Statistical modeling of sprays using the droplet distribution function

Abstract

The theoretical foundations of a statistical spray modeling approach based on the droplet distribution function (ddf), which was originally proposed by Williams [Phys. Fluids 1, 541 (1958)], are established. The equation governing the ddf evolution is derived using an alternative approach. The unclosed terms in the ddf evolution equation are precisely defined, and the regime of applicability of current models is discussed. The theory of point processes 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 number of spray droplets in physical space, and the joint probability density function (jpdf) of velocity and radius conditional on physical location. Evolution equations for the spray intensity and the conditional jpdf of velocity and radius are derived. The intensity evolution equation contains a sink term corresponding to droplet vaporization, hitherto missing in previous derivations of this equation. This sink term is essential in order to correctly represent the vaporization phenomenon. Problems with numerical convergence of computed solutions to the ddf evolution are discussed, and criteria for establishing convergence are proposed. The study also shows how quantities predicted by ddf-based spray models can be compared to experimental measurements.