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.     

A BFG model for calculation of tidal current and diffusion of pollutants in nearshore areas

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Three-dimensional numerical simulation of solitary wave run-up using the IB method

Although the finite difference method is computationally efficient, it is acknowledged to be inferior when dealing with flow-over on structures with a complex geometry because of its rectilinear grid system. Therefore, we developed a numerical procedure that can cope with flow over structures with complex shapes while, at the same time, retaining the simplicity and efficiency of a rectilinear grid system. We used the immersed boundary method, which involves application of immersed boundary forces at solid boundaries rather than conventional boundary conditions, to investigate wave interactions with coastal structures in a three-dimensional numerical wave tank by solving the Navier–Stokes equations for two-phase flows. We simulated the run-up of a solitary wave around a circular island. Maximum run-up heights were computed around the island and compared with available laboratory measurements and previous numerical results. The three-dimensional features of the run-up process were analyzed in detail and compared with those of depth-integrated equations models.     

A fully nonlinear Boussinesq model in generalized curvilinear coordinates

Based on the fully nonlinear Boussinesq equations in Cartesian coordinates, the equations in generalized coordinates are derived to adapt computations to irregularly shaped shorelines, such as harbors, bays and tidal inlets, and to make computations more efficient in large near-shore regions. Contravariant components of velocity vectors are employed in the derivation instead of the normal components in curvilinear coordinates or original components in Cartesian coordinates, which greatly simplifies the equations in generalized curvilinear coordinates. A high-order finite difference scheme with staggered grids in the image domain is adopted in the numerical model. The model is applied to five examples involving curvilinear coordinate systems. The results of these cases are in good agreement with analytical results, experimental data, and the results from the uniform grid model, which shows that the model has good accuracy and efficiency in dealing with the computations of nonlinear surface gravity waves in domains with complicated geometries.     

A NUMERICAL PREDICTION MODEL FOR TYPHOON SURGE ALONG THE SOUTHEAST COAST OF CHINA

Based on the whole-current equations of typhoon surge and its definite difference equations, this paper develops a numerical prediction mode! for typhoon surge along the southeast coast of China by taking into account of the typhoon model of circular storm. On considering the complexities of the coast and boundaries, the development of the model was made use of a relative ideal method to deal with the boundary conditions according to the characters of the model and the boundary position of the calculation grid. The calculations fairly coincided with the field observations, that proves this numerical prediction model satisfactory.     

A fast convolution approach to the transformation of surface gravity waves: Linear waves in 1DH

The transformation of irrotational surface gravity waves in an inviscid fluid can be studied by time stepping the kinematic and dynamic surface boundary conditions. This requires a closure providing the normal surface particle velocity in terms of the surface velocity potential or its tangential derivative. A convolution integral giving this closure as an explicit expression is derived for linear 1D waves over a mildly sloping bottom. The model has exact linear dispersion and shoaling properties. A discrete numerical model is developed for a spatially staggered uniform grid. The model involves a spatial derivative which is discretized by an arbitrary-order finite-difference scheme. Error control is attained by solving the discrete dispersion relation a priori and model results make a perfect match to this prediction. A procedure is developed by which the computational effort is minimized for a specific physical problem while adapting the numerical parameters under the constraint of a predefined tolerance of damping and dispersion error. Two computational examples show that accurate irregular-wave transformation on the kilometre scale can be computed in seconds. Thus, the method makes up a highly efficient basis for a forthcoming extension that includes nonlinearity at arbitrary order. The relation to Boussinesq equations, mild-slope wave equations, boundary integral equations and spectral methods is briefly discussed.     

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.     

导热几何形状反演的变分原理及边界条件的确立

本文采用变域变分原理,建立了导热几何形状反演问题的变分原理,同时获得了该问题所需满足的边界条件和附加条件.该变分原理能将未知形状的几何变量及控制方程结合在一个变分泛函中,使得数学描述简洁、紧凑,且几何变量及控制方程的求解能耦合地进行.介绍了运用该变分原理并结合有限元方法进行数值计算的方法.     

Reducing horizontal diffusion errors in σ-coordinate coastal ocean models with a second-order Lagrangian-interpolation finite-difference scheme

When a steep bottom slope exists, it is well known that conventional methods for calculating horizontal diffusion in sigma-coordinate coastal ocean models causes spurious transport (e.g. salinity, temperature, and sediments) and currents. In this study, a second-order accurate finite-difference algorithm and program have been developed to reduce the spurious numerical diffusion errors. In the proposed algorithm, the finite differencing is performed in the x-z coordinate system to approximate the horizontal gradient. Each variable in the finite differential formation is calculated in the sigma-coordinate grid cells using a second-order Lagrangian interpolation polynomial. In conjunction with a stepwise bottom boundary condition, numerical experiments show that the proposed finite-difference scheme considerably reduces numerical errors compared to conventional approaches when dealing with horizontal diffusion over steep topography, which often occurs in coastal oceans and navigation channels.     

Moving shoreline boundary condition for nearshore models

The paper develops and analyzes two fully nonlinear boundary conditions that incorporate the motion of the shoreline in nonlinear time domain nearshore models. A moving shoreline essentially means the computational domain is changing with the solution of the flow. The problem is solved in two steps. The first is to establish an equation that determines the motion of the shoreline based on the local momentum balance. The second is to develop and implement into a shoreline model the capability of accommodating a changing computational domain. The two models represent two different ways of addressing this step: one is to track the position of the shoreline in a fixed grid by establishing a special shoreline point which generally is not a fixed grid point. The second is by a coordinate transformation that maps the changing domain onto a fixed domain and solves the basic equations in the mapped domain. The two shoreline conditions are tested against three known solution for nonlinear shoreline motion. Two are the 1-D solutions to the nonlinear shallow water (NSW) equations by Carrier and Greenspan [J. Fluid Mech. 4 (1958) 97], one representing the response to a transient change in the offshore water level, the other the motion due to a periodic standing wave, both on slopes steep enough to allow full reflection. The third is the 2-D horizontal (2DH) computational solution by Zelt [Coast. Eng. 15 (1991) 205] for the run-up of a solitary wave on a cusped beach. In all cases, both models are shown to behave well and give high accuracy results for suitably chosen grid and time spacings.     

