Experiments on, and discrete and continuum simulations of, the discharge of granular media from silos with a lateral orifice

Abstract

We compare laboratory experiments, contact dynamics simulations and continuum Navier–Stokes simulations with a $\unicode[STIX]{x1D707}(I)$ visco-plastic rheology, of the discharge of granular media from a silo with a lateral orifice. We consider a rectangular silo with an orifice of height $D$ which spans the silo width $W$, and we observe two regimes. For small enough aperture aspect ratio ${\mathcal{A}}=D/W$, the Hagen–Beverloo relation is obtained. For thin enough silos, ${\mathcal{A}}\gg {\mathcal{A}}_{c}$, we observe a second regime where the outlet velocity varies with $\sqrt{W}$. This new regime is also obtained in the continuum simulations when the friction on side walls is taken into account in a thickness-averaged version of $\unicode[STIX]{x1D707}(I)$ $+$ Navier–Stokes (in the spirit of Hele-Shaw flows). Moreover most of the internal details of the flow field observed experimentally are reproduced when considering this lateral friction. These two regimes are recovered experimentally for a cylindrical silo with a lateral rectangular orifice of height $D$ and arc length $W$. The dependency of the flow rate on the particle diameter is found to be reasonably described experimentally using two geometrical functions that depend respectively on the number of beads through the two aperture dimensions. This is consistent with two-dimensional discrete simulation results: at the outlet, the volume fraction and the velocities depend on the particle diameter and this behaviour is correctly described by those geometrical functions. A similar dependency is observed in the two-dimensional continuum simulations.