Numerical Investigation on the Generation and Propagation of Irregular Waves in A Two-Dimensional Wave Tank

The modeling of generation and subsequent propagation of irregular waves in a numerical wave flume is performed by mean of the boundary element method. Random waves are generated by a piston-type wave generator at one end of the flume with the Mitsuyasu-Bretschneider spectrum used as the target spectrum for the generation. An artificial absorbing beach is placed at the other end of the flume to minimize wave reflection. Surface fluctuations are described by use of the Lagrangian description, and finite difference is adopted for the approximation of time derivative. To monitor the developments of the waves, a number of pseudo wave gauges are installed along the tank. Through comparison of the spectra from those gauges with the target spectrum, satisfactory results can be obtained from the present numerical scheme.

Development of two-dimensional numerical wave tank based on lattice Boltzmann method

A 2-D numerical wave tank(NWT)is developed using the lattice Boltzmann method(LBM)and a multi-relaxation-time(MRT)collision model coupled with an algebraic volume of fluid(VOF)scheme for free surface tracking.An external force based on the momentum source function is used to generate the waves,and a zone of porous media is used to absorb the waves.Numerical simulations of the progressive and standing waves show that the NWT can generate stable wave trains in agreement with the analytical solutions and eliminate the re-reflection waves.The NWT is used to simulate two problems encountered in practice,namely:the wave transformation over a submerged breakwater and the wave runup on a sea dike.The numerical predictions are in good agreement with the measured data.

Numerical simulation for the two-dimensional nonlinear shallow water waves

This study deals with the general numerical model to simulate the two-dimensional tidal flow, flooding wave (long wave) and shallow water waves (short wave). The foundational model is based on nonlinear Boussinesq equations. Numerical method for modelling the short waves is investigated in detail. The forces, such as Coriolis forces, wind stress, atmosphere and bottom friction, are considered. A two-dimensional implicit difference scheme of Boussinesq equations is proposed. The low-reflection outflow open boundary is suggested. By means of this model,both velocity fields of circulation current in a channel with step expansion and the wave diffraction behind a semi-infinite breakwater are computed, and the results are satisfactory.

Linear and Nonlinear Stokes Waves Theory: Numerical Hydrodynamic and Energy Studies

The increase of wave energy in electricity production is an objective shared by many countries to meet growing demand and global warming. To analyze devices capable of converting the energy of sea waves into electrical energy, it is important to master the various theories of gravity waves and generation. We will in our work consider a numerical waves tank for an amplitude A=0.5, a wavelength λ=0.25 , an average height He=10 and a Froude number fixed at 1 × 105. Numerical wave channel analysis is used to reproduce the natural phenomenon of wave propagation in an experimental model. Wave makers are usually used to generate waves in the channel. In theory, the influence of an incident wave can be considered, as in the case of our study. In this study, the evolution of the hydrodynamic parameters and the energy transported in one wavelength can be determined by calculation. A change of variable will be done in this work to facilitate the writing of the boundary conditions at the free surface and at the bottom. The nonlinear Stokes theory will be studied in this case in order to provide hydrodynamic solutions through the Navier-Stokes equations to finally deduce the energetic results. To do this, the finite difference method will be used for the hydrodynamic results such as the velocity potential and the free surface elevation and the trapezium method of Newton for the energetic results. Thus, we will determine the energetic potential according to the decrease in the slope of the tank. To do this, we will take as values of beta representing the inverse of the slope of the tank, β=100, β=105, β=110 and β=105. .

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.

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.

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.

Air Gap Prediction for Floating Bodies Using a 3D Numerical Wave Tank Approach

Computations for air gap response of a semisubmersible platform based on a 3D numerical wave tank approach are presented.The developed method is in time domain and can consider nonlinearities associated with incident wave and hydrostatic forces exactly in determining the body response, but the interaction hydrodynamics of radiation and diffraction are based on simplified linearization assumptions. The incident wave can be defined by any suitable wave theory and here defined by a fully nonlinear numerical wave model. After verifying the present computations results in its degenerated linearized version against the usual linear 3D Green function–based frequency-domain results for air gap predictions, systematic comparative studies are undertaken between linear and the approximate nonlinear solutions. It is found that nonlinear computations can yield considerably conservative predictions as compared to fully linear calculations, amounting to a difference of up to 30%–40% in the minimum air gap in steep ambient incident waves at high and moderate frequencies.

