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

A turbulent flow is characterized by velocity fluctuations excited in an extremely broad interval of wave numbers $k>{\mathrm{\Lambda }}_f$ where ${\mathrm{\Lambda }}_f$ 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 ${\mathrm{\Lambda }}_f<k\le {\mathrm{\Lambda }}_0$, where $\eta =2\pi \mathrm{/}{\mathrm{\Lambda }}_0$ is the dissipation scale, leads to an infinite number of “relevant” scale-dependent coupling constants ( Reynolds numbers ) $R{e}_n(k)=O(1)$. It is shown that in the i.r. limit $k\rightarrow \Lambda_{f}$,theReynoldsnumbers$Re(k)\rightarrow Re_{tr}$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 $Re({\mathrm{\Lambda }}_f)=R{e}_{tr}$ “selects” the lowest - order non-linearity as the only relevant one. This means that in the infra-red limit $k\to {\mathrm{\Lambda }}_f$ all high-order nonlinearities generated by the scale-elimination sum up to zero.