We present two families of sub-grid scale (SGS) turbulence models developed for large-eddy simulation (LES) purposes. Their development required the formulation of physics-informed robust and efficient Deep Learning (DL) algorithms which, unlike state-of-the-art analytical modeling techniques can produce high-order complex non-linear relations between inputs and outputs. Explicit filtering of data from direct simulations of the canonical channel flow at two friction Reynolds numbers $Re_\tau\approx 395$ and 590 provided accurate data for training and testing. The two sets of models use different network architectures. One of the architectures uses tensor basis neural networks (TBNN) and embeds the simplified analytical model form of the general effective-viscosity hypothesis, thus incorporating the Galilean, rotational and reflectional invariances. The other architecture is that of a relatively simple network, that is able to incorporate the Galilean invariance only. However, this simpler architecture has better feature extraction capacity owing to its ability to establish relations between and extract information from cross-components of the integrity basis tensors and the SGS stresses. Both sets of models are used to predict the SGS stresses for feature datasets generated with different filter widths, and at different Reynolds numbers. It is shown that due to the simpler model's better feature learning capabilities, it outperforms the invariance embedded model in statistical performance metrics. In a priori tests, both sets of models provide similar levels of dissipation and backscatter. Based on the test results, both sets of models should be usable in a posteriori actual LESs.

Deep learning is increasingly becoming a promising pathway to improving the accuracy of sub-grid scale (SGS) turbulence closure models for large eddy simulations (LES). We leverage the concept of differentiable turbulence, whereby an end-to-end differentiable solver is used in combination with physics-inspired choices of deep learning architectures to learn highly effective and versatile SGS models for two-dimensional turbulent flow. We perform an in-depth analysis of the inductive biases in the chosen architectures, finding that the inclusion of small-scale non-local features is most critical to effective SGS modeling, while large-scale features can improve pointwise accuracy of the a-posteriori solution field. The filtered velocity gradient tensor can be mapped directly to the SGS stress via decomposition of the inputs and outputs into isotropic, deviatoric, and anti-symmetric components. We see that the model can generalize to a variety of flow configurations, including higher and lower Reynolds numbers and different forcing conditions. We show that the differentiable physics paradigm is more successful than offline, a-priori learning, and that hybrid solver-in-the-loop approaches to deep learning offer an ideal balance between computational efficiency, accuracy, and generalization. Our experiments provide physics-based recommendations for deep-learning based SGS modeling for generalizable closure modeling of turbulence.

Many practical combustion systems such as those in rockets, gas turbines, and internal combustion engines operate under high pressures that surpass the thermodynamic critical limit of fuel-oxidizer mixtures. These conditions require the consideration of complex fluid behaviors that pose challenges for numerical simulations, casting doubts on the validity of existing subgrid-scale (SGS) models in large-eddy simulations of these systems. While data-driven methods have shown high accuracy as closure models in simulations of turbulent flames, these models are often criticized for lack of physical interpretability, wherein they provide answers but no insight into their underlying rationale. The objective of this study is to assess SGS stress models from conventional physics-driven approaches and an interpretable machine learning algorithm, i.e., the random forest regressor, in a turbulent transcritical non-premixed flame. To this end, direct numerical simulations (DNS) of transcritical liquid-oxygen/gaseous-methane (LOX/GCH4) inert and reacting flows are performed. Using this data, a priori analysis is performed on the Favre-filtered DNS data to examine the accuracy of physics-based and random forest SGS-models under these conditions. SGS stresses calculated with the gradient model show good agreement with the exact terms extracted from filtered DNS. The accuracy of the random-forest regressor decreased when physics-based constraints are applied to the feature set. Results demonstrate that random forests can perform as effectively as algebraic models when modeling subgrid stresses, only when trained on a sufficiently representative database. The employment of random forest feature importance score is shown to provide insight into discovering subgrid-scale stresses through sparse regression.