 Subgrid-scale model

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 giveswhere repeated indices use the Einstein summation convention, $p$ is pressure, and the subgrid-scale stress in this sign convention isThe assumed filter commutes with the displayed derivatives. The definition of ${\tau }_{ij}$ is exact; a closure must approximate it using available resolved information.^[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: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.^[ref][ref]

For an incompressible flow, the model transfers resolved kinetic energy out at the rate per unit massA 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.^[ref]