Transcript Document

第四章 跨音速定常小扰动势流混合差分
方法及隐式近似因式分解法
chapter 4 The Mixed Finite Difference Method(FDM) for Velocity Potential
Function of Steady Small Perturbation and Implicit Approximate Factor
Decomposition Methods
 主要内容：
main contents
 混合差分解法
Mixed PD Method
 小扰动方程及小扰动激波差分式
perturbation relationship for shock flow
Small perturbation equation and small
 小扰动速势差分方程 The finite differential equation of small perturbation potential
function
 边界条件及边界条件的嵌入The initial condition and boundary
condition
 线松弛迭代解法Linear relaxation iteration method
 升力翼型的跨音速小扰动势流差分方法FD method of
velocity potential function for small perturbation
 隐式近似因子分解法Approximate factor decomposition method
 AF1方法
AF1 method
 AF2方法
AF2 method
 方法比较 Comparison of the method
重点： Focus
 混合差分方法Mixed FD Method
难点： Difficulty
 隐式近似因子分解法Implicit Approximate factory
decomposition
第四章 跨音速定常小扰动势流混合差分方法及隐式
近似因式分解法
chapter 4 The Mixed Finite Difference Method for Velocity
Potential Function of Steady Small Perturbation and Implicit
Approximate Factor Decomposition Methods
跨音速流：局部超音区与亚音速同时存在的流场
Transonic flow ：Local supersonic flow and supersonic flow exists meantime
 偏微分方程：混合型方程
The PDE：Mixed type equation

 混合差分方法：用不同的差分方程求解跨声速流场
Mixed Finite difference method is to solve transonic flow with different FDMs
 混合型方程及流场：采用迭代方法求解，求解之前不知道方程的类型
Mixed Equation and flow field, the iterative method is used because the type of the
equation is unknown before it was solved
 小扰动方程：小马赫（0.6~1.4）流过薄而微变的叶片（机翼或叶栅）时
全速势方程可简化为小扰动方程
Small perturbation equation(SPE): when mach number is small (ie 0.6~1.4)the full
velocity potential equation can be simplified to SPE
 混合差分:用混合差分格式求解小扰动方程
Mixed FDM :To solve equation using MFDM
 混合差分和松弛迭代法求解全速势方程
Mixed FDM and Relaxation iteration : To solve full velocity potential equation.
 优缺点：
Advantage/disadvantage
跨音速松弛法---速度快，有效 Transonic relaxation method faster efficient
时间推进法：适用范围广 Time matching methods, widely usage
近似因子分解法：快速 Approximate factor decomposition：faster
多层网格法：收敛性好 Multi-grid technique：good convergence
4.1 跨声速小扰动速度势方程 Equation of transonic small perturbation
velocity potential function
跨声速气流绕过薄翼的情况 For the case of transonic flow pass a thin
airfoil
 二维平面速势方程 2D velocity potential equation
(a 2  V 2 )xx  (a 2  V 2 )xy  2VxVyxy  0
r 1 2
其中a  a 
(V  V 2 )
2
2
2
气流绕过薄翼
适用范围：亚、跨、超音速无旋流动
Suitable case ：subsonic， transonic， supersonic irr-rotational flow.
将流动分解为两部分：未经扰动的流动、扰动流动To
decompose the flow into unperturbed flow and perturb flow
未经扰动的流动就是无穷远前方来流Flow at
unperturbed fields is far field flow
扰动运动速度势可以用  表示。速度可以用
表示 Vx ,Vy Potential function of perturbation flow is  ，
perturbation velocity components Vx ,Vy
Vx  V cos
V y  V sin 
   V ( x cos  y sin  )
Vx   x , V y   y
两部分的合速度势
The total velocity potential function
  V ( x cos  y sin  )  
代入速势方程可得小扰动速度  应满足的方程
Substitute the equation and then the small perturbation eq.
(a V )xx  (a V ) yy  2VxVyxy  0
2
2
x
2
2
y
求得速度场之后，可以得到压强及压强系数为
The pressure and pressure coefficient can be obtained from the following
equations.
 2  2
p  p  (v  v )
2
2
cp  p 
( p  p )
2
p

(  1)
2
1
 2v2 rM  p
2
再用等熵流动的关系式可得到其他参数
Then introduce the isentropy relation to get other parameters

p
T  1
( )
p
T
比热比
绝热指数
2
2
V
cos


V
sin

V

V
T
x
y
x
y
 1  (  1)M 2 [

]
2
T
V
2V

2
2

sin




2
y
x
y  1
2  x cos 
p  c p  2 {[1  (  1) M  (


)]  1}
2
rM 
v
vy
2v
小扰动条件下，扰动速度远小于自由来流速度
on small
perturbation condition, the perturbation velocity less than free stream
Vx , V y  V
a2  V2  V Vx
a2  V Vx
补充条件：
Supplement conditions
① 来流不能接近音速 incoming flow velocity does not approach
sonic
② 来流非高超声速 incoming flow velocity does not approach
hypersonic
为进一步简化扰动方程，忽略扰动速度一次项，可得到
下列关系：Simplified equation
a 2  Vx2  a2  V2
a 2  V y2  a2
VxV y  0
最后得到：
Final equation
(a2  V2 ) xx  a2  yy  0
(1  M 2 ) xx   yy  0
应用范围: 亚、超声速
Suitable for subsonic and supersonic
不适用于跨声速区域:
对于跨声速 M ≈1，必须取消补充假设条件，即取消来流不能接
近音速的假设，这时速势方程首项的系数一次项不能忽略 For
transonic flow field (M ≈1)， the supplement condition， the first
item of the potential function equation can not be neglected.
a  v  a  v  (  1)vvx
2
2

2

2

跨声速小扰动方程应为：The small perturbation equation of velocity
function
[a  v  (  1)VVx ] xx  a  xy  0
2