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 Numerical Wave Tank for Nonlinear Waves with Passive Absorption

A numerical wave tank with passive absorption for irregular waves is considered in this paper. Waves with spectral shapes corresponding to that of the Mitsuyasu- Bretschneider type are used as the initial condition at one end of the flume, An absorbing boundary is imposed at the other end of the wave flume to minimize reflection. By use of a Lagrangian description for the Surface elevation, and finite difference for approximation of the time derivative, the problem is then solved by the boundary element method, The effects of the absorbing boundary are investigated by varying the values of the absorption coefficient mu, and studying the time histories of the Surface elevations 'recorded' on pre-selected locations.

Numerical Wave Tank Based on A Conserved Wave-Absorbing Method

Recently the numerical wave tank has become a widely-used tool to study waves as well as wave-structure interactions, and the wave-absorbing method is very important as its effect on the quality of waves generated. The relaxation method and the derived momentum source method are often utilized, however, the damping weight is constant during calculation and repeated trials are required to obtain an acceptable wave-absorbing effect. To address the abovementioned issues, a conserved wave-absorbing method is developed. The damping weight is determined by solving the mass conservation equation of the absorbing region at every time step. Based on this method, a two-dimensional numerical wave tank is established by using the VB language to simulate various waves by which the validation of this method is evaluated.

Two Dimensional Fully Nonlinear Numerical Wave Tank Based on the BEM

The development of a two dimensional numerical wave tank （NWT） with a rocker or piston type wavemaker based on the high order boundary element method （BEM） and mixed Eulerian-Lagrangian （MEL） is examined. The cauchy principle value （CPV） integral is calculated by a special Gauss type quadrature and a change of variable. In addition the explicit truncated Taylor expansion formula is employed in the time-stepping process. A modified double nodes method is assumed to tackle the comer problem, as well as the damping zone technique is used to absorb the propagation of the free surface wave at the end of the tank. A variety of waves are generated by the NWT, for example; a monochromatic wave, solitary wave and irregular wave. The results confirm the NWT model is efficient and stable.

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.

Efficient computation method for two-dimensional nonlinear waves

The theory and simulation of fully-nonlinear waves in a truncated two-dimensional wave tank in time domain are presented. A piston-type wave-maker is used to generate gravity waves into the tank field in finite water depth. A damping zone is added in front of the wave-maker which makes it become one kind of absorbing wave-maker and ensures the prescribed Neumann condition. The efficiency of numerical tank is further enhanced by installation of a sponge layer beach (SLB) in front of downtank to absorb longer weak waves that leak through the entire wave train front. Assume potential flow, the space- periodic irrotational surface waves can be represented by mixed Euler-lagrange particles Solving the integral equation at each time step for new normal velocities, the instantaneous free surface is integrated following time history by use of fourth-order Runge- Kutta method. The double node technique is used to deal with geometric discontinuity at the wave- body intersections. Several precise smoothing methods have been introduced to treat surface point with high curvature. No saw-tooth like instability is observed during the total simulation. The advantage of proposed wave tank has been verified by comparing with linear theoretical solution and other nonlinear results, excellent agreement in the whole range of frequencies of interest has been obtained.

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.

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.

Numerical Simulation of Waves Generated by Seafloor Movements

Waves generated by vertical seafloor movements are simulated by use of a fully nonlinear two-dimensional numerical wave tank. In the souree region, the seafloor lifts to a designated height by a generation function. The numerical tests show that the linear theory is only valid for estimating the wave behaviors induced by the seafloor movements with a small amplitude, and the fully nonlinear numerical model should be adopted in the simulation of the wave generation by the large amplitude seafloor movements. Without the background surface waves, many numerical tests on the stable maximum elevations η0^max are carried out by beth the linear theory and the fully nonlinear model. The results of two models are compared and analyzed. For the fully nonlinear model, the influences of the amplitudes and the horizontal lengths on η^max are stronger than that of the characteristic duration times. Furthermore, results reveal that there are significant differences be- tween the linear theory and the fully nonlinear model. When the influences of the background surface waves are considered, the corresponding numerical analyses reveal that with the fully nonlinear model the η0^max near-linearly varies with the wave amplitudes of the surface waves, and the η0^max has significant dependences on the wave lengths and the wave phases of the surface waves. In addition, the differences between the linear theory and the fully nonlinear model are still obvious, and these differences are significantly affected by the wave parameters of the background surface waves, such as the wave amplitude, the wave length and the wave phase.

Numerical Investigation of the Effect of Current on Wave Focusing

In the present study, a numerical wave tank is developed to simulate the nonlinear wave-current interactions based on High Order Spectral （HOS） method. The influences of current on wave focusing are investigated by use of numerical model. The current is assumed to be constant in space. Focused waves with different amplitudes and frequency spectra are simulated with and without current. The focused wave characteristics, such as surface elevation, the maximum crest and frequency spectrum, with different current are compared. The results show that the opposing current increases the maximum crest and the energy transform during wave focusing process, and vice versa for the following current.

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.