Surface curvature and secondary vortices in steady dense shallow granular flows

Dense granular flows exhibit both surface deformation and secondary flows due to the presence of normal stress differences. Yet, a complete mathematical modelling of these two features is still lacking. This paper focuses on a steady shallow dense flow down an inclined channel of arbitrary cross-section, for which asymptotic solutions are derived by using an expansion based on the flow’s spanwise shallowness combined with a second-order granular rheology. The leading-order flow is uniaxial with a constant inertial number fixed by the inclination angle. The streamwise velocity then corresponds to a lateral juxtaposition of Bagnold profiles scaled by the varying flow depth. The correction at first order introduces two counter-rotating vortices in the plane perpendicular to the main flow direction (with downwelling in the centre), and an upward curve of the free surface. These solutions are compared with discrete element method simulations, which they match quantitatively. This result is then used together with laboratory experiments to infer measurements of the second-normal stress difference in dense dry granular flow.