Large Eddy simulation of turbulence: A subgrid scale model including shear, vorticity, rotation, and buoyancy

Abstract

The Reynolds numbers that characterize geophysical and astrophysicalturbulence (Re approximately equals 108 for the planetaryboundary layer and Re approximately equals 1014 for the Sun'sinterior) are too large to allow a direct numerical simulation (DNS) ofthe fundamental Navier-Stokes and temperature equations. In fact, thespatial number of grid points N approximately Re9/4 exceedsthe computational capability of today's supercomputers. Alternativetreatments are the ensemble-time average approach, and/or the volumeaverage approach. Since the first method (Reynolds stress approach) islargely analytical, the resulting turbulence equations entail manageablecomputational requirements and can thus be linked to a stellarevolutionary code or, in the geophysical case, to general circulationmodels. In the volume average approach, one carries out a large eddysimulation (LES) which resolves numerically the largest scales, whilethe unresolved scales must be treated theoretically with a subgrid scalemodel (SGS). Contrary to the ensemble average approach, the LES+SGSapproach has considerable computational requirements. Even if thisprevents (for the time being) a LES+SGS model to be linked to stellar orgeophysical codes, it is still of the greatest relevance as an'experimental tool' to be used, inter alia, to improve theparameterizations needed in the ensemble average approach. Such amethodology has been successfully adopted in studies of the convectiveplanetary boundary layer. Experienc e with the LES+SGS approach fromdifferent fields has shown that its reliability depends on thehealthiness of the SGS model for numerical stability as well as forphysical completeness. At present, the most widely used SGS model, theSmagorinsky model, accounts for the effect of the shear induced by thelarge resolved scales on the unresolved scales but does not account forthe effects of buoyancy, anisotropy, rotation, and stablestratification. The latter phenomenon, which affects both geophysicaland astrophysical turbulence (e.g., oceanic structure and convectiveovershooting in stars), has been singularly difficult to account for inturbulence modeling. For example, the widely used model of Deardorff hasnot been confirmed by recent LES results. As of today, there is no SGSmodel capable of incorporating buoyancy, rotation, shear, anistropy, andstable stratification (gravity waves). In this paper, we construct sucha model which we call CM (complete model). We also present a hierarchyof simpler algebraic models (called AM) of varying complexity. Finally,we present a set of models which are simplified even further (calledSM), the simplest of which is the Smagorinsky-Lilly model. Theincorporation of these models into the presently available LES codesshould begin with the SM, to be followed by the AM and finally by theCM.