iit-madras, momentum transfer: july 2005-dec 2005 perturbation: background n algebraic n...

IIT-Madras, Momentum Transfer: July 2005-Dec 2005

Perturbation: Background

Algebraic Differential Equations


Y vs X

-30

-25

-20

-15

-10

-5

0

5

10

15

-8 -6 -4 -2 0 2 4 6 8

X

Y

Epsilon=0.0

Y vs X

-30

-25

-20

-15

-10

-5

0

5

10

15

-8 -6 -4 -2 0 2 4 6 8

X

Y

Epsilon=0.5

Y vs X

-30

-25

-20

-15

-10

-5

0

5

10

15

-8 -6 -4 -2 0 2 4 6 8

X

Y Epsilon=0.8

Perturbation252 XXY

0252 X

Original Equation

0252 XX Perturbed equation

10


Perturbation

Change in result (absolute values) vs Change in equation


Simple (Regular) Perturbation

Answer can be in the form

...)( 200 XXX

Root -1

=0.01=-0.01

=-0.1

=0.1

0

0.01

0.02

0.03

0.04

0.05

0.06

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Epsilon (perturbation)

Per

turb

atio

n i

n t

he

resu

lt (

roo

t)

Root -2

=0.01=-0.01

=0.1

=-0.1

0

0.01

0.02

0.03

0.04

0.05

0.06

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Epsilon (perturbation)

Per

turb

atio

n i

n t

he

resu

lt


Perturbation05X

Original Equation


Y vs X

-10

-5

0

5

10

15

20

25

30

35

40

-6 -4 -2 0 2 4 6

X

Y

Epsilon=0

Y vs X

-10

-5

0

5

10

15

20

25

30

35

40

-6 -4 -2 0 2 4 6

X

Y

Epsilon=1

Y vs X

-10

-5

0

5

10

15

20

25

30

35

40

-6 -4 -2 0 2 4 6

X

Y

Epsilon=0.8

52 XXY

Two roots instead of one

Roots are not close to the original root


Perturbation52 XXY

Other root varies from the original root dramatically, as epsilon approaches zero!

Root-1

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1 1.2

Epsilon

Pe

rtu

rba

tion

in R

esu

lt

Change in result (absolute values) vs Change in equation

Singular perturbation

Answer may NOT be in the form

...)( 200 XXX

Root-2

0

200

400

600

800

1000

1200

0 0.2 0.4 0.6 0.8 1 1.2

Epsilon

Pe

rtu

rba

tion

in R

esu

lt


Differential Equations

0 xdx

dy 0,1 xaty

21

2xy Solution

xdx

dy Perturbation-1

Solution

0,1 xaty

xx

y 2

12

21,0

2xy Regular Perturbation


Differential Equationsa

dx

dy 1a

xay Solution

yadx

dy Perturbation-1 0,0 xaty

Solution 1 xea

y

axy ,0

Regular Perturbation

0,0 xaty

1...

21

22xx

a

...

2

22xx

a

...

21

xax



Another Regular Perturbation

xadx

dy Perturbation-1a 0,0 xaty

2

2xaxy

axy ,0

Solution

Perturbation-1b

Solution

adx

dy0,0 xaty

xaxy

axy ,0



adx

dy

dx

yd

2

2

Perturbation-2 0,0 xaty

Exact Solution (eg using Integrating factors method)

1

1

11

e

eaaxy

x

Singular Perturbation

1,1 xaty

axy ,0

1a


Differential Equations Singular Perturbation

Y vs X

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

X

Y Y=ax

Can the solution be of the following form? (to satisfy the extra boundary condition?)

example)(for5.0a

No! Based on the perturbed equation

1at x 1,y

adx

dy

dx

yd

2

2



Y vs X

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

X

Y Y=ax

Y vs X

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

X

Y

Epsilon 0.1

Y vs X

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

X

Y

Epsilon 0.01

10),1(,0 xforaaxy

At the limit

Method to find solution

Transform variables (x,y,) Called “Stretching Transformation” Zooms in the ‘rapidly varying

domain’ Obtain “inner solution”

0),1(0,0 xforay



Y vs X

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

X

Y Y=ax


Inner solution: Let =0 and simplify eqn 2nd order equation, satisfying only

one boundary condition (x=0) one constant remains arbitrary Valid only near x=0

Outer SolnInner solutions

Obtain outer solution, for first order equation, satisfying one Boundary Condition (x=1) Valid everywhere, except near

x=0



Y vs X

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

X

Y Y=ax


Match the two solutions in the segment in between, by choosing the remaining constant Match the value and the slopes

Outer SolnInner solution

1

11

x

eaaxy

Close to the exact solution


Numerical Solution (to BL) Grid generation

Structured grid vs unstructured grid Uniform vs non-uniform grid

“Real” solution

What about placing more grid everywhere?

