PDF model of settling particle transport in a turbulent boundary layer using an asymptotic closure approximation

The transport of settling, inertial particles in a turbulent boundary layer can be modeled using a phase-space probability density function (PDF) equation, from which one can derive a hierarchy of moment equations. In these moment equations, closure problems are associated with both the particle velocity moments and a flux associated with the preferential sampling of the underlying fluid field. In previous work by Zhang et al. (Phys. Rev. Fluids 8, 014301, 2023), the particle velocity moments were closed using an asymptotic closure approximation (ACA), which achieves comparable results to the Quasi-normal approximation (QNA) in the weak-inertia regime but significantly outperforms QNA in the strong-inertia regime. In the model by Zhang et al., the preferential sampling flux was closed using a gradient-diffusion approximation. In the presence of gravity, however, such a closure is insufficient because it does not capture the preferential sweeping mechanism. In this study, we derive a closed PDF-based model for settling particles by extending the ACA to account for settling, and by using a model for the sampling flux that accounts for the effect of preferential sweeping. The latter is achieved by modifying the model of Stafford and Swailes (Phys. Rev. E 103, 063101, 2021) that was derived for homogeneous, isotropic turbulence to make it suitable for wall-bounded turbulence. The performance of the model is assessed by comparison with results from direct numerical simulations.