Euler-like modelling of dense granular flows: application to a rotating drum

General conservation equations are derived for 2D dense granular flows from the Euler equation within the Boussinesq approximation. In steady flows, the 2D fields of granular temperature, vorticity and stream function are shown to be encoded in two scalar functions only. We checked such prediction on steady surface flows in a rotating drum simulated through the Non-Smooth Contact Dynamics method. This result is non trivial because granular flows are dissipative and therefore not necessarily compatible with Euler equation. Finally, we briefly discuss some possible ways to predict theoretically these two functions using statistical mechanics.

Geophysical granular flows, like avalanches and debris flows, are hazardous phenomena for infrastructures and human activities. The complex dynamics of granular systems, involved in these phenomena, poses important challenges to their reliable mathematical modelling. In this context, laboratory investigations on granular flows and non-invasive optical measuring methods are irreplaceable tools for getting a deeper insight into the granular dynamics, as they provide useful information for model calibration and validation. Frictional and collisional momentum flux exchanges coexist in dense granular flows. These resistance mechanisms and their complex interplay with the boundary conditions generate non-negligible fluxes of translational and fluctuation energies along the flow domain, which also influence the kinematics of the free surface. In this work we study steady granular flows in a chute geometry over a fixed bed, specifically focusing on the spatial fluctuations of the free surface. Several tests have been carried out by systematically varying the mass flow rate, the chute inclination and the basal roughness. The experimental apparatus, consisting of a narrow Plexiglas flume, is equipped with a load cell to measure the mass flow rate and a high-speed camera to get non-invasive optical measurements of the free surface fluctuations. A harmonic analysis, employing the Welch method, is performed on a time-series signal obtained from suitably binarized digital images of the free surface fluctuations, recorded at a sampling frequency of 1 kHz. Subsequently, the power spectral densities and the power content of different frequency intervals are analysed. Meaningful correlations are observed between these power components and other relevant measurements, such as the mass flow rates and the flow depths. These preliminary results may be useful for a better understanding of the momentum exchange mechanisms in granular flows and the related fluctuation energy fluxes.

The current paper focusses on the characterization of gravity-driven dry granular flows in cylindrical tubes. With a motive of using dense particulate media as heat transfer fluids (HTF), the main focus was to address the characteristics of flow regimes with a packing fraction of ∼60%. In a previous work [1], experimental and computational studies were conducted to understand the effects of different geometrical parameters on the flow physics. The current paper is an extension of that work to gain more insights into the granular flow physics. The three-dimensional computer simulations were conducted by implementing the Discrete Element Method (DEM) for the Lagrangian modelling of particles. Hertz-Mindilin models were used for the soft-particle formulations of inter-particulate contacts. Simulations were conducted to examine the particulate velocities and flow rates to understand the rheology in the dense flow regime. Past studies suggested the existence of a Gaussian mean velocity profile for dense gravity-driven granular flows. These observations were further analyzed by studying the influence of geometrical parameters on the same. The current work thus focusses on studying the rheology of dense granular flows and obtaining a better understanding of the velocity profiles, the wall friction characteristics, and the particle-wall contact behavior.

To model submarine flows of granular materials we propose an extension of the drift-flux approach. The extended model is able to represent dilute suspensions as well as dense granular flows. The dense granwular flow is modelled as a Herschel–Bulkley fluid, with a yield stress that depends on the dispersed phase pressure. Qualitative numerical experiments show that the model is able to correctly reproduce the stability of submerged sand heaps with different internal angles of friction and initial slopes. When initially starting with heaps with an angle smaller than the internal angle of friction, the heaps are stable. When starting with heaps with angles larger than the internal angle of friction, a flow of solid material is initiated. The flow later stops when the bed is at an angle smaller than the internal angle of friction.

