End-wall effects in granular tumblers: From quasi-two-dimensional flow to three-dimensional flow

The flow of granular material in rotating tumblers is confined to a thin flowing layer at the free surface in which the particle velocity is primarily streamwise, perpendicular to the axis of rotation, with minimal axial motion. Particle tracking velocimetry was used to measure the surface velocity for 1 and 2 mm glass particles and sand in cylindrical tumblers of various diameters, lengths, and rotation rates for a flat continuously flowing surface. End-wall friction slows the streamwise surface velocity adjacent to the walls, yet material just inward from the end wall flows faster than the material in the center of a long tumbler. An axial velocity occurs near the end walls both upstream and downstream of the midlength of the flowing layer. Increasing the tumbler diameter and corresponding flowing layer length causes the magnitude of the axial and streamwise velocity components, as well as the axial thickness of the end-wall region, to increase. An increase of end-wall friction slows particles directly adjacent to the end wall, further enhancing the axial flow near the end wall. Increasing the rotation rate, while still maintaining a flat continuously flowing surface, causes a corresponding increase in both the streamwise and axial velocities in the tumbler. The boundary flow effects are localized to end walls when the axial length of the tumbler (L) is greater than the diameter (D), so that the flow in the center region of the tumbler is independent of the end walls and hence similar to the conditions at the center of an infinitely long tumbler. Decreasing the axial length of the tumbler so that L/D<1 causes the axial velocity to decrease as the regions affected by the end walls merge. This also leads to a streamwise velocity in quasi-two-dimensional tumblers that can be more than twice that at the center of three-dimensional tumblers.