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IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Stream Function & Velocity Potential Stream lines/ Stream Function ()
Concept Relevant Formulas Examples Rotation, vorticity
Velocity Potential() 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
0
zyx Vz
Vy
Vxt
0x yV Vx y
Vx and Vy are related Can you write a common function for both?
xy
VV dy
x
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Stream Function Assume
xV y
Instead of two functions, Vx and Vy, we need to solve for only one function Stream Function
Order of differential eqn increased by one
Then2
xy
VV dy dy
x x y
2
dyy x x
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
x y z
dx dy dz
V V V
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation Definition of rotation
Time=t
x
y
ROTATION2z
d
dt
x
y
y x xV
y xV
x yV
x y yV
Assume Vy|x < Vy|x+x
and Vx|y > Vx|y+y
Time = t + t
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation To Calculate Rotation
1tany
x
Time = t + t
y1
x
1 y yx x xy V t V t
arctan
y yx x xV V t
x
arctanx xy y yV V t
y
Similarly
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation To Calculate Rotation
ROTATION2z
d
dt
0
1lim
2t t t
t t
For very small time and very small element, x, y and t are close to zero
0 0
arctan arctan
1 1lim lim
2 2
y y x xy y yx x x
t t
V V t V V t
x y
t t
0x 0y 0x
0y
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
0 0
arctan
lim lim
y y y yx x x x x x
t t
V V t V V t
x x
t t
0x 0y
0x 0y
Rotation To Calculate Rotation
For very small i.e.
cos 1 tan
arctan
arctan
y y y yx x x x x xV V t V V t
x x
sin
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation To Calculate Rotation
0
limy y yx x x
x
V V V
x x
1
2y x
z
V V
x y
Simplifies to
0 0
arctan arctan
1 1lim lim
2 2
y y x xy y yx x x
zt t
V V t V V t
x y
t t
0x
0y 0x 0y
1
2z V
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
To write rotation in terms of stream functions
2 2
2 2
1 1
2 2y x
z
V V
x y x y
xV y
yV x
21
2
2 2 0z
That is
For irrotational flow (z=0)
2 0
Rotation in terms of Stream Function
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
This equation is “similar” to continuity equation Vx and Vy are related
Can we find a common function to relate both Vx and Vy ?
For irrotational flow (z=0)
10
2z V
0V
0y xV V
x y
Rotation and Potential
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Velocity Potential
In 3D, similarly it can be shown that
Assume
xV x
y xV V
x y
0y xV V
x y
2
y x
2
x y
yV y
zV z
Then
is the velocity potential
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Velocity Potential vs Stream Function
Stream Function () Velocity Potential ()only 2D flow all flows
viscous or non-viscous flowsIrrotational (i.e. Inviscid or zero viscosity) flow
Incompressible flow (steady or unsteady)
Incompressible flow (steady or unsteady state)
compressible flow (steady state only)
compressible flow (steady or unsteady state)
Exists for
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 2D (viscous or inviscid) flow (incompressible flow OR steady state compressible flow), = constant represents the streamline.
Proof
d dx dyx y
0
y xV dx V dy
If = constant, then d
y
x
Vdy
dx V
If = constant, then
Vx
Vy