A Theory of Hydrodynamic Turbulence Based on Non-equilibrium Statistical Mechanics

Abstract

In earlier papers, we have studied the turbulent flow exponents $\zeta _p$, where $\langle |\Delta \mathbf{v}|^p\rangle \sim \ell ^{\zeta _p}$ and $\Delta \mathbf{v}$ is the contribution to the fluid velocity at small scale $\ell$. Using ideas of non-equilibrium statistical mechanics we have found

$$\begin{aligned} \zeta _p={p\over 3}-{1\over \ln \kappa }\ln \Gamma \left( {p\over 3}+1\right) \end{aligned}$$

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.