Shallow Granular Flows Research Articles

Shallow granular flows down flat frictional channels: Steady flows and longitudinal vortices

Granular flows down inclined channels with smooth boundaries are common in nature and industry. Nevertheless, flat boundaries have been much less investigated than bumpy ones, which are used by most experimental and numerical studies to avoid sliding effects. Using numerical simulations of each grain and of the side walls we recover quantitatively experimental results. At larger angles we predict a rich behavior, including granular convection and inverted density profiles suggesting a Rayleigh-Bénard type of instability. In many aspects flows on a flat base can be seen as flows over an effective bumpy base made of the basal rolling layer, giving Bagnold-type profiles in the overburden. We have tested a simple viscoplastic rheological model [Nature (London) 441, 727 (2006)] in average form. The transition between the unidirectional and the convective flows is then clearly apparent as a discontinuity in the constitutive relation.

Frictional dependence of shallow-granular flows from discrete particle simulations

A shallow-layer model for granular flows down inclines is completed with a closure relation for the macroscopic bed friction obtained from micro-scale, discrete particle simulations of steady flows over geometrically rough bases with contact friction. Microscopic friction can be different between bulk particles and with particles at the base, where the latter is systematically varied. When extending the known friction closure relation to be a function of both bulk flow and bed properties, surprisingly, we find that the macroscopic bed friction is only weakly dependent on the contact friction of the bed particles and is predominantly determined by the properties of the flowing particles. Implications for constitutive modelling and possible experiments to better understand the bulk rheology are discussed.

Closure relations for shallow granular flows from particle simulations

The Discrete Particle Method (DPM) is used to model granular flows down an inclined chute. We observe three major regimes: static piles, steady uniform flows and accelerating flows. For flows over a smooth base, other (quasi-steady) regimes are observed where the flow is either highly energetic and strongly layered in depth for small inclinations, or non-uniform and oscillating for larger inclinations. For steady uniform flows, depth profiles of density, velocity and stress have been obtained using an improved coarse-graining-method, which allows accurate statistics even at the base of the flow. A shallow-layer model for granular flows is completed with macro-scale closure relations obtained from micro-scale DPM simulations of steady flows. We thus obtain relations for the effective basal friction, shape factor, mean density, and the normal stress anisotropy as functions of layer thickness, flow velocity and basal roughness. For collisional flows, the functional dependencies are well determined and have been obtained.

Boundary stresses due to impacts from dry granular flows

Field data and laboratory experiments suggest that bedrock wear from debris flows is largely due to particle–bed impacts, rather than solely due to abrasion by sliding, and that the associated bedrock erosion rates are dependent on the particle size distribution in the debris flow. Here we use Discrete Element Method (DEM) simulations with an established contact mechanics model to explore grain‐size influences on contact forces associated with particle–bed impacts in sheared granular mixtures. We first compare DEM simulations with experimental observations obtained from shallow granular flows in rotating drums of diameters 0.56 m and 4.0 m. Our simulations reproduce, without parameter tuning, experimentally measured segregation, boundary pressures, and height profiles. We perform additional simulations systematically varying particle size distributions in binary mixtures. We show that local time‐averaged boundary pressures in thin flows are essentially the normal component of the weight of the flow, independent of particle size distribution. However, other statistical measures of boundary forces scale with mass‐averaged particle size. We demonstrate that this is because individual particle–bed impacts, rather than impacts from multiple particle collisions, dominate the largest contact forces. We show that these largest impact forces vary as the square of grain size and the 1.2 power of impact velocity as predicted from the contact mechanics model underlying the DEM. These results support the particle size dependence of a recently proposed bedrock incision model and suggest that next steps for a predictive bedrock incision model require the statistics of the largest impact velocities.

Simulations of granular flow along an inclined plane using the Savage–Hutter model

In this work a velocity-dependent friction is introduced into a depth-averaged Savage–Hutter dynamical model for shallow granular flows. The process of granular material flowing along an inclined plane and then depositing on a horizontal plane is simulated. The surface profiles and evolution of various types of energy are investigated and compared when using the standard Coulomb-type friction versus velocity-dependent friction. Interestingly, there is a small difference between the two different types of friction.

