求解二维浅水流动方程的Godunov格式

采用网格变换和Strang算子分裂,以准确Riemann解为基础。建立了求解非平底浅水流动方程的Godunov格式,用“水位方程法（Water Level Formulation)”求Riemann解,结合中心差分和Riemann解离散底坡项,保证了计算格式的和谐性,经验证,方法健全。通用,且分辨率高。     

浅水问题动边界数值模拟

采用基于准确Riemann解的Godunov格式求解浅水流动方程，将仅适用于平底的干底Riemann解推广到处理非平底动边界问题．在计算干底溃坝波的基础上，将该方法应用到有大片滩地的钱塘江河口涌潮计算中，能较好地模拟钱塘江涌潮在滩地上的传播过程．     

Numerical simulation of wave transformation,breaking and runup by a contravariant fully non-linear Boussinesq equations model

In this paper we propose a new model based on a contravariant integral form of the fully non-linear Boussinesq equations(FNBE) in order to simulate wave transformation phenomena, wave breaking, runup and nearshore currents in computational domains representing the complex morphology of real coastal regions. The above-mentioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. A high-order upwind weighted essentially non-oscillatory(WENO) finite volume scheme that involves an exact Riemann solver is implemented. The wave breaking is represented by discontinuities of the weak solution of the integral form of the non-linear shallow water equations(NSWE). On the basis of the shock-capturing high order WENO scheme a new procedure, for the computation of the structure of the solution of a Riemann problem associated with a wet/dry front, is proposed in order to simulate the run up hydrodynamics in swash zone. The capacity of the proposed model to correctly represent wave propagation, wave breaking, run up and wave induced currents is verified against test cases present in literature. The results obtained are compared with experimental measures, analytical solutions or alternative numerical solutions. The proposed model is applied to a real case regarding the simulation of wave fields and nearshore currents in the coastal region opposite San Mauro Cilento(Italy).     

AN UNSTRUCTURED FINITE-VOLUME ALGORITHM FOR NONLINEAR TWO-DIMENSIOAL SHALLOW WATER EQUATION

An unstructured finite-volume numerical algorithm was presented for solution of the two-dimensional shallow water equations, based on triangular or arbitrary quadrilateral meshes. The Roe type approximate Riemann solver was used to the system. A second-order TVD scheme with the van Leer limiter was used in the space discretization and a two-step Runge-Kutta approach was used in the time discretization. An upwind, as opposed to a pointwise, treatment of the slope source terms was adopted and the semi-implicit treatment was used for the friction source terms. Verification for two-dimension dam-break problems are carried out by comparing the present results with others and very good agreement is shown.     

基于有限体积法的二维浅水流动数值模拟程序开发

应用近年来浅水动力学理论的研究成果,参考国内外在软件设计方面几十年积累的经验,开发了基于有限体积法的二维浅水流动数值模拟程序CFD-FVM2D.程序以Roe格式的近似Riemann解为基础,离散浅水方程并将源项按特征方向进行特征分解,建立带源项浅水方程的通量平衡的Godunov求解格式,并对摩擦源项采用隐式、半隐式求解格式,以解决浅水方程的间断、动边界和河床起伏问题.     

Study of total variation diminishing (TVD) slope limiters in dam-break flow simulation

A two-dimensional (2D) dam-break flow numerical model was developed based on the finite-volume total variation diminishing (TVD) and monotone upstream-centered scheme for conservation laws (MUSCL)-Hancock scheme, which has second-order accuracy in both time and space. A Harten-Lax-van Leer-contact (HLLC) approximate Riemann solver was used to evaluate fluxes. The TVD MUSCL-Hancock numerical scheme utilizes slope limiters, such as the minmod, double minmod, superbee, van Albada, and van Leer limiters, to prevent spurious oscillations and maintain monotonicity near discontinuities. A comparative study of the impact of various slope limiters on the accuracy of the numerical flow model was conducted with several dam-break examples including wet and dry bed cases. The numerical results of the superbee and double minmod limiters agree better with the theoretical solution and have higher accuracy than other limiters in one-dimensional (1D) space. The ratio of the downstream water depth to the upstream water depth was used to select the proper slope limiter. For the 2D numerical model, the superbee limiter should not be used, owing to significant numerical dispersion.     

