Discontinuous Galerkin Methods for Multiphase Flow
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Tormod Ravnanger Landet Discontinuous Galerkin methods for multiphase flow Thesis submitted for the degree of Philosophiae Doctor Department of Mathematics Faculty of Mathematics and Natural Sciences University of Oslo 2020 © Tormod Ravnanger Landet, 2020 Series of dissertations submitted to the Faculty of Mathematics and Natural Sciences, University of Oslo No. XYZX ISSN XYZY-XYZYX All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. Cover: Hanne Baadsgaard Utigard. Print production: Reprosentralen, University of Oslo. Abstract This thesis describes a finite element method for simulation of free-surface flows, such as ocean waves, using the discontinuous Galerkin method. Free-surface flows where there is a large difference in density between two immiscible fluids will have a large jump in the magnitude of the momentum field at the interface between the two fluids. Such discontinuities create problems for higher-order discretisations, which are more computationally efficient where the solution is smooth, but need careful handling near discontinuities in order to avoid Gibbs oscillations. This thesis shows how slope limiting can be used to stabilise the momentum equation in an exactly mass conserving discontinuous Galerkin finite element discretisation of the Navier–Stokes equation. The stabilised method is tested on a range of well known 2D and 3D free-surface-flow test cases. The results are in good agreement with published experimental results, and also give the expected higher-order convergence rates for smooth solutions. i Preface I have been incredibly lucky to be able to work on whatever I wanted for more or less the entirety of my scholarship. I want to give my sincerest thanks to the Department of Mathematics at the University of Oslo for the opportunity to have such a free position. There may have been times when I would gladly have traded some of that academic freedom for a position in a team—doing a more collaborative, shared-frustrations type of research—but in the end, maybe the humbling experience of going it alone is an integral part of the PhD experience. What I do know is that I am incredibly proud of what I have accomplished, though I sometimes question how on earth it could take so long. On the topic of efficiency, it took me six years of working as a wave-loads consultant to finally come to the realisation that the time was right for further studies. I had always said that I would eventually apply for a PhD position, but I think my boss, Olav Rognebakke, may have almost given up on encouraging me to get on with it when I finally sat down and wrote the application letter. It was seeing how much fun and personal growth can be had from working on a PhD project that made me finally want to do it, and I must thank my wife, Ann Kristin Sperrevik, for being so good at leaving most of the frustrations and insecurities inherent in the life of a researcher behind her before coming home every evening—something I have since learned is vital to the mental health of any PhD student. Thank you for showing me the all the good sides of research, and also for acknowledging the less good sides as I discovered them. I would like to thank all my colleagues at DNV GL for keeping me sharp all those years away from university. Life as a consultant was both enjoyable and rewarding, mostly due to the great team spirit and all the nice people I worked with. To all my UiO colleagues: thank you for the open social and academic atmosphere; from the ionosphere ninjas and down to the planet core deep thinkers, with everything from oceans to Kolmogorov scales in between, thank you! There are so many of you with brilliant ideas and can-do attitudes. I would love to have some alternative time lines to work with all of you and learn more about your academic interests, but at least we got to share the most important part of each day together, the lunch break! Special thanks to Miro and Diako who were there when I started and made our shared office a welcoming and productive place, and Erika, Susanne and Lisa for initiating and helping organise social and sometimes academically relevant activities. Not every group has such a great positive dynamic, and I am grateful that I ended up in the mechanics group, easily provable to be the best group on campus. Outside the mechanics group I would also like to thank Lucy, Biljana, Terje and the rest of the administration for your support and nice conversations. Sometimes it really helps to walk up and down a few stairs and get away from iii Preface research for a few minutes. I would also like to thank the Ocean Modeling Group in the Department of Marine and Coastal Sciences at Rutgers University for welcoming me into