Whether a machine-learned subgrid-scale (SGS) stress closure for large-eddy simulation (LES) can be made simultaneously symmetry-invariant, general across turbulent-flow regimes, and computationally affordable — and how much physical domain knowledge must be embedded in it rather than learned from data alone. The question arises from Wang, Luo, Li, Tan and Fan (Physics of Fluids 30, 125101, 2018), whose artificial-neural-network closure was trained and tested only on homogeneous isotropic turbulence with input features lacking exact frame invariance, and who left its extension to wall-bounded flows, shear-driven turbulence, high-Reynolds-number regimes, and anisotropic grids unexplored. Later work has resolved each component in isolation — invariant eigenframe/tensor-basis formulations, wall-bounded channel-flow closures, anisotropic-grid models, Reynolds-number transfer, cheap algebraic forms — but no single learned closure has been demonstrated a posteriori across all the named regimes with exact invariance at a cost comparable to algebraic closures, and the domain-knowledge-versus-data design question remains actively contested. The problem is open.

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. The velocity field is spatially filtered, and the Navier–Stokes equations for the filtered velocity u~i\tilde u_i acquire an extra term, the subgrid-scale (SGS) stress tensor τij=uiuj~−u~iu~j\tau_{ij}=\widetilde{u_i u_j}-\tilde u_i\tilde u_j, which represents momentum transport by the unresolved scales and must be closed, i.e. expressed approximately in terms of the resolved field. The classical closure is the Smagorinsky eddy-viscosity model, which assumes the SGS stress is aligned with the resolved strain-rate tensor with a coefficient fixed either statically or, in the widely used dynamic Smagorinsky variant, computed on the fly from the resolved scales. A data-driven closure instead trains a regression model — here an artificial neural network (ANN), a parametric nonlinear function fitted to data by gradient descent — on pairs of resolved flow fields and true SGS stresses extracted from filtered direct numerical simulations (DNS, which resolve all scales).

Two evaluation modes are standard and must be distinguished. An a priori test compares predicted and true SGS stresses on saved filtered-DNS fields; an a posteriori test runs an actual LES with the closure in place and compares the resulting turbulence statistics with DNS. The a posteriori test is the practically decisive one, since a closure can correlate well a priori and still destabilize or degrade a simulation.

A sound closure must also respect the symmetries of the underlying equations. Galilean invariance means the modeled stress must not change under a constant-velocity translation of the reference frame: features built from the filtered velocity itself violate this, while derivatives such as the velocity-gradient tensor ∂u~i/∂xj\partial\tilde u_i/\partial x_j satisfy it. Rotational invariance (objectivity) is a stronger, separate requirement: the model must transform correctly under rotations and reflections of the frame, which forces the closure to be built from scalar invariants and irreducible tensor combinations rather than raw tensor components; 'unit invariance' additionally demands invariance under changes of measurement units via proper non-dimensionalization. Homogeneous isotropic turbulence (HIT — turbulence, e.g. in a periodic box, with no preferred direction or position) satisfies rotation invariance only in a statistical sense, so a closure trained on HIT with non-objective features can look adequate there yet fail in anisotropic situations such as near-wall or shear-driven turbulence, where these symmetries are physically essential.

Wang, Luo, Li, Tan and Fan (Physics of Fluids 30, 125101, 2018) trained ANN SGS closures for homogeneous isotropic turbulence using 30 candidate input features, proposed a uniform ANN model covering all six stress components, and showed, in a priori and a posteriori tests on isotropic turbulence, better agreement with DNS than the Smagorinsky and dynamic Smagorinsky models. The demonstrated closure has these documented limitations: (i) it was examined only for forced isotropic turbulence, and it is unknown whether it performs well for other flow physics such as wall-bounded flows, shear-driven turbulence, high-Reynolds-number regimes, or on anisotropic grids (characterization by Prakash, Jansen and Evans 2022, DOI: 10.1016/j.cma.2022.115457); (ii) it is neither rotationally, reflectionally, nor unit invariant and does not depend on filter width (same source); (iii) its ANN evaluation carries a per-time-step computational overhead — later benchmarking reports a cost of approximately 1.8 times that of the dynamic Smagorinsky model (Xu et al. 2023, DOI: 10.1017/jfm.2023.179) — which needs algorithmic simplification for production LES. Separately, the data-driven-turbulence-modeling literature documents as open the general design question (iv) of how much domain knowledge — invariance properties, tensor representations, symmetry constraints — should be embedded in machine-learned turbulence models rather than discovered from data alone (Duraisamy, Iaccarino and Xiao 2019, DOI: 10.1146/annurev-fluid-010518-040547; Girimaji 2024, DOI: 10.1088/1367-2630/ad6689). Items (i)–(iii) define one concrete engineering objective — a single learned closure, valid across flow regimes, cheap enough for practical LES — and item (iv) is the design principle that governs how such a closure should be built.

Problem Statement

Determine whether a single trained data-driven SGS stress closure for LES can simultaneously satisfy all of the following, and, if so, construct and validate such a closure: (1) exactly embeds the symmetry requirements of the Navier–Stokes equations — Galilean, rotational, and reflectional invariance, with consistent non-dimensionalization (unit invariance) — by construction; (2) generalizes a posteriori, without retraining, from its training data to wall-bounded flows, shear-driven turbulence, Reynolds numbers substantially above the training range (within the window for which filtered-DNS validation data are available), and anisotropic grids or filtering, with turbulence statistics (mean profiles, Reynolds stresses, spectra, energy budgets) in quantified agreement with the corresponding filtered DNS — agreement metrics, tolerances, and reference datasets reported explicitly — and at least matching the dynamic Smagorinsky model on the same statistics in each tested regime, and with stable long-time integration; and (3) incurs a per-time-step computational cost comparable to classical algebraic closures, benchmarked against a dynamic Smagorinsky implementation in the same solver and on the same grid. Within the same framework, determine by controlled comparison how much of this performance is owed to the embedded domain knowledge (invariance constraints, tensor or eigenframe representations, symmetry restrictions) as opposed to structure learned from data alone — that is, whether items (1)–(2) can be achieved by data alone under a bounded data budget (a single filtered-DNS training dataset from one flow regime together with a fixed training-compute ceiling). These deliverables form one overarching objective — a deployable, symmetry-consistent, regime-general learned SGS closure — and the controlled comparison is the mechanism by which the domain-knowledge-versus-data design question (item (iv) in Background) is answered, not a separate problem.

References

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