# Statistical Lyapunov Theory Based on Bifurcation Analysis of Energy Cascade in Isotropic Homogeneous Turbulence: A Physical–Mathematical Review

## Abstract

**:**

## 1. Introduction

- (i)
- The bifurcation rate associated with the velocity gradient is shown to be much larger than the maximal Lyapunov exponent of the velocity gradient.
- (ii)
- As the consequence of (i), the energy cascade can be viewed as a succession of stretching and folding of fluid particles which involves smaller and smaller length scales, where the particle folding happens at the frequency of the bifurcation rate.
- (iii)
- As the consequence of (i), the central limit theorem provides reasonable argumentation that the finite time Lyapunov exponent is distributed following a gaussian distribution function.
- (iv)
- The proposed closures generate correlations self–similarity in proper ranges of variation of the separation distance which is directly caused by the continuous fluid particles trajectories divergence.
- (v)
- A specific bifurcation analysis of the closed von Kármán–Howarth equation is proposed which allows to estimate the critical Taylor scale Reynolds number in isotropic turbulence.
- (vi)
- A statistical decomposition of velocity and temperature is presented which is based on stochastic variables distributed following extended distribution functions. Such decomposition leads to the statistics of velocity and temperature difference, where the intermittency of these latter increases as Reynolds number and Péclet number rise.

## 2. Background

## 3. Navier–Stokes Bifurcations

## 4. Qualitative Analysis of the Route Toward the Chaos

## 5. Kinematic Bifurcations. Bifurcation Rate

## 6. Lyapunov Kinematic Analysis

## 7. *Turbulent Energy Cascade, Material Vorticity and Link with Classical Kinematic Lyapunov Analysis

## 8. Distribution Functions of $\mathbf{u}$, $\mathit{\vartheta}$, x, $\mathit{\xi}$ and $\tilde{\mathit{\lambda}}$

## 9. *Finite Time Lyapunov Exponents and Their Distribution in Fully Developed Turbulence

## 10. Closure of von Kármán–Howarth and Corrsin Equations

- (1)
- In fully developed chaos, the Navier–Stokes bifurcations determine a continuous distribution of velocity, temperature and of length scales, where one single bifurcation causes doubling of velocity, temperature, length scale and of all the properties associated with the velocity and temperature fields according to Equations (15) and (16). This leads to nonsmooth spatial variations of velocity field and very frequent kinematic bifurcations.
- (2)
- The huge kinematic bifurcations rate generates in turn continuous distributions of $\tilde{\lambda}$ and $\mathit{\xi}$, while fluid incompressibility and the mentioned alignment property of $\mathit{\xi}$ make $\tilde{\lambda}$ unsymmetrically distributed with $\overline{\lambda}\left(r\right)$≡${\u2329\tilde{\lambda}\u232a}_{\xi}>$ 0 and the relative particles trajectories to be chaotic.
- (3)
- The tendency of the material vorticity to follow direction and variations of the Lyapunov vectors gives the phenomenon of the kinetic energy cascade.

**Remark**

**1.**

## 11. Properties of the Proposed Closures

## 12. *Self–Similarity and Developed Correlations of the Proposed Closures

## 13. *Bifurcation Analysis of Closed von Kármán–Howarth Equation: From Fully Developed Turbulence Toward Non–Chaotic Regimes

**Remark**

**2.**

## 14. Velocity and Temperature Fluctuations

## 15. *Statistics of Velocity and Temperature Difference

## 16. Conclusions

- -
- The bifurcation rate of velocity gradient, calculated along fluid particles trajectories is shown to be much larger than the maximal Lyapunov exponent of the kinematic field.
- -
- On the basis of the previous item, the energy cascade is viewed as a stretching and folding succession of fluid particles which gradually involves smaller and smaller scales.
- -
- The central limit theorem, in the framework of the bifurcation analysis, provides reasonable argumentation that the finite time Lyapunov exponent can be approximated by a gaussian random variable if $\tau \approx 1/\mathsf{\Lambda}$.
- -
- The closures of von Kármán–Howarth and Corrsin equations given by this theory determine velocity and temperature correlations which exhibit local self–similarity directly linked to the continuous particles trajectories divergence.
- -
- The proposed bifurcation analysis of the closed von Kármán–Howarth equation studies the route from developed turbulence toward non–chaotic regimes and leads to an estimation of the critical Taylor scale Reynolds number in isotropic turbulence in agreement with the various experiments.
- -
- Finally, a specific statistical decomposition of velocity and temperature is presented. This decomposition, adopting random variables distributed following extended distribution functions, leads to the statistics of velocity and temperature difference which agrees with the data of experiments.

