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Jian Guo Zhou Lattice Boltzmann Methods for Shallow Water Flows

Springer-Verlag Berlin Heidelberg GmbH

Jian Guo Zhou Lattice Boltzmann Methods for Shallow Water Flows with 50 Figures Springer

DR. JIAN GUO ZHOU 7 Cherry Lane SALE M334NF United Kingdom Email: jgzhou77@yahoo.co.uk ISBN 978-3-642-07393-9 DOI 10.1007/978-3-662-08276-8 Cataloging-in-Publication Data applied for ISBN 978-3-662-08276-8 (eBook) Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at . This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights oC translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this _publication or parts thereof is permitted only under the provisions of the German Copyright Law oCSeptember 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law. http://www.springer.de © Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of the hardcover I st edition 2004 The use of general descriptive names, registered names, trademarks, 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. Product liability: The publishers cannot guarantee the accuracy of any information about the application of operative techniques and medications contained in this book. In every individual case tlie user must dieck such information by consulting the relevant literature. Camera ready by authors Cover design: E. Kirchner, Heidelberg Printed on acid-free paper 32/3141fas 54 3 2 1 0

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Preface The lattice Boltzmann method (LBM) is a modern numerical technique, very efficient, flexible to simulate different flows within complex/varying geome tries. It is evolved from the lattice gas automata (LGA) in order to overcome the difficulties with the LGA. The core equation in the LBM turns out to be a special discrete form of the continuum Boltzmann equation, leading it to be self-explanatory in statistical physics. The method describes the micro scopic picture of particles movement in an extremely simplified way, and on the macroscopic level it gives a correct average description of a fluid. The av eraged particle velocities behave in time and space just as the flow velocities in a physical fluid, showing a direct link between discrete microscopic and continuum macroscopic phenomena. In contrast to the traditional computational fluid dynamics (CFD) based on a direct solution of flow equations, the lattice Boltzmann method provides an indirect way for solution of the flow equations. The method is characterized by simple calculation, parallel process and easy implementation of boundary conditions. It is these features that make the lattice Boltzmann method a very promising computational method in different areas. In recent years, it receives extensive attentions and becomes a very potential research area in computational fluid dynamics. However, most published books are limited to the lattice Boltzmann methods for the Navier-Stokes equations. On the other hand, shallow water flows exist in many practical situations such as tidal flows, waves, open channel flows and dam-break flows. The ba sic feature of the flows is that the vertical effect can be neglected compared with the horizontal one with a good approximation. This allows a considerable simplification in the mathematical formulation by replacing the vertical mo mentum equation with the hydrostatic pressure distribution. As a result, such flows are usually described with the shallow water equations. A numerical so lution of the shallow water equations turns out to be a very successful tool in studying a wide range of flow problems occurring in ocean, environmental and hydraulic engineering, for instance, tidal flows in estuary and coastal regions, river, reservoir and open channel flows. In literature, there are many compu-

VIII Preface tational methods available for solutions of the shallow water equations such as finite difference method, finite volume method, finite element method and Godunov-type method. Usually, a special treatment is required in these nu merical procedures for either convective term, depth computations or source terms. All of these methods are developed on the basis of direct solutions to the shallow water equations. Since the lattice Boltzmann method is a modern numerical technique, it is necessary and natural to investigate how to use the method for solving the shallow water equations. The author studied the problem and developed a well-defined lattice Boltzmann model for shallow water flows with or without flow turbulence, namely LABS WE and LABSWE™. It follows out that the lattice Boltzmann method is simple, efficient and accurate for solution of the shallow water equations. Therefore, it is timely to write a book in order to introduce this elegant method into research field, educational area, engineering sector and consultancy organisations so that the method may be used to solve real life flow problems efficiently and accurately. This book may be used as a research reference for scientist, a practical method for engineers and consultancy organisations, and a text book for both undergraduate and postgraduate students. Peterborough, June 2003 Jian Guo Zhou

Contents 1 2 Introduction............................................... 1 1.1 Outline of the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.1 Cellular Automata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Lattice Gas Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.3 Lattice Boltzmann Method . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Shallow Water Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 General Flow Equations.................................. 7 2.3 Subgrid-Scale Stress Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Shallow Water Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Various Numerical Methods.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Lattice Boltzmann Method ................................ 19 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Lattice Boltzmann Equation .............................. 19 3.3 Lattice Pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Local Equilibrium Distribution Function . . . . . . . . . . . . . . . . . . . 23 3.5 Macroscopic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.6 Recovery of the Shallow Water Equations. . . . . . . . . . . . . . . . . . . 27 3. 7 Stability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.8 Relation to Continuum Boltzmann Equation . . . . . . . . . . . . . . . . 30 3.9 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.9.1 Two Variants of the LBE. . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.9.2 Solution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.9.3 Choice of the LBE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.10 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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