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 $C(h)\ge 1-3h$, which is consistent with the four-fifths law. Moreover, $h$ must also be bounded from below such that $h\ge (1-d)/3$. When $d=3$ the allowed range of $h$ is given by $h\ge -2/3$ thereby bounding $h$ away from $h=-1$. The implications of this are discussed.

arXiv comment9 pages, 1 figure

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Cited by in corpus (3)

• Identifying the multifractal set on which energy dissipates in a turbulent Navier-Stokes fluid
• An analytical and computational study of the incompressible Toner-Tu Equations
• How to extract a spectrum from hydrodynamic equations