基于三角形网格求解二维浅水方程的改进的HLL方法

该文采用HLL格式,在三角形网格下采用有限体积离散建立了求解二维浅水方程的数值模型。基于三角形网格,底坡源项采用简单的斜底模型离散,为保证计算格式的和谐性,对经典的HLL格式计算的数值通量中的静水压力项进行了修正,并证明了修正后的HLL格式具备和谐性。算例证明本文提出的方法是和谐性的并具有良好的间断捕捉能力与稳定性。     

A TVD-WAF-based hybrid finite volume and finite difference scheme for nonlinearly dispersive wave equations

A total variation diminishing-weighted average flux(TVD-WAF)-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer(HLL) Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourthorder monotone upstream-centered scheme for conservation laws(MUSCL). The time marching scheme based on the third-order TVD RungeKutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.     

在浅水方程中利用Riemann问题精确解的数值方法

先从一维Riemann问题解的结构入手,分别对湿区和干区两种情况进行分析,得到精确解的解析形式,并讨论其存在性和唯一性,然后采用物理方程旋转不变量的原理,将全局坐标旋转至界面法相的局部坐标,从而将二维浅水方程中的Riemann问题转化为一维问题进行求解。并通过对三个溃坝案例进行数值模拟,证明了在浅水方程中利用Riemann精确解的可行性,验证了该方法在干/湿区问题上的合理性与有效性,也证明此算法具有精确捕捉溃坝波问题的能力。     

一维浅水方程的高精度GODUNOV格式

以HLLE的近似Riemann解为基础，时间积分采用改进的二步RUNGE—KUTTA法，空间通过重构和通量限制获得时间和空间均为二阶精度的一维浅水方程的Godunov离散格式。此格式具有TVD的性质，保证了数值解收敛于弱解，在TVD意义下无假振，保持初值的单调性，解自动满足熵条件。通过实例比较了各种通量限制因子在求解方程中的优缺点，验证了此方法具有守恒性、鲁棒性、无虚假振荡并能高分辨率地捕捉间断等优点。     

MODELING DAM-BREAK FLOOD OVER NATURAL RIVERS USING DISCONTINUOUS GALERKIN METHOD

A well-balanced numerical model is presented for two-dimensional, depth-averaged, shallow water flows based on the Discontinuous Galerkin (DG) method. The model is applied to simulate dam-break flood in natural rivers with wet/dry bed and complex topography. To eliminate numerical imbalance, the pressure force and bed slope terms are combined in the shallow water flow equations. For partially wet/dry elements, a treatment of the source term that preserves the well-balanced property is presented. A treatment for modeling flow over initially dry bed is presented. Numerical results show that the time step used is related to the dry bed criterion. The intercell numerical flux in the DG method is computed by the Harten-Lax-van Contact (HLLC) approximate Riemann solver. A two-dimensional slope limiting procedure is employed to prevent spurious oscillation. The robustness and accuracy of the model are demonstrated through several test cases, including dam-break flow in a channel with three bumps, laboratory dam-break tests over a triangular bump and an L-shape bend, dam-break flood in the Paute River, and the Malpasset dam-break case. Numerical results show that the model is robust and accurate to simulate dam-break flood over natural rivers with complex geometry and wet/dry beds.     

考虑涌潮作用的钱塘江二维泥沙输移数值模拟

钱塘江涌潮前后水位、流速和含沙量等物理量存在突变,给数值模拟带来了很大的困难,常规的数值模型往往不能准确模拟涌潮作用下的泥沙输移。在已建立的二维涌潮数值模型的基础上,应用基于准确Riemann解的Godunov格式建立了二维泥沙数值模型,模型首先检验了纯对流问题,然后模拟了钱塘江涌潮作用下的泥沙输移,计算结果反映了涌潮到达时刻含沙量的突变过程以及涌潮对泥沙输移的影响,揭示了钱塘江河口高含沙区的成因,以及涌潮是钱塘江河口大冲大淤的机理之一。     

大名泛区洪水演进数值模拟

利用结构化网格的有限体积法,建立了二维浅水方程的高精度、低计算负荷的数值计算模型。对结构化网格的处理采用一种自适应网格加密方法,提出四叉树的网格构造,以提高网格对地形和水力变化的敏感度。推导出一种具有激波捕捉能力的二维浅水方程的守恒形式,采用单元中心式有限体积离散。方程的离散采用Godunov格式,利用近似Riemann解求解界面通量,并介绍和采用了具有二阶精度的HLLC算法。对在此基础上,建立大名泛区二维洪水演进模型,获取了不同时段的淹没范围栅格图像、不同时刻淹没水深、洪泛区的蓄水量变化过程以及泛区内点的水力要素信息,为泛区的防洪规划和实时洪水预报提供理论参考。     

基于和谐性离散格式求解带源项的二维浅水方程

本文采用基于HLL近似Riemann解的Godunov格式,在三角形网格下,建立了二维浅水方程的有限体积数值离散模型.底坡源项采用与静压梯度项统一的矩阵散度处理方式,并证明了格式具备和谐性.通过算例验证了格式的有效性.     

