Transcript Stream Function & Velocity Potential

Stream Function & Velocity Potential

Stream lines/ Stream Function (Y)
 Concept
 Relevant
Formulas
 Examples
 Rotation, vorticity

Velocity Potential(f)
 Concept
 Relevant
Formulas
 Examples
 Relationship between stream function and velocity
potential
 Complex velocity potential
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Stream Lines


Consider 2D incompressible flow
Continuity Eqn
 


 Vx   Vy   Vz   0
t x
y
z


Vx   Vy   0
x
y


 Vx
Vy    
 x
Vx and Vy are related
Can you write a common function for both?
IIT-Madras, Momentum Transfer: July 2005-Dec 2005

dy

Stream Function
Assume


Then



Vx 
y
2

 

dy     xy dy



  2 
  
 
dy   


y

x

x




 Vx
Vy    
 x
Instead of two functions, Vx and Vy, we need to solve
for only one function   Stream Function
Order of differential eqn increased by one
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Stream Function

What does Stream Function  mean?

Equation for streamlines in 2D are given by
 = constant

Streamlines may exist in 3D also, but stream function
does not
 Why?
(When we work with velocity potential, we may
get a perspective)
 In 3D, streamlines follow the equation
dx dy dz
 
Vx Vy Vz
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation

Definition of rotation

Vx
y Dy
Time=t
Dy
y
Vx

Dx
y
Vy
Vy
x
x
Assume Vy|x < Vy|x+Dx
and Vx|y > Vx|y+Dy
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
x Dx
d    
ROTATION   z 


dt  2 
Rotation

To Calculate Rotation
Dy1
tan  
Dx

Dy1  Vy
x Dx
V

  arctan

Similarly
  arctan
 
Dt  Vy
y x Dx
Vy
Dx
 Vx y Dy Vx

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Dy
Dt
x
x

 Dt
y
 Dt

Dx
Dy1
Rotation

To Calculate Rotation
    







d    
1
t Dt
t



lim
ROTATION   z 



2 Dt 0 
Dt
dt  2 



V
 y x Dx Vy
arctan 
Dx

1
 

   lim
Dt
 2  DDxt 0 0
Dy  0

x
 Dt 


V
 x y Dy Vx
arctan 
Dy
  1 

   lim
Dt
 2  Dt 0
Dx  0
Dy  0
For very small time and very small element, Dx, Dy
and Dt are close to zero
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
y
 Dt 


Rotation

To Calculate Rotation

For very small , i.e. ~ 0
sin   
 tan   
cos   1
 arctan   

V
 y
 arctan 

x Dx
Vy
x
Dx

V
 y
arctan 


lim
x Dx
Vy
Dx
Dt
Dt 0
Dx  0
Dy  0
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
 Dt   V


x
y x Dx
 Dt 
x
 Dt
Dx

V
 y


  lim 

Dt 0
Dx  0
Dy  0
Vy
x Dx
Vy
Dx
Dt
x
 Dt 


Rotation

To Calculate Rotation
V

lim
Dx 0
Vy
y x Dx
Dx
x
  V
y
x

V
 y x Dx Vy
arctan 
Dx

1
 

 z    lim
Dt
 2 DxDt00
Dy  0
x
 Dt 

V
 x y Dy Vx
arctan 
Dy
  1 

   lim
0
Dt
 2 DDx t 
0
Dy  0
Simplifies to
 1   Vy Vx 
z    


2

x

y
 

1
z       V 
2
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
y
 Dt 

Rotation in terms of Stream Function

To write rotation in terms of stream functions
Vx 

y
  
Vy   


x


2
2
 1   Vy Vx   1       
z    

     2  2 
y   2   x
y 
 2   x
1
     2
2

That is
 2  2 z  0

For irrotational flow (z=0)
2  0
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation and Potential

For irrotational flow (z=0)


1
z    V  0
2
V  0
Vy
Vx

0
x
y



This equation is “similar” to continuity equation
Vx and Vy are related
Can we find a common function to relate both Vx
and Vy ?
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Vy
Vx

0
x
y
Velocity Potential

Assume
f
Vx 
x
Vy 


Then
Vy
Vx

x
y
f
y
In 3D, similarly it can be shown that
f
Vz 
z

 2f

yx
f is the velocity potential
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
 2f

xy
Velocity Potential vs Stream
Function
Stream Function () Velocity Potential (f)
only 2D flow
all flows
Irrotational (i.e. Inviscid or
viscous or non-viscous flows zero viscosity) flow
Exists
Incompressible flow (steady or Incompressible flow (steady
for
unsteady)
or unsteady state)
compressible flow (steady
compressible flow (steady or
state only)
unsteady state)


In 2D inviscid flow (incompressible flow OR steady
state compressible flow), both functions exist
What is the relationship between them?
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Stream Function- Physical
meaning

Statement: In

Proof

2D (viscous or inviscid) flow
(incompressible flow OR steady state compressible
flow),  = constant represents the streamline.
If  = constant, then d0
 
d  
 x
 

 dx  

 y

 dy

  Vy  dx  Vx  dy
0

Vy
If  = constant, then
dy Vy

dx Vx
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Vx