Anthony Richard Thornton

Modeling of particle size segregation: Calibration using the discrete particle method

Over the last 25 years a lot of work has been undertaken on constructing continuum models for segregation of particles of different sizes. We focus on one model that is designed to predict segregation and remixing of two differently sized particle species. This model contains two dimensionless parameters, which in general depend on both the flow and particle properties. One of the weaknesses of the model is that these dependencies are not predicted; these have to be determined by either experiments or simulations. We present steady-state simulations using the discrete particle method (DPM) for bi-disperse systems with different size ratios. The aim is to determine one parameter in the continuum model, i.e., the segregation Peclet number (ratio of the segregation velocity to diffusion) as a function of the particle size ratio. Reasonable agreement is found; but, also measurable discrepancies are reported; mainly, in the simulations a thick pure phase of large particles is formed at the top of the flow. In the DPM contact model, tangential dissipation was required to obtain strong segregation and steady states. Additionally, it was found that the Peclet number increases linearly with the size ratio for low values, but saturates to a value of approximately 7.35.

Coarse-grained local and objective continuum description of three-dimensional granular flows down an inclined surface

Dry, frictional, steady-state granular flows down an inclined, rough surface are studied with discrete particle simulations. From this exemplary flow situation, macroscopic fields, consistent with the conservation laws of continuum theory, are obtained from microscopic data by time-averaging and spatial smoothing (coarse-graining). Two distinct coarse-graining length scale ranges are identified, where the fields are almost independent of the smoothing length w. The smaller, sub-particle length scale, w ≪ d, resolves layers in the flow near the base boundary that cause oscillations in the macroscopic fields. The larger, particle length scale, w ≈ d, leads to smooth stress and density fields, but the kinetic stress becomes scale-dependent; however, this scale-dependence can be quantified and removed. The macroscopic fields involve density, velocity, granular temperature, as well as strain-rate, stress, and fabric (structure) tensors. Due to the plane strain flow, each tensor can be expressed in an inherently anisotropic form with only four objective, coordinate frame invariant variables. For example, the stress is decomposed as: (i) the isotropic pressure, (ii) the “anisotropy” of the deviatoric stress, i.e., the ratio of deviatoric stress (norm) and pressure, (iii) the anisotropic stress distribution between the principal directions, and (iv) the orientation of its eigensystem. The strain rate tensor sets the reference system, and each objective stress (and fabric) variable can then be related, via discrete particle simulations, to the inertial number, I. This represents the plane strain special case of a general, local, and objective constitutive model. The resulting model is compared to existing theories and clearly displays small, but significant deviations from more simplified theories in all variables – on both the different length scales.

Understanding and exploiting competing segregation mechanisms in horizontally rotated granular media

The axial segregation of granular and particulate media is a well-known but little-understood phenomenon with direct relevance to various natural and industrial processes. Over the past decades, many attempts have been made to understand this phenomenon, resulting in a significant number of proposed mechanisms, none of which can provide a full and universally applicable explanation. In this paper, we show that several mechanisms can be simultaneously active within a single system, and that by considering all relevant mechanisms, it is possible to understand and explain a systems segregative behaviours over a wider range of parameter space than is possible by considering any one, single process. We explore the interrelation and competition between the individual mechanisms present within a given system and demonstrate that by understanding these interactions, we can predict and even, through carefully designed systems, control their behaviour. In particular, we demonstrate that it is possible to deliberately direct segregation, allowing an arbitrary number of pre-determined segregation patterns to be induced in a system. We also illustrate a manner in which the competition between two opposing segregation mechanisms may be exploited in order to enhance the mixing of two dissimilar species of particle - a much sought after ability.

From discrete particles to continuum fields in mixtures

We present a novel way to extract continuum fields from discrete particle systems that is applicable to flowing mixtures as well as boundaries and interfaces. The mass and momentum balance equations for mixed flows are expressed in terms of the partial densities, velocities, stresses and interaction terms for each constituent. Expressions for these variables in terms of the microscopic quantities are derived by coarse-graining the balance equations, and thus satisfy them exactly. A simple physical argument is used to apportion the interaction forces to the constituents. Discrete element simulations of granular chute flows are presented to illustrate the strengths of the new boundary/mixture treatment. We apply the mixture formulation to confirm two assumptions on the segregation dynamics in particle simulations of bidispersed chute flows: Firstly, the large constituent supports a fraction of the stress that is higher than their volume fraction. Secondly, the interaction force between the constituents follows a drag law that causes the large particles to segregate to the surface. Furthermore, smaller particles support disproportionally high kinetic stress, which is a prediction of the theory on shear-induced segregation.

From discrete elements to continuum fields: Extension to bidisperse systems

Micro–macro transition methods can be used to, both, calibrate and validate continuum models from discrete data obtained via experiments or simulations. These methods generate continuum fields such as density, momentum, stress, etc., from discrete data, i.e. positions, velocity, orientations and forces of individual elements. Performing this micro–macro transition step is especially challenging for non-uniform or dynamic situations. Here, we present a general method of performing this transition, but for simplicity we will restrict our attention to two-component scenarios. The mapping technique, presented here, is an extension to the micro–macro transition method, called coarse-graining, for unsteady two-component flows and can be easily extended to multi-component systems without any loss of generality. This novel method is advantageous; because, by construction the obtained macroscopic fields are consistent with the continuum equations of mass, momentum and energy balance. Additionally, boundary interaction forces can be taken into account in a self-consistent way and thus allow for the construction of continuous stress fields even within one element radius of the boundaries. Similarly, stress and drag forces can also be determined for individual constituents of a multi-component mixture, which is critical for several continuum applications, e.g. mixture theory-based segregation models. Moreover, the method does not require ensemble-averaging and thus can be efficiently exploited to investigate static, steady and time-dependent flows. The method presented in this paper is valid for any discrete data, e.g. particle simulations, molecular dynamics, experimental data, etc.; however, for the purpose of illustration we consider data generated from discrete particle simulations of bidisperse granular mixtures flowing over rough inclined channels. We show how to practically use our coarse-graining extension for both steady and unsteady flows using our open-source coarse-graining tool MercuryCG. The tool is available as a part of an efficient discrete particle solver MercuryDPM (www.MercuryDPM.org).