2

[1  M 
2

2

 1
v
M 2 x ] xx   yy  0
可以证明：当M∞→1时，
It’s proved ,when M∞→1,
[1  M 
2

 1
v
M 2 x ]  1  M 2
 因此跨声速条件下，小扰动方程可以写成 So that the
small perturbation equation at transonic flow can be written as
(1  M )xx   yy  0
2
此方程的类型取决于： Type of the equation depends on
=B2-4AC=4(M2-1)
1) 当M<1时, <0,不存在实特征根，没有特征线，为椭圆型 When
M<1,no real eigenvalue exists, that is no character line, the equ. is elliptic.
2) 当M>1时, >0,存在两个特征根，有两条特征线，为双曲型
When
M>1,there are two eigenvalue, two character lines, the equ. is hyperbolic eq.
3) 当M=1时, =0,存在一个特征根，有一条特征线，为抛物型When
M=1,there is one eigenvalue, one characteristic line ,the equ. is parabolic
 特征线(当M>1时)：斜率
The slope of characteristic line
dy
1
1 1
( )
 tg   sin
dx
M
M 2 1
是马赫角
is so call Mach angle
特征线与x轴夹角为局部马赫角，对称于x轴。 Local Mach angle is the
angle between velocity vector and the characteristic line
y
r’
依赖区
q’
o
q
p
影响区
r
x
 影响区：P点下游由两条特征线所夹的区域 Influence zone: upwind
zone between characteristic lines
 依赖区： P点上游由两条特征线所夹的区域
downstream zone between the characteristic lines
Depend zone
 扰动下的压强系数公式 The pressure coefficient on small perturbation condition
2Vx
x
p

V
V
§4-2小扰动激波关系式 The shock relations of small
perturbation .
等熵激波小扰动激波的熵增是三阶小量 For
small perturbation shock,
entropy increase is third order， so it is isentropy shock。
 激波的精确速度关系式：Accurate velocity relation of shock
V22y  (V1  V2 x )
V1V2 x  aCr2
2 2
V1  V1V2 x  aCr2
 1
 激波前后的速度关系式（几何关系）
front/rear-shock
V1V2 x  V1V2
V22y 
(V1V2 y ) 2
2
1
V

2
V1  V2
V12
(V1  V2 x )  (V1  V2 )  V
2
2
2
2y
即
(V1  V2 ) 2  (V1  V2 x ) 2  V22y
Velocity relations in
 对于直角坐标系 At Cartesian coordinates



V  V cos  Vx i  V sin  Vy  j
因此 so that
V1  V  2V cos  Vx  Vx21  2Vx sin Vy1  Vy21
2
2
 V  2V cos  Vy  Vx21  o(Vx2 )
2
V1V2 x  V  V (Vx1  Vx 2 )  o(Vx21 )
2
V22y  (V1y  Vy 2 )2  o(Vx31 )
(V1  V2 x )2  (Vx1  Vx 2 )2  o(Vx31 )
 由能量方程可得 From energy equation
2
2
1

M
2
acr2  V2 
V

  1 M 2
 由此得到 M∞→1时的方程（跨声速中）From where , the equation when
M∞→1,(transonic flow)
(1  M 2 
 1
V
 超声速中 At supersonic flow
M 2
Vx1  Vx 2
)(Vx1  Vx 2 ) 2  (V y1  V y 2 ) 2  0
2
v x1  v x 2
r 1
(1  M _
M
)(v x1  v x 2 ) 2  (v y1  v y 2 ) 2  0
V
2
2

适用范围：激波前后小扰动方程，适用于等熵波 Above eqs. are available for
small perturbation flow in front/behind of the shock, i.e. , iso-entropy flow
§4-3 跨声速小扰动速势差分方程
Small perturbation
equation for transonic flow
混合性方程，在同一流场中不同点所用的差分方程 不同。 Mixed
equation, different FDE is used for the scheme
一、中心差分格式 Centeral FDE scheme flow field
对速度势  For velocity potential function

 2 (x) 2
 ( x  x, y)   ( x, y) 
x  2

x
2!
x

 2 (x) 2
 ( x  x, y )   ( x, y ) 
x  2

x
2!
x
 一阶导数的差分格式
First order difference equation is obtained as
  ( x  x, y )   ( x  x, y )

 o(x) 2
x
2x
二阶导数的差分格式
Plus two equations, and get 2ed order PD
 2  ( x  x, y)  2 ( x, y)   ( x  x, y)
2


o
(

x
)
x 2
(x) 2
二阶精度
2nd order
二、一侧差分格式
One side FDE of the derivatives
 在超音速流中，气流参数只受上扰动游影响与下游扰动无关。At
supersonic flow, the parameters of flow are dependent on upwind
perturbation and independent on down flow perturbation
 需建立迎风一侧差分格式 The upwind one side FD scheme is needed to
built
 取上游一侧的点构成差分格式 Take the upwind point to construct FD
scheme
 一阶精度迎风格式 1st order upwind scheme
 
（x, y）-  ( x  x, y )

 o (x) 2 
x
x
 二阶精度迎风格式
2nd order upwind scheme
 2  ( x  2x, y )  2 ( x  x, y)   ( x)

 o ( x )
2
2
x
 x 
三、亚音速点的差分方程At subsonic flow equation
取网格点如图：正交等间距网格The space nodes are shown as
 x  2x, y 
y
 x  x, y 
 x, y   x  x, y 
v
x
 中心差分格式构成的差分方程

r  1 2 i 1, j  i 1, j
2
M
1  M  
V
2x

  i 1, j  2i , j  i 1, j
2
 .

