  This study presents a numerical procedure, which we call the macroscopic
forcing method (MFM), which reveals the differential operators acting on the
mean fields of quantities transported by underlying fluctuating flows.
Specifically, MFM can precisely determine the eddy diffusivity operator, or
more broadly said, it can reveal differential operators associated with
turbulence closure for scalar and momentum transport. We present this
methodology by considering canonical problems with increasing complexity.
Starting from the well-known problem of dispersion of passive scalars by
parallel flows we elucidate the basic steps in quantitative determination of
the eddy viscosity operators using MFM. Utilizing the operator representation
in Fourier space, we obtain a stand-alone compact analytical operator that can
accurately capture the nonlocal mixing effects. Furthermore, a cost-effective
generalization of MFM for analysis of nonhomogeneous and wall-bounded flows is
developed and is comprehensively discussed through a demonstrative example.
Extension of MFM for analysis of momentum transport is theoretically
constructed through the introduction of a generalized momentum transport
equation. We show that closure operators obtained through MFM analysis of this
equation provide the exact RANS solutions obtained through averaging of the
Navier-Stokes equation. We introduce MFM as an effective tool for quantitative
understanding of non-Boussinesq effects and assessment of model forms in
turbulence closures, particularly, the effects associated with anisotropy and
nonlocality of macroscopic mixing.