their group and providing a space for me to work for six months during the winter of 2018/2019. Lastly, I would like to thank my supervisors, always full of smiles and interesting ideas. Special thanks to Mikael Mortensen, my main supervisor, who got stuck with me and my free surface problems, which arguably may not be as explosive as combustion or as beautiful as spectral methods. I had the privilege of choosing you—I hope you do not regret too much your friendly explanation of university application procedures to the inexperienced young man who showed up in your office one day and declared that he was ready to start researching. Tormod Ravnanger Landet Oslo, Norway, February 2020 iv Contents Abstracti Preface iii Contentsv 1 Introduction1 1.1 Problem statement ...................... 1 1.2 Motivation........................... 2 1.3 Method selection ....................... 4 1.4 Literature review ....................... 9 1.4.1 Free-surface flow.................. 9 1.4.2 DG FEM for incompressible flow......... 10 1.4.3 Slope limiters in DG FEM............. 10 1.4.4 Recent developments................ 10 1.5 The discontinuous Galerkin method............. 15 1.5.1 The advection equation .............. 18 1.5.2 Elliptic operators.................. 20 1.6 Convective stability—flux limiters.............. 22 1.7 Convective stability—slope limiters............. 26 1.8 The volume-of-fluid method ................. 29 References ............................... 31 2 Summary of papers 43 Papers 46 I Slope limiting the velocity field in a two-phase flow solver 49 I.1 Introduction.......................... 50 I.2 The numerical method .................... 52 I.2.1 Instabilities..................... 52 I.2.2 Preliminaries .................... 52 I.2.3 Discretisation.................... 53 I.2.4 Hdiv projection of the velocity field . 58 I.3 Slope limiting ......................... 59 I.3.1 The hierarchical Taylor-polynomial-based slope limiter........................ 59 I.3.2 On slope limiting of solenoidal fields . 62 I.3.3 A split solenoidal slope-limiting algorithm . 67 v Contents I.4 Results............................. 68 I.4.1 Taylor-Green vortex ................ 69 I.4.2 Dam break ..................... 69 I.4.3 Tank filling ..................... 72 I.5 Discussion ........................... 74 I.6 Conclusion........................... 76 References ............................... 76 II Exactly incompressible DG FEM pressure-splitting schemes 83 II.1 Introduction.......................... 84 II.2 The DG FEM discretisation................. 87 II.2.1 Differential Poisson equation for the pressure . 90 II.3 The IPCS-D method ..................... 91 II.4 The IPCS-A method ..................... 92 II.5 The SIMPLE method..................... 93 II.6 Exact mass conservation ................... 94 II.7 Numerical experiments.................... 95 II.7.1 Taylor–Green 2D flow ............... 95 II.7.2 Ethier–Steinman 3D flow ............. 98 II.7.3 Efficiency ...................... 100 II.8 Conclusions .......................... 101 References ............................... 101 III High-density-ratio two-phase flow simulations in 3D 107 III.1 Introduction.......................... 108 III.2 Mathematical model of free-surface flow . 109 III.3 Discontinuous Galerkin discretisation . 110 III.3.1 Momentum equation . 111 III.3.2 Continuity equation . 112 III.3.3 Density transport.................. 112 III.3.4 Velocity slope limiter . 112 III.4 Solution algorithm....................... 113 III.5 Incoming waves and boundary reflections . 115 III.6 Implementation ........................ 118 III.6.1 Example input file . 118 III.7 Numerical examples...................... 122 III.7.1 3D dam breaking.................. 123 III.7.2 Cylinder in regular waves . 124 III.8 Discussion ........................... 128 References ............................... 129 IV Ocellaris: a DG FEM solver for free-surface flows 135 References ............................... 136 Conclusion 141 vi Contents Novelty 141 Suggestions for further work 142 Appendices 147 A Example: Lid-driven cavity flow 149 A.1 Introduction.......................... 149 A.2 Results............................. 150 A.3 Input file............................ 150 References ............................... 152 B Example: Taylor–Green vortex 153 B.1 Introduction.......................... 153 B.2 Assessing the numerical viscosity . 153 B.3 Results............................. 155 B.4 Input file............................ 157 References ............................... 159 C Example: flow around 2D cylinder 161 C.1 Introduction.......................... 161 C.2 Results............................. 162 C.3 Input file............................ 163 References ............................... 168 vii Chapter 1 Introduction This thesis