presentation

Transcription

presentation

DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS
Volume 28, Number 1, September 2010
doi:10.3934/dcds.2010.28.259
pp. 259–274
MODELS & MEASURES OF MIXING & EFFECTIVE DIFFUSION
Zhi Lin
Institute for Mathematics & Its Applications
University of Minnesota, Minneapolis, MN 55455, USA
Katarı́na Boďová
Department of Applied Mathematics & Statistics
Faculty of Mathematics, Physics and Informatics
Comenius University, 84248 Bratislava, Slovakia
Charles R. Doering
Department of Mathematics, University of Michigan
Ann Arbor, MI 48109-1043, USA
and
Department of Physics and Michigan Center for Theoretical Physics
University of Michigan, Ann Arbor, MI 48109-1040, USA
and
Center for the Study of Complex System, University of Michigan
Ann Arbor, MI 48109-1107, USA
Abstract. Mixing a passive scalar field by stirring can be measured in a
variety of ways including tracer particle dispersion, via the flux-gradient rela-
Big questions:
How can we gauge the effectiveness of a stirrer as a mixer?
How might we parameterize stirring as diffusion?
Outline:
- Models
- Conflicts
- Resolution
- More models
- Reconciliation
Mathematical models of mixing
 

Given flow field u( x,t) with ∇ ⋅ u = 0, consider


 
Stochastic Diff Eq : dX (t) = u( X ,t)dt + 2κ dW (t)
€
•  X(t) is passive tracer particle position
€
•  κ is the molecular diffusion coefficient

Advection - Diffusion Eq : ∂ tθ + u ⋅ ∇θ = κΔθ + s
•  θ(x,t) is passive scalar density, concentration
€
•  s(x,t) is passive scalar source-sink distribution
•  plus appropriate initial and boundary conditions
Mathematical measures of mixing
Temptation & tradition suggest characterizing
stirring as an "effective" diffusion

u ⋅ ∇ − κΔ → - ∂ i K ijeff ∂ j
Three questions:
€
•  Which aspects of mixing should be encoded in Keff ?
•  Do different criteria produce different Keff ?
•  Transferable among applications?
Measure 1: K PD = K ijeff
€
€
Measure 2 : K FG = K11eff
eff
Measure 3: κ VR
=
κ
for p = +1, 0, −1
p
p
p = +1, 0, -1 ~ “small”, “intermediate”, or “large” scale variance reduction
Measure 1: K PD = K ijeff
via tracer particle dispersion
E{( X i (t) − X i (0))( X j (t) − X j (0))} ~ 2K ijeff t
as t → ∞.
€
via flux - gradient relation, T = −Gx + θ ⇒


ˆ
∂tθ + u ⋅ ∇θ = κΔθ + G i ⋅ u
( )
€
€
everything mean zero & periodic on a cell ⇒
€
eff
K11
 2
#
| ∇θ | &
u1θ
(
=κ +
= κ %1+
2
%
(
G
G
$
'
eff
Measure 3 : κ VR
=
κ
0
via concentration variance reduction



For s( x ) mean 0 and ∂ tθ + u ⋅ ∇θ = κΔθ + s( x )
(Δ s)
−1
κ eff =
€
θ2
2
VR
eff
Measure
3(a)
:
κ
=
κ
Multiscale
p
p
via concentration (inverse) gradient variance reduction



For s( x ) mean 0 and ∂tθ + u ⋅ ∇θ = κΔθ + s( x )
±1 −1 2
eff
κ ±1
=
∇ Δ s
±1
∇ θ
2
Measure 1: K
PD
=K
eff
ij
1
~ E{( X i (t) − X i (0))( X j (t) − X j (0))}
2t

%
| ∇θ |2 (
*
= κ '1+
2
'
*
G
&
)
p
eff
Measure 3 : κ VR
=
κ
p
p =
−1 2
∇ Δ s
∇ pθ
2
for p = +1,0,−1
Strength of stirring
Ul
Dimensionless Péclet number : Pe ≡
κ
U ~ velocity scale
€
... l ~ length scale
Dimensionless Enhancement or Efficacy factor:
PD,FG
κ VR
K
E(Pe) ≡ p or
κ
κ
Fact:
In terms of tracer dispersion or flux-gradient relation, there are
flows for which the enhancement may be as large as
K FG
E(Pe) =
~ Pe 2 as Pe → ∞.
κ
Fact:
In terms of concentration variance reduction in presence of steady
sources & sinks the enhancement cannot be that big. κ VR
Theorem : E(Pe) = p ≤ Pe1 as Pe → ∞.
κ
Resolution
•  What length scale l is used in Pe = Ul/κ?
In examples where
K FG
κ
~ Pe 2 as Pe → ∞,
Ul flow
Pe ≡
.
κ
€
κ p VR
In theorem where E(Pe) =
≤ Pe1 as Pe → ∞,
κ
Ul source ' l source * Ul flow
Pe ≡
= ))
.
,, ×
κ
κ
( l flow +
Example: Basic Two-scale Model
A single-scale flow stirring a single-scale source-sink distribution

