= Test filter
{title2=$\widehat{\bar f}=H*\bar f$}
A <test filter> is an additional <spatial averaging filter> applied to already resolved data in a <dynamic subgrid-scale model>. If $\bar f=G*f$ is the field after the <grid filter>, a test kernel $H$ gives
$$
\widehat{\bar f}=H*\bar f=(H*G)*f.
$$
It probes interactions within the available resolved motion at a coarser analysis level. It does not measure the original subgrid motion directly.<A dynamic subgrid-scale eddy viscosity model by Massimo Germano and others (1990)>[§2, grid and test filter construction and two-level stress definitions]{ref}
For example, suppose two resolved values of one <velocity> component are $1$ and $3\,\mathrm{m\,s^{-1}}$. A test average gives $\widehat{\bar u}=2\,\mathrm{m\,s^{-1}}$ and $\widehat{\bar u^2}=5\,\mathrm{m^2\,s^{-2}}$, so the resolved difference
$$
L=\widehat{\bar u^2}-\widehat{\bar u}^{\,2}
=1\,\mathrm{m^2\,s^{-2}}
$$
is nonzero. This quantity is computable from the resolved samples; a <dynamic subgrid-scale model> uses such information to constrain a closure.
The combined filter is $H*G$, so its width cannot always be identified with the width of $H$ alone. A centred <box averaging filter> of width $\Delta$ has second moment $\Delta^2/12$. Combining it with one of width $2\Delta$ gives second moment $5\Delta^2/12$, because centred convolution kernels add their second moments. A single box of width $\sqrt5\,\Delta$ has the same second moment, but not the same kernel shape.
Choosing a width convention is therefore part of the model. The <Germano identity> holds for the filters actually defined; an assumed relation between modeled stresses at those levels is an additional approximation. Near boundaries, the test operation also needs a specified treatment of data outside the available region.
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