[PPT] - Mathematical Models and Measures of Mixing Part II: More Models and PowerPoint Presentation

SLIDE 1

Mathematical Models and Measures of Mixing

Part II: More Models and Reconciliation Zhi George Lin1 Katarina Bodova1 Charles R. Doering1,2,3

1Department of Mathematics

University of Michigan

2Department of Physics and Michigan Center for Theoretical Physics

University of Michigan

3Center for the Study of Complex Systems

University of Michigan

SIAM Conference on Applications of Dynamical Systems, 2009

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 1 / 24

SLIDE 2

Outline

1

Recap of Part I Models, Measures and Conflicts Resolution Questions

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 2 / 24

SLIDE 3

Outline

1

Recap of Part I Models, Measures and Conflicts Resolution Questions

2

More Modeling Kinetics Dispersion-Diffusion Theory

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 2 / 24

SLIDE 4

Outline

1

Recap of Part I Models, Measures and Conflicts Resolution Questions

2

More Modeling Kinetics Dispersion-Diffusion Theory

3

Reconciliation

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 2 / 24

SLIDE 5

Outline

1

Recap of Part I Models, Measures and Conflicts Resolution Questions

2

More Modeling Kinetics Dispersion-Diffusion Theory

3

Reconciliation

4

Conjecture Richardson Turbulence

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 2 / 24

SLIDE 6

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 7

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 8

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 9

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 10

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 11

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 12

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 13

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 14

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 15

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 16

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 17

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 18

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 19

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 20

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 21

Recap of Part I Models, Measures and Conflicts

Recap of Part I

Advection-Diffusion Equation for a Passive Scalar

∂tθ +

u·∇θ = κ∆θ +s, ∇· u = 0, 〈s〉 = 0 Method Measure Scaling Homogenization Flux-Gradient Ansatz Effective Diffusivity

∼ κPe2

Variational Methods Internal Layer Theory Variance Suppression

∼ κPe

=

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

SLIDE 22

Recap of Part I Resolution

Resolution

Exact HT: 1+Pe2 ILT: r7/6Pe5/6

100 10 2 104 100 102 104 106 108 1010

E0 =

〈θ02〉/〈θ2〉

Pe = U

κku

r

=

k

u

/

k

s

=

5 6 2

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 4 / 24

SLIDE 23

Recap of Part I Resolution

Resolution

Exact HT: 1+Pe2 ILT: r7/6Pe5/6

100 10 2 104 100 102 104 106 108 1010

→ ↑

r 1 r 1 Pe 1 Pe 1 r = P e

HT: r 1 r > Pe ILT: Pe 1 Pe > r

Pe r

E0 =

〈θ02〉/〈θ2〉

Pe = U

κku

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 4 / 24

SLIDE 24

Recap of Part I Questions

Questions

1

HT fails to predict the scalar variance sustained by steady sources and sinks when Pe > r. Why?

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 5 / 24

SLIDE 25

Recap of Part I Questions

Questions

1

HT fails to predict the scalar variance sustained by steady sources and sinks when Pe > r. Why?

2

How can information about particle dispersion predict variance suppression at high P´ eclet numbers without assuming the scale separation?

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 5 / 24

SLIDE 26

More Modeling Kinetics

Kinetics

HT fails to predict the scalar variance sustained by steady sources and sinks when Pe > r. Why?

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 6 / 24

SLIDE 27

More Modeling Kinetics

Kinetics

HT fails to predict the scalar variance sustained by steady sources and sinks when Pe > r. Why? Answer: Physically, a particle needs to “live long enough" to experience the long excursion from the flow, which explains the curves branched at r = Pe;

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 6 / 24

SLIDE 28

More Modeling Kinetics

Kinetics

HT fails to predict the scalar variance sustained by steady sources and sinks when Pe > r. Why? Answer: Physically, a particle needs to “live long enough" to experience the long excursion from the flow, which explains the curves branched at r = Pe; Mathematically, large r and large Pe asymptotics do not commute.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 6 / 24

SLIDE 29

More Modeling Kinetics

Kinetics

HT fails to predict the scalar variance sustained by steady sources and sinks when Pe > r. Why? Answer: Physically, a particle needs to “live long enough" to experience the long excursion from the flow, which explains the curves branched at r = Pe; Mathematically, large r and large Pe asymptotics do not commute.

