# 2017_Book_TheLatticeBoltzmannMethod

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- the lattice boltzmann method 格子玻尔兹曼 LBM 流体模拟

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- Graduate Texts in Physics Timmuni00A0Krüger Halimuni00A0Kusumaatmaja Alexandruni00A0Kuzmin Orestuni00A0Shardt Goncalouni00A0Silva Erlenduni00A0Magnusuni00A0Viggen The Lattice Boltzmann Method Principles and Practice Graduate Texts in Physics Series editors Kurt H. Becker, Polytechnic School of Engineering, Brooklyn, USA Jean-Marc Di Meglio, Université Paris Diderot, Paris, France Sadri Hassani, Illinois State University, Normal, USA Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan Richard Needs, University of Cambridge, Cambridge, UK William T. Rhodes, Florida Atlantic University, Boca Raton, USA Susan Scott, Australian National University, Acton, Australia H. Eugene Stanley, Boston University, Boston, USA Martin Stutzmann, TU München, Garching, Germany Andreas Wipf, Friedrich-Schiller-Univ Jena, Jena, Germany Graduate Texts in Physics Graduate Texts in Physics publishes core learning/teaching material for graduate- and advanced-level undergraduate courses on topics of current and emerging fields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledge of, a research field. More information about this series at http://www.springer.com/series/8431 Timm KrRuger Halim Kusumaatmaja Alexandr Kuzmin Orest Shardt Goncalo Silva Erlend Magnus Viggen The Lattice Boltzmann Method Principles and Practice 123 Timm KrRuger School of Engineering University of Edinburgh Edinburgh, United Kingdom Halim Kusumaatmaja Department of Physics Durham University Durham, United Kingdom Alexandr Kuzmin Maya Heat Transfer Technologies Westmount, Québec, Canada Orest Shardt Department of Mechanical and Aerospace Engineering Princeton University Princeton, NJ, USA Goncalo Silva IDMEC/IST University of Lisbon Lisbon, Portugal Erlend Magnus Viggen Acoustics Research Centre SINTEF ICT Trondheim, Norway ISSN 1868-4513 ISSN 1868-4521 (electronic) Graduate Texts in Physics ISBN 978-3-319-44647-9 ISBN 978-3-319-44649-3 (eBook) DOI 10.1007/978-3-319-44649-3 Library of Congress Control Number: 2016956708 © Springer International Publishing Switzerland 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Preface Interest in the lattice Boltzmann method has been steadily increasing since it grew out of lattice gas models in the late 1980s. While both of these methods simulate the flow of liquids and gases by imitating the basic behaviour of a gas— molecules move forwards and are scattered as they collide with each other—the lattice Boltzmann method shed the major disadvantages of its predecessor while retaining its strengths. Furthermore, it gained a stronger theoretical grounding in the physical theory of gases. These days, researchers throughout the world are attracted to the lattice Boltzmann method for reasons such as its simplicity, its scalability on parallel computers, its extensibility, and the ease with which it can handle complex geometries. We, the authors, are all young researchers who did our doctoral studies on the lattice Boltzmann method recently enough that we remember well how it was to learn about the method. We remember particularly well the aspects that were a little difficult to learn; some were not explained in the literature in as clear and straightforward a manner as they could have been, and some were not explained in sufficient detail. Some topics were not possible to find in a single place: as the lattice Boltzmann method is a young but rapidly growing field of research, most of the information on the method is spread across many, many articles that may follow different approaches and different conventions. Therefore, we have sought to write the book that the younger versions of ourselves would have loved to have had during our doctoral studies: an easily readable, practically oriented, theoretically solid, and thorough introduction to the lattice Boltzmann method. As the title of this book says, we have attempted here to cover both the lattice Boltzmann method’s principles, namely, its fundamental theory, and its practice, namely, how to apply it in practical simulations. We have made an effort to make the book as readable to beginners as possible: it does not expect much previous knowledge except university calculus, linear algebra, and basic physics, ensuring that it can be used by graduate students, PhD students, and researchers