# About Universality and Thermodynamics of Turbulence

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## 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 $\beta \left(Re\right)$

#### 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

- Dubrulle, B. Beyond Kolmogorov cascades. J. Fluid Mech. Perspect.
**2019**. [Google Scholar] [CrossRef] - Castaing, B. Conséquences d’un principe d’extremum en turbulence. J. Phys. Fr.
**1989**, 50, 147–156. [Google Scholar] [CrossRef] - Frisch, U.; Parisi, G. On the singularity structure of fully developed turbulence. In Turbulence and Predictability in Geophysical Fluid Dynamics and ClimateDynamics; Gil, M., Benzi, R., Parisi, G., Eds.; Elsevier: Amsterdam, The Netherlands, 1985; pp. 84–88. [Google Scholar]
- Eyink, G.L. Turbulence Theory. Course Notes, The Johns Hopkins University. 2007–2008. Available online: http://www.ams.jhu.edu/~eyink/Turbulence/notes/ (accessed on 24 September 2018).
- Paladin, G.; Vulpiani, A. Anomalous scaling laws in multifractal objects. Phys. Rev.
**1987**, 156, 147–225. [Google Scholar] [CrossRef][Green Version] - Frisch, U.; Vergassola, M. A Prediction of the Multifractal Model: The Intermediate Dissipation Range. Europhys. Lett. (EPL)
**1991**, 14, 439–444. [Google Scholar] [CrossRef] - Castaing, B.; Gagne, Y.; Marchand, M. Log-similarity for turbulent flows? Phys. D Nonlinear Phenom.
**1993**, 68, 387–400. [Google Scholar] [CrossRef] - Muzy, J.F.; Bacry, E.; Arneodo, A. Wavelets and multifractal formalism for singular signals: Application to turbulence data. Phys. Rev. Lett.
**1991**, 67, 3515. [Google Scholar] [CrossRef] [PubMed] - Saw, E.W.; Debue, P.; Kuzzay, D.; Daviaud, F.; Dubrulle, B. On the universality of anomalous scaling exponents of structure functions in turbulent flows. J. Fluid Mech.
**2018**, 837, 657–669. [Google Scholar] [CrossRef][Green Version] - Debue, P.; Shukla, V.; Kuzzay, D.; Faranda, D.; Saw, E.W.; Daviaud, F.; Dubrulle, B. Dissipation, intermittency, and singularities in incompressible turbulent flows. Phys. Rev. E
**2018**, 97, 053101. [Google Scholar] [CrossRef] [PubMed] - Kolmogorov, A.N. The local structure of turbulence in incompressible viscous fluids for very large Reynolds number. Dokl. Akad. Nauk SSSR [Sov. Phys.-Dokl.]
**1941**, 30, 913. [Google Scholar] [CrossRef] - Kolmogorov, A.N. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech.
**1962**, 13, 82. [Google Scholar] [CrossRef] - Touchette, H. The large deviation approach to statistical mechanics. Phys. Rep.
**2009**, 478, 1–69. [Google Scholar] [CrossRef][Green Version] - Frisch, U. Turbulence: The Legacy of A.N. Kolmogorov; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Benzi, R.; Ciliberto, S.; Tripiccione, R.; Baudet, C.; Massaioli, F.; Succi, S. Extended self-similarity in turbulent flows. Phys. Rev. E
**1993**, 48, R29–R32. [Google Scholar] [CrossRef] - Bohr, T.; Rand, D. The entropy function for characteristic exponents. Phys. D Nonlinear Phenom.
**1987**, 25, 387–398. [Google Scholar] [CrossRef] - Rinaldo, A.; Maritan, A.; Colaiori, F.; Flammini, A.; Rigon, R.; Rodriguez-Iturbe, I.; Banavar, J.R. Thermodynamics of fractal networks. Phys. Rev. Lett.
**1996**, 76, 3364. [Google Scholar] [CrossRef] [PubMed] - Arneodo, A.; Benzi, R.; Berg, J.; Biferale, L.; Bodenschatz, E.; Busse, A.; Calzavarini, E.; Castaing, B.; Cencini, M.; Chevillard, L.; et al. Universal Intermittent Properties of Particle Trajectories in Highly Turbulent Flows. Phys. Rev. Lett.
**2008**, 100, 254504. [Google Scholar] [CrossRef] [PubMed] - Biferale, L.; Boffetta, G.; Celani, A.; Devenish, B.J.; Lanotte, A.; Toschi, F. Multifractal Statistics of Lagrangian Velocity and Acceleration in Turbulence. Phys. Rev. Lett.
**2004**, 93, 064502. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Von Kármán swirling flow generator. (

