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.     

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.     

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。     

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.     

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.     

Experimental Study of Nonlinear Behaviors of A Free-Floating Body in Waves

Nonlinear behaviors of a free-floating body in waves were experimentally investigated in the present study. The experiments were carried out for 6 different wave heights and 6 different wave periods to cover a relatively wide range of wave nonlinearities. A charge-coupled device (CCD) camera was used to capture the real-time motion of the floating body. The measurement data show that the sway, heave and roll motions of the floating body are all harmonic oscillations while the equilibrium position of the sway motion drifts in the wave direction. The drift speed is proportional to wave steepness when the size of the floating body is comparable to the wavelength, while it is proportional to the square of wave steepness when the floating body is relatively small. In addition, the drift motion leads to a slightly longer oscillation period of the floating body than the wave period of nonlinear wave and the discrepancy increases with the increment of wave steepness.     

Experimental study of nonlinear behaviors of a free-floating body in waves

Nonlinear behaviors of a free-floating body in waves were experimentally investigated in the present study. The experiments were carried out for 6 different wave heights and 6 different wave periods to cover a relatively wide range of wave nonlinearities. A charge-coupled device (CCD) camera was used to capture the real-time motion of the floating body. The measurement data show that the sway, heave and roll motions of the floating body are all harmonic oscillations while the equilibrium position of the sway motion drifts in the wave direction. The drift speed is proportional to wave steepness when the size of the floating body is comparable to the wavelength, while it is proportional to the square of wave steepness when the floating body is relatively small. In addition, the drift motion leads to a slightly longer oscillation period of the floating body than the wave period of nonlinear wave and the discrepancy increases with the increment of wave steepness.     

The initialization of nonlinear water waves in a numerical wave flume

This study investigates the initialization of nonlinear free-surface simulations in a numerical wave flume.Due to the mismatch between the linear input wavemaker motion and the kinematics of fully nonlinear waves,direct numerical simulations of progressive waves,generated by a sinusoidally moving wavemaker,are prone to suffering from high-frequency wave instability unless the flow is given sufficient time to adjust.A time ramp is superimposed on the wavemaker motion at the start that allows nonlinear free-surface simulations to be initialized with linear input.The duration of the ramp is adjusted to test its efficiency for short waves and long waves.Numerical results show that the time ramp scheme is effiective to stabilize the wave instability at the start of the simulation in a wave flume.     

Progress on Nonlinear-Wave-Forced Sediment Transport Simulation

In this paper, we report on the use of a numerical wave tank (NWT), based on fully nonlinear potential flow (FNPF) equations, in driving simulations of flow and sediment transport around partially buried obstacles. The suspended sediment transport is modeled in the near-field in a Navier-Stokes (NS) model using an immersed-boundary method and an attached sediment transport simulation module. Turbulence is represented by large eddy simulation (LES). The NWT is based on a higher order boundary element method (BEM), with an explicit second-order time stepping. Hence, only the NWT boundary is discretized. The solution for the velocity potential and its derivatives along the boundary is obtained in the BEM, which subsequently provides a solution at any required internal point within the domain. At initial time, the NS-LES model domain is initialized with the 3-D velocity field provided by the NWT and driven for later time by the pressure gradient field obtained in the NWT. Incident wave fields, as specified in the NWT to drive sediment transport, can be arbitrary. Applications are presented here for single frequency waves, such as produced by a harmonic piston wavemaker in the laboratory, and modulated frequency wave groups. The feasibility of coupling the irrotational flow and NS solutions is demonstrated.     

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...     

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.     

Time domain simulation of side-by-side floating bodies using a 3D numerical wave tank approach

