 Germano identity

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.