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)≥1-3h, which is consistent with Kolmogorov's four-fifths law; and (ii) h≥-23. 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)'.