Code for: *Bryngelson, S. H., *Schäfer, F., Liu, J., & Mani, A. (2024). Fast Macroscopic Forcing Method. Journal of Computational Physics, 499, 112721. *Authors contributed equally.
The macroscopic forcing method (MFM) of Mani and Park and similar methods for obtaining turbulence closure operators, such as the Green’s function-based approach of Hamba, recover reduced solution operators from repeated direct numerical simulations (DNS). MFM has already been used to successfully quantify Reynolds-averaged Navier–Stokes (RANS)-like operators for homogeneous isotropic turbulence and turbulent channel flows. Standard algorithms for MFM force each coarse-scale degree of freedom (i.e., degree of freedom in the RANS space) and conduct a corresponding fine-scale simulation (i.e., DNS), which is expensive. We combine this method with an approach recently proposed by Schäfer and Owhadi to recover elliptic integral operators from a polylogarithmic number of matrix–vector products. The resulting fast MFM introduced in this work applies sparse reconstruction to expose local features in the closure operator and reconstructs this coarse-grained differential operator in only a few matrix–vector products and correspondingly, a few MFM simulations. For flows with significant nonlocality, the algorithm first "peels" long-range effects with dense matrix–vector products to expose a more local operator. We demonstrate the algorithm’s performance for scalar transport in a laminar channel flow and momentum transport in a turbulent channel flow. For these problems, we recover eddy–diffusivity- and eddy–viscosity-like operators, respectively, at 1% of the cost of computing the exact operator via a brute-force approach for the laminar channel flow problem and 13% for the turbulent one. We observe that we can reconstruct these operators with an increase in accuracy by about a factor of 100 over randomized low-rank methods. Applying these operators to compute the averaged fields of interest has visually indistinguishable behavior from the exact solution. Our results show that a similar number of simulations are required to reconstruct the operators to the same accuracy under grid refinement. Thus, the accuracy corresponds to the physics of the problem, not the numerics. We glean that for problems in which the RANS space is reducible to one dimension, eddy diffusivity and eddy viscosity operators can be reconstructed with reasonable accuracy using only a few simulations, regardless of simulation resolution or degrees of freedom.
@article{bryngelsonfmfm24,
author = {{*}Bryngelson, S. H. and {*}Sch{\"a}fer, F. and Liu, J. and Mani, A.},
title = {Fast macroscopic forcing method},
journal = {Journal of Computational Physics},
year = {2024},
volume = {499},
pages = {112721},
doi = {10.1016/j.jcp.2023.112721},
note = {{*}Authors contributed equally}
}MIT