**a**) normal view, bottom (

**b**) and top (

**c**) impellers rotating -both seen from the center of the cylinder, and (

**d**) sketch with the relevant measures. A device not shown here maintains the temperature constant during the experiment. Both impellers are counter-rotating.

**Figure 2.**Test of K41 universality Equation (4). (

**a**) Numerical data (

**b**) Experimental data. The structure functions have been shifted by arbitrary factors for clarity and are coded by color: $p=1$: blue symbols; $p=2$: orange symbols; $p=3$: yellow symbols; $p=4$: magenta symbols; $p=5$: green symbols; $p=6$: light blue symbols; $p=7$: red symbols; $p=8$: blue symbols; $p=9$: orange symbols. For K41 universality to hold, all the function should be constant, for a given p.

**Figure 3.**Test of K62 universality Equation (7). (

**a**) Numerical data (

**b**) Experimental data. The structure functions are shifted by arbitrary factors for clarity and are coded by color: $p=1$: blue symbols; $p=2$: orange symbols; $p=3$: yellow symbols; $p=4$: magenta symbols; $p=5$: green symbols; $p=6$: light blue symbols; $p=7$: red symbols; $p=8$: blue symbols; $p=9$: orange symbols. The dashed lines are power laws with exponents $\tau \left(p\right)=\zeta \left(p\right)-\zeta \left(3\right)p/3$, with $\zeta \left(p\right)$ shown in Figure 4a.

**Figure 4.**(

**a**) Determination of $\zeta \left(3\right)$ by best collapse using both DNS (open symbols) and experiments (filled symbols). The black dashed line is ${\ell}^{0.8}$. (

**b**) Scaling exponents $\zeta \left(p\right)$ of the wavelet structure functions of $\delta W$ as a function of the order, from Table 4, for DNS (blue circle) and experiments (red square). The red dotted line is the function ${min}_{h}(hp+C\left(h\right))$ with $C\left(h\right)$ given by $C\left(h\right)={(h-a)}^{2}/2b$, with $a=0.35$ and $b=0.045$. The black stars correspond to ${\zeta}_{\mathrm{SAW}}\left(p\right)/{\zeta}_{\mathrm{SAW}}\left(3\right)$ (see Table 4), while the black squares correspond to ${\zeta}_{\mathrm{EXP}}\left(p\right)/{\zeta}_{\mathrm{EXP}}\left(3\right)$.

**Figure 5.**Test of general universality Equation (20) using both DNS (open symbols) and experiments (filled symbols). The functions are coded by color. (

**a**) $p=1$: blue symbols; $p=2$: orange symbols; $p=4$: magenta symbols; $p=5$: green symbols; (

**b**) $p=6$: light blue symbols; $p=7$: red symbols; $p=8$: blue symbols; $p=9$: orange symbols. The functions have been shifted by arbitrary factors for clarity. The dashed lines are power laws with exponents $\tau \left(p\right)=\zeta \left(p\right)-\zeta \left(3\right)p/3$, with $\zeta \left(p\right)$ shown in Figure 4a.

**Figure 6.**(

**a**) Variation of $1/\beta \left(\mathrm{Re}\right)$ versus $log\left({R}_{\lambda}\right)$ in experiments (red square) and DNS (blue circle) when using the DNS at ${R}_{\lambda}=138$ as the reference case. Black stars correspond to the values found by Gagne and Castaing in [7] shifted by an arbitrary factor to coincide the values at large Reynolds. The black dashed line is $(4/3)log({R}_{\lambda}/5)$. (

**b**) Multifractal spectrum $C\left(h\right)$ for the experiments. The spectrum is obtained by taking inverse Legendre transform of the scaling exponents $\zeta \left(p\right)$ shown in Figure 4. The dotted line is a parabolic fit $C\left(h\right)={(h-a)}^{2}/2b$ with $a=0.35$ and $b=0.045$.

