Techniques for improved statistical convergence in quantification of eddy diffusivity moments

TL;DR

This work tackles slow statistical convergence in measuring nonlocal, anisotropic eddy diffusivity moments via the macroscopic forcing method (MFM) by advocating a single-donor DNS approach and introducing decomposition MFM to treat the forcing semi-analytically with consistent boundary conditions. The methodology is extended from scalar to momentum transport, and applied to a 2D Rayleigh–Taylor instability case, where decomposition MFM achieves convergence of higher-order moments (e.g., $D^{01}$) with far fewer realizations than standard MFM using separate donors. Results show that the single-donor plus decomposition framework yields more accurate, stable eddy-diffusivity moments, enabling more reliable construction of nonlocal closure operators (e.g., via matched-moment inverse) and reducing computational cost by roughly an order of magnitude in the tested RT case. Overall, the paper demonstrates a practical, scalable approach to quantify nonlocal eddy diffusivity moments and to inform improved nonlocal closure models for scalar and momentum transport in complex flows.

Abstract

While recent approaches, such as the macroscopic forcing method (MFM) or Green's function-based approaches, can be used to compute Reynolds-averaged Navier--Stokes closure operators using forced direct numerical simulations, MFM can also be used to directly compute moments of the effective nonlocal and anisotropic eddy diffusivities. The low-order spatial and temporal moments contain limited information about the eddy diffusivity but are often sufficient for quantification and modeling of nonlocal and anisotropic effects. However, when using MFM to compute eddy diffusivity moments, the statistical convergence can be slow for higher-order moments. In this work, we demonstrate that using the same direct numerical simulation (DNS) for all forced MFM simulations improves statistical convergence of the eddy diffusivity moments. We present its implementation in conjunction with a decomposition method that handles the MFM forcing semi-analytically and allows for consistent boundary condition treatment, which we develop for both scalar and momentum transport. We demonstrate that for a two-dimensional Rayleigh--Taylor instability case study, using the same DNS for all forced MFM simulations results in convergence with O(100) simulations rather than O(1000) simulations. We then demonstrate the impacts of improved convergence on the quantification of the eddy diffusivity.

Figures (12)

• Figure 1: Diagrams outlining MFM implementation with separate donors vs. single donor and the decomposition method in this work, presented in one dimension for simplicity. Superscripts denote variables ($c^i$, $\mathscr{c}^i$, etc.) and operators ($\mathcal{L}^i$) belonging to the receiver equations solved to obtain $D^i$.
• Figure 2: Root mean square errors (RMSE) of MFM with separate donors (solid lines) vs. single donor (dotted lines) as percent of maximum magnitude of each eddy diffusivity moment at each time. Errors are computed at 20 realizations with respect to 200 realizations for each method.
• Figure 3: Mixing half-width $h$ measured from separate donor simulations in Ares for the same initial conditions. Inset plot is the percent differences of the last three donors with respect to the first donor. $\tau$ is a nondimensional time, defined as $\frac{t}{t_0}$, where $t_0=\sqrt{\frac{h_0}{Ag}}$ and $h_0$ is the dominant lengthscale determined by the peak of the initial perturbation spectrum.
• Figure 4: Normalized moments of the eddy diffusivity kernel of RT instability measured using the standard MFM with separate donors. Data is averaged over 1,000 realizations and homogeneous direction $x_1$.
• Figure 5: Normalized moments of the eddy diffusivity kernel of RT instability measured using the decomposition MFM. Data is averaged over 200 realizations and homogeneous direction $x_1$.