x



 i , j 1  2i , j  i , j 1
0
 
2
 y 

即

r  1 2 i1, j  i1, j   i1, j  i 1, j   i , j 1  i , j 1  
2
M



1  M  
2
2

V
2

x

   x     y   

i , j 

r  1 i1, j  i1, j  2
2
2
1

M


2
2



V
2

x


  x   y 

 i , j受周围四点的影响，这是亚声速流动的特点  is effect by
i, j
around four points , this is subsonic feature
j 1
j
j 1
i 1
i
i 1
四、超声速点的差分方程FDE for supersonic flow
 当计算点为超音速（M大于1）时，方程为双曲线
型When local supersonic flow appear ,the equation is
hyperbolic
 存在依赖区（上游马赫锥内部）The dependence zone
exists ,(up mach core)
 对y的差分可以用中心格式The centurial difference is
used for the derivative with sped to y
 对x的差分要用迎风格式Upwind scheme is used for X-
direction
 显示格式： yy差分式取
scheme
 yy 
i线法Explicit
i ，而不用
1
i 1, j 1  2i 1, j  i 1, j 1
 y 
2
 每次都用i网格线上的已知值，可以从左到右逐点计
算The known value is used to calculate the value at every
node sequently
 隐式格式：利用当前网格线上的值构筑差分方程
Implicit scheme : using present value to construct FDE
 yy



i , j 1
 2i , j  i . j 1 
 y 
2
具有三个未知量（在网格线i上）
Where there are 3 unknown points
显式比隐式方便Explicitly scheme is more convenient than implicit
scheme
 显式格式稳定区域小The stability zone of explicit is smaller than that
of implicitly
稳定性和收敛性
Stability and convergence
 收敛性：当步长趋于零时，差分方程解趋于微分方
程解Convergence: when step length tends to zero, the solution of the PDF tends
to the solution of PDE
 稳定性：差分误差在传播过程中有界且逐渐减小
Stability :the error is limited or decreased
 对波动方程（双曲型）：稳定性条件是差分方程依
赖区不小于微分方程的依赖区For viberation Eq ,the stability
condition is that the dependent zone of PDE less than that of PDE
对超声速势函数
For potential velocity fuction
差分方程依赖区半顶角 The half conical angle The dependent
zone of the FDE
y
tg 
x
y

x
 微分方程的半顶角the angle of the dependent zone
tg  
1
M 2 1
差分方程稳定条件为

  tg
y
tg  
x
1
M 2 1
对于跨声速势流，不满足稳定条件，因为For transonic flow, the
stability condition is not satisfied
y
 , 

2
x
M 1
1
 跨声速势流不能用显示格式
so transonic potential
function can not solve with explicit method
隐式格式的依赖范围大于微分方程的依赖范围The
dependent zone of implicit scheme is great than that of PED
J+1


2
 tg
1
1
M 1
2
J
J-1
 双曲方程差分采用一侧隐式格式For hyperbolic equation ,one
side implicitly scheme is used

r+1 2 i , j  i2, j  i , j  2i1, j  i2, j i , j1  2i , j  i , j 1
2
M

0
2
2
1  M  

V
2x 
 x 
 y 

五、音速点的差分方程The finite diffence at sonic points
 当M=1时，方程为抛物性，存在一族特征线When M=1,the equation
is parabolic, there exist a series of characterist line
 y 

 
 x c
 速度势方程化为potential equation become
 yy  0
 采用差分方程可以写成Using FDE
i , j1  2i , j  i , j1  0
六、速度判别式Velocity critical condition
四种情况:
Four cases
I.
亚声速sub
M
1
Subsonic
亚声速sub
II.
超声速supe
超声速super
III.
亚声速sub
超声速super
IV.
超声速 super
亚声速sub
airfoil
sup
er
superso
nic
Ⅰ Ⅱ Ⅲ：过渡连续 continually changes
Ⅳ：出现激波 参数不连续 the shock appears, parameters are
discontinous
Ⅲ：有音速线存在There exists sonic points
逐点判别：根据  xx 系数进行判别
Judge according to the coefficient of
Axx  C yy  0
 中心差分
r+1 2 i1, j  i1, j
Ai  1  M  
M
V
2x
1
 一侧差分
2
r+1 2 i , j  i2, j
Ai-1  1  M  
M
V
2x
 2
2
情况
A1 的值
Ⅰ
>0
>0
亚-亚声速
subsonic
Ⅱ
<0
<0
超-超声速
supersonic
Ⅲ
>0
<0
亚-超声速
sonic
Ⅳ
<0
>0
超-亚声速
subsonic
A  2  的值
差分方程形式
PDE form
(i,j)点性
质
A
0
any
0
亚音速
subsonic
超音速
supersonic
0
0
音速点
sonic
对应的差分方程

r+1 2 i1, j  i1, j  i1, j  2i , j  i 1, j
2
M
2
1  M  

V
2x
 x 



  2i , j  i , j 1
 i , j 1
2
 x 

r+1 2 i , j  i2, j  i , j  2i1, j  i2, j
2
1

M

M
2



V
2

x

x





i , j1  2i , j  i , j1

0
2
 y 
i , j1  2i , j  i , j1  0
七.跨声速小扰动激波的差分方程
PDE for transonic small perturbation shock flow
 激波处：速度由超声速过渡到亚声速
At shock, the flow transfer from supersonic to subsonic
 激波前流场均匀（近似）
shock
i-1
i
i+1
In front of the shock ,the flow is uniform supersonic flow
Vx1  (i , j  i 1, j ) x
Vy1  (i , j 1  i , j ) y
i-1
j
i,j
j+1
i+1,j
i+1,j+1
i-1
i,j+1
 激波后流场均匀（近似）
After the shock ，the flow is also uniform
Vx 2  (i 1, j  i , j ) x
Vy 2  (i , j  i , j 1 ) y
 差分方程（跨声速小扰动方程的差分形式）
FDE (Transonic small perturbation flow)
r  1 2 Vx 2  Vx1 Vx 2  Vx1 Vy 2  Vy1
(1  M 
M
)

