IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Stream Function & Velocity Potential Stream lines/...

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


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


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


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


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


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



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


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


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



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


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


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


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


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?


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

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Stream Function & Velocity Potential Stream lines/...

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