用TVD显隐格式模拟一维溃坝洪水波的演进与反射

应用空气动力学中的TVD格式构造了用于溃坝水流计算的显隐格式，模拟瞬间全溃时洪水波的运动，给出了与分析解的比较及多种数值试验，结果表明这类格式精度高，稳定性好，不产生虚假的数值振荡，对溃坝涌波具有较强的捕捉能力。     

具有复杂计算域和地形的二维浅水流动数值模拟

本文采用非结构化网格的有限体积方法，对具有复杂计算域和地形的二维河道进行数值模拟。采用Roe格式的近似Riemann解计算界面通量，对地形变化源项采用特征分解平衡界面通量以保证格式的和谐性，对摩擦力源项采用隐式或半隐式求解以增加格式的稳定性。通过与混合流算例理论解比较，验证了格式是和谐性的并具有良好的间断捕捉能力与稳定性。应用此方法对地形复杂，存在多处险滩和深潭的某河段进行了实际模拟，数值计算结果和模型试验结果吻合较好。     

阀门关闭速度对TVD格式数值模拟水击影响分析

采用总变差减小（TVD）格式模拟了一维有压管道水击过程,对管道中阀门突然关闭和缓慢关闭引起的水击压强进行了数值计算。通过改变阀门关闭速度,分析TVD格式对水击数值模拟精度的影响。结果表明,与传统经典差分二阶MC格式相比,阀门关闭速度越快,TVD格式捕捉激波间断的能力越强,模拟水击波效果越好;当阀门关闭速度缓慢到一定程度,TVD格式与MC计算方法所得结果基本吻合,与解析解最大误差不超过2%。说明TVD格式是数值模拟水击现象行之有效的方法之一。     

一维水击波的高精度数值模拟

用本质无振荡ENO、TVD格式以及MacCormark和Lax-Friedrichs格式对一维水击模型进行数值模拟。结果表明,Lax-Friedrichs和二阶MacCormark格式在间断处有较大的数值耗散,计算精度较低,后者还存在明显的虚假数值振荡;TVD格式在光滑区具有高阶精度,在间断附近能在2个计算网格步长的范围内被光滑化,数值耗散较小;ENO格式数值解在间断附近无虚假数值振荡,数值解耗散很小,对间断具有较强的分辨能力和捕捉能力,能够很好地模拟水击波的变化过程。另外,ENO格式对计算网格的要求较TVD格式低,数值计算性能稳定。     

用TVD格式预测溃坝洪水波的演进

溃坝洪水波通常为迅变或间断流动,同时兼有急流和缓流、压缩波和稀疏波等流态。常用的求解圣维南方程组的差分方法常遭失败,或不稳定,或数值解含有振动,或间断被抹平而分辨率低。本文利用气体动力学中新近提出的TVD格式求解获得成功。 文中介绍了一阶及二阶、显及隐、对称及逆风TVD格式的有关算法。以简单的一阶TVD显格式为对象,结合洪水演进问题,详细分析了其内在机制及优良性能,说明它既具有逆风格式及中心格式的性能,又能及时对水位或流量过程的峰谷作出响应,保持解在间断处陡峭而在其两侧无振动。文中最后通过一个概化为黎曼问题的瞬时溃坝模型,以及一座高坝的假想迅溃实例,将该格式与LF格式、LW格式的计算结果作比较,充分显示了该格式在预测一般洪水及溃坝洪水波演进方面的前景。     

基于自适应网格的二维水动力模型

为准确地模拟流域洪水传播和淹没情况,基于自适应网格技术,以有限体积法为离散格式构建数值模型.模型以二维浅水方程为控制方程,采用HLL格式近似黎曼解计算界面通量,采用WAF格式对计算的通量进行加权修正,保证时间和空间上的二阶精度,并使其满足TVD条件,减小数值振荡.底坡源项采用水深差分重构,摩阻源项采用半隐式格式,保证数...     

