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

## TL;DR

This paper reviews a turbulence theory based on Lyapunov bifurcation analysis, providing new insights into energy cascade mechanisms, turbulence regimes, and statistical properties of velocity and temperature differences in isotropic homogeneous turbulence.

## Contribution

It introduces a novel Lyapunov-based approach for turbulence closure, bifurcation analysis of the von Kármán-Howarth equation, and detailed statistical descriptions of turbulence phenomena.

## Key findings

- Bifurcation rate exceeds maximal Lyapunov exponent.
- Velocity gradient statistics follow a normal distribution.
- Critical Reynolds number estimated for transition to non-chaotic regimes.

## 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 K\'arm\'an-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 K\'arm\'an--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.