THE COMPACT SECOND-ORDER UPWIND FINITE DIFFERENCE SCHEMES FOR THE CONVECTION-DIFFUSION EQUATIONS

The compact second-order upwind finite difference schemes free of ceil Reynoldsnumber limitation are developed in this paper for the one-to three-dimensional steady convection-diffusion equations,using a perturbational technique applied to the classical first-order upwindschemes.The present second-order schemes take essentially the same form as those of the first-order schemes,but involve a simple modification to the diffusive coefficients.Numerical exam-ples including one-to three-dimensional model equations of fluid flow and a problem of naturalconvection with boundary-layer effect are given to illustrate the excellent behavior of the presentschemes.     

A NEW FINITE DIFFERENCE SCHEME FOR 2-D FLOWS AT LARGE Re

Based on the conventional ADI method and SOR method for solving the Navier-Stokes equations, averaging those linkage terms which also are the terms of vary-ing coefficients of the equations, a new finite difference scheme-Averaging Finite Diffe-rence (AFD) scheme for 2-D flows was proposed. A 2-D driven cavity flow was calculated numerically as an example with the presented scheme at Re= 100, 1000, 2000, 3200, 10000. The results were discussed and compared to those obtained with the conventional methods as well as experimental data. It showed that a slight change of the approximation pattern of the conventional scheme in the terms of varying-coefficients of the governing equations seems to have an obvious influence on the solutions at high Re which will be erro- neous if the conventional schemes was employed.     

对流扩散方程的变步长摄动有限差分格式

摄动有限差分(PFD)方法是构造高精度差分格式的一种新方法。变步长摄动有限差分方法是等步长摄动有限差分方法的发展和推广。对需要局部加密网格的计算问题，变步长PFD格式不需要对自变量进行数学变换，且和等步长PFD格式一样，具有如下的共同特点：从变步长一阶迎风格式出发，通过把非微商项(对流系数和源项)作变步长摄动展开，展开幂级数系数通过消去摄动格式修正微分方程的截断误差项求出，由此获得高精度变步长PFD格式。该格式在一、二和三维情况下分别仅使用三、五和七个基点，且具有迎风性。文中利用变步长PFD格式对对流扩散反应模型方程，变系数方程及Burgers方程等进行了数值模拟，并与一阶迎风和二阶中心格式及其问题的精确解作了比较。数值试验表明，与一阶迎风和二阶中心格式相比，变步长PFD格式具有精度高，稳定性与收敛性好的特点。变步长PFD格式与等步长PFD格式相比，变步长PFD解在薄边界层型区域的分辨率得到了明显的提高。     

高雷诺数圆柱绕流的数值模拟

本文试图通过不同精度的差分格式在较高Re数下的计算结果的对照比较,以期了解在高Re数下不同格式的优劣,达到合理选择计算格式,使其达到既计算精确,又相对节省时间的目的。     

一种求解不可压粘性流动的高阶精度迎风格式的应用

本文应用人工可压缩性方法和高阶迎风差分格式相结合的方法，进行不可压缩粘性流动的数值计算，通过对典型算例的研究，表明高阶格式可以在网络较少的情形下获得较高的精度，且有较好的网格收敛性。     

A STUDY ON NUMERICAL METHOD OF NAVIER-STOKES EQUATION AND NON-LINEAR EVOLUTION OF THE COHERENT STRUCTURES IN A LAMINAR BOUNDARY LAYER

1. INTRODUCTION A high accuracy compact finite difference scheme has been extensively applied to direct simulation of turbulence coherent structure’s complicated and remote physical phenomena [1-5]. The Fourier spectralexpansion was previously commonly …     

长腔体内混合流体行进波对流的高精度数值模拟

该文构造了含源项的对流-扩散方程的高精度紧致格式,并应用于求解描述混合流体行进波对流系统的控制方程.其中,对流项采用四阶精度的紧致格式离散,扩散项用四阶对称格式离散.首先对长高比为46腔体内的混合流体行进波对流系统进行验证性计算,得到了瞬态的对传波(counterpropagating wave)以及两种类型的调制行进波(modulated traveling wave).验证结果与其他前人的数值模拟结果非常吻合.该文计算得到了延伸行进波(extend traveling wave),在此基础上通过减小瑞利数,分析了局部行进波的稳定性对Rayleigh数r的关系,并确定了局部行进波的稳定范围为r=1.12～1.22.     

HIGH ACCURATE SOLUTION OF INCOMPRESSIBLE VISCOUS FLOW WITH PRIMATIVE VARIABLE

This paper presents a higher order difference scheme for the computationof the incompressible viscous flows.The discretization of the two-dimensional incompress-ible viscous Navier-Stokes equations,in generalized curvilinear coordinates and tensor for-mulation,is based on a non-ataggered grid.The momentum equations are integrated intime using the four-stage explicit Runge-Kutta algorithm [1]and discretized in space us-ing the fourth-order accurate compact scheme[2]The pressure-Poisson equation is dis-cretized using the nine-point compact scheme.In order to satisfy the continuity constraintand ensure the smoothness of pressure field,an optimum procedure to derive a discretepressure equation is proposed [9][3]The method is applied to calculate the driven cavityflow on a stretched grid with the Reynolds numbers from 100 to 10000.The numerical re-sults are in very good agreement with the results obtained by Ghia et al [7]and includethe periodic solutions for high Reynolds numbers.     

