LAGFD-WAM numerical wave model-Ⅰ. Basic physical model

The LAGFD-WAM wave model is a third generation wave model. In the present paper the physical aspect of the model was shown in great detail including energy spectrum balance equation, complicated characteristics equations and source functions.

Numerical Wave Channel with Absorbing Wave-Maker

The numerical wave channel has been developed based on the volume of fluid method (VOF) in conjunction with the Navier-Stokes equations. The absorbing wave-maker boundary on the left side of the channel is presented by prescribing velocity reference to linear wave-maker theory. The principle of which is that the numerical wave-maker is designed to move in a way that generates the required incident wave and cancels out any reflected wave that reach it at the same time. On the right side of the channel, the open boundary is set to permit incident waves to be transmitted freely. The parametric studies have been carried out at a range of ratios of water depth to wave length d/ L from 0.124 to 0.219, with wave height in the front of paddle/water depth ratio (H0 / d) from 0.1 to 0.3. Wave height, wave pressure distribution along the channel and velocity field are obtained for both open boundary condition and reflective boundary condition at the other end of the channel. For a reflective case, it is shown that the absorbing wave-maker is very effective in canceling out the reflected wave that reaches the numerical paddle and highly repeatable waves can be generated.

LAGFD-WAM numerical wave model- Ⅱ. Characteristics inlaid scheme and its application

In this paper the parameterizational approach of nonlinear source function and the implicit scheme of the model are discussed in detail. The matching problem is solved between time and space steps using the characteristics inlaid scheme with very strong physical meaning. The computational comparison in typical winds shows some improvements to the WAM model. That the hindcast results of the model for typhoon cases are in good agreement with real data illustrates its applicability to wave forecast and engineering study.

A hybrid model for numerical wave forecasting and its implementation-Ⅰ.The wind wave model

The authors make an endeavor to explain why a new hybrid wave model is here proposed when several such models have already been in operation and the so- called third generation wave modej is proving attractive. This part of the paper is devoted to the wind wave model. Both deep and shallow water models have been developed, the former being actually a special case of the latter when water depth is great. The deep water model is exceptionally simple in form. Significant wave height is the only prognostic variable. In comparison with the usual methods to compute the energy input and dissipations empirically or by 'tuning', the proposed model has the merit that the effects of all source terms are combined into one term which is computed through empirical growth relations for significant waves, these relations being, relatively speaking, easier and more reliable to obtain than those for the source terms in the spectral energy balance equation. The discrete part of the model and the implementation of the model as a whole will be discussed in the second part of the present paper.

Numerical Study of Air Chamber for Oscillating Water Column Wave Energy Convertor

Oscillating Water Column （OWC） wave energy converting system is one of the most widely used facilities all over the world. The air chamber is utilized to convert the wave energy into the pneumatic energy. The numerical wave tank based on the two-phase VOF model is established in the present study toinvestigate the operating performance of OWC air chamber. The RANS equations, standard k-ε turbulence model and dynamic mesh technology are employed in the numerical model. The effects of incident wave conditions and shape parameters on the wave energy converting efficiency are studied and the capability of the present numerical wave tank on the corresponding engineering application is validated.

Numerical study of resonance induced by wave action on multiple rectangular boxes with narrow gaps

By introducing a source term into the Laplace equation, a two-dimensional fully nonlinear time-domain numerical wave flume （NWF） is developed to investigate the resonance induced by the interaction between waves and multiple objects with narrow gaps. In the numerical model, the fully nonlinear kinematic and dynamic boundary conditions are satisfied on the instantaneous free surface and the constant artificial damping is employed in the gaps to approximate the viscous dissipation due to vortex motion and flow separation. The computational domain is discretized using a higher-order boundary element method （HOBEM）. The proposed model is firstly validated against the published experimental data and numerical results of the wave height in the narrow gap between two boxes, the wave heights in the two gaps of three boxes, and wave loads on the boxes. Then, the extensive numerical experiments are performed to study the influences of the number of the boxes and the gap spacing on the resonant frequency, reflected and transmitted wave heights and wave loads on the boxes.