s( x ) =
 
u( x ) = iˆ 2 U sin ku y
2 S sin k s x
€
€
U
Two parameters : Pe ≡
κk u
l source
ku
and r =
=
l flow
ks
Dispersion/Flux-gradient mixing measure
•  a.k.a. Homogenization Theory (HT) …
•  … presumably good for r = ku/ks >> 1:
K FG
•
= 1+Pe 2
κ
0=K
FG
⇒ HT approximation is
d 2θ HT (x)
+ s(x)
2
dx
(Δ s)
−1
€ HT appx of κ VR =
p
θ
2
HT
⇒
2
2 S sin ks x
θ HT (x) =
κku2 (1 + Pe 2 )
±1
−1
∇ Δ s
=
∇±1θ HT
2
2
2
= κ (1+ Pe )
Exact solution (for r = 1)
Pe = 0 Pe = 100
Pe = 300
High-Pe (fixed r) asymptotic analysis: Internal-layer theory (ILT)
E0 = κ0 VR/κ ~ r 7/6 Pe 5/6
E+1 = κ+1VR/κ ~ r 1/2 Pe 1/2
E–1 = κ–1VR/κ ~ r Pe
r = 562
r = 106
r = 105
r = 104
r = 103
r = 102
r = 101
r = 100
r = 10-1
r = 106
r = 105
r = 104
r = 103
r = 102
E −1
∇ −1θ 0
∇ −1θ
r = 101
2
2
r = 100
r = 10-1
E +1
∇θ 0
∇θ
2
2
r = 106
r = 105
r = 104
r = 103
r = 102
r = 101
r = 100
r = 10-1
Stirring strength–scale separation phase diagram
Outline
•  Models
•  Conflicts
•  Resolution
•  More models
•  Reconciliation
Questions:
•  HT fails to predict the scalar variance sustained by
steady sources & sinks when Pe > r >> 1. Why?
•  Can information about particle dispersion predict
variance supression at high Péclet numbers?
•  Particle dispersion is time and initial-location dependent …
E{( X i (t) − X i (0))( X j (t) − X j (0))} ~ 2K i,PDj t

K
PD
i, j

d 1
(t; X (0)) ≡
2 E{( X i (t) − X i (0))( X j (t) − X j (0))}
dt
•  KPD~ κ Pe2 = O (κ –1) takes O (lflow2/κ) time to develop
€
2

l
... but K i,PDj (t; X (0)) ~ κ + U 2 t (at most) for t << flow κ .
Effective diffusion K11PD (t,y0) vs. time  
u( x ) = ˆi 2 U sin k u y
€
More modeling
•  Concentration variance for stirred scalars sustained by
inhomogeneous sources and sinks is dominated by the
“latest” stuff introduced or deleted from the system.
•  “Old” particles are relatively well mixed and so don’t
contribute substantially to the observed variance.
•  Variance supression is controlled by particle dispersion rate
on relatively short, rather than long, time scales at high Pe.
•  In the presence of sustainted sources & sinks, even as t → ∞
we cannot neglect transient behavior of KPD …
Dispersion-diffusion theory (DDT)
Given a stirring flow u(x,t) and its associated KijPD (t-t0 | x0, t0),
density due to stuff injected at x0, t0 may best be described by
 

PD
∂t ρ ( x,t | x 0 ,t 0 ) = ∂ iK ij (t − t 0 | x 0 ,t 0 )∂ j ρ
 
 
lim ρ ( x,t | x 0 ,t 0 ) = δ ( x,t | x 0 )
t↓t 0
G. K. Batchelor, Diffusion in a field of homogeneous turbulence I. Eulerian analysis, Aust. J. Sci. Res. Series A, Phys. Sci., 2 (1949), 437–450.
Then the total density in presence of sources and
€
sinks is at best described by
t

  

θ DDT ( x,t) = ∫ dt 0 ∫ dx 0 ρ ( x,t | x 0 ,t 0 )s( x 0 ,t 0 )
−∞
… which does not satisfy an inhomogeneous diffusion equation!
€
On a periodic domain [0, L]d
 
