一种基于Roe格式的有限体积法在二维溃坝问题中的应用

溃坝属于典型的非恒定含间断的浅水问题。应用有限体积法离散二维浅水控制方程的守恒型方程组,将基于近似黎曼解的Roe格式用于数值计算溃坝问题,并利用MUSCL方法构造二阶空间积分格式和预测-校正二步法构造二阶时间格式,从而使数值解的整体达到二阶,提高了精度。文中算法在一维溃坝的Stoker问题的数值结果与解析解进行对比,结果证明了此方法的可行性。应用此方法在二维溃坝问题上的结果,说明了此算法可有效模拟溃坝水流的演进过程。     

一维全非线性Green-Naghdi水波方程数值模型

建立了求解一维全非线性Green-Naghdi水波方程的中心有限体积/有限差分混合数值格式。采用结构化网格对守恒形式的控制方程进行离散和积分,界面数值通量采用有限体积法计算,剩余项则采用中心有限差分格式求解。其中,采用中心迎风有限体积格式计算控制体界面数值通量,并结合界面变量的线性重构方法,使其在空间上具有四阶精度,通过引入静压重构技术和波浪破碎指标使模型具备处理海岸水-陆动边界及波浪破碎的能力。时间积分则采用具有总时间变差减小(Total Variation Diminishing,TVD)性质的三阶龙格-库塔法进行。应用该模型对孤立波在常水深和斜坡海岸上的传播过程及规则波跨越潜堤传播的实验进行了数值模型研究,数值计算同解析解及实验数据吻合良好。     

Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure

A numerical method for non-hydrostatic, free-surface, irrotational flow governed by the nonlinear shallow water equations including the effects of vertical acceleration is presented at the aim of studying surf zone phenomena. A vertical boundary-fitted grid is used with the water depth divided into a number of layers. A compact finite difference scheme is employed for accurate computation of frequency dispersion requiring a limited vertical resolution and hence, capable of predicting the onset of wave breaking. A novel wet–dry algorithm is applied for a proper handling of moving shoreline. Mass and momentum are strictly conserved at discrete level while the method only dissipates energy in the case of wave breaking. The numerical results are verified with a number of tests and show that the proposed model using two layers without ad-hoc assumptions enables to resolve propagating nonlinear shoaling, breaking waves and wave run-up within the surf and swash zones in an efficient manner.     

A numerical method for the two layer shallow water equations with dry states

A numerical method is proposed for solving the two layer shallow water equations with variable bathymetry in one dimension based on high-resolution f-wave-propagation finite volume methods. The method splits the jump in the fluxes and source terms into waves propagating away from each grid cell interface. It addresses the required determination of the system’s eigenstructure and a scheme for evaluating the flux and source terms. It also handles dry states in the system where the bottom layer depth becomes zero, utilizing existing methods for the single layer solution and handling single layer dry states that can exist independently. Sample results are shown illustrating the method and its handling of dry states including an idealized ocean setting.     

The applicability of the shallow water equations for modelling violent wave overtopping

The shallow water equations (SWE) have been used to model a series of experiments examining violent wave overtopping of a near-vertical sloping structure with impacting wave conditions. A finite volume scheme was used to solve the shallow water equations. A monotonic reconstruction method was applied to eliminate spurious oscillations and ensure proper treatment of bed slope terms. Both the numerical results and physical observations of the water surface closely followed the relevant Rayleigh probability distributions. However, the numerical model overestimated the wave heights and suffered from the lack of dispersion within the shallow water equations. Comparisons made on dimensionless parameters for the overtopping discharge and percentage of waves overtopping between the numerical model and the experimental observations indicated that for the lesser impacting waves, the shallow water equations perform satisfactorily and provide a good alternative to computationally more expensive methods.     

迎风有限元法在三维潮流数值模拟中的应用

为研究和开发迎风有限元法在河口海岸三维波数值模拟中的应用，并为建立三维物质扩散模型，最终为解决实际工程问题打下基础，以非线性浅水波方程为基础，采用一咱平面迎风有奶元与垂向工有限差分相结合地数值计算方法建立三维潮流数学模型，基地实际的物理过程，在计算中将三维流动分成外重力波和内重力波，耦联求解潮位和流速的空间分布。应用本模式计算了天津新港附近渤海海域的三维潮流运动，以较高的分辩率揭示了潮流空间结构特     

Unsteady ship waves in shallow water of varying depth based on Green–Naghdi equation

Based on Green–Naghdi equation this work studies unsteady ship waves in shallow water of varying depth. A moving ship is regarded as a moving pressure disturbance on free surface. The moving pressure is incorporated into the Green–Naghdi equation to formulate forcing of ship waves in shallow water. The frequency dispersion term of the Green–Naghdi equation accounts for the effects of finite water depth on ship waves. A wave equation model and the finite element method (WE/FEM) are adopted to solve the Green–Naghdi equation. The numerical examples of a Series 60 (C_B=0.6) ship moving in shallow water are presented. Three-dimensional ship wave profiles and wave resistance are given when the ship moves in shallow water with a bed bump (or a trench). The numerical results indicate that the wave resistance increases first, then decreases, and finally returns to normal value as the ship passes a bed bump. A comparison between the numerical results predicted by the Green–Naghdi equation and the shallow water equations is made. It is found that the wave resistance predicted by the Green–Naghdi equation is larger than that predicted by the shallow water equations in subcritical flow , and the Green–Naghdi equation and the shallow water equations predict almost the same wave resistance when , the frequency dispersion can be neglected in supercritical flows.     