1 Uks

Time for the flow to carry a particle from a source to a sink

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 6 / 24

SLIDE 30

More Modeling Kinetics

Kinetics

HT fails to predict the scalar variance sustained by steady sources and sinks when Pe > r. Why? Answer: Physically, a particle needs to “live long enough" to experience the long excursion from the flow, which explains the curves branched at r = Pe; Mathematically, large r and large Pe asymptotics do not commute.

1 Uks

Time for the flow to carry a particle from a source to a sink

1

κk2

u

Time for a particle to diffuse across two op- posite streamlines

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 6 / 24

SLIDE 31

More Modeling Kinetics

Kinetics

HT fails to predict the scalar variance sustained by steady sources and sinks when Pe > r. Why? Answer: Physically, a particle needs to “live long enough" to experience the long excursion from the flow, which explains the curves branched at r = Pe; Mathematically, large r and large Pe asymptotics do not commute.

1 Uks

Time for the flow to carry a particle from a source to a sink

1

κk2

u

Time for a particle to diffuse across two op- posite streamlines

1 Uks > 1

κk2

u Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 6 / 24

SLIDE 32

More Modeling Kinetics

Kinetics

HT fails to predict the scalar variance sustained by steady sources and sinks when Pe > r. Why? Answer: Physically, a particle needs to “live long enough" to experience the long excursion from the flow, which explains the curves branched at r = Pe; Mathematically, large r and large Pe asymptotics do not commute.

1 Uks

Time for the flow to carry a particle from a source to a sink

1

κk2

u

Time for a particle to diffuse across two op- posite streamlines

1 Uks > 1

κk2

u ⇐

⇒ r > Pe: Homogenization Theory applies;

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 6 / 24

SLIDE 33

More Modeling Kinetics

Kinetics

HT fails to predict the scalar variance sustained by steady sources and sinks when Pe > r. Why? Answer: Physically, a particle needs to “live long enough" to experience the long excursion from the flow, which explains the curves branched at r = Pe; Mathematically, large r and large Pe asymptotics do not commute.

1 Uks

Time for the flow to carry a particle from a source to a sink

1

κk2

u

Time for a particle to diffuse across two op- posite streamlines

1 Uks > 1

κk2

u ⇐

⇒ r > Pe: Homogenization Theory applies;

1 Uks ≤ 1

κk2

u ⇐

⇒ r ≤ Pe: Homogenization Theory fails.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 6 / 24

SLIDE 34

More Modeling Dispersion-Diffusion Theory

Remedy

How can information about particle dispersion predict variance suppression at high P´ eclet numbers without assuming the scale separation?

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 7 / 24

SLIDE 35

More Modeling Dispersion-Diffusion Theory

Remedy

How can information about particle dispersion predict variance suppression at high P´ eclet numbers without assuming the scale separation? Answer: Since the homogenization limit may not be reached, why not just keep all the transient information of the effective diffusivity?

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 7 / 24

SLIDE 36

More Modeling Dispersion-Diffusion Theory

Remedy

How can information about particle dispersion predict variance suppression at high P´ eclet numbers without assuming the scale separation? Answer: Since the homogenization limit may not be reached, why not just keep all the transient information of the effective diffusivity?