Shallow granular flows

Many processes in geophysical and industrial settings involve the flow of granular materials down a slope. In order to investigate the granular dynamics, we report a series of laboratory experiments conducted by releasing grains at a steady rate from a localized source on a rough inclined plane. Different types of dense granular flow are observed by varying the flow rate at the source and the slope of the inclined plane. The two cases of steady flow confined by levees and the flow of avalanches down the plane are examined. The width of the steady flow increases linearly with the prescribed flow rate, which does not appreciably affect the characteristic depth or surface velocity of the bulk flow. When the flow rate is just below that required for sustaining the steady flow, avalanches are triggered at regular intervals. The avalanches maintain their shape, size, and speed down the inclined plane. We propose a simple model of steady flow that is consistent with our observations and discuss the challenges associated with the theoretical treatment of avalanche dynamics.

A Roe-type scheme for two-phase shallow granular flows over variable topography

We study a depth-averaged model of gravity-driven flows made of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. The model system consists of mass and momentum balance equations for the solid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the unknowns, and by interphase drag source terms. The system is hyperbolic at least when the difference between solid and fluid velocities is sufficiently small. We solve numerically the one-dimensional model equations by a high-resolution finite volume scheme based on a Roe-type Riemann solver. Well-balancing of topography source terms is obtained via a technique that includes these contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problems of perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions.

Stable solutions of a scalar conservation law for particle-size segregation in dense granular avalanches

Dense, dry granular avalanches are very efficient at sorting the larger particles towards the free surface of the flow, and finer grains towards the base, through the combined processes of kinetic sieving and squeeze expulsion. This generates an inversely graded particle-size distribution, which is fundamental to a variety of pattern formation mechanisms, as well as subtle size-mobility feedback effects, leading to the formation of coarse-grained lateral levees that create channels in geophysical flows, enhancing their run-out. In this paper we investigate some of the properties of a recent model [Gray, J. M. N. T. & Thornton, A. R. (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. 461, 1447–1473]; [Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. (2006) A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid. Mech. 550, 1–25] for the segregation of particles of two sizes but the same density in a shear flow typical of shallow avalanches. The model is a scalar conservation law in space and time, for the volume fraction of smaller particles, with non-constant coefficients depending on depth within the avalanche. It is proved that for steady flow from an inlet, complete segregation occurs beyond a certain finite distance down the slope, no matter what the mixture at the inlet. In time-dependent flow, dynamic shock waves can develop; they are interfaces separating different mixes of particles. Shock waves are shown to be stable if and only if there is a greater concentration of large particles above the interface than below. Constructions with shocks and rarefaction waves are demonstrated on a pair of physically relevant initial boundary value problems, in which a region of all small particles is penetrated from the inlet by either a uniform mixture of particles or by a layer of small particles over a layer of large particles. In both cases, and under a linear shear flow, solutions are constructed for all time and shown to have similar structure for all choices of parameters.

Supercritical shallow granular flow through a contraction: experiment, theory and simulation

Supercritical granular flow through a linear contraction on a smooth inclined plane is investigated by means of experiments, theoretical analysis and numerical simulations. The experiments have been performed with three size classes of spherical glass beads, and poppy seeds (non-spherical). Flow states and flow regimes are categorized in the phase space spanned by the supercritical Froude number and the minimum width of the contraction. A theoretical explanation is given for the formation of steady reservoirs in the contraction observed in experiments using glass beads and water. For this purpose, the classical, one-dimensional shallow-water theory is extended to include frictional and porosity effects. The occurrence of the experimentally observed flow states and regimes can be understood by introducing integrals of acceleration. The flow state with a steady reservoir arises because friction forces in the reservoir are much smaller than in other parts of the flow. Three-dimensional discrete-particle simulations quantitatively agree with the measured granular flow data, and the crucial part of the theoretical frictional analysis is clearly confirmed. The simulations of the flow further reveal that porosity and frictional effects interact in a complicated way. Finally, the numerical database is employed to investigate the rheology in a priori tests for several constitutive models of frictional effects.

