Adnan Khan Department of Mathematics Lahore University of Management Sciences.
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Transcript of Adnan Khan Department of Mathematics Lahore University of Management Sciences.
THEORY OF HOMOGENIZATION WITH
APPLICATIONS TO TURBULENT TRANSPORT
Adnan KhanDepartment of MathematicsLahore University of Management Sciences
2
Outline
Introduction
Theory of Periodic Homogenization
The Advection Diffusion Equation – Eulerian and Lagrangian Pictures
Non Standard Homogenization Theory
SummaryInternational Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8
2010
3
What are Multi Scale Problems
Many physical systems involve more than one time/space scales
Usually interested in studying the system at the large scale
Multiscale techniques have been developed for this purpose
We would like to capture the information at the fast/small scales in some statistical sense
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
4
Some Areas of Application Heterogeneous Porous Media
Bhattacharya et.al, Asymptotics of solute dispersion in periodic porous media, SIAM J. APPL. MATH 49(1):86-98, 1989
Plasma Physics Soward et.al, Large Magnetic Reynold number dynmo action in spatially periodic flow
with mean motion, Proc. Royal Soc. Lond. A 33:649-733
Ocean Atmospheric Science Cushman-Roisin et.al, Interactions between mean flow and finite amplitude mesoscale
eddies in a baratropic ocean Geophys. Astrpophys. Fkuid Dynamics 29:333-353, 1984
Astrophysics Knobloch et.al, Enhancement of diffusive transport in Oscillatory Flows, Astroph. J.,
401:196-205, 1992
Fully Developed Turbulence Lesieur. M., Turbulence in Fluids, Fluid Mechanics and its Applications 1, Kluwer,
Dordrecht, 1990
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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A Picture is Worth ……
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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Theory of Periodic Homogenization
To smooth out small scale heterogeneities
Assume periodicity at small scales for mathematical simplification
Capture the behavior of the small scales in some ‘effective parameter’
Obtain course grained ‘homogenized’ equation at large scale
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
7
A ‘Spherical Cow’
As a ‘toy’ problem consider the following Dirichlet Problem
D is periodic in the second ‘fast’ argument
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
)()()x
D(x,. xfxu
x
xxgxu )()(
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Procedure of Homogenization
Using the ‘ansatz’
Where are periodic functions
We obtain
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
yx 1
)(),(),( 210 Oyxuyxuu
)(2 xfuD xyxy
iu )1(O
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The Asymptotic Hierarchy
Collecting terms with like powers of ε we obtain the following asymptotic hierarchy
O(1):
O(ε):
O(ε2):
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
0),(. 0 uyxD yy
01 .. uDuD xyyy
)()().().(. 0112 xfuDuDuDuD xxyxxyyy
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Solution Process Applying periodicity and zero mean conditions
O(1)
O(ε)where → The ‘Cell Problem’
O(ε2) on Homogenized
on Equation
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
)(),( 00 xuyxu
)().(),( 01 xcuyayxu x
DaD iiyy .
)()()(. 0 xfxuxD xx )()(0 xgxu
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The Homogenized Equation
We have obtained an ‘homogenized’ equation
The effective diffusivity is given by
Where the average over a period is
a is obtained by solving the ‘cell problem’
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
)()(. 0 xfuxD xx
pjyijpij aDDDi
00
1dAp
DaDiyiyy .
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A Model for Turbulent Transport
Transport is governed by the following non dimensionalized Advection Diffusion Equation
There are different distinguished limits
Weak Mean Flow
Equal Strength Mean Flow Strong Mean Flow
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
TPeaTtx
avtxVt
Tl
1),(),(
1a
)1(Oa
1a
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Simplifying Assumptions
We study the simplest case of two scales with periodic fluctuations and a mean flow
The case of weak and equal strength mean flows has been well studied
For the strong mean flow case standard homogenization theory seems to break down
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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Previous Work
For the first two cases we obtain a coarse grained effective equation
is the effective diffusivity given by
is the solution to the ‘cell problem’
The goal is to try an obtain a similar effective equation for the strong mean flow case
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
)),((),(_
*_
_
txTKTtxVt
T
*K
jiijlij vPeK 1
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Our Work
We study the transport using Monte Carlo Simulations for tracer trajectories
We compare our MC results to numerics obtained by extrapolating homogenization code
We develop a non standard homogenization theory to explain our results
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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Monte Carlo Simulations for Tracer Trajectories
We use Monte Carlo Simulations for the particle paths to study the problem
The equations of motion are given by
The enhanced diffusivity is given by
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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111 2),(),( dWaPedttx
avtxVdX l
21
222 2),(),( dWPeadttx
avtxVdX l
))0()())(0()((2
1jjiiij XTXXTX
TK
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Effective Diffusivity from MC and Homogenization
Some MC runs with Constant Mean Flow & CS fluctuations
MC and homogenization results agree
Need a modified Homogenization theoryInternational Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June
7-8 2010
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Non Standard Homogenization Theory
We consider one distinguished limit where we take
We develop a Multiple Scales calculation for the strong mean flow case in this limit
We get a hierarchy of equations (as in standard Multiple Scales Expansion) of the form
is the advection operator, is a smooth function with mean zero over a cell
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
a
fgL 0
0L f
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The Solvability Condition
We develop the correct solvability condition for this case
We want to see if becomes large on time scales
This is equivalent to estimating the following integral
The magnitude of this integral will determine the solvability condition
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
g
1
O
1
0 Ot
t
dssYsXf0
))(),((
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Geometry of the problem
has mean zero over a ‘cell’
Two cases Low order rational ratio High Order rational ratio
Magnitude of Integral in both these cases
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
f
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Characteristic Coordinates
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
Change to coordinates ‘s’ & ‘t’ along and perpendicular to the characteristics
Estimate magnitude of the integral in traversing the cell over the characteristics
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Analysis Analysis of the integral gives the following
Hence the magnitude of the integral depends on the ratio of and
For low order rational ratio the integral gets in time
For higher order rational ratio the integral stays small over timeInternational Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad,
June 7-8 2010
t
OVV
VVCdssf
022
21
22
21 )()(
1V 2V
)1(
O
)1(
O
)1(
O
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The Asymptotic Expansion
We develop the asymptotic expansion in both the cases
We have the following multiple scales hierarchy
We derive the effective equation for the quantity International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8
2010
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10 TTTT
000 TL
00110 TLTL
0021120 TLTLTL003122130 TLTLTLTL
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The Homogenized Equation
For the low order rational case we get
For the high order rational ratio case we get
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
0)( 32
21 TRRR
)(~~21
1 vPeVt
R lx
jijijiijvPePevR lxxxlx
22221
211
2 2)(~~2)(
~~
))((~~
2221
121
3 xxlxxl vPePeR
021221
TPeTPeVt xllx
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Summary
Brief exposition of Periodic Homogenization
Toy Problem to illustrate the process
Advection Diffusion Equation Eulerian Approach – Homogenization Lagrangian Approach – Monte Carlo Simulation
Non Standard Homogenization TheoryInternational Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010