Particle dispersion is time and initial-location dependent Keff( X(0),t) ≡ 1

2 d dtE[(Xi(t)−Xi(0))(Xj(t)−Xj(0))]

⇓

Keff ∼ 1

2tE[(Xi(t)−Xi(0))(Xj(t)−Xj(0))]

t → ∞ Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 7 / 24

SLIDE 37

More Modeling Dispersion-Diffusion Theory

Remedy

How can information about particle dispersion predict variance suppression at high P´ eclet numbers without assuming the scale separation? Answer: Since the homogenization limit may not be reached, why not just keep all the transient information of the effective diffusivity?

Particle dispersion is time and initial-location dependent Keff( X(0),t) ≡ 1

2 d dtE[(Xi(t)−Xi(0))(Xj(t)−Xj(0))]

⇓

Keff ∼ 1

2tE[(Xi(t)−Xi(0))(Xj(t)−Xj(0))]

t → ∞

Keff ∼ κPe2 takes O(1/(κk2

u)) time to develop

But Keff( X(0),t) ∼ κ+U2t (at most) for t ≪

1

κk2

u Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 7 / 24

SLIDE 38

More Modeling Dispersion-Diffusion Theory

Remedy

How can information about particle dispersion predict variance suppression at high P´ eclet numbers without assuming the scale separation? Answer: Since the homogenization limit may not be reached, why not just keep all the transient information of the effective diffusivity?

For a single-scale shear u =ˆ i

2 Usin(kuy)

Keff( X0,t) = 1 2

E

X(t)−X0 2 E X(t)−X0 Y(t)−Y0

E

X(t)−X0 Y(t)−Y0

E

Y(t)−Y0 2

=

      κt +U2 k2 uκt−1+e−k2 uκt k4 uκ2 −cos(2kuY0) 3−4e−k2 uκt+e−4k2 uκt 12k4 uκ2

Ucos(kuY0)t
2ku

1−e−k2 uκt k2 uκt −e−k2 uκt Ucos(kuY0)t

2ku

1−e−k2 uκt k2 uκt −e−k2 uκt κt       Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 7 / 24

SLIDE 39

More Modeling Dispersion-Diffusion Theory

Time and Space Dependence of Keff

1 1(Y0,t)

10

−2

10 10

2

10

4

0.2 0.4 0.6 0.8 1

k2

uκt K11−κt

Pe

2κt

kuY0 = 0 kuY0 = π/2 kuY0 = π/4

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 8 / 24

SLIDE 40

More Modeling Dispersion-Diffusion Theory

What Does Sources and Sinks Do Actually?

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 9 / 24

SLIDE 41

More Modeling Dispersion-Diffusion Theory

What Does Sources and Sinks Do Actually?

1

Bulk concentration variance for stirred scalars sustained by inhomogeneous sources and sinks is dominated by the latest stuff introduced or deleted from the system.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 9 / 24

SLIDE 42

More Modeling Dispersion-Diffusion Theory

What Does Sources and Sinks Do Actually?

1

Bulk concentration variance for stirred scalars sustained by inhomogeneous sources and sinks is dominated by the latest stuff introduced or deleted from the system.

2

“Old" particles are relatively well mixed and so don’t contribute substantially to the observed variance.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 9 / 24

SLIDE 43

More Modeling Dispersion-Diffusion Theory

What Does Sources and Sinks Do Actually?

1

Bulk concentration variance for stirred scalars sustained by inhomogeneous sources and sinks is dominated by the latest stuff introduced or deleted from the system.

2

“Old" particles are relatively well mixed and so don’t contribute substantially to the observed variance.

3

Variance suppression is controlled by particle dispersion rate on relatively short, rather than long, time scales at high Pe

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 9 / 24

SLIDE 44

More Modeling Dispersion-Diffusion Theory

What Does Sources and Sinks Do Actually?

1

Bulk concentration variance for stirred scalars sustained by inhomogeneous sources and sinks is dominated by the latest stuff introduced or deleted from the system.

2

“Old" particles are relatively well mixed and so don’t contribute substantially to the observed variance.

