Wang et al. (2018) closed the subgrid-scale (SGS) stress of large-eddy simulation (LES) with an artificial-neural-network model trained only on decaying homogeneous isotropic turbulence, using input features without full symmetry properties, at a per-time-step cost documented by subsequent studies as approximately 1.8 times that of the dynamic Smagorinsky model. These verified limitations pose as open problems: applicability to flow conditions beyond decaying isotropic turbulence, such as wall-bounded, shear-driven, high-Reynolds-number, and anisotropic-grid flows; and reduction of the per-time-step cost. A related design question - how much domain knowledge (invariance properties, tensor representations, symmetry constraints) to embed in machine-learning turbulence models rather than learn from data - is context for closure design, not a separate deliverable. The objective is a deployable, regime-general data-driven SGS closure: cross-regime a posteriori generalization at practical per-time-step cost.

Background

Large-eddy simulation (LES) computes turbulent flows by resolving only the large, energy-containing eddies and modeling the effect of the unresolved small scales. Spatially filtering the Navier-Stokes equations produces the subgrid-scale (SGS) stress tensor τij=uiuj‾−uˉiuˉj\tau_{ij}=\overline{u_i u_j}-\bar u_i\bar u_j, which must be supplied by a closure model. Classical algebraic closures are eddy-viscosity models built on the Boussinesq hypothesis, i.e. the assumption that the SGS stress is aligned with the resolved strain-rate tensor Sˉij\bar S_{ij}; the Smagorinsky model uses a fixed coefficient and the dynamic Smagorinsky variant computes it from the resolved field. Direct numerical simulation (DNS), which resolves all scales, provides ground truth. In the canonical test problem of decaying homogeneous isotropic turbulence (DHIT) - turbulence with no mean flow, statistically uniform and direction-independent, freely decaying - filtered DNS data yield the true SGS stresses for training; an a priori test then measures pointwise agreement of a model with filtered DNS, while an a posteriori test runs an actual LES with the model and compares statistics and spectra against DNS. Wang et al. (2018) trained random-forest and artificial-neural-network (ANN, a feed-forward regression network) regressions on 30 candidate input features from filtered DHIT, found filtered-velocity-gradient features dominant, proposed one ANN closure for all stress components (a priori correlation above 0.7), and obtained LES energy budgets better than Smagorinsky-type models. The limitations defining this problem are: training and testing confined to DHIT; input features lacking full symmetry properties; and a per-time-step computational cost of approximately 1.8 times that of the dynamic Smagorinsky model. A related design question is how much domain knowledge should be embedded into machine-learned turbulence models versus discovered from data alone. Two symmetry notions matter here. A closure is Galilean invariant if unchanged under translation at constant velocity (a boost); raw velocity is not boost-invariant, while the velocity gradient is. It is rotationally objective (equivariant) if, under a rotation RR of the frame, the predicted stress transforms as a tensor, τ′=RτRT\tau' = R\tau R^{\mathrm{T}}; the velocity gradient alone is not objective. Such symmetries can be embedded as inductive biases, i.e. structural assumptions built into the model architecture: Ling et al. (2016) predicted Reynolds stresses from an integrity basis (a minimal complete set of tensors and scalar invariants from which any objective tensor function can be built), and later SGS work modeled in the strain-rate eigenframe or with equivariant networks. The reviews of Duraisamy et al. (2019) and Brunton et al. (2020) frame both invariance embedding and cross-regime generalization as central, unsettled challenges for machine-learning closures.

Problem Statement

The problem is a deployable, regime-general data-driven SGS closure: establish data-driven closures for the SGS stress tensor that, trained on a limited set of flow configurations (such as decaying homogeneous isotropic turbulence or a single channel-flow database), remain accurate and stable in a posteriori LES of flows and discretizations outside the training data - for example wall-bounded or shear-driven turbulence, higher Reynolds numbers, or anisotropic grids - with a per-time-step computational cost of at most 1.2 times that of the dynamic Smagorinsky model, below the source closure's documented level of approximately 1.8 times, and with cost parity to classical algebraic closures such as the dynamic Smagorinsky model as a secondary target. The a posteriori validation is on flow conditions outside the training configuration(s), the wall-bounded, shear-driven, higher-Reynolds-number, and anisotropic-grid regimes named above being instances of that class of untrained conditions. On those untrained conditions, averaged over at least ten characteristic large-eddy turnover times (or the benchmark flow's equivalent statistical averaging interval), the closure should improve the a posteriori accuracy of the flow statistics relative to the dynamic Smagorinsky model - mean velocity profile, Reynolds-stress profiles, and resolved energy spectrum - and remain numerically stable with no divergence of the LES over that window. The closure is a genuine data-driven closure whose learned component is not vacuous and which does not reduce to the dynamic Smagorinsky model or another eddy-viscosity closure.

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