On symmetries and averaging of the G-equation for premixed combustion

It is demonstrated that the G-equation for premixed combustion admits a diversity of symmetries properties, i.e. invariance characteristics under certain transformations. Included are those of classical mechanics such as Galilean invariance, rotation invariance and others. Also a new generalized scaling symmetry has been established. It is shown that the generalized scaling symmetry defines the physical property of the G-equation precisely. That is to say the value of G at a given flame front is arbitrary. It is proven that beside the symmetries of classical mechanics, particularly the generalized scaling symmetry uniquely defines the basic structure of the G-equation. It is also proven that the generalized scaling symmetry precludes the application of classical Reynolds ensemble averaging usually employed in statistical turbulence theory in order to avoid non-unique statistical quantities such as for the mean flame position. Finally, a new averaging scheme of the G-field is presented which is fully consistent with all symmetries of the G-equation. Equations for the mean G-field and flame brush thickness are derived and a route to consistent invariant modelling of other quantities derived from the G-field is illustrated. Examples of statistical quantities derived from the G-field both in the context of Reynolds-averaged models as well as subgrid-scale models for large-eddy simulations taken from the literature are investigated as to whether they are compatible with the important generalized scaling symmetry.