The major contribution to the subgrid scale model brought about by Germano et al. (1991) is the identification that consistency between (34.1) and (34.2) depends on a proper choice of C . This is achieved by subtraction of the test-scale average of from T_ij (Lilly, 1992; Hoffmann and Benocci, 1994) to obtain

(34.6)

with

(34.7)

and

(34.8)

Equations (34.1), (34.3), (34.6), (34.7) and (34.8) are five independent equations which cannot be solved for the model constant C because It appears in a filter operation (Equation (34.6)). Lilly (1992) and Zang et al. (1993) suggested the following assumption

(34.9)

This enables Equation (34.6) to be written in the form

(34.10)

Eij is the residue of Equation (34.10). Application of the least square technique to minimize the residual gives the following expression for C

(34.11)

Here <.> means average over a plane in the model for which the flow is homogeneous. The least square minimization technique has been used by Piomelli (1993) to compute the flow in a plane channel at Reynolds numbers based on friction velocity and channel half width in the range between 200 and 2000. A limitation of the dynamic model is the plane averaging mentioned earlier. For an essentially three-dimensional flow like the rectangular impinging jet, there is no homogeneous space direction. Hence, instead of a plane averaging, we propose to use a local averaging over the test filter cell. Zang et al. (1993) performed this local averaging and also constrained the effective viscosity (molecular and eddy viscosity) to be non-negative for recirculating flows.