Probability Density Functions For Fluctuations In Turbulent Two-Phase Flames

Probability density function methods for characterizing various fluctuating quantities in the reacting gas phase of turbulent spray flames are presented, for the development of which O'Brien has been a pioneer. Balance equations for two-phase flows considered as a piecewise continuous medium are first presented with the interface conditions containing the vaporization-condensation equilibrium. From these, the equation for the joint probability density function of temperature and composition in the gas phase is derived, in which specific terms related to vaporization are identified. The terms that require modeling are then highlighted. The first one is the classical micromixing term, which has to include all the features already discussed for single phase reacting flows, but adaptation to two-phase flows has to be questioned. Mainly, several unclosed terms deal with vaporization, requiring specific attention. It is shown that proposals for their modeling can be obtained on the same basis as in the so-called Eulerian-Lagrangian models, which will directly induce changes in the shape of the probability density function. We finally discuss the model presented herein and comment on its connection with recently published literature for this subject. Specific attention is devoted to the combustion regimes and to the presence of diffusion flamelets surrounding fuel droplets or groups of droplets. The proposals presented here go one step further compared to the recent studies, and now, they have to be compared with well-defined experiments, numerical experiments with direct numerical simulations, or real experiments in conveniently controlled conditions.