Numerical simulation of nonlinear wave and force generated by a wedge-shape wave maker

Nonlinear waves and forces induced by a wedge-shape wave maker were simulated in a potential-theory-based fully nonlinear 2D Numerical Wave Tank (NWT). The NWT is developed in a time domain by using Boundary Element Method (BEM) including Mixed Eulerian–Lagrangian method (MEL) and Runge–Kutta 4th-order (RK4) integration as a time marching process. For ensuring accurate nonlinear free surface both material-node and semi-Lagrangian approach are independently developed for crosschecking. The acceleration-potential scheme is used for obtaining accurate time derivative of velocity potential. The developed NWT is utilized to calculate water particle velocity and a series of higher-harmonic force components on the wave maker. The added-mass and radiation-damping coefficients of the wave maker are also obtained from the least-square method. The simulation results are compared with the experimental and numerical results of other researchers. To compare the relative importance of free-surface and body-surface nonlinearities, a body nonlinear formulation is independently developed. Force by body nonlinear method is in good agreement with fully nonlinear result in case of low body-stroke frequency.     

Fully nonlinear wave-body interactions with surface-piercing bodies

Fully nonlinear wave-body interactions for stationary surface-piercing single and double bodies are studied by a potential-theory-based fully nonlinear 2D numerical wave tank (NWT). The NWT was developed in time domain by using boundary element method (BEM) with constant panels. MEL free surface treatment and Runge–Kutta fourth-order time integration with smoothing scheme was used for free-surface time simulation. The acceleration-potential scheme is employed to obtain accurate time derivative of velocity potential. Using the steady part of nonlinear force time histories, mean and a series of higher-harmonic force components are calculated and compared with the experimental and numerical results of other researchers. The slow-decaying second-harmonic vertical forces are investigated with particle velocities and corresponding body pressure. Typical patterns of two-body interactions, shielding effect, and the pumping/sloshing modes of water column in various gap distances are investigated. The pumping mode in low frequencies is demonstrated by the comparison of velocity magnitudes.     

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。     

Simulation of Fully Nonlinear 3-D Numerical Wave Tank

A fully nonlinear numerical wave tank (NWT) has been simulated by use of a three-dimensional higher order bouodary element method (HOBEM) in the time domain. Within the frame of potential flow and the adoption of simply Rankine source, the resulting boundary integral equation is repeatedly solved at each time step and the fully nonlinear free surface boundary conditions are integrated with time to update its position and boundary values. A smooth technique is also adopted in order to eliminate the possible saw-tooth numerical instabilities. The incident wave at the uptank is given as theoretical wave in this paper. The outgoing waves are absorbed inside a damping zone by spatially varying artificial damping on the free surface at the wave tank end. The numerical results show that the NWT developed by these approaches has a high accuracy and good numerical stability.     

Second-order Taylor expansion boundary element method for the second-order wave radiation problem

A novel Boundary Element Method (BEM) named the second-order Taylor Expansion Boundary Element Method (the 2nd order TEBEM) is developed for the solution of the second-order wave radiation velocity potential and sum-frequency wave loads for floating bodies. The radiation condition is enforced by a hybrid method of the multi-transmitting formula and damping zone. For the interior domain problem of a cube and a sphere, numerical results demonstrate that the 2nd order TEBEM can accurately solve the first and second-order gradients of velocity potential on the no-smoothed and smoothed boundary compared to the low-order BEM. The double frequency forces acting on the truncated cylinder are calculated under finite water depth. The agreement between the 2nd order TEBEM and others' numerical results is good. Moreover, all of the singular integrals in the 2nd order TEBEM can be solved analytically, so its implementation is much easier compared to the high-order BEM.     

非线性波浪波面追踪的一种新模式

基于Laplace方程的Green积分表达式和波面BemouUi方程所建立的非线性波动数学模型,是一个时域上具有初始值的边值问题,而精确地追踪自由表面的波动位置,给出波面运动瞬时的波面高度和波面势函数,是建立时域内非线性波浪数值模式的基础。本文采用0-1混合型边界元剖分计算域边界并离散Laplace方程的Green积分表达式,采用有限元剖分自由水面并推导满足自由表面非线性边界条件的波面有限元方程,联立计算域内以节点波势函数和波面位置高度的时间增量为未知量的线性方程组,通过时步内的循环迭代,给出每个时步上的波面位置和波面势函数,从而建立了一种新的非线性波浪波面追踪模式。数值造波水槽内的波浪试验表明,其数值模拟结果具有良好的计算精度。     

Nonlinear time–domain analysis of motion-restrained pneumatic floating breakwater

A pneumatic-type floating breakwater is simulated in the time–domain to evaluate wave blocking and wave energy absorption. For accurate nonlinear time–domain simulation, a fully nonlinear numerical wave tank (NWT) technique has been used. In the present study, the NWT for the pneumatic breakwater is extended to the case of restrained body motion using the mode-decomposition method in the acceleration potential field. In particular, the effect of individual body motion coupled with pneumatic damping in the chamber is investigated for the case in which the breakwater is only allowed to move one degree-of-freedom: for instance, using a heave-only allowable body. The present results are compared with various motion cases as well as a box-shaped breakwater.     

