Statistical modeling of sprays using the droplet distribution function

The theoretical foundations of a statistical spray modeling approach based on the droplet distribution function (ddf), which was originally proposed b y Williams [Phys. Fluids 1, 541 (1958)], are established. The equation gove rning the ddf evolution is derived using an alternative approach. The unclo sed terms in the ddf evolution equation are precisely defined, and the regi me of applicability of current models is discussed. The theory of point pro cesses is used to rigorously establish the existence of a disintegration of the ddf in terms of a spray intensity, which is the density of expected nu mber of spray droplets in physical space, and the joint probability density function (jpdf) of velocity and radius conditional on physical location. E volution equations for the spray intensity and the conditional jpdf of velo city and radius are derived. The intensity evolution equation contains a si nk term corresponding to droplet vaporization, hitherto missing in previous derivations of this equation. This sink term is essential in order to corr ectly represent the vaporization phenomenon. Problems with numerical conver gence of computed solutions to the ddf evolution are discussed, and criteri a for establishing convergence are proposed. The study also shows how quant ities predicted by ddf-based spray models can be compared to experimental m easurements. (C) 2001 American Institute of Physics.