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A turbulent flow is characterized by velocity fluctuations excited in anextremely 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 fluidby external forces. Iterative averaging over small-scale velocity fluctuationsfrom the interval $\Lambda_{f}< k\leq \Lambda_{0}$, where$\eta=2\pi/\Lambda_{0}$ is the dissipation scale, leads to an infinite numberof "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 Reynoldsnumber of "smooth" transition from Gaussian to anomalous statistics of spatialvelocity derivatives. The calculated relation $Re(\Lambda_{f})=Re_{tr}$"selects" the lowest - order non-linearity as the only relevant one. This meansthat in the infra-red limit $k\rightarrow \Lambda_{f}$ all high-ordernonlinearities generated by the scale-elimination sum up to zero.

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