Reynolds number of transition and self-organized criticality of strong turbulence.

Abstract

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 infrared 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 nonlinearity as the only relevant one. This means that in the infrared limit k→Λf, all high-order nonlinearities generated by the scale elimination sum up to zero.