Modelling of laminar multiphase flow is extremely important in a wide range of engineering and scientific applications. The particle phases are often difficult to model, especially when particles display a range of sizes and velocities at each location in space. Lagrangian methods can be too expensive and many Eulerian methods, though often computationally more affordable, suffer from model deficiencies and mathematical artifacts that lead to non-physical results. For example, efficient Eulerian models that can accurately predict the crossing of multiple streams of non-interacting particles in laminar flow have traditionally been lacking. The predictive capabilities of modern techniques from the kinetic theory of gases to the treatment of disperse multiphase flows are investigated. In particular, several moment-methods, including a recently proposed fourteen-moment approximation to the underlying kinetic equation describing particle motion, are considered and their abilities to predict particle-stream crossing are assessed. Furthermore, a new polydisperse model has been proposed for treatment of flows that display a range of particles sizes. The proposed model is an extension of the well-known maximum-entropy ten-moment model from rarefied gas dynamics with an addition for the treatment of a range of particle diameters. This model allows for anisotropic variance of particle velocities in phase space and directly treats correlations between particle diameter and velocity. The derivation and mathematical structure, of the proposed models are presented. A fine-volume discretization solution procedure for the resulting moment equations is described and used for performing numerical experiments. Results for flow problems that are designed to demonstrate the fundamental behaviour of each model are presented. It is shown that the new models offer clear advantages in terms of accuracy as compared to traditional Eulerian models for multiphase flows.

Modelling multiphase flow, more specifically particle-laden flow, poses multiple challenges. These difficulties are heightened when the particles are differentiated by a set of “internal” variables, such as size or temperature. Traditional treatments of such flows can be classified in two main categories, Lagrangian and Eulerian methods. The former approaches are highly accurate but can also lead to extremely expensive computations and challenges to load balancing on parallel machines. In contrast, the Eulerian models offer the promise of less expensive computations but often introduce modelling artifacts and can become more complicated and expensive when a large number of internal variables are treated. Recently, a new model was proposed to treat such situations. It extends the ten-moment Gaussian model for viscous gases to the treatment of a dilute particle phase with an arbitrary number of internal variables. In its initial application, the only internal variable chosen for the particle phase was the particle diameter. This new polydisperse Gaussian model (PGM) comprises 15 equations, has an eigensystem that can be expressed in closed form and also possesses a convex entropy. Previously, this model has been tested in one dimension. The PGM was developed with the detonation of radiological dispersal devices (RDD) as an immediate application. The detonation of RDDs poses many numerical challenges, namely the wide range of spatial and temporal scales as well as the high computational costs to accurately resolve solutions. In order to address these issues, the goal of this current project is to develop a block-based adaptive mesh refinement (AMR) implementation that can be used in conjunction with a parallel computer. Another goal of this project is to obtain the first three-dimensional results for the PGM. In this thesis, the kinetic theory of gases underlying the development of the PGM is studied. Different numerical schemes and adaptive mesh refinement methods are described. The new block-based adaptive mesh refinement algorithm is presented. Finally, results for different flow problems using the new AMR algorithm are shown, as well as the first three-dimensional results for the PGM.