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.     

Effect of channel shape on selection of time marching scheme in the discontinuous Galerkin method for 1-D open channel flow

One-dimensional open channel flows are simulated using the discontinuous Galerkin finite element method. Three different explicit time marching schemes, including multistep/multistage schemes, are evaluated for different channel shapes for accuracy and efficiency. The Forward Euler, second-order Adam-Bashforth(multistep), and second-order total variation diminishing(TVD) Runge-Kutta(multistage) time marching schemes are utilized. The role of monotonized central, minmod, and zero TVD slope limiters for each of the time marching scheme is investigated. The numerical flux is approximated using HLL function. The accuracy and robustness of different time marching schemes are evaluated for steady and unsteady flows using analytical and measured data. The unsteady flows include dam break tests with wet and dry beds downstream of the dam in prismatic(rectangular, trapezoidal, triangular, and parabolic cross-sections) and non-prismatic(natural river) channels. The steady flow test involves simulation of hydraulic jump in a diverging rectangular channel. The various schemes are evaluated by comparing accuracy using statistical measures and efficiency using maximum possible time step size as well as CPU runtime. The second-order Adam-Bashforth time marching scheme is found to have the best accuracy and efficiency among the time stepping schemes tested.     

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。     

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。     

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。     

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.     

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.     

带源项的跨音速粘性流场的数值模拟方法

利用AF方法及采用系数分裂技术的全稳式TVD格式对带源型项的跨音速粘性流场进行了统计，获得了较好的计算效果。TVD格式与AF格式相比，需以较大的计算量为代价，分析了源项对计算收敛的影响，源项隐式或显式处理对收敛速度基本无影响，但是源项的显式处理会带来时间步长的减小，在此基础上，对JPL喷管内的两相流进行了数值模拟。     

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。     

瞬间全溃溃坝波的传播、反射和绕射的数值模拟

将二维浅水方程解耦并进行空间差分分裂，应用通量限制的组合型ＴＶＤ格式，分别求解分裂后的一维双曲型方程组，针对瞬间全溃情况下溃坝波的运动问题，对溃坝波的传播、反射和绕射进行数值模似，揭示出溃坝波复杂的运动特性，为溃坝灾害性分析和防灾减提供科学依据。     

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。     

二维溃坝波传播和绕流特性的高精度数值模拟

本文建立了二维浅水方程的高分辨率差分模型，即将浅水方程作空间差分分裂，分别沿x、y方向应用组合型的通量限制二阶TVD格式，采用Van Leer的MUSCL型限量函数，用以解决二维溃坝的计算问题.首先分别与一维理论解和其它二维结果的比较，表明本文的方法非常成功.然后对全溃和部分溃情况下的溃坝波传播和绕流问题进行了数值研究，给出了精细的数值结果，揭示了其复杂的运动特性.     

PERTURBATIONAL FINITE DIFFERENCE SCHEME OF CONVECTION-DIFFUSION EQUATION

The Perturbational Finite Difference (PFD) method is a kind of high-order-accurate compact difference method, But its idea is different from the normal compact method and the multi-nodes method. This method can get a Perturbational Exact Numerical Solution (PENS) scheme for locally linearlized Convection-Diffusion (CD) equation. The PENS scheme is similar to the Finite Analytical (FA) scheme and Exact Difference Solution (EDS) scheme, which are all exponential schemes, but PENS scheme is simpler and uses only 3, 5 and 7 nodes for 1-, 2- and 3-dimensional problems, respectively. The various approximate schemes of PENS scheme are also called Perturbational-High-order-accurate Difference (PHD) scheme. The PHD schemes can be got by expanding the exponential terms in the PENS scheme into power series of grid Renold number, and they are all upwind schemes and remain the concise structure form of first-order upwind scheme. For 1-dimensional (1-D) CD equation and 2-D incompressible Navier-Stokes equation, their PENS and PHD schemes were constituted in this paper, they all gave highly accurate results for the numerical examples of three 1-D CD equations and an incompressible 2-D flow in a square cavity.     

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。     

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…     

A NEW NONOSCILLATORY CHARACTERISTICS SCHEME FOR HYPERBOLIC CONVECTION EQUATIONS

For the second-order charachteristics schemes of hyperbolic convection e-quations, an analysis of the occurring factors of overshoots and undershoots is made, and the nonoscillatory conditions are found. Either the Lax-Wendroff scheme or the second-order upwind scheme is employed according to the value of the smooth parameter rj+-1/2 of the slope ratio of the solution. Numerical results show that the oscillation can be avoided and the high-order accuracy can be preserved. It is verified by a lot of numerical tests on typical examples of scalar convection equations. Further study is required for its extension to the system of hyperbolic equations.     

A HIGH RESOLUTION FINITE VOLUME METHOD FOR SOLVING SHALLOW WATER EQUATIONS

1 .　ＩＮＴＲＯＤＵＣＴＩＯＮＩｔｉｓｎｅｃｅｓｓａｒｙｔｏｃｏｎｄｕｃｔｆｌｕｉｄｆｌｏｗａｎａｌｙｓｅｓｉｎｍａｎｙａｒｅａｓ ,ｓｕｃｈａｓｉｎｅｎｖｉｒｏｎｍｅｎｔａｌａｎｄｈｙｄｒａｕｌｉｃｅｎｇｉｎｅｅｒｉｎｇ .Ｎｕｍｅｒｉｃａｌｍｅｔｈｏｄｂｅｃｏｍｅｓｇｒａｄｕａｌｌｙｔｈｅｍｏｓｔｉｍｐｏｒｔａｎｔａｐ ｐｒｏａｃｈ .Ｔｈｅｃｏｍｐｕｔａｔｉｏｎｆｏｒｇｅｎｅｒａｌｓｈａｌｌｏｗｗａｔｅｒｆｌｏｗ ｐｒｏｂｌｅｍｓａｒｅｓｕｃｃｅｓｓｆｕｌ,ｂｕｔｔｈｅｓｔｕｄｉｅｓｏｆｃｏｍｐｌｅ…     

NUMERICAL SIMULATION OF TWO-PHASE GRAVITY CURRENTS OVER A POROUS SUBSTRATE

1 .　INTRODUCTIONGravitycurrentsoccurinmanysituationsaris inginbothindustrialandnaturalsettings[1 2 ] .Commonlythecurrentisdrivenbycompositionalortemperaturedifference ,toleadtoahomoge nouscurrent ,orbysuspended particulatematter ,toleadtoaparticle drivencurrent.Combinationsbybothparticleandcompositionalortemperaturedifferencecanalsooccur[3] .Mostoftheresearchesareconcentratedonthesituationswherethecur rentsflowaboveanimpermeableboundary .How evertherearemanyimportantcaseswheregravitycurr…     

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.     

Minimal intervention to simulations of shallow-water equations

A minimal intervention (MI) strategy is employed to manage the unphysical oscillations in the classical finite volume (CFV) numerical solution of the shallow-water equations. Only the missing variables on the face of the finite volume are modified for Total Variation Diminishing (TVD). The MI strategy maintains long-term computational stability without compromising the accuracy of the numerical solution. Flux limiters are implemented by imposing downwind weighting factor constraints on the missing variables on a staggered grid as the shallow-water equations are updated by a fourth-order Runge-Kutta scheme. The shock capturing capability of this MI-CFV method is verified by comparing the simulations with the exact solutions. A series of turbulence simulations for sub-critical and super-critical flows demonstrates the long-term numerical stability of the computation scheme developed using the MI strategy.