Turbulence Modeling via the Fractional Laplacian

Herein, we derive the fractional Laplacian operator as a means to represent the mean friction force arising in a turbulent flow: $\rho \frac{D\bar{\bf u}}{Dt} = -\nabla p + \mu_\alpha \nabla^2\bar{\bf u} + \rho C_\alpha \iiint_{\!-\infty}^\infty \frac{ \bar{\bf u}{\scriptstyle(t,{\bf x}')} - \bar{\bf u}{\scriptstyle(t,{\bf x})} }{|{\bf x}'-{\bf x}|^{\alpha+3}} \,d{\bf x}'$, where $\bar{\bf u}{\scriptstyle(t,{\bf x})}$ is the ensemble-averaged velocity field, $\mu_\alpha$ is an enhanced molecular viscosity, and $C_\alpha$ is a turbulent mixing coefficient (with units (length)$^\alpha$/(time)). The derivation is grounded in Boltzmann kinetic theory, which presumes an equilibrium probability distribution $f_\alpha^{eq}(t,{\bf x},{\bf u})$ of particle speeds. While historically $f_\alpha^{eq}$ has been assumed to be the Maxwell-Boltzmann distribution, we show that any member of the family of L\'evy $\alpha$-stable distributions is a suitable alternative. If $\alpha=2$, then $f^{eq}_\alpha$ is the Maxwell-Boltzmann distribution, with large particle speeds very unlikely, and the Navier-Stokes equations are recovered (with $\mu_\alpha = \mu$ and $C_\alpha = 0$). If $0 < \alpha < 2$, then $f^{eq}_\alpha$ is a L\'evy $\alpha$-stable distribution, with "heavy tails" that permit large velocity fluctuations, as in turbulence. For shear turbulent flows, the choice of $\alpha = 1$ (Cauchy distribution for $f_\alpha^{eq}$) leads to the logarithmic velocity profile known as the Law of the Wall. We also present examples of 1D Couette flow and 2D boundary layer flow, and we discuss turbulent transport within this kinetic theory framework. This work lays out a new framework for turbulence modeling that may lead to new fundamental understanding of turbulent flows.