In this work, we present a novel data-based approach to turbulence modelling for Large Eddy Simulation (LES) by artificial neural networks. We define the exact closure terms including the discretization operators and generate training data from direct numerical simulations of decaying homogeneous isotropic turbulence. We design and train artificial neural networks based on local convolution filters to predict the underlying unknown non-linear mapping from the coarse grid quantities to the closure terms without a priori assumptions. All investigated networks are able to generalize from the data and learn approximations with a cross correlation of up to 47% and even 73% for the inner elements, leading to the conclusion that the current training success is data-bound. We further show that selecting both the coarse grid primitive variables as well as the coarse grid LES operator as input features significantly improves training results. Finally, we construct a stable and accurate LES model from the learned closure terms. Therefore, we translate the model predictions into a data-adaptive, pointwise eddy viscosity closure and show that the resulting LES scheme performs well compared to current state of the art approaches. This work represents the starting point for further research into data-driven, universal turbulence models.

SUMMARYIn this paper, we investigate the accuracy and efficiency of discontinuous Galerkin spectral method simulations of under‐resolved transitional and turbulent flows at moderate Reynolds numbers, where the accurate prediction of closely coupled laminar regions, transition and developed turbulence presents a great challenge to large eddy simulation modelling. We take full advantage of the low numerical errors and associated superior scale resolving capabilities of high‐order spectral methods by using high‐order ansatz functions up to 12th order. We employ polynomial de‐aliasing techniques to prevent instabilities arising from inexact quadrature of nonlinearities. Without the need for any additional filtering, explicit or implicit modelling, or artificial dissipation, our high‐order schemes capture the turbulent flow at the considered Reynolds number range very well. Three classical large eddy simulation benchmark problems are considered: a circular cylinder flow at ReD=3900, a confined periodic hill flow at Reh=2800 and the transitional flow over a SD7003 airfoil at Rec=60,000. For all computations, the total number of degrees of freedom used for the discontinuous Galerkin spectral method simulations is chosen to be equal or considerably less than the reported data in literature. In all three cases, we achieve an equal or better match to direct numerical simulation results, compared with other schemes of lower order with explicitly or implicitly added subgrid scale models. Copyright © 2014 John Wiley & Sons, Ltd.

We find that the strategy gives excellent results, but only when the resolution is such, that about 40% of the dissipation is resolved. For more realistic, coarser resolutions used in classical LES e.g. of industrial applications, the iLES DG strategy becomes quite in-accurate. We show that there is no obvious fix to this strategy, as adding for instance a sub-grid-scale models on top doesn't change much or in worst case decreases the fidelity even more. Finally, the core of this work is a novel LES strategy based on split form DG methods that are kinetic energy preserving. Such discretisations offer excellent stability with full control over the amount and shape of the added artificial dissipation. This premise is the main idea of the work and we will assess the LES capabilities of the novel split form DG approach. We will demonstrate that the novel DG LES strategy offers similar accuracy as the iLES methodology for well resolved cases, but strongly increases fidelity in case of more realistic coarse resolutions.

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