1
ρ( x,t | x 0 ,t 0 ) = d
L
Note : if K
€
t
&  
)

PD
∑ exp'(ik ⋅ ( x − x 0 ) − ki k j t∫ K ij (t'−t0 | x 0,t0 )dt'*+
0
k
PD
ij
$
&
&%
~ κ + U (t − t 0 ) δij as t − t 0 → 0, then

θ DDT ( x,t) =
€
'
)
)(
2
∫
t
−∞
  

dt 0 ∫ dx 0 ρ ( x,t | x 0 ,t 0 )s( x 0 )
⇓

 ∞ −κk 2τ − 1 k 2U 2τ 2
2
θˆDDT ( k ) ~ sˆ ( k ) ∫ e
dτ
0


sˆ ( k )
ˆ
⇒ as Pe → ∞, θ DDT ( k ) ~
so κ VR
=
0
€
kU
( Δ−1 s)
2
θ DDT
2
~ Ul source
Reconciliation
•  $64 question: Is DDT quantitatively accurate?
•  For single-scale flow stirring single-scale source …
r = 106
r = 105
r = 104
r = 103
r = 102
r = 101
r = 100
r = 10-1
More reconciliation
•  DDT respects the rigorous bounds on κ0VR .
•  For the single-scale source, the rigorous bound is
E(Pe) = κ0VR /κ ≤ [1 + r2 Pe2]1/2 ~ r Pe …
… for large r or for large Pe!
•  Plot E(Pe) as a function of (r Pe):
r = 101
r
= 102
r
= 103
r
= 104
r
= 105
r
= 106
Reconciliation, continued
•  How does DDT perform for variance supression at
large & small scales, i.e., for κ±1VR ?
r = 106
r = 105
r = 104
r = 103
E −1
r = 102
∇ −1θ 0
∇ −1θ
2
2
r = 101
E +1
∇θ 0
∇θ
2
2
r = 106
r = 105
r = 104
r = 103
r = 102
r = 101
Density pictures (r = 562)
DDT approximation for κ 0VR is uniformly accurate
Conjecture (potential application)
•  Single-scale source, sink & stirring is a special scenario …
… what about real turbulent mixing?
•  DDT hints how particle dispersion data may predict steady
state source-sink sustained variance suppression.
•  Homogeneous isotropic turbulence →
E[(Xi(t)–Xi(0))(Xj(t)–Xj(0))] = (2κ t + U2t2 + CR ε t3 + …) δij
… w/turbulent energy dissipation rate per unit mass ε ~ U3/lflow .
On a periodic domain [0, L]d
  
 
1
ρ ( x, t | x0 , t0 ) ≈ d ∑ exp ik ⋅ ( x − x0 ) − 12 k 2 $%2κ (t − t0 ) +U 2 (t − t0 )2 +&'
L k
{

θ DDT ( x, t) =
}
∫
t
−∞
  

dt0 ∫ dx0 ρ ( x, t | x0 , t0 )s( x0 )
⇓

 ∞ −κ k 2τ − 1 k 2U 2τ 2
2
ˆ
θ DDT (k ) = ŝ(k ) ∫ e
dτ
0
U l flow
l source
⇒ as Pe=
→ ∞ at fixed r=
,
κ
l flow

 ŝ(k )
ˆ
θ DDT (k ) ~
kU
Concrete conjecture:
Does Statistically Homogeneous
Isotropic Turbulence saturate
the upper bound on E (Pe)?
κ
eff
2
(Δ s)
−1
VR
approximated by κ 0 =
2
θ DDT
~ κ r Pe
#l
&
source
((U l flow
= %%
$ l flow '
with l source =
#
%
%
%
%
%
%
$
&1/2
(
(
(
(
2 (
−1/2
Δ
s
(
'
2
Δ−1s
( )
(
)
i.e., "mixing length" ~ l source
↵
Test
UConjecture
l flow Richardson Turbulence l source
⇒ as Pe=
→ ∞ at fixed r=
,
κ
l flow

 ŝ(k )
ˆ
θ DDT (k ) ~
kU
Last words
•  Different definitions of effective diffusion may indeed
yield different effective diffusivities.
•  We cannot generally use long-time transient dispersion
results for source-sink problems.
•  There may not be an effective diffusion equation to
describe source-sink stirring.
•  Flux-gradient model does not contain all the relevant
information for source-sink stirring.
•  Transient mixing and source-sink stirring are different
phenomena using different features of the flow:
Scalar source-sink stirring is all about transport
Transient mixing is all about
shearing, stretching & straining
THE END

presentation

Transcription

Similar documents