both. Functional (eddy–viscosity) models Functional models are simpler than structural models, focusing only on dissipating energy at a rate that is physically correct. These are based on an artificial eddy viscosity approach, where the effects of turbulence are lumped into a turbulent viscosity. The approach treats dissipation of kinetic energy at sub-grid scales as analogous to molecular diffusion. In this case, the deviatoric part of is modeled as: where is the turbulent eddy viscosity and is the rate-of-strain tensor. Based on dimensional analysis, the eddy viscosity must have units of . Most eddy viscosity SGS models model the eddy viscosity as the product of a characteristic length scale and a characteristic velocity scale. Smagorinsky–Lilly model The first SGS model developed was the Smagorinsky–Lilly SGS model, which was developed by Smagorinsky and used in the first LES simulation by Deardorff. It models the eddy viscosity as: where is the grid size and is a constant. This method assumes that the energy production and dissipation of the small scales are in equilibrium - that is, . The Dynamic Model (Germano et al. and beyond) Germano et al. identified a number of studies using the Smagorinsky model that each found different values for the Smagorinsky constant for different flow configurations. In an attempt to formulate a more universal approach to SGS models, Germano et al. proposed a dynamic Smagorinsky model, which utilized two filters: a grid LES filter, denoted , and a test LES filter, denoted for any turbulent field . The test filter is larger in size than the grid filter and adds an additional smoothing of the turbulence field over the already smoothed fields represented by the LES. Applying the test filter to the LES equations (which are obtained by applying the "grid" filter to Navier-Stokes equations) results in a new set of equations that are identical in form but with the SGS stress replaced by . Germano {\it et} al. noted that even though neither nor can be computed exactly because of the presence of unresolved scales, there is an exact relation connecting these two tensors. This relation, known as the Germano identity is Here can be explicitly evaluated as it involves only the filtered velocities and the operation of test filtering. The significance of the identity is that if one assumes that turbulence is self similar so that the SGS stress at the grid and test levels have the