3

Variance suppression is controlled by particle dispersion rate on relatively short, rather than long, time scales at high Pe

4

In the presence of sources and sinks, even as t → ∞ we cannot neglect the transient behavior of Keff.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 9 / 24

SLIDE 45

More Modeling Dispersion-Diffusion Theory

Dispersion-Diffusion Theory (DDT)

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 10 / 24

SLIDE 46

More Modeling Dispersion-Diffusion Theory

Dispersion-Diffusion Theory (DDT)

For a homogeneous diffusion equation with diffusivity K

∂tρ = ∂i

Kij∂jρ
ρ(
x,t = t0)

= δ(

x−

x0) finds the kernel.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 10 / 24

SLIDE 47

More Modeling Dispersion-Diffusion Theory

Dispersion-Diffusion Theory (DDT)

For a homogeneous diffusion equation with diffusivity K

∂tρ = ∂i

Kij∂jρ
ρ(
x,t = t0)

= δ(

x−

x0) finds the kernel.

θ(

x,t) =

t

dt0

Rd d
x0ρ(
x,t|
x0,t0)s(
x0,t0).

solves the inhomogeneous diffusion equation with source-sink s(

x,t)

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 10 / 24

SLIDE 48

More Modeling Dispersion-Diffusion Theory

Dispersion-Diffusion Theory (DDT)

Now replace K with Keff(

x0,t)

∂tρDDT = ∂i

Keff

ij (

x0,t0)∂jρDDT
ρDDT(
x,t = t0)

= δ(

x−

x0) finds the effective kernel.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 10 / 24

SLIDE 49

More Modeling Dispersion-Diffusion Theory

Dispersion-Diffusion Theory (DDT)

Now replace K with Keff(

x0,t)

∂tρDDT = ∂i

Keff

ij (

x0,t0)∂jρDDT
ρDDT(
x,t = t0)

= δ(

x−

x0) finds the effective kernel. BUT

θDDT(

x,t) =

t

dt0

Rd d
x0ρDDT(
x,t|

x0,t0)s(

x0,t0).

does NOT solve an inhomogeneous diffusion equation. Yet this is the best “effective diffusion" approximation possible.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 10 / 24

SLIDE 50

More Modeling Dispersion-Diffusion Theory

Reality Check

On a periodic domain [0,L]d

ρDDT(

x,t|
x0,t0) = 1

Ld

k

exp

i
k(
x−

x0)−kiku

t

t0

Keff

ij (s−t0 |

x0,t0)ds

Lin, Bodova, Doering (U. Michigan)

Mixing Models and Measures SIAM DS09 11 / 24

SLIDE 51

More Modeling Dispersion-Diffusion Theory

Reality Check

On a periodic domain [0,L]d

ρDDT(

x,t|
x0,t0) = 1

Ld

k

exp

i
k(
x−

x0)−kiku

t

t0

Keff

ij (s−t0 |

x0,t0)ds

Note: If Keff

ij ∼ [κ+U2(t −t0)]δij as t → t0, then as Pe → ∞

θDDT(

x,t) =

t dt0

Rd d
x0ρDDT(
x,t|

x0,t0)s(

x0,t0)

⇒ ˆ θDDT(

x,t) ∼ ˆ

s(

k)

∞

0 e−|

k|2κτ− 1

2 |

k|2U2τ2dτ

∼ ˆ

s(

k)

U|

k|

⇒ κeff

DDT =

〈(∆−1s)2〉

〈θ2

DDT〉

∼ U

ku

∝ Pe

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 11 / 24

SLIDE 52

More Modeling Dispersion-Diffusion Theory

Reality Check

On a periodic domain [0,L]d

ρDDT(

x,t|
x0,t0) = 1

Ld

k

exp

i
k(
x−

x0)−kiku

t

t0

Keff

ij (s−t0 |

x0,t0)ds

Note: If Keff

ij ∼ [κ+U2(t −t0)]δij as t → t0, then as Pe → ∞

θDDT(

x,t) =

t dt0

Rd d
x0ρDDT(
x,t|

x0,t0)s(

x0,t0)

