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

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 D_LL(r,Re) for different values of Re is bounded by an envelope. In one generic scenario, close to the envelope, D _LL(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 C_K and corrections to Kolmogorov's linear relation for the third-order structure function D_LLL(r) are proportional to (ln Re)^-1. Recent experimental data of Praskovsky and Oncley appear to show a definite dependence of C_K on Re, which, if confirmed, would be consistent with the arguments given here.