Monodisperse and bidisperse granular flows are studied in rotating tumblers using DEM. In spherical tumblers, flowing particles’ trajectories do not follow straight lines but are curved. At the same time particles near the surface drift toward the pole, inducing two global recirculation cells. Combined with radial segregation, drift and curvature compete to impose the axial segregation pattern: Small-Large-Small (SLS) or Large-Small-Large (LSL). Fill level, rotation speed and wall roughness influence drift and curvature, and modify the resulting segregation pattern. In cylindrical tumblers, equivalent recirculation cells occur next to the end walls. A second pair of recirculation cells with a weak drift in the opposite direction appears at the center for long enough tumblers. Unlike the sphere case, curvature and drift in the primary cells combine to push large particles toward the end walls, explaining why large particle bands appear at the end walls for axial segregation in cylinder.