iit-madras, momentum transfer: july 2005-dec 2005 perturbation: background n algebraic n...
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IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Perturbation: Background
Algebraic Differential Equations
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Perturbation
Change in result (absolute values) vs Change in equation
0252 XX Perturbed 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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Perturbation05X
Original Equation
052 XX Perturbed 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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Differential Equations
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Differential Equations
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Differential Equations
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Differential Equations
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
Method to find solution
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Differential Equations
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
Method to find solution
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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 :
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
~
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
~
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Prandtl BL eqn : flow over Flat plate
� 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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Prandtl BL eqn : flow over Flat plate
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Prandtl BL eqn : flow over Flat plate
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’’...)
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Prandtl BL eqn : flow over Flat plate
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
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