Hydrodynamic modeling of dilute and dense granular flow

We study numerically a continuum model for granular flow, which covers the regime of fast dilute flow as well as slow dense flow up to vanishing velocity. The constitutive relations at small and intermediate densities are equivalent to those derived from kinetic theory of granular flow. The existence of an inherent instability due to the vanishing kinetic or collisional pressure for small granular temperatures requires a cross over from a collisional pressure to an a thermal yield pressure at densities close to random close packing. Contrary to a kinetic viscosity, the viscosity turns into a function diverging for small temperatures analogous to the diverging viscosities of liquids close to the glass transition. In this respect the presented model is a simplified version of a model of Savage (J Fluid Mech 377:1–26, 1998), which nevertheless recovers many aspects of dense granular flow. As examples we show simulations of sandpiles with predictable slopes, hopper simulations with mass and core flow and angle dependent critical sand heights in flows down an inclined plane. We solve the system of the strongly nonlinear singular hydrodynamic equations with the help of a newly developed nonlinear time stepping algorithm together with a finite volume space discretization. The numerical algorithm is implemented using a finite volume solver framework developed by the authors which allows discretization on cell-centred bricks in arbitrary domains.     

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

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.With 12 Figures     

Study of the origin of shear zones in quasi-static vertical chute flows by using discrete particle simulations

We perform discrete-particle simulations of vertical chute flows in the quasi-static regime using disk-like particles. The velocity profiles show a plug flow in the central region and shear zones next to the walls approximately 6 particle diameters wide regardless of bin width, as was observed experimentally. The stress distributions are in good agreement with the predictions of the continuum mechanics equations even for small systems (15 and 20 particle diameters wide) as those studied in the present work. Large stress fluctuations in space and time have been observed, these are mainly due to the inhomogeneity of the force network. It is observed in the simulations that the wall friction does not act homogeneously but it is concentrated at certain points on the wall depending on the local arrangement of the packing. Large stress zones or arches appear at these points of the wall. It is the formation and the way these arches collapse that seems to generate the shear zones. Based on this, a simple mechanism to explain the formation of the shear zone is proposed. The simulations have revealed other interesting features of the flow, particularly the presence of macroscopic fluctuations of velocity, in which large blocks of material move together showing sudden accelerations (corresponding to the collapse of the arches) and sudden decelerations (corresponding to the formation of the arches).     

Segregation pattern of binary-size mixtures in a double-walled rotating drum

Size-induced granular segregation was performed systematically and experimentally in an almost fully filled double-walled rotating drum at 10 different rotation speeds and two different side wall types. The motion of the granular materials was recorded using a high-speed camera for image analysis of particle segregation development in the drum. With continual tracking of the particle movements, the velocity, fluctuations, and granular temperatures were measured. The experimental results indicate that both rotation speeds and friction coefficient of side walls significantly affect segregation phenomena in binary-size mixture granular flows. The results demonstrate similar situations to the Brazil-nut effect and its reverse in the radial direction at either high or a low rotational speed (where the Froude number (Fr) is far from 1). At these instances, the maximum granular temperature occurs near the side walls. Specifically, a double segregation effect (DSE) is found at Froude number (Fr) close to 1. These results can be used in many industrial processes, for example, size grading of materials, screening of impurities, and different structures of functionally graded materials. Moreover, the maximum granular temperature occurs in the middle of the ring space. It causes small particles to move toward both side walls as it pushes bigger particles to accumulate in the middle of the ring space of rotating drum.     

Mathematical simulation of the dynamics of a dispersed phase in free gravitational convection of a viscous incompressible liquid in a square cavity

The problem of motion of monodisperse spherical particles in a heterogeneous medium in nonisothermal free convection of a carrying incompressible liquid in a square cavity with inhomogeneous distribution of temperature on the walls is considered. The problem is solved by the finite-difference method via joint solution of equations for the carrying phase in Euler variables and the equation for a disperse particle in Lagrange variables. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 81, No. 1, pp. 81–89, January–February, 2008.     

