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.     

模拟水质突跃问题的三种二阶高性能格式

在有限体积法和黎曼近似解理论框架下建立三种二阶TVD格式(Total Variation Diminishing Scheme全变差消减格式)。通过理想条件下方形浓度场在均匀流场中的输移这一算例，分析了在水质间断情况下三种格式的计算精度。与一阶逆风格式的计算结果进行了比较，结果表明，所建的三种二阶TVDS格式的计算精度显著高于一阶逆风格式，与理论值吻合较好，说明所建三种格式适用于水质突跃(包括水质间断)情况的水质预测。     

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。     

浅水流动与污染物输运的高分辨率计算模型

将基于流向量分裂技术的两步有限体积分量方向TVD格式应用于求解浅水方程和污染物输运方程,建立了二者耦合求解的高分辨率计算模型。实际算例表明,该模型能较好地计算浅水流动和污染物输运问题,为浅水流动和水环境模拟提供了精度高、稳定性好的数值计算模型。     

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.     

Numerical study of steady-state acoustic oscillations in semi-closed straight channel

The influence of properties of finite-difference schemes of Godunov, Lax-Wendroff and TVD on prediction of amplitudefrequency responses of pressure oscillations is investigated as a part of a problem of steady-state oscillations in a semi-closed channel. Problems of Riemann and pipe draining are solved as a test. It is shown that the dissipative properties of Godunov scheme underestimate the amplitude of high-frequency oscillations relative to the experimental data. Lax-Wendroff scheme predicts an amplitude-frequency response with sufficient degree of accuracy. TVD scheme leads to nonmonotonic amplitude-frequency response and overestimates values of resonance frequencies.     

基于高阶和高分辨率格式的自然空泡流数值模拟

基于有限体积通量修正,应用了一类高阶迎风对流格式:二阶迎风格式、二次迎风插值格式、三次迎风插值格式及带TVD限制器的高分辨率格式,对于自然空泡湍流流动的数值计算方法进行了研究.计算模型为轴对称圆盘,计算结果表明所有高阶格式都能有效抑制一阶迎风格式产生的假扩散,并得到了更薄,更"透明"的汽-液界面,其中高分辨率TVD格式最好地捕捉到了空泡界面附近物理量的阶跃特性关键词: 自然空泡;高阶迎风对流格式;汽-液界面     

FINITE-VOLUME TVD ALGORITHM FOR DAM-BREAK FLOWS IN OPEN CHANNELS

1 .　INTRODUCTIONThepropagationofborewavesorthemove mentoffreesurfaceflowscausedbydambreak ing ,orsuddenopeningorclosingofasluiceinachannelhasincreasedinterestinenvironmentalandhydraulicengineering .Generallyitcanbede scribedbytheShallowWaterEquations (SWE )throughtheintegrationoftheNavier Stokesequationsoverdepthwiththeassumptionsofhydrostaticpressuredistributionandmildchannel bedslope .ManyclassicaltechniquesthatwereusedforsolvingtheSWEcanbefoundinthepaststudies .However ,whendiscon…     

浅水流动的模拟方法比较

本文研究浅水模拟中几种具有代表性的计算格式,包括不同阶数的简单格式、迎风格式、利用不同通量控制器的TVD格式以及基于分子运动理论的KFVS格式。通过对具有多种不同水流以及间断的一维溃坝的模拟,这些格式的优劣从各种典型数值现象中体现出来。最后用筛选后的格式对一维溃坝物理实验以及二维扩展渠道水流进行模拟。由于KFVS格式符合基本的流体分子运动规律,虽然引入了一定的耗散误差,但此格式无论对一维问题还是二维问题总体模拟效果都较佳。     

A SPLIT-CHARACTERISTIC FINITE ELEMENT MODEL FOR 1-D UNSTEADY FLOWS

An efficient and accurate solution algorithm was proposed for 1-D unsteady flow problems widely existing in hydraulic engineering. Based on the split-characteristic finite element method, the numerical model with the Saint-Venant equations of 1-D unsteady flows was established. The assembled finite element equations were solved with the tri-diagonal matrix algorithm. In the semi-implicit and explicit scheme, the critical time step of the method was dependent on the space step and flow velocity, not on the wave celerity. The method was used to eliminate the restriction due to the wave celerity for the computational analysis of unsteady open-channel flows. The model was verified by the experimental data and theoretical solution and also applied to the simulation of the flow in practical river networks. It shows that the numerical method has high efficiency and accuracy and can be used to simulate 1-D steady flows, and unsteady flows with shock waves or flood waves. Compared with other numerical methods, the algorithm of this method is simpler with higher accuracy, less dissipation, higher computation efficiency and less computer storage.     

