Modelling dry granular flows over topography

Dry granular flows are common and important in the environment and industry, and yet their behaviour is very poorly understood. The dynamics of individual grains are governed by the simple and well-known laws of Newtonian mechanics, but how do these 'microscopic' particle-level laws translate into the 'macroscopic' collective motion of thousands or millions of grains, which flow like a liquid? Various rheological models of granular flows have been developed to facilitate a continuum approach, but hitherto they have only been applied to flows in very simple geometries such as parallel shear flow. In these applications, the flows are assumed to be quasi-steady and to vary only over very long distances in the streamwise direction. This approximation, related to the 'shallow water' model of hydraulics, greatly simplifies the equations of motion. However, the assumption is inappropriate for modelling flows interacting with basal features that vary over lengthscales comparable to the depth of the current, or for flows with abrupt time-dependence that cannot be assumed to be quasi-steady. We refer to these spatial and temporal inhomogeneities collectively as topography. In this thesis, we apply a common rheological model to problems involving various types of spatial or temporal topography. One problem that we shall particularly study concerns a flow down a chute that experiences a sudden increase in basal roughness, either spatially or in time. This change induces an evolution of the depthwise velocity profile that begins near the base but eventually spreads throughout the current. We introduce an adaptation of the $\mu(I)$ rheology and find the velocity profile that this rheological model predicts, using a technique similar to the Blasius boundary layer theory for Newtonian fluids flowing past an aerofoil. We validate the predictions of the rheological model by comparing them against the results of discrete particle model (DPM) simulations. We review existing techniques for DPM, and present a number of novel ways of employing these techniques. These methods allow us to reduce the computational cost of simulations while maintaining their realism. The internal profile of a granular flow, and its response to a change in basal conditions, are difficult to observe in experiments or in real life, since grains are opaque. However, the models studied here can help to make predictions about the depth and speed of the flows, or conversely to make inferences about the nature of the base, given measurements on the surface of the flow.EPSRC Studentshi