Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems 9789819912636, 9789819912643

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Mengtao Han · Ryozo Ooka

Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems

Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems

Mengtao Han · Ryozo Ooka

Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems

Mengtao Han School of Architecture and Urban Planning Huazhong University of Science and Technology Wuhan, Hubei, China

Ryozo Ooka Institute of Industrial Science The University of Tokyo Tokyo, Japan

ISBN 978-981-99-1263-6 ISBN 978-981-99-1264-3 (eBook) https://doi.org/10.1007/978-981-99-1264-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Since the lattice Boltzmann method was introduced in the 1980s, researchers have become increasingly interested in it. For more than 30 years after development, it gained a robust theoretical physical basis. The simplicity of the algorithm, ease of processing complex geometries, and scalability in parallel computation have attracted researchers globally. The development of this method has begun to play a role in many fields, including building environments. Currently, many scholars have made achievements in applying the lattice Boltzmann method to built environments. However, they have been scattered across various excellent studies. As researchers in the field of architecture environment, we entered the field of the lattice Boltzmann method several years ago and applied it to our doctoral research on built environment simulation. We hope to provide a solution to the lattice Boltzmann method applied to the built environment for scientific and engineering personnel based on our own experience, understanding, and collection of past results. We hope to present a simple, complete, and practical book to readers in the field of built environments. This is also what we were searching for when we began our research. The readers of this book are researchers and engineering technicians in the fields of architecture, building technology science, urban planning, HVAC, built environment engineering, and civil engineering. We hope this book will serve as a reference theory book and a technical guide for research and engineering practice using the lattice Boltzmann method to simulate and solve built environment problems. It can also be used as a reference textbook for teachers, students, and engineering technicians to study this exciting method and conduct architectural and urban wind environment simulations. Wuhan, China Tokyo, Japan

Mengtao Han Ryozo Ooka

v

Acknowledgments

In writing this book and related research, many people have provided selfless help, and we would like to express our gratitude to them. First, we thank Associate Professor Hideki Kikumoto at the Institute of Industrial Science at the University of Tokyo (I.I.S.), who provided us with many valuable suggestions since the initial stage of our study. His wisdom shines through many parts of this book. We would also thank Prof. Fujihiro Hamba, Prof. Shinichi Sakamoto at I.I.S., and Prof. Takeshi Ishihara at Tokyo University for their helpful comments on our research. Professor Hamba has put forward many constructive suggestions in Chap. 3 of this book. Personally, Mengtao would also like to thank all colleagues of the Ooka and Kikumoto Labs at I.I.S., particularly Dr. Keigo Nakajima, Dr. Chao Lin, Dr. Bingchao Zhang, and Dr. Hongyuan Jia. They provided various forms of help including suggestions, support, discussions, and encouragement. Mengtao also thanks Prof. Hong Chen at the School of Architecture and Urban Planning at the Huazhong University of Science and Technology for guiding him in the field of the built environment. Moreover, Mengtao also thanks Ms. Xu Wang, who provided selfless support throughout the entire research and writing process. Mengtao also thanks the Japan Society for the Promotion of Science (JSPS) and National Natural Science Foundation of China (NSFC). The former provided financial support for Mengtao’s doctoral and postdoctoral research period, while the latter supported the book’s final publication (NSFC Grant No. 52208059). The Fundamental Research Funds for the Central Universities HUST (Grant No. 2021XXJS053) also partially supported the publication of this book. Finally, we thank the editors in Springer for their support and patience.

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Contents

Part I

Fundamental Theory and Implementation of the Lattice Boltzmann Method

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Review of Navier–Stokes Equations (NSE)-Based Computational Fluid Dynamics (CFD) in the Built Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Continuity Assumption of Fluid . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Important Physical Quantities in CFD . . . . . . . . . . . . . . . . . . 1.2.4 Dimensional Analysis and Dimensionless Form of the NSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 NSE-Based CFD Simulation Methods . . . . . . . . . . . . . . . . . 1.3 Development of the Lattice Boltzmann Method (LBM) . . . . . . . . . 1.3.1 Lattice Gas Automata (LGA) . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 From LGA to LBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 LBM Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 LBM Application in the Built Environment . . . . . . . . . . . . . 1.4 Purpose and Outline of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fundamental Theory of the Lattice Boltzmann Method . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fluid From a Mesoscopic Perspective . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Equilibrium Distributions and Maxwell–Boltzmann Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3

