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Non-Fickian diffusion and Minimal Tau Approximation from numerical turbulence. …and why First Order Smoothing seemed always better than it deserved to be. Brandenburg 1 , P. K äpylä 2,3 , A. Mohammed 4 1 Nordita, Copenhagen, Denmark 2 Kiepenheuer Institute, Freiburg, Germany - PowerPoint PPT Presentation
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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
Brandenburg: non-Fickian diffusion 5
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!)
Brandenburg: non-Fickian diffusion 6
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:
Brandenburg: non-Fickian diffusion 7
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
Brandenburg: non-Fickian diffusion 8
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
Brandenburg: non-Fickian diffusion 9
Comparison with Fickian diffusionComparison with Fickian diffusion
No agreement whatsoever
Brandenburg: non-Fickian diffusion 10
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
Brandenburg: non-Fickian diffusion 12
Test 3: imposed mean C gradientTest 3: imposed mean C gradient
>>1 (!)
Convergence to St=3for different Re
Brandenburg: non-Fickian diffusion 13
kf=5kf=5
Brandenburg: non-Fickian diffusion 14
kf=1.5kf=1.5
Brandenburg: non-Fickian diffusion 15
Comment on the bottleneck effect Comment on the bottleneck effect Dobler et al (2003) PRE 68, 026304
Brandenburg: non-Fickian diffusion 16
Bottleneck effect: Bottleneck effect: 1D vs 3D spectra1D vs 3D spectra
Brandenburg: non-Fickian diffusion 17
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:
Brandenburg: non-Fickian diffusion 18
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
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