**Figure 7.**Multifractal equation of state of turbulence. Multifractal pressure as a function of the volume for $\phantom{\rule{4pt}{0ex}}{R}_{\lambda}=25$ (line), $\phantom{\rule{4pt}{0ex}}{R}_{\lambda}=53$ (dashed-dotted line). The functions are coded by color. (

**a**) $p=1$: blue symbols; $p=2$: orange symbols; $p=4$: magenta symbols; $p=5$: green symbols; (

**b**) $p=6$: light blue symbols; $p=7$: red symbols; $p=8$: blue symbols; $p=9$: orange symbols. The colored dotted line (resp. dashed dotted line) are values corresponding to $P(p,V)={\tau}_{\mathrm{EXP}}\left(p\right)$ (resp. ${\tau}_{\mathrm{DNS}}\left(p\right)$), that are reported in Table 4.

**Figure 8.**Same as Figure 7 for $\phantom{\rule{4pt}{0ex}}{R}_{\lambda}=90$ (line), $\phantom{\rule{4pt}{0ex}}{R}_{\lambda}=138$ (dotted line). Note the inflexion point appearing in the curves.

Symbol | Mathematical Definition | Interpretation |
---|---|---|

$\mathbf{u}(\mathbf{x},t)$ | $\in {\mathbb{R}}^{3}\times \mathbb{R}\to {\mathbb{R}}^{3}$ | Velocity field |

k | $\in {\mathbb{R}}_{+}$ | Wavenumber |

$E\left(k\right)$ | FT$\left({\langle {u}_{i}(\mathbf{x}+\mathbf{r},t){u}_{i}(\mathbf{x},t)\rangle}_{\mathbf{x},\parallel \mathbf{r}\parallel =\ell ,t}\right)$ | Energy spectrum |

${k}_{f}$ | $\in {\mathbb{R}}_{+}^{*}$ | Forcing wavenumber |

${N}_{x}$ | $\in \mathbb{N}$ | Grid size in direction x |

$\nu $ | $\in {\mathbb{R}}_{+}^{*}$ | Kinematic viscosity |

$\u03f5$ | $\in {\mathbb{R}}_{+}^{*}$ | Mean dissipation power per unit mass |

$\eta $ | ${\left(\frac{{\nu}^{3}}{\u03f5}\right)}^{\frac{1}{4}}$ | Kolmogorov scale |

${u}_{\mathrm{K}}$ | ${\left(\nu \u03f5\right)}^{\frac{1}{4}}$ | Kolmogorov velocity |

${u}_{0}$ | $\in {\mathbb{R}}_{+}^{*}$ | Characteristic velocity |

${L}_{0}$ | $\in {\mathbb{R}}_{+}^{*}$ | Characteristic length |

Re | $\frac{{u}_{0}{L}_{0}}{\nu}$ | Reynolds number |

$\lambda $ | $\sqrt{\frac{{\langle {\mathbf{u}}^{2}\rangle}_{\mathbf{x},t}}{{\langle \nabla {\mathbf{u}}^{2}\rangle}_{\mathbf{x},t}}}$ | Taylor length |

${u}^{\mathrm{rms}}$ | $\sqrt{{\langle {\mathbf{u}}^{2}\rangle}_{\mathbf{x},t}-{\langle \mathbf{u}\rangle}_{\mathbf{x},t}^{2}}$ | Root mean squared velocity |

${R}_{\lambda}$ | $\frac{\lambda {u}^{\mathrm{rms}}}{\nu}$ | Taylor Reynolds number |

$\Delta x$ | $\in {\mathbb{R}}_{+}^{*}$ | SPIV spatial resolution |

p | $\in [1,9]$ | Power |

ℓ | $\in {\mathbb{R}}_{+}^{*}$ | Scale |

L | $\in {\mathbb{R}}_{+}^{*}$ | Inertial large scale |

${\delta}_{\ell}u(\mathbf{x},t)$ | $\langle \parallel \mathbf{u}(\mathbf{x}+\mathbf{r},t)-{\mathbf{u}(\mathbf{x},t)\parallel \rangle}_{\parallel \mathbf{r}\parallel =\ell}$ | Velocity increment at scale ℓ |

