Realizability conditions for the turbulent stress tensor in large-eddy simulation

TLDR

The turbulent stress tensor in large‑eddy simulation is examined theoretically, revealing that the commonly used spectral cut‑off filter is not positive. The study aims to ensure that subgrid models satisfy realizability conditions by examining several models under this requirement. The authors recommend using Gaussian or top‑hat filters with subgrid models that require positive turbulent kinetic energy, rather than the spectral cut‑off filter. The authors derive realizability conditions that hold only for positive filters, showing that spectral cut‑off filters violate these conditions and produce negative turbulent kinetic energy, while Gaussian or top‑hat filters satisfy them and yield a positive k, and they further establish a lower bound on k that yields an inequality for Smagorinsky‑type model constants in compressible flows.

Abstract

The turbulent stress tensor in large-eddy simulation is examined from a theoretical point of view. Realizability conditions for the components of this tensor are derived, which hold if and only if the filter function is positive. The spectral cut-off, one of the filters frequently used in large-eddy simulation, is not positive. Consequently, the turbulent stress tensor based on spectrally filtered fields does not satisfy the realizability conditions, which leads to negative values of the generalized turbulent kinetic energy k . Positive filters, e. g. Gaussian or top-hat, always give rise to a positive k . For this reason, subgrid models which require positive values for k should be used in conjunction with e. g. the Gaussian or top-hat filter rather than with the spectral cutoff filter. If the turbulent stress tensor satisfies the realizability conditions, it is natural to require that the subgrid model for this tensor also satisfies these conditions. With respect to this point of view several subgrid models are discussed. For eddy-viscosity models a lower bound for the generalized turbulent kinetic energy follows as a necessary condition. This result provides an inequality for the model constants appearing in a ‘Smagorinsky-type’ subgrid model for compressible flows.