A computer solution for two-dimensional fluid-particle flows

A computer solution for two-dimensional fluid-particle flows using the cellular approach is presented. The euqations describing the flow of the continuous (fluid) phase are formulated using stream function, vorticity and enthalpy as the dependent variables. The effect of the disperse (particle) phase on the continous phase is represented by vorticity and energy sources in each cell. To illustrate the capability of the solution scheme, the flow of gas laden with hot particles issuing into a sudden expansion is analysed.     

Free-surface granular flows down heaps

The present paper extends the granular-flow constitutive model of Savage (1998 J Fluid Mech 377:1–26) to treat spherical particles. Savage accounted for both quasi-static and collisional stresses by considering: (i) strain-rate fluctuations embodied in a critical state plasticity model, as well as, (ii) individual particle velocity fluctuations modelled by granular-flow kinetic theory. In the present work, the governing equations of the kinetic theory of Jenkins (1998 In: Hermann HJ, Luding S (eds) Physics of Dry Granular Media. Kluwer Academic pp. 353–370) for identical spherical, smooth, inelastic particles are supplemented with additional quasi-static terms that have forms patterned after the corresponding terms in the equations of Savage for two-dimensional disk-like particles. The resulting equations along with side-wall and free-surface boundary conditions are applied to examine free-surface granular flow down a heap contained between two frictional vertical side walls. Width-averaged equations of motion are integrated to obtain depth profiles of mean velocity, granular temperature, solids fraction and the Savage–Jeffrey parameter. Detailed comparisons are made with particle-tracking experiments. When the gap between the vertical side walls is fairly narrow, good agreement is found between the predicted and the measured profiles of mean velocity and granular temperature.     

Surface curvature of steady granular flows

Laboratory experiments have shown that the steady flow of granular material down a rough inclined plane has a surface that is not parallel to the plane, but has a curvature across the slope with the height increasing toward the middle of the flow. We study this observation by postulating a new granular rheology, similar to that of a second order fluid. This model is applied to the experiments using a shallow water approximation, given that the depth of the flow is much smaller than the width. The model predicts that a second normal stress difference allows cross-slope height variations to develop in regions with considerable cross-slope velocity shear, consistent with the experiments. The model also predicts the development of lateral eddies, which are yet to be observed.     

Agglomeration and refragmentation in microscale granular flows

A model is developed to describe dendritic agglomeration in microscale granular flows. The individual particulate grains under consideration are approximated as being spheres that remain spherical after impact. The spheres may adhere to one another, forming branched aggregates (dendrites), based upon an empirical contact pressure relation. The possibility for fragmentation is also included in the analysis. The computational model developed is used to demonstrate agglomeration behavior in granular flows for a range of control parameters. The results indicate that there is a transition from size-unstable agglomeration to size-stable agglomeration; which is controlled by the velocity field and the material properties.     

Density-driven sinking dynamics of a granular ring in sheared granular flows

This study reports experimental findings on the sinking dynamics of a heavy granular ring caused by the density-driven segregation effect in sheared granular flows. Specifically, this study systematically investigates the influences of the density ratio, shear rate, and solid fraction of the granular material on the sinking behavior of a heavy granular ring. The parameters of the dimensionless sinking depth and sinking rate, respectively, describe the change in the granular ring position and quantify the particle sinking speed. Experimental results show that both the dimensionless sinking depth and the sinking rate increase as the bottom wall velocity (shear rate) and solid fraction increase. The dimensionless sinking depth and the sinking rate also exhibit a linear relation. The dimensionless sinking depth does not increase monotonically as the density ratio increases. The sinking rate increases linearly with the final steady-state sinking depth for the same heavy granular ring structure, regardless of the wall velocity (shear rate), solid fraction, and density ratio.     