$\Phi \left(\mathbf{x}\right)$ | ${exp(-\parallel \mathbf{x}\parallel}^{2}{/2)/\left(2\pi \right)}^{\frac{3}{2}}$ | Wavelet filter |

${\Phi}_{\ell}\left(\mathbf{x}\right)$ | ${\ell}^{-3}\Phi (\mathbf{x}/\ell )$ | Wavelet filter at scale ℓ |

${G}_{ij}(\mathbf{x},\ell ,t)$ | $\int {\nabla}_{j}{\Phi}_{\ell}\left(\mathbf{r}\right){u}_{i}(\mathbf{x}+\mathbf{r},t)\mathrm{d}\mathbf{r}$ | Wavelet transform of $\nabla \mathbf{u}$ |

$\delta W(\mathbf{x},\ell ,t)$ | $\ell {max}_{ij}\left|{G}_{ij}(\mathbf{x},\ell ,t)\right|.$ | Wavelet velocity increment |

${S}_{p}(\ell )$ | $\left\{\begin{array}{cc}{\langle {\left({\delta}_{\ell}u\right)}^{p}\rangle}_{\mathbf{x},t}& \mathrm{In}\phantom{\rule{4.pt}{0ex}}\mathrm{theory}\\ {\langle {\left(\delta W(\mathbf{x},\ell ,t)\right)}^{p}\rangle}_{\mathbf{x},t}& \mathrm{For}\phantom{\rule{4.pt}{0ex}}\mathrm{data}\phantom{\rule{4.pt}{0ex}}\mathrm{analysis}\end{array}\right.$ | Velocity structure function |

${\tilde{S}}_{p}(\ell )$ | $\frac{{S}_{p}}{{S}_{3}^{p/3}}$ | Relative structure function |

$h(\mathbf{x},t)$ | $\in {\mathbb{R}}^{3}\times \mathbb{R}\to [-1,1]$ | Local Hölder exponent |

$C\left(h\right)$ | $\mathbb{P}\left(log\left(\right|{\delta}_{\ell}u|/{u}_{0})=hlog\left(\ell /{L}_{0}\right)\right)\sim {\left(\ell /{L}_{0}\right)}^{C\left(h\right)}$ | Multifractal Spectrum |

${\eta}_{h}$ | ${L}_{0}{\mathrm{Re}}^{-\frac{1}{1+h}}$ | Multifractal regularization scale |

$\kappa $ | $\in {\mathbb{R}}_{+}^{*}$ | Intermittency parameter |

$\tau \left(p\right)$ | $\kappa p(3-p)$ | Lognormal Intermittency correction |

$\zeta \left(p\right)$ | $\frac{p}{3}+\tau \left(p\right)$ | Scaling exponent |

$\theta (\ell )$ | $\frac{log(L/\ell )}{log\left(\mathrm{Re}\right)}$ | Rescaled length |

$\tau (p,\theta )$ | $\left\{\begin{array}{ccc}\tau \left(p\right)& \mathrm{if}& \theta \le \frac{1}{1+{h}_{\mathrm{max}}}\\ p(\theta -\frac{1}{3})+C(-1+\frac{1}{\theta})& \mathrm{if}& \frac{1}{1+{h}_{max}}\le \theta \le \frac{1}{1+{h}_{min}}\end{array}\right.$ | General intermittency correction |

$\tau (p,\ell )$ | $\tau (p,\theta (\ell \left)\right)$ | General intermittency correction |

$\gamma \left(\mathrm{Re}\right)$,$\beta \left(\mathrm{Re}\right)$ | ${\mathbb{R}}_{+}\to \mathbb{R}$ | Fitting functions |

