Statistical Mechanics (S.M.) on Turbulence*

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Statistical Mechanics (S.M.) on Turbulence

• Sunghwan (Sunny) Jung
• Harry L. Swinney

Physics Dept. University of Texas at Austin
Supported by ONR.
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Contents

• Revise Castaings method

• Introduce another method under transformation

• Stochastic Model from statistical mechanics

• Revise Kolmogorov 1962 (K62) in terms of
  statistical mechanics

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Couette-Taylor Exp.
In turbulence regime
At moderate rotation rate
Data Out
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Observed quantities
Velocity Difference
Extensive variable
Energy dissipation rate
Intensive variable
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Statistical Universality
Coarse-Grained Quantity
Physical Quantity
Temporal information
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Separation(r) Dependence
Castaings model

r L Gaussian Dist. Delta function
r ltlt L Gaussian Dist. Log-normal Dist.
We can rewrite its as
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Cascade to the smaller scale(r)
r L
r ltlt L
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Transform Castaing Model
Gaussian Dist.
Log-normal Dist.
Transform
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Statistical Universality
Coarse-Grained Quantity
Physical Quantity
Temporal information
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Probability of beta
where
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Conditioned Probability
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Separation(r) Dependence

d N Non-Gaussian Delta function
d ltlt N Gaussian Dist. Log-normal Dist.
Where N is the total number of data sets.
We can rewrite it as
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Cascade to large coarse-grain cell
d ltlt N
d L
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Compared the predicted PDF
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Lebesgue Measure
Changes from Delta function to Log-normal Dist.
x
Gaussian Dist.
x
K62
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S.M. Interpretation on K62
Thermodynamic variable
Taylor Expansion
Probability of velocity differences
If we assume that
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Conclusion

• Castaings method and Beck-Cohens method are
  the same under the transformation.

• Beck-Cohens method represents a cascade from a
  small coarse-grain to a large one.

• We revised Kolmogorovs 1962 theory in terms of
  the thermodynamic fluctuation of physical
  variables.

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Conditional PDF
(Stolovitzky et. al, PRL, 69)
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Thanks all