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 (−Δ)α 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 α=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 Lévy jumps that lead to the superdiffusion. For other values of α, 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 first-order accurate and unconditionally stable. 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.