# A two-dimensional depth-averaged μ(I)-rheology for dense granular avalanches

### What is it about?

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 μ(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 μ(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.

### Why is it important?

This was the first introduction of a two-dimensional generalisation of the equations for the depth-averaged μ(I)-rheology. The paper also provided the first test case for the model - that of flow on a frictional inclined plane confined between parallel lateral sidewalls. Accurate modelling of these chute flows is important in many industrial and geophysical contexts and it's simplicity allows certain analytic treatments to be made. One important finding was the scaling of the shear bands, that are localised close to the walls of wide chutes. The model also predicts the mass flow from a chute given a certain inclination angle and depth of flowing material. This suggests that it could be used in debris flow hazard mitigation modelling and to control granular mass injection for industrial fabrication. The success of the formulation was first confirmed through comparison with numerical solutions of the full steady 2D μ(I) equations. Small-scale laboratory experiments were also performed using spherical glass beads, a rough bed and adjustable perspex sidewalls. For shallow flows the results from all three tools (experimental, full numerical and depth-averaged) agreed very well. For thicker flows, with taller aspect ratios, the 2D depth average model deviated from the other data. This is to be expected due to the necessary assumption of shallowness made in the depth-averaging. As well as being much less computationally expensive than the full equations, this model is also well-posed for the full range of slope angles in which steady uniform flows are observed. These strengths and validations inspire confidence that the model is well placed to handle transient flow problems that may include complex terrain and boundaries as well as capturing the necessary physics to explain wave-like phenomena and other physical instabilities of granular flow.

### Perspectives

http://dx.doi.org/10.1017/jfm.2015.684

The following have contributed to this page: Professor John M N T Gray, Thomas Barker, and Dr James Lindsay Baker

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