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

A numerical model is developed to simulate fully nonlinear extreme waves in finite and infinite water-depth wave tanks. A semi-mixed Eulerian-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 boundary 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.     

Second-order theory for coupling 2D numerical and physical wave tanks: Derivation,evaluation and experimental validation

A full second-order theory for coupling numerical and physical wave tanks is presented. The ad hoc unified wave generation approach developed by Zhang et al. [Zhang, H., Schäffer, H.A., Jakobsen, K.P., 2007. Deterministic combination of numerical and physical coastal wave models. Coast. Eng. 54, 171–186] is extended to include the second-order dispersive correction. The new formulation is presented in a unified form that includes both progressive and evanescent modes and covers wavemaker configurations of the piston- and flap-type. The second order paddle stroke correction allows for improved nonlinear wave generation in the physical wave tank based on target numerical solutions. The performance and efficiency of the new model is first evaluated theoretically based on second order Stokes waves. Due to the complexity of the problem, the proposed method has been truncated at 2D and the treatment of regular waves, and the re-reflection control on the wave paddle is also not included. In order to validate the solution methodology further, a series of nonlinear, periodic waves based on stream function theory are generated in a physical wave tank using a piston-type wavemaker. These experiments show that the new second-order coupling theory provides an improvement in the quality of nonlinear wave generation when compared to existing techniques.     

Second-order coupling of numerical and physical wave tanks for 2D irregular waves. Part II: Experimental validation in two-dimensions

This paper provides an experimental validation of the second-order coupling theory outlined by Yang et al. (Z. Yang, S. Liu, H.B. Bingham and J. Li., 2013. Second-order coupling of numerical and physical wave tanks for 2D irregular waves. Part I: Formulation, implementation and numerical properties, submitted for publication) using 2D irregular waves. This work provides a second-order dispersive correction for the physical wavemaker signal which improves the nonlinear transfer of information between the numerical and physical models compared to the first-order method of Zhang et al. (2007). The important nonlinear parameters and numerical performance were theoretically investigated in Part I. In the present Part II, careful experimental validation is carried out using a sequence of progressively more complex analytical and numerical target waves. The results demonstrate clearly that improved performance is achieved by using the second-order correction. When controlling with a second-order coupling signal, two key points are notable: (i) The higher harmonics underlying the numerical waves are accurately captured and transferred into the physical model. (ii) The second-order behavior leads to an unwanted spurious freely propagating second harmonic that is substantially reduced when compared to an identical wave paddle operating with a first-order coupling signal. Using nonlinear regular (monochromatic), bi-chromatic and irregular wave cases as well as varying coupled wave tank bathymetries, both these aspects are verified over a broad range of wave frequencies and shown to be extensively applicable to physical wave tanks.     

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 Eulerian-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 boundary 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 appropr...     

潜堤后高阶自由谐波的研究

基于高阶边界元方法的完全非线性数值水槽模型模拟潜堤地形上波浪的传播变形,通过与实验值进行比较,考察数学模型的正确性.采用两点法分离得到堤后高倍频自由波来研究入射波参数、水深对堤后高倍频自由波的影响.研究发现:基频波、二阶和三阶自由波幅值分别与入射波波幅成线性、二次和三次函数关系,基频波幅值基本不随波浪周期变化,而二阶和...     

Simulation of directional waves in a numerical basin by a desingularized integral equation approach

A numerical solution is developed to investigate the generation and propagation of small-amplitude water waves in a semi-infinite rectangular wave basin. The three-dimensional wave field is produced by the prescribed “snake-like” motion of an array of segmented wave generators located along the wall at one end of the tank. The solution technique is based on the boundary element approach and uses an appropriate three-dimensional Green function which explicitly satisfies the tank-wall boundary conditions. The Green function and its derivatives which appear in the integral equation formulation can be shown to be slowly convergent when the source and field points are in close proximity. Therefore, when computing the velocity potentials on the wave generators, the source points are chosen outside the fluid domain, thereby ensuring the rapid convergence of these functions and rendering the integral equations non-singular. Numerical results are shown which illustrate the influence of the various wavemaker and basin parameters on the generated wave field. Finally, the complete wave field produced by the diffraction of oblique waves by a vertical circular cylinder in a basin is presented.     

Nonlinear numerical simulation on extreme-wave kinematics

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.     

Three-dimensional liquid sloshing in a tank with baffles

A numerical model has been developed to study three-dimensional (3D) liquid sloshing in a tank with baffles. The numerical model solves the spatially averaged Navier-Stokes equations, which are constructed on a non-inertial reference frame having six degree-of-freedom (DOF) of motions. The large-eddy-simulation (LES) approach is employed to model turbulence by using the Smagorinsky sub-grid scale (SGS) closure model. The two-step projection method is employed in the numerical solutions, aided by the Bi-CGSTAB technique to solve the pressure Poisson equation for the filtered pressure field. The second-order accurate volume-of-fluid (VOF) method is used to track the distorted and broken free surface. The baffles in the tank are modeled by the concept of virtual boundary force (VBF) method. The numerical model is first validated against the available analytical solution and experimental data for two-dimensional (2D) liquid sloshing in a tank without baffles. The 2D liquid sloshing in tanks with baffles is then investigated. The numerical results are compared with other results from available literatures. Good agreement is obtained. Finally, the model is used to study 3D liquid sloshing in a tank with vertical baffles. The effect of the baffle is investigated and discussed.     

