Avalanche Dynamics

Abstract

This article reviews recent work on avalanche, landslide and rockfall dynamics. Two limiting cases of these flows exist, the so-called flow avalanche, i. e., the dense gravity driven “laminar type flow” in which the role of the solid particles dominates, while that of the interstitial fluid is negligible — these flows are typical for most sturzstroms, debris flows, landslides, rockfalls and snow avalanches — and the less dense powder avalanche, i. e., the turbulent flow of air borne particles in a mixture, in which the role of the fluid dominates, while that of the particles is less significant — these flows are typical for density and turbidity currents such as dust clouds occuring in the desert, in pyroclastic volcanic eruptions, in submarine slope instabilities and in snow and ice avalanches. The latter application will be our focus. The state of the art in the description of both these phenomena is given. In the Introduction, after some historical remarks, we turn our attention to the characterization of the physical behaviour of the two limiting flow types and then discuss the laws of similitude and model theory relevant to modelling the flows in the laboratory.

Flow avalanches, landslides and rockfalls are discussed first. It is argued that these flows can be described as a continuum consisting of a cohesionless granular material. Such materials exhibit dilatancy effects and large energy dissipation, and under quasi static or dynamic shear deformation, they exhibit the property that the ratio of the shear stress to the normal stress on any interior plane is nearly constant, a fact reflected in the constancy of the internal angle of friction. In shear cell tests under quasi static and rapid deformation at constant normal load, the internal stress is practically independent of the rate of shear; when the shear deformation is performed at constant volume however, the dependence of the stress on the rate of shear is quadratic. Three different flow regimes which partly interact can be distinguished: (i), Dry Coulomb, rubbing frictional behaviour, typical when particles are in contact and ride one over the other; (ii), collisional interactions when particles bounce against each other, contact is short, and the mean free path is of the order of the particle diameter; (iii), translational transport when the mean free path is large and particle concentrations correspondingly small. For the second and the third regime, statistical theories along the lines of the kinetic theory of a dense gas have been developed, but some of these show ill behaviour in steady chute flow problems. An adequate formulation must incorporate quasistatic and collisional contributions; the emerging theories, however, are patched together from alien components. Furthermore, simple chute flow problems turn out to be very difficult to solve. In the flow regime where most of the moving granular mass rides more or less passively on a fluidized bed, Savage and Hutter (1989) have proposed to treat the material as being of the rate independent Coulomb type, both in the interior and at the bed, with constant internal and bed friction angle ø and δ, respectively. In this formulation, the bed friction angle δ is a measure of the resistance of the bed to a fluidized thin layer, a first approximation to a more general bed friction law that may also include viscous components. The depth averaged field equations of Savage and Hutter reproduce laboratory chute flow of a cohesionless granular material quite well even in cases when a localized bump in an otherwise convexly curved bed separates an initially single mass into two separate piles. The equations also permit similarity solutions, i.e., solutions of permanent shape (but not size). These solutions are not quantitatively corroborated in general by laboratory experiments, but they permit better analytic insight and thus help in understanding the model equations physically. Extensions of both theory and experiments to avalanche flows that spread in two dimensions indicate promising results.

Powder avalanches have been treated in the literature by essentially two different concepts: (i), a simple binary mixture of turbulent air and suspended particles with two balance laws for each constituent (or the mixture as a whole and one constituent: the particles) and a momentum balance for the mixture as a whole and (ii), a two-phase mixture with balance laws of mass and linear momentum for each constituent.

The first class of models concentrates on integral, or global representations. Long turbulent gravity currents from a steady continuous source on inclined planes and short, finite mass “clouds” or “thermals” are studied from the viewpoint of depth integrated mass, buoyancy and momentum balance laws. Typical mean velocity and averaged density variations are determined for idealized conditions, and corresponding results compared or matched with data obtained from laboratory experiments. Air entrainment through the upper free surface of the turbulent gravity current is accounted for, but entrainment of the particles from the bed or deposition of particles at the bed are often ignored when asymptotic behaviour for large time or large distances from the source are considered. Such earlier and simpler models of powder avalanches do not quantify the level of turbulence and cannot, therefore, properly describe the physical conditions of autosuspension or re-sedimentation of the particles. We present, however, a brief summary of formulations for long and short gravity currents in which the balance laws of mass, buoyancy and momentum are complemented by a balance of the averaged turbulent fluctuation energy that is coupled with these by closure statements on basal friction and snow entrainment Such models can be computationally handled, but have not been verified in detail by laboratory experiments, even though an ad hoc application of the model equations to a real avalanche event indicates promising results. Proposals of theoretical formulations for simple mixtures of a turbulent fluid carrying suspended particles have also been put forward on the basis of κ-ε closure and using local equations; very little experience, however, seems to have been done with this model.

Two phase turbulent mixtures that use the balance laws of mass and momentum for both constituents have also been proposed; they also use second order (κ-ε) closure conditions. The novel theoretical feature in these models is, however, not so much the set of field equations than the handling of the free boundary as a singular surface, a concept that is briefly touched upon here. Finally we also present experimental results on the distribution of the streamwise particle velocity and density, both in the tail and the head of a laboratory powder avalanche of polystyrene particles suspended in water and moving down a straight and curved chute.