More grid points near surface Similar to “stretching

transformation”

Approx solution

“Real” solution

Approx solution


Boundary Layer theory Situations we have seen so far

Laminar flow in cylinder Fully developed (entrance

effects are negligible) Steady State

Unsteady State Again, entrance effects are

negligible Movement of infinite plate, in

a semi-infinite mediumV0


Boundary Layer theory

Flow over cylinder (inviscid) Flow over sphere (2D, viscous

flow) (tutorial problem)

Flow over any other shape (while accounting for no-slip condition

and not assuming fully developed flow) is treated with “boundary layer theory”

Inviscid flow (irrotational) Will NOT satisfy ‘no slip’

condition at the plateFluid Velocity V0

Semi-infinite plate


Boundary Layer theory Away from plate, inviscid

solution is valid (and will satisfy the boundary condition). This is “outer solution”

Near the plate, different solution (including viscosity) will be found using ‘stretching transformation’. [Inner Solution]

Inner solution will satisfy the boundary condition (no slip)

Match both solutions to find the other constant

Fluid Velocity V0

Semi-infinite plate


Boundary Layer theory Solid Boundary

No Slip

Velocity 0

Velocity V0

Velocity V0

0

INF

0

INF

INVISCID FLOWASSUMPTION OK HERE

FRICTION CANNOT BENEGLECTED HERE


Boundary Layer theory Solid Boundary

0

INF

0

INF

x

B L thickness99% Free Stream Velocity

B L thicknessincreases with x

What happens to when you move in x?

Momentum Transfer


Boundary Layer theory Draw vs x

x

B L thickness increases with x

Analytical Expression, for velocity vs (x,y), below BL:

Continuity

Navier Stokes Equation

y


Boundary Layer theory

Steady,incompressible, two dimensional (semi infinite plate)

0.

Vt

0

zyx Vz

Vy

Vxt

0

y

V

x

V yx Hence 1


N-S Eqn

Consider only X and Y equations (2D assumption)

gVPDt

DV 2

xxxxx

zx

yx

xx g

z

V

y

V

x

V

x

P

z

VV

y

VV

x

VV

t

V

2

2

2

2

2

2

yyyyy

zy

yy

xy g

z

V

y

V

x

V

y

P

z

VV

y

VV

x

VV

t

V

2

2

2

2

2

2

Steady flow, gravity can be incorporated in Pressure term (or assume gravity is in Z direction, for example)

Vz=0, Vx and Vyare not functions of z


N-S Eqn Obtain “order of magnitude” idea Can be used to ignore small terms (simplify eqn by removing ‘regular’ perturbations) Can be used to non-dimensionalize equations example:

1

'

L

xx

2

'

L

yy

1

'

U

VV xx

2

'

U

VV yy

21

'

U

PP

1

1'

L

tUt

Steady State


N-S Eqn Write the NS-eqn in “usual” form, for steady state

2

2

2

2

y

V

x

V

x

P

y

VV

x

VV xxx

yx

x

2

2

2

2

y

V

x

V

y

P

y

VV

x

VV yyy

yy

x


N-S Eqn What are the relevant scales for the lengths (eg what are L1, L2 in this particular case?

x

y

L

xx '

L

y

y '

LL 1 2L xoffunctionaisNote :


N-S Eqn What are the relevant scales for the velocity?

x

y

Vx varies from 0 to Vo (or we can call it VINF)

VVx ~ ~ means “Order of ”

Note: Some books show it as VOVx ~

2~UVy Similarly

2

2

2

2

y

V

x

V

x

P

y

VV

x

VV xxx

yx

x


N-S Eqn What are the relevant scales for the derivatives?

