Physically consistent subgrid-scale models for large-eddy simulation: significance, compatibility, and realizability of model constraints

Large-eddy simulation (LES) of incompressible turbulent flows requires modeling the subgrid-scale (SGS) stress tensor. Silvis, Remmerswaal & Verstappen (Phys. Fluids 29, 015105, 2017) consolidated a systematic, constraint-based approach to constructing SGS models, assembling requirements from the symmetries of the Navier-Stokes equations, near-wall scaling, realizability, and dissipation properties of the turbulent stresses, and showed that no existing velocity-gradient-based model satisfies all of them, partly because some constraints are provably incompatible. The paper explicitly posed the follow-up questions that define this problem: determine the practical importance and relative significance of each constraint across flow classes; analyze symmetry preservation (especially scaling invariance) by the dynamic procedure and by flow-dependent subgrid length-scale definitions, including on anisotropic grids; construct invariant-dependent coefficient functions that make nonlinear velocity-gradient models realizable and thermodynamically consistent while retaining nondissipative transport terms; and devise new constraints for the construction of subgrid-scale models. Substantial progress has since been made on individual pieces (new anisotropic-grid length scales, mixed minimum-dissipation-Bardina models, Lie-symmetry-preserving invariant model families, energy-transfer-optimal closures, physics-constrained machine-learning closures), but none of the four questions has been answered systematically; the problem remains open.

Background

In large-eddy simulation (LES), the incompressible Navier-Stokes equations are spatially filtered so that only large eddies are computed directly; the effect of the unresolved small eddies appears through the subgrid-scale (SGS) stress tensor $\tau_{ij}=\overline{u_i u_j}-\bar u_i\bar u_j$, which must be modeled. Eddy-viscosity models (Smagorinsky, Vreman, WALE (wall-adapting local eddy-viscosity), the $\sigma$ model, and the anisotropic minimum-dissipation model) model only the net dissipative effect of the small scales by aligning the modeled stress with the resolved rate-of-strain tensor $S_{ij}$; nonlinear velocity-gradient models express the modeled stress as a tensor polynomial in $S$ and the rate-of-rotation tensor $\Omega$ with scalar coefficient functions of the five independent invariants $I_1,\dots,I_5$ of $S$ and $\Omega$, and can in addition represent nondissipative energy transport (backscatter). Silvis, Remmerswaal & Verstappen (2017) consolidated a framework of model constraints demanding that an SGS model respect: the symmetries of the Navier-Stokes equations (time shifts, pressure shifts, generalized Galilean invariance, rotations and reflections, scaling invariance, two-dimensional material frame indifference, and time reversal, labeled S1-S7); the proper near-wall scaling behavior (eddy viscosity of order $(y^+)^3$ at a distance $y^+$ in wall units from a wall, labeled N); realizability (R), meaning the modeled stress is symmetric positive semidefinite, expressible as nonnegativity of its principal invariants, since Vreman et al. showed the true filtered stress of a positive filter has this property; and dissipation/production properties, namely Vreman's requirements P1a (zero subgrid dissipation for laminar-type flows) and P1b (nonzero dissipation for nonlaminar flows), derived from the mathematical properties of the turbulent stress tensor, Nicoud et al.'s physical requirements P2a (vanishing subgrid dissipation in all two-component flows) and P2b (vanishing dissipation for the pure axisymmetric strain), consistency with the second law of thermodynamics P3 (nonnegative total dissipation, i.e. model dissipation plus viscous dissipation $2\nu I_1$, hence time irreversibility; the subgrid-scale production of turbulent kinetic energy may itself become negative), and Verstappen's minimum-dissipation condition P4 ensuring scale separation between resolved and unresolved motions. Applying this framework, they found that no existing velocity-gradient-based model satisfies all constraints: P2a is incompatible with P1b, P2b is incompatible with P4, scaling invariance S5 cannot be satisfied by any model whose only length scale is grid-based, and no velocity-gradient expression for the eddy viscosity was found satisfying both P1a and P1b. Their nonlinear example model (their Eq. 54), with coefficients depending on invariants of $S$, is not realizable and, with nondynamic constants, is time-reversible, thereby violating P3 - although they argued 'a more complex definition of the model coefficients could be found to solve this problem'. The dynamic procedure (in which model coefficients are computed during the simulation from a Germano identity relating stresses at two filter widths) can supply a flow-dependent coefficient, but its symmetry-preservation properties were left unanalyzed. On anisotropic grids the choice of the subgrid characteristic length $\delta$ (e.g. Deardorff's cube root of the cell volume) is known to affect results strongly. A further flow-dependent route is the integral-length-scale approximation of Piomelli, Rouhi & Geurts (2015) and Rouhi, Piomelli & Geurts (2016), which estimates the turbulent kinetic energy and the energy-dissipation rate from the resolved field through an integral length scale of the resolved turbulence, so that the subgrid length scale reflects the local turbulence activity and is decoupled from the grid (the later work determines the model coefficient dynamically). Within this framework, the paper then constructed and tested a new vortex-stretching-based eddy-viscosity model with proper near-wall scaling in decaying homogeneous isotropic turbulence and plane-channel flow, and posed the unresolved questions listed below.

Problem Statement

For large-eddy simulation of incompressible turbulent flows, complete the constraint-based program of constructing physically consistent subgrid-scale models consolidated by Silvis, Remmerswaal & Verstappen (2017). The common objective uniting all requested results is to turn their assembled framework of model constraints (symmetries S1-S7, near-wall scaling N, realizability R, and dissipation/production properties P1-P4) into a reliable, principled model-design methodology, specifically by answering: (1) Significance - which of the assembled constraints genuinely matter, for which classes of flows, in realistic a priori and a posteriori LES settings, given the documented mutual incompatibilities (P2a with P1b; P2b with P4) and the fact that successful models violate some constraints? (2) Symmetry and length scales - how can the lack of built-in scaling invariance (S5) in velocity-gradient-based models be overcome; in particular, what are the symmetry-preservation properties of the dynamic procedure, of integral-length-scale-type estimates of turbulent kinetic energy and dissipation, and of flow-dependent definitions of the subgrid characteristic length scale, including on anisotropic grids? (3) Realizable nonlinear models - can the scalar coefficient functions of the general nonlinear velocity-gradient model class (their Eqs. 8-11, coefficients depending on the five invariants $I_1,\dots,I_5$ of the rate-of-strain and rate-of-rotation tensors) be chosen so that the model is realizable and consistent with the second law of thermodynamics while retaining terms that describe nondissipative energy transport, in contrast to their Eq. (54) example, which is not realizable and, being time-reversible with nondynamic constants, violates P3? (4) New constraints - what new constraints for the construction of subgrid-scale models can be devised to close the gap between the analytical desiderata and robust closures? These four threads jointly support the single objective.

References

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