Discretize first, filter next: learning divergence-consistent closure models for large-eddy simulation

TL;DR

It is shown that using a divergence-consistent LES formulation coupled with a convolutional neural closure model produces stable and accurate results for both a-priori and a-posteriori training, while a general (divergence-inconsistent) LES model requires a-posteriori training or other stability-enforcing measures.

Abstract

We propose a new neural network based large eddy simulation framework for the incompressible Navier-Stokes equations based on the paradigm "discretize first, filter and close next". This leads to full model-data consistency and allows for employing neural closure models in the same environment as where they have been trained. Since the LES discretization error is included in the learning process, the closure models can learn to account for the discretization. Furthermore, we employ a divergence-consistent discrete filter defined through face-averaging and provide novel theoretical and numerical filter analysis. This filter preserves the discrete divergence-free constraint by construction, unlike general discrete filters such as volume-averaging filters. We show that using a divergence-consistent LES formulation coupled with a convolutional neural closure model produces stable and accurate results for both a-priori and a-posteriori training, while a general (divergence-inconsistent) LES model requires a-posteriori training or other stability-enforcing measures.

Figures (23)

• Figure 1: Proposed route (in green) to a discrete LES model, based on "discretize first, then filter" instead of "filter first, then discretize" (in red). The term $\mathcal{F}(u)$ contains the convective and diffusive terms.
• Figure 2: Finite volume discretization on a staggered grid. The pressure is defined in the volume center, and the velocity components on the volume faces.
• Figure 3: Alternative view of figure \ref{['fig:equations']} to highlight the effect of differentiating the constraint. The red arrows show the traditional route of filtering first and then discretizing. The solid green arrows show our proposed route of discretizing first, then differentiating the constraint, then filtering, and finally reintroducing a pressure term (if the filter is divergence-consistent). This is done to circumvent the pressure problems of the dashed green route (discretizing first, then filtering).
• Figure 4: Four coarse volumes (blue) and their fine grid sub-grid volumes (red) in 2D. For each of the coarse volume faces, the discrete filter $\Phi^\text{FA}$ combines the DNS velocities $u$ into one LES velocity $\bar{u}$ using averaging. The interior sub-grid velocities are not present in $\bar{u}$. The coarse grid pressure $\bar{p}$ is defined in the coarse volume centers, but is not obtained by filtering $p$. Instead, it is computed from $\bar{u}$. Left: Structured grid, used in this work. Right: Unstructured grid.
• Figure 5: DNS velocity components $u$ contributing to a single filtered DNS component $\bar{u}$ for two filters. Both filters have a filter width equal to the grid size. Both $u(t)$ and $\bar{u}(t)$ share the same dimension as the continuous velocity $u(x, t)$. Left: Volume-averaging filter $\Phi^\text{VA}$. Right: Face-averaging filter $\Phi^\text{FA}$.