1、12.4 常用的离散格式 在使用有限体积法建立离散方程时，很重要的一步是将控制体积界面上的物理量及其导数通过节点物理量插值求出。引入插值方式的目的就是为了建立离散方程，不同的插值方式对应于不同的离散结果。插值方式常称为离散格式。2离散格式n假设速度场已知，则为求解离散方程，需计算广义未知量在边界e和w处的值。n为完成这一任务，必须决定界面物理量如何通过节点物理量的插值表示。n各种不同的插值方法就构成了不同的离散格式。n中心差分格式n一阶迎风格式n混合格式n指数格式n乘方格式32.4.1术语的约定n对离散格式的讨论以一维稳态对流扩散方程为例，不涉及瞬态项。n定义两个新的物理量：nF：通过界面上单
2、位面积的对流质量通量，简称对流质量流量；nD：界面的扩散传导性n在此基础上定义一个非常重要的参数Peclet数对流与扩散的强度之比xuDFPe/4Peclet数(佩克莱数)n1976年Roache提出，网格或单元Peclet数可以用来度量某点处的对流和扩散的强度比例。n从物理意义来说，Peclet数表示的是对流与扩散强度之比。而Re雷诺数表示作用于流体微团的惯性力与粘性力之比。Pr数是动量扩散厚度与热量扩散厚度之比。(粘性与导热性之比）n佩克莱特数(Peclet)是普朗特数(Prandtl)与雷诺数(Reynolds)的乘积，实际上反映了惯性力与紊动强度的比值。5PeE(x)PeECentra
3、l differencing scheme中心差分格式nWe determine the value of at the face by linear interpolation between the cell centered values.就是界面上的物理量采用线性插值公式来计算。n具有二阶精度，扩散占优时计算具有较高精度。但流动为强对流时，计算的收敛性和精度较差。interpolated value6中心差分格式的特点及适用性nPe小于2时，中心差分格式的计算结果与精确解基本吻合。nPe大于2时，数值解完全失去物理意义。n原因：离散方程系数出现负值，负的系数会导致物理上不真实的解。n解
4、决办法：将计算网格加密，使网格间距变小，降低Pe数的大小。（计算量随之增大）n基于此限制，中心差分格式不能作为对于一般流动问题的离散格式，需创建其它更合适的格式（对纯扩散稳态，如热传导是适用的）。7对流扩散方程的精确解8精确解随Pe数的变化（Pe0纯扩散，Pe增大对流增强）9具体算例（不同计算工况意味着不同Pe数）10第一种工况Pe=0.2n尽管网格粗糙，但数值解与精确解非常接近。11第二种工况Pe=5n数值解在精确解周围振荡，计算精度不可接受。n离散方程中系数有负值。12第三种工况（加密网格Pe=1.25）13First order upwind scheme界面上的未知量恒取上游节点的值P
5、eE(x)PeEFlow directionnThis is the simplest numerical scheme.It is the method that we used earlier in the discretization example.nWe assume that the value of at the face is the same as the cell centered value in the cell upstream of the face.nThe main advantages are that it is easy to implement and
6、that it results in very stable calculations,but it also very diffusive.Gradients in the flow field tend to be smeared out,as we will show later.nThis is often the best scheme to start calculations with.interpolated value在确定界面物理量时考虑了流动方向14一阶迎风格式的数学描述及特点n考虑了流动方向影响n具有一阶截差nunconditionally stable(no osci
7、llations)（任何条件下都不会引起解的振荡，离散方程系数总是大于零）n不足：夸大了扩散项的影响，numerically diffusive15PwwEeeWwwPeeuuuu0for 0for 0for 0for diffusionnet convectionnet wewedxddxduu1617181920混合格式nCentral differencing scheme is more accurate than the first order upwind scheme,but it leads to oscillations in the solution or divergen
8、ce if the local Peclet number is larger than 2.nIt is common to then switch to first order upwind in cells where Pe2.Such an approach is called a hybrid scheme.n根据流体的Pe数在中心差分格式和迎风格式之间进行切换，综合两种格式的共同优点。n混合格式在CFD软件中广为采纳，是非常实用的格式，缺点是只有一阶精度。21指数格式n根据一维、稳态、无源的对流扩散方程的精确解建立的一种离散格式。n指数格式保证对任何佩克赖特数以及任何数量的网格点均
9、可以得到精确解n缺点：n指数运算是费时的n对于二维或三维问题，以及源项不为零的情况，这种离散方案是不准确的。22nThis is based on the analytical solution of the one-dimensional convection-diffusion equation.nThe face value is determined from an exponential profile through the cell values.The exponential profile is approximated by the following power law
10、equation:nPe is again the Peclet number.nFor Pe10,diffusion is ignored and first order upwind is used.Power law scheme乘方格式PEPePePe51.01PeE(x)PeExLinterpolated value参考精确解所做的近似，与指数格式的精度比较相近但比指数格式要省时与混合格式具有类似性质，可用作混合格式的替代格式。23假扩散假扩散在文献中，假扩散，人工粘性，数值粘性常常是作为同义词使用的。n假扩散是对流项离散过程中所引入的一个重要误差。由于对流扩散方程中一阶导数项的离散