0
V
2
x
y
2

 对无旋流动（无旋条件）
Condition of irrotational flow
Vx Vy

y
x
其差分形式 Its FD form
Vx 2  Vx1 Vy 2  Vy1

2y
2x
 考虑了无旋条件的扰动速度差分方程 After considering the
irrotatational condition the small perturbation equation becomes
r  1 2 Vx1  Vx 2
(1  M 
M
)(Vx 2  Vx1 )2  (Vy1  Vy 2 )2  0
V
2
2

 讨论：discussion：
跨声速区小扰动激波差分方程与小扰动激波关系相同
八、超音速点差分方程的人工粘性
artificial viscous for supersonic FDE
速势方法假设了流场均为等熵流
The velocity potential method assume that the flow is iso-entropy
导致流场间断解不唯一（可由亚-超，也可由超-亚）
It leads to non-unique solution
如果采用迎风格式 （单侧差分），则只适合压缩突跃（由
超-亚），不可能出现膨胀解。 Continuous solution，if the
upwind scheme is used, the solution only suitable for compressible
sharp increase (shock), not suitable for sharp decrease.
原因：
采用1阶迎风格式
x 
 xx 
i , j  i  2, j
1st order upwind scheme
  xxx  o(x 2 )
2x
ij  i 1, j  i 2, j
(x) 2
  xxxx  (x 2 )
 超声速点差分方程（迎风格式）
FDE of the potential
equation at supersonic flow
  1 2 i , j  i 2, j i , j  2i 1, j  i 2, j
(1  M 
M
)(
)
2
v
2x
x
i , j 1  2i , j  i , j 1
 1 2 2
2
2
2
2


(
1

M
)




(
1

M
)


x

M



x


(

x
,

y
)
xx
yy
xxx

xx
2
(y)
v
2

应用当地M数改成相对应的微分方程
Using local Mach number M to rewrite the PDE then
(1  M 2 ) xx   yy  (1  M 2 ) xxx x 
 1
v
M 2 xx2 x
2
(1

M
)xxx x 类似于跨音速小扰动粘性流方程中的粘
其中
2
(1

M
)xxx x is similar as the viscous
性项。称为人工粘性 Where
form of small pertubation equation, so called it artificial viscous
差分方程的解只含压缩突跃，即激波（是熵增过程）
PDE only includes compressed shape change(where the entropy creases )
不可能产生膨胀突跃（即熵减过程）
Not suitable for expanding shape change(where entropy decreases)
4.4 边界条件及其嵌入
Embeding of Boundary conditions
一、边界条件(Boundary Condition)
1.物面： 无粘，无穿透条件 on wall no normal velocity
V n0
F ( x, y )  0 则定常
对于翼型（叶栅），设物面方程为，
流动边界条件
V F  0
即：
Vx
F
F
 Vy
0
x
y
若翼型上下表面可表示为
F
y

,
x
x
y  f  ( x)
则
F
0
y
速度分量可写成
Vx  V cos   vx
Vy  V cos   vy
F ( x, y)  y  y ( x)  0
上表面的边界条件为
BC on up surface is
y上
(V cos   v上 )
 V sin   Vy  0
x
其中， v x , vy 为扰动速度
Where v x , vyis the perturbation velocity components
vx ，
v y  V

对于薄翼型
For thin wing
y上
 1
x
小迎角下， 
 0时
sin    ,co c  0
For small AOA, when
故上表面 (on up surface)
y上
 y  v y  V (
)
x
或写成 or be written as
 y上

 y  x, 0   V 
 
 x

同理，对于下表面
meantime for lower side
 y下

 y  x, 0   V 
 
 x

综合上下表面可以写成以下小扰动方程翼型上下表面边
界条件Consider upper and lower side of airfoil ,the small perturbations satisfy
following condition
 y

 y  x, 0   V 
 
 x

2.库塔条件（后缘边界条件）
Kutta condition (trailing edge condition )
 上下表面流线在后缘尖点平滑汇合
the streamlines on upside and Lower-side smoothly sinks at trailing edge
c
后
。
上
。
下
 在受气动载荷时，速度势在后缘不连续，形成间断面。
Under the aerodynamic loads ,velocity potential function at tracting edge is
discontinuous
 在这条间断面上必须满足
satisfy is
On the discontinuity surface，what must
(1)上下压强相等 the pressure on up and lower side of airfoil is equal
(2)速度方向相同，大小不同 the direction of velocity are consistent, but
the value of the velocity is not equal
P( x, 0)  P ( x, 0)
Vy ( x, 0)
Vx ( x, 0)

Vy ( x, 0)
Vx ( x, 0)
2Vx
 小扰动条件下 P 
V ，因此上述方程可写成：
for small perturbation, above equations can be written as:
 x ( x, 0)   x ( x, 0)
 y ( x, 0)   y ( x, 0)
 经间断面速度势变化称为环量
through the section surface the velocity potential function changes is circulation.
    d   ( x后，
 0) -  ( x后，
- 0)
c
3.远场条件Far field condition
 用有限远代替无限远场，扰动速度势的近似条件为：
using limited far field replace the real far field perturbation velocity potential function
BC can be written as:
 x  v x

0
 y  v y

0
二、边界条件的嵌入
Embeding of the boundary condition
边界点上速度势应同时满足边界条件和速势方程On boundary
the velocity potential function satisfy both the BC and the potential Eq.
1.物面边界嵌入
Embeding of wall boundary condition
 翼型上表面
On the airfoil surface
( y )i , j
y上
 V (
)
x
将速势拓延到边界的另一侧（i，j-1）
Extend the velocity potential function to other side of boundary
( y )i , j 
i , j 1  i , j 1
2y
 (y 2 )
即Or
i, j 1  i, j 1  2y  ( y )i, j  (y3 )
边界点的中心差分The central difference on boundary
( y )i , j 
i , j 1  2i , j  i , j 1
利用边界条件得到：
Using BC then get
(y)
2
 (y 2 )
( y )i , j
( yy )i , j
y上
 V (
)
x
2 i , j 1  i , j

