An approach to the autoignition of a turbulent mixture

This paper considers the turbulent homogeneous mixing of two reactants undergoing a one step, second order, irreversible, exothermic chemical reaction with a rate constant of the Arrhenius type. A statistically stationary turbulent velocity field is assumed given and unaffected by mass or heat production due to the chemical reaction. Relative density fluctuations are neglected. A Hopf-like functional formalism is presented, with application to both statistically inhomogeneous and statistically homogeneous flows. Single and double point probability density function differential equations are derived from those functional equations. The limit of very large activation energies is considered; a low degree of statistical correlation between temperature and concentration fields during the ignition period is hypothesized. After making use of the homogeneity assumption a closure problem is still present due to the nonlocalness of the molecular diffusion term. The problem is rendered closed by assuming a Gaussian conditional expected value for the temperature at a point given the temperature at a neighboring point. The closure is seen to preserve very important mathematical and physical properties. A linear first order hyperbolic differential equation with variable coefficients for the probability density function of the temperature field is obtained. A second Damköhler number based on Taylor's microscale turns out to be an important controlling parameter. A numerical integration for different values of the second Damköhler number and the initial stochastic parameters is carried out. The mixture is seen to evolve towards an eventual thermal runaway, the detailed behavior however being different for different systems. Some peculiarities during the ignition period evolution are uncovered.