Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept....
-
Upload
derick-andrews -
Category
Documents
-
view
215 -
download
0
Transcript of Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept....
![Page 1: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/1.jpg)
Statistical Mechanics (S.M.) on Turbulence*
Sunghwan (Sunny) Jung
Harry L. Swinney
Physics Dept. University of Texas at Austin
*Supported by ONR.
![Page 2: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/2.jpg)
Contents
• Revise Castaing’s method
• Introduce another method under transformation
• Stochastic Model from statistical mechanics
• Revise Kolmogorov 1962 (K62) in terms of statistical mechanics
![Page 3: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/3.jpg)
Couette-Taylor Exp.
At moderate rotation rate In turbulence regime
Data Out
![Page 4: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/4.jpg)
Observed quantities
Extensive variable
Intensive variable
Velocity Difference
Energy dissipation rate
![Page 5: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/5.jpg)
Statistical UniversalityCoarse-Grained Quantity
Physical Quantity Temporal information
![Page 6: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/6.jpg)
Separation(r) Dependence
r ~ L Gaussian Dist. Delta function
r << L Gaussian Dist. Log-normal Dist.
Castaing’s model
We can rewrite its as
![Page 7: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/7.jpg)
Cascade to the smaller scale(r)
r ~ L
r << L
![Page 8: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/8.jpg)
Transform Castaing Model
Gaussian Dist. Log-normal Dist.
Transform
![Page 9: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/9.jpg)
Statistical UniversalityCoarse-Grained Quantity
Physical Quantity Temporal information
![Page 10: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/10.jpg)
Probability of beta
where
![Page 11: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/11.jpg)
Conditioned Probability
![Page 12: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/12.jpg)
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
![Page 13: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/13.jpg)
Cascade to large coarse-grain cell
d << N
d = L
![Page 14: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/14.jpg)
Compared the predicted PDF
![Page 15: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/15.jpg)
Lebesgue Measure
x
x
Changes fromDelta function toLog-normal Dist.
Gaussian Dist.
K62 :
![Page 16: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/16.jpg)
S.M. Interpretation on K62
Taylor Expansion
Probability of velocity differences
Thermodynamic variable
If we assume that
![Page 17: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/17.jpg)
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.
![Page 18: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/18.jpg)
Conditional PDF
(Stolovitzky et. al, PRL, 69)
![Page 19: Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.](https://reader038.fdocuments.us/reader038/viewer/2022103022/56649cea5503460f949b5d19/html5/thumbnails/19.jpg)
Thanks all