Abstract
Steady uniform granular chute flows are common in industry and provide an important test case for new theoretical models. This paper introduces depth-integrated viscous terms into the momentum-balance equations by extending the recent depth-averaged ${\it\mu}(I)$-rheology for dense granular flows to two spatial dimensions, using the principle of material frame indifference or objectivity. Scaling the cross-slope coordinate on the width of the channel and the velocity on the one-dimensional steady uniform solution, we show that the steady two-dimensional downslope velocity profile is independent of scale. The only controlling parameters are the channel aspect ratio, the slope inclination angle and the frictional properties of the chute and the sidewalls. Solutions are constructed for both no-slip conditions and for a constant Coulomb friction at the walls. For narrow chutes, a pronounced parabolic-like depth-averaged downstream velocity profile develops. However, for very wide channels, the flow is almost uniform with narrow boundary layers close to the sidewalls. Both of these cases are in direct contrast to conventional inviscid avalanche models, which do not develop a cross-slope profile. Steady-state numerical solutions to the full three-dimensional ${\it\mu}(I)$-rheology are computed using the finite element method. It is shown that these solutions are also independent of scale. For sufficiently shallow channels, the depth-averaged velocity profile computed from the full solution is in excellent agreement with the results of the depth-averaged theory. The full downstream velocity can be reconstructed from the depth-averaged theory by assuming a Bagnold-like velocity profile with depth. For wide chutes, this is very close to the results of the full three-dimensional calculation. For experimental validation, a laser profilometer and balance are used to determine the relationship between the total mass flux in the chute and the flow thickness for a range of slope angles and channel widths, and particle image velocimetry (PIV) is used to record the corresponding surface velocity profiles. The measured values are in good quantitative agreement with reconstructed solutions to the new depth-averaged theory.
Highlights
Steady granular chute flows, where a mass of material flowing down an inclined plane is laterally confined between two rigid vertical plates, are common in industrial set-ups and are interesting to study in their own right
The viscous terms in the depth-averaged equations are closely related to this rheology but only in the steady uniform regime where cross-slope derivatives are once again neglected
This paper presents a two-dimensional depth-averaged μ(I)-rheology for shallow granular free-surface flows
Summary
Steady granular chute flows, where a mass of material flowing down an inclined plane is laterally confined between two rigid vertical plates, are common in industrial set-ups and are interesting to study in their own right. The two-dimensional (downslope and normal) results were used to propose a simple local rheology, where the effective-friction coefficient was a function of non-dimensional inertial number I (da Cruz et al 2005; Jop, Forterre & Pouliquen 2005). This number is the square root of the Savage or Coulomb number (Savage 1984; Ancey, Coussot & Evesque 1999). This number is the square root of the Savage or Coulomb number (Savage 1984; Ancey, Coussot & Evesque 1999). Jop, Forterre & Pouliquen (2006) generalized this scalar rheology into a full tensor constitutive law, referred to as the μ(I)-rheology
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