= Smagorinsky model
{c}
{title2=$\nu_t=(C_s\Delta)^2|\bar S|$}
The <Smagorinsky model> approximates the deviatoric <subgrid-scale stress> using an <eddy viscosity> determined by the resolved <strain-rate tensor>:
$$
\tau_{ij}-\frac13\tau_{kk}\delta_{ij}=-2\nu_t\bar S_{ij},\qquad
\nu_t=(C_s\Delta)^2|\bar S|,\qquad
|\bar S|=\sqrt{2\bar S_{ij}\bar S_{ij}}.
$$
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.<A dynamic subgrid-scale eddy viscosity model by Massimo Germano and others (1990)>[Introduction, equations 1–2, standard Smagorinsky closure and strain magnitude]{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 $\gamma=10\,\mathrm{s^{-1}}$ yields
$$
\nu_t=4\times10^{-5}\,\mathrm{m^2\,s^{-1}},\qquad
\tau_{12}=-\nu_t\gamma=-4\times10^{-4}\,\mathrm{m^2\,s^{-2}}.
$$
These 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 is
$$
\Pi=-\tau_{ij}\bar S_{ij}=2\nu_t\bar S_{ij}\bar S_{ij}\geq0.
$$
A 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.<General Circulation Experiments with the Primitive Equations: I. The Basic Experiment by Joseph Smagorinsky>[Abstract and discussion of exchange coefficient proportional to squared grid length times deformation magnitude]{ref}
Back to article page
