Several Applications of a Model for Dense Granular Flows
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Several applications of a model for dense granular flows Christopher John Cawthorn Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy Corpus Christi College and Institute of Theoretical Geophysics Department of Applied Mathematics and Theoretical Physics University of Cambridge This document is the result of my own work and includes nothing which is the outcome of work done in collaboration, except where specifically indicated in the text. No part of this dissertation has been submitted for any other qualification. Christopher John Cawthorn 1 Several applications of a model for dense granular flows Christopher John Cawthorn Abstract This dissertation describes efforts to evaluate a recently proposed continuum model for the dense flow of dry granular materials (Jop, Forterre & Pouliquen, 2006, Nature, 441, 167-192). The model, based upon a generalisation of Coulomb sliding friction, is known to perform well when modelling certain simple free surface flows. We extend the application of this model to a wide range of flow configurations, beginning with six simple flows studied in detailed experiments (GDR MiDi, 2004, Eur. Phys. J. E, 14, 341-366). Two-dimensional shearing flows and problems of linear stability are also addressed. These examples are used to underpin a thorough discussion of the strengths and weaknesses of the model. In order to calculate the behaviour of granular material in more complicated configura- tions, it is necessary to undertake a numerical solution. We discuss several computational techniques appropriate to the model, with careful attention paid to the evolution of any shear-free regions that may arise. In addition, we develop a numerical scheme, based upon a marker-and-cell method, that is capable of modelling two-dimensional granular flow with a moving free surface. A detailed discussion of our unsuccessful attempt to construct a scheme based upon Lagrangian finite elements is presented in an appendix. We apply the marker-and-cell code to the key problem of granular slumping (Balmforth & Kerswell, 2005, J. Fluid Mech., 538, 399-428), which has hitherto resisted explanation by modelling approaches based on various reduced (shallow water) models. With our numerical scheme, we are able to lift the assumptions required for other models, and make predictions in good qualitative agreement with the experimental data. An additional chapter describes the largely unrelated problem of contact between two objects separated by a viscous fluid. Although classical lubrication theory suggests that two locally smooth objects converging under gravity will make contact only after infinite time, we discuss several physical effects that may promote contact in finite time. Detailed calculations are presented to illustrate how the presence of a sharp asperity can modify the approach to contact. 2 Acknowledgements Several people have offered suggestions, comments, and ideas related to the work con- tained in this thesis. I am particularly indebted to John Hinch for his advice, support, and seemingly infinite patience. I would also like to express my gratitude to Neil Balmforth, who was a wonderful person to work with during my stay in Woods Hole. In addition, I thank my supervisor, Herbert Huppert, for his support and advice throughout my doctoral studies. My work has been funded by a variety of stipends and grants. I am especially grateful for a doctoral research grant from the Natural Environment Research Council (NERC). In addition, I would like to express gratitude to the organisers of the Geophysical Fluid Dynamics Program at the Woods Hole Oceanographic Institution for allowing me to attend their prestigious program in Summer 2008. I would also like to thank the Oxford Centre for Collaborative Applied Mathematics (OCCAM) for funding a three-month visit in Autumn 2009. I would like to express my appreciation for those who have helped to keep me sane during my research. I am grateful to the members of the Institute of Theoretical Geophysics for their companionship, support, and stimulating discussions. A special mention should be made of Maurice Blount, Dominic Vella, and Andrew Wells, whose friendship has been an invaluable asset; and of Doris Allen, who kept everything running smoothly. Additional thanks ought to go to those people in the CMS and in Corpus with whom I have played games – it was always nice to ponder over something other than broken code! Finally, I am eternally grateful to Jenny, who has been an unwavering source of support and encouragement throughout my studies. No matter how bleak things have seemed, she has encouraged me to retain my optimism and press onwards. The very existence of the document before you is proof positive of her success. 