Using singular values to build a subgrid-scale model for large eddy simulations

TLDR

The paper derives an eddy‑viscosity subgrid‑scale model for large eddy simulations based on singular values of the resolved velocity gradient tensor. The σ‑model automatically vanishes in two‑dimensional, two‑component, pure shear, solid rotation, axisymmetric, or isotropic contraction/expansion flows, behaves cubically near solid boundaries, and is implemented with low cost and no need for homogeneous directions. Analytically, the model shows correct cubic boundary behavior and, in test cases, performs as well or better than the Dynamic Smagorinsky model while being cheaper, easier to implement, and suitable for non‑homogeneous, wall‑bounded flows.

Abstract

An eddy-viscosity based, subgrid-scale model for large eddy simulations is derived from the analysis of the singular values of the resolved velocity gradient tensor. The proposed σ-model has, by construction, the property to automatically vanish as soon as the resolved field is either two-dimensional or two-component, including the pure shear and solid rotation cases. In addition, the model generates no subgrid-scale viscosity when the resolved scales are in pure axisymmetric or isotropic contraction/expansion. At last, it is shown analytically that it has the appropriate cubic behavior in the vicinity of solid boundaries without requiring any ad-hoc treatment. Results for two classical test cases (decaying isotropic turbulence and periodic channel flow) obtained from three different solvers with a variety of numerics (finite elements, finite differences, or spectral methods) are presented to illustrate the potential of this model. The results obtained with the proposed model are systematically equivalent or slightly better than the results from the Dynamic Smagorinsky model. Still, the σ-model has a low computational cost, is easy to implement, and does not require any homogeneous direction in space or time. It is thus anticipated that it has a high potential for the computation of non-homogeneous, wall-bounded flows.