Tangential velocity profiles of granular material within horizontal rotating cylinders modelled using the DEM

Flow regimes of granular materials in horizontal rotating cylinders are industrially important since they have a strong influence on the rates of heat and mass transfer within these systems. The tangential velocity profile, which describes how the average particle velocity in the direction parallel to the surface of the bed varies along a radius perpendicular to the surface of the bed, has been examined for many experimental and simulated systems. This paper is concerned with tangential velocity profiles within rotating cylinders simulated using the discrete element method. For high fill levels good agreement is found between the simulated velocity profiles and the equation proposed by Nakagawa et al. (Exp Fluids 16:54–60, 1993) based on magnetic resonance measurements. At lower fill levels slip is observed between the cylinder wall and the particles in contact with it and also between the outer layer of particles and the bulk of the bed. It is demonstrated that this slip occurs when the particles in contact with the wall are able to rotate and that it may be prevented either by using non-spherical particles or by attaching “lifters” to the cylinder wall.     

A fast first order model of a rough annular shear cell using cellular automata

The motivation of the current work is to simulate granular phenomena in a rough annular shear cell without load using a physics-based cellular automata (CA) modeling approach. A simple yet powerful two-dimensional (2D) cellular automata model was developed to model granular flows inside a 2D annular shear cell from a tribological perspective. Physics-based equations developed from first-principles to model collisions have been adopted to form the cellular automata local rules of interaction. This combines the computational efficiency of CA modeling with the accuracy and generalization of first-principle physics modeling. The local flow properties—solid fraction, velocity and granular temperature profiles- were predicted from the CA simulation and compared to the granular shear cell experiments. The resulting steady-state CA model profiles show good agreement with the experimental results. The ability of the CA model to probe the effect of each granular flow property is demonstrated by performing parametric studies of the particle–particle coefficient of restitution and roughness factor.     

Falling process of a rectangular granular step

This study experimentally investigates the falling process of a dry granular step in a transparent plexiglass chute by particle image analysis. Three types of uniform spherical beads and one type of quartz sand were piled up with various bed slopes and widths to elucidate their flow characteristics. The surface angles during the early slipping phase are close to the failure angles that are associated with the active earth pressure, based on the Mohr-Coulomb friction law. For a given size of particles (d) and slope (θ), the retreating upper granular surface follows a theoretical curve, and dimensionless mobile length decreases as the dimensionless time parameter t* increases. Velocity profiles measured at the side wall exhibit an exponential-like tail close to the static region at the bottom of the chute. As determined by the conservation of mass and momentum, the relationship between the characteristic velocity and the characteristic depth is linear in the transient flow.     

Effects of Material Properties on Granular Flow in a Silo Using DEM Simulation

In order to test the effect of material properties on flowability of particulate materials, discharge procedures of spherical particles within a flat-bottomed model silo with three sets of material properties, i.e., soft and hard without adhesion and adhesive hard, were simulated using the Discrete Element Method. For each system, three particles on the center line were selected and their instant vertical velocity components were traced. In addition, both discharge and the rate were recorded throughout the procedure. The predicted results show that, for both the systems without adhesion, though the soft has a material modulus only 1/1000 of the hard, there are no significant differences in f low pattern and discharge rate. This suggests that a soft system can be used to predict the behavior of a hard one to save CPU time in a gravity-driven granular flow. On the other hand, comparison between both hard systems shows that adhesion can significantly reduce the flowability in granular flow. By analyzing the velocity plot for the traced particles, free fall was clearly detected above the decompression zone, indicating the motion of a particle in a granular flow can be resolved as free fall together with the movement due to particle collision. In addition, select dynamic behavior related to the kinetic fluctuations affecting flow was observed. discrete element method silo granule flow particulate material     

Simulated pulsed flow of gas and particles in a horizontal oppose-pulsed gas jets of bubbling fluidized bed

Flow behavior of gas and particles with a horizontal oppose-pulsed gas jets are simulated by means of a three dimensional Computational Fluid Dynamics (CFD) model with the kinetic theory of granular flow in a gas-particles bubbling fluidized bed. The effects of amplitudes and frequencies on the hydrodynamics of gas and particles are analyzed. The simulation results are presented in terms of phase velocity vector plot, volume fraction of phases, granular temperature, power spectrum and Reynolds stresses in the bed. Results show that the impingement caused by the oppose-pulsed gas jets oscillates with the variation of pulsed gas velocity. The impingement zone with the high solid volume fraction reciprocates from the left side to the right side through the bed center with the variation of pulsed jet gas velocities. The lateral velocity and gas turbulent kinetic energy, granular temperature and Reynolds stresses of gas and particles are larger near the pulsed gas jets than that at the center of the bed. The large dispersion coefficients of particles using the horizontal oppose-pulsed gas jets enhance the mixing of particles in gas-solid fluidized bed.     

