Multiscale preferential sweeping of particles settling in turbulence

Abstract

In a seminal article, Maxey (J. Fluid Mech., vol. 174, 1987, pp. 441–465) presented a theoretical analysis showing that enhanced particle settling speeds in turbulence occur through the preferential sweeping mechanism, which depends on the preferential sampling of the fluid velocity gradient field by the inertial particles. However, recent direct numerical simulation (DNS) results in Ireland et al. (J. Fluid Mech., vol. 796, 2016b, pp. 659–711) show that even in a portion of the parameter space where this preferential sampling is absent, the particles nevertheless exhibit enhanced settling velocities. Further, there are several outstanding questions concerning the role of different turbulent flow scales on the enhanced settling, and the role of the Taylor Reynolds number $R_{\unicode[STIX]{x1D706}}$. The analysis of Maxey does not explain these issues, partly since it was restricted to particle Stokes numbers $St\ll 1$. To address these issues, we have developed a new theoretical result, valid for arbitrary $St$, that reveals the multiscale nature of the mechanism generating the enhanced settling speeds. In particular, it shows how the range of scales at which the preferential sweeping mechanism operates depends on $St$. This analysis is complemented by results from DNS where we examine the role of different flow scales on the particle settling speeds by coarse graining the underlying flow. The results show how the flow scales that contribute to the enhanced settling depend on $St$, and that contrary to previous claims, there can be no single turbulent velocity scale that characterizes the enhanced settling speed. The results explain the dependence of the particle settling speeds on $R_{\unicode[STIX]{x1D706}}$, and show how the saturation of this dependence at sufficiently large $R_{\unicode[STIX]{x1D706}}$ depends upon $St$. The results also show that as the Stokes settling velocity of the particles is increased, the flow scales of the turbulence responsible for enhancing the particle settling speed become larger. Finally, we explored the multiscale nature of the preferential sweeping mechanism by considering how particles preferentially sample the fluid velocity gradients coarse grained at various scales. The results show that while rapidly settling particles do not preferentially sample the fluid velocity gradients, they do preferentially sample the fluid velocity gradients coarse grained at scales outside of the dissipation range. This explains the findings of Ireland et al., and further illustrates the truly multiscale nature of the mechanism generating enhanced particle settling speeds in turbulence.