[
 ( y )i , j ]  (y j )
y j
y j
y上
2 i , j 1  i , j

[
 V (
  )]  (y j )
y j
y j
x
2.库塔条件的嵌入
Embedding of Kutta condition
增加新方程使上下表面上 y 相同，即
Additional new equation to make
consistent
on up and lower surface
y
 ( x,0)   ( x,0)   ( x后,0)   ( x后,0)
 y ( x, 0)   y ( x, 0)
3.远场条件的嵌入
Embedding of far field condition
根据具体问题特点建立运动场  的计算方法
To found the computation method according to the character of certain problem
对于自由绕流，运动速度为 v  ，自由来流的速度势为
for a free flow around the airfoil， the far field velocity is
function of free flow is
，and
v  the velocity potential
  v x cos  v y sin 
扰动速度势应满足
Therefore the perturbation velocity potential satisfy
 x  vx  0 
 y  vy  0 
i 1, j  i 1, j
2(x)
i , j 1  i , j 1
2(x)
0
0
§4.5 线松弛迭代解法
The line relaxation iteration method
一、非线性代数方程的迭代解法 Iterative method for non-linear
equations
 跨声速小扰动速势方程是非线性的 Transonic small perturbation
equation is nonlinear PDE
 其差分方程为非线性代数方程，即系数是与函数值或其导数有关
Its FDE is also nonlinear equation that is its coefficients are related to the
variables
 迭代求解： Iteration method
把系数假设成已知量，每次求解之后再重新计算系数,再次求
解直到得出收敛解为止.Assume the coefficient are known at first
iteration, then recalculate the coefficients again after once iteration,
repeat iteration until the iteration convergences
二、高阶代数方程的线松弛解法
The line relaxation iteration method for High order arithmetic linear equations
 高阶线性方程组，线性化后的差分方程 High order linear equations,
linearized FDE
 阶数为
Mn
, M为网格点数, n为问题的维数. 或阶数M*N*L
（M,N,L为空间三坐标方向的网格点数） The order of linearalgebra equation is
Mn
, where M is the number
of the grids, n is
the number of dimension. The order of linear- algebra equation is
M*N*L, where M,N,L are number of grids in coordinates x,y and z
 松弛迭代： Relaxation iteration 轮流放松流场中的的部分速势，
将其假设为未知，其余部分看成已知的, 利用线性方程组联立
求解 Relaxate the potential function sequently, assume that the
present point is unknown, and others are known.
点松弛：每次把一个点作为未知点
Point relaxation: only one point is assumed to be unknown
松弛迭代
线松弛：每次把一条网格线上的所有点作为未知
Line relaxation :all point on one grid line are assumed
to be unknown
j
线松弛
line relaxation
i
线松弛
line relaxation
i
j
点松弛 point relaxation
 线松弛法：要求内存较多，方程组的个数减少到一维点数Line
relaxation :require more memory source, the number of equation equals to
the number of 1D points
 逐点松弛：要求内存较少（为线性松弛的
1
2
倍）, 扫描流场
中的各个网格点，把周围点均看成是已知点。Sequent point:
require less memory resource ,only
1
2
times of line elaxation .Scan
all the grid points sequently.
 线松弛方程组可采用三对角矩阵快速解法For line relaxation
method ,the tri-diagonal array can be solve with quick method.
三、简单迭代和改进迭代
Improve method of simple iteration method
 简单迭代：迭代公式右端的速度势全部采用前次迭代结果
Simple iteration,all parameters on right hand are old value of last
iteration.
 改进迭代：每次迭代都用最新速度势值代替右端项。速度判别
式要用简单迭代方式计算，则会导致超临界气流计算振荡发散。
The improved iteration method always uses the newest value, and
the velocity criteria must be calculated according to the simple
iteration way, otherwise the divergence will occur at critical state.
四、追赶法
The chase method
 求解三对角矩阵线性方程快速方法 It is a fast method to solve
tri-diagonal matrix
a ji , j 1  b ji , j  c ji , j 1  d j

i,j+1
线松弛方法求解方程组
Equation for line relaxation method
i-1,j

对于边界点：for boundary points

上边界 up

下边界 lower boundary c  0
N
boundary
j=1,2……N
a1  0
i,j
i,j-1
i+1,j
 对应的系数矩阵为三对角矩阵
 b1
 a
 2
 0


 0

 0
c1
0
0
b2
a3
c2
b3
0
c3
0
an1 bn1
0
an






cn1 

bn 
追赶法：顺着消去，逆着带入。从上至下消去首项，从下
而上代入末项。The chase method：eliminating sequently，
substituting inversely. Eliminate from top to down，
substitute from down to up.
五、初场Initial field
 初始值分布：影响收敛速度
influence convergence
Initial field distribution of parameters will
 对亚音速流场：可以选全场速度势为0，即未经扰动 Subsonic
flow field：globe field can be put as 0， that is the flow is not disturbed
i, j  0
 对跨音速流场：初值选取需谨慎，合理初场能加速收敛 For
transonic flow initial value must given carefully，the reasonable initial value
might accelerate convergence
 一般应选用与流场相近的速度势分布 Usually select a near
solution of potential function
 可以用相近的亚声速计算结果
result can be used
The approximately subsonic
六、收敛标准
Criterion of convergence
 所有点相邻两次计算所得的速势差别的最大值 The
maximum difference between two immediate vicinity iterative
i(,nj)  i(,nj1)
max
  ( n )   (103  105 )
)
)
 (1比值判别收敛
 可以用  (n与初始
the ratio of current
and initial  can be the criterion
(1)
 ( n )
3
5

(
10
~
10
)
(1)

