The statistical theory of turbulent flames

Abstract

The theory stemming from the statistical representation of turbulent flames is presented and developed, the major aim being to describe the basic processes in relatively simple flames. Starting from conservation equations, with the assumption of low Mach number and high Reynolds number, it is shown that the properties at any point in the flame can be determined from the transport equations for the velocity U and a set of scalars $\phi $: $\phi $ represents the species mass fractions and enthalpy. However, the solution of these equations with initial conditions and boundary conditions appropriate to turbulent flames is prohibitively difficult. Statistical theories attempt to describe the behaviour of averaged quantities in terms of averaged quantities. This requires the introduction of closure approximations, but renders a more readily soluble set of equations. A closure of the Reynolds-stress equations and the equation for the joint probability density function of $\phi $ is considered. The use of the joint probability density function (p.d.f.) equation removes the difficulties that are otherwise encountered due to non-linear functions of the scalars (such as reaction rates). While the transport equation for the joint p.d.f. provides a useful description of the physics, its solution is feasible only for simple cases. As a practical alternative, a general method is presented for estimating the joint p.d.f. from its first and second moments: transport equations for these quantities are also considered therefore. Modelled transport equations for the Reynolds stresses, the dissipation rate, scalar moments and scalar fluxes are discussed, including the effects of reaction and density variations. A physical interpretation of the joint p.d.f. equation is given and the modelling of the unknown terms is considered. A general method for estimating the joint p.d.f. is presented. It assumes that the joint p.d.f. is the statistically most likely distribution with the same first and second moments. This distribution is determined for any number of reactive or non-reactive scalars.