Time-domain simulation of wave–structure interaction based on multi-transmitting formula coupled with damping zone method for radiation boundary condition

Based on the Rankine source, this paper proposed a time-domain method for analyzing the three-dimensional wave–structure interaction problem in irregular wave. A stable integral form of the free-surface boundary condition (IFBC) is employed to update the velocity potential on the free surface. A multi-transmitting formula, with an artificial wave speed, is used to eliminate the wave reflection for radiation condition on the artificial boundary. An effective multi-transmitting formula, coupled with damping zone method, is further used to analyze the irregular wave diffraction at the artificial boundary. We investigate hydrodynamic forces on floating structure and compare our solution to the frequency-domain solution. It is shown that long time simulation can be done with high stability and the numerical results agree well with the solution obtained under the frequency domain. The efficiency of the proposed multi-transmitting formula and the coupled methods for radiation boundary make them promising candidates in studying the irregular water wave problem in time domain.     

An experimental study on the flow characteristics of a standing wave: application of FLDV measurements

The determination of the characteristics of a standing wave boundary layer, such as water particle velocities and shear stresses, is a vital issue in the prediction of sediment transport rate. Thus, an accurate measure of boundary layer characteristics is a key factor in the study of the wave boundary layer. In this study, a fiber-optic laser Doppler velocimeter (FLDV) is applied to directly measure the velocity profile in the standing wave boundary layer. As the experimental data presented in this paper, the antinode points of standing waves are found to move in the temporal domain. The previous second-order solution for a standing wave boundary layer is insufficient for the prediction of larger Ursell numbers.     

Fully nonlinear numerical wave tank (NWT) simulations and wave run-up prediction around 3-D structures

A finite-difference scheme and a modified marker-and-cell (MAC) algorithm have been developed to investigate the interactions of fully nonlinear waves with two- or three-dimensional structures of arbitrary shape. The Navier–Stokes (NS) and continuity equations are solved in the computational domain and the boundary values are updated at each time step by the finite-difference time-marching scheme in the framework of a rectangular coordinate system. The fully nonlinear kinematic free-surface condition is implemented by the marker-density function (MDF) technique developed for two fluid layers.To demonstrate the capability and accuracy of the present method, the numerical simulation of backstep flows with free-surface, and the numerical tests of the MDF technique with limit functions are conducted. The 3D program was then applied to nonlinear wave interactions with conical gravity platforms of circular and octagonal cross-sections. The numerical prediction of maximum wave run-up on arctic structures is compared with the prediction of the Shore Protection Manual (SPM) method and those of linear and second-order diffraction analyses based on potential theory and boundary element method (BEM). Through this comparison, the effects of non-linearity and viscosity on wave loading and run-up are discussed.     

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.     

海底管道冲刷势流模型的改进

对Li&Cheng的势流模型在数值方法、床面平衡条件及冲刷床面调整技术做出了3方面的改进。采用边界元法代替差分法求解Laplace方程,前者可以准确地拟合地形与管道边界,因而可以准确反映固壁边界对流态的影响,此外,还具有数据准备简单,降低计算维数,计算速度快等优点。Li&Cheng模型以床面水流切应力等于泥沙起动切应力,τb=τc,作为床面平衡条件,这只适用于清水冲刷。以沿程输沙相同作为平衡剖面条件,理论上,将模型推广到了动床冲刷。此外,为了提高模型的收敛性,提出了最速下降法与牛顿迭代法相结合的床面调整技术。采用实验资料对模型进行了检验,计算的冲刷深度与实验结果符合良好。     

基于高阶边界元的三维数值波浪港池--波浪破碎的模拟

在势流理论的框架内,采用高阶边界元方法和混合欧拉-拉格朗日法,实现了对三维波浪破碎过程的数值模拟.数值模型使用可调节时间步长的基于二阶显式泰勒展开的混合欧拉-拉格郎日时间步进来求解自由表面的演化过程.在所使用的边界元方法中,采用16节点三次滑移四边形单元来表示,这种单元在单元内具有高阶的精度同时在单元之间具有良好的连续性.给出了孤立波的传播和周期性非线性波浪沿缓坡传播的计算结果,表明数值模型具有良好的稳定性.     

