Fractional and fractal derivatives modeling of turbulence

Abstract

This study makes the first attempt to use the 2/3-order fractional Laplacian modeling of enhanced diffusing movements of random turbulent particle resulting from nonlinear inertial interactions. A combined effect of the inertial interactions and the molecule Brownian diffusivities is found to be the bi-fractal mechanism behind multifractal scaling in the inertial range of scales of moderate Reynolds number turbulence. Accordingly, a stochastic equation is proposed to describe turbulence intermittency. The 2/3-order fractional Laplacian representation is also used to construct a fractional Reynolds equation for nonlinear interactions of fluctuating velocity components, underlying turbulence spacetime fractal structures of Levy 2/3 stable distribution. The new perspective of this study is that the fractional calculus is an effective approach modeling of chaotic fractal phenomena induced by nonlinear interactions.