Numerical analysis of particle-laden flows with the finite element method

In this work we study the numerical simulation of particle-laden fluids, with a focus on Newtonian fluids and spherical, rigid particles. We are thus dealing with a multi-phase (more precisely, a multi-component) problem, with two phases: the fluid (continuous phase) and the the particles (disperse phase). Our general strategy consists in using the discrete element method (DEM) to model the particles and the finite element method (FEM) to discretize the Navier-Stokes equations, which model the continuous phase. The interaction model between both phases is (must be) based on a multiscale concept, since the smallest scales resolved of the continuous phase are considered much bigger than the particles. In other words, the resolution of the numerical model for the particles is finer than that used for the fluid. Consequently, whether implicit or explicit, there must be a filtering or averaging operation involved in the interaction between both phases, where the details of their motions smaller than the smallest resolution scale of the fluid are soothed out, since the latter is the coarsest of the two different resolutions considered. The spatial discretization of the continuous phase is performed with the FEM, using equal-order spaces of shape functions for the velocity and for the pressure. It is a well-known fact that this type of combination involves the violation of the Ladyzenskaja-Babuska-Brezzi (LBB) condition, resulting in an unstable numerical method. Moreover, the presence of the convective term in Eulerian description of the flow also leads to numerical instabilities. Both effects are treated with the sub-grid scale stabilization methods here. About the disperse phase, the trajectory of each particle is calculated based both on the fluid-interaction forces and on the contact forces between them and the surrounding rigid boundaries. The differential equation that describes the motion of particles in between successive collisions, given the mean (averaged) far field and for particles much smaller than the smallest scales of the flow (the Kolmogorov scale in turbulence) is the Maxey-Riley equation (MRE). This equation is the subject of chapter 2. The objective of this theoretical study is to establish quantitative (up to order-of-magnitude accuracy) limits to its range of validity and to the relative importance of its various terms. The method employed is dimensional analysis, which is systematically applied to derive the 'first effects' of a series of phenomena that are neglected in the derivation of the MRE. Chapter 3 is dedicated to the numerical resolution of the MRE. Here we present improvements to the method of van Hinsberg et al. (2011) for the calculation of the history term and analyse the method thoroughly. We include several tests to show the efficiency and utility of the proposed approach. The MRE is directly applicable to flows where the particle-based Reynolds number is Re << 1. But its relevance reaches further, as its structure is the basis for the majority of extensions that model the movement of suspended particles outside the range of validity of the MRE. Chapter 4 is markedly more applied than the two preceding ones. It treats various industrial flux types with particles where we employ several extensions of the MRE of the type mentioned above. In the first part of this chapter we review the most important of these extensions and study the process of derivative recovery, necessary to calculate several terms in the equation of motion. The tests examples considered include bubble trapping in 'T'-junction tubes, the simulation of drilling systems of the oil industry based on the bombardment of steel particles and fluidized beds. For the latter we use a discrete filtering-based coupling approach, that mirrors the continuous theory sketched above. This set of three chapters (2, 3, 4) is the core of the Thesis, which is completed with an introduction (chapter 1) and the conclusions (chapter 5).