ALBERT

Notes: [Auszug] The experimental set-up is shown in Fig. 1. A 2-m-long and 70-cm-wide glass surface is roughened by gluing onto its surface a single layer of 0.5-mm-diameter glass beads. The rough bed is transparent and allows the system to be viewed from below. A double gate system (Fig. la) can be opened ...

Notes: Abstract This paper continues a series of studies on the plane flow of a pile of cohesionless granular material down a rough inclined plane. Internal and basal friction laws are assumed to obey the Mohr-Coulomb yield criterion but in contrast to previous investigations the angle of friction at the bottom of the pile is considered to depend on the position or on the velocity or on both. Similarity, i.e. shape preserving solutions are constructed. The depth of the pile and the profile of the total minus the centre of mass velocity are determined analytically, but the total length and the position of the centre of mass are calculated numerically. If the basal friction angle is constant, the centre of mass moves with constant acceleration and the length of the pile extends monotonically. These motions change, when the angle of friction varies along the pile — the length of the pile may extend, contract or remain stationary and the centre of mass motion may decelerate or even reach steady state. Eight special cases are exhibited which demonstrate the influence of the friction law on the speed and spread of the pile.

Notes: Summary Following the granular flow kinetic theory of Lun, Savage, Jeffrey and Chepurniy, a moment method is used to obtain the approximate form for the single particle velocity distribution function for the case of smooth, slightly inelastic, uniform spherical particles in which the coefficient of restitutione depends upon the particle impact velocity. Constitutive equations for stress are derived and the theory is applied to the case of a simple shear flow. Theoretical predictions of stresses are compared with experimental results. The effect of the impact velocity dependente is to cause the stresses to vary with the shear rate raised to a power less than two; this is consistent with the experimental observations. On the basis of the present theory and comparisons with experimental data it is concluded that theoretical models which include both surface friction and an impact velocity dependente will lead to improved agreement between the theoretical predictions and the measurements.

Notes: Summary This paper describes a model to predict the flow of an initially stationary mass of cohesionsless granular material down a rough curved bed and checks it against laboratory experiments that were conducted with two different kinds of granular materials that are released from rest and travel in a chute consisting of a straight inclined section, a curved segment that is followed by a straight horizontal segment. This work is of interest in connection with the motion of landslides, rockfalls and ice and dense flow snow avalanches. Experiments were performed with two different granular materials, nearly spherical glass beads of 3 mm nominal diameter, Vestolen particles (a light plastic material) of lense type shape and 4 mm nominal diameter and 2,5 mm height. Piles of finite masses of these granular materials with various initial shapes and weight were released from rest in a 100 mm wide chute with the mentioned bent profile. The basal surface consisted of smooth PVC, but was in other experiments also coated with drawing paper and with sandpaper. The granular masses under motion were photographed and partly video filmed and thus the geometry of the avalanche was recorded as a function of position and time. For the two granular materials and for the three bed linings the angle of repose and the bed friction angle were determined. The experimental technique with which the laboratory avalanches were run are described in detail as is the reliability of the generated data. We present and use the depth-averaged field equations of balance of mass and linear momentum as presented by Savage and Hutter [28]. These are partial differential equations for the depth averaged streamwise velocity and the distribution of the avalanche depth and involve two phenomenological parameters, the internal angle of friction, ø, and a bed friction angle, δ, both as constitutive properties of Coulomb-type behaviour. We present the model but do not derive its equations. The numerical integration scheme for these equations is a Lagrangian finite difference scheme used earlier by Savage and Hutter [27],[28]. We present this scheme for completeness but do not discuss its peculiarities. Comparison of the theoretical results with experiments is commenced by discussing the implementation of the initial conditions. Observations indicate that with the onset of the motion a dilatation is involved that should be accomodated for in the definition of the initial conditions. Early studies of the temporal evolution of the trailing and leading edges of the granular avalanche indicate that their computed counterparts react sensitively to variations in the bed friction angle but not to those of the internal angle of friction. Furthermore, a weak velocity dependence of the bed friction angle, δ, is also scen to have a small, but negligible influence on these variables. We finally compare the experimental results with computational findings for many combinations of the masses of the granular materials and bed linings. It is found that the experimental results and the theoretical predictions agree satisfactorily. They thus validate the simple model equations that were proposed in Savage and Hutter [28].

