Two-way coupling Eulerian numerical simulations of particle clouds settling in a quiescent fluid

To get a deeper understanding of our laboratory experiments [Q. Kriaa et al., Phys. Rev. Fluids 7, 124302 (2022)], we numerically model settling clouds produced by localized instantaneous releases of heavy particles in a quiescent fluid. By modeling particles as a field of mass concentration in an equilibrium Eulerian approach, our two-way coupling simulations recover our original experimental observation of a maximum growth rate for clouds laden with particles of finite Rouse number R ti 0.22, where R is the ratio of the individual particle settling velocity to the typical cloud velocity based on its initial radius and total buoyancy. Consistent with the literature on buoyant vortex rings, our clouds verify the relation & alpha; oc Poo-2 between the clouds' growth rate & alpha;, as firstly defined by Morton et al. [Proc. R. Soc. London A 234, 1 (1956)], and their eventually constant circulation Poo. As the Rouse number approaches R ti 0.22, the baroclinic forcing of the circulation reduces down to a minimum, thus optimizing the cloud growth rate & alpha;. This analysis highlights the role of the mean flow in the enhanced entrainment of ambient fluid by negatively buoyant clouds. Our results also validate, on the basis of direct comparison with experimental results, the use of a one-fluid two-way coupling numerical model to simulate particle clouds in the limit of weak particle inertia.