11、格式的截断误差小于二阶而引起较大数值计算误差的现象称为假扩散。在一个离散格式中，假扩散的存在会使数值解的结果偏离真解的程度加剧。假扩散系数越大，偏离得越严重。以上所指的假扩散是由于截断误差而引起的，也是假扩散这一概念的最初含义。现在，在计算传热学领域中，这一概念的含义有所扩大。能引起较大的数值计算误差的原因有三个：1.非稳态项或对流项采用一阶截差的格式；2.流动方向与网格线呈倾斜交叉（多维问题）；3.建立差分格式时没有考虑到非常数的源项影响。现在一般把由这三种原因而引起的数值计算误差都归在假扩散的名称下。24低阶格式的假扩散特性低阶格式的假扩散特性n迎风格式,指数格式,混合格式及乘方格式等一阶
12、格式应用于实际问题时都可能引起较严重的假扩散,这在HVAC领域的高大空间流体流动及传热计算中尤为明显.因此,为了有效地克服或减轻假扩散所带来的计算误差,空间导数应当采用二阶或更高阶的格式(如QUICK格式,二阶迎风差分格式等).25Accuracy of numerical schemesnEach of the previously discussed numerical schemes assumes some shape of the function.These functions can be approximated by Taylor series polynomials:nTh
13、e first order upwind scheme only uses the constant and ignores the first derivative and consecutive terms.This scheme is therefore considered first order accurate.nFor high Peclet numbers the power law scheme reduces to the first order upwind scheme,so it is also considered first order accurate.nThe
14、 central differencing scheme and second order upwind scheme do include the first order derivative,but ignore the second order derivative.These schemes are therefore considered second order accurate.QUICK does take the second order derivative into account,but ignores the third order derivative.This i
15、s then considered third order accurate.)(!)(.)(!2)()(!1)()()(2nPePnPePPePPexxnxxxxxxxxx26Hot fluidCold fluidT=100CT=0CDiffusion set to zerok=0Accuracy and false diffusion(1)nFalse diffusion is numerically introduced diffusion and arises in convection dominated flows,i.e.high Pe number flows.nConside
16、r the problem below.If there is no false diffusion,the temperature will be exactly 100 C everywhere above the diagonal and exactly 0 C everywhere below the diagonal.nFalse diffusion will occur due to the oblique flow direction and non-zero gradient of temperature in the direction normal to the flow.
17、nGrid refinement coupled with a higher-order interpolation scheme will minimize the false diffusion as shown on the next slide.假设热传导系数为零，冷热水之间应该没有混合现象。研究不同格式和不同网格密度对计算结果的影响278 x 864 x 64First-order UpwindSecond-order UpwindAccuracy and false diffusion(2)28Solution accuracynHigher order schemes will
18、be more accurate.They will also be less stable and will increase computational time.nIt is recommended to always start calculations with first order upwind and after 100 iterations or so to switch over to second order upwind.nThis provides a good combination of stability and accuracy.nThe central di
19、fferencing scheme should only be used for transient calculations involving the large eddy simulation(LES)turbulence models in combination with grids that are fine enough that the Peclet number is always less than one.nIt is recommended to only use the power law or QUICK schemes if it is known that those are somehow especially suitable for the particular problem being studied.