High effective WENO scheme for transonic inviscid flow computation on a helicopter rotor in hover

A high-order upwind scheme with high efficiency has been developed to compute the inviscid flow of a helicopter rotor in hover. For rotary-wing wake capturing, an improved fifth-order weighted essentially non-oscillatory (WENO) scheme is adopted to interpolate higher order left and right states across a cell interface with Roe Riemann solver updating inviscid flux. To improve the efficiency and convergence to steady state, three-level V-cycle multigrid relaxation scheme is used. The performance of the schemes is investigated in a transonic inviscid flow around hovering rotor. The results reveal that WENO has the great capabilities to capture shock with high resolution and has a lower numerical dissipation than MUSCL for capturing wake vorticity. Moreover, multigrid scheme can accelerate convergence to a great extent.

     

应用二维数值方法模拟蓄滞洪区洪水运动

针对蓄滞洪区内洪水运动的特点,对二维浅水方程的守恒形式进行非结构离散,建立了 高精度的数学模型。该模型通过Roe格式的近似Riemann解计算界面数值通量,采用数值重构的 方法获得时空二阶离散精度。通过方形溃坝算例和超临界流算例验证了该模型具有较高的数值精 度和可靠的稳定性,能够捕捉水面间断和处理干滩问题。应用该模型对胖头泡蓄滞洪区的溃堤洪 水进行了数值模拟,结果表明,该模型很好地模拟了溃堤洪水在蓄滞洪区内的运动过程,所得结 果可以为防洪决策提供依据     

Modelling of 2-D extended Boussinesq equations using a hybrid numerical scheme

In this paper, a hybrid finite-difference and finite-volume numerical scheme is developed to solve the 2-D Boussinesq equations. The governing equations are the extended version of Madsen and Sorensen＇s formulations. The governing equations are firstly rearranged into a conservative form. The finite volume method with the HLLC Riemann solver is used to discretize the flux term while the remaining terms are discretized by using the finite difference method. The fourth order MUSCL-TVD scheme is employed to reconstruct the variables at the left and right states of the cell interface. The time marching is performed by using the explicit second-order MUSCL-Hancock scheme with the adaptive time step. The developed model is validated against various experimental measurements for wave propagation, breaking and runup on three dimensional bathymetries.     

Two-dimensional shallow water equations with porosity and their numerical scheme on unstructured grids

In this study, porosity was introduced into two-dimensional shallow water equations to reflect the effects of obstructions, leading to the modification of the expressions for the flux and source terms. An extra porosity source term appears in the momentum equation. The numerical model of the shallow water equations with porosity is presented with the finite volume method on unstructured grids and the modified Roe-type approximate Riemann solver. The source terms of the bed slope and porosity are both decomposed in the characteristic direction so that the numerical scheme can exactly satisfy the conservative property. The present model was tested with a dam break with discontinuous porosity and a flash flood in the Toce River Valley. The results show that the model can simulate the influence of obstructions, and the numerical scheme can maintain the flux balance at the interface with high efficiency and resolution.     

基于三角形网格的浅水间断流动泥沙输移数值模拟

浅水间断流动条件下泥沙输移具有广阔的工程应用背景.采用基于准确Riemann解的Godunov格式求解法向数值通量,应用干底Riemann解计算动边界问题,建立了基于无结构三角形网格空间二阶精度的二维水流和泥沙输移有限体积法数值模型.模型在纯对流问题检验的基础上,模拟了孤立波在滩面上传播变形以及对圆柱周围的局部冲刷,复演了孤立波变形后流场和泥沙场突变的过程,并且圆柱周围局部冲刷坑计算结果与实验基本吻合.     

桥渡对河道水流影响的二维无结构网格模型

采用有限体积法建立平面二维水流运动的无结构网格数学模型,模型采用Roe格式的近似Riemann解计算界面通量和特征分解法处理底坡源项.通过无结构网格对桥墩形状进行精细拟合克服了局部糙率修正、局部地形修正等方法的困难.为研究桥墩、桥形等对水位、局部流速等建桥前后的变化提供了新的计算模式.最后,应用模型对河段上的圆形墩和圆端矩形墩壅水进行了模拟,结果表明模型能够较好的模拟建桥前后桥墩附近的水流变化.