A two-dimensional numerical wave flume based on SA-MPLS method

A spatially adaptive (SA) two-dimensional (2-D) numerical wave flume is presented based on the quadtree mesh system,in which a new multiple particle level set (MPLS) method is proposed to solve the problem of interface tracking,in which common intersection may be traversed by multiple interfaces.By using the adaptive mesh technique and the MPLS method,mesh resolution is updated automatically with time according to flow characteristics in the modeling process with higher resolution around the free surface and the solid boundary and lower resolution in less important area.The model has good performance in saving computer memory and CPU time and is validated by computational examples of small amplitude wave,second-order Stokes wave and cnoidal wave.Computational results also indicate that standing wave and wave overtopping are also reasonably simulated by the model.

Numerical Simulation of Irregular Wave Overtopping Against A Smooth Sea Dike

Based on the filtered Navier-Stokes equations and Smagorinsky turbulence model, a numerical wave flume is developed to investigate the overtopping process of irregular waves over smooth sea dikes. Simulations of fully nonlinear standing wave and regular wave＇s run-up on a sea dike are carried out to validate the implementation of the numerical wave flume with wave generation and absorbing modules. To model stationary ergodic stochastic processes, several cases with different random seeds are computed for each specified irregular wave spectrum. It turns out that the statistical mean overtopping discharge shows good agreement with empirical formulas, other numerical results and experimental data.

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.

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 boundary 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.

Development of A Fully Nonlinear Numerical Wave Tank

A fully nonlinear numerical wave tank based on the solution of the σ-transformed Navier-Stokes equation is developed in this study. The numerical wave is generated from the inflow boundary, where the surface elevation and/or velocity are specified by use of the analytical solution or the laboratory data. The Sommerfeld/Orlanski radiation condition in conjunction with an artificial damping zone is applied to reduce wave reflection from the outflow boundary. The whole numerical solution procedures are split into three steps, i.e., advection, diffusion and propagation, and a new method, the Lagrange-Euler Method, instead of the MAC or VOF method, is introduced to solve the free surface elevation at the new time step. Several typical wave cases, including solitary waves, regular waves and irregular waves, are simulated in the wave tank. The robustness and accuracy of the NWT are verified by the good agreement between the numerical results and the linear or nonlinear analytical solutions. This research will be further developed by study of wave-wave, wave-current, wave-structure or wave-jet interaction in the future.

Generation and Properties of Freak Waves in A Numerical Wave Tank

Freak waves are generated based on the mechanism of wave focusing in a 2D numerical wave tank. To set up the nonlinear numerical wave tank, the Boundary Element Method is used to solve potential flow equations incorporated with fully nonlinear free surface boundary conditions. The nonlinear properties of freak waves, such as high frequency components and wave profile asymmetry, are discussed. The kinematic data, which can be useful for the evaluation of the wave forces exerted on structures to avoid underestimation of linear predictions, are obtained, and discussed, from the simulated results of freak waves.

A hybrid model for numerical wave forecasting and its implementation-Ⅱ .The discrete part and implementation of the model

In the first part of the present paper we have explained why we manage to formulate another wave prediction model when so many of them, including the so-called third generation model, have already been in use. The wind-wave part of the proposed model has also been given. Now we proceed to discuss the swell part,the implementation of the model as a prediction method,mumerical experiments done with ideal wind fields and hindcasts made in the Bohai Sea,in the neighboring seas adjacent to China and in the Northwest Pacific.

Determining the Onset and Strength of Unforced Wave Breaking in A Numerical Wave Tank

A numerical wave tank is used to investigate the onset and strength of unforced wave breaking, and the waves have three types of initial spectra： constant amplitude spectrum, constant steepness spectrum and Pierson-Moscowitz spectrum. Numerical tests are performed to validate the model results. Then, the onset of wave breaking is discussed with geometric, kinematic, and dynamic breaking criteria. The strength of wave breaking, which is always characterized by the fractional energy loss and breaking strength coefficient, is studied for different spectra. The results show how the energy growth rate is better than the initial wave steepness on estimating the fractional energy losses as well as breaking strength coefficient.