The coupled system of two side-by-side fixed and/or floating bodies interacting with a large amplitude nonlinear wave is studied using a direct time domain solution method. The numerical model is based on a three-dimensional mixed Eulerian–Lagrangian (MEL) method under certain simplifying approximations permitting Rankine panel scheme to be implemented over a time-invariant boundary surface to solve the boundary value problem for the unknown velocity potentials. A 4th order Adams–Bashforth–Moulton scheme is used for time marching of rigid-body motion histories of the individual bodies and evolution of the free-surface including the gap region in which large resonant fluid motions occur. A systematic study has been carried out to evaluate the performance of the developed time domain method in simulating the forces and motions as well as the fluid motion in the gap region for the two body system under various arrangements and in different wave-headings. At first, the computed numerical results have been validated and verified with computational and experimental results available in literature for standard geometries such as vertical truncated cylinders and rectangular boxes. Secondly, effectiveness of the damping lid model which is introduced to suppress wave resonance in the gap region is investigated including its influence on maximum sway forces on fixed and floating rectangular barges in side-by-side configurations. Thirdly, comparative studies on absolute and relative motion response for two cases (two rectangular barges, and a FLNG-FPSO + shuttle tanker) in side-by-side arrangement are detailed to bring out the importance of nonlinearities arising due to steep nonlinear incident waves. Finally, coupled motions of the two-body system of an FPSO and a shuttle tanker floating in side-by-side configuration in a steep nonlinear wave field are studied in which the two bodies are connected through hawsers, and also the FPSO is moored to the ground. Additionally there is a fender between the two bodies.     

Boundary integral equation solutions for solitary wave generation, propagation and run-up

The boundary integral equation method (BIEM) is developed as a tool for studying two-dimensional, nonlinear water wave problems, including the phenomena of wave generation, propagation and run-up. The wave motions are described by a potential flow theory. Nonlinear free-surface boundary conditions are incorporated in the numerical formulation. Examples are given for either a solitary wave or two successive solitary waves. Special treatment is developed to trace the run-up and run-down along a shoreline. The accuracy of the present scheme is verified by comparing numerical results with experimental data of maximum run-up.     

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.     

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...     

Nonlinear simulations of wave-induced motions of a freely floating body using WCSPH method

Nonlinear interactions between waves and floating bodies are investigated using the weakly compressible Smoothed Particle Hydrodynamic (WCSPH) method. An improved algorithm based on the dynamic boundary particles (DBPs) is proposed to treat the moving boundary of the floating body. The force exerted on the floating body boundary particle by the particles surrounding it is evaluated using the volume integration of the stress tensors obtained from the momentum equation in its compact support. The improved WCSPH model is validated by the experimental results. The numerical test cases of the vertical oscillation of a rectangular box, the damped rolling oscillation of a floating box and the wave forces on a fixed rectangular box are then carried out to demonstrate the performance of the proposed model. Finally the evolution in time of the dynamic response of the freely floating body under nonlinear waves are discussed and compared with experimental results.     

Direct time domain analysis of floating structures with linear and nonlinear mooring stiffness in a 3D numerical wave tank

In this paper, motion response of a moored floating structure interacting with a large amplitude and steep incident wave field is studied using a coupled time domain solution scheme. Solution of the hydrodynamic boundary value problem is achieved using a three-dimensional numerical wave tank (3D NWT) approach based upon a form of Mixed-Eulerian–Lagrangian (MEL) scheme. In the developed method, nonlinearity arising due to incident wave as well as nonlinear hydrostatics is completely captured while the hydrodynamic interactions of radiation and diffraction are determined at every time step based on certain simplifying approximations. Mooring lines are modelled as linear as well as nonlinear springs. The horizontal tension for each individual mooring line is obtained from the nonlinear load-excursion plot of the lines computed using catenary theory, from which the linear and nonlinear line stiffness are determined. Motions of three realistic floating structures with different mooring systems are analyzed considering various combinations of linear and approximate nonlinear hydrodynamic load computations and linear/nonlinear mooring line stiffness. Results are discussed to bring out the influence and need for consideration of nonlinearities in the hydrodynamics and hydrostatics as well as the nonlinear modelling of the line stiffness.     

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.     

Bragg reflection in a fully nonlinear numerical wave tank based on boundary integral equation method

We investigated the phenomenon of Bragg reflection of submerged structures in a 2D fully nonlinear numerical wave tank (NWT) based on the boundary integral equation method (BIEM). This model was validated by comparing not only the free surface elevations with that of the analytic solution of Stokes’ second-order wave theory, but also the reflection coefficients of submerged bars with that from other sources. The results of the present model show that the free surface nonlinear effect on the reflection coefficient of the primary resonance reduces significantly for all of the submerged bars considered. Finally, a case study is presented to demonstrate the reflecting capacity and overall performance of various submerged bars. Results indicate that sinusoidal bar has the maximum reflection capacity at the primary resonance, but the trapezoidal submerged bar is suggested as the better option for the practical convenience of coastal underwater construction.