⇒ ˆ θDDT(

x,t) ∼ ˆ

s(

k)

∞

0 e−|

k|2κτ− 1

2 |

k|2U2τ2dτ

∼ ˆ

s(

k)

U|

k|

⇒ κeff

DDT =

〈(∆−1s)2〉

〈θ2

DDT〉

∼ U

ku

∝ Pe

The homogenization scaling Pe2 can also be trivially obtained

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 11 / 24

SLIDE 53

Reconciliation

$64 question: Is DDT quantitatively accurate?

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 12 / 24

SLIDE 54

Reconciliation

$64 question: Is DDT quantitatively accurate?

For single-scale flow stirring and single-scale source ...

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 12 / 24

SLIDE 55

Reconciliation

$64 question: Is DDT quantitatively accurate?

For single-scale flow stirring and single-scale source ...

Exact HT: 1+Pe2 ILT: r7/6Pe5/6

100 10 2 104 100 102 104 106 108 1010

E0 =

〈θ02〉/〈θ2〉

Pe = U

κku

r = 106 r = 105 r = 104 r = 103 r = 100 r = 10 r = 1 r = 0.1 Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 12 / 24

SLIDE 56

Reconciliation

$64 question: Is DDT quantitatively accurate?

For single-scale flow stirring and single-scale source ...

Exact DDT HT: 1+Pe2 ILT: r7/6Pe5/6

100 10 2 104 100 102 104 106 108 1010

E0 =

〈θ02〉/〈θ2〉

Pe = U

κku

r = 106 r = 105 r = 104 r = 103 r = 100 r = 10 r = 1 r = 0.1 Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 12 / 24

SLIDE 57

Reconciliation

More Reconciliation

DDT respects the rigorous bounds on κeff

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 13 / 24

SLIDE 58

Reconciliation

More Reconciliation

DDT respects the rigorous bounds on κeff For the single-source, the rigorous bound is

E0(r,Pe) = κeff κ ≤

1+r2Pe2 ∼ rPe ...

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 13 / 24

SLIDE 59

Reconciliation

More Reconciliation

DDT respects the rigorous bounds on κeff For the single-source, the rigorous bound is

E0(r,Pe) = κeff κ ≤

1+r2Pe2 ∼ rPe ...

... for large r OR large Pe

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 13 / 24

SLIDE 60

Reconciliation

More Reconciliation

DDT respects the rigorous bounds on κeff For the single-source, the rigorous bound is

E0(r,Pe) = κeff κ ≤

1+r2Pe2 ∼ rPe ...

... for large r OR large Pe

Plot E0(r,Pe) as a function of rPe ...

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 13 / 24

SLIDE 61

Reconciliation

More Reconciliation

r = 10 r = 100 r = 103 r = 104 r = 105 r = 106 Exact Efficacy DDT Efficacy Bound 1+(rPe)2

102 104 106 108 1010 102 103 104 105 106 107 108 109 1010 1011

E0 =

〈θ02〉/〈θ2〉

rPe = U

κks

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 13 / 24

SLIDE 62

Reconciliation

More Reconciliation

How does DDT perform for variance supression at large & small scales, i.e., for κeff

± ≡ κE±1?

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 14 / 24

SLIDE 63

Reconciliation

More Reconciliation

How does DDT perform for variance supression at large & small scales, i.e., for κeff

± ≡ κE±1?

1

Exact DDT HT: 1+Pe2 ILT: ~rPe/ln(Pe)

100 102 104 106 100 102 104 106 108 1010 1012

E−1 =

〈∇−1θ02〉

〈∇−1θ2〉

Pe = U

κku

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 14 / 24

SLIDE 64

Reconciliation

More Reconciliation

How does DDT perform for variance supression at large & small scales, i.e., for κeff

± ≡ κE±1?

