A <subgrid-scale model> represents the influence of unresolved motion on the quantities retained in a <large eddy simulation>. Its need follows directly from the nonlinear velocity product in the <Navier-Stokes equations>: averaging a product generally differs from multiplying the averages.

Let a bar denote a <spatial averaging filter>. With constant <mass density> $\rho$, constant <kinematic viscosity> $\nu$, and a time-independent translation-invariant filter on an unbounded or periodic domain, filtering the incompressible momentum equation gives
$$
\partial_t\bar u_i+\partial_j(\bar u_i\bar u_j)
=-\rho^{-1}\partial_i\bar p+\nu\partial_j\partial_j\bar u_i-\partial_j\tau_{ij},
$$
where repeated indices use the <Einstein summation convention>, $p$ is <pressure>, and the <subgrid-scale stress> in this sign convention is
$$
\boxed{\tau_{ij}=\overline{u_i u_j}-\bar u_i\bar u_j.}
$$
The assumed filter commutes with the displayed derivatives. The definition of $\tau_{ij}$ is exact; a closure must approximate it using available resolved information.<A dynamic subgrid-scale eddy viscosity model by Massimo Germano and others (1990)>[Section2, equations3–7: filtering, filtered momentum and the unresolved stress]{ref}

For an elementary example, average two equally weighted values of one velocity component, $a$ and $-a$. The mean is zero but the mean square is $a^2$, so the corresponding stress component is $a^2$. A resolved mean alone has lost the information needed to reconstruct that <covariance>.

The <Smagorinsky model> closes the deviatoric part with an <eddy viscosity>:
$$
\tau_{ij}-\frac13\tau_{kk}\delta_{ij}=-2\nu_t\bar S_{ij},\qquad
\nu_t=(C_s\Delta)^2\sqrt{2\bar S_{ij}\bar S_{ij}},
$$
where $\delta_{ij}$ is the <Kronecker delta>, $\bar S_{ij}=(\partial_j\bar u_i+\partial_i\bar u_j)/2$ is the resolved <strain-rate tensor>, $C_s$ a dimensionless model coefficient and $\Delta$ a characteristic filter width. The isotropic stress can be included in a modified <pressure>. This strain-dependent viscosity has its historical origin in <Joseph Smagorinsky>'s atmospheric circulation modelling.<General Circulation Experiments with the Primitive Equations: I. The Basic Experiment by Joseph Smagorinsky>[Abstract: lateral exchange coefficient based on horizontal strain and a length scale]{ref}<A dynamic subgrid-scale eddy viscosity model by Massimo Germano and others (1990)>[Introduction, equations1–2: the standard Smagorinsky expression]{ref}

For an <incompressible flow>, the model transfers resolved kinetic energy out at the rate per unit mass
$$
\Pi=-\tau_{ij}\bar S_{ij}=2\nu_t\bar S_{ij}\bar S_{ij}\ge0.
$$
A fixed nonnegative <eddy viscosity> therefore permits dissipation but excludes local <turbulent backscatter>. A <dynamic subgrid-scale model> can instead estimate its coefficient from resolved stresses at two filter levels. This addresses a modelling choice, rather than changing the exact definition of the unresolved stress.<A dynamic subgrid-scale eddy viscosity model by Massimo Germano and others (1990)>[Introduction and Section2: backscatter limitations and the dynamic two-filter construction]{ref}


 Subgrid-scale model (source code)