Table Of Content1122337755__99778899881111224400551155__TTPP..iinndddd 11 1122//1100//2211 66::2222 PPMM
B1948 Governing Asia
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World Scientific
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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
and
Maruzen Publishing Co., Ltd.
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Chiyoda-ku, Tokyo 101-0051, Japan
Library of Congress Control Number: 2021050556
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
AN INTRODUCTION TO THE LATTICE BOLTZMANN METHOD
A Numerical Method for Complex Boundary and Moving Boundary Flows
Copyright © 2022 Takaji INAMURO, Masato YOSHINO and Kosuke SUZUKI
All rights reserved. No part of this book may be reproduced in any form or be any electronic or
technical means, including information storage and retrieval systems, without permission in writing
from publisher, except by reviewers, who quote brief passages in a review.
ISBN 978-981-124-051-5 (hardcover)
ISBN 978-981-124-052-2 (ebook for institutions)
ISBN 978-981-124-053-9 (ebook for individuals)
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/12375#t=suppl
Printed in Singapore
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Preface
This book presents the fundamentals and applications of the lattice Boltz-
mann method, one of the numerical methods for incompressible viscous
fluid flows.
Oneofthefeaturesofthenumericalmethodsforincompressibleviscous
fluid flows is how to find the pressure that satisfies the continuity equa-
tion because there is no time evolution equation for finding the pressure.
Looking at the history of this topic, numerical methods for incompress-
ible viscous fluid flows have been developed since the 1960s, as represented
by the MAC method, and the basic scheme was almost completed in the
1980s. In the basic scheme, the pressure satisfying the continuity equa-
tion is obtained by solving the Poisson equation of pressure at each time.
Finding this solution consumes most of the calculation time. An alterna-
tive method, also proposed in the 1960s, is the artificial compressibility
method. Instead of solving the Poisson equation of pressure, this method
adds a derivative term of pressure by pseudo-time to the continuity equa-
tion. Thebasicschemeofthismethodincludingunsteadyflowcalculations
was almost completed in the early 1990s.
Since the 1990s, the lattice Boltzmann method (hereafter referred to
as the LBM) has been proposed as a numerical method for incompressible
viscous fluid flows. In the LBM, a modeled fluid, which is composed of
identical fictitious particles whose velocities are restricted to a finite set
of vectors (called a lattice gas model), is considered. The streaming and
collision processes of each particle are computed with a particle velocity
distribution function, and the macroscopic quantities such as mass density
andmomentumdensityareobtainedfromthemomentsofthevelocitydis-
tribution function. In other words, the LBM is a numerical method that
uses an analogy to the kinetic theory of gases to study fluid flows from
v
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vi An Introduction to the Lattice Boltzmann Method
a microscopic standpoint. To understand the LBM, researchers therefore
require some knowledge of the kinetic theory of gases, that is, the Boltz-
mann equation. For this reason (among others), the LBM is regarded as a
particular numerical scheme that is difficult to understand for researchers
familiar with the Navier–Stokes equations. However, once the basic ideas
are grasped, the LBM is a very simple and efficient numerical scheme. In
particular, it removes the need for solving the Poisson equation of pressure
and has a simple algorithm for complicated flow fields. It also has good
conservation of mass and momentum and is suitable for parallel computa-
tion. Thus far, the LBM has been successfully applied to flow fields with
complex boundaries such as flows in porous media, and to flow fields with
complicated interfacial changes such as moving boundary flows and gas–
liquid two-phase flows. Unfortunately, owing to rapid development, meth-
ods are commonly applied without understanding the basic ideas, leading
to misevaluations of the LBM.
This book gives the fundamentals and applications of the LBM. Most
of the contents are based on the authors’ results of previous studies. The
first author has also written several reviews of the LBM [61,65,72,73,75],
butthisbookiswrittenforeasyunderstandingofboththereviewsandthe
latest research results. The book is organized as follows. In Chap. 1, we
formulatetheLBMforincompressibleviscousfluidflowsandclarifythere-
lationshipbetweentheLBMandtheNavier–Stokesequationsinanasymp-
totic analysis. We also present a numerical example of complex boundary
flows. Chapter 2 introduces the lattice kinetic scheme developed from the
LBM.Chapter3describestheimmersedboundaryLBMapplicabletomov-
ing boundary flows. Chapter 4 describes the two-phase LBM applicable to
gas–liquid and liquid–liquid two-phase fluid flows. Regarding a two-phase
interfaceasaboundary, atwo-phasefluidflowisoneofthemovingbound-
ary flows. To assist readers, the contents of each chapter are also sum-
marized in a section on computational algorithms. Appendix F contains
the download URL of simple program examples (source codes) and their
explanations. Referring to these, readers are encouraged to program the
computationalalgorithmsandcomputeasimpleexampletoappreciatethe
advantages of the LBM. We hope to inspire interest and further develop-
mentoftheLBMinfuturestudies. Wewouldappreciatereadersusingthis
book as a guide to the LBM.
NotethatgoodbooksandreviewsontheLBMhavebeenpublishedelse-
where[2,15,114,130,154]. TheJapaneseversionofthisbookwaspublished
by Maruzen Publishing in January 2020.
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Preface vii
Finally,wesincerelythankthepeopleofMaruzenPublishingandWorld
Scientific Publishing who engaged in the publishing of this book. We also
thank Crimson Interactive Pvt. Ltd. for the English proofreading of the
manuscript.
October, 2021
Authors
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Contents
Preface v
1. Lattice Boltzmann Method (LBM) 1
1.1 Lattice Gas Model . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Basic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Formulation of the Collision Term. . . . . . . . . . . . . . . 5
1.4 Local Equilibrium Distribution Function . . . . . . . . . . . 7
1.5 Derivation of the Fluid Dynamic Equations . . . . . . . . . 8
1.6 LBM with an External Force Term . . . . . . . . . . . . . . 15
1.7 Incompressible Local Equilibrium Distribution Function . 17
1.8 Flow with Heat Transfer . . . . . . . . . . . . . . . . . . . . 17
1.9 Equation of an Acoustic Field . . . . . . . . . . . . . . . . . 19
1.10 LBM-derived Method . . . . . . . . . . . . . . . . . . . . . . 20
1.11 Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . 20
1.12 Computational Algorithm of LBM . . . . . . . . . . . . . . 26
1.13 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 27
1.13.1 Flows in a porous structure . . . . . . . . . . . . . . 27
1.13.2 Rayleigh–B´enard convection . . . . . . . . . . . . . 29
2. Lattice Kinetic Scheme 33
2.1 LKS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 LWACM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Improved LKS . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Bulk Viscosity Coefficient . . . . . . . . . . . . . . . . . . . . 39
2.5 Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 Computational Algorithm of Improved LKS . . . . . . . . . 41
ix