无结构网格二维河口海岸水动力数值模式的建立及其应用

为完全拟合河口近海复杂岸线和工程结构以及有效局部加密,设计并建立了一个无结构三角形网格二维河口海岸水动力数值模式。空间离散主要基于有限体积法以保证守恒性,时间积分采用预估修正法以提高精度。水位在三角形网格中心通过连续方程求解;水平x方向和y方向的流速U和V均在网格边中点上通过动量方程求解。流速平流项的求解中采用了TVD格式。TVD流速平流通量为一个一阶迎风格式通量和一个二阶格式通量的组合,一阶格式通量和二阶格式通量根据流速的局部分布情况得出配比,最终组合得到TVD通量。TVD格式具有低耗散和无频散的优点,提高了模式的稳定性。应用实测资料验证建立的模式,结果显示水位、流速和流向的计算值与实测值均符合良好。     

An Explicit High Resolution Scheme for Nonlinear Shallow Water Equations

The present study develops a numerical model of the two-dimensional fully nonlinear shallow water equations （NSWE） for the wave run-up on a beach. The finite volume method （FVM） is used to solve the equations, and a second-order explicit scheme is developed to improve the computation efficiency. The numerical fluxes are obtained by the two dimensional Roe＇ s flux function to overcome the errors caused by the use of one dimensional fluxes in dimension splitting methods. The high-resolution Godunov-type TVD upwind scheme is employed and a second-order accuracy is achieved based on monotonic upstream schemes for conservation laws （MUSCL） variable extrapolation; a nonlinear limiter is applied to prevent unwanted spurious oscillation. A simple but efficient technique is adopted to deal with the moving shoreline boundary. The verification of the solution technique is carried out by comparing the model output with documented results and it shows that the solution technique is robust.     

Free surface flow impacting on an elastic structure: Experiment versus numerical simulation

The purpose of this study is to investigate the phenomenon of free surface flow impacting on elastic structures, which is a research topic of great interest in ship and ocean engineering. A series of quasi two-dimensional experiments on dam-break with an elastic plate are conducted. The main features of free surface flow impacting on elastic structures including large impacting force, structural vibration, violent free surface flow, are investigated. The coupled FDM–FEM method developed by the authors is applied for numerical simulation of such dam-break problem. Extensive analysis and discussion based on the comparisons between experimental data and numerical results are made and presented in this paper.     

二维浅水波方程的间断-连续Galerkin方法

将基于二维简化浅水波模型的间断Galerkin有限元与连续Galerkin有限元耦合方法推广至形式更为复杂的浅水波方程,并给出了误差分析以及模型问题的数值算例。     

The motions of a ship in swallow water

A general formulation is given of the hydrodynamic forces on a ship, oscillating about a state of rest in 6df in response to excitation by a harmonic wave in shallow water. A method is described to obtain a numerical approximation of the velocity potential, describing the flow around the moving ship by means of a distribution of discrete three-dimensional sources.With this method it is possible to take the influence of a quay into account.Calculated values of wave excited forces, hydrodynamic coefficients and motions of a 200,000 tdw tanker in shallow water are presented and compared with experimental results.     

A Eulerian–Lagrangian method for the simulation of wave propagation

A Eulerian–Lagrangian method (ELM) is employed for the simulation of wave propagation in the present research. The wave action conservation equation, instead of the wave energy balance equation, is used. The wave action is conservative and the action flux remains constant along the wave rays. The ELM correctly accounts for this physical characteristic of wave propagation and integrates the wave action spectrum along the wave rays. Thus, the total derivative for wave action spectrum may be introduced into the numerical scheme and the complicated partial differential wave action balance equation is simplified into an ordinary differential equation. A number of test cases on wave propagation are carried out and show that the present method is stable, accurate and efficient. The results are compared with analytical solutions and/or other computed results. It is shown that the ELM is superior to the first-order upwind method in accuracy, stability and efficiency and may better reflect the complicated dynamics due to the complicated bathymetry features in shallow water areas.     

