Physics-constrained and data-driven correction of the Smagorinsky model

Turbulence is ubiquitous in nature and engineering applications and remains one of the most complex problems in fluid mechanics. In recent years, data-driven approaches have achieved substantial progress in turbulence modeling. Large Eddy Simulation (LES) is a high-fidelity turbulence simulation methodology: it resolves eddies larger than the grid scale directly, while the effects of subgrid-scale (SGS) motions smaller than the grid scale are represented through SGS models. SGS modeling therefore plays a critical role in LES. Conventional SGS models are typically built upon empirical assumptions, making it difficult to simultaneously ensure physical consistency and predictive accuracy; under complex flow conditions, they often exhibit pronounced grid dependence. Symbolic Regression (SR), a white-box machine learning technique, provides explicit analytical expressions that are easier to inspect and implement than black-box models for turbulence modeling. In this work, a corrected Smagorinsky-type SGS eddy-viscosity model is constructed by combining an analytically constrained mixed-rate formulation with a symbolic-regression-based near-wall damping function. The regression target is extracted from filtered DNS channel-flow data through an equivalent SGS eddy viscosity. The resulting model preserves cubic near-wall decay and vanishes in the pure-strain/pure-rotation limits of the proposed formulation. A posteriori tests on circular- and square-cylinder flows show that, relative to the baseline Smagorinsky model, the proposed model yields clearer improvements in selected pressure-coefficient and wake-velocity distributions and exhibits markedly reduced degradation under grid coarsening. Although its integrated-force predictions are not uniformly more accurate than those of WALE and DSM, SRSM shows distinct advantages in selected distributional quantities and maintains stronger robustness as the grid is coarsened.