Statistical Lyapunov theory based on bifurcation analysis of energy cascade in isotropic homogeneous turbulence: a physical -- mathematical review

Abstract

This work presents a review of previous articles dealing with an original turbulence theory proposed by the author, and provides new theoretical insights into some related issues. The new theoretical procedures and methodological approaches confirm and corroborate the previous results. These articles study the regime of homogeneous isotropic turbulence for incompressible fluids and propose theoretical approaches based on a specific Lyapunov theory for determining the closures of the von Kármán-Howarth and Corrsin equations, and the statistics of velocity and temperature difference. Furthermore, novel theoretical issues are here presented among which we can mention the following ones. The bifurcation rate of the velocity gradient, calculated along fluid particles trajectories, is shown to be much larger than the corresponding maximal Lyapunov exponent. On that basis, an interpretation of the energy cascade phenomenon is given and the statistics of finite time Lyapunov exponent of the velocity gradient is shown to be represented by normal distribution functions. Next, the self--similarity produced by the proposed closures is analyzed, and a proper bifurcation analysis of the closed von Kármán--Howarth equation is performed. This latter investigates the route from developed turbulence toward the non--chaotic regimes, leading to an estimate of the critical Taylor scale Reynolds number. A proper statistical decomposition based on extended distribution functions and on the Navier--Stokes equations is presented, which leads to the statistics of velocity and temperature difference.

physical--mathematical review of previous works and new theoretical insights into some relates issues

References in corpus (9)

• Interpretations of Negative Probabilities
• Self-Similarity in Fully Developed Homogeneous Isotropic Turbulence Using the Lyapunov Analysis
• Lyapunov Analysis for Fully Developed Homogeneous Isotropic Turbulence
• Finite Scale Lyapunov Analysis of Temperature Fluctuations in Homogeneous Isotropic Turbulence
• von Kàrmàn--Howarth and Corrsin equations closure based on Lagrangian description of the fluid motion
• Statistics of finite scale local Lyapunov exponents in fully developed homogeneous isotropic turbulence
• Extended Probabilities: Mathematical Foundations
• Bifurcations analysis of turbulent energy cascade
• Refinement of a previous hypothesis of the Lyapunov analysis of isotropic turbulence

Cited by in corpus (3)

• A Novel Approach for Estimating Largest Lyapunov Exponents in One-Dimensional Chaotic Time Series Using Machine Learning
• von Kármán--Howarth and Corrsin equations closures through Liouville theorem
• Liouville Spectral Gap and Bifurcation Driven Lagrangian Eulerian Decoupling with Nondiffusive Turbulence Closures