# Smagorinsky-Lilly Subgrid Scale Model

The Smagorinsky-Lilly SGS model is based on the Prandtl’s mixing length model and assumes that a kinematic SGS viscosity can be expressed in terms of the length scale and the strain rate magnitude of the resolved flow.

$${\tau}_{ij}^{\text{'}}=\rho \tilde{{u}_{i}{u}_{j}}-\rho \tilde{{u}_{i}}\tilde{{u}_{j}}=-2{\mu}_{s}\tilde{{S}_{ij}}$$

where

- ${\mu}_{s}=\rho {\left({C}_{S}\text{\Delta}\right)}^{2}\sqrt{2\tilde{{S}_{ij}}\tilde{{S}_{ij}}}$ is the subgrid turbulent viscosity,
- Model constant 0.17 < ${C}_{s}$ < 0.21 (Lilly, 1996)
- $\tilde{{S}_{ij}}=\frac{1}{2}\left(\frac{\partial \tilde{{u}_{i}}}{\partial {x}_{j}}+\frac{\partial \tilde{{u}_{j}}}{\partial {x}_{i}}\right)$ is the filtered (resolved) strain rate tensor

A major drawback of this model is that the model constant ( ${C}_{s}$ ) does not vary in space and time. Furthermore, this model has no correct wall behavior and it is too dissipative for laminar turbulent transition cases. These limitations led to the development of a dynamic model for which the model constant is allowed to vary depending on the grid resolution and flow regime.