定常不可压势流数值计算Numerical

1、定常不可压势流数值计算Numerical在柱坐标系中 In cylindrical coordinate ,it is 计算出 ，即可得到 和 After gained ,the velocity and pressure can be obtained as follow 定常不可压势流数值计算Numerical若流场中有源或者汇，则 If there exist a source or sink in the flow ,then The equation becomes 定常不可压势流数值计算Numerical假设流场中源的分布（平面问题）， Assume: the distributi

2、on of the source and sink is followingthe field within a cycle regime 则PDE为Therefore the PDE is 2.2 由分布的源，汇引起的径向流动计算The computation of the flow introduced by the distributed source and sink .定常不可压势流数值计算Numerical无量纲化参数 choose the dimensionless parameters as follows,无量纲方程 the dimensionless equation be

3、comes 定常不可压势流数值计算Numerical流场区域 Flow field regained is limited as 此方程为二阶线性齐次方程，存在精确解This eq is a two order linear equation ,it has a accurate solution对于 则，精确解为The accurate solution is定常不可压势流数值计算Numerical对应的方程阶为The corresponding PDE is 边界条件 R=1 时 B.C R=4 时速度解The solution of velocity定常不可压势流数值计算Numerica

4、l下面讨论其数值解The numerical value solution will be discussed as following 一般线性二项齐次常微分方程边值问题： In general case , the 2D liner PDE can be written in to express ， using centeral difference scheme, which has 2 order accuracy.定常不可压势流数值计算Numerical将方程中 和 用中心差分格式表示（具有二阶精度）微分方程可化为差分方程：Then ,the PDE can be written

5、into FDE form.x定常不可压势流数值计算NumericalThat is其中：where定常不可压势流数值计算Numerical对于n各节点（i=1,2,3,n）,上式构成一个线性方程组，可写为一个三对角矩阵For node number n , the series equations can be written as a linear equations, also can be express as a triangle array as following.定常不可压势流数值计算Numerical此线性方程组可用追赶法求解，也可用高斯法求解，还可以采用迭代法求解This

6、linear equations can be solved using gauss method ,or Saidel iterative method对于源汇问题：For above distributed source problem可以求出：The potential function can be solved ,and the velocity can be calculated . （i=1,2,3,n）作业：用书上的程序计算出数值解，并与精确解进行比较。Question : using the fortran program provided in the text book

7、(in p13) to get the numerical solution ,and compare the results with the accuracy solution.定常不可压势流数值计算Numerical用点源（汇）分布在对称轴上来模拟流体绕过任意旋转体 Source and sink are used to simulate the flow past a rotational body.轴对称不可压流动流函数方程为： The steam function equation for axis-symmetric flow is :该方程是线性方程，其通解为： It is a

8、 linear PDE , and the general solution is 旋转体绕流的数值解法（源、汇、偶极子）Numerical method for a Rotational Body Flow(source ， sink and doublets)定常不可压势流数值计算Numerical 上式代表点源 It denotes a flow introduced by a source.可以用基本解叠加构成绕旋转体的解。 采用在Z轴上分布的源 ，使这些点源在物体表面各点的扰动速度与自由来流叠加后在 法线方向都为0. To distribute source and sink on

9、the axis of the body and let the normal component of velocity on body surface zero. 各点处线源的流函数为 The stream function of the line source element on 定常不可压势流数值计算Numerical其中 为源的线密度线元 和 上的源流函数 ：stream function on linear source from toP(zi,ri)P(zj, rj)d(zj+1,rj+1)Zrd(zj, rj)定常不可压势流数值计算Numerical把AB分成n段，其总的流函

10、数为：divide AB into n segments and the stream function is : 在旋转体子午线上任意一点 上的流函数为：for a point on the surface the stream function is :定常不可压势流数值计算Numerical 是一条流线上的点，通常称为零流线Since is on the surface ,it is defined as zero streamline 令 ， 表面为零流线定常不可压势流数值计算Numerical 则 then 上式是关于 的线性方程组，同时，要使 的总和为0，即 which denot

11、es a linear equations about ,and the total net volume flax inside the body must be zero, so:总源强为零定常不可压势流数值计算Numerical在物面上其n-1个点 ，构成n-1组线性方程，再加上面的方程即可构成线性方程： n-1 points on surface and above equation construct a closed linear equations as following:定常不可压势流数值计算Numerical求解此方程可得到 ，从而得到流线数解 和 solving this

12、 equation ,the source can be got .定常不可压势流数值计算Numerical利用伯努利方程可得： Using Bernoulli equation , then 定常不可压势流数值计算Numerical2.4 椭圆型偏微分方程的数值解 Numerical solution of the ellipse PPE 二阶偏微分方程的一般形式 General form of 2nd order PPE af xx+bf xy +cfyy=F(x,y,z,f,fx,fy)具有三种可能的类型 It can be three following types (1)椭圆型（方程

13、），当 b2-4ac0 Hyperbola when定常不可压势流数值计算Numerical以泊松方程为例说明椭圆型方程的数值解法To explain the numerical solution using a Possion equation f x x + f y y=q (x , y) 2D Possion equation用i代表x方向节点序号i denote the sequence where in x direction j代表y方向节点序号j denote the sequence where in y direction左边界（1, j）left右边界（m, j）Right

14、内点 （i, j）inner (1im)上边界点（i, n）(i=1,2,m) b2-4ac=0-4*1*10定常不可压势流数值计算Numerical若x= y=h,则上式可解为Give x= y=h,then将Possion 方程写成差分方程 To write the Possion equation into FDEUp boundary下边界点（i, 1）(i=1,2,m)Low boundary内点 （i, j）(i=2,3,m-1,j=2,3, ,n-1)Inner point 定常不可压势流数值计算NumericalLaplace调和函数的平均值定理 Average of the

15、Laplace function此式在给定边界值时构成一个(m-2)*(n-2)阶的代数方程组。可以用多种方法。 When the boundary value is known, it constructs a (m-2)*(n-2)order linear equations直接法direct method矩阵求逆 ArrayLU分解 decompose迭代法 iterative method定常不可压势流数值计算Numerical前三种方法要求的计算内存和计算时间长 The first three methods ned more memory and CPU time 迭代法：对计算机

16、资源要求低（逐点迭代） Iteration method requires less resource of computer每一点都同周围四点的最新值平均和当前点的原项值求解 Value on every points is the average of surroundings定常不可压势流数值计算Numerical可以证明，当 n 时 ， 将接近有限差分方程的解 It is proved that when n , will approach the DE solution.定常不可压势流数值计算Numerical当节点数比较多时，迭代收敛很慢 When the number of node is large convergence will be slowly超松弛迭代： Super-Relaxation iteration 把迭代计算结果作为中间值， The iterativation solution is give as median定常不可压势流数值计算Numerical将 与 进行加权平均得到 The wrighted average of the and 或写成：or定常不可压势流数值计算