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

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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

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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


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A Picture is Worth ……


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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


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A ‘Spherical Cow’

As a ‘toy’ problem consider the following Dirichlet Problem

D is periodic in the second ‘fast’ argument


)()()x

D(x,. xfxu

x

xxgxu )()(

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Procedure of Homogenization

Using the ‘ansatz’

Where are periodic functions

We obtain


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):


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


)(),( 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’


)()(. 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


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


14

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


)),((),(_

*_

_

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


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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


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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

18

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


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


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


f

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Characteristic Coordinates


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

22

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


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

adnan khan department of mathematics lahore university of management sciences

Documents

periodic flow

homogenized equation

mean motion

cell problem o

mean conditions o1 o

periodic porous media

periodic functions

toy problem