter weather window,closer installation distance and lower maintenance cost.However,impact loads generated by depth variation from offshore to nearshore and wave-current interaction,may lead to a catastrophic damage or complete destruction to wave energy converters(WECs).This objective of this paper is to investigate slamming response of a coastal oscillating wave surge converter(OWSC)entering or leaving water freely.Based on fully nonlinear potential flow theory,a time-domain wave-current-structure interaction model combined with higher-order boundary element method(HOBEM),is developed to analyze the coupled hydrodynamic problem.The variable-depth seabed is considered in the model to illustrate the shallow water effect on impact loads and free surface profiles in coastal zone.A domain decomposition approach is utilized to simulate the overlapping phenomenon generated by a jet falling into water under gravity effect.Through a series of Lagrangian interpolation methods,the meshes on boundaries are rearranged to avoid the mismatch between element size on free surface and body surface.The present model is validated against the existing experimental and numerical results.Simulations are also provided for the effects of wave-current interaction and uneven local seabed on the slamming responses.It is found that the length of the splash jet increases for a following current and decreases for an opposing current,and that the slamming response of the OWSC device is sensitive to the geometric features of the uneven seabed.

Numerical analysis of added mass and damping of floating production,storage and offloading system

An integral equation approach is utilized to in- vestigate the added mass and damping of floating produc- tion, storage and offloading system （FPSO system）. Finite water depth Green function and higher-order boundary ele- ment method are used to solve integral equation. Numeri- cal results about added mass and damping are presented for odd and even mode motions of FPSO. The results show ro- bust convergence in high frequency range and can be used in wave load analysis for FPSO designing and operation.

布拉格共振条件下的高阶谐波特性研究

为了考虑波浪在周期性地形上传播时诱发的布拉格共振现象和高阶谐波之间的相互影响,本文基于时域高阶边界元方法,建立二维完全非线性数值波浪水槽,数值研究布拉格共振条件下周期性地形诱发的高阶谐波特性。研究发现:随着潜体高度增加,二阶自由波透射系数呈现先增大后减小的趋势,而其反射系数则持续增大。同时,随着潜体长度增加,二阶自由波透反射系数均呈现增大的趋势,并最终各自稳定于一定值附近振荡。在布拉格共振条件下,除了普遍认为的自由波,二阶锁定波也将发生共振反射现象;此外,在布拉格共振条件下,周期性地形迎浪侧的二阶谐波波幅会出现“拍”和“寄生拍”的组合现象,使得迎浪侧波浪场更为复杂,并增加极端大波出现的概率。

Nonlinear numerical simulation on extreme-wave kine-matics

A fully nonlinear numerical model based on a time-domain higher-order boundary element method （HOBEM） is founded to simulate the kinematics of extreme waves. In the model, the fully nonlinear free surface boundary conditions are satisfied and a semi-mixed Euler-Lagrange method is used to track free surface; a fourth-order Runga-Kutta technique is adopted to refresh the wave elevation and velocity potential on the free surface at each time step; an image Green function is used in the numerical wave tank so that the integrations on the lateral surfaces and bottom are excluded. The extreme waves are generated by the method of wave focusing. The physical experiments are carried out in a wave flume. On the horizontal velocity of the measured point, numerical solutions agree well with experimental results. The characteristics of the nonlinear extreme-wave kinematics and the velocity distribution are studied here.

An Efficient Model for Transient Surface Waves in Both Finite and Infinite Water Depths

A numerical model is developed to simulate fully nonlinear extreme waves in finite and infinite water-depth wave tanks. A semi-mixed Enlerian-Lagrangian formulation is adopted and a higher-order boundary element method in conjunction with an image Green function is used for the fluid domain. The botmdary values on the free surface are updated at each time step by a fourth-order Runga-Kutta time-marching scheme at each time step. Input wave characteristics are specified at the upstream boundary by an appropriate wave theory. At the downstream boundary, an artificial damping zone is used to prevent wave reflection back into the computational domain. Using the image Green function in the whole fluid domain, the integrations on the two lateral walls and bottom are excluded. The simulation results on extreme wave elevations in finite and infinite water-depths are compared with experimental results and second-order analytical solutions respectively. The wave kinematics is also discussed in the present study.

Iterative Rankine HOBEM analysis of hull-form effects in forward-speed diffraction problem

A time-domain numerical algorithm based on the higher-order boundary element method and the iterative time-marching scheme is proposed for seakeeping analysis. The ship waves generated by a hull advancing at a constant forward speed in incident waves and the resultant diffraction forces acting on the hull are computed to investigate the hull-form effects on the hydrodynamic forces. A rectangular computational domain travelling at ship＇s speed is considered. An artificial damping beach for satisfying the radiation condition is installed at the outer portion of the free surface except the downstream side. An iterative time-marching scheme is employed for updating both kinematic and dynamic free-surface boundary conditions for numerical accuracy and stability. The boundary integral equation is solved by distributing higher-order boundary elements over the wetted body surface and the free surface. The hull-form effects on the naval hydrodynamics are investigated by comparing three different Wigley models. Finally, the corresponding unsteady wave patterns and the wave profiles around the hulls are illustrated and discussed.