 (n )
七、超松弛法
Super relaxation method
 加速速度势函数的修正步伐
To accelerate the convergence
i(,nj)  i(,nj1)  (i(,nj)  i(,nj1) )
 超松弛 super relaxation
2   1
 弱松弛 weak relaxation
 1
  1.5
 1
八、加密网格法
Mesh refine method
计算精度增加 , 计算网格数增加 To increase the precision , to increase
the mesh No.
问题复杂度增加 The increase of complexity
to increase the mesh No.
计算机时与网格总点数以正比增加 computation time increase as the
mesh No.
采用疏密结合的方法可以减少计算时间 Using coarse/fine mesh may
decrease computation time
 加密网格法：先用疏网格数算初始场，加密之后获得精确解
meshes refining method
 多重网格：先疏后密、再疏；交替使用疏密相间的网格
multiple grid
*§4-6 绕升力翼型的跨声速小扰动势流差分
计算方法
FDM for potential function of transonic small perturbation flow around airfoil
一、绕升力翼型的跨声速小扰动方程势流的差分方程
The equation for potential function of transonic small perturbation flow around airfoil
4—7 隐式近似因式分解法的基本思想
The basic concept of Approximate Decomposition Method
 求解速度势方程的快速收敛解法
It is a fast working method for
Potation equation
 SLOR 是显式迭代方法，因此收敛慢
SLOR is full explicit method,
thus it works slowly
 全隐式松弛算法：每次迭代中流场中的任意一点能受到它的依赖
区中全部其点的影响 Full implicit relaxation method ,any point
in flow field can be influenced by all other points
 ADI(Alternating Direction Implicit)交替方向隐式迭代，分为AF1和
AF2
Using AF1 and AF2
 基本差分算子：
Some basic finite difference calculator
 迎风差分（前差） upwind FD
x
i , j
1

x
 
i, j

i 1, j 
 顺风差分（后差）backward/rearward FD
 x () i , j
1

x
 
i 1, j

i , j 
 二阶中心差分： 2nd order central FD
1
 xx  2
x
 
i 1, j
 2
i , j   i 1, j 
 二阶一侧迎风差分：upwind 2nd one side FD
1
 xx  2
x
 
i, j
 2
i 1, j   i 2, j 
 位移算子：displacement (FD) calculator
Ex1 ()  ()i 1, j
Ex1    
i1, j
 用位移算子表示差分算子 The FD expressed using displacement
calculators
1
1
1


x 
1  E x ,  x 
E x1  1
x
x
1
1
1

 xx 
E

2

E
x
x 
2
x
1
1
1
1

 xx 
1

2
E

E
E
x
x
x 
2
x
 差分算子位移：The displacement of FD calculator


1
1
( E x1  1) 
1  E x1  
x
x
1
1
1 1
1
1
1
1 1
E x  xx  E x
E

2

E

1

2
E

E
  xx
x
x
x
x Ex
2
2
x
x
E x1 x  E x1




 差分算子的分解与组合：The decomposition and combination of FDs
1
1
1
1
E

1
1

E
1

E
E
1
1 1
x
x
 xx  2 ( Ex  2  Ex1 )  x .

. x
  x x   x x
x
x
x
x
x
1
1
1
1

 xx 
1

2
E

E
E
x
x
x 
2
x
1  E x1 1  E x1

.
x
x
  x x
 差分方程可以用算子表示The expression of FDE using FD calculator
Li , j  0
 L代表未经松弛的差分算子 FD calculator relaxation
The FD calculator of
relaxation iteration, the correction of n th iteration.
 松弛差分算子N，第n次迭代的修正值为
i,nj  i,nj1  Ci,nj
 算子表达式:
 Calculator expression
NCi,nj  Li,nj1  0
 当松弛迭代收敛时，C n 
i, j
 0，即 Li,nj  0 两者相同
iteration converged, both calculator are the same .
When the

(n)
当 ci(,nj)  0 时， Li, j  0表示其不是差分方程的解，因此 c表
i, j
( n)
( n1)
示差分方程满足微分方程的程度。When ci , j 
，0 Li, j denotes
0
( n1)
the solution of FDE is not the solution of the original PDE ， therefore
n ) dose the FDE satisfy the PDE.
the degree of
c (how
denotes
i, j

差分松弛迭代算子的选取原则 The principle for seleeting FD calculator
 便于求解，线性，有快速解法
method， fast solver
convenience for solving equation，linear
 稳定，能达到收敛标准 stable
 高效，N尽可能接近L。 higher efficiency， N approaching L
 差分算子的用途：可以清晰的显示差分方程的结构
Usage of the PD calculator， it makes the FDE simple and evident
近似因式分解的基本思路
The basic consideration of approximate decompose
 Laplace方程的差分格式（简单迭代法）
The FD scheme of Laplace equation （simple iteration method）
i(n1,1j)  2i(,nj)  i(n1,1j)
(x)
2

i(,nj11)  2i(,nj)  i(,nj11)
(y)
2
0
 改进的迭代法 improved iteration method
i(n1,) j  2i(,nj)  i(n1,1)j
(x)
2

i(,nj)1  2i(,nj)  i(,nj1)1
(y)
2
0
 松弛迭代格式 Relaxation iteration scheme
中间值

( n 1)
i 1, j
~
 2i , j  
(x)

( n)
i, j
( n 1)
i 1, j
2


( n 1)
i , j 1
~
 2i , j  i(,nj11)
(y)
2
~
 i , j  (1   )i(,nj1)
由此then
~
i , j 
1

[i(,nj )  i(,nj 1)  i(,nj ) ]
0
 还原为（n）和（n-1）表达式后差分方程还原为 Express the FDE
using （n）and （n-1）
(n)
i-1,
j 

