A data-driven study on implicit LES using a spectral difference method

In this paper, we introduce a data-driven filter to analyze the relationship between Implicit Large-Eddy Simulations (ILES) and Direct Numerical Simulations (DNS) in the context of the Spectral Difference method. The proposed filter is constructed from a linear combination of sharp-modal filters where the weights are given by a convolutional neural network trained to replicate ILES results from filtered DNS data. In order to preserve the compactness of the discretization, the filter is local in time and acts at the elementary cell level. The neural network is trained on the data generated from the Taylor-Green Vortex test-case at Re=1600. In order to mitigate the temporal effects and highlight the influence of the spatial discretization, the ILESs are periodically restarted from DNS data for different time windows. Smaller time windows result in higher cross-correlations between ILES and the filtered DNS snapshots using the data-driven filters. The modal decay of the filter for the smallest time window considered aligns with classical eigenanalysis, showing better energy conservation for higher orders of approximation. Similarly, an analysis of the filter's kernel in the Fourier space confirms that higher polynomial orders are less dissipative compared to lower orders. As large time windows are considered, the trained filter encounters difficulties in representing the data due to significant non-linear effects. Additionally, the impact of the data-driven filter on the resolved kinetic energy has been assessed through the evaluation of the sub-grid production term which results in both direct and inverse cascades with the former being more likely on average. The presence of backscatter suggests that ILESs based on Discontinuous Spectral Element Methods might be equipped with an intrinsic mechanism to transfer energy in both directions with a predominance of direct kinetic energy cascade. Finally, it was shown that the learned filters can be used as test-filters within a self-similarity context for a-posteriori computations. The models have been evaluated at a Reynolds number (Re=5000) and grid resolution not included in the training data.