Particle-size segregation and diffusive remixing in shallow granular avalanches

Segregation and mixing of dissimilar grains is a problem in many industrial and pharmaceutical processes, as well as in hazardous geophysical flows, where the size-distribution can have a major impact on the local rheology and the overall run-out. In this paper, a simple binary mixture theory is used to formulate a model for particle-size segregation and diffusive remixing of large and small particles in shallow gravity-driven free-surface flows. This builds on a recent theory for the process of kinetic sieving, which is the dominant mechanism for segregation in granular avalanches provided the density-ratio and the size-ratio of the particles are not too large. The resulting nonlinear parabolic segregation–remixing equation reduces to a quasi-linear hyperbolic equation in the no-remixing limit. It assumes that the bulk velocity is incompressible and that the bulk pressure is lithostatic, making it compatible with most theories used to compute the motion of shallow granular free-surface flows. In steady-state, the segregation–remixing equation reduces to a logistic type equation and the ‘S’-shaped solutions are in very good agreement with existing particle dynamics simulations for both size and density segregation. Laterally uniform time-dependent solutions are constructed by mapping the segregation–remixing equation to Burgers equation and using the Cole–Hopf transformation to linearize the problem. It is then shown how solutions for arbitrary initial conditions can be constructed using standard methods. Three examples are investigated in which the initial concentration is (i) homogeneous, (ii) reverse graded with the coarse grains above the fines, and, (iii) normally graded with the fines above the coarse grains. Time-dependent two-dimensional solutions are also constructed for plug-flow in a semi-infinite chute.

A three-phase mixture theory for particle size segregation in shallow granular free-surface flows

Particle-size segregation within granular materials is of great technological significance yet it is still very poorly understood. There are several causes of segregation, but this paper focuses on kinetic sieving which is the dominant mechanism in dense gravity-driven shallow free-surface flows, or, granular avalanches. The segregation model is derived from a three-phase mixture theory composed of large particles, small particles and a passive interstitial fluid. Steady-state solutions are constructed for a normally graded inflow in a steady uniform flow field. This problem is of fundamental interest, because it shows how an unstably stratified layer readjusts into a stable configuration. Expansion fans and concentration shocks are generated and sufficiently far downstream inversely graded segregated layers form, with the larger particles overlying the finer ones. This a good approximation for segregation in flows with weak diffusive remixing. The distance for complete segregation to occur is shown to increase with rising fluid density and tends to infinity as its density approaches that of the grains. If the particles are buoyant then the initial configuration is stable. An exact time-dependent two-dimensional solution is constructed for plug flow, which exploits the uncoupling of material columns of grains in the absence of shear. This yields insight into the nature of more complex numerical solutions for strong shear, which are computed with a high-resolution shock-capturing numerical scheme.

Time-dependent solutions for particle-size segregation in shallow granular avalanches

Rapid shallow granular free-surface flows develop in a wide range of industrial and geophysical flows, ranging from rotating kilns and blenders to rock-falls, snow slab-avalanches and debris-flows. Within these flows, grains of different sizes often separate out into inversely graded layers, with the large particles on top of the fines, by a process called kinetic sieving. In this paper, a recent theory is used to construct exact time-dependent two-dimensional solutions for the development of the particle-size distribution in inclined chute flows. The first problem assumes the flow is initially homogeneously mixed and is fed at the inflow with homogeneous material of the same concentration. Concentration shocks develop during the flow and the particles eventually separate out into inversely graded layers sufficiently far downstream. Sections with a monotonically decreasing shock height, between these layers, steepen and break in finite time. The second problem assumes that the material is normally graded, with the small particles on top of the coarse ones. In this case, shock waves, concentration expansions, non-centred expanding shock regions and breaking shocks develop. As the parameters are varied, nonlinearity leads to fundamental topological changes in the solution, and, in simple-shear, a logarithmic singularity prevents a steady-state solution from being attained.

A theory for particle size segregation in shallow granular free-surface flows

Abstract 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.

Stability properties of shallow granular flows

We study the stability of shallow layers of granular materials considered as Bagnold fluids (the fluid with power law exponent n = 2) in two cases: (i) inertia terms are negligible and (ii) inertia terms are important. The linear analysis of the governing equations in the two cases demonstrates stable and unstable regimes, respectively. The evolution of some finite amplitude disturbances is solved numerically.