Does fully developed turbulence exist? Reynolds number independence versus asymptotic covariance

Abstract

By analogy with recent arguments concerning the mean velocity profile of wall-bounded turbulent shear flows, we suggest that there may exist corrections to the 2/3 law of Kolmogorov, which are proportional to (ln Re)−1 at large Re. Such corrections to K41 are the only ones permitted if one insists that the functional form of statistical averages at large Re be invariant under a natural redefinition of Re. The family of curves of the observed longitudinal structure function DLL(r,Re) for different values of Re is bounded by an envelope. In one generic scenario, close to the envelope, DLL(r,Re) is of the form assumed by Kolmogorov, with corrections of O[(ln Re)−2]. In an alternative generic scenario, both the Kolmogorov constant CK and corrections to Kolmogorov’s linear relation for the third-order structure function DLLL(r) are proportional to (ln Re)−1. Recent experimental data of Praskovsky and Oncley appear to show a definite dependence of CK on Re, which, if confirmed, would be consistent with the arguments given here.