Reynolds number of transition and large-scale properties of strong turbulence

TL;DR

This paper investigates the transition to turbulence by analyzing scale-dependent Reynolds numbers, revealing a universal transition Reynolds number where flow statistics change, and highlighting the dominance of the lowest-order nonlinearity in this process.

Contribution

It introduces a scale-dependent framework showing that the transition Reynolds number is universal and that higher-order nonlinearities vanish at the transition point.

Findings

Reynolds numbers approach a universal transition value at large scales.

Higher-order nonlinearities become negligible near the transition.

The transition is governed primarily by the lowest-order nonlinearity.

Abstract

A turbulent flow is characterized by velocity fluctuations excited in an extremely broad interval of wave numbers $k> \Lambda_{f}$ where $\Lambda_{f}$ is a relatively small set of the wave-vectors where energy is pumped into fluid by external forces. Iterative averaging over small-scale velocity fluctuations from the interval $\Lambda_{f}< k\leq \Lambda_{0}$, where $\eta=2\pi/\Lambda_{0}$ is the dissipation scale, leads to an infinite number of "relevant" scale-dependent coupling constants ( Reynolds numbers ) $Re_{n}(k)=O(1)$. It is shown that in the i.r. limit $k\rightarrow \Lambda_{f}$, the Reynolds numbers $Re(k)\rightarrow Re_{tr}$ where $Re_{tr}$ is the recently numerically and experimentally discovered universal Reynolds number of "smooth" transition from Gaussian to anomalous statistics of spatial velocity derivatives. The calculated relation $Re(\Lambda_{f})=Re_{tr}$ "selects" the lowest - order non-linearity as the only relevant one. This means that in the infra-red limit $k\rightarrow \Lambda_{f}$ all high-order nonlinearities generated by the scale-elimination sum up to zero.