from a wide variety of scientific backgrounds. Of course, one textbook cannot cover everything, and for the lattice Boltzmann topics beyond the scope of this book, we refer to the literature. v vi Preface The lattice Boltzmann method has become a vast research field in the past 25 years. We cannot possibly cover all important applications in this book. Examples of systems that are often simulated with the method but are not covered here in detail are turbulent flows, phase separation, flows in porous media, transonic and supersonic flows, non-Newtonian rheology, rarefied gas flows, micro- and nanofluidics, relativistic flows, magnetohydrodynamics, and electromagnetic wave propagation. We believe that our book can teach you, the reader, the basics necessary to read and understand scientific articles on the lattice Boltzmann method, the ability to run practical and efficient lattice Boltzmann simulations, and the insights necessary to start contributing to research on the method. How to Read This Book Every textbook has its own style and idiosyncrasies, and we would like to make you aware of ours ahead of time. The main text of this book is divided into four parts. First, Chaps. 1 and 2 provide background for the rest of the book. Second, Chaps. 3–7 cover the fundamentals of the lattice Boltzmann method for fluid flow simulations. Third, Chaps. 8–12 cover lattice Boltzmann extensions, improvements, and details. Fourth, Chap. 13 focuses on how the lattice Boltzmann method can be optimised and implemented efficiently on a variety of hardware platforms. Complete code examples accompany this book and can be found at https://github.com/lbm-principles-practice. For those chapters where it is possible, we have concentrated the basic practical results of the chapter into an “in a nutshell” summary early in the chapter instead of giving a summary at the end. Together, the “in a nutshell” sections can be used as a crash course in the lattice Boltzmann method, allowing you to learn the basics necessary to get up and running with a basic LB code in very little time. Additionally, a special section before the first chapter answers questions frequently asked by beginners learning the lattice Boltzmann method. Our book extensively uses index notation for vectors (e.g. u ˛ ) and tensors (e.g. ESC ˛ˇ ), where a Greek index represents any Cartesian coordinate (x, y,orz)and repetition of a Greek index in a term implies summation of that term for all possible values of that index. For readers with little background in fluid or solid mechanics, this notation is fully explained, with examples, in Appendix A.1. The most important paragraphs in each chapter are highlighted, with a few keywords in bold. The purpose of this is twofold. First, it makes it easier to know which results are the most important. Second, it allows readers to quickly and easily pick out the most central concepts and results when skimming through a chapter by reading the highlighted paragraphs in more detail. Instead of gathering exercises at the end of each chapter, we have integrated them throughout the text. This allows you to occasionally test your understanding as you Preface vii read through the book and allows us to quite literally leave certain proofs as “an exercise to the reader”. Acknowledgements We are grateful to a number of people for their help, big and small, throughout the process of writing this book. We are indebted to a number of colleagues who have helped us to improve this book by reading and commenting on early versions of some of our chapters. These are Emmanouil Falagkaris, Jonas Latt, Eric Lorenz, Daniel Lycett-Brown, Arunn Sathasivam, Ulf Schiller, Andrey Ricardo da Silva, and Charles Zhou. For various forms of help, including advice, support, discussions, and encour- agement, we are grateful to Santosh Ansumali, Miguel Bernabeu, Matthew Blow, Paul Dellar, Alex Dupuis, Alejandro Garcia, Irina Ginzburg, Jens Harting, Oliver Henrich, Ilya Karlin, Ulf Kristiansen, Tony Ladd, Taehun Lee, Li-Shi Luo, Miller Mendoza, Rupert Nash, Chris Pooley, Tim Reis, Mauro Sbragaglia, Ciro Sempre- bon, Sauro Succi, Muhammad Subkhi Sadullah, and Alexander Wagner. On a personal level, Alex wants to thank his family which allowed him to spend some family time on writing this book. Erlend wants to thank Joris Verschaeve for helping him get started with the LBM and his friends and family who worried about how much time he was spending on this book; it all worked out