Numerical analysis of particulate flows with the finite element method Guillermo Casas González Supervisors: Eugenio Oñate Riccardo Rossi Doctorate Program in Structural Analysis Department of Civil and Environmental Engineering BarcelonaTech (UPC) This dissertation is submitted for the degree of Doctor of Philosophy March 2019 Dedico aquest treball, el més gran que he fet mai, als meus pares, Ana i Juan Ignacio, i a la Vera, el meu amor. Declaration I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other university. This dissertation is my own work and contains nothing which is the outcome of work done in collaboration with others, except as specified in the text and Acknowledgements. Guillermo Casas González March 2019 Acknowledgements Writing this doctoral dissertation often felt like writing a book, perhaps a long one. A somewhat lonely activity, a fight to keep a focus for a long time chasing a quality standard that keeps getting away. And as so often writers explain, this is best achieved in solitude: it is a lonely job. On the other hand, one is not ready to write it the first day. I arrived at CIMNE knowing very little about programming, or physics, or about how to do research. I must thank Eugenio Oñate for, in spite of this, giving me the opportunity to enter the world of numerical methods and science in general. Eugenio has taken care of my career and has cultivated my confidence in a very generous way over these years. I am especially grateful for his welcoming me a second time after I came back from my failed entrepreneurial adventures. A year into my doctorate, Riccardo Rossi kindly took me in as his candidate as co-director. He has been the one to listen to my worries and patiently help me every time I got stuck. An advantage of working in CIMNE is that there is no shortage of smart researchers, always ready to discuss your problems. And I certainly took advantage of this during the course of this work: First of all, thank you Jordi Cotela. I owe most of my work on the extension of the finite element code to you (and also my understanding of subscale stabilization). You provided me with code, lessons and fruitful discussions. I am sorry I could not finish the work on turbulence we were collaborating on in time. Imustalso thank my other lab-mates: Julio Marti, Pavel Ryzhakov and Ricardo Reyes for their many discussions and advice. Even before I began my doctorate, I had entered the DEMPack team. I owe a lot to my colleagues there too: Miguel Ángel Celigueta, with whom I have collaborated in many developments and held numerous stimulating discussions on programming and on the discrete element method; to Salvador Latorre, with whom I have shared many jobs; and Ferran Arrufat, who helped me with crucial developments at the end of this work. I am indebted to these people, as well as to Mercè López and Joaquin Irazábal for covering my back so many times when I was working on this manuscript. It has been a pleasure sharing all those good times during the lunch breaks with these friends. viii I must thank several other researchers who have offered their help. I do not want to forget any of them: Roberto Flores, a friend and a such a talented scientist, who helped me sort out my ideas in several occasions; Enrique Ortega, who gave me valuable ideas on how to use polynomial smoothers; Ramón Codina and Joan Baiges, who helped me understand concepts on subscales stabilization and other questions; Prashanth Nadukandi, who helped me with matrix manipulations and with whom I enjoyed discussing possible research topics; and Pooyan Dadvant and Carlos Roig, who gave me programming solutions I would have taken an eternity to find. I am convinced that the best research is done in collaboration. I must thank the very fruitful (and pleasant) collaborations I have had with Àlex Ferrer, who played a crucial role in the optimization process described in the third chapter and Ignasi Pouplana, with whom we worked on particle impact drilling. On occasions, I encountered difficulties when reading crucial sections of the literature. Thankfully, most times the authors were kind to answer my questions. I want to thank Benoît Pouliot, Ksenia Guseva and Andrew Bragg for their patient and elaborate answers. I spent two extremely productive months at UP Berkeley thanks to Tarek Zohdi, who kindly hosted me two times and collaborated with me in writing a paper. I thank him and Eugenio for making that exchange possible. My research would have not been possible without financial support from Generalitat de Catalunya, under Doctorat Industrial program 2013 (DI 024). I also thank the support