G | ${\mathbb{R}}^{2}\to \mathbb{R}$ | General function from Castaing [2] |

${A}_{p}$, ${K}_{0}$ | $\gamma \left(\mathrm{Re}\right)log\left(\frac{{S}_{p}}{{A}_{p}{u}_{\mathrm{K}}^{p}}\right)=G\left(p,\gamma \left(\mathrm{Re}\right)log(\ell {K}_{0}/\eta )\right)$ | Universal parameters |

H | ${\mathbb{R}}^{2}\to \mathbb{R}$ | New general function |

${S}_{0p}$ | $\beta \left(\mathrm{Re}\right)\left(\frac{log({\tilde{S}}_{p}/{S}_{0p})}{log({L}_{0}/\eta )}\right)=H\left(p,\beta \left(\mathrm{Re}\right)\frac{log(\ell /\eta )}{log({L}_{0}/\eta )}\right)$ | Universal parameter |

a, b | $C\left(h\right)=\frac{{(h-a)}^{2}}{2b}$ | Parabolic fit |

${\beta}_{0}$ | $1/\beta \left({R}_{\lambda}\right)\sim {\beta}_{0}/log\left({R}_{\lambda}\right)$ | Parameter |

${\tau}_{p,\mathrm{univ}}$ | $\frac{\tau (p,\ell )}{log(\ell /L)}$ for ℓ in Inertial range | Intermittency correction from general rescaling |

${\mu}_{\ell}\left(\mathbf{x}\right)$ | $\frac{\delta W{(\mathbf{x},\ell )}^{3}}{<\delta W{(\mathbf{y},\ell )}^{3}{>}_{\mathbf{y}}}$ | Spatial scale dependent measure |

$S\left(E\right)$ | $\mathbb{P}\left[log\left({\mu}_{\ell}\right)=Elog(\ell /\eta )\right]\sim {e}^{log(\ell /\eta )S\left(E\right)}$ | Large deviation function of $log\left({\mu}_{\ell}\right)$ |

${k}_{\mathrm{B}}$ | $\in {\mathbb{R}}_{+}^{*}$ | Boltzmann constant |

T | $1/{k}_{\mathrm{B}}p$ | Temperature |

E | $log\left({\mu}_{\ell}\right)$ | Energy |

N | $log\left(\mathrm{Re}\right)$ | Number of degrees of freedom |

V | $log(\ell /\eta )$ | Volume |

P | $\tau (p,\ell )$ | Pressure |

F | $log\left({\tilde{S}}_{3p}\right)$ | Free energy |

**Table 2.**Parameters for the 5 experiments realized $(\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}$ and $\mathrm{E})$. F is the rotation frequency of the discs, Re refers to the Reynolds number based on the diameter of the tank, ${R}_{\lambda}$ is the Reynolds based on the Taylor micro-scale. $\eta $ gives the estimated Kolmogorov length according to the experiment and $\Delta x$ refers to the spatial resolution of SPIV measurements. The second last column gives the number of frames over which are calculated the statistics. Except for (E), the Reynolds are much larger than those available with DNS. Table adapted from [10].

Case | Frequency (Hz) | Glycerol Part | Re | ${\mathit{R}}_{\mathit{\lambda}}$ | $\mathit{\eta}$ (mm) | $\mathbf{\Delta}\mathit{x}$ | Frames | Symbol |
---|---|---|---|---|---|---|---|---|

A | 5 | 0% | $3\times {10}^{5}$ | $1.9\times {10}^{3}$ | 0.02 | 2.4 | $3\times {10}^{4}$ | ○ |

B | 5 | 0% | $3\times {10}^{5}$ | $2.7\times {10}^{3}$ | 0.02 | 0.48 | $3\times {10}^{4}$ | □ |

C | 5 | 0% | $3\times {10}^{5}$ | $2.5\times {10}^{3}$ | 0.02 | 0.24 | $2\times {10}^{4}$ | ◊ |

D | 1 | 0% | $4\times {10}^{4}$ | $9.2\times {10}^{2}$ | 0.08 | 0.48 | $1\times {10}^{4}$ | △ |

E | 1.2 | 59% | $6\times {10}^{3}$ | $2.1\times {10}^{2}$ | 0.37 | 0.24 | $3\times {10}^{4}$ | ⋆ |

**Table 3.**Parameters for the DNS. ${R}_{\lambda}$ is the Reynolds based on the Taylor micro-scale. $\eta $ is the Kolmogorov length. The third column gives resolution of the simulation through ${k}_{max}\eta $, where ${k}_{max}={N}_{x}/3$ is the maximum wavenumber. The fourth column gives the grid size; notice that the dimensionless length of the box is $2\pi $. Here, ${\ell}_{min}$ is the smallest scale available for the calculations of the wavelets. ${k}_{f}$ is the forcing wavenumber. The Sample columns gives the number of points (frames × grid size) over which the statistics are computed.

