SGS eddy viscosity

The most widely used model for the SGS stresses is due to Smagorinsky (1963) and involves a SGS eddy viscosity, ve, which is related to the local resolved deformation rate S ... 

Note that the Eqs. (1), (2), and (8) are really and essentially different due to the absence or presence of different turbulent transport terms. Only by incorporating dedicated formulations for the SGS eddy viscosity can one attain that LES yield the same flow field as DNS. RANS-based simulations with their turbulent viscosity coefficient, however, essentially deliver steady flow fields and as such are never capable of delivering the same velocity fields as the inherently transient LES or DNS, irrespectively of the refinement of the computational grid ... 

Thereafter, the closure consists in determining the SGS eddy viscosity parameter. By analogy to the mixing length hypothesis, the Smagorinsky [150] model is defined as ... 

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