Reduced subgrid scale terms in three-dimensional turbulence

Large eddy simulation (LES) has become a central technique for simulating turbulent flows in engineering and applied sciences, offering a compromise between accuracy and computational cost by resolving large-scale motions and modeling the effects of smaller, unresolved scales through a subgrid scale (SGS) model. The fidelity and robustness of LES depends critically on the SGS model, particularly in coarse simulations where much of the turbulence spectrum remains unresolved. In this work, we extend the tau-orthogonal (TO) method, a data-driven SGS modeling framework, to three-dimensional turbulent flows. The method reformulates the high-dimensional SGS closure problem as a low-dimensional prediction task focused on spatially-integrated, scale-aware quantities of interest (QoIs). We extend the model to incorporate QoI-state dependence and temporal correlations by combining regularized least-squares regression with a multivariate Gaussian residual model. This yields a simple yet effective stochastic time-series prediction model, with orders-of-magnitude fewer parameters than typical deep learning approaches which try to directly learn the high-dimensional SGS closure. We demonstrate the effectiveness of the new TO model in three-dimensional forced isotropic turbulence, turbulent channel flow and on a Taylor-Green vortex. The model achieves accurate long-term QoI distributions, robust performance across hyperparameter settings, and good reproduction of key flow features such as kinetic energy spectra and coherent structures, despite being trained solely on QoI trajectories. Comparisons against classical SGS models, including Smagorinsky and WALE formulations, highlight the new model's balance of accuracy and computational efficiency.

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