Dilatant double shearing theory applied to granular chute flow

Summary Although the steady flow of a granular material down a plane inclined slope has been exhaustively examined from both theoretical and experimental points of view, there is still no general agreement concerning the basic flow properties such as density and velocity profiles. The majority of studies assume that the velocity component of the material perpendicular to the inclined plane is sufficiently small to assume that it is everywhere zero. However, recent dynamical modelling of granular chute flow indicates that this component of velocity, although small, is actually non-zero. In this paper, we examine a dilatant double shearing theory for chute flow assuming that the perpendicular component of velocity is non-zero. An explicit analytical form for the perpendicular velocity profile is deduced which gives rise to an integral expression for the chute stream velocity. Assuming a linear decreasing density profile, numerical integration for the chute stream velocity predicts a non-linear profile which is concave in shape and which is in agreement with recent results from computer simulation and existing experimental data in the literature.     

Granular chute flow regimes:mass flowrates,flowrate limits and clogging

It has previously been shown that the transition from supercritical to subcritical behavior in granular chute flows often results in the formation of stagnant zones which limit the mass flow and eventually clog the chute. This paper reports an analysis of previously unpublished data for the flow of cohesionless granular material in inclined open channels. Three regimes of mass flow behavior, supercritical and two regimes of subcritical flow, were observed and are characterized by their dependence on chute inclination and geometry. (This paper represents the first report of the third flow regime.) Analysis of the experimental findings lead into empirical/mathematical expressions for the mass flow in the first two regimes as well as a prediction of the regime transition points for all three regimes. This last gives insight into the effect of chute geometry on regime transition and the subsequent clogging of chutes.     

Flow–obstacle interaction in rapid granular avalanches: DEM simulation and comparison with experiment

This paper investigates the interaction between rapid granular flow and an obstacle. The distinct element method (DEM) is used to simulate the flow regimes observed in laboratory experiments. The relationship between the particle properties and the overall flow behaviour is obtained by using the DEM with a simple linear contact model. The flow regime is primarily controlled by the particle friction, viscous normal damping and particle rotation rather than the contact stiffness. Rolling constriction is introduced to account for dispersive flow. The velocity depth-profiles around the obstacles are not uniform but varying over the depth. The numerical results are compared with laboratory experiments of chute flow with dry granular material. Some important model parameters are obtained, which can be used to optimize defense structures in alpine regions.     

Effect of external factors on segregation of different granular mixtures

Segregation is a common problem faced by different pharmaceutical, chemical, and food processing industries due to non-uniformity in the end-products. Our aim is to minimize the segregation of binary granular mixture by changing the external chute-related factors rather than internal factors like material properties which is often not possible in industries. We investigate the effect of inclination angle, friction, fill volume and channel geometry in a steady, gravity driven flow of granular mixture in an inclined plane. We perform the numerical simulation using an open-source Discrete Element Method code - LIGGGHTS. We observe that the segregation of dry granular particles in stream-wise direction of the chute is minimum at low stream-wise velocity i.e. by keeping the chute at a low inclination and adding the wall roughness. The segregation in cross-stream and vertical direction is at a minimum when the chute is filled to at least 40% of its height. We also investigated the optimal conditions for minimum segregation in different granular mixtures and found that a mixture with size or density ratio up to 4 can have minimum segregation, if we fill the chute to 75% height. For a greater size or density ratio, it is difficult to minimize the segregation. The optimal segregation conditions for mixtures with different elastic modulus ratio were generally constant.     

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     

Non-dilatant double-shearing theory applied to dynamical granular chute flow

Summary The non-dilatant double-shearing theory for granular flow is applied to the problem of dynamical chute flow down an inclined slope. Four distinct families of solutions are examined and a number of simple exact solutions are determined which apply to the special case of a granular material with an angle of internal friction equal to ninety degrees. Some of the solutions obtained are illustrated graphically.     

Density variations in dry granular avalanches

Dry granular avalanches exhibit bulk density variations. Understanding the physical mechanisms behind these density variations is especially important in the study of geophysical flows such as snow and rock avalanches. We performed small-scale chute experiments with glass beads to investigate how bulk density changes, measuring velocity profiles, flow height and basal normal stress in an Eulerian measurement frame. The chute inclination and the starting volume of glass beads were systematically varied. From the flow height and basal normal stress data, we could compute the depth-averaged density at the measurement location during the passing of the avalanches. We observed that the depth-averaged density is not constant, varying with chute inclination and starting volume. Furthermore, the depth-averaged density varies from the head to the tail within a single avalanche. We model changes in density by accounting for the energy associated with the velocity fluctuations of the grains, the density and the velocity fluctuations being related by the constitutive relation for the normal stress. We propose expressions for the conduction and decay coefficients of the fluctuation energy which allow us to model the observed density variations in the experiments.     

