 Test filter

A test filter is an additional spatial averaging filter applied to already resolved data in a dynamic subgrid-scale model. If $\bar{f}=G\ast f$ is the field after the grid filter, a test kernel $H$ givesIt probes interactions within the available resolved motion at a coarser analysis level. It does not measure the original subgrid motion directly.^[ref]

For example, suppose two resolved values of one velocity component are $1$ and $3\mathrm{m}{\mathrm{s}}^{-1}$. A test average gives $\hat{\bar{u}}=2\mathrm{m}{\mathrm{s}}^{-1}$ and $\widehat{{\bar{u}}^2}=5{\mathrm{m}}^2{\mathrm{s}}^{-2}$, so the resolved differenceis 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 $H\ast G$, 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 $\sqrt{5}\Delta$ 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.