Capturing the mechanisms for gravitational settling of inertial particles in turbulence using the kinetic PDF approach

The modification of average settling velocity for inertial particles in turbulence is a striking feature of dispersed particle flows that has received continued attention since being first addressed, resulting in a variety of theories concerning the underlying physical mechanisms that act at different scales. Yet to date, the unification of these theories remains elusive. Whilst the preferential sweeping of particles in downward flowing regions occurs across the range of Stokes number (St) [1], the manifestation of this effect through the centrifuge mechanism [2] has been shown to be dominant only for St << 1 [3], with the path history playing an increasingly important role for more inertial particles [4]. This highlights the need for an approach which is able to collectively describe these different mechanisms.Previously, it has been demonstrated that the kinetic probability density function (PDF) approach provides an exact representation for the increase in settling velocity of particles subject to a Stokes drag force within a Gaussian flow field [5], and is able to account for the distinct physical mechanisms that occur across the entire range of St. In this work an asymptotic analysis of the kinetic PDF approach to O(St) is contrasted with that presented in [2], and it is shown using kinematic simulation that a quantitatively greater proportion of the increase in settling velocity is accounted for as the particle inertia is increased away from the low St regime. In particular, even under the reduced description at O(St), the kinetic PDF approach is able to accurately capture the peak increase in settling velocity that occurs for St </< 1. The key difference in the modelling procedure compared to [2] is that supposition of a particle velocity field is not necessary, meaning that important phenomena such as the crossing trajectories effect are respected in this framework, and the inertial dependence of particles can therefore be more faithfully accounted for in the model description. These observations suggest that the instantaneous centrifuging of particles from vortical structures within the flow field that occurs at low St is the limiting case of a more general mechanism underlying the increase in settling velocity, which can be interpreted from the kinetic PDF approach as arising due to the correlation of the fluid velocity gradient tensor history with the spatial structure of the Eulerian flow field sampled along trajectories. This is consistent with the arguments put forth in [4], and crucially the appearance of the path history is not only retained, but is in fact the dominant contribution to the settling velocity increase at O(St). This emphasises the multiscale nature of the description provided by the kinetic PDF approach, and illustrates the implications that extending this framework to the consideration of full dynamical turbulence could have for the modelling and simulation of dispersed particle flows.[1] Wang, L.-P. & Maxey, M. R. Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence., Journal of Fluid Mechanics. 256, 27–68 (1993).[2] Maxey, M. R. The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields., Journal of Fluid Mechanics. 174, 441–465 (1987).[3] Bragg, A. D. & Collins, L. R. New insights from comparing statistical theories for inertial particles in turbulence: I. Spatial distribution of particles, New Journal of Physics. 16, 055013 (2014).[4] Bragg, A. D., Ireland, P. J. & Collins, L. R. On the relationship between the non-local clustering mechanism and preferential concentration, Journal of Fluid Mechanics. 780, 327–343 (2015).[5] Stafford, C. P. & Swailes, D. C. Mass flux of dispersed particles in turbulence: Representations and the influence of correlation structure in gravitational settling., Physical Review E. 103, 063101 (2021)