in the end. Halim wants to thank his family for the continuous and unwavering support and Julia Yeomans for introducing him to the wonder of the LBM. Goncalo thanks his family for their unlimited support, Alberto Gambaruto for introducing him to the LBM, and Viriato Semiao for the chance to start working in the field; special thanks to Irina Ginzburg for the opportunity to work with her and the countless discussions and teachings. Orest thanks his family, friends, and colleagues for their valuable support during his doctoral and postdoctoral studies. Timm thanks Aline for her understanding and support and Fathollah Varnik for introducing him to the LBM. Finally, we would like to thank our editor Angela Lahee for her support and patience. Edinburgh, UK Timm Krüger Durham, UK Halim Kusumaatmaja Brossard, QC, Canada Alexandr Kuzmin Princeton, NJ, USA Orest Shardt Lisbon, Portugal Goncalo Silva Trondheim, Norway Erlend Magnus Viggen The authors can be contacted at authors@lbmbook.com. Contents Part I Background 1 Basics of Hydrodynamics and Kinetic Theory 3 1.1 Navier-Stokes and Continuum Theory 3 1.1.1 Continuity Equation . 4 1.1.2 Navier-Stokes Equation 5 1.1.3 Equations of State 8 1.2 Relevant Scales . 11 1.3 Kinetic Theory 15 1.3.1 Introduction 15 1.3.2 The Distribution Function and Its Moments 16 1.3.3 The Equilibrium Distribution Function . 19 1.3.4 The Boltzmann Equation and the Collision Operator 21 1.3.5 Macroscopic Conservation Equations. 23 1.3.6 Boltzmann’sH-Theorem 27 References 29 2 Numerical Methods for Fluids 31 2.1 Conventional Navier-Stokes Solvers 32 2.1.1 Finite Difference Method 34 2.1.2 Finite Volume Method . 38 2.1.3 Finite Element Methods . 41 2.2 Particle-Based Solvers . 42 2.2.1 Molecular Dynamics . 42 2.2.2 Lattice Gas Models 43 2.2.3 Dissipative Particle Dynamics. 47 2.2.4 Multi-particle Collision Dynamics 48 2.2.5 Direct Simulation Monte Carlo 51 2.2.6 Smoothed-Particle Hydrodynamics . 52 2.3 Summary 53 2.4 Outlook: Why Lattice Boltzmann? 54 References 56 ix x Contents Part II Lattice Boltzmann Fundamentals 3 The Lattice Boltzmann Equation . 61 3.1 Introduction. 61 3.2 The Lattice Boltzmann Equation in a Nutshell . 62 3.2.1 Overview . 63 3.2.2 The Time Step: Collision and Streaming . 65 3.3 Implementation of the Lattice Boltzmann Method in a Nutshell 66 3.3.1 Initialisation 67 3.3.2 Time Step Algorithm 67 3.3.3 Notes on Memory Layout and Coding Hints . 68 3.4 Discretisation in Velocity Space . 70 3.4.1 Non-dimensionalisation . 71 3.4.2 Conservation Laws. 73 3.4.3 Hermite Polynomials 74 3.4.4 Hermite Series Expansion of the Equilibrium Distribution . 77 3.4.5 Discretisation of the Equilibrium Distribution Function 80 3.4.6 Discretisation of the Particle Distribution Function 82 3.4.7 Velocity Sets 84 3.5 Discretisation in Space and Time 94 3.5.1 Method of Characteristics . 94 3.5.2 First- and Second-Order Discretisation . 96 3.5.3 BGK Collision Operator . 98 3.5.4 Streaming and Collision . 101 References 103 4 Analysis of the Lattice Boltzmann Equation 105 4.1 The Chapman-Enskog Analysis . 106 4.1.1 The Perturbation Expansion . 106 4.1.2 Taylor Expansion, Perturbation, and Separation . 108 4.1.3 Moments and Recombination . 109 4.1.4 Macroscopic Equations 111 4.2 Discussion of the Chapman-Enskog Analysis 113 4.2.1 Dependence of Velocity Moments 113 4.2.2 The Time Scale Interpretation . 114 4.2.3 Chapman-Enskog Analysis for Steady Flow . 116 4.2.4 The Explicit Distribution Perturbation 118 4.2.5 Alternative Multi-scale Methods 119 4.3 Alternative Equilibrium Models . 120 4.3.1 Linear Fluid Flow 121 4.3.2 Incompressible Flow . 123 4.3.3 Alternative Equations of State . 124 4.3.4 Other Models . 126 Contents xi 4.4 Stability . 127 4.4.1 Stability Analysis 128 4.4.2 BGK Stability 130 4.4.3 Stability for Advanced Collision Operators 133 4.4.4 Stability Guidelines 134 4.5 Accuracy 136 4.5.1 Formal Order of Accuracy. 136 4.5.2 Accuracy Measure . 138 4.5.3 Numerical Errors . 139 4.5.4 Modelling Errors . 143 4.5.5 Lattice Boltzmann Accuracy 145 4.5.6 Accuracy Guidelines . 146 4.6 Summary 149 References 150 5 Boundary and Initial Conditions . 153 5.1 Boundary and Initial Conditions in LBM in a Nutshell 154 5.1.1 Boundary Conditions 155 5.1.2 Initial Conditions. 156 5.2 Fundamentals . 157 5.2.1 Concepts in Continuum Fluid Dynamics . 157 5.2.2 Initial Conditions in Discrete Numerical Methods . 159 5.2.3 Boundary Conditions in Discrete Numerical Methods .
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