EnergyConserving Neural Network for Turbulence Closure Modeling
Abstract
In turbulence modeling, and more particularly in the LargeEddy Simulation (LES) framework, we are concerned with finding closure models that represent the effect of the unresolved subgrid scales on the resolved scales. Recent approaches gravitate towards machine learning techniques to construct such models. However, the stability of machinelearned closure models and their abidance by physical structure (e.g. symmetries, conservation laws) are still open problems. To tackle both issues, we take the `discretize first, filter next' approach, in which we apply a spatial averaging filter to existing energyconserving (finegrid) discretizations. The main novelty is that we extend the system of equations describing the filtered solution with a set of equations that describe the evolution of (a compressed version of) the energy of the subgrid scales. Having an estimate of this energy, we can use the concept of energy conservation and derive stability. The compressed variables are determined via a datadriven technique in such a way that the energy of the subgrid scales is matched. For the extended system, the closure model should be energyconserving, and a new skewsymmetric convolutional neural network architecture is proposed that has this property. Stability is thus guaranteed, independent of the actual weights and biases of the network. Importantly, our framework allows energy exchange between resolved scales and compressed subgrid scales and thus enables backscatter. To model dissipative systems (e.g. viscous flows), the framework is extended with a diffusive component. The introduced neural network architecture is constructed such that it also satisfies momentum conservation. We apply the new methodology to both the viscous Burgers' equation and the KortewegDe Vries equation in 1D and show superior stability properties when compared to a vanilla convolutional neural network.
 Publication:

arXiv eprints
 Pub Date:
 January 2023
 DOI:
 10.48550/arXiv.2301.13770
 arXiv:
 arXiv:2301.13770
 Bibcode:
 2023arXiv230113770V
 Keywords:

 Mathematics  Numerical Analysis;
 Physics  Fluid Dynamics;
 65Mxx (Primary)
 EPrint:
 26 pages, 15 figures, source code can be found at https://github.com/tobyvg/ECNCM_1D