About Universality and Thermodynamics of Turbulence
Abstract
:1. Introduction
2. Experimental and Numerical Setup
2.1. Experimental Facilities and Parameters
2.2. Direct Numerical Simulation
3. Theoretical Background
3.1. Velocity Increments vs. Wavelet Transform (WT) of Velocity Gradients
3.2. K41 and K62 Universality
3.3. Multifractal and Fluctuating Dissipation Length
3.4. General Universality
4. Check of Universality Using Data Analysis
4.1. Check of K41 Universality
4.2. Check of K62 Universality
4.3. Check of General Universality
4.4. Function
4.5. Scaling Exponents
4.6. Multifractal Spectrum
5. Thermodynamics and Turbulence
5.1. Thermodynamical Analogy
5.2. Multifractal Pressure and Phase Transition
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Symbol | Mathematical Definition | Interpretation |
---|---|---|
Velocity field | ||
k | Wavenumber | |
FT | Energy spectrum | |
Forcing wavenumber | ||
Grid size in direction x | ||
Kinematic viscosity | ||
Mean dissipation power per unit mass | ||
Kolmogorov scale | ||
Kolmogorov velocity | ||
Characteristic velocity | ||
Characteristic length | ||
Re | Reynolds number | |
Taylor length | ||
Root mean squared velocity | ||
Taylor Reynolds number | ||
SPIV spatial resolution | ||
p | Power | |
ℓ | Scale | |
L | Inertial large scale | |
Velocity increment at scale ℓ | ||
Wavelet filter | ||
Wavelet filter at scale ℓ | ||
Wavelet transform of | ||
Wavelet velocity increment | ||
Velocity structure function | ||
Relative structure function | ||
Local Hölder exponent | ||
Multifractal Spectrum | ||
Multifractal regularization scale | ||
Intermittency parameter | ||
Lognormal Intermittency correction | ||
Scaling exponent | ||
Rescaled length | ||
General intermittency correction | ||
General intermittency correction | ||
, | Fitting functions | |
G | General function from Castaing [2] | |
, | Universal parameters | |
H | New general function | |
Universal parameter | ||
a, b | Parabolic fit | |
Parameter | ||
for ℓ in Inertial range | Intermittency correction from general rescaling | |
Spatial scale dependent measure | ||
Large deviation function of | ||
Boltzmann constant | ||
T | Temperature | |
E | Energy | |
N | Number of degrees of freedom | |
V | Volume | |
P | Pressure | |
F | Free energy |
Case | Frequency (Hz) | Glycerol Part | Re | (mm) | Frames | Symbol | ||
---|---|---|---|---|---|---|---|---|
A | 5 | 0% | 0.02 | 2.4 | ○ | |||
B | 5 | 0% | 0.02 | 0.48 | □ | |||
C | 5 | 0% | 0.02 | 0.24 | ◊ | |||
D | 1 | 0% | 0.08 | 0.48 | △ | |||
E | 1.2 | 59% | 0.37 | 0.24 | ⋆ |
Samples | Symbol | |||||
---|---|---|---|---|---|---|
25 | 0.079 | 3.35 | 0.635 | 5000 | ⋆ | |
53 | 0.034 | 8.5 | 0.31 | 105,000 | △ | |
80 | 0.020 | 1.68 | 1.22 | 270,000 | □ | |
90 | 0.017 | 5.7 | 0.36 | 10,000 | ◊ | |
138 | 0.009 | 1.55 | 1.37 | 12,000 | ○ |
Exponent\Order | |||||||||
---|---|---|---|---|---|---|---|---|---|
0.36 | 0.69 | 1 | 1.29 | 1.55 | 1.78 | 1.98 | 2.17 | 2.33 | |
0.31 | 0.58 | 0.80 | 0.98 | 1.12 | 1.23 | 1.26 | 1.25 | 1.23 | |
0.32 | 0.58 | 0.80 | 0.98 | 1.12 | 1.23 | 1.32 | 1.39 | 1.44 | |
0.04 | 0.05 | 0 | |||||||
0.05 | 0.05 | 0 |
Thermodynamics | Turbulence | |
---|---|---|
Temperature | ||
Energy | E | |
Number of d.f. | N | |
Volume | V | |
Pressure | P | |
Free energy | F |
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Geneste, D.; Faller, H.; Nguyen, F.; Shukla, V.; Laval, J.-P.; Daviaud, F.; Saw, E.-W.; Dubrulle, B. About Universality and Thermodynamics of Turbulence. Entropy 2019, 21, 326. https://doi.org/10.3390/e21030326
Geneste D, Faller H, Nguyen F, Shukla V, Laval J-P, Daviaud F, Saw E-W, Dubrulle B. About Universality and Thermodynamics of Turbulence. Entropy. 2019; 21(3):326. https://doi.org/10.3390/e21030326
Chicago/Turabian StyleGeneste, Damien, Hugues Faller, Florian Nguyen, Vishwanath Shukla, Jean-Philippe Laval, Francois Daviaud, Ewe-Wei Saw, and Bérengère Dubrulle. 2019. "About Universality and Thermodynamics of Turbulence" Entropy 21, no. 3: 326. https://doi.org/10.3390/e21030326