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An alternative derivation of the Germano identity as the residual of the LES equation

The Note presents an alternative derivation and interpretation of the Germano identity and its error, showing that the Germano identity error directly estimates the residual of the LES equation, i.e., the misfit when evaluating the inexact equation for the exact solution, and therefore represents the source of errors in LES. This has many applications, including for optimal output-based grid/filter-adaptation and uncertainty quantification in LES.