Many contributions on the modelling of granular flow and the solving of such models numerically restrict themselves either to fast, dilute or to dense, quasi‐static granular flow. In many engineering applications both regimes appear at the same time in different parts of the simulations or consecutively at different stages of flow. We numerically investigate a continuum model for granular flow, which covers the regime of fast dilute flow as well as slow dense flow down to vanishing velocity. Our model is at small and intermediate densities equivalent to the model used by Bocquet et al.. [1], An inherent instability in this model for vanishing granular temperature and vanishing velocities is removed by a cross over from a kinetic pressure to an athermal yield pressure at densities close to random close packing. Also the model for the viscosity is modified such that it diverges for small granular temperatures analogous to the diverging viscosities of liquids close to the glass transition. The presented model is a simplified version of a model of Savage [2], which nevertheless recovers many aspects of dense granular flow. We solve the system of the strongly nonlinear singular hydrodynamic equations with the help of a newly developed nonlinear predictor corrector algorithm together and a finite volume space discretisation. To validate our numerical approach we solve a Couette problem to demonstrate the existence of shear bands and simulate the formation of sandpiles with angle of repose consistent with the analytically estimated internal friction angle of the model.

The applicability of continuum granular flow models relies on the characterization of the macroscopic constituting relations of such models. Recent works have presented relations especially for the granular viscosity in certain well‐defined experiments. Here, a different, highly dynamic, yet dense granular flow experiment using a vertically rotating feeding device with an off‐centered rotating bottom opening together with an inverse problem approach of model essential parameters is described. For validation purposes, glass beads are used, even though the method is developed for a broader scope of materials. The same workflow is applied to a continuum and a discrete element method (DEM) model and results with a relative error in a comparable range are obtained.

Dense granular flows are well described by several continuum models; however, their internal dynamics remain elusive. This study explores the contact force distributions in simulated steady and homogenous shear flows. The results demonstrate the existence of high magnitude contact forces in faster flows with stiffer grains. A proposed physical mechanism explains this rate-dependent force transmission. This analysis establishes a relation between contact forces and grain velocities, providing an entry point to unify a range of continuum models derived from either contact forces or grain velocity.

Granular kinetic theories are valid in both kinetic (dilute) flow regime and dense collisional (intermediate) flow regime. This is true as long as contacts between two colliding particles are instantaneous. Empirical theories derived from soil mechanics have been traditionally used in continuum modeling for dense granular flows dominated by enduring contact between particles. This study focuses on validating two continuum theories commonly used to model dense frictional granular flows. The first validation example is conducted for a granular bin discharge, and model predictions are compared with a well-known empirical correlation for the discharge rate. The second example involves a more detailed comparison of flow variables with predictions obtained using a discrete technique for granular flow in a simple shear cell. The frictional flow theory that is only activated at the quasi-static flow regime above maximum packing does not show accurate predictions. Better predictions are obtained using a frictional theory that extends in the intermediate flow regime below packing where both collisional and enduring contact between particles occur.

A continuum description of granular flows would be of considerable help in predicting natural geophysical hazards or in designing industrial processes. However, the constitutive equations for dry granular flows, which govern how the material moves under shear, are still a matter of debate. One difficulty is that grains can behave like a solid (in a sand pile), a liquid (when poured from a silo) or a gas (when strongly agitated). For the two extreme regimes, constitutive equations have been proposed based on kinetic theory for collisional rapid flows, and soil mechanics for slow plastic flows. However, the intermediate dense regime, where the granular material flows like a liquid, still lacks a unified view and has motivated many studies over the past decade. The main characteristics of granular liquids are: a yield criterion (a critical shear stress below which flow is not possible) and a complex dependence on shear rate when flowing. In this sense, granular matter shares similarities with classical visco-plastic fluids such as Bingham fluids. Here we propose a new constitutive relation for dense granular flows, inspired by this analogy and recent numerical and experimental work. We then test our three-dimensional (3D) model through experiments on granular flows on a pile between rough sidewalls, in which a complex 3D flow pattern develops. We show that, without any fitting parameter, the model gives quantitative predictions for the flow shape and velocity profiles. Our results support the idea that a simple visco-plastic approach can quantitatively capture granular flow properties, and could serve as a basic tool for modelling more complex flows in geophysical or industrial applications.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation V. Kumaran; The hard‐particle model for a dense granular flow down an inclined plane. AIP Conference Proceedings 5 May 2010; 1227 (1): 58–71. https://doi.org/10.1063/1.3435421 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

To study the energy dissipation mechanism of nonobstructive particle dampers (NOPDs) and provide guidance to the application of NOPDs, the dense granular flow theory was introduced to establish a quantitative energy dissipation model for NOPDs. The convection movement of the particles under vibrational excitations was studied using the discrete element method, and the Prandtl mixing length theory was adopted to modify the constitution law of dense granular flows. The pressure of the granular flow was obtained by equivalenting the vibrational excitation to a body force acted on particles. Theoretical results showed that the energy dissipation rate of the NOPD was increased with the vibration intensity and decreased with the granular diameter. It also indicated that particles near the side wall and the bottom of the damper dissipated more energy than those particles in other regions. The theoretical model was verified by simulation and experimental result. The results may provide a new approach to studying the energy dissipation mechanism of NOPD and give some guidance to enhancing the damping performance of NOPD in engineering practices.