(n)
i , j 1

2

2

[i(,nj )  i(,nj1)  i(,nj1) ]  i(n1,1)j
( x )2
[i(,nj )  i(,nj1)  i(,nj1) ]  i(,nj1)1
( y )
2
0
 改进的差分格式为
Improvement of FDE
1
1
(n)
( n 1)
[


2



]

[i , j 1  2i , j  i , j 1 ]  0
i 1, j
i, j
i 1, j
2
2
(x)
(y )
 或（隐式）
or (Implicit Form)
1
2 (n)
(n)
( n 1)
( n 1)
( n 1)
{


[





]


i 1, j
i, j
i, j
i, j
i 1, j }
2
( x )

1 1 n
1 2 n
( n 1)
( n 1)
( n 1)
( n 1)

[





]

[





]
i , j 1
i , j 1
i , j 1
i, j
i, j
i, j
2
2
( y ) 
( y ) 
1 1 n
( n 1)
( n 1)

[





i , j 1
i , j +1
i , j 1 ]}  0
2
( y ) 
 引入差分算子    xx  ，采用差分算子表示，并令
yy
Introduce the FD calculator , using FD calculator expression.
Ci , j  i(,nj )  i(,nj 1)
 则松弛迭代法的差分格式为：
The FD scheme of relaxation iteration is
1
1
2 1
1
(n)
(n)
(n)
( n 1)
C

C

[

]
C


L

i 1, j
i , j 1
i, j
(x)2
(y)2
 (x) 2 (y) 2 i , j
 线松弛迭代对应的差分格式
FDS related to line relaxation iteration is
[ xx
2

1
n
( n 1)


E
]
C


L

0
x
i, j
i, j
2
2
(x)
(x)
 因此超松弛差分算子
Thus ,the super-relaxation FD calculator is
N   xx 
2

1

E
x
(x) 2 (x) 2
NCin, j   Li(,nj 1)  0
 N分解成两个因式的乘积，则 If factorize N into two factors ,then
1
N 
N1 N 2

N1N2Ci(,nj)  Li(,nj 1)  0
令
N 2 Cin, j  f i ,nj ,
则
N1 f i ,nj   Li(,nj 1)
 f i ,( nj )
N 2 Cin, j  f i ,nj
 Cin, j
 i(,nj )  i(,nj 1)  Cin, j
4-8 AF1方法
AF1 method
小扰动速势方程
Equation of potential function for small perturbation flow
(1  M 2 )xx  yy  0
或
Axx  yy  0
，对亚音速小扰动 A  0 。可用中心
差分格式或隐式方程
其中 A  1  M
2
Where A  1  M 2 ,for subsonic point A  0 ,the central scheme can be used
1
1
[


2



]

[i , j 1  2i , j  i , j 1 ]  0
i 1, j
i, j
i 1, j
2
2
( x )
( y )
 其中 where
i , j 
1

i 1, j 
i 1, j 
i , j 1 
[i(,nj)  i(,nj1)  i(,nj1) ]
1

1

1

[i(n1,) j  i(n1,1)j  i(n1,1)j ]
[i(n1,) j  i(n1,1)j  i(n1,1)j ]
[i(,nj)1  i(,nj1)1  i(,nj1)1 ]
 令Let
L  A xx   yy
 则 小扰动速势方程的隐式差分格式为
Then ，the implicit FDE scheme of the small perturbation potential is
( A xx   yy )Ci(,nj)  Li(,nj 1)
分解第一项系数
To factorize the first coefficient term
N  A xx   yy  
1
(   xx )( A   yy )

 A xx   yy  A 
原系数
origin
1

 yy yy
误差
error
 松弛差分算子N可分解成为N1和N2，
 为加速收敛参数，
求解可分两步，
Relaxation PDE calculator N can be factorized Into N1 and N2 ,the solving can be
decompose into two steps
N  N1 N 2
(   xx ) f i,(nj)  Li(,nj1)
(A   yy )ci(,nj)  f i,(nj)
f i,(nj) 代表中间结果 ~i(,nj)
( n)
Where f i, j
is a middle result ~i(,nj)
交替方向隐式差分格式（ADI, or, Approximate
Factorization）
 Step1:全场沿X方向线松弛，解三对角矩阵
X direction line relaxation, to solve triangle matrix
 Step2:全场方向沿y方向线松弛，也解三对角矩阵
Y direction line relaxation , to solve a triangle matrix
 全隐式格式，对亚声速区适用，称为AF1，
full implicit scheme ,For subsonic it’s call AFI
 对超音速点，采用迎风格式
For supersonic points the upwind scheme is used
NCi(,nj)  Li(,nj1)
N 
1

(   xx )(A   yy )
对应两步
Corresponding two steps are
1.
(   xx ) f i ,( nj )   L i(,nj1)
2.
(A   yy )i(,nj)  f i,( nj)
二、AFI的收敛性
The convergence of AFI
 亚音速点（中心差分格式）等价于时间相依方程（将
迭代过程看成时间推进）
For subsonic ( central PDE scheme ),the solving preceding is equivalent to a timedependent problem
t[(1   )( A xx   yy )  A t 
1

 xxyyt ]   ( A xx   yy )  0
 和   0，当 t 与  xx系数异号时，差分方程的
解收敛于微分方程的解。
t and  xxhave different sign, then the PDE converges two the PDE
设
三、AF1的稳定性
Stability of AFI
采用Von neumann 方法分析误差
Using Voneumann analytic method
 ( x, y, t )  e t e iax eiby
代入AF1差分方程
NCi(,nj)  Li(,nj1)
Substitute into the PDE of AFI
e
t
其中
where
a1  (1   )a2

a1  a1
a1  A 
a2 
4
(1  cosax)(1  cosby)
2
2
x y
2A
2
(
1

cos
a

x
)

