Reynolds Number of Transition as a Dynamic Constraint on Statistical Theory of Turbulence

TL;DR

This paper uses an iterative coarse-graining approach to connect the fixed-point Reynolds number with the transition to anomalous scaling in turbulence, suggesting high-order nonlinearities are irrelevant near the integral scale.

Contribution

It demonstrates that the fixed-point Reynolds number aligns with the transition point to anomalous scaling, providing a dynamic constraint for turbulence theories.

Findings

Fixed-point Reynolds number matches transition to anomalous scaling

High-order nonlinearities are irrelevant near the integral scale

Infra-red divergencies are contained in large-scale flow equations

Abstract

Iterative coarse-graining procedure based on Wyld's perturbation expansion is applied to the problem of Navier-Stokes turbulence. It is shown that the low-order calculation gives the fixed-point Reynolds number $ Re_{fp}$ (coupling constant) almost identical to the Reynolds number of the recently discovered transition to anomalous scaling of the moments of {\bf "velocity derivatives"}. Using this result as a dynamic constraint, it is argued that in the vicinity of the fixed point (integral scale) the high-order non-linearities, generated by the procedure, are irrelevant. The infra-red divergencies do not disappear but are are contained in the derived equations for the symmetry-breaking large-scale flows (turbulence models or "condensates"), which are source of the small-scale turbulence.