Non-Fickian diffusion and Minimal Tau Approximation from numerical turbulence

Non-Fickian diffusion and Non-Fickian diffusion and Minimal Tau Approximation Minimal Tau Approximation from numerical turbulencefrom numerical turbulence

A. Brandenburg1, P. Käpylä2,3, A. Mohammed4

1 Nordita, Copenhagen, Denmark2 Kiepenheuer Institute, Freiburg, Germany

3 Dept Physical Sciences, Univ. Oulu, Finland4 Physical Department, Oldenburg Univ., Germany

…and why First Order Smoothing seemed always better than it deserved to be

2

MTA - the minimal tau approximationMTA - the minimal tau approximation

1) replace triple correlation by quadradatic

2) keep triple correlation

3) instead of now:

4) instead of diffusion eqn: damped wave equation

uc

cuu uUU

ccCct

cuuuU

neglected!not

cct uu /F 'd)'( ttcc uuF

cCC

Ct

C

t

C 2231

2

2 1

u

i) any support for this proposal??ii) what is tau??

(remains to be justified!)

Brandenburg: non-Fickian diffusion 3

Purpose and backgroundPurpose and background

• Need for user-friendly closure model• Applications (passive scalar just benchmark)

– Reynolds stress (for mean flow)

– Maxwell stress (liquid metals, astrophysics)

– Electromotive force (astrophysics)

• Effects of stratification, Coriolis force, B-field• First order smoothing is still in use

– not applicable for Re >> 1 (although it seems to work!)

4

Testing MTA: passive scalar “diffusion”Testing MTA: passive scalar “diffusion”

>>1 (!)

Ct

C

U

ccCt

c

uuu

cCt

c

uuuuu

c

Ct

c uuu

u

primitive eqn

fluctuations

Flux equation triple moment

MTA closure

0U


System of mean field equationsSystem of mean field equations

>>1 (!)

F

t

C

FF

C

tuu

mean concentration

flux equation

Damped wave equation, wave speed

Ct

C

t

C 2231

2

2 1

u

231 u

(causality!)


Wave equation: consequencesWave equation: consequences

>>1 (!)

small tau

i) late time behavior unaffected (ordinary diffusion)ii) early times: ballistic advection (superdiffusive)

large tauintermediate tau

Illustration of wave-like behavior:


Comparison with DNSComparison with DNS

• Finite difference– MPI, scales linearly– good on big Linux clusters

• 6th order in space, 3rd order in time

• forcing on narrow wavenumber band

• Consider kf/k1=1.5 and 5


Test 1: initial top hat functionTest 1: initial top hat function

Monitor width and kurtosis

black:closure model

red:turbulence sim.

Fit results:kf/k1 St=ukf

1.5 1.8 2.2 1.8 5.1 2.4


Comparison with Fickian diffusionComparison with Fickian diffusion

No agreement whatsoever


Spreading of initial top-hat functionSpreading of initial top-hat function

Test 2: finite initial flux experimentTest 2: finite initial flux experiment

0CInitial state: but with

black:closure model

red:turbulence sim.

0F

direct evidence for oscillatory behavior!

DispersionRelation:Oscillatoryfor k1/kf<3


Test 3: imposed mean C gradientTest 3: imposed mean C gradient

>>1 (!)

Convergence to St=3for different Re


kf=5kf=5


kf=1.5kf=1.5


Comment on the bottleneck effect Comment on the bottleneck effect Dobler et al (2003) PRE 68, 026304


Bottleneck effect: Bottleneck effect: 1D vs 3D spectra1D vs 3D spectra


Relation to ‘laboratory’ 1D spectraRelation to ‘laboratory’ 1D spectra2222

3 )(4)( kuku kdkE kD yxkyxkE zzD d d ),,(2)(

2

1 u

kkkkkkkzk

z d )(4d ),(42

0

2

uu

kk

E

zk

D d 3

Parseval

222zkkk

used:


ConclusionsConclusions

• MTA viable approach to mean field theory• Strouhal number around 3

– FOSA not ok (requires St 0)

• Existence of extra time derivative confirmed– Passive scalar transport has wave-like properties– Causality

• In MHD, <j.b> contribution arrives naturally• Coriolis force & inhomogeneity straightforward

Non-Fickian diffusion and Minimal Tau Approximation from numerical turbulence

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