计算流体力学第3章 定常不可压势流的数值解法

For point 4,the characteristics equation and compatibility equations are
r 1 2 a a V 2
2 *2
由于其最高阶导数是线性的，故也称拟线性
It is so called quasi-linear equation because the coefficient of its highest order PD is linear
可改写为标准形式：
rewrite as a general standard expression
dy B B AC ( ) dx A
2
dy B B 2 AC ( ) dx A
x , y 沿特征线的变化规律 物理面特征线确定 The characteristic lines determine the change regulation of the velocity components
2 2 V A 1 2x 1 x2 a a x y VxVy B 2 2 a a 2 2 V y y C 1 2 1 2 a a y Vy D y y
因此特征线的斜率为
The slope of the characteristics line
set of the 1st order ordinary equations.
二、特征线的定义
Definition of characteristic line
超音速流中特征线在物理上相当于马赫线，代表空
间不连续的间断面（线），是流场中的弱间断线。
In physics, the Mach line of supersonic flow corresponds to Mach line on which the flow parameters are not continuous
五. 二维平面和轴对称流的特征线方程和相 容方程
The characteristic line equation and compatibility equations for 2D and axis- symmetry flow
由速度势方程可得知
Using the velocity potential function
由此得相容性方程：
The consult equation
V
2 x
dVy a dVx 2 V V V a dX 0 x y y
2 2 x 2
a Vy
2
讨论：
Discussion
1.利用特征线方法可以把求解速度势的二阶偏微分方程问
A dx 0
0 dx d y
C 0 dy
C 0 dy
(A)
当分母为零时二阶导数将不确定。对应这样的曲线称为特征 线，特征线在（x-y)平面上的投影叫做物理面特征线
When the denominator is zero ,the 2nd order mix derivatives will be undetermined ,the corresponding curves are so called characteristic line, its projection on xoy plane is called physics plane characteristic line .
To solve the supersonic irrotional/rotational 2D/Axisymmetric flow.
适用于双曲型偏微分方程
It is a method for hyperbolic PDE
3-2 特征线法理论
The theory of characteristic line method
整理得
rewritten
dy tg ( ) c dx c dy tg ( ) c dx c
c 和 c 表示特征线的斜率。
c and
the slopes of the characteristic curves denote c
当B2-AC>0有两个解，即代表两条特征线，称为第一特族征
线和第族二特征线。
When B2-AC>0, the characteristic line equation have two roots, which denote the two family of characteristic lines.
一、速度势方程
Equation of velocity potential function
二维轴对称流动的速度势方程
The equation of velocity potential function for 2D and Axis-symmetric flow.
2 y x y y (1 )xx 2 2 (1 2 ) yy 0 a a a y 2 x 2
题化为求解速度分量的一阶常微分方程
The 2nd order PDE was transferred to 1st order ordinary different equations.
2.利用特征线方法求出 Vx ,Vy之后，相当于求解 x , y ，因
此不用直接求解 x, y To solve V and using characteristic line V x y
在此线上速度是连续的，而速度导数可能不连续。
The velocity on this line is continuous ,but the derivatives of the velocity is not continuous.
满足方程速度 the general equation of velocity potential function
xxdx xydy d x xydx yydy d y A 2 B C 0 xx xy yy
由上面的方程组可得
The solution of above linear equation for
xy
xy

A 2B dx dy 0 dx Adydx Ddydx Cdxd y Ady2 2 Bdxdy Cdx2
dy B dx
B AC A
2
特征线方程规定了物理面特征线斜率的变化规律
The characteristic line equation restricts the slope of characteristic lines
只有当B2-AC>0有意义，即只有对超音速流动才有意义。
Only when B2-AC>0, the root will be real number , and the corresponding solution will be meaningful only for supersonic.
四、相容性方程
compatibility equation
在特征线上，（A）式分母为零，要求其数学上有意义
（ xy存在），则必须使分子也为零，即：
On the characteristics line, the denominator is aero， so the numerator must be zero to insure the solution is meaningful
第三章 特征线法
Chapter3 Characteristic Line Method
内容
特征线理论 特征线定义 特征线方程 相容性方程 特征线数值方法
重点
特征线理论 特征线方程
相容性方程 特征线数值方法
3-1 引言
Introduction
求解超音速流动中无旋/有旋二维及轴对称流场
For supersonic flow (V>a), since
v2 a2 2 B AC ( 2 ) (1 2 )(1 2 ) Ma 1 0 2 a a a a
2 2
xy
x2
2 y
可见M>1时是双曲型的
Evidently, when M>1, it is hyperbolic PDE,.
或写为
or
dy tg dx
任意一条特征线与速度矢量的夹角为马赫角


denotes the angle between velocity and the characteristic line
速度矢量与
X 轴的夹角为倾角

The angle between velocity vector and X axis
ຫໍສະໝຸດ Baidu
三、特征线方程
The equation of characteristic line
分母为零： The denominator is zero:
dy 2 dy A( ) 2 B( ) C 0 dx dx
其解为物理面特征方程 Its solution denotes the characteristic equations in physic plane
对二维平面流，取
for 2D flow
0
0
对轴对称流动，取
for Axisymmetric flow
1 1
对于气体
k 1 2 V 或 2 方程是二阶非线性偏微分方程 a 2 a *2
The potential flow equation is 2nd PDE For gas
Axx 2Bxy Cyy D
方程的类型
The type of the equation are
B 2 AC 0 B 2 AC 0 B 2 AC 0
双曲型 Hyperbolic 抛物型 Parabolic 椭圆型 Elliptic
对于超音速流动（V>a），因为：
B 2 AC 0
双曲型方程可以用特征线方法求解
Hyperbolic PDE can be solved using characteristic line method
特征线法的基本思路：将二阶偏微分方程问题转化为
沿特征线方向的一阶常微分方程组问题。
The idea of characteristic line method: To transfer the 2nd order PDE into a
method instead of solving
directly
3-3 特征线数值解法
过1点的右行特征线C 与过2点的左行特征线 C 相交于4
点
Two characteristic curves 1-4 and 2-4 intersect at point 4
对交点4应用相容性方程
Adydx Ddydx Cdxd y 0
A dy D dy 即( ) ( )c ( )c d x C dx C dx B B 2 AC D dy ( )c A C d x
d y
特征线的相容方程规定了沿特征线dVX和dVy必须满足的关
系。
The compatibility equations of characteristic line restricts the difference of Vx and Vy
dy dx c

VxVy a
2

Vx2 Vy2 a
2
1
Vx2 1 2 a
利用速度三角形关系及马赫角定义
Using the velocity triangle relations and the mach angle of supersonic flow
Vx V cos Vy V sin 1 sin Ma