Stuart B. Savage

Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield

The flow of an idealized granular material consisting of uniform smooth, but nelastic, spherical particles is studied using statistical methods analogous to those used in the kinetic theory of gases. Two theories are developed: one for the Couette flow of particles having arbitrary coefficients of restitution (inelastic particles) and a second for the general flow of particles with coefficients of restitution near 1 (slightly inelastic particles). The study of inelastic particles in Couette flow follows the method of Savage & Jeffrey (1981) and uses an ad hoc distribution function to describe the collisions between particles. The results of this first analysis are compared with other theories of granular flow, with the Chapman-Enskog dense-gas theory, and with experiments. The theory agrees moderately well with experimental data and it is found that the asymptotic analysis of Jenkins & Savage (1983), which was developed for slightly inelastic particles, surprisingly gives results similar to the first theory even for highly inelastic particles. Therefore the ‘nearly elastic’ approximation is pursued as a second theory using an approach that is closer to the established methods of Chapman-Enskog gas theory. The new approach which determines the collisional distribution functions by a rational approximation scheme, is applicable to general flowfields, not just simple shear. It incorporates kinetic as well as collisional contributions to the constitutive equations for stress and energy flux and is thus appropriate for dilute as well as dense concentrations of solids. When the collisional contributions are dominant, it predicts stresses similar to the first analysis for the simple shear case.

The motion of a finite mass of granular material down a rough incline

Rock, snow and ice masses are often dislodged on steep slopes of mountainous regions. The masses, which typically are in the form of innumerable discrete blocks or granules, initially accelerate down the slope until the angle of inclination of the bed approaches the horizontal and bed friction eventually brings them to rest. The present paper describes an initial investigation which considers the idealized problem of a finite mass of material released from rest on a rough inclined plane. The granular mass is treated as a frictional Coulomb-like continuum with a Coulomb-like basal friction law. Depth-averaged equations of motion are derived; they bear a superficial resemblance to the nonlinear shallow-water wave equations. Two similarity solutions are found for the motion. They both are of surprisingly simple analytical form and show a rather unanticipated behaviour. One has the form of a pile of granular material in the shape of a parabolic cap and the other has the form of an M -wave with vertical faces at the leading and trailing edges. The linear stability of the similarity solutions is studied. A restricted stability analysis, in which the spread is left unperturbed shows them to be stable, suggesting that mathematically both are possible asymptotic wave forms. Two numerical finite-difference schemes, one of Lagrangian, the other of Eulerian type, are presented. While the Eulerian technique is able to reproduce the M -wave similarity solution, it appears to give spurious results for more general initial conditions and the Lagrangian technique is best suited for the present problem. The numerical predictions are compared with laboratory experiments of Huber (1980) involving the motion of gravel released from rest on a rough inclined plane. Although in these experiments the continuum approximation breaks down at large times when the gravel layer is only a few particle diameters thick, the general features of the development of the gravel mass are well predicted by the numerical solutions.

Gravity flow of cohesionless granular materials in chutes and channels

A constitutive equation appropriate for flow of cohesionless granular materials at high deformation rates and low stress levels is proposed. It consists of an extension and a reinterpretation of the theory of Goodman & Cowin (1972), and accounts for the non-Newtonian nature of the flow as evidenced by Bagnolds (1954) experiments. The theory is applied to analyses of gravity flows in inclined chutes and vertical channels. Experiments were set up in an attempt to generate two-dimensional shear flows corresponding to these analyses. Velocity profiles measured by a technique which makes use of fibre optic probes agree qualitatively with the theoretical predictions, but direct comparison is inappropriate because of unavoidable side-wall friction effects in the experiments. The existing measure of agreement suggests that the most prominent effects have been included in the proposed constitutive relations. Tests in the inclined chute revealed the possible existence of surge waves and granular jumps analogous to hydraulic jumps.