3 Contents 1 Introduction 9 1.1 Motivation ..................................... 9 1.2 A brief review of granular phenomenology . 11 1.3 Theoretical developments . 15 1.4 The µ(I)constitutivelaw ............................. 16 1.5 Aimsofthesis.................................... 25 2 Unidirectional flows 27 2.1 Steadyflows..................................... 27 2.1.1 Linearshearflow.............................. 28 2.1.2 Annularshearflow............................. 30 2.1.3 Flowinaverticalchute .......................... 35 2.1.4 Flow down an inclined plane . 39 2.1.5 Flowoveradeeppile............................ 41 2.1.6 Flowinarotatingdrum.......................... 43 2.2 Unsteadyflows ................................... 45 2.2.1 The‘draggedplate’ ............................ 46 2.2.2 Calculating yield surfaces in two dimensions . 58 2.2.3 Results: Split-top shear . 61 2.2.4 Results: Split-bottom shear . 63 2.2.5 Regularisation: A numerical simplification . 66 2.2.6 Results: Flow in an inclined channel . 69 3 Collapse of a granular column 75 3.1 Previouswork.................................... 76 3.2 Numerical approaches . 80 3.2.1 Lagrangian finite elements . 82 3.2.2 Finite difference marker-and-cell . 83 3.3 Results........................................ 92 5 CONTENTS 3.3.1 Collapseofshortcolumns . .. 92 3.3.2 Collapse of tall columns . 98 3.3.3 Discussion..................................102 4 Stability 107 4.1 Longitudinal vortices in inclined plane flow . 109 4.1.1 Governing equations . 111 4.1.2 Linearisation ................................112 4.1.3 Long-wave asymptotic result . 114 4.1.4 Numericalapproach ............................115 4.1.5 Discussion..................................117 4.2 Growthofsanddunes ...............................119 4.2.1 Two-layermodel ..............................121 4.2.2 Linearisation ................................125 4.2.3 Asymptotic solutions in the long wave limit . 127 4.2.4 Numericalsolution .............................129 4.2.5 Discussion..................................132 5 Contact in a viscous fluid 135 5.1 The effect of asperities . 137 5.1.1 The Stokes problem for a falling wedge . 138 5.1.2 Smoothing out the corner at a smaller scale . 144 5.1.3 Consequences for sedimentation and contact . 146 5.2 Other physical effects . 151 6 Conclusions and extensions 155 6.1 Successes and limitations of the µ(I) rheological law . 155 6.2 Lessons for numerical solutions . 157 6.3 Extensions and outlook . 159 A Lagrangian finite element approaches for simulating the collapse of a gran- ular column 165 6 CONTENTS A.1 Asimplebeginning.................................165 A.2 Keyproblems....................................169 A.3 Pressuresmoothing.................................174 A.4 Differentelements .................................179 A.5 Mesh quality and refinement . 184 A.6 Epilogue.......................................189 7 1 Introduction 1.1 Motivation Granular materials have played an important role in daily life throughout history. For centuries, mankind has used the motion of sand through hourglasses to mark the passage of time. Cereals or rice form a staple part of almost every cultural diet on Earth. In the modern world, vast numbers of people live in dwellings and travel on roads constructed using granular aggregates. Furthermore, few children (at least in developed countries) will not, at some point in their lives, have sculpted castles, faces, or animals out of sand or snow. There are countless engineering applications involving granular materials, some of which are illustrated by Figure 1.1. The construction industry deals with grains on a daily basis, not only for use as building materials, but also when assessing the quality and safety of ground on which new structures are to be built. Maritime engineers seek to improve and extend harbours by dredging deep channels into the sandy ocean bed. Pharmaceutical manufacturing is dominated by powders and pills, all of which need to be stored, transported, and mixed safely and efficiently. Similar problems exist in the farming of cereal grains, for which Figure 1.1(c) gives an indication that even the simple task of storing grains in a silo is perhaps not sufficiently understood. Extending our view to include geophysical phenomena reveals many more applications. 9 CHAPTER 1. INTRODUCTION Figure 1.1: Some applications involving granular material: (a) Cereal and (b) pharmaceu- tical materials; (c) Failure and buckling of a grain silo; (d) An hourglass; (e) Sand dunes in the Sahara desert; (f) An avalanche on the Liberty Wall Range, USA; (g) The slumping of a steep hillside following the collapse of a confining wall. 10 CHAPTER 1. INTRODUCTION Avalanches and rockslides are a fact of life in mountainous areas. Although many efforts are taken to predict, divert,