Gravity-driven free surface flow of granular avalanches over complex basal topography

A two-dimensional depth-integrated theory is derived for the gravity-driven free surface flow of cohesionless granular avalanches over complex shallow basal topography. This is an important extension of the one-dimensional Savage-Mutter theory. A simple curvilinear coordinate system is adopted, which is fitted to the 'mean' downslope chute topography. This defines a quasi-two-dimensional reference surface on top of which shallow three-dimensional basal topography is superposed. The governing equations are expressed in the curvilinear coordinate system and the mass- and momentum-balance equations are integrated through the avalanche depth. An ordering argument and a Mohr-Coulomb closure model are used to obtain a simple reduced system of equations. Laboratory experiments have been performed on a partly confined chute to validate the theory. An avalanche is released on a section inclined at 400 to the horizontal, on which there is a concave parabolic cross-slope profile, and runs out through a smooth transition zone onto a horizontal plane. A comparison of the experiment with numerical solutions shows that the avalanche tail speed is under-predicted, A modification to the bed-friction angle is proposed, which brings theory and experiment into very good agreement. The partly confined chute channels the flow and results in significantly longer maximum run-out distances than on an unconfined chute. A simple shallow-water avalanche model is also derived and tested against the experimental results.