statistical mechanics (s.m.) on turbulence* sunghwan (sunny) jung harry l. swinney physics dept....
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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.
Contents
• Revise Castaing’s method
• Introduce another method under transformation
• Stochastic Model from statistical mechanics
• Revise Kolmogorov 1962 (K62) in terms of statistical mechanics
Couette-Taylor Exp.
At moderate rotation rate In turbulence regime
Data Out
Observed quantities
Extensive variable
Intensive variable
Velocity Difference
Energy dissipation rate
Statistical UniversalityCoarse-Grained Quantity
Physical Quantity Temporal information
Separation(r) Dependence
r ~ L Gaussian Dist. Delta function
r << L Gaussian Dist. Log-normal Dist.
Castaing’s model
We can rewrite its as
Cascade to the smaller scale(r)
r ~ L
r << L
Transform Castaing Model
Gaussian Dist. Log-normal Dist.
Transform
Statistical UniversalityCoarse-Grained Quantity
Physical Quantity Temporal information
Probability of beta
where
Conditioned Probability
Separation(r) Dependence
d ~ N Non-Gaussian Delta function
d << N Gaussian Dist. Log-normal Dist.
Where N is the total number of data sets.
We can rewrite it as
Cascade to large coarse-grain cell
d << N
d = L
Compared the predicted PDF
Lebesgue Measure
x
x
Changes fromDelta function toLog-normal Dist.
Gaussian Dist.
K62 :
S.M. Interpretation on K62
Taylor Expansion
Probability of velocity differences
Thermodynamic variable
If we assume that
Conclusion• Castaing’s method and Beck-Cohen’s method are the same under the transformation.
• Beck-Cohen’s method represents a cascade from a small coarse-grain to a large one.
• We revised Kolmogorov’s 1962 theory in terms of the thermodynamic fluctuation of physical variables.
Conditional PDF
(Stolovitzky et. al, PRL, 69)
Thanks all