颗粒在斜槽中流动的实验研究

用示踪颗粒法对细玻璃球在斜槽中的低速流动进行了实验研究。结果发现 ,颗粒的流动行为与斜槽表面特征密切相关 ,对光滑表面存在明显的壁面滑移 ,而对粗糙壁面 ,在低速流情况下不存在滑移。实验测量了不同条件下的速度分布和流层厚度 ,并分析了斜槽倾角、流率及壁面状况对流动的影响     

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.     

Position dependent velocity profiles in granular avalanches

We present velocity profile measurements in granular avalanches flowing down a flat chute with wide rectangular cross section. The flow is recorded through a transparent side-wall by a high-speed camera, which is able to capture 1,825 pictures in a second. Due to the high frame rate of the camera, several flow features can be observed. Quantitative statements can be made by analysing the images with a pattern matching algorithm. This provides us with flow-normal velocity profiles with a very high temporal and spatial resolution. We find that even on flat surfaces, velocity profiles are strongly changing through the flow and for the range of investigated chute angles (from 26° to 36°) clear trends can be recognised. In the head of the avalanche the velocity is highest, decreasing continuously over the length of the avalanche. Thus, the investigated granular avalanches stretch through the flow. The experimental method allows us to study the evolution of characteristic flow properties such as depth averaged velocity, slip velocity, surface velocity, shear rates or flow depth. Side-wall friction effects are estimated.     

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.     

A numerical study of steady plane granular chute flows using the Jenkins–Savage model and its extension

The granular flow model proposed by Jenkins and Savage and extended by us is used here to construct numerical solutions of steady chute flows thought to be typical of granular flow behaviour. We present the governing differential equations and discuss the boundary conditions for two flow cases: (i) a fully fluidized layer of granules moving steadily under rapid shear and (ii) a fluidized bottom-near bed covered by a rigid slab of gravel in steady motion under its own weight. The boundary value problem is transformed into a dimensionless form and the emerging system of non-linear ordinary differential equations is numerically integrated. Singularities at the free surface and (in one case) also at an unknown point inside the solution interval make the problem unusual. Since the non-dimensionalization is performed with the maximum particle concentration and the maximum velocity, which are both unknown, these two parameters also enter the formulation of the problem through algebraic equations. The two-point boundary value problem is solved with the aid of the shooting method by satisfying the boundary conditions at the end of the soluton interval and these normalizing conditions by means of a minimization procedure. We outline the numerical scheme and report selective numerical results. The computations are the first performed with the exact equations of the Jenkins–Savage model; they permit delineation of the conditions of applicability of the model and thus prove to be a useful tool for the granular flow modeller.     

DEM simulation of oblique shocks in gravity-driven granular flows with wedge obstacles

Deflecting wedge obstacles are often built to divert hazardous flows away from residential areas that are in the way of harm. When a rapid avalanche flow is deflected by an obstacle, this usually causes abrupt changes in the flow thickness and velocity and exhibits characteristics like oblique shock waves in the aerodynamic system or oblique hydraulic jumps in the open channel flows. In this study, the Discrete Element Method (DEM) is employed to simulate the motion of granular materials impinging on a wedge obstacle in an adjustable inclination chute. We use chutes with four different inclined angles combined with wedge obstacles having different angles to investigate the overall flow behavior. Both the flow velocity and the flow depth are obtained by averaging the numerical simulation data, and then the granular temperature and the shock angle are calculated. The results of the DEM simulation are compared with that of the classical oblique shock theory. We find that there is good agreement between the classical oblique granular shock theoretical calculations and the DEM simulation results. Moreover, the microdynamic variables related to flow structure such as the packing density and the coordination number are also discussed in the present study.     

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].     

