Analysis and approximation of a fractional Laplacian-based closure model for turbulent flows and its connection to Richardson pair dispersion

Abstract: We study a turbulence closure model in which the fractional Laplacian $(-\Delta)^\alpha$ of the velocity field represents the turbulence diffusivity. We investigate the energy spectrum of the model by applying Pao's energy transfer theory. For the case $\alpha=1/3$, the corresponding power law of the energy spectrum in the inertial range has a correction exponent on the regular Kolmogorov -5/3 scaling exponent. For this case, this model represents Richardson's particle pair-distance superdiffusion of a fully developed homogeneous turbulent flow as well as L\'evy jumps that lead to the superdiffusion. For other values of $\alpha$, the power law of the energy spectrum is consistent with the regular Kolmogorov -5/3 scaling exponent. We also propose and study a modular time-stepping algorithm in semi-discretized form. The algorithm is minimally intrusive to a given legacy code for solving Navier-Stokes equations by decoupling the local part and nonlocal part of the equations for the unknowns. We prove the algorithm is unconditionally stable and unconditionally, first-order convergent. We also derive error estimates for full discretizations of the model which, in addition to the time stepping algorithm, involves a finite element spatial discretization and a domain truncation approximation to the range of the fractional Laplacian.