(1  cosby)
2
2
x
y
假设   0   0
a1 , a1  0 ，
t
e 为实数
t
Assume   0   0 a1 , a1  0， eisreal
number
收敛条件：
The condition of convergence
e  t  1
即
or
a1  (1   )a 2
1 
1
a1  a1
 稳定性条件：
stability condition
0  2
2a1
 0,   0
a2
 两个可选参数  和

，适当选取可以加快收敛
can be chosen carefully to get quick
Two parameters 
and
convergence
t
e
 0 得到最短迭代次数（   2）
取
Take et  0
,then get minim iteration times
 对应的最佳选择是：
Corresponding optical choice is
2
  2 1  cosax 
x
2
 AFI中所有的误差Fourier分量均可以同速度下降
AFI error components of Fourier series may decrease

t
2 

2
 超声速点的AF1差分方程等价于（4-8-12），但没有阻尼项
The AFI for supersonic point it equal to equation（4-8-12）
 亚，超声速流混合问题，AF1第一个算子需四对角矩阵求逆
For sub-super-sonic mixed problem, AFI has to solve four-diagonal matrix ,thus the
efficiency is lower
 对此类流动，AF1不是最有效的方法
For such flow ,AFI is not the most efficiency one
§ 4-9 AF2方法
method of AF2
 对超声速流增加时间阻尼项，取
For supersonic flow ,the tune damping is traduced
 差分算子
FD calculator with upwind scheme
N 
1

     A
yy
x
  A
xx
  yy    x 
A

 yy  x
 对超声速点，取：
For supersonic flow point
N 

      A   A

1
yy
xx
  yy    x 
A

 yy  x
 等价的一阶微分方程（时间依存）
Equivalent 1st order PDE(time dependent)
A


1     Axx  yy   xt   yyxt     Axx  yy   0
 xt 为超声速阻尼项，当
 其中 ，
  0,   0, A  0 时，  xt
与  xx 系数同号，大小取决于 
Where ,  xtis supersonic damping, when the cofficent of
havethe
xx same sign and the quantity depends on
and xt

 更加高效（有阻尼）
It is higher efficiency
 AF2格式对亚声速流收敛性比AF1差
AF2 convergence for subsonic is worst than AF1
 亚，超声速点的两步AF2格式如下：
For sub-supersonic flow, two steps, AF2 have following scheme
亚：
sub
 

n 
 n 1 （y方向线松弛x方向迎风格式）


f


L

x
yy
i, j
i, j
Line relaxation in Y upwind scheme in X
  A C
x
n 
i, j
 f
（x向线松弛x方向顺风格式）
n 
i, j
Line relaxation in X bear wind scheme in X

 超 ：  x
Super

  yy f i ,nj    L i,nj(y1方向线松弛，x方向迎风格式)
  A C
x
line relaxation in Y ，upwind in X
n 
i, j
 f i ,nj 
(X方向线松弛，X方向顺风格式)
line relaxation scheme in X, rear ward FD in X
n 
f
 其中， i , j 是中间结果，为全隐式差分
 immediate value ,it is full implicit PD
Where f i ,njis
 AF2格式稳定性
The stability of AF2
 令误差 let error
  x, y, t   et eiax eiby
则 then
et 
b1 
b1  (1  w)a2
b1  a2

x


1  eiax 

2A
1  cos by  1  eiax
xy

 稳定条件 et  1 等价于Which is equalent to
1 
Re(b1 )  (1  w)a2
1
Re(b1 )  a2
设   0 则AF2的稳定条件Assume   0 then the stability
condition of AF2 is
0 w 2
其中Where
Re(b1 ) 

x
Re(b1 )
a2
1  cos(ax)  
 t
取w和  值使 e
最小得Take w and to
H

2A
iax
1

cos
b

y
1

e


xy 2
let

e t minimized
2
2 2

   (1  A)( )   L
x
L
其中 H、 L 分别为最短波和最长波 are the shortest and longest
wave
由于A是变量，因此上述用线性理论分析得到的结果不能直接用，
要经过试算来确定
Since A is a variable above analyzing based on linear
w， H， L
theory can not be used directly there
must
w，chose
 H，empirically
L
超声速点的AF2稳定分析只考虑误差长波分量，在
时不增长，
0 not
w increase
2
即为稳定线For supersonic points，the longest wave will
when
0 w2
§4-10
SLOR AF1和AF2的算例对比
Comparison of SLOR, AF1 and AF2
 例子：双圆弧型， 相对厚度10% 迎角 0 double arc airfoil,
relative thickness 10%, angle of attack 0
 网格91×21， 物面网点47个grids, 91*21, 47points on
surface
 远场 , Far field
x  5, y  6
c
c
 初场 Initial field
 ij  0
M
 收敛标准
ij（n） 106
 0.7 亚subsonic

0.84 超临界transonic
比较结果：亚临界流，AF1比SLOR 计算效率提高100倍以上Result
of comparison : subsonic the efficiency of AF1 is 100
超临界流，AF2比AF1好，比SLOR提高10倍For supersonic case,
AF2 is better than AF1, and 10 times higher efficiency than that of
SLOR
10
2
ij
AF1
AF2
102
SLO
R
103
105
n
100
200
300
400
小结
Brief summary
本章讲解内容：
Contents discussed
 跨声速小扰动方程 Transonic small perturbation equation
 小扰动激波方程 Equation for small perturbation shock
 小扰动差分方程 FDE of small perturbation flow
 边界条件及其嵌入 The boundary condition and it’s embedding
 隐式近似因式分解法思路（差分表示法） The concept of
implicit approximate factorization （PD calculator）
 AF1
 AF2
重点：
keystone/emphases
小扰动差分方程
PDE of small perturbation flow
边界处理方法
The treatment of boundary conditions
差分表示方法
Expression of PD
难点：
difficult
AF1及其稳定性、收敛性分析
AF1 and it’s stable and convergence analysis
AF2及其稳定性、收敛性分析
AF2 and it’s stable and convergence analysis