2/3-order fractional and fractal derivatives modeling of turbulence

This study makes the first attempt to use the 2/3-order fractional Laplacian modeling of Kolmogorov -5/3 scaling of fully developed turbulence and enhanced diffusing movements of random turbulent particles. A combined effect of inertial interactions induced diffusivity and the molecular Brownian diffusivity is considered the bi-fractal mechanism behind multifractal scaling of moderate Reynolds number turbulence in the inertial range of scales. Accordingly, a stochastic equation is proposed to describe turbulence intermittency. The 2/3-order fractional Laplacian representation is also used to model nonlinear interactions of fluctuating velocity components, and then we construct a fractional Reynolds equation, underlying fractal spacetime structures of Lévy 2/3 stable distribution and the Kolmogorov scaling at small scales. The new perspective of this study is that the fractional calculus is an effective approach modeling of chaotic fractal phenomena induced by nonlinear interactions.