A generalized kinetic framework for coagulating particles in turbulent flows: beyond mean-field theory

Predicting particle collisions in turbulent flows is critical for cloud physics, aerosol transport, and industrial sprays. Classical models, like the Smoluchowski equation, neglect spatial correlations and can predict unphysical outcomes such as finite-time gelation. We present a generalized kinetic framework that models the evolving particle number within a fixed-size ensemble by assigning zero mass to surplus, inactive particles. This formulation enables a BBGKY-type hierarchy for many-particle probability densities and leads, through marginalization, to a population balance equation with a time-dependent collision kernel. The kernel depends on small-scale observables, such as the radial distribution function, obtainable from direct numerical simulation. In the absence of correlations, the framework recovers classical results including the Saffman-Turner collision kernel for tracers and Telford's lucky-droplet mechanism for settling drops. Beyond these limits, the approach supports the systematic inclusion of correlation effects, such as local depletion around clusters, providing a consistent foundation for modeling particle coagulation in turbulent flows.