Plane steady shear flow of a cohesionless granular material down an inclined plane. A model for flow avalanches Part II: numerical results

Summary The granular flow model proposed by Jenkins and Savage., [2], and extended by us [1] is used here to construct numerical solutions of steady chute flows thought to be typical of such flows.We briefly state the equations and boundary conditions and present numerical solutions when the following model parameters of the Senkins and Savage model are varied: (a) the coefficient of restitution of the particles under binary collision, (b) the number of particles per layer, (c) the inclination angle of the chute, and (d) the basal and free surface boundary conditions. We demonstrate that the Jenkins and Savage model may yield physically questionable results, that those of its extension differ markedly from them and are physically more reasonable in certain cases, but yield equally questionable results in others. The results are apt to redefine research directions which granular flow modellers might want to pursue in the near future.With 18 Figures     

Effects of Material Properties on Granular Flow in a Silo Using DEM Simulation

In order to test the effect of material properties on flowability of particulate materials, discharge procedures of spherical particles within a flat-bottomed model silo with three sets of material properties, i.e., soft and hard without adhesion and adhesive hard, were simulated using the Discrete Element Method. For each system, three particles on the center line were selected and their instant vertical velocity components were traced. In addition, both discharge and the rate were recorded throughout the procedure. The predicted results show that, for both the systems without adhesion, though the soft has a material modulus only 1/1000 of the hard, there are no significant differences in f low pattern and discharge rate. This suggests that a soft system can be used to predict the behavior of a hard one to save CPU time in a gravity-driven granular flow. On the other hand, comparison between both hard systems shows that adhesion can significantly reduce the flowability in granular flow. By analyzing the velocity plot for the traced particles, free fall was clearly detected above the decompression zone, indicating the motion of a particle in a granular flow can be resolved as free fall together with the movement due to particle collision. In addition, select dynamic behavior related to the kinetic fluctuations affecting flow was observed.

discrete element method silo granule flow particulate material     

Thermal slip for a gas with collision frequency proportional to the velocity of molecules

Exact solutions are obtained for the problem of the thermal slip of an inhomogeneously heated gas along a plane surface in a half-space. Two classes of the model kinetic Boltzmann equations are applied: the equation with a collision operator in the BGK (Bhatnagar, Gross, and Krook) form and in the form of an ellipsoidalstatistical model (this equation is constructed for the first time). The collision frequency in both models is proportional to the velocity of molecules. The results of numerical calculations are presented. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 2, pp. 353–359, March–April, 1998.     

Analysis of the evolution of granular stress-strain and voidage states based on DEM simulations

We review and discuss the results of our granular-dynamics simulations of the time evolution of the microstructure of compact granular beds as found in pouring, in hopper filling and discharge, and in a shear cell. These systems are mainly quasi-static. However, it is also common to encounter localized 'shear zones' with significant velocity/voidage fluctuations and high bulk-strain gradients. These narrow-banded zones are separated from near-static regions by sharp, discontinuous changes of bulk stress and voidage. Within these bands the granular assembly undergoes a transition from the quasi-static to the inertial state, where enduring particle contacts are increasingly replaced by collisional ones. We focus on the discrete particle origins of this inhomogeneous yield/flow behaviour. We show the usefulness of analysing the local evolution in terms of relative rotation of the grains which is observed to cause rapid local bulk dilation responsible for setting off avalanches near free-surface boundaries and protracted bulk-failure planes in confined static assemblies. We also present some evidence to suggest that allowing for effective continuous particle-particle interactions could approximate observed effects attributable to particle shape and surface roughness. Wavelet analyses have been applied successfully to generate the variations in periodicity and the relative sequence of evolution of the stress, strain-rate and voidage states in avalanching granular heaps and in the wall region of axially symmetric hopper flows.     

The dynamics of avalanches of granular materials from initiation to runout. Part II. Experiments

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, , 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, , 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].     

