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

Abstract

We study a correspondence between the multifractal model of turbulence and the Navier-Stokes equations in $d$ spatial dimensions by comparing their respective dissipation length scales. In Kolmogorov's 1941 theory the key parameter $h$, which is an exponent in the Navier-Stokes invariance scaling, is fixed at $h=1/3$ but is allowed a spectrum of values in multifractal theory. Taking into account all derivatives of the Navier-Stokes equations, it is found that for this correspondence to hold the multifractal spectrum $C(h)$ must be bounded from below such that $C(h) \geq 1-3h$, which is consistent with the four-fifths law. Moreover, $h$ must also be bounded from below such that $h \geq (1-d)/3$. When $d=3$ the allowed range of $h$ is given by $h \geq -2/3$ thereby bounding $h$ away from $h=-1$. The implications of this are discussed.