Particle size segregation in inclined chute flow of dry cohesionless granular solids

If granular materials comprising particles of identical material but different sizes are sheared in the presence of a gravitational field, the particles are segregated according to size. The small particles fall to the bottom and the larger ones drift to the top of the sheared layer. In an attempt to isolate and study some of the essential segregation mechanisms, the paper considers a simplified problem involving the steady two-dimensional flow of a binary mixture of small and large spherical particles flowing down a roughened inclined chute. The flow is assumed to take place in layers that are in motion relative to one another as a result of the mean shear. For relatively slow flows, it is proposed that there are two main mechanisms responsible for the transfer of particles between layers. The first mechanism, termed the ‘random fluctuating sieve’, is a gravity-induced, size-dependent, void-filling mechanism. The probability of capture of a particle in one layer by a randomly generated void space in the underlying layer is calculated as a function of the relative motion of the two layers. The second, termed the ‘squeeze expulsion’ mechanism, is due to imbalances in contact forces on an individual particle which squeeze it out of its own layer into an adjacent one. It is assumed that this mechanism is not size preferential and that there is no inherent preferential direction for the layer transfer. This second physical mechanism in particular was proposed on the basis of observations of video recordings that were played back at slow speed. Since the magnitude of its contribution is determined by the satisfaction of overall mass conservation, the exact physical nature of the mechanism is of less importance. By combining these two proposed mechanisms the net percolation velocity of each species is obtained. The mass conservation equation for fines is solved by the method of characteristics to obtain the development of concentration profiles with downstream distance. Although the theory involves a number of empirical constants, their magnitude can be estimated with a fair degree of accuracy. A solution for the limiting case of dilute concentration of fine particles and a more general solution for arbitrary concentrations are presented. The analyses are compared with experiments which measured the development of concentration profiles during the flow of a binary mixture of coarse and fine particles down a roughened inclined chute. Reasonable agreement is found between the measured and predicted concentration profiles and the distance required for the complete separation of fine from coarse particles.

The stress tensor in a granular flow at high shear rates

The stress tensor in a granular shear flow is calculated by supposing that binary collisions between the particles comprising the granular mass are responsible for most of the momentum transport. We assume that the particles are smooth, hard, elastic spheres and express the stress as an integral containing probability distribution functions for the velocities of the particles and for their spatial arrangement. By assuming that the single-particle velocity distribution function is Maxwellian and that the spatial pair distribution function is given by a formula due to Carnahan & Starling, we reduce this integral to one depending upon a single non-dimensional parameter R : the ratio of the characteristic mean shear velocity to the root mean square of the precollisional particle-velocity perturbation. The integral is evaluated asymptotically for R [Gt ] 1 and R [Lt ] 1 and numerically for intermediate values. Good agreement is found between the stresses measured in experiments on dry granular materials and the theoretical predictions when R is given the value 1·7. This case is probably the one for which the present analysis is most appropriate. For moderate and large values of R , the theory predicts both shear and normal stresses that are proportional to the square of the particle diameter and the square of the shear rate, and depend strongly on the solids volume fraction. A provisional comparison is made between the stresses predicted in the limit R → ∞ and the experimental results of Bagnold for shear flow of neutrally buoyant wax spheres suspended in water. The predicted stresses are of the correct order of magnitude and yield the proper variation of stress with concentration. When R [Lt ] 1, the shear stress is linear in the shear rate, and the analysis can be applied to shear flow in a fluidized bed, although such an application is not developed further here.

The Mechanics of Rapid Granular Flows

Publisher Summary This chapter discusses the mechanics of rapid granular flows. A “bulk solid” or “granular fluid” may be defined as an assembly of discrete solid components dispersed in a fluid, such that the solid constituents are in contact or near contact with their neighbors. Bulk solids comprise one member of a larger class of two-phase disperse systems made up of solids and fluids; dilute suspensions form another related member, which is more familiar to fluid mechanicists. The section II of this chapter presents a review of the classical papers of Bagnold and a discussion of the various modes and regimes of granular flow. A dimensional analysis provides a physical background to the detailed review of experimental and theoretical work that follows. Section III reviews flows in vertical channels and inclined chutes; it deals primarily with experimental observations of stress, velocity, and bulk-density fields. Laboratory devices and viscometric type experiments designed to determine the stress-strain-rate behavior are discussed in Section IV. Quasi-static testers, suspension viscometers, and high-shear-rate devices are described. Section V reviews theories for high-shear-rate granular flows. It includes continuum models as well as analytical and numerical microstructural models that consider the details of collisions between particles.

