Three-dimensional simulation of unstable gravity-driven infiltration of water into a porous medium

Infiltration of water in dry porous media is subject to a powerful gravity-driven instability. Although the phenomenon of unstable infiltration is well known, its description using continuum mathematical models has posed a significant challenge for several decades. The classical model of water flow in the unsaturated flow, the Richards equation, is unable to reproduce the instability. Here, we present a computational study of a model of unsaturated flow in porous media that extends the Richards equation and is capable of predicting the instability and captures the key features of gravity fingering quantitatively. The extended model is based on a phase-field formulation and is fourth-order in space. The new model poses a set of challenges for numerical discretizations, such as resolution of evolving interfaces, stiffness in space and time, treatment of singularly perturbed equations, and discretization of higher-order spatial partial–differential operators. We develop a numerical algorithm based on Isogeometric Analysis, a generalization of the finite element method that permits the use of globally-smooth basis functions, leading to a simple and efficient discretization of higher-order spatial operators in variational form. We illustrate the accuracy, efficiency and robustness of our method with several examples in two and three dimensions in both homogeneous and strongly heterogeneous media. We simulate, for the first time, unstable gravity-driven infiltration in three dimensions, and confirm that the new theory reproduces the fundamental features of water infiltration into a porous medium. Our results are consistent with classical experimental observations that demonstrate a transition from stable to unstable fronts depending on the infiltration flux.United States. Dept. of Energy (Early Career Award Grant DE-SC0003907

Multiscale modelling of twin roll casting

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel UniversityTwin roll casting (TRC) is an emerging material processing technique to manufacture thin metal strips directly from liquid metal. Formation of unfavourable microstructural features during TRC, such as centreline segregation and columnar grains, prompted experimental and modelling studies on the effects of various casting conditions on the as-cast microstructures of different alloys. Previous modelling work focuses on low speed TRC with large rolling effect. By increasing nuclei density in the melt, via addition of grain refiners or melt conditioning, the effect of solidification can be significantly improved. This thesis concerns the development and application of multiscale multiphysics modelling techniques to provide an insight into the evolution of microstructure during solidification processing of metals, with a focus on twin-roll casting of thin strips of light alloys. The effects of various casting conditions on the as-cast microstructures are investigated using the multiscale model. The first part of the multiscale model is a macroscale thermal-mechanical model, which predicts the temperature and stress distributions developed in the metal strips during TRC, under the influence of various casting conditions, including casting speed, roll temperature and air convection. The theoretical maximum casting speed is calculated from a quasi-1D solidification model, which can be used to give a guideline for casting conditions used in models and experiments. The second part is a microscale model which predicts the as-cast microstructure, via a phase field-Potts model which simulates the evolution of phase field, alloy concentration and grain orientation during solidification, coupled with a Bingham lattice Boltzmann model which simulates the velocity field in fluid flow. The temperature profile obtained from the macroscale model is used as the thermal boundary condition of the microscale model. To validate the model, temperature data obtained from experiments of Al and Mg TRC is compared with the results of the macroscale model. More experimental data of the as-cast microstructure and texture is required to validate the microscale model

THE FOURIER SPECTRAL METHOD FOR THE CAHN-HILLIARD EQUATION 1

Abstract In this paper, a Fourier spectral method for numerically solving Cahn-Hilliard equation with periodic boundary conditions is developed. We establish their semi-discrete and fully discrete schemes that inherit the energy dissipation property and mass conservation property from the associated continuous problem. we prove existence and uniqueness of the numerical solution and derive the optimal error bounds. we perform some numerical experiments which confirm our results. Keywords: Cahn-Hilliard Equation, Spectral Method.