 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.