Smagorinsky model (${\nu }_t=(C_s\Delta {)}^2|\bar{S}|$)

The Smagorinsky model approximates the deviatoric subgrid-scale stress using an eddy viscosity determined by the resolved strain-rate tensor:Here ${\bar{S}}_{ij}=({\partial }_j{\bar{u}}_i+{\partial }_i{\bar{u}}_j)/2$, $\Delta$ is the filter length, $C_s$ is dimensionless, and repeated indices use the Einstein summation convention. The Kronecker delta selects the isotropic part. The construction estimates a missing effect from resolved deformation rather than calculating each unresolved eddy.^[ref]

For simple shear flow $\bar{u}=(\gamma y,0,0)$, the only nonzero strain components are ${\bar{S}}_{12}={\bar{S}}_{21}=\gamma /2$, giving $|\bar{S}|=|\gamma |$. Taking $C_s=0.1$, $\Delta =0.02\mathrm{m}$ and $\gamma =10{\mathrm{s}}^{-1}$ yieldsThese are illustrative parameter choices, not a universal fitted constant. The dimensions follow from ${\Delta }^2|\bar{S}|$.

For an incompressible flow, the resolved-to-unresolved kinetic energy transfer isA nonnegative eddy viscosity thus removes resolved kinetic energy and cannot represent local turbulent backscatter. Coefficients and near-wall treatment require physical and numerical judgment.

Joseph Smagorinsky's 1963 atmospheric-circulation experiment used a lateral exchange coefficient based on horizontal deformation and a length scale. The modern three-dimensional expression above extends that strain-dependent modeling idea; it should not be mistaken for the complete original atmospheric model.^[ref]