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 [Formula: see text], which appears naturally in the Large Deviation formulation of the MFM, and on [Formula: see text] the standard scaling parameter. These bounds respectively take the form: (i) [Formula: see text], which is consistent with Kolmogorov's four-fifths law ; and (ii) [Formula: see text]. 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)'.
Highlights
In a volume in which a significant number of papers are devoted to turbulent intermittency, it is a moot question whether any correspondence exists between the results derived from fractal theories of turbulence and those derived using the theory of weak solutions of the Navier–Stokes equations
The power of the multifractal method lies in the fact that a spread of values of the parameter h are employed
Whereas the discussion so far has been restricted to the lower bounds on h and C(h), we wish to discuss the physical manifestations of these bounds
Summary
A brief and mainly descriptive summary of both ideas are given in the rest of this section and the whole of §2. Which becomes h ≥ −2/3 when d = 3 It can be directly verified that the Fn,m(t) defined in (2.2) are invariant under the scale invariance property (2.5) for every finite value of the dimensionless parameter λ = 0 and of h This invariance at every length and time scale in the flow makes the set of Fn,m invaluable as a tool for investigating a cascade of energy through the system. The r−1 lower bound suggests a minimal rate of approach to the CKN singular set for which the corresponding value of h is h = −1. An alternative and more general way of expressing this result in d-dimensions is to say that for any d < 4, which implies that d)
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