Decompaction waves in falling granular flows

We study the falling of a 2D column of granular material placed initially into a container formed by two lateral vertical walls with a positive friction coefficient by using molecular dynamics simulations. The time evolution of the mean velocity shows an intermediary region where the grains velocity is relatively constant. Follows a decompaction of the granular flow propagating from the bottom to the top of the column with a velocity of approximately 450 cm/s. Just before the flow decompaction an analysis of the forces exhibits a peculiar behavior which can be assimilated to stick-slip. Finally, after the passage of the decompaction wave low and high density regions coexist both descending with relatively the same velocity and the density exhibits a power law spectrum in 1/f ^a , with a≈ 1.51. Received: 9 May 1999     

Shear dispersion in dense granular flows

We formulate and solve a model problem of dispersion of dense granular materials in rapid shear flow down an incline. The effective dispersivity of the depth-averaged concentration of the dispersing powder is shown to vary as the Péclet number squared, as in classical Taylor–Aris dispersion of molecular solutes. An extension to generic shear profiles is presented, and possible applications to industrial and geological granular flows are noted.     

Influence of obstacles on rapid granular flows

Summary. One means of preventing areas from being hit by avalanches is to divert the flow by appropriately constituted obstacles. Thus, there arises the question how a given avalanche flow is modified by obstructions and how the diverted flow depth and direction emerge. In this paper rapid gravity-driven dense granular flows, partly blocked by obstacles with different shapes, sizes and positions, are numerically investigated by solving the hyperbolic Savage-Hutter equations with an appropriate integration technique. The influences of the obstructions on the granular flows are graphically demonstrated and discussed for a finite mass and a steady inflow of granular material down an inclined plane, respectively. These flows are accompanied by shocks induced by both the presence of the obstacles and the transition of granular flows from an inclined surface into a horizontal run-out zone when the velocity transits from its supercritical to its subcritical state. The numerical results show that the theory is capable of capturing key qualitative features, such as shocks wave and particle-free regions.     

Cellular automata model of gravity-driven granular flows

A cellular automata model is used to simulate a variety of granular chute flows. The model is tested against several case studies: flow down a chute, flow past an obstacle, chute flow in which complex, counter-rotating vortices result in streamwise surface stripes and flow near a boundary. The model successfully reproduces experimental observations in all of these cases. These results lead us to propose that simple, rule-based, models such as this can improve our detailed understanding of dynamics and flow within an opaque granular bed.     

Energy characteristics of simple shear granular flows

This work uses a 3-D discrete element simulation to calculate the elastic and kinetic energy for a nonuniform granular shear flow to determine whether the ratio of these energies is sufficient to identify specific flow regimes of granular materials in a fashion to other dimensionless parameters such as inertial number and dimensionless stiffness. We first obtain the critical packing fraction under isostatic compression, then analyze the mean and fluctuating parts of the elastic and kinetic energy as the granular flow reaches a steady state. External work performed on a system during granular flow partially dissipates into heat, while the remaining work is stored in particles as elastic and kinetic energy; thus processes occurring at a particle level not only control the energy transformation, but also affect the bulk behavior of a granular flow. The effective frictions are correlated with the mean elastic energy to mean kinetic energy ratio and it is interesting to find a power law function with an index of $-0.16$ for the systems used in this work. Analysis of this ratio’s ability to classify flow shows that its determination is quite sufficient to identify specific flow regimes of granular materials, even though energy has a scalar expression. Therefore, these energetics studies can provide a theoretical basis for unifying the mechanics of granular flows over the entire range of regimes.     

Electrostatics of granules and granular flows: A review

In past decades, the electrostatics of granules and granular flows has obtained more and more attention due to many industrial problems and the associated development of new technologies. Granule-wall collision causes electrification, where charge transfer can be characterized by work function, electron transfer, ion transfer, and material transfer. Electrification is affected by many factors and increases with granule processing, and the charge amount can reach a saturated state where electrification no longer increases, which has been confirmed by single granule and granule conveying systems. In addition, the presence of electrostatic charges has profound influences in relevant areas, including chemistry, chemical engineering, energy, pharmaceuticals, and so on. The measurement technology of electrostatics used in granule conveying systems has been improved with the continuous progress of industry. Furthermore, electrostatics of granules and granular flows will be developed into a more accurate area together with other subjects as an interdisciplinary problem to be concerned. In addition, in the pneumatic conveying system, granule-wall and granule-granule collision or friction can cause material transfer due to material breakage. The working mechanism of the material transfer due to collision or friction has never been fully understood. Such problems will be solved gradually in the future.     