Frictional dependence of shallow-granular flows from discrete particle simulations.

A shallow-layer model for granular flows down inclines is completed with a closure relation for the macroscopic bed friction obtained from micro-scale, discrete particle simulations of steady flows over geometrically rough bases with contact friction. Microscopic friction can be different between bulk particles and with particles at the base, where the latter is systematically varied. When extending the known friction closure relation to be a function of both bulk flow and bed properties, surprisingly, we find that the macroscopic bed friction is only weakly dependent on the contact friction of the bed particles and is predominantly determined by the properties of the flowing particles. Implications for constitutive modelling and possible experiments to better understand the bulk rheology are discussed.Graphical abstract

Free cooling phase-diagram of hard-spheres with short- and long-range interactions

We study the stability, the clustering and the phase-diagram of free cooling granular gases. The systems consist of mono-disperse particles with additional non-contact (long-range) interactions, and are simulated here by the event-driven molecular dynamics algorithm with discrete (short-range shoulders or wells) potentials (in both 2D and 3D). Astonishingly good agreement is found with a mean field theory, where only the energy dissipation term is modified to account for both repulsive or attractive non-contact interactions. Attractive potentials enhance cooling and structure formation (clustering), whereas repulsive potentials reduce it, as intuition suggests. The system evolution is controlled by a single parameter: the non-contact potential strength scaled by the fluctuation kinetic energy (granular temperature). When this is small, as expected, the classical homogeneous cooling state is found. However, if the effective dissipation is strong enough, structure formation proceeds, before (in the repulsive case) non-contact forces get strong enough to undo the clustering (due to the ongoing dissipation of granular temperature). For both repulsive and attractive potentials, in the homogeneous regime, the cooling shows a universal behaviour when the (inverse) control parameter is used as evolution variable instead of time. The transition to a non-homogeneous regime, as predicted by stability analysis, is affected by both dissipation and potential strength. This can be cast into a phase diagram where the system changes with time, which leaves open many challenges for future research.

From the granular Leidenfrost state to buoyancy-driven convection

Grains inside a vertically vibrated box undergo a transition from a density-inverted and horizontally homogeneous state, referred to as the granular Leidenfrost state, to a buoyancy-driven convective state. We perform a simulational study of the precursors of such a transition and quantify their dynamics as the bed of grains is progressively fluidized. The transition is preceded by transient convective states, which increase their correlation time as the transition point is approached. Increasingly correlated convective flows lead to density fluctuations, as quantified by the structure factor, that also shows critical behavior near the transition point. The amplitude of the modulations in the vertical velocity field are seen to be best described by a quintic supercritical amplitude equation with an additive noise term. The validity of such an amplitude equation, and previously observed collective semiperiodic oscillations of the bed of grains, suggests a new interpretation of the transition analogous to a coupled chain of vertically vibrated damped oscillators. Increasing the size of the container shows metastability of convective states, as well as an overall invariant critical behavior close to the transition.

Resonance effects on the dynamics of dense granular beds: achieving optimal energy transfer in vibrated granular systems

Using a combination of experimental techniques and discrete particle method simulations, we investigate the resonant behaviour of a dense, vibrated granular system. We demonstrate that a bed of particles driven by a vibrating plate may exhibit marked differences in its internal energy dependent on the specific frequency at which it is driven, even if the energy corresponding to the oscillations driving the system is held constant and the acceleration provided by the base remains consistently significantly higher than the gravitational acceleration, g. We show that these differences in the efficiency of energy transfer to the granular system can be explained by the existence of resonances between the beds bulk motion and that of the oscillating plate driving the system. We systematically study the dependency of the observed resonant behaviour on the systems main, controllable parameters and, based on the results obtained, propose a simple empirical model capable of determining, for a given system, the points in parameter space for which optimal energy transfer may be achieved.

Hele-Shaw beach creation by breaking waves: a mathematics-inspired experiment

Fundamentals of nonlinear wave-particle interactions are studied experimentally in a Hele-Shaw configuration with wave breaking and a dynamic bed. To design this configuration, we determine, mathematically, the gap width which allows inertial flows to survive the viscous damping due to the side walls. Damped wave sloshing experiments compared with simulations confirm that width-averaged potential-flow models with linear momentum damping are adequately capturing the large scale nonlinear wave motion. Subsequently, we show that the four types of wave breaking observed at real-world beaches also emerge on Hele-Shaw laboratory beaches, albeit in idealized forms. Finally, an experimental parameter study is undertaken to quantify the formation of quasi-steady beach morphologies due to nonlinear, breaking waves: berm or dune, beach and bar formation are all classified. Our research reveals that the Hele-Shaw beach configuration allows a wealth of experimental and modelling extensions, including benchmarking of forecast models used in the coastal engineering practice, especially for shingle beaches.