= Subgrid-scale stress
{title2=$\tau_{ij}$}
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 convention
$$
\tau_{ij}=\overline{u_i u_j}-\bar u_i\bar u_j.
$$
It 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.<A dynamic subgrid-scale eddy viscosity model by Massimo Germano and others (1990)>[§2, filtered momentum equation and definition of the unresolved stress]{ref}
For a normalized nonnegative kernel, define $v_i(r)=u_i(x-r)-\bar u_i(x)$. Expanding the product gives
$$
\tau_{ij}(x)=\int G_\Delta(r)v_i(r)v_j(r)\,dr.
$$
It is therefore a weighted <covariance> at the chosen point. For every real <vector> $b$, $b_i\tau_{ij}b_j=\int G_\Delta(b_iv_i)^2\,dr\geq0$, 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 $\tau=\operatorname{diag}(a^2,0,0)$. With $a=2\,\mathrm{m\,s^{-1}}$, $\tau_{11}=4\,\mathrm{m^2\,s^{-2}}$. 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.
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