A correspondence between the multifractal model of turbulence and the Navier–Stokes equations

Abstract

The multifractal model of turbulence (MFM) and the three-dimensional Navier–Stokes equations are blended together by applying the probabilistic scaling arguments of the former to a hierarchy of weak solutions of the latter. This process imposes a lower bound on both the multifractal spectrum C ( h ) , which appears naturally in the Large Deviation formulation of the MFM, and on h the standard scaling parameter. These bounds respectively take the form: (i) C ( h ) ≥ 1 − 3 h , which is consistent with Kolmogorov’s four-fifths law ; and (ii) h ≥ − 2 3 . The latter is significant as it prevents solutions from approaching the Navier–Stokes singular set of Caffarelli, Kohn and Nirenberg. This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’.