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–Hutter 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 40 degrees 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 channel 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.

Granular materials composed of particles with differing grain sizes, densities, shapes, or surface properties may experience unexpected segregation during flow. This review focuses on kinetic sieving and squeeze expulsion, whose combined effect produces the dominant gravity-driven segregation mechanism in dense sheared flows. Shallow granular avalanches that form at the surface of more complex industrial flows such as heaps, silos, and rotating drums provide ideal conditions for particles to separate, with large particles rising to the surface and small particles percolating down to the base. When this is combined with erosion and deposition, amazing patterns can form in the underlying substrate. Gravity-driven segregation and velocity shear induce differential lateral transport, which may be thought of as a secondary segregation mechanism. This allows larger particles to accumulate at flow fronts, and if they are more frictional than the fine grains, they can feedback on the bulk flow, causing flow fingering, levee formation, and longer runout of geophysical mass flows.

Data from large‐scale debris‐flow experiments are combined with modeling of particle‐size segregation to explain the formation of lateral levees enriched in coarse grains. The experimental flows consisted of 10 m³ of water‐saturated sand and gravel, which traveled ∼80 m down a steeply inclined flume before forming an elongated leveed deposit 10 m long on a nearly horizontal runout surface. We measured the surface velocity field and observed the sequence of deposition by seeding tracers onto the flow surface and tracking them in video footage. Levees formed by progressive downslope accretion approximately 3.5 m behind the flow front, which advanced steadily at ∼2 m s⁻¹ during most of the runout. Segregation was measured by placing ∼600 coarse tracer pebbles on the bed, which, when entrained into the flow, segregated upwards at ∼6–7.5 cm s⁻¹. When excavated from the deposit these were distributed in a horseshoe‐shaped pattern that became increasingly elevated closer to the deposit termination. Although there was clear evidence for inverse grading during the flow, transect sampling revealed that the resulting leveed deposit was strongly graded laterally, with only weak vertical grading. We construct an empirical, three‐dimensional velocity field resembling the experimental observations, and use this with a particle‐size segregation model to predict the segregation and transport of material through the flow. We infer that coarse material segregates to the flow surface and is transported to the flow front by shear. Within the flow head, coarse material is overridden, then recirculates in spiral trajectories due to size‐segregation, before being advected to the flow edges and deposited to form coarse‐particle‐enriched levees.

Shock waves, dead zones and particle-free regions form when a thin surface avalanche of granular material flows around an obstacle or over a change in the bed topography. Understanding and modelling these flows is of considerable practical interest for industrial processes, as well as for the design of defences to protect buildings, structures and people from snow avalanches, debris flows and rockfalls. These flow phenomena also yield useful constitutive information that can be used to improve existing avalanche models. In this paper a simple hydraulic theory, first suggested in the Russian literature, is generalized to model quasi-two-dimensional flows around obstacles. Exact and numerical solutions are then compared with laboratory experiments. These indicate that the theory is adequate to quantitatively describe the formation of normal shocks, oblique shocks, dead zones and granular vacua. Such features are generated by the flow around a pyramidal obstacle, which is typical of some of the defensive structures in use today.

Granular materials composed of a mixture of grain sizes are notoriously prone to segregation during shaking or transport. In this paper, a binary mixture theory is used to formulate a model for kinetic sieving of large and small particles in thin, rapidly flowing avalanches, which occur in many industrial and geophysical free-surface flows. The model is based on a simple percolation idea, in which the small particles preferentially fall into underlying void space and lever large particles upwards. Exact steady-state solutions have been constructed for general steady uniform velocity fields, as well as time-dependent solutions for plug-flow, that exploit the decoupling of material columns in the avalanche. All the solutions indicate the development of concentration shocks, which are frequently observed in experiments. A shock-capturing numerical algorithm is formulated to solve general problems and is used to investigate segregation in flows with weak shear.

The $\mu (I)$ -rheology is a nonlinear viscous law, with a strain-rate invariant and pressure-dependent viscosity, that has proved to be effective at modelling dry granular flows in the intermediate range of the inertial number, $I$ . This paper shows how to incorporate the rheology into depth-averaged granular avalanche models. To leading order, the rheology generates an effective basal friction, which is equivalent to a rough bed friction law. A depth-averaged viscous-like term can be derived by integrating the in-plane deviatoric stress through the avalanche depth, using pressure and velocity profiles from a steady-uniform solution to the full $\mu (I)$ -rheology. The resulting viscosity is proportional to the thickness to the three halves power, with a coefficient of proportionality that is angle dependent. When substituted into the depth-averaged momentum balance this term generates second-order derivatives of the depth-averaged velocity, which are multiplied by a small parameter. Its inclusion therefore represents a singular perturbation to the equations. It is shown that a granular front propagating down a rough inclined plane is completely unaffected by the rheology, but, discontinuities, which naturally develop in inviscid roll-wave solutions, are smoothed out. By comparison with existing experimental data, it is shown that the depth-averaged $\mu (I)$ -rheology accurately predicts the growth rate of spatial instabilities to steady-uniform flow, as well as the dependence of the cutoff frequency on the Froude number and inclination angle. This provides strong evidence that, in the steady-uniform flow regime, the predicted decrease in the viscosity with increasing slope is correct. Outside the range of angles where steady-uniform flows develop, the viscosity becomes negative, which implies that the equations are ill-posed. This is a signature of the ill-posedness of the full $\mu (I)$ -rheology at both high and low inertial numbers. The depth-averaged $\mu (I)$ -rheology therefore cannot be used outside the valid range of angles without additional regularization.