Characterization of nonlocal eddy diffusivity in homogeneous shear flow

Downgradient models are often used to estimate or “close” terms such as the Reynolds stresses and turbulent scalar fluxes in the Reynolds-averaged Navier-Stokes equations. These models often invoke assumptions of isotropy and locality when relating the closure terms to mean gradients. In this work, we present a quantitative characterization of the nonlocal eddy diffusivity by incorporating unimodal harmonic forcing to the passive scalar equations, which allow the mean scalar gradient to vary in space while satisfying periodic boundary conditions. We assess how the nonlocal eddy diffusivity depends on the forcing wavenumber by varying it in the k₁, k₂ plane (streamwise and shear directions). Lastly, we contrast the eddy diffusivity operators between homogeneous shear flow and isotropic turbulence.