Sub–grid Scale Modeling: Correct Models and Others

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Abstract
One approach to studying turbulent convection in stars is through the use of large computers with a technique called Large Eddy Simulation (LES). Since no computer presently available, or even projected in the foreseeable future, can resolve all of the scales that characterize a fully developed turbulent flow, the LES technique resolves the largest scales, while it models the "unresolved" scales with a sub-grid scale (SGS) model. In the astrophysical literature, two such models have been used. It is the purpose of this paper to show that the first of the SGS models (1) contradicts Galilei invariance, (2) employs an incorrect timescale (Eulerian instead of Lagrangean), (3) contradicts the Kolmogorov inertial law and the Richardson diffusion law, (4) misinterprets dynamical and kinematic effects, and (5), from the numerical view point, gives rise to a turbulent viscosity much larger than it ought to be. The second SGS model has none of the above problems, the only shortcoming being its incompleteness, since it was originally devised for shear rather than buoyancy dominated flows. Thus, it must be improved—a process that present knowledge of turbulence allows us to carry out quite systematically. In conclusion, the first SGS should be avoided, while the second should be used and improved.