A compatibility principle, analytical formulation, and the G- and Σ-equations in the theory of turbulent premixed combustion

Abstract

We analyze the nature of the known difficulties that arise when trying to apply the kinematic G-equation and the surface average Σ-equation to the Reynolds-Averaged Navier–Stokes modeling of turbulent premixed combustion, based on the use of the Favre average equations of combustion and hydrodynamics. We consider this issue in the context of an analytical formulation that includes an asymptotically closed infinite system of successively derived unclosed moment combustion and hydrodynamics equations of the problem. We show that the cause of these difficulties is the incompatibility of the G-equation and Σ-equation with the other instantaneous and averaged equations of the problem. This leads us to formulate a compatibility principle, i.e., a rule for the identification of misconceptions about the use of instantaneous and averaged equations that are correct by themselves, due to their incompatibility with the equations of the corresponding analytical formulation. Special attention is devoted to a comparison of two approaches to a description of instantaneous and average premixed combustion, based on the use of the progress variable c and the scalar G. We state that these approaches are equivalent. To prove this, we derive the equations missing from these two approaches and show that the following two groups of instantaneous and Favre-averaged equations, formulated in terms of c and G, respectively, are equivalent. The main result is the proposed compatibility principle. The conclusion following from our analysis is that the concepts of the scalar G and flame surface density Σ useful in applications could be considered superfluous from the theoretical viewpoint.