Cross-stream oscillations in the granular flow through a vertical channel

Abstract

The gravity flow of a granular material between two vertical walls separated by a width $2W$ is simulated using the discrete element method (DEM). Periodic boundary conditions are applied in the flow (vertical) and the other horizontal directions. The mass flow rate is controlled by specifying the average solids fraction $\bar {\phi }$ , the ratio of the volume of the particles to the volume of the channel. A steady fully developed state can be achieved for a narrow range of $\bar {\phi }$ , $\bar {\phi }_{max} \geq \bar {\phi } \geq \bar {\phi }_{cr}$ , and the material is in free fall for $\bar {\phi } < \bar {\phi }_{min}$ . For an intermediate range of $\bar {\phi }$ ( $\bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$ ), there are oscillations in the horizontal coordinate of the centre of mass, velocity components and stress. As $\bar {\phi }$ decreases in the range $\bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$ , the amplitude of the oscillations increases proportional to $\sqrt {\bar {\phi }_{cr} - \bar {\phi }}$ and the frequency appears to tend to a non-zero value as $\bar {\phi } \rightarrow \bar {\phi }_{cr}$ , indicating a supercritical Hopf bifurcation. The relation between the dominant frequency and the higher harmonics of the position, velocity and stress fluctuations are explained using the momentum balance. It is found that dissipation in the inter-particle and particle–wall contacts is critical for the presence of an oscillatory state.