极坐标变换变边界模型及其应用

Ｊｏｈｎｓ变边界模型用于海湾海域风暴潮漫滩计算是，由于岸界变曲较大，而影响了计算的稳定性和精度，本文针对这个问题，导出了极坐标下的连续移动边界模型，从而改善了对岸界弯曲较大海域的风暴潮漫滩计算，通过对１９６９年黄河口沿岸一带水域风暴潮漫滩的模拟，证明这种极坐标变边界模型用于海湾海域的风暴潮漫滩计算，优于Ｊｏｈｎｓ变边界模型。     

波浪沿斜面传播过程的数值模拟

精确模拟非线性波沿斜面传播过程非常困难,为此论文从势函数的边界积分方程出发,建立了一种时域内二维波浪模拟的数值模型,主要用来模拟完全非线性波浪的传播变形过程。论文的数值模型使用高阶二维边界元方法,采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带自由表面的线性或完全非线性波浪传播问题。在计算区域一端造出线性或非线性的周期性波浪,另一端采用消除反射波的人工粘性吸收边界。通过与现有理论比较证明了论文数值方法所得结果是准确可靠的。     

Implementation of variable time stepping in an ocean general circulation model

Ocean general circulation models (OGCMs) which represent the governing equations on a finite difference grid require shorter time steps with increasing resolution. Thus, until now, in the absence of filtering, the time step length has been determined by the smallest grid spacing within the model domain. Here we present a method for reducing the time step length (and increasing the number of time steps taken) at selected points in the grid, so as to minimise the computational cost of integrating the OGCM, whilst achieving numerical stability throughout the model domain without filtering. This variable time stepping method can be used to overcome numerical constraints associated with the convergence of longitude–latitude grids at the poles, and also to allow efficient integration of model domains with variable resolution. Examples of the computational saving are given.     

Reducing horizontal diffusion errors in σ-coordinate coastal ocean models with a second-order Lagrangian-interpolation finite-difference scheme

When a steep bottom slope exists, it is well known that conventional methods for calculating horizontal diffusion in sigma-coordinate coastal ocean models causes spurious transport (e.g. salinity, temperature, and sediments) and currents. In this study, a second-order accurate finite-difference algorithm and program have been developed to reduce the spurious numerical diffusion errors. In the proposed algorithm, the finite differencing is performed in the x–z coordinate system to approximate the horizontal gradient. Each variable in the finite differential formation is calculated in the sigma-coordinate grid cells using a second-order Lagrangian interpolation polynomial. In conjunction with a stepwise bottom boundary condition, numerical experiments show that the proposed finite-difference scheme considerably reduces numerical errors compared to conventional approaches when dealing with horizontal diffusion over steep topography, which often occurs in coastal oceans and navigation channels.     

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.     

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.     

Refined seaward boundary conditions for modelling shallow water waves in semi-enclosed water bodies

Based on the theory of characteristics, this research elaborates on the numerical treatment of two types of seaward boundary conditions for modelling long-wave dynamics in truncated estuarine and coastal domains. These seaward boundary conditions are devised for the solution of the fully non-linear shallow water equations in the time domain. The first type is the clamped boundary, at which the water level variation is given and the velocity is computed along the characteristic line going out of the domain. The second type is the non-reflecting boundary, where the incident wave information is introduced and the reflected waves from inside the computational domain are allowed to escape at the same time. The essence of its numerical implementation is to distinguish the inward and outward characteristics and to disconnect the incoming characteristic relation from the actual flow inside the domain. Compared with previous techniques, the present method includes extra terms in the derivation to account for the effects of the uneven bed, bottom friction and shape of the characteristic lines. A shock-capturing finite difference method is used to solve the shallow water equations in the deviatoric format, but the seaward boundary algorithms constructed herein are generic and applicable to other solvers. The necessity of these refinements is highlighted by simulating the tidal oscillation in the Persian/Arabian Gulf, periodic wave runup on the coastline and the wave resonance in a narrow harbour. It is found that neglecting the bed slope at the boundary may result in biased mean water levels in the prediction.     

Modeling of ocean dynamics with large variations in sea level

The formulation and the algorithm of solving an ocean model for the prediction and assimilation of the observed data which makes it possible to reconstruct the circulation in the deep-water parts of the sea and at a shallow water shelf, as well as to describe the large time–space variability in the surface level, are considered. The model uses a vertical hybrid σ–z coordinate system: the several upper tens of meters of the ocean are described in the σ-coordinate system and the rest of the water column is described in the z coordinates. Such hybridization extends the possibilities of models for reconstructing thermo-hydrodynamic processes in different sea basins and the World Ocean. The differential formulation of the model in the σ–z coordinate system is presented; the simplified records of several operators that are allowable in the case of a small thickness of the ocean σ-layer are described. The construction of a computational grid, approximation of the bottom topography on it, and discretization of equations and boundary conditions of the models are considered; an approach to describing the bottom friction at shallow waters is offered. The results of the comparative experiments in the z and σ–z coordinate models are analyzed.