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