Plane steady shear flow of a cohesionless granular material down an inclined plane: A model for flow avalanches Part I: Theory

Summary A continuum mechanical model describing rapid shear flow of granular materials as deduced by Jenkins and Savage (1983) [11] from considerations of statistical mechanics is applied to steady plane shear flows down an inclined chute. Depending on the type and form of the physically suggested boundary conditions that are imposed at the base and the free surface, respectively, the emerging boundary value problems permit or prohibit existence of mathematical solutions. For instance, the model does not permit incorporation of an aerodynamic drag and requires special sliding boundary conditions at the base. Cause for the singular behavior is the fact that granular pressure and fluctuation energy vanish simultaneously. Rectification is e.g. possible by including particle density gradients in the constitutive relation of granular stress, but this requires postulation of additional boundary conditions.We present the differential equations and boundary conditions and suggest a procedure of non-dimensionalization which yields the dimensionless parameters governing the problem. Construction of local solutions close to the boundaries by means of Frobenius expansions discloses the singular behavior and yields the basis for the non-existence proof under limiting conditions. Adding to the particle stress a Newtonian viscous contribution is not sufficient to regularize the problem and neither is the form of the stress tensor resulting from Lun et al.'s statistical model that incorporates kinetic terms. The stress tensor must have a term proportional to the dyadic product of the particle concentration gradient with itself. Numerical solution techniques and computational results are given in a companion paper (Hutter, Szidarovszky, Yakowitz, 1986 [9]).With 3 Figures     

POWDER FLOW MONITORING USING MICROWAVE DOPPLER RADAR

A computerised, microwave Doppler effect flowmeter has been developed and applied to the flow of model powders down a chute. The instrument enabled the velocity and distribution of velocities of moving powders to be determined simultaneously.

The velocity profiles on a 1.8m stainless steel chute of rectangular cross-section have been obtained for sand in the size range 355-425µm at chute inclinations ranging from 30 to 65 degrees. Freely accelerating flows, which occurred at high chute inclinations, led to beds of open structure while flow at lower angles led to much denser beds being formed. A greater range of velocities existed within the denser beds at low angles than was present in the more open beds at higher angles.     

Measuring the velocity fields of granular flows – Employment of a multi-pass two-dimensional particle image velocimetry (2D-PIV) approach

Measuring velocity fields plays a crucial role in investigating the dynamics of granular flows, which can improve the modeling of hazardous geophysical flows (e.g. avalanches and debris flows) and the control of powder flows in industrial applications. Non-invasive optical methods are invaluable tools for estimating this physical quantity at the laboratory scale. Despite the recent improvements of particle image velocimetry (PIV) algorithms, the employment of PIV to granular flows is still a non-trivial application, where there are several specific aspects to be carefully addressed. Here, we address the main challenges of granular PIV applications and systematically test the open-source window deformation multi-pass code, PIVlab [Thielicke and Stamhuis, J. Open Res. Soft., 2014], for dry granular flows in rotating drum and chute flow experiments. Three granular media (glass spheres, Ottawa sand and acetalic resin beads) with different optical properties are used as a broad test bench for validating the PIV approach. As well, comparisons between the estimations by PIVlab and those obtained by the commercial code, IDT ProVision-XS, are reported, where the advantages of the multi-pass approach are highlighted. This extensive experimental investigation allowed the evaluation of the accuracy of PIVlab in granular flow applications and also helped to assess the reliability of measurements of second-order statistics, such as the granular temperature. Finally, a guideline for setting a reliable PIV arrangement is suggested.     

Application of a cellular automaton to simulations of granular flow in silos

A cellular automaton based on a gas model of hydrodynamics was used to calculate the kinematics of non-cohesive granular materials during confined flow in a mass flow and funnel flow model silo. In the model, collisions of particles were taken into account during granular flow. In addition, a simplified automaton was used wherein granular flow was assumed as an upward propagation of holes through a lattice composed of cells representing single particles. The advantages and disadvantages of both cellular automata were outlined.     

A flow intensification model for granular filter applications

A flow intensification model was proposed for calculation of the initial collector efficiency of a granular filter. In this model, the flow acceleration within voids of granular media is taken into account with an intensification factor β. Simple physical argument gives an estimation of β, which should be close to 1/ε. Creeping flow is assumed and Happel model is used to represent the granular media. After obtaining the flow field, the initial collector efficiency η₀ is calculated from trajectory analysis and compared with experimental data. The reasonably good agreement between the theory and experiments suggests that the current model is physically plausible and potentially useful for granular filter performance prediction if further theoretical development is implemented.