Dynamic subgrid-scale model

A dynamic subgrid-scale model estimates its closure coefficient from the resolved flow as the calculation proceeds. In a common construction, a second, coarser test filter probes interactions among resolved scales, providing information for a subgrid-scale model below the original resolution.^[ref]

Write a bar for the original spatial averaging filter and a hat for the test filter. DefineDirect subtraction gives the Germano identityThe rightmost expression uses only resolved quantities. The identity is exact for these definitions; extrapolating a closure between the filter levels is the modeling step.^[ref]

For instance, suppose a chosen closure writes the deviatoric stresses as ${\tau }^d=CA$ and $T^d=CB$, and assumes $C$ can be taken outside the test filtering. Then $L^d\approx CM$, where $M=B-\hat{A}$. One way to choose $C$ is to minimize ${\sum }_{ij}(L_{ij}^d-C{M}_{ij}{)}^2$. Differentiation giveswhen the denominator is nonzero. This is a least squares fit of a scalar to tensor data. In a one-component illustrative fit, $L^d=0.012$ and $M=0.3$ give $C=0.04$.

The coefficient need not remain positive. Depending on the closure, a negative value can permit turbulent backscatter, but a small denominator can also make an estimate unreliable. Averaging, bounds and treatment of weak strain are additional choices. Dynamically fitting a coefficient does not remove these assumptions or make unresolved motion observable.

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.

The Germano identity is an exact relation between stresses at two filtering levels and a stress computable from the resolved field. Let a bar denote a grid filter and a hat a test filter. With the subgrid-scale stress conventionthe identity statesIt follows by substituting ${\hat{\tau }}_{ij}=\widehat{\overline{u_i{u}_j}}-\widehat{{\bar{u}}_i{\bar{u}}_j}$ and cancelling the common term. Massimo Germano and collaborators used this relation to construct a dynamic subgrid-scale model.^[ref]

A scalar block-average example makes every term visible. Start with four velocity values grouped as $(0,2)$ and $(2,4)$. Averaging within each group gives $\bar{u}=(1,3)$ and $\overline{u^2}=(2,10)$, hence $\tau =(1,1)$. A test average over the two groups then givesThus $L=T-\hat{\tau }$ reads $1=2-1$. The resolved group averages contain enough information for $L$, although they alone do not determine the original within-group variances.

The identity has no fitted coefficient. To estimate a missing subgrid-scale stress, a closure assumes some model form at both filtering levels and tries to make their difference agree with the known $L_{ij}$. Extracting a varying coefficient from a filter, choosing a filter width and stabilizing a fit are modeling decisions; they do not enter the cancellation proof.