A theory of hydrodynamic turbulence based on non-equilibrium statistical mechanics

TL;DR

This paper presents a theoretical derivation of turbulence exponents using non-equilibrium statistical mechanics, providing a formula for velocity scaling exponents and estimating the Reynolds number for turbulence onset.

Contribution

It offers a more physical derivation of the turbulence scaling exponents formula based on non-equilibrium statistical mechanics.

Findings

Derived a formula for turbulence exponents $\

,

,

Abstract

In earlier papers we have studied the turbulent flow exponents $\zeta_p$, where $\langle|\Delta{\bf v}|^p\rangle\sim\ell^{\zeta_p}$ is the contribution to the fluid velocity at small scale $\ell$. Using ideas of non-equilibrium statistical mechanics we have found $$ \zeta_p={p\over3}-{1\over\ln\kappa}\ln\Gamma({p\over3}+1) $$ where $1/ln\kappa$ is experimentally $\approx 0.32\pm 0.01$. The purpose of the present note is to propose a somewhat more physical derivation of the formula for $\zeta_p$. We also present an estimate $\approx 100$ for the Reynolds number at the onset of turbulence.