Comparison of constitutive relationships based on kinetic theory of granular gas for three dimensional vibrofluidized beds

Granular materials exist in many forms in nature ranging from space debris to sand dunes and from breakfast cereals to pharmaceutical tablets. They can behave like a solid or a viscous fluid or a gas. The gas-like nature of granular materials in rapid flows allows the use of models based on kinetic theory thus revealing in depth complex physics and phenomena. However unlike conventional fluids here the energy balance requires additional dissipation terms as a consequence of inelasticity. The complexity of their interaction and diversity in application has led to numerous studies using experimental methods and numerical simulations in order to determine the most appropriate constitutive relationships for granular gases. With large dissipation the form of the constitutive relationship becomes particularly important, especially in the presence of non-equipartition and anisotropy. This thesis is focused on constitutive models of simple granular flows. A vibrated bed is often used as an idealisation of granular flows, providing a convenient approximation to the simplest type of flow: binary and instantaneous collisions with no rotations. Using finite element method (FE) based COMSOL modules we solve conservation of mass, momentum and energy resulting from granular kinetic theory in axi-symmetric form to generate time and spatial resolved solutions of packing fraction, velocity and granular temperature and compare the predictions to numerical simulation and experiment. At first we show the comparison for two closure sets, one based on a simple near elastic approach while the second based on revised Enskog theory for dense inelastic flows. The results for the second approach show good agreement with the results of previously validated near elastic models and experimental results. The observed differences between the two closure sets are small except for the observation of temperature upturn in a dilute region of the cell away from base. One cause of this is the presence of additional constitutive terms in the balance equations and are a consequence of inelasticity. The models also consider time varying effects at low frequency of excitation. These solutions show existence of wave-like effects in the cell with associated temperature upturn within the hydrodynamic applicability region. Presence of instantaneous cyclic rolling is also seen in both approaches. Evidence from MD simulations and experiments qualitatively support the findings of hydrodynamic models in phase resolved as well as time average behaviour. Subsequently, the frequency of vibration was varied to unlink the wave motion from the bulk temperature. Lack of agreement between experiment and the model predictions are shown to be due to lack of separation of time scale between the grain-base interaction and the base frequency. A sharp decrease of heat flux is measured showing that the energy input is frequency dependent. Analysis of the bulk behaviour shows that at high frequency, hard sphere based models are able to capture the steady state behaviour reasonably well. Further investigations that modulate the driving with a low frequency amplitude change revealed the dynamic nature of flow with the low frequency component. No significant influence of high frequency signal is noted except the reduction of base heat flux. Independent analysis of bulk behaviour for modulated wave excitation using MD simulations and hydrodynamic models showed wave motion in a pattern similar to non-modulated low frequency vibration. A one-dimensional inviscid model was used to determine the underlying scaling relationships for near elastic granular flows. A form of non-dimensionalisation predicts scaling behaviour for the granular flow. The predictions show good results for the dilute flows using hard sphere MD simulations. Results from MD simulations confirm dilute limit scaling of base temperature, packing fractions and heat flux coefficients. At higher inelasticity and loading condition the model fails to capture the real physics suggesting the need for a more accurate model. This simplified model does, however, set the basis for describing the main scalings for vibrofluidized granular beds, and in the future we anticipate that effects of further inelasticity and enhanced density could be incorporated.