The present work is intended as a proposition for a new research program for rigorous physical subgrid-scale (SGS) modeling on the combined basis of (i) the Germano identity (ii) and Lie symmetries, which are the axiomatic foundation of classical mechanics. First, new results are presented in this regard. The basic idea here is based on the Germano identity and the fundamental assumption in that the SGS model is just a functional of the resolved scales U¯, that is, in the usual notation τijSGS=τijSGS(U¯), although this can of course also be generalized.

This alone defines a new functional equation of the form τik[U˜˜]−τik[U¯]˜−U¯iU¯k˜+U¯˜iU¯˜k≡0 for the SGS model, if the residual error in the Germano identity is set exactly to zero. This is in contrast to the usual dynamic procedure, where a given SGS model is introduced into the Germano identity and the residual error is minimized according to a given norm. The resulting functional equation for τijSGS(U¯) defines a new class of model equations. The solution of the aforementioned equation for the SGS model τijSGS using homogenization transform and Fourier transform shows an extremely large variety of potential solutions, that is, SGS models, which at the same time addresses the classical question of how the shape of the test filter as well as the SGS model are related to each other. The analysis quite naturally shows that the proposed analysis focuses solely on the nonlinear term of the Navier–Stokes equations. For physically realizable SGS models, the very large variety of solutions is restricted by means of the classical as well as statistical Lie symmetries of the filtered Navier–Stokes equations. The latter describes the intermittency and non-Gaussian behavior of turbulence. The symmetries can be used decidedly here, that is, it can be selected quite specifically which symmetries are to be fulfilled. A number of models are presented as examples, some of which have similarities to classical models, and also new nonlocal models emerge. As an additional new result we find that the Germano identity can be extended by a divergence-free tensor. The physical meaning of this previously overlooked term needs to be further investigated, but in the classical dynamical procedure the term does not vanish and may be employed profitably, for example, for model optimization. We conclude the presented formulation of a mathematical work program for the development of SGS models based on Lie symmetries and the Germano identity with an extensive outlook for potential further research directions.