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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 $k> Λ_{f}$ where $Λ_{f}$ 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 $Λ_{f}< k\leq Λ_{0}$, where $η=2π/Λ_{0}$ 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 $k\rightarrow Λ_{f}$, 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 $Re(Λ_{f})=Re_{tr}$ "selects" the lowest - order non-linearity as the only relevant one. This means that in the infra-red limit $k\rightarrow Λ_{f}$ all high-order nonlinearities generated by the scale-elimination sum up to zero.