Effect of fluid inertia on the orientation of a small prolate spheroid settling in turbulence

Abstract

We study the angular dynamics of small non-spherical particles settling in a turbulent flow, such as ice crystals in clouds, aggregates of organicmaterial in the oceans, or fibres settling in turbulent pipe flow. Most solid particles encountered inNature are not spherical, and their orientations affect their settling speeds, aswell as their collision and aggregation rates in suspensions.Whereas the randomaction of turbulent eddies favours an isotropic distribution of orientations, gravitational settling breaks the rotational symmetry.The precise nature of the symmetry breaking, however, is subtle.We demonstrate here that the fluid-inertia torque plays a dominant role in the problem. As a consequence rod-like particles tend to settle in turbulencewith horizontal orientation, themore so the larger the settling number Sv (a dimensionlessmeasure of the settling speed). For large Sv we determine the fluctuations around this preferential horizontal orientation for prolate particles with arbitrary aspect ratios, assuming small Stokes number St (a dimensionlessmeasure of particle inertia).Our theory is based on a statistical model representing the turbulent velocity fluctuations byGaussian randomfunctions. This overdamped theory predicts that the orientation distribution is very narrowat large Sv,with a variance proportional to Sv-4. By considering the role of particle inertia,we analyse the limitations of the overdamped theory, and determine its range of applicability.Our predictions are in excellent agreementwith numerical simulations of simplifiedmodels of turbulent flows. Finallywe contrast our resultswith those of an alternative theory predicting that the orientation variance is proportional to Sv-2 at large Sv.