WAVE-CURRENT INTERACTIONS WITH THREE-DIMENSIONAL FLOATING BODIES

A Time-domain Higher-Order Boundary Element Method（THOBEM） is developed for simulating wave-current interactions with 3-D floating bodies.Through a Taylor series expansion and a perturbation procedure,the model is formulated to the first-order in the wave steepness and in the current velocity,respectively.The boundary value problem is decomposed into a steady double-body flow problem and an unsteady wave problem.Higher-order boundary integral equation methods are then used to solve the proposed problems with a fourth-order Runge-Kutta method for the time marching.An artificial damping layer is adopted to dissipate the scattering waves.Different from the other time-domain numerical models,which are often focused on the wave-current interaction with restrained bodies,the present model deals with a floating hemisphere.The numerical results of wave forces,wave run-up and body response are all in a close agreement with those obtained by frequency-domain methods.The proposed numerical model is further applied to investigate wave-current interactions with a floating body of complicated geometry.In this work,the regular and focused wave combined with current interacting with a truss-spar platform is investigated.

海上风机基础高阶水动力荷载数值模拟研究

在实际海洋环境下,海上风机结构常常承受非线性波浪和系泊系统的联合作用,探究海上浮式风机基础高阶水动力荷载及其运动响应非常重要。该文基于非线性势流理论和摄动展开技术,建立了海上浮式风机基础与波浪和系泊系统耦合作用的二阶时域高阶边界元数值分析模型,研究了OC4-DeepCwind半潜浮式风机基础在波浪作用下的动力响应特性。并通过与已发表数据进行对比,验证了模型准确性。分析计算得到的波浪力和平台运动响应随时间变化的演变特性及其与线性结果的区别,发现某些波频作用下非线性效应影响显著。该研究成果可为海上风电系统及海洋结构物设计提供一定参考。

风对二维极值波作用的数值研究

实际海域中风浪往往共存,当极值波浪出现时,风会对其特性产生很大影响。为了研究风对极值波浪特性的影响规律,该文基于势流理论建立自由水面满足完全非线性边界条件的风浪混合作用数值水槽模型,通过实时模拟造波板运动产生入射波,并基于波群聚焦原理定点产生极值波浪,利用改进的杰弗里斯遮蔽原理(Jeffreys'sheltering mechanism)将风压引入到自由水面动力边界条件中。在时域模拟中,采用混合欧拉-拉格朗日方法追踪瞬时水面,利用格林第二定理建立了边界积分方程,采用高阶边界元方法进行离散求解。通过与已发表实验结果对比验证模型的准确性,进而开展数值实验研究风对极值波浪水动力特性的影响,包括聚焦幅值、聚焦位置和极值波高随风速的变化以及波群聚焦和解焦过程波浪演变特性等。

Application of the generalized body-fixed coordinate system for the wave-body interaction problem of a small-depth elastic structure in head seas

The body-fixed coordinate system is applied to the wave-body interaction problem of a small-depth elastic structure which has both rigid and elastic body motions in head waves.In the weakly non-linear assumption,the perturbation scheme is used and the expansion is conducted up to second-order to consider several non-linear quantities.To solve the boundary value problem,linearization is carried out based not on inertial coordinate but on body-fixed coordinate which could be accelerated by a motion of a body.At first,the main feature of the application of body-fixed coordinate system for a seakeeping problem is briefly described.After that the transformation of a coordinate system is extended to consider an elastic body motion and several physical variables are re-described in the generalized mode.It has been found that the deformation gradient could be used for the transformation of a coordinate system if several conditions are satisfied.Provided there are only vertical bending in elastic modes and the structure has relatively small depth,these conditions are generally satisfied.To calculate an elastic motion of a body,the generalized mode method is adopted and the mode shape is obtained by solving eigen-value problem of dynamic beam equation.In the boundary condition of the body-fixed coordinate system,the motion effect reflected to free-surface boundary is considered by extrapolating each mode shape to the horizontal direction from a body.At last,simple numerical tests are implemented as a validation process.The second-order hydrodynamic force of a freely floating hemisphere is first calculated in zero forward speed condition.Next,motion and added resistance of a ship with forward speed are considered at different flexibility to confirm the effect of an elastic body motion in body-fixed coordinate system.