A mixed one-point PDF and two-point correlation function approach for two-step parallel/consecutive chemical reactions in homogeneous turbulence

A procedure is introduced which combines a one-point joint scalar probability density function (pdf) description with the use of two-point scalar correlation functions in order to calculate concentration statistics for an isothermal multi-species chemical reaction carried by stationary isotropic turbulence. The particular chemistry addressed in this paper is the parallel/consecutive reaction, ;Three concurrent processes take place in such a system; chemical kinetics, molecular diffusion and turbulent advection. Time splitting is invoked to separate these processes in each short time interval. Chemical reaction terms are treated without approximation by the one-point pdf format. Molecular diffusion plays two roles. It causes an evolution of the one-point pdf, which is treated by an LMSE (Linear Mean Square Estimation) closure, and it also contributes to the evolution of the scalar correlation functions. The latter contribution is treated exactly, without need of a closure approximation. Turbulent advection has no effect on the evolution of the one-point pdf in statistically homogeneous systems such as this one. But it does contribute to the evolution of scalar correlation functions; the well studied EDQNM (Eddy Damped Quasi Normal Markovian) closure approximation is used to represent this effect.;Although the reaction terms are exactly treated by the one-point pdf formulation, they also incidentally alter the scalar correlation functions. This effect is treated by a similarity assumption, which argues on the basis of DNS data and the full two-point pdf simulation of Jiang (31) that chemical reaction does not significantly modify the length scales of the scalar correlations and cross-correlations.;The method we propose is validated first by comparisons with DNS data of Gao (1990) and the full two-point pdf simulation of Jiang, for a bilinear reaction, such as (1) in the absence of (2). The DNS data and Jiang's model predictions are accurately reproduced by our formulation over a range of simulation parameters. For the parallel competitive reaction (1) and (2) DNS data is also available. An exact comparison with the simulation is not possible in this case, in part because numerical representation of a delta function is necessarily approximate. Nevertheless the results satisfactorily mimic DNS data. We also investigate the role of Reynolds numbers higher than can be reached with the DNS method, as well as a range of Schmidt numbers.