Stresses developed by dry cohesionless granular materials sheared in an annular shear cell

Experimental results obtained during rapid shearing of several dry, coarse, granular materials in an annular shear cell are described. The main purpose of the tests was to obtain information that could be used to guide the theoretical development of constitutive equations suitable for the rapid flow of cohesionless bulk solids at low stress levels. The shear-cell apparatus consists of two concentric disk assemblies mounted on a fixed shaft. Granular material was contained in an annular trough in the bottom disk and capped by a lipped annular ring on the top disk. The bottom disk can be rotated at specified rates, while the top disk is loaded vertically and is restrained from rotating by a torque arm connected to a force transducer. The apparatus was thus designed to determine the shear and normal stresses as functions of solids volume fraction and shear rate. Tests were performed with spherical glass and polystyrene beads of nearly uniform diameters, spherical polystyrene beads having a bimodal size distribution and with angular particles of crushed walnut shells. The particles ranged from about ½ to 2 mm in size. At the lower concentrations and high shear rates the stresses are generated primarily by collisional transfer of momentum and energy. Under these conditions, both normal and shear stresses were found to be proportional to the particle density, and the squares of the shear rate and particle diameter. At higher concentrations and lower shear rates, dry friction between particles becomes increasingly important, and the stresses are proportional to the shear rate raised to a power less than two. All tests showed strong increases in stresses with increases in solids concentrations. The ratio of shear to normal stresses showed only a weak dependence upon shear rate, but it increased with decreasing concentration. At the very highest concentrations with narrow shear gaps, finite-particle-size effects became dominant and differences in stresses of as much as an order of magnitude were observed for the same shear rate and solids concentration.

The dynamics of avalanches of granular materials from initiation to runout. Part I: Analysis

SummaryThis 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.

Analyses of slow high-concentration flows of granular materials

A theory intended for slow, dense flows of cohesionless granular materials is developed for the case of planar deformations. By considering granular flows on very fine scales, one can conveniently split the individual particle velocities into fluctuating and mean transport components, and employ the notion of granular temperature that plays a central role in rapid granular flows. On somewhat larger scales, one can think of analogous fluctuations in strain rates. Both kinds of fluctuations are utilized in the present paper. Following the standard continuum approach, the conservation equations for mass, momentum and particle translational fluctuation energy are presented. The latter two equations involve constitutive coefficients, whose determination is one of the main concerns of the present paper. We begin with an associated flow rule for the case of a compressible, frictional, plastic continuum. The functional dependence of the flow rule is chosen so that the limiting behaviours of the resulting constitutive relations are consistent with the results of the kinetic theories developed for rapid flow regimes. Following Hibler (1977) and assuming that there are fluctuations in the strain rates that have, for example, a Gaussian distribution function, it is possible to obtain a relationship between the mean stress and the mean strain rate. It turns out, perhaps surprisingly, that this relationship has a viscous-like character. For low shear rates, the constitutive behaviour is similar to that of a liquid in the sense that the effective viscosity decreases with increasing granular temperature, whereas for rapid granular flows, the viscosity increases with increasing granular temperature as in a gas. The rate of energy dissipation can be determined in a manner similar to that used to derive the viscosity coefficients. After assuming that the magnitude of the strain-rate fluctuations can be related to the granular temperature, we obtain a closed system of equations that can be used to solve boundary value problems. The theory is used to consider the case of a simple shear flow. The resulting expressions for the stress components are similar to models previously proposed on a more ad hoc basis in which quasi-static stress contributions were directly patched to rate-dependent stresses. The problem of slow granular flow in rough-walled vertical chutes is then considered and the velocity, concentration and granular temperature profiles are determined. Thin boundary layers next to the vertical sidewalls arise with the concentration boundary layer being thicker than the velocity boundary layer. This kind of behaviour is observed in both laboratory experiments and in granular dynamics simulations of vertical chute flows.

The effects of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials

SummaryFollowing 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.