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

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A turbulent flow is characterized by velocity fluctuations excited in an extremely broad interval of wave numbers k> Λf where Λf 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 Λf< k≤ Λ₀, where η=2π/Λ₀ is the dissipation scale, leads to an infinite number of "relevant" scale-dependent coupling constants (Reynolds numbers) Reₙ(k)=O(1). It is shown that in the i.r. limit k→ Λf, the Reynolds numbers Re(k)→ Retr where Retr 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(Λf)=Retr "selects" the lowest - order non-linearity as the only relevant one. This means that in the infra-red limit k→ Λf all high-order nonlinearities generated by the scale-elimination sum up to zero.