A two-dimensional depth-averaged -rheology for dense granular avalanches

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 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.

Segregation-induced finger formation in granular free-surface flows

“…The system is then generalised to two spatial dimensions in an analogous way to Baker et al (2016) (for monodisperse flows) and numerical simulations of a propagating flow front are presented. The equations are able to predict the formation of a coarse-rich flow front that develops instabilities and breaks up into a series of finger-like structures, elongating over time.…”

“…The governing equations must therefore be extended to two spatial dimensions. Baker et al (2016) examined a similar problem for monodisperse material and generalised the one-dimensional depth-averaged µ(I)-rheology of to two dimensions. Using their work as a base and introducing a cross-slope coordinate y and depth-averaged velocityv, the two-dimensional segregation-mobility feedback equations (reverting to dimensional variables) are…”

“…Baker, Barker & Gray (2016) recently proposed a two-dimensional extension of the equations to account for lateral variation, and applied the model to steady uniform channel flows. The generalised viscous terms give rise to downslope velocities with cross-slope profiles, another physical feature not captured by classical shallow-water models.…”

“…This paper therefore describes how to generalise their work into a bidisperse set-up, and shows that the resulting model is mathematically well posed. A two-dimensional (downslope and lateral) extension of the system of equations, based on the work of Baker et al (2016), admits numerical solutions showing the formation of fingering instabilities on an inclined plane, with the key finger characteristics being independent of the grid resolution and controlled by the newly introduced physical viscosity.…”

Formation of levees, troughs and elevated channels by avalanches on erodible slopes

“…This contrasts with inviscid avalanche models that incorrectly predict the growth of granular roll waves at all frequencies above the critical Froude number, Fr c > 2/3. Baker, Barker & Gray (2016a) have generalized the depth-averaged µ(I)-rheology to two dimensions by using the principle of frame invariance. Their extended model is able to capture the depth-averaged downslope velocity profile that develops across a fully developed chute flow with either wall slip or zero velocity at the rough side walls.…”