Chapter 2 numerical computation of steady incompressible potential flow
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第二章 定常不可压势流的数值计算 Chapter 2 Numerical computation of Steady Incompressible Potential Flow PowerPoint PPT Presentation

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第二章 定常不可压势流的数值计算 Chapter 2 Numerical computation of Steady Incompressible Potential Flow. 2.1 定常不可压势流的基本方程 The Basic Equation of Incompressible Potential Flow 当流场无旋时,存在速度势函数 , 应满足 Laplace 方程

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第二章 定常不可压势流的数值计算 Chapter 2 Numerical computation of Steady Incompressible Potential Flow

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第二章 定常不可压势流的数值计算Chapter 2 Numerical computation of Steady Incompressible Potential Flow

2.1 定常不可压势流的基本方程

The Basic Equation of Incompressible Potential Flow

  • 当流场无旋时,存在速度势函数 , 应满足Laplace方程

    When the flow is irrotational ,there must be exists the potential function ,and it is satisfies the Laplace Equation .

  • 在直角坐标系中可写成

    In Cartesian coordinate ,the Laplace Equation is



  • 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


If there exist a source or sink in the flow ,then

The equation becomes


Assume: the distribution of the source and sink is following

the field within a cycle regime


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 .


choose the dimensionless parameters as follows,


thedimensionless equation becomes

  • 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

  • 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

  • 下面讨论其数值解The numerical value solution will be discussed as following


  • In general case , the 2D liner PDE can be written in to express ,using centeral difference scheme, which has 2 order accuracy.


  • 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 into FDE form.

That is



  • 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.

  • 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 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 (in p13) to get the numerical solution ,and compare the results with the accuracy solution.


Source and sink are used to simulate the flow past a rotational body.


The steam function equation for axis-symmetric flow is :


It is a linear PDE , and the general solution is

2.3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)


It denotes a flow introduced by a source.


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 the axis of the body and let the normal component of velocity on body surface zero.


The stream function of the line source element on

其中 为源的线密度线元 和 上的源流函数 :

stream function on linear source from to



d(zj, rj)


P(zj, rj)



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 :

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 ,



上式是关于 的线性方程组,同时,要使 的总和为0,即which denotes a linear equations about ,and the total net volume flax inside the body must be zero, so:


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:

求解此方程可得到 ,从而得到流线数解 和

solving this equation ,the source can be got .


Using Bernoulli equation , then

§2.4 椭圆型偏微分方程的数值解

Numerical solution of the ellipse PPE


General form of 2nd order PPE

af xx+bf xy +cfyy=F(x,y,z,f,fx,fy)


It can be three following types

(1)椭圆型(方程),当 b2-4ac<0

Ellipse, when

(2)抛物型(方程),当 b2-4ac=0

Parabola ,when

(3)双曲型(方程),当 b2-4ac>0

Hyperbola when


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

  • 内点 (i, j)

  • inner (1<i<m)

  • 上边界点(i, n)(i=1,2…,m)

  • Up 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

  • 将Possion 方程写成差分方程

  • To write the Possion equation into FDE

若△x= △y=h,则上式可解为

Give △x= △y=h,then

  • Laplace调和函数的平均值定理

  • Average of the 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 Array

    • LU分解 decompose

    • 迭代法 iterative method

  • 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

  • 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.

    • 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


    • The iterativation solution is give as median

    • 将 与 进行加权平均得到

    • The wrighted average of the and

    • 或写成:or

    • 把两次迭代得到的差别(利用松弛因子对修正量进行放大或缩小)

    • To apply the difference between the cuidial and calculated value

    • 当 时为弱松弛

    • When it is so called weak relaxation

    • 当 时为超松弛

    • When it is so called weak relaxation

    • 最佳超松弛因子

    • Optical best value of the Relaxation factor

    • 例:不可压平面流通过二维容器(如图)

    • Example 2D incompressible flow in a conduct

    • 差分方程的迭代公式

    • iteration of the equation formula

    • 边界条件:

    • BC

    • 容器的进口的体积流量为AB线上的任意一点与其表面上点的连线上

    • 的体积流量为1








    在 线上 on

    在其余边界上 on other B

    小 结

    • 本章内容(contents)

      • 定常不可压势流差分方法(FDM of elliptic PDE for steady incompressible potential flow)

      • 定常不可压势流源汇法(Source and sink method for axis-symmetric incompressible flow)

      • 椭圆型偏微分方程数值方法(Numerical method for elliptic PDEs)

  • 本章重点(focus)

    • 椭圆型偏微分方程数值方法(Numerical method for elliptic DEs)

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