Numerical Model of Cnoidal Wave Flume

The volume of fluid (VOF) method is used to set up a wave flume with an absorbing wave maker of cnoidal waves. Based on the transfer function between wave surface and paddle velocity obtained by the shallow water wave theory, the velocity boundary condition of an absorbing wave maker is introduced to absorb reflected waves that reach the numerical wave maker. For Hid ranging from 0.1 to 0.59 and T root g/d from 7.9 to 18.3, the parametric studies have been carried out and compared with experiments.

Second Harmonic Generation of Lamb Wave in Numerical Perspective

The influences of phase and group velocity matching on cumulative second harmonic generation of Lamb waves are investigated in numerical perspective. Finite element simulations of nonlinear Lamb wave propagation are performed for Lamb wave mode pairs with exact and approximate phase velocity matching, with and without group velocity matching, respectively. The evolution of time-domain second harmonic Lamb waves is analyzed with the propagation distance. The amplitudes of primary and second harmonic waves are calculated to characterize the acoustic nonlinearity. The results verify that phase velocity matching is necessary for generation of the cumulative second harmonic Lamb wave in numerical perspective, while group velocity matching is demonstrated to not be a necessary condition.

Numerical investigation of water-entry of flatted-bottom seafloor mining tool in ocean waves

A numerical wave load model based on two-phase(water-air) Reynolds-Averaged Navier Stokes(RANS) type equations is used to evaluate hydrodynamic forces exerted on flatted-bottom seafloor mining tool during its entering ocean waves of deploying process.The discretization of the RANS equations is achieved by a finite volume approach(FV).The volume of fluid method(VOF) is employed to track the complicated free surface.A numerical wave tank is built to generate the ocean waves which are suitable for deploying seafloor mining tool.A typical deploying condition is employed to reflect the process of flatted-bottom body impacting with waves,and the pressure distribution of bottom is also presented.Four different lowering velocities are applied to obtain the time histories of maximum pressure of bottom,and it can be concluded that the pressure coefficient decreases with water velocity increasing,which is similar with ordinary water entry case.The numerical results clearly demonstrate the characteristics of flatted-bottom body entering ocean waves.

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.

Generation of linear and nonlinear waves in numerical wave tank using clustering technique-volume of fluid method

A two-dimensional （2D） numerical model is developed for the wave sim- ulation and propagation in a wave flume. The fluid flow is assumed to be viscous and incompressible, and the Navier-Stokes and continuity equations are used as the governing equations. The standard k-e model is used to model the turbulent flow. The Navier- Stokes equations are discretized using the staggered grid finite difference method and solved by the simplified marker and cell （SMAC） method. Waves are generated and propagated using a piston type wave maker. An open boundary condition is used at the end of the numerical flume. Some standard tests, such as the lid-driven cavity, the constant unidirectional velocity field, the shearing flow, and the dam-break on the dry bed, are performed to valid the model. To demonstrate the capability and accuracy of the present method, the results of generated waves are compared with available wave theories. Finally, the clustering technique （CT） is used for the mesh generation, and the best condition is suggested.

A simple method of depressing numerical dissipation effects during wave simulation within the Euler model

Numerical wave tanks are widely-acknowledged tools in studying waves and wave-structure interactions. They can generate waves under realistic scales and offers more information on the fluid field. However, most numerical wave tanks suffer from issues known as the numerical dissipation and numerical dispersion. The former causes wave energy to be slowly dissipated and the latter shifts wave frequencies during wave propagation. This paper proposes a simple method of depressing numerical dissipation effects on the basis of solving Euler equations using the finite difference method(FDM). The wave propagation solutions are solved analytically taking into account the influence of the damping terms. The main idea of the method is to append a source term to the momentum equation, whose strength is determined by how strong the numerical damping effect is. The method is verified by successfully depressing numerical effects during the simulation of regular linear waves, Stokes waves and irregular waves. By applying the method, wave energy is able to be close to its initial value after long distance of travel.