The collapse and decrepitation of a lava dome at the summit of a volcano generally results in the generation of dense granular flows, often referred to as block and ash flows. As the dome particles propagate from the source, they break apart by internal pressure as well as collision. The propagation of block and ash flows can be simulated to some accuracy with a depth averaged numerical model of the equations of continuity and momentum for a material with a frictional resistance. However, important features of such flows, such as the influence of remote stress through force chains, erosion of the volcano substrate, and shock formation and pressurization upon particle break up are poorly understood. In the near future, the influence of these factors will be incorporated into depth averaged models. Various numerical techniques based on particles will some day yield results that can be compared not only with bulk flow properties, but to the internal layering of block and ash flow deposits.

The granular flow down an inclined plane is simulated using the discrete element (DE) technique to examine the extent to which the dynamics of an unconfined dense granular flow can be well described by a hard particle model. First, we examine the average coordination number for the particles in the flow down an inclined plane using the DE technique using the linear contact model with and without friction, and the Hertzian contact model with friction. The simulations show that the average coordination number decreases below 1 for values of the spring stiffness corresponding to real materials, such as sand and glass, even when the angle of inclination is only 1° larger than the angle of repose. Additional measures of correlations in the system, such as the fraction of particles with multibody contact, the force ratio (average ratio of the magnitudes of the largest and the second largest force on a particle), and the angle between the two largest forces on the particle, show no evidence of force chains or other correlated motions in the system. An analysis of the bond-orientational order parameter indicates that the flow is in the random state, as in event-driven (ED) simulations [V. Kumaran, J. Fluid Mech. 632, 107 (2009); J. Fluid Mech. 632, 145 (2009)]. The results of the two simulation techniques for the Bagnold coefficients (ratio of stress and square of the strain rate) and the granular temperature (mean square of the fluctuating velocity) are compared with the theory [V. Kumaran, J. Fluid Mech. 632, 107 (2009); J. Fluid Mech. 632, 145 (2009)] and are found to be in quantitative agreement. In addition, we also conduct a comparison of the collision frequency and the distribution of the precollisional relative velocities of particles in contact. The strong correlation effects exhibited by these two quantities in event-driven simulations [V. Kumaran, J. Fluid Mech. 632, 145 (2009)] are also found in the DE simulations.

The flow around a fixed cylinder immersed in a uniform granular flow is studied experimentally. Experiments are performed in a tall vertical chute producing a quasi two-dimensional granular flow. A storage bin at the top of the chute feeds glass particles into the channel while the mean velocity of the flow is controlled by varying the exit width of a hopper located at the channel bottom. Measurements of the drag force acting on a fixed cylinder are made using a strain gauge force measurement system. The flow velocity field is measured through a transparent wall using a particle image velocimetry analysis of high speed video recordings of the flow. Experiments are performed for a range of upstream particle velocities, cylinder diameters, and two sizes of glass particles. For the range of velocities studied, the mean drag force acting on the cylinder is independent of the mean flow velocity, contrary to what is expected from any ordinary fluid. The drag force increases with cylinder diameter and decreases with particle diameter. The drag force scales with the asymptotic static stress state in a tall granular bed. The drag coefficient, defined in terms of a dynamic pressure and an effective cylinder diameter, scales with the flow Froude number based on the hydraulic diameter of the channel. This analysis indicates that the drag acting on the cylinder is strongly affected by the surrounding channel geometry. Although the drag force on the cylinder does not change with the upstream flow velocity, the flow streamlines do change with velocity. A large stagnation zone forms at the leading edge of the cylinder while at the trailing edge an empty wake is observed. The wake size increases with flow velocity. Measurements of the flow vorticity and granular temperature are also presented and discussed.

Experimental investigation of the overall residence time of pebbles in a pebble bed reactor (PBR) using radioactive pebble