Using PIV to measure granular temperature in saturated unsteady polydisperse granular flows

The motion of debris flows, gravity-driven fast moving mixtures of rock, soil and water can be interpreted using the theories developed to describe the shearing motion of highly concentrated granular fluid flows. Frictional, collisional and viscous stress transfer between particles and fluid characterizes the mechanics of debris flows. To quantify the influence of collisional stress transfer, kinetic models have been proposed. Collisions among particles result in random fluctuations in their velocity that can be represented by their granular temperature, T. In this paper particle image velocimetry, PIV, is used to measure the instantaneous velocity field found internally to a physical model of an unsteady debris flow created by using “transparent soil”—i.e. a mixture of graded glass particles and a refractively matched fluid. The ensemble possesses bulk properties similar to that of real soil-pore fluid mixtures, but has the advantage of giving optical access to the interior of the flow by use of plane laser induced fluorescence, PLIF. The relationship between PIV patch size and particle size distribution for the front and tail of the flows is examined in order to assess their influences on the measured granular temperature of the system. We find that while PIV can be used to ascertain values of granular temperature in dense granular flows, due to increasing spatial correlation with widening gradation, a technique proposed to infer the true granular temperature may be limited to flows of relatively uniform particle size or large bulk.     

Impulse strengths in rapid granular shear flows

Summary This paper describes measurements of the impulses that particles experience while undergoing rapid shear. These were performed with an eye towards understanding the processes that lead to particle attrition and fracture. The measurements were taken from a discrete particle computer simulation of a simple shear flow of spheres. Special attention is paid to the strongest impulses as these will do the most damage. The results indicate that the largest impulses arise, not from the mean shear flow, but from the random particle velocities that are characterized by the so-called granular temperature. Measurements of the largest impulses are presented as functions of particle properties and solid concentration. Histograms of the impulse strengths illustrate the effect of concentration and particle surface friction. Finally, geometric distributions are presented that illustrate the shear induced anisotropy in the impulse strengths.     

Geophysical granular and particle-laden flows: review of the field

An introduction is given to the title theme, in general, and the specific topics treated in detail in the articles of this theme issue of the Philosophical Transactions. They fit into the following broader subjects: (i) dense, dry and wet granular flows as avalanche and debris flow events, (ii) air-borne particle-laden turbulent flows in air over a granular base as exemplified in gravity currents, aeolian transport of sand, dust and snow and (iii) transport of a granular mass on a two-dimensional surface in ripple formations of estuaries and rivers and the motion of sea ice.     

A two-phase model for dry density-varying granular flows

A granular flow is normally comprised of a mixture of grain-particles (such as sand, gravel or rocks) of different sizes. In this study, dry granular flows are modeled utilizing a set of equations akin to a two-phase mixture system, in which the interstitial fluid is air. The resultant system of equations for a two-dimensional configuration includes two continuity and two momentum balance equations for the two respective constituents. The density variation is described considering the phenomenon of air entrainment/extrusion at the flow surface, where the entrainment rate is assumed to be dependent on the divergent or convergent behavior of the solid constituent. The density difference between the two constituents is extremely large, so, as a consequence scaling analysis reveals that the flow behavior is dominated by the solid species, yielding small relative velocities between the two constituents. A non-oscillatory central (NOC) scheme with total variation diminishing (TVD) limiters is implemented. Three numerical examples are investigated: the first being related to the flow behaviors on a horizontal plane with an unstable initial condition; the second example is devoted to simulating a dam-break problem with respect to different initial conditions; and in the third one investigates the behavior of a finite mass of granular material flowing down an inclined plane. The key features and the capability of the equations to model the behavior are illustrated in these numerical examples.     