Notes: Summary Following the approach of the kinetic theory for mixtures of dense gases, the general conservation equations for the rapid flow of a binary mixture of smooth, inelastic, spherical granular particles are derived. Explicit constitutive relations for stress and rate of energy dissipation are obtained by making simple approximations for the particle velocity distribution functions. These approximations are appropriate for cases where collisional interactions are the dominant mechanism for momentum and energy exchange in the system. The theory is applied to the case of simple shear flow. In general, the theory predicts that stresses decrease with increasing concentration of the small particles and decreasing diameter ratio of small to large particles. Theoretical predictions of stresses are compared with experimental results and reasonable agreement is found.

Notes: Summary This paper describes a model to predict the flow of an initially stationary mass of cohesion-less granular material down rough curved beds. This work is of interest in connection with the motion of rock and ice avalanches and dense flow snow avalanches. The constitutive behaviour of the material making up the pile is assumed to be described by a Mohr-Coulomb criterion while the bed boundary condition is treated by a similar Coulomb-type basal friction law assumption. By depth averaging the incompressible conservation of mass and linear momentum equations that are written in terms of a curvilinear coordinate system aligned with the curved bed, we obtain evolution equations for the depthh and the depth averaged velocityū. Three characteristic length scales are defined for use in the non-dimensionalization and scaling of the governing equations. These are a characteristic avalanche lengthL, a characteristic heightH, and a characteristic bed radius of curvatureR. Three independent parameters emerge in the non-dimensionalized equations of motion. One, which is the aspect ratio ε-H/L, is taken to be small. By choosing different orderings for the other two, the tangent of the bed friction angle δ and the characteristic non-dimensional curvature λ=L/R, we can obtain different sets of equations of motion which appropriately display the desired importance of bed friction and bed curvature effects. The equations, correct to order ε for moderate curvature, are discretized in the form of a Lagrangian-type finite difference representation which proved to be successful in the earlier studies of Savage and Hutter [24] for granular flow down rough plane surfaces. Laboratory experiments were performed with plastic particles flowing down a chute having a bed made up of a straight, inclined portion, a curved part and a horizontal portion. Numerical solutions are presented for conditions corresponding to the laboratory experiments. It is found that the predicted temporal-evolutions of the rear and front of the pile of granular material as well as the shape of the pile agree quite well with the laboratory experiments.

Notes: Summary This paper is concerned with the motion of an unconfined finite mass of a granular material released from rest on an inclined plane. The granular mass is treated as a frictional Coulomb-like continuum with a Coulomb-like basal friction law. Depth averaged equations are deduced from the three-dimensional dynamical equations by scaling the equations and imposing the shallowness assumption that the moving piles are long and wide but not deep. Several distinguished limits for small depth to length and depth to width ratios can be analysed. We develop an approximate theory based upon the full dynamical equations parallel to the inclined plane and imposed hydrostatic pressure conditions perpendicular to it. The resulting model equations are then applied to construct either yet simpler model equations or else solutions for particular cases. In a first application the transverse distributions of the velocity fields and of the depth profile are prescribed, while representative values of these functions (such as the cross sectional averages or maxima) as functions of time and the downhill coordinate are left unspecified. For these quantities evolution equations are obtained from a lateral averaging of the vertically averaged equations. In a second application approximate similarity solutions of the spatially two-dimensional equations are derived. The depth and velocity profiles for the moving mass are determined in analytical form, and the evolution equation for the total length and the total width of the pile is integrated numerically. A parameter study illustrates the performance of the model.

Notes: Summary The present paper investigates similarity solutions for the two-dimensional flow of a mass of cohesionless granular material down rough curved beds having gradually varying slopes. The work is relevant to the motion of rockfalls and loose snow flow avalanches. The depth and velocity profiles for the moving mass are determined in analytical form and the evolution equation for the total length of the pile is integrated numerically using a Runge-Kutta technique. Although similarity solutions can occur for general bed shapes (as long as the curvature is small), specific computations are performed here for two families of bed profiles, one which is in the shape of a circular arc and the other in which the slope decays exponentially with downstream distance. The pile of granular material starts from rest, initially accelerates and then decelerates, finally coming to rest as a result of bed friction and the gradually decreasing bed slope. Depending upon the frictional parameters, the shape of the bed and the initial depth to length of the pile, it is found that the variation of total length with time can exhibit different behaviours. The pile can grow monotonically, it can asymptote to a constant length, it can grow to a maximum and then decrease or it can decrease to a minimum and then increase with time. Furthermore, there are regions in parameter space for which the pile moves as a rigid body either for the whole time of travel or for portions of it.