Motion of a granular avalanche in an exponentially curved chute: experiments and theoretical predictions

Abstract

This paper describes a model to predict the flow of an initially stationary mass of cohesionless granular material down a rough curved bed, and checks it against laboratory experiments that were conducted with several cohesionless granular materials that are released from rest and travel in an exponentially curved chute. We use the depth-averaged field equations of balance of mass and linear momentum as presented by Savage & Hutter (1990). These equations are evolution equations for the transversely averaged stream wise velocity and the distribution of avalanche depths and involve two phenomenological parameters, the internal angle of friction, ϕ, and a bed friction angle, δ, both as constitutive properties of Coulomb-type behaviour. We present the model but do not derive its equations, which are presented in two variants that incorporate weak and strong curvature effects. For granular avalanches which start as parabolic piles, the governing equations (incorporating weak curvature effects) permit similarity solutions. These solutions preserve the parabolic shape and have simple velocity distributions. We present the equations again without detailed explanations. Experiments were performed with seven different granular materials (two classes of glass beads, Vestolen plastic particles, two samples of quartz granules and two types of crunched marmor particles). Piles of finite masses of such granular materials with various initial geometries were released from rest in a 100 mm wide chute having an exponentially curved bed that was lined with Makrolon (a plexiglass), drawing-paper and sandpaper. The granular masses under motion were photographed and video filmed and thus the geometry of the avalanche was recorded as a function of position and time. With all materials and for all the bed linings, the angle of repose and the bed friction angle were determined. The former was identified with the static internal angle of friction. Using a second measuring technique, the effects of the chute walls on the bed friction angle was experimentally determined and incorporated in an effective bed friction angle which thus showed a linear dependence on the pile depth. Coefficients of restitution were also estimated for the particles on the different bed linings. The numerical integration scheme for the general model that was proposed earlier by Savage & Hutter (1989) is a lagrangian finite difference scheme which incorporates numerical diffusion. We present this scheme and analyse its reliability when the numerical diffusion is varied. We also discuss the integration procedure for the similarity solutions. Comparison of the theoretical results with experiments pertain to the similarity model (SM) and the general equation model (GM). Crucial in such comparisons is the identification of initial condition which is not unique from the observational data. For SM it is shown that no initial condition can be found, in general, that would yield computational predictions of the evolution of the position of the leading and trailing edges of the granular avalanche in sufficient agreement with observations. When depth-to-length ratios of the initial pile geometry and the curvature of the bed are sufficiently small, however, then the SM solutions may be used for diagnostic purposes. We finally compare experimental results with computational findings of the GM equations for many combinations of masses of the granular materials and bed linings. It is found that experimental results and theoretical predictions agree satisfactorily if the internal angle of friction, ϕ, exceeds the total bed friction angle, δ, or is not close to it. Limited variations of the bed friction angle along the bed do not seem to have a sizeable effect on the computational results, but it is important that dynamic values rather than static values for ϕ and δ are used in the computations. When δ is very close to ϕ and δ < ϕ , the computational travel time of the granular avalanche exceeds the travel time of experiments considerably. Furthermore, when avalanche masses are reasonably small and coefficients of restitution of the granules on the bed relatively high, again the predictions of the theory overestimate travel times and underestimate avalanche lengths. Thus the theory does seem to be reasonable when the bed friction angle is definitely smaller than the internal angle of friction.