A Two-Fluid Conditional Averaging Paradigm for the Theory and Modeling of Turbulent Premixed Combustion

Abstract

This paper extends a recent theoretical study that was previously presented in the form of a brief communication (Zimont, C&F, 192, 2018, 221-223), in which we proposed a simple splitting method for the derivation of two-fluid conditionally averaged equations of turbulent premixed combustion in the flamelet regime, formulated more conveniently for applications involving unclosed equations without surface-averaged unknowns. This two-fluid conditional averaging paradigm avoids the challenge in the Favre averaging paradigm of modeling the countergradient scalar transport phenomenon and the unusually large velocity fluctuations in a turbulent premixed flame. It is a more suitable conceptual framework that is likely to be more convenient in the long run than the traditional Favre averaging method. In this article, we further develop this paradigm and pay particular attention to the problem of modeling turbulent premixed combustion in the context of a two-fluid approach. We formulate and analyze the unclosed differential equations in terms of the conditions of the Reynolds stresses τij,u, τij,b and the mean chemical source ρW¯, which are the only modeling unknowns required in our alternative conditionally averaged equations. These equations are necessary for the development of model differential equations for the Reynolds stresses and the chemical source in the advanced modeling and simulation of turbulent premixed combustion. We propose a simpler approach to modeling the conditional Reynolds stresses based on the use of the two-fluid conditional equations of the standard “k-ε” turbulence model, which we formulate using the splitting method. The main problem arising here is the appearance in these equations of unknown terms describing the exchange of the turbulent energy k and dissipation rate ε in the unburned and burned gases. We propose an approximate way to avoid this problem. We formulate a simple algebraic expression for the mean chemical source that follows from our previous theoretical analysis of the transient turbulent premixed flame in the intermediate asymptotic stage, in which small-scale wrinkles in the instantaneous flame surface reach statistical equilibrium, while the large-scale wrinkles remain in statistical nonequilibrium.