用密度函数法对自由表面进行数值模拟

本文从二维非定常N-S方程出发，提出了一种处理自由表面问题的数值方法。引入密度函数法(density-function method)追踪自由表面，在密度函数的传输方程中采用TVD格式达到了二阶精度。对均匀来流中的非定常不可压方柱绕流进行了数值模拟。数值结果成功地捕捉了自由表面波动以及漩涡的生成、分离的时间发展历程。同时验证了本方法的有效性。     

STUDIES ON THE NUMERICAl SIMULATION OF UNSTEADY FLOWS

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Numerical solution of the Saint-Venant equations by an efficient hybrid finite-volume/finite-difference method

A computationally efficient hybrid finite-volume/finite-difference method is proposed for the numerical solution of SaintVenant equations in one-dimensional open channel flows. The method adopts a mass-conservative finite volume discretization for the continuity equation and a semi-implicit finite difference discretization for the dynamic-wave momentum equation. The spatial discretization of the convective flux term in the momentum equation employs an upwind scheme and the water-surface gradient term is discretized using three different schemes. The performance of the numerical method is investigated in terms of efficiency and accuracy using various examples, including steady flow over a bump, dam-break flow over wet and dry downstream channels, wetting and drying in a parabolic bowl, and dam-break floods in laboratory physical models. Numerical solutions from the hybrid method are compared with solutions from a finite volume method along with analytic solutions or experimental measurements. Comparisons demonstrates that the hybrid method is efficient, accurate, and robust in modeling various flow scenarios, including subcritical, supercritical, and transcritical flows. In this method, the QUICK scheme for the surface slope discretization is more accurate and less diffusive than the center difference and the weighted average schemes.     

以波尔兹曼方法模拟潮流

基于BGK波尔兹曼分子运动理论建立的二维明渠非恒定水流新数值模型,较传统的基于圣维南方程的数值模型有很多优点(Xu,1995,1996,Deng,2000),新模型满足熵原理,因而在计算中不会出现非物理性震荡,且模型精度高、运算速度快。然而,分子运动计算不同于自然河流水流计算,主要区别在     

Lattice Boltzmann simulation of the open channel flow connecting two cascaded hydropower stations

This paper examines the feasibility and the efficiency of a multiple-relaxation-time lattice Boltzmann model(MRT-LBM) for simulating open channel flows in engineering practice. A MRT-LBM scheme for 2-D shallow water flows taking into account of the bed slope and the friction is proposed. The scheme's reliability is verified by benchmark problems and the simulation capability is improved by implementing the scheme on a graphic processing unit(GPU). We use the method to analyze the flow characteristics in the connecting open channel of two cascaded hydropower stations. The flow fields and parameters such as the water depth, the flow rate, and the side-weir discharge, under different operating conditions, are analyzed. The factors affecting the accuracy and the efficiency are discussed. The results are found to be reasonable and may be used as a guidance in the project design. It is shown that the GPU-implemented MRT-LBM on a fine mesh can efficiently simulate two-dimensional shallow water flows in engineering practice.     

NUMERICAL SIMULATION OF LEVEE BREACH FLOWS UNDER COMPLEX BOUNDARY CONDITIONS

Levee or dam failure can cause a significant disaster in most cases. A good prediction of the flood process especially in a real complex terrain is necessary for working out emergency plans for levee or dam breaches. Numerical simulations of levee or dam breach flow were carried out often with constant flow parameters and in relatively simple channels rather than in natural rivers with complex boundaries. This article presents our dedicated studies on the 2-D numerical model of levee or dam breach hydraulics with finite difference schemes. The good performance of the model is demonstrated by comparisons with the theoretical solution of an idealized dam-break flow over a frictionless flat rectangular channel. The model is also validated through its stability and conservation properties. The model is applied to simulate the flood propagation under complex boundary conditions, and the unsteady flood process in a river and in the dry floodplain with a complex bed terrain simultaneously. Furthermore, with respect to engineering practice, the numerical solutions can give special guidance to the effects of parameters such as the flood depth at different sites and the inundated area at different time periods after the levee breach and the travel time of the flood waves, which may be very important for practicing engineers in an efficient flood management.     

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.     

NUMERICAL SIMULATION OF TURBULENT SPOTS IN INCLINED OPEN-CHANNEL FLOW

The generation and evolution of turbulent spots in the open-channel flow are simulated numerically by using the Navier-Stokes equations. An effective numerical method with high accuracy and high resolution is developed.The fourth order time splitting methods with high accuracy is proposed. Thre-dimensional coupling difference methods are presented for the spatial discretization of the Poisson equation of pressure and Hemholtz equations of velocity, therefore, the fourth-order thre-dimensional coupling central difference schemes are constituted. The fourth-order explicit upwind-biased compact difference schemes are designed to overcome the difficulty for the general higher-order central differences cheme which is inadaptable in the boundary neighborhood.The iterative algorithm and overall time marching is used to enhance efficiency. The method is applied in the numerical simulation of turbulent spots at various complex boundary conditions and flow domains. The generation and the developing process of turbulent spots are given, and the basic char acteristics of turbulent spots are shown by simulating the evolution of the wall pulse in inclined open-channel flow.     

A HIGH RESOLUTION FINITE VOLUME METHOD FOR SOLVING SHALLOW WATER EQUATIONS

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渠网非恒定流图论原理

基于明渠非恒定流普里斯曼的隐式差分方法,采用矩阵线性变换建立了渠道进出口流量与水深的线性方程。提出了新的渠网非恒定流图论原理,包括适应渠网非恒定流特征的节支关联矩阵A的构建方法及性质分析,基于A的节点连续性矩阵方程和节点水深矩阵方程等。应用矩阵分析方法,推导出渠网非恒定流的节点分析法数学模型,最后给出应用实例。