A coupled-mode model for the refraction–diffraction of linear waves over steep three-dimensional bathymetry

A consistent coupled-mode model recently developed by Athanassoulis and Belibassakis [1], is generalized in 2+1 dimensions and applied to the diffraction of small-amplitude water waves from localized three-dimensional scatterers lying over a parallel-contour bathymetry. The wave field is decomposed into an incident field, carrying out the effects of the background bathymetry, and a diffraction field, with forcing restricted on the surface of the localized scatterer(s). The vertical distribution of the wave potential is represented by a uniformly convergent local-mode series containing, except of the ususal propagating and evanescent modes, an additional mode, accounting for the sloping bottom boundary condition. By applying a variational principle, the problem is reduced to a coupled-mode system of differential equations in the horizontal space. To treat the unbounded domain, the Berenger perfectly matched layer model is optimized and used as an absorbing boundary condition. Computed results are compared with other simpler models and verified against experimental data. The inclusion of the sloping-bottom mode in the representation substantially accelerates its convergence, and thus, a few modes are enough to obtain accurately the wave potential and velocity up to and including the boundaries, even in steep bathymetry regions. The present method provides high-quality information concerning the pressure and the tangential velocity at the bottom, useful for the study of oscillating bottom boundary layer, sea-bed movement and sediment transport studies.     

Numerical simulation of unsteady free-surface motions using a boundary integral formulation

A numerical time simulation method is described to solve fluid flow problems including unsteady free surface motion. The method is based on potential flow theory. At every time step, the problem is solved using a boundary integral formulation of the fluid domain. The linearized free surface conditions are integrated in time and the solution is marched forward. Computational results simulating the free surface motion for the cases of a linear progressive wave, wave propagating into a region of calm water and the wave maker motion are presented. Comparison with theoretical results demonstrate the feasibility of the proposed simulation scheme.     

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.     

Internal wave generation in an improved two-dimensional Boussinesq model

A set of Boussinesq-type equations with improved linear frequency dispersion in deeper water is solved numerically using a fourth order accurate predictor-corrector method. The model can be used to simulate the evolution of relatively long, weakly nonlinear waves in water of constant or variable depth provided the bed slope is of the same order of magnitude as the frequency dispersion parameter. By performing a linearized stability analysis, the phase and amplitude portraits of the numerical schemes are quantified, providing important information on practical grid resolutions in time and space. In contrast to previous models of the same kind, the incident wave field is generated inside the fluid domain by considering the scattered wave field in one part of the fluid domain and the total wave field in the other. Consequently, waves leaving the fluid domain are absorbed almost perfectly in the boundary regions by employment of damping terms in the mass and momentum equations. Additionally, the form of the incident regular wave field is computed by a Fourier approximation method which satisfies the governing equations accurately in water of constant depth. Since the Fourier approximation method requires an Eulerian mean current below wave trough level or a net mass transport velocity to be specified, the method can be used to study the interaction of waves and currents in closed as well as open basins. Several computational examples are given. These illustrate the potential of the wave generation method and the capability of the developed model.     

Internal wave generation in an improved two-dimensional Boussinesq model

A set of Boussinesq-type equations with improved linear frequency dispersion in deeper water is solved numerically using a fourth order accurate predictor-corrector method. The model can be used to simulate the evolution of relatively long, weakly nonlinear waves in water of constant or variable depth provided the bed slope is of the same order of magnitude as the frequency dispersion parameter. By performing a linearized stability analysis, the phase and amplitude portraits of the numerical schemes are quantified, providing important information on practical grid resolutions in time and space. In contrast to previous models of the same kind, the incident wave field is generated inside the fluid domain by considering the scattered wave field in one part of the fluid domain and the total wave field in the other. Consequently, waves leaving the fluid domain are absorbed almost perfectly in the boundary regions by employment of damping terms in the mass and momentum equations. Additionally, the form of the incident regular wave field is computed by a Fourier approximation method which satisfies the governing equations accurately in water of constant depth. Since the Fourier approximation method requires an Eulerian mean current below wave trough level or a net mass transport velocity to be specified, the method can be used to study the interaction of waves and currents in closed as well as open basins. Several computational examples are given. These illustrate the potential of the wave generation method and the capability of the developed model.     

Interactions between nonlinear water waves and non-wall-sided 3D structures

Interactions between water waves and non-wall-sided cylinders are analyzed based on velocity potential theory with fully nonlinear boundary conditions on the free surface and the body surface. The finite element method (FEM) is adopted together with a 3D mesh generated through an extension of a 2D Delaunay grid on a horizontal plane along the depth. The linear matrix equation for the velocity potential is constructed by imposing the governing equation and boundary conditions through the Galerkin method and is solved through an iterative method. By imposing the gradient of the potential equal to the velocity, the Galerkin method is used again to obtain the velocity field in the fluid domain. Simulations are made for bottom mounted and truncated cylinders with flare in a numerical tank. Periodic waves and wave groups are generated by a piston type wave maker mounted on one end of the tank. Results are obtained for forces, wave profiles and wave runups. Further simulations are made for a cylinder with flare subjected to forced motion in otherwise still open water. Results are provided for surge and heave motion in different amplitudes, and for a body moving in a circular path in the horizontal plane. Comparisons are made in several cases with the results obtained from the second order solution in the time domain.