Dynamic Model of LES

Dynamic modeling of the sub grid-scale stresses was introduced by Germano et al. (1991). In such an approach, the model coefficients are computed dynamically as the calculation progresses. The dynamic model of Germano et al. (1991) is based on the introduction of two filters. In addition to the grid filter (denoted by an overbar), which defines the resolved and subgrid scales, a test filter (denoted by circumflex) is used. The width of the test filter is larger than that of the grid filter. When the grid filter is applied, the stress terms that appear in the Navier-Stokes equations are the subgrid scale (SGS) stresses. In a similar manner, the test filter defines a new set of stresses, the test-level subgrid-scale stresses. Invoking (33.11) in (33.9) yields

(34.1)

The quantity C is the Smagorinsky coefficient (basically this is square of original quantity) depends on the type of the flow consideration (Canuto and Cheng, 1997). Germano et al. (1991) and Lilly (1992) suggested a method to calculate C for each time step and grid point dynamically from flow field data. The width of the test filter is larger than the grid filter width. The test filter defines a new set of stresses. The test-level subgrid scale stresses or subtest-scale stresses, τ_ij (see Najjar and Tafti, 1996) are given by

(34.2)

Equation (34.2) can also be expressed in terms of Smagorinsky closure as

(34.3)

where

(34.4)

and

(34.5)