Numerical simulation of non-linear regular and focused waves in an infinite water-depth

Inviscid three-dimensional free surface wave motions are simulated using a novel quadratic higher order boundary element model (HOBEM) based on potential theory for irrotational, incompressible fluid flow in an infinite water-depth. The free surface boundary conditions are fully non-linear. Based on the use of images, a channel Green function is developed and applied to the present model so that two lateral surfaces of an infinite-depth wave tank can be excluded from the calculation domain. In order to generate incident waves and dissipate outgoing waves, a non-reflective wave generator, composed of a series of vertically aligned point sources in the computational domain, is used in conjunction with upstream and downstream damping layers. Numerical experiments are carried out, with linear and fully non-linear, regular and focused waves. It can be seen from the results that the present approach is effective in generating a specified wave profile in an infinite water-depth without reflection at the open boundaries, and fully non-linear numerical simulations compare well with theoretical solutions. The present numerical technique is aimed at efficient modelling of the non-linear wave interactions with ocean structures in deep water.     

A 3D fully coupled analysis of nonlinear sloshing and ship motion

This study investigates the coupling effects of six degrees of freedom in ship motion with fluid oscillation inside a three-dimensional rectangular container using a novel time domain simulation scheme. During the time marching, the tank-sloshing algorithm is coupled with the vessel-motion algorithm so that the influence of tank sloshing on vessel motions and vice versa can be assessed. Several factors influencing the dynamic behavior of tank–liquid system due to moving ship are also investigated. These factors include container parameters, environmental settings such as the significant wave height, current velocity as well as the direction of wind, wave and flow current acting on the ship. The nonlinear sloshing is studied using a finite element model whereas nonlinear ship motion is simulated using a hybrid marine control system. Computed roll response is compared with the existing results, showing fair agreement. Although the two hull forms and the sea states are not identical, the numerical result shows the same trend of the roll motion when the anti-rolling tanks are considered. Thus, the numerical approach presented in this paper is expected to be very useful and realistic in evaluating the coupling effects of nonlinear sloshing and 6-DOF ship motion.     

无限水深聚焦波完全非线性数值模拟

基于势流理论提出一种新的高阶边界元方法对无限水深的聚焦波浪进行完全非线性数值模拟.自由水面满足完全非线性边界条件,模拟波浪的非线性效果可以达到更高阶.利用镜像原理,建立一种全新的格林函数应用到无限水深的数值波浪水槽中,以致于两无限深水槽侧壁的积分可以被排除.为了产生相应的入射波和吸收出流波浪,一个由点源组成的造波装置被布置于计算域内,同时人工阻尼层被用来吸引出流波浪,由波浪聚焦的方法得到极限波浪.通过开展线性和完全非线性聚焦波浪的数值实验及与理论解对比,验证本数值模型可以用来模拟无限深水域的极限波浪,且在出流边界没有反射.     

A modal approach to second-order analysis of sloshing using boundary element method

This paper aims at developing a modal approach for the non-linear analysis of sloshing in an arbitrary-shape tank under both horizontal and vertical excitations. For this purpose, the perturbation technique is employed and the potential flow is adopted as the liquid sloshing model. The first- and second-order kinematic and dynamic boundary conditions of the liquid-free surface are used along with a boundary element model which is formulated in terms of the velocity potential of the liquid-free surface. The boundary element model is used to determine the natural mode shapes of sloshing and their corresponding frequencies. Using the modal analysis technique, a non-linear model is presented for the calculation of the first- and second-order potential which can be used to obtain a reduced-order model for the sloshing dynamics. The results of the presented model are verified with the analytical solution for the second-order analysis of sloshing in a rectangular tank and very good results were obtained. Also, the second-order sloshing in some other example tanks with complex bed shapes is studied. The second-order resonance conditions of liquid sloshing in the example tanks are investigated and some conclusions are drawn.     

Extreme wave run-up and pressure on a vertical seawall

The performance of coastal vertical seawalls in extreme weather events is studied numerically, aiming to provide guidance in designing and reassessing coastal structures with vertical wall. The extreme wave run-up and the pressure on the vertical seawall are investigated extensively. A time-domain higher-order boundary element method (HOBEM) is coupled with a mixed Eulerian-Lagrangian technique as a time marching technique. Focused wave groups are generated by a piston wave-maker in the numerical wave tank using a wave focusing technique for accurately reproducing extreme sea states. An acceleration-potential scheme is used to calculate the transient wave loads. Comparisons with experimental data show that the extended numerical model is able to accurately predict extreme wave run-ups and pressures on a vertical seawall. The effects of the wave spectrum bandwidth, the wall position and the wave nonlinearity on the wave run-up and the maximum wave load on the vertical seawall are investigated by doing parametric studies.     

