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Adnan KhanLahore University of Management Sciences
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES OCTOBER 8, 2012
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FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
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To study turbulent transport of passive scalars
Transport occurs via two mechanisms Advection (at Macro and Meso Scales) Diffusion (Eddy Diffusivity)
The advective field being turbulent has waves at a continuum of wavelengths
We study simplified models with widely separated scales to understand these issues
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
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Want to study the problem at scales of observation (Macro Scales – Mean Flow)
Use homogenization theory to capture effects of the meso scales where we assume periodic fluctuations
The idea is to capture the effects of the flow as a lumped parameter, in this case as an enhancement in diffusivity
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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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As a ‘toy’ problem consider the following Dirichlet Problem
D is periodic in the second ‘fast’ variable
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
)()()xD(x, xfxu =
∇∇
εΩ∈x
Ω∂∈= xxgxu )()(
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Using the ‘ansatz’
We obtain
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
εε∇+∇→∇
1x
)(),(),( 210 εε Oyxuyxuu ++≅
( ) ( )( ) )(2 xfuD xyxy εεε =∇+∇∇+∇
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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 +∇∇−∇∇−∇−∇=∇∇
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
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Applying periodicity and no secularity conditions
O(1)
O(ε)where
O(ε2) onon
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
)(),( 00 xuyxu ≡⇒
)().(),( 01 xcuyayxu x +∇=⇒
( ) )()()( 0 xfxuxD xx =∇∇⇒ Ω)()(0 xgxu = Ω∂
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FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
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
>∂<+>=< δ
∫ΩΩ
=><0
),(1),(0
dAyxvyxv p
( ) DaDiyiyy −∂=∇∇
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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
TPeaTtxavtxVtT
l ∆=∇⋅
++
∂∂ −1),(),( δ
ηδ
1>>a)1(Oa ≈
1<<a
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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
)),((),(_
*_
_
txTKTtxVtT
∇⋅∇=∇⋅+∂∂
*K><−= −
jiijlij vPeK χδ1χ
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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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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
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
11
111 2),(),( dWaPedttxavtxVdX l−+
+=
ηδ
21
222 2),(),( dWPeadttxavtxVdX l−+
+= δ
ηδ
>−−<= ))0()())(0()((21
jjiiij XTXXTXT
K
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Some MC runs with Constant Mean Flow & CS fluctuations
MC and homogenization results agree
Need a modified Homogenization theory
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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
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
δ=a
fgL =0
0L f
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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
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
g
δ1O
<<δ10 Ot
∫δt
dssYsXf0
))(),((
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FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
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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 time
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
∫ ++
≤δ
δδ
t
OVVVVCdssf
02
22
1
22
21 )()(
1V 2V
)1(δ
O)1(
δO
)1(δ
O
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We develop the asymptotic expansion in both the cases
We have the following multiple scales hierarchy
We derive the effective equation for the quantity
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
22
10 TTTT δδ ++=
000 =TL00110 =+ TLTL
0021120 =++ TLTLTL003122130 =+++ TLTLTLTL
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For the low order rational case we get
Where the operators are given by
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
0)( 32
21 =++ TRRR δδ
αα α ∇⋅+∇−∇⋅+∂∂
= − )(~~211 vPeV
tR lx
jijijiijvPePevR lxxxlx ααααααααα χαχχρρα ∂∂∂+∇⋅∇+∇⋅∇+∂∂−∇⋅−∇⋅∇−∇⋅= −−
22221
211
2 2)(~~2)(~~
))((~~22
211
213 xxlxxl vPePeR ∇+∇∇⋅+∇∇+∇⋅+∇−= −−
ααα αχχρ
iR
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For the high order rational ratio case we get the following homogenized equation
FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012
021221 =∇+
∇−∇⋅+∂∂ −− TPeTPeVt xllx δξ
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Random Homogenization – No restriction of periodicity on fluctuations
Lagrangian Inference Problem – In the Lagrangian formulation estimate the ‘drift’ and ‘diffusivity’ from large scale data
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