Parameterization of subgrid-scalestress by the velocity gradient tensorBy

In large eddy simulation (LES), the large scale motions are directly computed and the eeects of the small scales are modeled. The term to be modeled is the subgrid-scale stress tensor, ij = u i u j ? u i u j ; (1) which arises when the Navier-Stokes equations are spatially ltered (denoted by overbar) to remove the small scale information. Most large eddy simulations make use of the Smagorinsky (1963) eddy-viscosity model, ij ? 1 3 kk ij = ?2(C 2 j Sj) S ij ; (2) where C is a non-dimensional constant, is the grid spacing, and S ij is the resolved strain-rate. Although the Smagorinsky model has been in use for nearly thirty years, for roughly half that period it has been known that the model provides only a crude estimate for the stresses. This fact was rst demonstrated by Clark et al. (1979), where direct numerical simulation (DNS) data for homogeneous isotropic turbulence was used to evaluate model predictions. Clark et al. found a correlation coeecient of approximately 0:2 when comparing predictions of the Smagorinsky model with the exact stresses. McMillan et al (1979) found that the correlation coeecient was even lower in homogeneous shear ow, being close to 0:1. Later, Piomelli et al. (1988) found similar results in turbulent channel ow. When contemplating these extremely low correlations, it may seem striking that the Smagorinsky model is successful when used in a large eddy simulation. The reason for the seemingly unwarranted accuracy is that, by construction, the Smagorin-sky model insures a net drain of energy from the large scales to the subgrid-scale motions. This is the primary objective of the subgrid-scale model, and as long as this requirement is met, reasonable results are evidently obtained. On the other hand, the Smagorinsky model provides poor predictions of the individual elements of the stress tensor. It is natural to expect that superior results could be obtained with a model that predicts the stress tensor more accurately. The aim of this work is to seek out potentially more accurate models. The Smagorinsky model is based on a molecular transport analogy where the stress is proportional to the rate of strain. The molecular analogy is a rather crude