Large eddy simulations of free-falling perforated disks with small inertias

Abstract

The dynamics of free-falling perforated disks of porosity $\chi =0.2$ are numerically investigated by the large eddy simulation (LES) within the range $100\le Ar\le 1000$ and $7\times 10^{-4}\le I^{\ast }\le 2.7\times 10^{-3}$. Three falling styles are identified, namely spiral motion, spiral irregular motion and Hula-Hoop motion. A linear relationship of the Archimedes number $Ar$ and the Reynolds number $Re$ is observed within the intermediate Reynolds number regime. The mean values of crucial kinematic and dynamic variables are also given, and some scaling laws related to the perforated disk thickness $h$ and diameter $D$ are determined. For the mean descent velocity $\left(U_z\right)$, the gravitational velocity $U_g$ is a suitable characteristic velocity scale; this is not the case for the terminal velocity $U_t$, which is proportional to $h^{\frac{1}{5}}{D}^{\frac{3}{10}}$. The characteristic timescale $t_v$ for vortex shedding is proportional to $D^{\frac{4}{5}}/h^{\frac{3}{10}}$, which indicates thin perforated disks facilitate vortex shedding. The mean normal force $\left(F_N\right)$ is proportional to $h^{\frac{8}{5}}{D}^{\frac{7}{5}}$, and irrespective of falling styles. It indicates different falling styles are related to the velocity fluctuations $U\left(t\right)-\left(U_z\right)e_z$. Vortex structures of three falling styles are provided. For spiral motion and spiral irregular motion, a vertical vortex appears inside the large-scale helical vortex. For Hula-Hoop motion, small-scale vortexes are omnipresent, and a new independent vortex is identified. The results of this paper is limited, and much work remains to be done, including the effects of the solid-to-fluid density ratio $\rho$ and holes topology.