NonFickian diffusion and Minimal Tau Approximation from numerical turbulence

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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

• Brandenburg1, P. Käpylä2,3, A. Mohammed4
• 1 Nordita, Copenhagen, Denmark
• 2 Kiepenheuer Institute, Freiburg, Germany
• 3 Dept Physical Sciences, Univ. Oulu, Finland
• 4 Physical Department, Oldenburg Univ., Germany

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MTA - 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

(remains to be justified!)
i) any support for this proposal?? ii) what is
tau??
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Purpose 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 gtgt 1 (although it seems to
  work!)

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Testing MTA passive scalar diffusion
primitive eqn
fluctuations
Flux equation ? triple moment
MTA closure
gtgt1 (!)
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System of mean field equations
mean concentration
flux equation
Damped wave equation, wave speed
(causality!)
gtgt1 (!)
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Wave equation consequences

• late time behavior unaffected (ordinary
  diffusion)
• early times ballistic advection (superdiffusive)

Illustration of wave-like behavior
small tau
large tau
intermediate tau
gtgt1 (!)
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Comparison 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/k11.5 and 5

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Test 1 initial top hat function
Monitor width and kurtosis
black closure model red turbulence sim.
Fit results kf/k1 Sttukf 1.5
1.8 2.2 1.8 5.1 2.4
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Comparison with Fickian diffusion
No agreement whatsoever
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Spreading of initial top-hat function
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Test 2 finite initial flux experiment
but with
Initial state
black closure model red turbulence sim.
Dispersion Relation Oscillatory for k1/kflt3
? direct evidence for oscillatory behavior!
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Test 3 imposed mean C gradient
Convergence to St3 for different Re
gtgt1 (!)
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kf5
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kf1.5
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Comment on the bottleneck effect
Dobler et al (2003) PRE 68, 026304
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Bottleneck effect 1D vs 3D spectra
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Relation to laboratory 1D spectra
Parseval
used
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Conclusions

• 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, ltj.bgt contribution arrives naturally
• Coriolis force inhomogeneity straightforward