水击波数值模拟中高分辨率TVD格式优选

TVD方法特别适合数值模拟双曲型偏微分方程间断波问题,选用4种典型的TVD格式应用于水击方程的数值模拟。通过水击实例计算,将数值结果与理论解比较,揭示4种不同构造形式的TVD格式在捕捉水击波方面耗散性和压制性数值性能的差异,且TVD格式均优于常规的差分方法;改变计算库朗数条件下的计算结果表明:Harten修正通量TVD格式、TVD-MC格式以及全离散TVD格式在大范围库朗数条件下计算稳定性较差,优选Sweby反扩散TVD格式模拟水击,其计算稳定性良好,精度较高。     

长江口北支涌潮的一维数值模拟

本文利用二阶TVD格式对长江口北支的涌潮现象进行了一维的数值模拟,成功捕捉了涌潮现象。高阶TVD格式原用于计算气动力学中激波的自动捕捉,具有高分辨率、高精度、不产生假振而且适应性强等性能。本文涌潮现象的计算结果与实际观测结果基本相符,说明了这类格式也适用于求解水动力学中的间断问题。     

非结构化网格下求解二维浅水方程的和谐Roe格式

为保证底坡源项和重力梯度项的平衡离散,采用把底坡源项分解为两不同部分单独处理的方法,在非结构化网格上建立了求解带复杂地形的二维浅水方程数值模型.采用Roe格式的计算界面通量,隐式求解摩擦力源项以增加格式的稳定性,并给出计算格式在非结构化网格上满足和谐性条件的证明.通过实例验证了此格式是和谐的,并具有良好的间断捕捉能力和稳定性.     

污染物对流扩散方程的几种新的高阶QUICK组合显格式比较研究

本文运用组合式的有限差分QUICK格式,将对流扩散方程进行了高精度离散,通过对流项、时间项、扩散项几种高阶差分格式的优化组合,最终建立了一种时间三阶、对流三阶、扩散二阶的显式差分格式,通过经典的数值算例验证了本格式具有精度高、编程简单、计算速度快的特点。本文还详细介绍了由有限体积法建立的经典QUICK格式和通过有限差分法建立的QUICK格式的区别以及各自的精度,澄清了某些文章作者对QUICK格式的认识偏差。     

Time stepping in discontinuous Galerkin method

The time discretization in the Discontinuous Galerkin (DG) scheme has been traditionally based on the Total Variation Diminishing (TVD) second-order Runge-Kutta (RK2) scheme. Computational efficiency and accuracy with the Euler Forward (EF) and the TVD second-order RK2 time stepping schemes in the DG method are investigated in this work. Numerical tests are conducted with the scalar Burgers equation, 1-D and 2-D shallow water flow equations. The maximum Courant number or time step size required for stability for the EF scheme and RK2 scheme with different slope limiters are compared. Numerical results show that the slope limiters affect the stability requirement in the DG method. The RK2 scheme is generally more diffusive than the EF scheme, and the RK2 scheme allows larger time step sizes. The EF scheme is found to be more efficient and accurate than the RK2 scheme in the DG method in computation.     

A LEVEL SET METHOD FOR SIMULATION OF RISING BUBBLE

A level set method, the TVD scheme of second-order upwind procedure coupled with flux-limiter, and SIMPLE algorithm were incorporated to simulate the flow and interracial motion of immiscible two-fluids with large density ratio and viscosity ratios, large topology distortion and surface tension. As a numerical example axi-symmetric rising bubbles were investigated. It is found that the method is numerically stable and has good convergence property and the results are in good agreement with other works.     

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.     

A New Total Variation Diminishing Predictor Corrector Approach for Two-Dimensional Shallow Water Flow

This paper presents a new efficient and robust hydrodynamic model for simulation of unsteady shallow water flow. The governing equations of shallow water flows in two dimensional forms are solved using a new total variation diminishing (TVD) MacCormack predictor corrector scheme. In this numerical technique an additional TVD term is added after the traditional predictor corrector steps. The advantage of the present TVD term is that it is very simple and gives accurate results at the same time removing the numerical oscillations. Further, application of semi implicit treatment of the friction slope term helps in flow simulation even with very low water depth. Finally the model is used to analyze a wide variety of hydraulic problems including quiescent water above irregular bed, steady flow over irregular bed, steady flow over irregular bed with a shock, dam break flow over dry bed and dam break flow over wet bed. For each of the cases numerical results are compared with available analytical solution and known experimental data. The agreements between the results are satisfactory.     

曲线坐标网格下二维涌波数值模拟的TVD型格式

针对水动力学中的涌波现象导出二维曲线坐标系下的守恒型方程，然后利用算子分裂法和TVD差分格式思想构造出计算格式，在此基础上建立起由实际流场区域至方形计算区域的全局同胚映射，对急流过缩窄河道产生的涌波结构进行数值模拟，结果表明，本文的数值方法能够很好地模拟附体涌波、交叉涌波等复杂结构现象，所得波系结构和有关物理量与理论分析结果非常吻合，表明本文的数值方法对于模拟水动力学中的涌波现象是可行的。     

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.