Anisotropic eddy diffusivity field associated with a two-dimensional vortex shedding flow

Eddy viscosity models are often used to close turbulent momentum flux terms in the RANS equations. Analogously, eddy diffusivity models close turbulent scalar flux terms in Reynolds-averaged scalar transport equations. Such models commonly invoke the Boussinesq hypothesis, which assumes isotropy – that the magnitude of eddy diffusivity is agnostic to direction of diffusion of the mean scalar field – and locality – that turbulent fluxes are sensitive only to the local mean scalar gradient. In this work, we use the macroscopic forcing method (Mani and Park, Phys. Rev. Fluids, 2021) to compute anisotropic eddy diffusivity for scalar flux of a passive scalar subject to a two-dimensional vortex shedding flow. Specifically, macroscopic forcings that result in polynomial mean fields are used to quantify the leading coefficients of the Kramers-Moyal expansion of the eddy diffusivity operator. The leading order eddy diffusivity is found to exhibit substantial anisotropy in the wake flow with non-orthogonal principal axes. Results from this study provide insights into challenges of steady RANS modeling of shedding-dominated wake flows.