Exact DDT HT: 1+Pe2 ILT: rPe

100 102 104 106 100 102 104 106 108 1010 1012

E+1 =

〈∇θ02〉

〈∇θ2〉

Pe = U

κku

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 14 / 24

SLIDE 65

Reconciliation

Scalar Fields Comparison at r = 562

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 15 / 24

SLIDE 66

Reconciliation

Scalar Fields Comparison at r = 562

Pe=10 Pe=100 Pe=1000 Pe=10000

Exact DDT HT

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 15 / 24

SLIDE 67

Reconciliation

→ ↑

r 1 r 1 Pe 1 Pe 1 r = P e

HT: r 1 r > Pe ILT: Pe 1 Pe > r

Pe r

... DDT is valid for E0 for arbitrary r and Pe.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 16 / 24

SLIDE 68

Reconciliation

→ ↑

r 1 r 1 Pe 1 Pe 1 r = Pe

HT: r 1 r > Pe ILT: Pe 1 Pe > r

Pe r

... DDT is valid for E0 for arbitrary r and Pe.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 16 / 24

SLIDE 69

Conjecture Richardson Turbulence

Richardson Turbulence

Ongoing research

Single-scale source, sink & stirring is a special scenario ...

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 17 / 24

SLIDE 70

Conjecture Richardson Turbulence

Richardson Turbulence

Ongoing research

Single-scale source, sink & stirring is a special scenario ...

... what about real turbulent mixing?

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 17 / 24

SLIDE 71

Conjecture Richardson Turbulence

Richardson Turbulence

Ongoing research

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

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 17 / 24

SLIDE 72

Conjecture Richardson Turbulence

Richardson Turbulence

Ongoing research

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 → Richardson diffusion:

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 17 / 24

SLIDE 73

Conjecture Richardson Turbulence

Richardson Turbulence

Ongoing research

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 → Richardson diffusion: E[(Xi(t)−Xi(0))(Xj(t)−Xj(0))] ∼ 2κt +CR ǫt3

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 17 / 24

SLIDE 74

Conjecture Richardson Turbulence

Richardson Turbulence

Ongoing research

On a periodic domain [0,L]d

ρDDT(

x,t|
x0,t0) ≈ 1

Ld

k

exp

i
k(
x−

x0)− 1 2|

k|2[2κ(t −t0)+U2t2 +CR ǫt3]
Lin, Bodova, Doering (U. Michigan)

Mixing Models and Measures SIAM DS09 18 / 24

SLIDE 75

Conjecture Richardson Turbulence

Richardson Turbulence

Ongoing research

On a periodic domain [0,L]d

ρDDT(

x,t|
x0,t0) ≈ 1

Ld

k

exp

i
k(
x−

x0)− 1 2|

k|2[2κ(t −t0)+U2t2 +CR ǫt3]
As Pe = U

κku → ∞ at fixed r = ku

ks

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 18 / 24

SLIDE 76

Conjecture Richardson Turbulence

Richardson Turbulence

Ongoing research

On a periodic domain [0,L]d

ρDDT(

x,t|
x0,t0) ≈ 1

Ld

k

exp

i
k(
x−

x0)− 1 2|

k|2[2κ(t −t0)+U2t2 +CR ǫt3]
As Pe = U

κku → ∞ at fixed r = ku

ks

θDDT(

x,t) =

t dt0

Rd d
x0ρDDT(
x,t|

x0,t0)s(

x0,t0)

⇒ ˆ θDDT(

x,t) ∼ ˆ

s(

k)

