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Granular materials represent the most abundant form of matter on earth and are most simply described as a collection of a large numbers of interacting solid particulates (often referred to as grains or simply particles). Such materials are represented in the construction industry (e.g. concrete powder), as powders in the pharmaceutical industry and make up a large proportion of agricultural processing. Beyond industrial motivations, countless natural phenomena are manifested as granular materials/flows, such as avalanches, volcanic eruptions and landslides. Evidently, the ubiquity of granular materials means that being able to predict their rheological properties is essential for both optimising industrial processes and understanding important natural phenomena. In this thesis, the canonical micro-to-macro transition is followed, primarily, in the context of non-spherical particles using the Multi-Sphere Discrete Element Method (MS-DEM). A brief overview of the motivations of this thesis, as well as a cursory introduction to some of the most important concepts explored is provided in Chapter 1. In Chapter 2, the validity of contact models used for the MS-DEM is investigated. Five sources of critical error are identified, three errors are found to be algorithmic in nature, with two shown to occur due to erroneous fundamental physics. Interestingly, the foundational source of error is independent of the contact model, making the findings in Chapter 2 applicable to a wide range of problems. All of the errors are shown to be rectified with the proposals put forward in Chapter 2 and should substantially improve the quality of not only MS-DEM simulations, but related methods for simulating non-spherical particles. In Chapter 3, Lees-Edwards boundary conditions are implemented for the MS-DEM. It is shown that a traditional approach to implementing Lees-Edwards conditions will result in significant microstructural and macroscopic errors when using the MS-DEM. By proposing a new algorithm, Lees-Edwards conditions are successfully implemented for the MS-DEM, allowing one to perform accurate simulations of large strain simple shear flows to study the rheology of non-spherical particulate systems. In Chapters 4 to 6, a new kinetic energy based dimensionless number is proposed, which is shown to form a power-law relationship with the inertial number. Extensive volume-controlled discrete element method simulations show that this power law scaling successfully collapses simple shear flow data, spanning from dilute systems to beyond the jamming point. The constitutive equations derived from this scaling are valid across a broad range of inter-particle friction coefficients and are insensitive to finite stiffness effects. Additionally, the constitutive equations remain valid for highly dilute systems, a wide range of restitution coefficient as well as for elongated particles. Moreover, it is also shown that the traditional µ(I) rheology can be recovered from the proposed framework. An extensive analysis of the differences between the granular kinetic energy and temperature is then performed to understand the utility of the kinetic energy for constitutive modelling. Finally, a brief summary of the findings in this thesis and suggestions for future work are provided in Chapter 7.

The flow around a cylinder is a crucial engineering approximation and is well studied for a range of Reynolds numbers. However, the research into the effects of compressibility has lagged behind as a recent paper notes (Nagata et al., 2020) that the low sub-critical Reynolds numbers are unexplored. Furthermore, the investigation of the interactions between a cylinder and wall has been gaining further attention in the last 50 years, however, compressible research in this area is extremely limited. With the advent of thin-atmosphere flight where Reynolds numbers are low and flight speed is appreciable, there is a growing need for this fundamental engineering approximation to be properly resolved. This thesis tackles the range of 400 < ReD < 800 in a subsonic flow, 0.20 ≤ Ma ≤ 0.45, for both the free and near-wall cases to fill the gaps identified in the literature for this critical range of Reynolds numbers. This is achieved using a high-order, quasi-spectral finite difference method to solve the compressible Navier-Stokes equations. Firstly, the case of a 3D cylinder in a free flow is investigated with good agreement with the incompressible literature at Ma = 0.20. At ReD = 400 it was found that compressibility has a limited impact on the flow and the drag followed the 2D model. At ReD = 800 however, increasing Mach number caused 2D flow features to strengthen and resulted in a 3 fold increase in shedding intensity as well as a significant increase in drag over what the model predicts. This is followed by a larger parametric study of the near-wall case for gap heights of 3 to 1 exposed to a boundary layer with thicknesses greater than 1 diameter. For ReD < 700, the effects due to Mach number were limited and these effects were removed entirely upon immersion in the boundary layer. However, at G/D = 1.5, ReD = 800, increasing the Mach number led to an 8.5% increase in cylinder drag, a strong negative lift and an 8% increase in the shedding frequency. It was shown that in this case, the wall had the combined effect with the Mach number of inducing strong circulation due to accelerated flow in the gap, further decreasing the base pressure and shortening the formation region. These effects were significant enough to cause the excitation of vortex shedding as well as the increased drag. Finally, this unique case was studied in 3D and it was found that there was a drag increase, a drop in base pressure and a shortening of the formation region due to wall proximity. However, these changes were smaller in comparison to the 2D results. Furthermore, there was no reversal in the lift, rather a stronger positive lift was generated. Although shedding intensity increased due to wall proximity, there was no discernible trend in the shedding frequency. It was demonstrated that the longer formation region in 3D was preventing larger pressure decreases. The longer shear layers move high speed fluid further downstream, keeping the circulation lower and preventing the low pressure in wake from interacting strongly with the rear of the cylinder.