A temporally second-order accurate Godunov-type scheme for solving the extended Boussinesq equations

A numerical scheme for solving the class of extended Boussinesq equations is presented. Unlike previous schemes, where the governing equations are integrated through time using a fourth-order method, a second-order Godunov-type scheme is used thus saving storage and computational resources. The spatial derivatives are discretised using a combination of finite-volume and finite-difference methods. A fourth-order MUSCL reconstruction technique is used to compute the values at the cell interfaces for use in the local Riemann problems, whilst the bed source and dispersion terms are discretised using centred finite-differences of up to fourth-order accuracy. Numerical results show that the class of extended Boussinesq equations can be accurately solved without the need for a fourth-order time discretisation, thus improving the computational speed of Boussinesq-type numerical models. The numerical scheme has been applied to model a number of standard test cases for the extended Boussinesq equations and comparisons made to physical wave flume experiments.     

用含高色散性Boussinesq方程模拟潜堤上的波浪变形

基于一种高阶Boussiensq方程(刘忠波等,2004),采用预报-校正格式的有限差分法对该方程进行了数值离散,建立了数值模型。针对动量方程中三阶项的差分形式,采用了迎风格式和五点格式。通过数值模拟常水深下不同周期波浪传播变形,指出迎风格式在计算小周期波浪时存在的问题。为进一步验证数值模型的适用性,模拟了淹没潜堤上的传播变形。从数值结果与实验值的对比结果上看,该数值模型能较好地模拟波浪变形,可用于模拟实际中的波浪场问题。     

A Eulerian–Lagrangian method for the simulation of wave propagation

A Eulerian–Lagrangian method (ELM) is employed for the simulation of wave propagation in the present research. The wave action conservation equation, instead of the wave energy balance equation, is used. The wave action is conservative and the action flux remains constant along the wave rays. The ELM correctly accounts for this physical characteristic of wave propagation and integrates the wave action spectrum along the wave rays. Thus, the total derivative for wave action spectrum may be introduced into the numerical scheme and the complicated partial differential wave action balance equation is simplified into an ordinary differential equation. A number of test cases on wave propagation are carried out and show that the present method is stable, accurate and efficient. The results are compared with analytical solutions and/or other computed results. It is shown that the ELM is superior to the first-order upwind method in accuracy, stability and efficiency and may better reflect the complicated dynamics due to the complicated bathymetry features in shallow water areas.     

EFFECTS OF WIND ON THE WAVE ADVANCING TOWARDS SHORE

The effects of wind on wave propagation in shallow water are approached in this paper. The equations for calculating wave elements in shallow water are derived by the law of wave refraction and the equation of wave energy flux. The numerical results calculated by the electronic computer show that under the influence of wind, the wave elements, wave direction angle and the conditions of wave breaker change greatly. Consequently, the effects of wind on the water wave propagating in shallow water need to be considered in practice.     

On the numerical modeling of tsunami wave generation and propagation

Abstract

In this article three main stages of tsunami wave evolution are investigated. At first, the development of disturbances from a given patched elevation of the bottom surface in an incompressible nonviscous fluid of the uniform depth is considered. Then, a tsunami wave diffraction by underwater bottom elevation or cavity is investigated. In this case the shallow water equations are already used, and it is supposed that a cylindrical wave is spread from patched water elevation over the epicentrum. Last, the tsunami propagation and transformation in a shallow water region and its run‐up on a beach are investigated on the basis of the improved shallow water theory, taking into consideration the nonlinear and dispersive terms of higher order. The proposed theory is tested in a problem of collisions of two solutions. Solutions of the first and the second problems are obtained by the method of integral Laplace's transformation with following numerical inversion of transformations. A finite difference method for a solution of the last problem is used.     

An efficient shock capturing algorithm to the extended Boussinesq wave equations

A hybrid finite-volume and finite-difference method is proposed for numerically solving the two-dimensional (2D) extended Boussinesq equations. The governing equations are written in such a way that the convective flux is approximated using finite volume (FV) method while the remaining terms are discretized using finite difference (FD) method. Multi-stage (MUSTA) scheme, instead of commonly used HLL or Roe schemes, is adopted to evaluate the convective flux as it has the simplicity of centred scheme and accuracy of upwind scheme. The third order Runge–Kutta method is used for time marching. Wave breaking and wet–dry interface are also treated in the model. In addition to model validation, the emphasis is given to compare the merits and limitations of using MUSTA scheme and HLL scheme in the model. The analytical and experimental data available in the literature have been used for the assessment. Numerical tests demonstrate that the developed model has the advantages of stability preserving, shock-capturing and numerical efficiency when applied in the complex nearshore region. Compared with that using HLL scheme, the proposed model has comparable numerical accuracy, but requires slightly less computation time and is much simpler to code.     

极浅海域潮流数值模型

通过对以往的浅水环流数值模型进行改进，建立适用于极浅水域的且能达到较高分辨率的潮汐环流数值模型。模型的主要特点是：（１）通过改进海底摩擦项的表达式来克服传统的二次方律在极浅的潮滩区所产生的不稳定问题；（２）采用逆风格式处理动量方程中的平流项，有效地抑制由于岸边界移动引起的数值短波的扩散。结合老虎滩湾海上工程的需要，利用本模型进行一系列的数值模拟试验。经过实测资料验证表明，改进后的模型具有良好的稳定