0

0~

L

V

x

Vx Note: The sign is not important here

xy

VVx

0x Lx

0xV

L

V

x

Vx

~

2

2

2

2

y

V

x

V

x

P

y

VV

x

VV xxx

yx

x

?~y

Vy

0

y

V

x

V yx Continuity 1

L

V

y

Vy

~ L

VVy

~


N-S Eqn What are the relevant scales for the derivatives?

xy

L

2

2

2

2

y

V

x

V

x

P

y

VV

x

VV xxx

yx

x

V

y

Vx ~

22

2

~L

V

x

Vx

22

2

~

V

y

Vx

LThin Boundary Layer assumption

2

2

2

2

y

V

x

V xx


N-S Eqn Can we approximate pressure drop? Assume that pressure drop is similar to inviscid flow

xy

L

x

VV

x

P

From Bernoulli’s eqn

L

V

x

VV

x

P 2

~

2

2

y

V

x

P

y

VV

x

VV xx

yx

x

L

V

x

VV xx

2

~

L

VV

L

V

y

VV xy

2

~~

22

2

~

V

y

Vx

Claim: as -->0,

zerononV

y

Vx

22

2

~


N-S Eqn

For the Y component of N-S equation

L

V

y

V

x

VV

y

VV

x

VV xx

yx

x

2

2

2

~

2

2

2

2

y

V

x

V

y

P

y

VV

x

VV yyy

yy

x

322

2

~~L

V

LLV

x

Vy

22

2

~~

L

VLV

L

V

y

VV yy

L

V

LLV

Vx

VV yx

2

~~

L

VLV

y

Vy

~~22

2

L

V

y

Vy

~2

2 Each term is small compared to the

equivalent in X-eqn ==> y

P

0


Prandtl BL eqn (steady state)

2

2

y

V

x

VV

y

VV

x

VV xx

yx

x

0

y

V

x

V yx

2

2

y

V

x

VV

y

VV

x

VV

t

V xxy

xx

x

Unsteady State


Prandtl BL eqn : flow over Flat plate

2

2

y

V

x

VV

y

VV

x

VV xx

yx

x 0

y

V

x

V yx

� No pressure Drop� Steady State� 2D-flow (Stream Function)

xy

L

yVx

xVy

ax

� Stretching Transformation (near the boundary)

b c

V

12

1

V

x

21

21

2~~

x

Vy

x

Vy

y

� Non-dimensionalize y



� Another perspective for the choice of

xy

L

L

V

y

V

y

VV

x

VV xx

yx

x

2

2

2

~

L

VV 2

2~

V

L ~2

21

~

V

x

�Boundary Conditions

0,0

y

Vy x

0,0

y

Vy y

Vy

Vy x

,

Vy

Vx x

,0

3

3

2

22

yyxyxy

� If we write the BL eqn in stream function



2

2

y

V

x

VV

y

VV

x

VV xx

yx

x 0

y

V

x

V yx

xy

L

21

),()(

Vx

yxf

fVx 21

y

fVx

yVx

2

1 y

fVx

21

21

2

1

x

V

y

xx

Vy

xx 2

1

22

1 21

21

2

x

Vy



xy

L

fV

Vx

2

fx

V

x

Vx

4

xVy

ffx

VVy

2

1

2

1

fx

V

y

Vx

8

2

2

2

0 fff

2

2

y

V

x

VV

y

VV

x

VV xx

yx

x

Some books may have -ve sign, or a factor of 2, in the equation, depending on the definition of Stream function and transformations used

fx

VV

y

Vx

21

4

x

V

8

2



Boundary Conditions:0 fff

No solution in ‘usual’ form Blasius Solution: Series solution, valid for small

Plot of Vx/VINF vs

0,0 ff 2, f

Note: definition of may be slightly different in various books (usually by a factor of 2)

V

Vx

1

030

0 !

nn

n

Af

For large , asymptotic series that matches with the boundary condition Numerical values tabulated (f,f’,f’’...)


Prandtl BL eqn : flow over Flat plate Blasius Solution

Valid for high Reynolds Number Re

Local:( V/) More useful (convenient): (X V ) (sometimes, this is referred to

as “local” Reynolds number) 105 or more

Not valid very near x= 0 (at the point x=0,y=0) Another way to express boundary layer thickness

212

12

1

Re

1~~~

xVLV

x

Reynolds number high ==> Boundary layer is thin



Boundary layer thickness Drag estimate Other definitions (for thickness) Similarity Effect of pressure variation (Loss of similarity and separation) Thermal vs momentum Boundary Layer von Karman method


References:

Introduction to Mathematical Fluid Dynamics by Richard E Meyer

Perturbation methods in fluid dynamics, by Van Dyke BSL 3W&R Fluid flow analysis by Sharpe Introduction to Fluid Mechanics by Fox & McDonald

iit-madras, momentum transfer: july 2005-dec 2005 perturbation: background n algebraic n...

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