Some aspects of wedge impact on granular layers and attendant flows

We observed and measured impacts of steel wedges on layers of 200 μ m solid glass spheres and the subsequent response with the ultimate aim to establish the effectiveness of these materials for blast panels and energy absorption. The success of the layers to absorb and dissipate energy is quantified, especially as regards the unconfined free surface of the granular material and its ability to dissipate and redirect momentum laterally through wave-like motion. Experiments were conducted with two granular layer depths; two drop elevations; and three wedge angles. As expected, as the wedge included-angle increases the peak force increases significantly, whereas the period associated with the response is much-reduced. It was shown that with a reduction in layer thickness, the bed can be forced to vacate locally, and subsequently the wedge strikes the reservoir bottom causing a rapid rise to a very large peak force. It was seen that the included-angle 180° wedge caused predominantly solid-like response (i.e. compaction); the 102° wedge generated mostly viscous/fluid-like behavior (i.e. similar to a Newtonian fluid); and as might be expected the 141° wedge exhibited both fluid-like and solid-like behavior. High-speed imaging of the surface response for the viscous-like motion showed breaking wavelike characteristics. In addition granular material was forced upward in a thin free sheet with hexagonal structure. Lastly we quantified the peak force ratio, the impulse, and the energy dissipation in the underlying load cells for the 102° wedge impacting the 16.36 cm bed versus the same test with no bed in place. For this last comparison the peak force ratio exceeded 15; the impulse was about a third less for the granular bed; and the ratio of energy dissipation in the load cells was more than 200.     

Closure relations for shallow granular flows from particle simulations

The discrete particle method (DPM) is used to model granular flows down an inclined chute with varying basal roughness, thickness and inclination. We observe three major regimes: arresting flows, steady uniform flows and accelerating flows. For flows over a smooth base, other (quasi-steady) regimes are observed: for small inclinations the flow can be highly energetic and strongly layered in depth; whereas, for large inclinations it can be non-uniform and oscillating. For steady uniform flows, depth profiles of density, velocity and stress are obtained using an improved coarse-graining method, which provides 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 obtain functional relations for effective basal friction, velocity shape factor, mean density, and the normal stress anisotropy as functions of layer thickness, flow velocity and basal roughness.     

Towards a theoretical picture of dense granular flows down inclines

Unlike most fluids, granular materials include coexisting solid, liquid or gaseous regions, which produce a rich variety of complex flows. Dense flows down inclines preserve this complexity but remain simple enough for detailed analysis. In this review we survey recent advances in this rapidly evolving area of granular flow, with the aim of providing an organized, synthetic review of phenomena and a characterization of the state of understanding. The perspective that we adopt is influenced by the hope of obtaining a theory for dense, inclined flows that is based on assumptions that can be tested in physical experiments and numerical simulations, and that uses input parameters that can be independently measured. We focus on dense granular flows over three kinds of inclined surfaces: flat-frictional, bumpy-frictional and erodible. The wealth of information generated by experiments and numerical simulations for these flows has led to meaningful tests of relatively simple existing theories.     

Texture of granular surface flows: experimental investigation and biphasic non-local model

We investigate here the rheology of dense granular surface flow. First, steady surface flows in a rotating drum are studied experimentally and a large number of clusters of `jammed' grains embedded in the flowing layer is evidenced. The clusters size is power-law distributed, from the grain size scale up to the thickness of the flowing layer. Theoretical implications are then discussed and a non-local biphasic rheological law is proposed. The resulting model succeeds quantitatively to account for the unusual shape of the velocity profile within granular surface flows as well as for the different scalings observed in rotating drum experiments.     

Jamming in granular shear flows of frictional,polydisperse cylindrical particles

In this work, frictional, cylindrical particle shear flows with different size distributions (monodisperse, binary, Gaussian, uniform) are simulated using the Discrete Element Method (DEM). The influences of particle size distribution and interparticle friction coefficient on the solid phase stresses, bulk friction coefficient, and jamming transition are investigated. In frictional dense flows, shear stresses rise rapidly with the increasing solid volume fraction when jamming occurs. The results suggest that at the jamming volume fraction, stress fluctuation and granular temperature achieve the maximum values, and the rate of the stress increase with increasing solid volume fraction approaches the peak value. Meanwhile, the degree of cylindrical particle alignment approaches a valley value. In the polydisperse flows, the jamming volume fraction exhibits significant dependences on the fraction of the longer particles and the particle size distribution. Two models considering the effect of particle size distribution are discussed for predicting the jamming volume fractions of polydisperse flows with frictional, cylindrical particles.