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Qualitative scheme of the route toward the turbulence. (

**a**,

**b**): velocity and length scale in terms of kinematic viscosity. (

**c**–

**e**): symbolic representation of solutions in the velocity fields set. (

**f**,

**g**): U and L in terms of kinematic viscosity.

**Figure 2.**Qualitative scheme of the route toward the turbulence: Reynolds number in terms of kinematic viscosity.

**Figure 3.**Qualitative scheme of fluid particle trajectory ${l}_{t}$, bifurcation line ${l}_{b}$ and their intersections over ${\mathsf{\Sigma}}_{1}$.

**Figure 4.**Taylor and Corrsin microscales and root mean square of classical Lyapunov exponent in function of the dimensionless time.

**Figure 5.**Longitudinal velocity correlations (

**left**) and energy spectra (

**right**) at different Taylor scale Reynolds numbers ${R}_{T}$ = 100, 200, 300, 400, 500, 600.

**Figure 6.**Triple longitudinal velocity correlations (

**left**) and the corresponding spectra (

**right**) at different Taylor scale Reynolds numbers ${R}_{T}$ = 100, 200, 300, 400, 500, 600.

**Figure 7.**Spectra for Pr = 10

^{−3}, 10

^{−2}, 0.1, 1.0 and 10, at different Reynolds numbers.

**Top**: kinetic energy spectrum $E\left(\kappa \right)$ (dashed line) and temperature spectra $\mathsf{\Theta}\left(\kappa \right)$ (solid lines).

**Bottom**: velocity transfer function $T\left(\kappa \right)$ (dashed line) and temperature transfer function $\mathsf{\Gamma}\left(\kappa \right)$ (solid line).

**Figure 9.**Left: PDF of $\partial {u}_{r}/\partial r$ for different values of ${R}_{T}$. (

**a**) Dotted, dash–dotted and continuous lines are for ${R}_{T}$ = 15, 30 and 60, respectively. (

**b**,

**c**) PDFs for ${R}_{T}$ = 255, 416, 514, 1035 and 1553. (

**c**) represents an enlarged part of the diagram (

**b**). Right–bottom: Data from Reference [47].

**Figure 10.**Left: Statistical moments of ${u}_{r}$ in terms of separation distance, for ${R}_{T}$ = 600. Right: Scaling exponents of $\partial {u}_{r}/\partial r$ at different ${R}_{T}$. Solid symbols are for the data calculated with the present analysis. Dashed line is for Kolmogorov K41 data [44]. Dotted line is for Kolmogorov K62 data [45]. Continuous line is for She–Leveque data [46].

**Figure 11.**Left: Distribution function of the longitudinal temperature derivatives, at different values of ${\mathsf{\Psi}}_{\theta}$. Right: Dimensionless statistical moments, ${H}_{\theta}^{\left(4\right)}$ and ${H}_{\theta}^{\left(6\right)}$ in function of ${\mathsf{\Psi}}_{\theta}$.

**Figure 12.**Comparison of the results: Kurtosis of temperature dissipation in function of ${\mathsf{\Psi}}_{\theta}$. The symbols represent the results by [73].

**Table 1.**Comparison of the results: Skewness of $\partial {u}_{r}/\partial r$ at diverse Taylor–scale Reynolds number ${R}_{T}\equiv u{\lambda}_{T}/\nu $ following different authors.

Reference | Simulation | ${\mathit{R}}_{\mathit{T}}$ | ${\mathit{H}}_{3}\left(0\right)$ |
---|---|---|---|

Present analysis | - | - | −3/7 = −0.428… |

[59] | DNS | 202 | −0.44 |

[60] | DNS | 45 | −0.47 |

[61] | DNS | 64 | −0.40 |

[62] | LES | <71 | −0.40 |

[63] | LES | ∞ | −0.40 |

[64] | LES | 720 | −0.42 |

${\mathit{R}}_{\mathit{T}}$ | C |
---|---|

100 | 1.8860 |

200 | 1.9451 |

300 | 1.9704 |

400 | 1.9847 |

500 | 1.9940 |

600 | 2.0005 |

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de Divitiis, N.
Statistical Lyapunov Theory Based on Bifurcation Analysis of Energy Cascade in Isotropic Homogeneous Turbulence: A Physical–Mathematical Review. *Entropy* **2019**, *21*, 520.
https://doi.org/10.3390/e21050520

**AMA Style**

de Divitiis N.
Statistical Lyapunov Theory Based on Bifurcation Analysis of Energy Cascade in Isotropic Homogeneous Turbulence: A Physical–Mathematical Review. *Entropy*. 2019; 21(5):520.
https://doi.org/10.3390/e21050520

**Chicago/Turabian Style**

de Divitiis, Nicola.
2019. "Statistical Lyapunov Theory Based on Bifurcation Analysis of Energy Cascade in Isotropic Homogeneous Turbulence: A Physical–Mathematical Review" *Entropy* 21, no. 5: 520.
https://doi.org/10.3390/e21050520