received from the project PRECISE - Numerical methods for PREdicting the behaviour of CIvil StructurEs under water natural hazards (MINECO - BIA2017-83805-R - 01/01/2018 – 31/12/2020) Finally, I want to thank my family for their extreme patience and constant love and support. Abstract In this work we study the numerical simulation of particle-laden fluids, with an emphasis on Newtonian fluids and spherical, rigid particles. Our general strategy consists in using the discrete element method (DEM) to model the particles and the finite element method (FEM) to discretize the continuous phase, suchthat the fluid is not resolved around the particles, but rather averaged over them. The effectofthe particles on the fluid is taken into account by averaging (filtering) their individual volumes and particle-fluid interaction forces. In the first part of the work we study the Maxey–Riley equation of motion for anisolated particle in a nonuniform flow; the equation used to calculate the trajectory of theDEM particles. In particular, we perform a detailed theoretical study of its range of applicability, reviewing the initial effects of breaking its fundamental hypotheses, such as small Reynolds number, sphericity of the particle, isolation etc. The output of this study is a set of tables containing order-of-magnitude inequalities to assess the validity of the method in practice. The second part of the work deals with the numerical discretization of the MRE and, in particular, the study of different techniques for the treatment of the history-dependent term, which is difficult to calculate efficiently. We provide improvements on an existing method, proposed by van Hinsberg et al. (2011), and demonstrate its accuracy and efficiency in a sequence of tests of increasing complexity. In the final part of the work we give three application examples representative of different regimes that may be encountered in the industry, demonstrating the versatility of our numerical tool. For that, we describe necessary generalizations to the MRE to cover problems outside its range of applicability. Furthermore, we give a detailed account of the stabilized FEM algorithm used to discretize the fluid phase and compare several derivative recovery tools necessary to calculate some of the interphase coupling terms. Finally, we generalize the algorithm to include the backward-coupling effects according to the theory of multicomponent continua, allowing the code to deal with arbitrarily dense flow regimes. Table of contents List of figures xv List of tables xix Nomenclature xxiii 1 Introduction1 1.1 Background and Motivation . .1 1.1.1 Numerical simulations . .4 1.1.2 Industrial Doctorate . .9 1.2 Objectives and methodology . .9 1.3 Outline of this document . 11 2 The Maxey–Riley equation 15 2.1 Introduction . 15 2.2 Range of validity . 17 2.2.1 Inertial effects: first finite-Rep effects . 21 2.2.2 Finite radius effects . 28 2.2.3 Nonsphericity effects . 29 2.2.4 Effect of neighbouring particles . 31 2.2.5 Small-size effects . 49 2.2.6 Other effects . 62 2.3 Scaling analysis . 64 2.3.1 Analysis of a simplified MRE . 65 2.3.2 Importance of the Faxén terms . 72 2.4 Summary . 74 3 The numerical solution of the Maxey–Riley equation 77 3.1 Introduction . 77 xii Table of contents 3.2 Overview of approaches for the treatment of the Boussinesq–Basset term . 79 3.2.1 Fractional derivative approach . 80 3.2.2 Hybrid polynomial interpolation/analytic approach . 82 3.2.3 Comparing the accuracies of the different quadrature methods . 83 3.2.4 Addressing memory requirements: window methods . 83 3.3 Improvements on the MAE . 86 3.3.1 Introduction of quadrature substepping . 89 3.3.2 How to choose the ti parameters . 96 3.4 Overall algorithm . 104 3.5 The fractional calculus perspective . 107 3.5.1 Exploring an idea: Richardson’s extrapolation . 112 3.6 Performance of the methodology . 114 3.6.1 First benchmark: an integral with analytical solution . 117 3.6.2 Second benchmark: Candelier’s solution . 121 3.6.3 Third benchmark: Sedimentation through synthetic vortices . 128 3.7 Summary . 131 4 Forward and backward-coupled particulate flows 137 4.1 Introduction . 137 4.2 Beyond the MRE . 141 4.2.1 Unperturbed fluid and added mass forces . 143 4.2.2 Drag force . 144 4.2.3 History force . 144 4.2.4 Lift force . 145 4.2.5 Torque . 145 4.3 The continuous-phase problem . 146 4.3.1 Variational form of the problem . 146 4.3.2 VMS-stabilized finite element formulation . 148 4.3.3 The overall algorithm . 155 4.4 Derivative recovery . 156 4.4.1 Overview of existing approaches . 158 4.4.2 Comparison of the different recovery approaches .