Internal Waves Control Bulk Flow in Silos

Abstract

We experimentally measure paticle acceleration within the bulk during the discharge of a granular silo. We highlight the existence of a deceleration wave emerging at the outlet level near the dead zone and propagates toward the top of the medium. The wave emission frequency is extracted from spatiotemporal diagrams of the Eulerian instantaneous acceleration profiles. Surprisingly, we find that this frequency decreases with the cohesion of the medium and is independent of the outlet size.

Figures (6)

• Figure 1: (a) Flow near the silo outlet with the corresponding velocity profile (shown in red). Black dashed lines indicate the boundaries of the dead zones, defined by $v/\sqrt{gd} < 0.05$. (b) Schematic of the experimental setup.
• Figure 2: (a) Flow rates with Beverloo law for $C=0.65$, $k=1.5$ and $\phi_b=0.58$ and (b) normalized density $\phi_c(x)/\phi_b$ for all values of aperture size $D/d$ and Bond number $Bo$ ($\alpha=0.66$, $\beta=0.11$ and $\phi_b=0.58$). Normalized profiles of (c) velocity and (d) density at the outlet level normalized by their fitted center value $v_{c}$ and $\phi_c$.
• Figure 3: Spatiotemporal diagrams and corresponding power spectrum of the instantaneous acceleration $a_y(x=0)$ for $Bo\,=\,0$ (a,b) and $Bo\,=\,51.1$ (c,d). $D\,=\,24$ mm in both cases.
• Figure 4: (a) Example of a spatiotemporal diagram with (b-f) corresponding instantaneous acceleration profiles $a_y(x,y,t)/g$ for $D\,=\,24$ mm and $Bo\,=\,0$.
• Figure 5: (a) Profiles of the normalized velocity fluctuations, $v'(x,y)/\sqrt{gd}$, for $Bo = 0$ and different aperture ratios $D/d$. (b) Dependence of the transient arch height, $h_{\cap}/d$, on the silo opening. (c) Dependence of the iso-acceleration threshold of transient arches, $a_{\cap}/g$, on the silo opening. The star symbol indicates the limiting case for which the iso-acceleration equals $g$. Solid lines in panels (b) and (c) are guides to the eye.