PDF-based model for inertial particles settling in a turbulent boundary layer using an asymptotic closure approximation

In a recent study, Zhang et al. (Phys. Rev. Fluids 8, 014301, 2023) considered closures for the transport equations governing inertial particle motion in turbulent boundary layers that are derived from an underlying phase-space, probability density function (PDF) equation. These equations are usually closed using the quasi-Normal approximation (QNA), however, this leads to large quantitative and qualitative errors, except for particles with weak inertia. Zhang et al. derived a new closure approximation based on an asymptotic solution to the transport equations in regions where the effect of particle inertia is significant. The closure captures the strong non-Gaussianity of the particle velocities, and comparisons with DNS show that the new closure performs much better than the QNA. In this study, we extend the model of Zhang et al. to the case of settling inertial particles, for which the system of equations is more complex due to the presence of a finite mean vertical particle velocity. The additional mathematical complexity introduces challenges when solving the system of equations numerically, and careful treatment of the boundary conditions is required. We will present the new model along with comparisons to direct numerical simulations.