A probabilistic approach to turbulent flame theory

Abstract

A given statistically stationary field of wind-tunnel grid-generated turbulence acting as the carrier flow of a combustible species is considered. The turbulence is assumed dynamically unaffected by the chemical reaction, i.e. the heat release is taken to be small compared with the initial thermal enthalpy of the mixture. Consequently, the density may be treated as a constant. Reaction rates with explicit scalar-property dependence are used but no restriction on their functional form is imposed, i.e. linear, nonlinear or transcendental reaction rates can be exactly studied. A hierarchy of turbulent probability density functions is presented. The closure problem is discussed. The advantages of working with probabilistic equations rather than in terms of traditional moment formulations are outlined. A closure is proposed at the second order level for the joint probability density function of the scalar and fluctuating velocity equation. After some simplifying assumptions a second Damköhler number based on the diffusive time computed using the turbulence microscale and a dimensionless relaxation parameter, namely, the ratio of a turbulence relaxation time and a characteristic chemical time are seen to be the two controlling parameters. A linear reaction rate is finally chosen for possible comparison with previous results. The limit of very low turbulence intensity can be approximated by the statistically homogeneous study and is analytically solved for three different initial conditions. The general inhomogeneous case is numerically solved for different values of the two parameters mentioned above and various sets of initial scalar fluctuations and turbulence intensities. The effect of the initial scalar-fluctuating velocity correlation is also explored. The homogeneous turbulence limit is for all purposes identical to the numerical solution for turbulence intensity equal to 1%.