Comparison of kinetic theory predictions with experimental results for vibrated three-dimensional granular beds

In recent years there has been a great deal of interest in understanding the fundamental behaviour of granular materials. Granular materials are ubiquitous in natural and industrial settings however, their flow behaviour cannot be described using classical ideas of fluid flows as they stand. Of particular interest are theories which have been developed over the last 25 years. These ideas develop the analogy between granular flows and the kinetic theory of gases, but unlike thermal fluids, kinetic energy in granular systems is dissipated into heat during collisions and hence is not conserved; one must balance the energy input rate with the dissipation rate due to collisions in the system to achieve a steady state. Vibrofluidised granular beds are often used as an idealisation of granular flows as they provide a convenient approximation to the simplest type of flow: steady state, binary and instantaneous collisions with no rotation. In this research work, we explore the behaviour of vibrofluidised three-dimensional granular beds by developing various models based on the granular kinetic theory approach. A finite element (FE) based software, Comsol Multiphysics, was used as a toolkit to numerically compute solutions to the three-dimensional conservation equations resulting from granular kinetic theory and the results are shown in dimensionless units. In the first case, an inviscid model for a vibrofluidised granular bed is developed using only observable system parameters such as particle number, size, mass and coefficients of restitution. Two closures based on granular kinetic theory are described, one the standard Fourier law relating heat flux to temperature gradient, the other including an additional concentration gradient term. Each prediction of the twodimensional axisymmetric granular temperature and packing fraction fields was compared against a previously validated one-dimensional model and threedimensional experimental results, acquired using the technique of Positron Emission Particle Tracking (PEPT). Both closures result in solutions that are in reasonable agreement with the experimental results without any fitting parameters, but it was found that differences between the predictions of each of the closures were relatively small in comparison to the anisotropy of the experimentally determined temperature distribution. The models resulting from both theories predict the existence of a small non-zero radial pressure gradient due to a net radial force on any given volume element in the cell, which is not balanced by the gravitational body force since gravity acts parallel to the z axis. Subsequently, considering the viscous effects on the system, a full NavierStokes like viscous model was developed using the Standard Fourier type heat flux based on granular kinetic theory. The resulting granular temperature and packing fraction profiles compare well against the inviscid model and the PEPT experimental results suggesting that the viscous effects are small. The mean velocity profiles from the viscous model show the presence of asymmetric toroidal convection rolls in the system that match well with the shape of the roll observed in the experiments. Quantitatively, the mean velocity profiles show good agreement with the experimental results at relatively low altitudes for a range of experimental values. However, unlike the experimental results the viscous model results show trends in the relationship between angular velocity at the centre of the convection roll and base amplitude of vibration. Additionally, the wall effects are explored in the model which shows that the convection rolls are influenced by the sidewall restitution coefficient, a result that was earlier confirmed using the molecular dynamics simulations. The viscous model was extended to predict the behaviour of an annular vibrated three-dimensional granular bed. The results from the model are compared with the molecular dynamics simulations and experimental data obtained using PEPT. The predictions from the kinetic theory model for mean velocity, granular temperature and packing fraction fields show good agreements despite the presence of anisotropy in molecular dynamics simulations and experimental results. Subsequently the particle-inner wall, particle-outer wall coefficients of restitution phase diagrams generated from the model and the simulation results from molecular dynamics are seen to be in excellent agreement. A comprehensive analysis to probe other key factors that control the direction and magnitude of convection rolls was carried out. This involved a study on five critical variables namely, the inner and outer wall coefficients of restitution, number of grains, ratio of surface areas of the inner and outer cylinders and base amplitude. The results from a systematic study indicate that all the five variables examined can influence the direction and magnitude of the convection rolls in the system. However, it is determined that to initiate convection rolls the presence of energy dissipation at the walls is required. Finally, a comparison between the double convection rolls previously observed experimentally and in simulation shows excellent agreement suggesting that the model may further be used to study the transition from single convection to double convection roll motion of the grains and to explore the precise experimental conditions under which double rolls occur.