Smagorinsky eddy-viscosity model

For LES performed in physical space, the basic sub-grid stress model is the eddy-viscosity model proposed by Smagorinsky . The Smagorinsky model is based on the gradient transport hypothesis and the sub-grid viscosity concept, just as the Reynolds stress models based on the Boussinesq eddy viscosity hypothesis, and expressed as ... 

Germano et al. [56] and Germano [57] have devised a new concept that they describe as a Dynamic SGS Model. Their formulation is also based on the Smagorinsky eddy-viscosity approximation. However, rather than fixing the value of Cs a priori, they permit it to be computed as the LES proceeds. This is accomplished by using two filters. 

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 ... 

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 ... 

The Smagorinsky Model (cf. Ref. [51]) is an algebraic model in the same spirit as the Prandtl mixing length model discussed in section RANS Turbulence Modeling. In the Smagorinsky model, the SGS stresses are assumed to be proportional to the rate of strain, that is, = VtS, and the kinematic eddy viscosity is determined from the expression... 

The model proposed by Smagorinsky [ 16] is one of the most used in LES. In this model, the eddy viscosity is assumed to be proportional to the sub-grid length scale (A) and the characteristic turbulent velocity (La = A S ) as... 

Substituting the expressions for the eddy viscosity given by the Smagorinsky model, the Germano identity (4.121) is obtained ... 

Among the models for the turbulent viscosity commonly used in Large-Eddy Simulation (LES), we present those of Smagorinsky and Germano. 

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