Macroscopic Forcing Method: A computational approach for evaluation of turbulence closure operators*

We have recently developed a numerical procedure, the macroscopic forcing method (MFM), which probes the closure operators acting upon the mean fields of quantities transported by underlying fluctuating flows. Specifically, MFM can reveal differential operators associated with turbulent transport of scalars and momentum. At its core, MFM is similar to methods that compute the Green’s functions associated with the mean-space transport. However, by utilizing forcings that allow economical access to the leading order moments of desired closure operators MFM offers a cost-effective generalization of the Green’s function approach in diagnosis of closures. We present this methodology by considering canonical problems with increasing complexity, starting with spatially homogeneous and statistically stationary systems and transitioning to problems with inhomogeneity such as Rayleigh-Taylor instability and wall-bounded flows. For this presentation, we will primarily focus on passive scalar transport and briefly review advancements in the application of MFM in analysis of momentum transport. We will discuss anisotropy and nonlocality of the generalized eddy diffusivity operator through the lens of MFM. Additionally, we will outline a strategy for accurate construction of closure equations through utilization of the obtained leading-order space-time moments of the eddy diffusivity kernel.