基于高阶边界元的三维数值波浪港池

初步建立了一个基于高阶边界元的三维数值波浪港池,港池具有造波和消波功能。采用高阶边界元16节点四边形单元和基于二阶显式泰勒展开的混合欧拉-拉格朗日时间步进求解带自由表面的完全非线性势流方程。模型中对于影响数值精度的问题作了细致的处理。数值计算结果表明本港池可以用来模拟非线性波浪的传播,具有很高的数值精度和稳定性。     

三维完全非线性波浪水槽的数值模拟

用有限元求解拉普拉斯方程,建立了三维完全非线性数值波浪水槽.跟踪流体自由表面的方法为满足完全非线性自由表面条件的半拉格朗日法,对离散单元采用20节点的六面体二次等参数单元.并把数值计算结果与水面初始升高产生箱体内流体运动解析解和二阶斯托克斯波理论解进行了对比,结果表明该模型是稳定的、守恒的,能精确模拟非线性波浪的产生和传播.     

Consistent expressions for the free-surface Green function in finite water depth

Fast and accurate computation of the free-surface Green function is of key importance for the numerical solution of linear and second-order wave-structure interaction problems in three dimensions. Integral and series expressions for the Green function are derived for which the limiting values for zero and infinite frequency are consistent with the zero and infinite frequency Green function defined in terms of infinite series of Rankine image sources. The integral expressions presented here have the advantage that they are slowly varying with the non-dimensional wave frequency, making them more efficient to approximate compared with previous expressions.     

Fully nonlinear analysis of near-trapping phenomenon around an array of cylinders

The wave diffraction around an array of fixed vertical circular cylinders is simulated in a numerical wave tank by using a fully nonlinear model in the time domain. The emphasis of the paper lies in the insightful investigation of the nonlinear properties of the near-trapping phenomenon associated with the multiple cylinders. The numerical model is validated by analytical solutions as well as experimental data for waves propagating past two and four vertical cylinders in certain arrangements. An array of four identical circular cylinders at the corners of a square with an incident wave along the diagonal of the square is the main focus here for investigating the near-trapping phenomenon. When near-trapping occurs, the present study shows that an extremely high wave elevation near the cylinders can be observed. At the same time, the hydrodynamic forces on different cylinders are found to be either in phase or out of phase, leading to some characteristic force patterns acting on the whole structure. Due to the nature of the numerical model adopted, nonlinearity at different orders can be captured using a harmonic analysis. In addition to first- and second-order near-trapping, the third-order (triple-frequency) nonlinear component is presented for the first time. For the configuration selected, it is found that at one specific incident wave frequency and direction one trapped mode is excited by second-order effects, while a different trapped mode (having similar symmetries) is excited by the third harmonic of the incident wave frequency.     

三维弹性侧壁液舱内液体晃动波面的实验研究

通过物理模型实验,对弹性侧壁液舱和刚性液舱内液体晃荡问题进行了研究。由于流固耦合的影响,弹性侧壁液舱内液体晃荡的最低阶固有频率稍小于同尺寸的刚性液舱内液体晃荡的最低阶固有频率。液舱模型处于纵向简谐激励作用下,其中激励频率在最低阶固有频率附近。实验分析两种相对液深比h/L=0.167和h/L=0.333,在二阶模态的次共振和一阶模态的共振状况下,对弹性侧壁液舱与刚性液舱内不同测点的波面、振幅谱和晃动波高进行对比分析。结果表明:在浅液深(h/L=0.167)一阶共振下,流固耦合对波面形态的影响比较明显,弹性侧壁液舱内测点晃动波高明显大于刚性液舱内对应测点波高;而在一般液深(h/L=0.333)一阶共振下,水弹性效应减弱,弹性侧壁液舱与刚性液舱内对应测点处波高差异较小。     

Numerical study of nonlinear freak wave impact underneath a fixed horizontal deck in 2-D space

Freak waves are extreme and unexpected surface waves with huge wave heights that may lead to severe damage to ships and offshore structures. However, few researches have been conducted to investigate the impact underneath fixed horizontal decks caused by freak waves. To study these phenomena, a 2-D numerical wave tank is built in which nonlinear freak waves based on the Peregrine breather solution are generated. As a validation, a regular-wave-induced underneath impact is simulated and compared to the existing experimental measurements. Then the nonlinear freak-wave-induced impact is investigate with different values of deck clearance above the mean free surface. In addition, a comparative simulation of a “large” regular wave based on the 2nd-order Stokes wave theory with the same crest height and wave length of the nonlinear freak wave is carried out to reveal the unique features of the nonlinear freak-wave-induced impact. By applying a fluid–structure interaction (FSI) algorithm in which the bottom deck and front side wall are simplified as Euler beams in 2-D and discretized by the finite element method (FEM), the hydroelastic effects are considered during the impact event. The vertical force acting underneath the bottom deck, the transversal force acting on the front side wall, the structural displacements of the elastic deck and wall are analyzed and discussed respectively, from which meaningful conclusions are drawn.