Surface curvature and secondary vortices in steady dense shallow granular flows

TL;DR

This paper develops an asymptotic mathematical model for steady dense shallow granular flows, capturing surface curvature and secondary vortices, validated by simulations and experiments to understand stress differences.

Contribution

It introduces a novel asymptotic solution for shallow granular flows that includes secondary vortices and surface curvature, advancing the modeling of normal stress effects.

Findings

Model accurately predicts secondary vortices and surface shape.

Quantitative agreement with DEM simulations.

Experimental data used to measure normal stress differences.

Abstract

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 shallowness combined with a second-order granular rheology. The leading order flow is uniaxial, with a streamwise velocity corresponding 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 to DEM 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.