热分层流动计算中若干差分格式的比较

本文应用常规的守恒方程及湍流模型，对二维池内热分层流动建立了数学模型，并在Ｐａｔａｎｋａｒ的压力校正法基础上对该数学模型进行了数值计算，计算中的差分格式为乘方格式、一阶迎风、二阶迎风和ＱＵＩＣＫ。通过与实验的比较分析，建议计算应采用ＱＵＩＣＫ格式，并在进行风格独立性试验的基础上使用加密网络。     

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.     

水沙河口衔接数学模型的研究及其应用

在河口水流、泥沙数值计算时提出一、二维衔接的数学模型.根据一、二维水流连续方程、运动方程、悬沙扩散方程、两相流底沙运动方程、床面演变方程,对一维河道采用河网三级联解法、迎风差分格式,对二维海区采用有限元加权集中质量法、三角差分格式,建立水沙河口衔接数学模型.将此模型应用于天津泰达永定新河河口工程的研究上,经实测水沙资料验证,计算值与实测值基本吻合.为河口改造时引起的河道与海区的水流及泥沙稳定性问题提供了可行的研究方法.     

求解泊松方程的紧致修正法

将紧致格式与低阶格式结合,构造紧致格式的修正项,并将修正项加入到源项中进行求解,得到了一种基于非均分网格求解泊松方程的紧致修正法,且将该方法应用于二维和三维泊松方程的数值求解中。数值计算结果表明:紧致修正方法的精度高于经典方法的精度,但四阶紧致修正方法比二阶经典方法对网格的依赖性强。     

A NEW HIGH ORDER SCHEME FOR CONVECTION EQUATION AND ITS APPLICATION

1. INTRODUCTIONThe choice of computational scheme for the convection term is the most difficult problem in numerical simulation of flow. Due to the development of modern numerical methods and practical needs for more efficient simulation, higher order sch…     

二维非定常不可压涡量-速度Navier-Stokes方程组的高精度紧致差分格式

提出了数值求解二维非定常不可压涡量-速度变量Navier-Stokes方程组的一种高精度全隐式紧致差分格式,其空间为四阶精度,时间为二阶精度,并且是无条件稳定的。为了验证本文方法的精确性和可靠性,进行了数值实验,数值实验结果与精确解或文献中的结果吻合得很好。     

二维溃坝波遇障碍物的水流泥沙数值模拟

本文通过对浅水方程进行Godunov差分离散和对泥沙运动方程进行一阶迎风差分离散，建立了平面l二维溃坝水流泥沙数值模型。在算例中计算了溃坝波遇障碍物后的水流反射、绕射及泥沙冲淤，计算结果表明，溃坝波遇障碍物后会产生明显的反射和绕射，并在障碍物前形成壅水、障碍物的侧面会发生剧烈的冲刷，障碍物前后则会发生局部淤积。本模型能够较好地模拟溃坝间断水流的运动，结果符合水流泥沙运动的基本规律。     

通量差分分裂上风差分格式求解船舶粘性流动

本文阐述了采用人工可压缩方法计算不可压缩粘性流体的流动问题。对一般曲线坐标系下的流动控制方程采用了通量差分分裂上风差分格式和二阶中心差分格式离散。在隐式求解格式下，代数方程组用线松弛方法求解。本文计算了Ｍ维圆柱绕流和Ｗｉｇｌｅｙ船的绕流问题，计算结果同试验相当吻合，这说明本文所采用的方法是处理不可压缩流动的可靠方法。     

二维不可压缩平板边界层的紧致有限差分法

针对二维不可压缩平板边界层N-S扰动方程,采用高精度紧致有限差分格式,应用空间模式,直接数值模拟了二维不稳定T-S波传播的过程。该算法的时间离散采用三阶精度混合显隐分裂格式,空间离散采用高精度紧致有限差分逼近。法向采用非等距网格方法,出口边界条件采用嵌边函数法。实例验证,该方法计算结果与流动稳定性分析的结果一致。     

非定常不可压绕流的有限元-有限差分计算

本文用一种有限元-有限差分混合法解不可压非定常Navier-Stokes方程来模拟非定常绕流。在同一时间步的不同计算步中,用三阶迎风差分格式离散动量方程中的对流项,压力修正方程用有限元离散,离散后的方程用直接法求解。本文给出了雷诺数(Re)为1200时的圆柱绕流结果。模拟了突然起动过程中对称主涡和二次涡的发展过程,以及柱后涡街的发展过程中主涡和二次涡的形成、合并、脱落过程。     

在正常网格下解Navier—Stokes方程的有限分析法

以往在用有限分析法来解Navier-Stokes方程时,速度从动量方程解出,采用有限分析法,压力从连续方程解出,实际上采用了差分法。因此不得不使用交叉网格处理。这使收敛速度较慢,并降低了计算精度。本文提出一种新的处理连续方程的方式,使压力可由有限分析法解出,并且采用正常网格。计算实践表明,这种处理方式提高了计算精度,加快了收敛速度。     

高雷诺数流动的平均化差分格式

本文在N—S方程涡量一流函数形式的经典ADI格式和SOR松驰迭代公式的基础上,对其“外来项”进行平均化处理,构造出了平均化差分格式及迭代求解程序。对雷诺数Re=100,1000,10000的空腔流动分别进行了数值计算,并与各数值方法的结果及实验资料进行了比较,结果令人满意。