${\mathit{R}}_{\mathit{\lambda}}$ | $\mathit{\eta}$ | ${\mathit{k}}_{max}\mathit{\eta}$ | ${\mathit{N}}_{\mathit{x}}\phantom{\rule{-0.166667em}{0ex}}\times {\mathit{N}}_{\mathit{y}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{\mathit{N}}_{\mathit{z}}$ | ${\mathbf{\ell}}_{min}/\mathit{\eta}$ | Samples | Symbol |
---|---|---|---|---|---|---|

25 | 0.079 | 3.35 | ${128}^{3}$ | 0.635 | 5000 | ⋆ |

53 | 0.034 | 8.5 | ${768}^{3}$ | 0.31 | 105,000 | △ |

80 | 0.020 | 1.68 | ${256}^{3}$ | 1.22 | 270,000 | □ |

90 | 0.017 | 5.7 | ${1024}^{3}$ | 0.36 | 10,000 | ◊ |

138 | 0.009 | 1.55 | ${512}^{3}$ | 1.37 | 12,000 | ○ |

**Table 4.**Scaling exponents $\tau \left(p\right)$ and $\zeta \left(p\right)$ found by the collapse method based on K62 universality for experimental data (subscript EXP) or numerical data (subscript DNS). The subscript SAW refers to the values obtained by [9]. The exponents ${\tau}_{\mathrm{EXP}}\left(p\right)$ (red square) and ${\tau}_{\mathrm{DNS}}$ (blue circle) have been computed through a least square algorithm.

Exponent\Order | $\mathit{p}=1$ | $\mathit{p}=2$ | $\mathit{p}=3$ | $\mathit{p}=4$ | $\mathit{p}=5$ | $\mathit{p}=6$ | $\mathit{p}=7$ | $\mathit{p}=8$ | $\mathit{p}=9$ |
---|---|---|---|---|---|---|---|---|---|

${\zeta}_{\mathrm{SAW}}/{\zeta}_{\mathrm{SAW}}\left(3\right)$ | 0.36 | 0.69 | 1 | 1.29 | 1.55 | 1.78 | 1.98 | 2.17 | 2.33 |

${\zeta}_{\mathrm{DNS}}$ | 0.31 | 0.58 | 0.80 | 0.98 | 1.12 | 1.23 | 1.26 | 1.25 | 1.23 |

${\zeta}_{\mathrm{EXP}}$ | 0.32 | 0.58 | 0.80 | 0.98 | 1.12 | 1.23 | 1.32 | 1.39 | 1.44 |

${\tau}_{\mathrm{DNS}}$ | 0.04 | 0.05 | 0 | $-0.09$ | $-0.21$ | $-0.37$ | $-0.61$ | $-0.88$ | $-1.17$ |

${\tau}_{\mathrm{EXP}}$ | 0.05 | 0.05 | 0 | $-0.09$ | $-0.21$ | $-0.36$ | $-0.54$ | $-0.74$ | $-0.96$ |

**Table 5.**Summary of the analogy between the multifractal formalism of turbulence and thermodynamics.

Thermodynamics | Turbulence | |
---|---|---|

Temperature | ${k}_{\mathrm{B}}T$ | $1/p$ |

Energy | E | $log\left({\mu}_{\ell}\right)$ |

Number of d.f. | N | $log\left(\mathrm{Re}\right)=log({L}_{0}/\eta )/{\beta}_{0}$ |

Volume | V | $log(\ell /\eta )$ |

Pressure | P | $\tau (p,\ell )$ |

Free energy | F | $log\left({\tilde{S}}_{3p}\right)$ |

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Geneste, 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