Wang, Luo, Li, Tan and Fan (Physics of Fluids 30, 125101, 2018) showed that an artificial-neural-network subgrid-scale (SGS) closure trained on decaying homogeneous isotropic turbulence can beat Smagorinsky-type models in a posteriori large-eddy simulation (LES), while leaving open how such data-driven closures should be made invariant, general, and affordable. The open problem is to determine how much physical domain knowledge - invariance properties, tensor representations, symmetry constraints - must be embedded in machine-learning SGS closures, versus discovered from data alone, for the closure to remain stable and accurate in a posteriori LES of wall-bounded flows, shear-driven turbulence, higher-Reynolds-number regimes, and anisotropic grids, at computational cost competitive with classical algebraic closures. No existing work establishes which embedded constraints are necessary for cross-regime generalization, and a posteriori stability and cost competitiveness remain unresolved.

Background

Large-eddy simulation (LES) computes the energy-carrying large scales of a turbulent flow directly and models the effect of the unresolved small scales. In the spatially filtered Navier-Stokes equations this unresolved effect enters through the subgrid-scale (SGS) stress tensor τij=uiuj‾−uˉiuˉj\tau_{ij}=\overline{u_i u_j}-\bar u_i\bar u_j, where the overbar denotes spatial filtering. The classical closure is the Smagorinsky model, an eddy-viscosity assumption that aligns τij\tau_{ij} with the resolved strain-rate tensor (the symmetric part of the resolved velocity gradient); the dynamic Smagorinsky model (DSM) computes the model coefficient on the fly from the resolved flow. Closures are evaluated in two modes: a priori tests, in which the model regresses τij\tau_{ij} against filtered direct numerical simulation (DNS) data offline, and a posteriori tests, in which the model runs inside an actual LES and its errors feed back on the evolving flow - the stricter and practically decisive test.

Wang, Luo, Li, Tan and Fan (Physics of Fluids 30, 125101, 2018) proposed a data-driven closure in which an artificial neural network (ANN, a trainable parametric regression function) maps single-point filtered flow variables to all components of the SGS stress. Trained and tested on decaying homogeneous isotropic turbulence (HIT, turbulence with no mean shear and statistically identical in all directions), it reached a priori correlation coefficients above 0.7 and gave a posteriori LES in better agreement with DNS than the Smagorinsky and dynamic Smagorinsky models. The authors' stated limitations define the open problem: (i) the inputs - the filtered velocity vector and its first and second spatial derivatives - are not Galilean invariant (a closure is Galilean invariant if its predictions do not change under a uniformly translating reference frame), and their components are not invariant under rotations of the coordinate frame; for isotropic turbulence this deficiency is masked statistically, but exact invariance is required for general turbulent flows; (ii) the model was trained and tested only on decaying HIT, leaving wall-bounded flows, shear-driven turbulence, higher Reynolds numbers (the ratio of inertial to viscous forces), and anisotropic computational grids unexplored; (iii) the ANN added a large per-time-step computational overhead relative to classical algebraic closures; and (iv) more broadly, it remained unresolved how much domain knowledge - invariance properties, tensor representations, symmetry constraints - should be built into machine-learning turbulence models rather than left for the algorithm to discover from data.

The embed-invariance route predates the source paper in Reynolds-averaged Navier-Stokes (RANS) modeling, which predicts statistically averaged turbulence rather than its resolved large scales: Ling, Kurzawski and Templeton (Journal of Fluid Mechanics 807, 155-166, 2016) built the anisotropic Reynolds stress from an integrity basis - a minimal set of tensors and scalar invariants, constructed from the strain-rate and rotation tensors (the symmetric and antisymmetric parts of the velocity gradient), in which any frame-indifferent tensor function can be represented exactly - so that Galilean and rotational invariance hold by construction rather than by learning. A complementary embedded representation, adopted by later SGS work discussed below, expresses the closure in the filtered strain-rate eigenframe - the orthonormal basis of eigenvectors of the resolved strain-rate tensor - in which tensor-valued models reduce to scalar functions of the eigenvalues and rotation invariance again holds by construction. Reviews such as Duraisamy, Iaccarino and Xiao (Annual Review of Fluid Mechanics 51, 357-377, 2019) frame the field-wide question as how to combine such foundational physical constraints with data-driven fitting. The problem asks this question specifically for SGS closures in LES, where it is sharpened by the a posteriori stability requirement and by the cost of evaluating the learned model at every grid point at every time step.

Problem Statement

Determine how much physical domain knowledge must be embedded in machine-learning-based SGS closures for LES - which invariance properties (Galilean and rotational invariance, as flagged by Wang et al. (2018) for their inputs, together with reflectional and unit invariance as later identified in the invariant SGS-modeling literature, the last meaning invariance under changes of the measurement units), tensor representations (tensor bases; strain-rate-eigenframe forms, the strain-rate eigenframe being the orthonormal basis of eigenvectors of the resolved strain-rate tensor), symmetry constraints, and normalization schemes are required by construction, and which can be left to be discovered from data alone - such that the resulting closure is stable and accurate in a posteriori LES of turbulent flows beyond decaying homogeneous isotropic turbulence, with the a posteriori demonstration spanning all of the following regimes: wall-bounded flows, shear-driven turbulence, higher-Reynolds-number regimes, and anisotropic grids, while reducing the per-time-step computational overhead of the learned closure relative to classical algebraic closures (Smagorinsky, dynamic Smagorinsky) to a competitive level. Whether that cross-regime capability is achieved through embedded structure alone or also through multi-regime training data, transfer, or retraining is part of the domain-knowledge question, not a hard precondition. The single common objective is a physically consistent, generalizable, and affordable data-driven SGS closure together with the design principles that explain which embedded constraints made it so; its invariance structure, cross-regime a posteriori validation, and cost competitiveness are aspects of that one closure rather than independent questions. Because no consensus thresholds exist for 'stable', 'accurate', 'higher-Reynolds-number', and 'cost competitive', establishing their quantitative meanings is itself part of determining the necessity and sufficiency of embedded domain knowledge. The research target is a demonstrated closure satisfying all of these conditions, together with a systematic account of which embedded constraints are responsible for its a posteriori stability and cross-regime generalization.

References

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