∞

0 e−|

k|2κτ− 1

2 |

k|2CR ǫt3dτ

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 18 / 24

SLIDE 77

Conjecture Richardson Turbulence

Richardson Turbulence

Ongoing research

On a periodic domain [0,L]d

ρDDT(

x,t|
x0,t0) ≈ 1

Ld

k

exp

i
k(
x−

x0)− 1 2|

k|2[2κ(t −t0)+U2t2 +CR ǫt3]
As Pe = U

κku → ∞ at fixed r = ku

ks

θDDT(

x,t) =

t dt0

Rd d
x0ρDDT(
x,t|

x0,t0)s(

x0,t0)

⇒ ˆ θDDT(

x,t) ∼ ˆ

s(

k)

∞

0 e−|

k|2κτ− 1

2 |

k|2CR ǫt3dτ

⇒ ˆ θDDT(

x,t) ∼

ˆ

s(

k)

U|

k|2/3k1/3

u

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 18 / 24

SLIDE 78

Conjecture Richardson Turbulence

Concrete Conjecture

κeff approximated by κeff

DDT

=

〈(∆−1s)2〉

〈θ2

DDT〉

∼ κr

4 3 Pe

= ku

ks

4

3 U

ku with ks =

〈(∆−1/3s)2〉

〈(∆−1s)2〉

8

3

, i.e.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 19 / 24

SLIDE 79

Conjecture Richardson Turbulence

Concrete Conjecture

κeff approximated by κeff

DDT

=

〈(∆−1s)2〉

〈θ2

DDT〉

∼ κr

4 3 Pe

= ku

ks

4

3 U

ku with ks =

〈(∆−1/3s)2〉

〈(∆−1s)2〉

8

3

, i.e. “Mixing Length" ∼

ku

k4

s

1

3 Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 19 / 24

SLIDE 80

Conjecture Richardson Turbulence

Test

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 20 / 24

SLIDE 81

Last Words

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 21 / 24

SLIDE 82

Last Words

Different definitions of effective diffusion may indeed yield different effective diffusivities.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 21 / 24

SLIDE 83

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.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 21 / 24

SLIDE 84

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.

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 21 / 24

SLIDE 85

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. Transient mixing and source-sink stirring are different phenomena exploiting different features of the flow:

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 21 / 24

SLIDE 86

Last Words

They May Look Simiar, But ...

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 22 / 24

SLIDE 87

Last Words

They May Look Simiar, But ...

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 22 / 24

SLIDE 88

Last Words

They May Look Simiar, But ...

Scalar source-sink stirring is all about transport

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 22 / 24

SLIDE 89

Last Words

They May Look Simiar, But ...

Scalar source-sink stirring is all about transport

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining

Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

Roberto Camassa, Neil Martinsen-Burrell, and Richard M. McLaughlin Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA PHYSICS OF FLUIDS 19, 117104 2007

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 22 / 24

SLIDE 90

Last Words

They May Look Simiar, But ...

Scalar source-sink stirring is all about transport

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 22 / 24

SLIDE 91

Last Words

They May Look Simiar, But ...

Scalar source-sink stirring is all about transport

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining

Scalar source-sink stirring is all about transport about transport Transient mixing is all about shearing, stretching & straining Transient mixing is all about shearing, stretching & straining

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 22 / 24

SLIDE 92

Acknowledgments

Thanks to Roberto Camassa, Pete Kramer, Richard McLaughlin, Bill Young, ... Supported in part by NSF Awards PHY–555324 and DMS–0553487

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 23 / 24

SLIDE 93

References

ZL, Bodova, K. and Doering, C. R. (2009) Mixing Measures and Scaling Laws for Passive Scalars, Preprint Majda, A. J. and Kramer, P. R. (1999) Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena, Physics Reports 314: 237-574 Shaw, T. A., Thiffeault, J. L. and Doering, C. R. (2007) Stirring up trouble: Multi-scale mixing measures for steady scalar sources, Physica D 231 (2):143–164 Young, W. R. and Jones, S. (1991) Shear Dispersion,

Phys. Fluids A, 3 (5): 1087–1101

Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 24 / 24