On the calculation of turbulent fan jets

The self-similar equations for the dynamic and temperature fields of a forced free fan jet have been numerically integrated within the framework of the standard k-ε model of turbulence. The tables of solutions obtained for the velocity, temperature, and kinetic turbulent energy of the jet, as well as the mean square of the jet-temperature fluctuations at different turbulent Prandtl numbers, are presented. The quantitative parameters of the evolution of the average and turbulent characteristics of the jet flow were determined. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 81, No. 1, pp. 62–67, January–February, 2008.     

IS SEGREGATION-BY-PARTICLE-TYPE A GENERIC MECHANISM UNDERLYING FINGER FORMATION AT FRONTS OF FLOWING GRANULAR MEDIA?

The fronts of flowing granular media often manifest fingered patterns. For the case of a granular flow down an inclined plane, Pouliquen, Delour and Savage (1997) have proposed that such patterns are induced by the segregation of coarse irregularly shaped particles. Here, we consider a thin layer of a granular medium flowing within a cylinder that is rotated about its horizontal axis of symmetry. Our results show that—even when the medium is well-sieved and consists of nearly spherical grains—its leading edge may develop fingers. This suggests that, in general flow configurations, mechanisms other than segregation by particle type may be active in the instability of a straight front.     

A 3D numerical model for computing non-breaking wave forces on slender piles

In this paper a numerical model for water-wave-body interaction is validated by comparing the numerical results with laboratory data. The numerical model is based on Euler’s equation without considering the effects of energy dissipation. The Euler equations are solved by a two-step projection finite-volume scheme and the free-surface displacements are tracked by the volume-of-fluid method. The numerical model is used to simulate solitary waves as well as periodic waves and their interaction with a vertical slender pile. A very good agreement between the experimental data and numerical results is observed for the time history of free-surface displacement, fluid-particle velocity, and dynamic pressure on the pile.     

Study of powder flow patterns in a Couette cell with axial flow using tracers and solid fraction measurements

We present different aspects of dense granular flows in a Couette geometry using a variety of particulate materials with shape and size distributions. Tracer studies point to an apparent coupling of particle size with flow and stress field gradients. While there is a clear industrial motivation to use “real” materials as a means to expand basic physical and engineering research in granular dynamics, the current study suggests additional academic motivations. Indeed, particles with distributed characteristics uncover rich interactions between flow and stress fields that might otherwise go un-noticed with model materials such as spherical glass beads. Distribution of size and shape play a strong role in how stress is transmitted in granular media (Kheiripour Langroudi et al. in Powder Technol 203:23–32, 2010) and how particle pattern arrangements evolve. Direct solid fraction measurements, using a capacitance probe, show that dense particle flows exhibit significant variations in solid fraction in both sheared and stagnant layers. Furthermore, these measurements also show different dependence of the solid fraction on shearing rate: solid fraction decreases in sheared layers and increases in stagnant layers as the shear rate is increased. From these results the thickness of the shear band could be estimated and was found to vary as a function of particle shape and the roughness of the container walls. The main result is that shear stress (or torque) (see also Kheiripour Langroudi et al. in Powder Technol 197:91–101, 2010) and solid fraction profiles depend on particle shape and whether or not an extra degree of freedom in their movement is provided so that the system can dilate under various shear states in the Couette cell. This extra degree of freedom is assured in the present experimental work by allowing a slight axial outflow from the Couette device while the driven shear fields are in the radial and tangential directions.     

Characterization of granular silicon,powders, and agglomerates from a fluidized bed reactor

Growth of polycrystalline silicon from fluidized bed reactors (FBR) produces two general types of silicon products: granular material (diameters on the order of mm) and homogeneously nucleated material often called nanopowder (diameters in the range 10–100 nm). Nanopowder particles tend to be amorphous and have a spherical morphology with an average particle diameter of ~80 nm. Granular material is generally spherical, highly twinned, polycrystalline with crystallite sizes that can reach 200 nm, and includes regions of porosity. The porosity is ~1–4 volume percent, and only the smallest pores exhibit evidence of amorphous regions along the pore surface. The amount of nanopowder that agglomerates on the granular material has been identified using transmission electron microscopy, but agglomeration plays only a minor role in the overall growth process. Therefore, it is proposed that the primary mechanism for granular formation in commercial FBR is chemical vapor deposition, and the pores are associated with nanopowder agglomeration and incomplete sintering.