Subgrid-scale stress (${\tau }_{ij}$)

The subgrid-scale stress records velocity-product information missing from a filtered velocity field. For constant mass density and a specified spatial averaging filter, use the conventionIt enters the filtered momentum equation through $-{\partial }_j{\tau }_{ij}$. This ${\tau }_{ij}$ has units of squared speed; multiplication by mass density gives units of mechanical stress. Some sources use the opposite stress sign, so the momentum equation fixes the convention.^[ref]

For a normalized nonnegative kernel, define $v_i(r)=u_i(x-r)-{\bar{u}}_i(x)$. Expanding the product givesIt is therefore a weighted covariance at the chosen point. For every real vector $b$, $b_i{\tau }_{ij}{b}_j=\int G_{\Delta }(b_i{v}_i{)}^{2}dr\ge 0$, so the stress matrix is positive semidefinite. This argument depends on a nonnegative kernel; it cannot be applied unchanged to filters with negative weights.

For example, average equally weighted velocities $(a,0,0)$ and $(-a,0,0)$. The mean velocity vanishes but $\tau =\mathrm{diag}(a^2,0,0)$. With $a=2\mathrm{m}{\mathrm{s}}^{-1}$, ${\tau }_{11}=4{\mathrm{m}}^2{\mathrm{s}}^{-2}$. A zero mean has hidden nonzero motion.

The definition is exact. A subgrid-scale model instead approximates this tensor using available resolved data. In an incompressible flow, the isotropic part ${\tau }_{kk}{\delta }_{ij}/3$ can be absorbed into a modified pressure; the remaining deviatoric stress controls the modeled directional transfer.