# Macroscopic forcing method: a tool for turbulence modeling and analysis of closures

## TL;DR

The paper introduces the macroscopic forcing method (MFM), a numerical technique to identify differential operators governing turbulence closures, enabling precise analysis of eddy diffusivity and momentum transport in complex flows.

## Contribution

This work develops MFM as a novel, quantitative tool for revealing turbulence closure operators, including nonlocal and anisotropic effects, across various flow configurations.

## Key findings

- MFM accurately determines eddy diffusivity operators in canonical flows.
- The method captures nonlocal mixing effects through Fourier space operator representation.
- Extension of MFM to wall-bounded flows and momentum transport provides exact RANS solutions.

## Abstract

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.