5 5 6 6 8 9 9 10 10 11 12 13 15 16 21 21 21 21 23

ix

x

Contents

2.3

Lattice Boltzmann Equation (LBE) . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Stream and Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 LBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Discrete Velocity Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Discrete Velocity and DdQq Schemes . . . . . . . . . . . . . . . . . . 2.4.2 D2Q9 Scheme for 2D Problems . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 D3Q19 and D3Q27 Schemes for 3D Problems . . . . . . . . . . 2.4.4 Equilibrium Distribution Function in the DdQq Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Collision Function: Relaxation Time Scheme . . . . . . . . . . . . . . . . . . 2.5.1 Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Single-Relaxation Time (SRT) Scheme and the BGK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Multi-Relaxation Time (MRT) Scheme . . . . . . . . . . . . . . . . . 2.5.4 Two-Relaxation Time (TRT) Scheme . . . . . . . . . . . . . . . . . . 2.5.5 Other Advanced Collision Function Schemes . . . . . . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26 26 27 27 28 30 30

3 Implementation of the Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 First-Type BC: The Dirichlet BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Second-Type BC: The Neumann BC . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Periodic BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Symmetric BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Free-Slip BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Straight Solid-Wall BC: The Bounce-Back Model . . . . . . . . . . . . . . 3.7.1 Fullway Bounce-Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Halfway Bounce-Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Fullway Versus Halfway Bounce-Back . . . . . . . . . . . . . . . . . 3.8 Bounce-Back Improvement: The Wall-Function Bounce (WFB) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 WFB Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Shear Drag τw Calculation: Spalding’s Law . . . . . . . . . . . . . 3.9 Bounce-Back for Curved-Wall BC: The BouzidiFirdaouss-Lallemand (BFL) Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Extrapolation Method for Curved-Wall BC: The Guo Scheme . . . . 3.11 Other Solid-Wall BCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 58 59 60 62 63 64 66 67

4 From the Lattice Boltzmann Equation to Fluid Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Taylor Expansion of the LBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Chapman-Enskog Multi-scale Analysis . . . . . . . . . . . . . . . . . . . . . . .

32 35 35 36 37 50 52 52 53

68 69 70 72 74 75 76 77 81 81 82 82

Contents

From Mesoscopic Temporal-Spatial Scale to Macroscopic Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Definition of Macroscopic Quantities and the Equilibrium Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Derivation of the Continuity Equation (Mass Conservation) . . . . . 4.7 Derivation of the NSE (Momentum Conservation) . . . . . . . . . . . . . 4.8 Detailed Mathematical Operations in the Derivation Process . . . . . 4.8.1 Derivation of (4.21) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Derivation of (4.41)–(4.44) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Derivation of (4.22) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

4.4

5 Turbulence Models and LBM-Based Large-Eddy Simulation (LBM-LES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 LES Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 LES for the NSE-Based Method . . . . . . . . . . . . . . . . . . . . . . 5.2.2 LES for LBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Smagorinsky SGS Model and Its Development . . . . . . . . . . . . . . . . 5.3.1 Smagorinsky-Lilly SGS Model . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Dynamic Smagorinsky SGS Model (DSM) . . . . . . . . . . . . . 5.4 Advanced SGS Models in LBM-LES for Built Environment Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Wall-Adapting Local Eddy-Viscosity (WALE) SGS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Coherent Structure SGS Model (CSM) . . . . . . . . . . . . . . . . . 5.5 LBM-LES Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 From LBE to LBM: Using the LBM to Solve Built Environment Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Discretization and Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Normalization of Physical Quantities . . . . . . . . . . . . . . . . . . 6.3 LBM Simulation Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Common User-Induced Simulation Errors in the LBM . . . . . . . . . . 6.4.1 Grid Discretization Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Compressibility Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Over-Relaxation and Numerical Oscillations . . . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 86 87 89 89 90 97 99 99 101 101 103 103 104 104 105 107 108 108 109 110 111 115 115 116 116 116 118 119 122 122 124 125 126 127

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Part II

Contents

Practice of LBM-LES in Built Environment

7 LBM-LES in Ideal 3D Lid-Driven Cavity Flow Problems . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Description of the Ideal 3D Lid-Driven Cavity Flow . . . . . . . . . . . . 7.3 Simulation Methodology and Boundary Conditions . . . . . . . . . . . . 7.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Instantaneous and Time-Averaged Velocities . . . . . . . . . . . . 7.4.2 Comparison Between LBM-LES and FVM-LES . . . . . . . . 7.4.3 Comparison Between Vortex Structures . . . . . . . . . . . . . . . . 7.5 Discussion on Computational Time and Parallel Computational Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 131 133 133 133 136 140

8 LBM-LES in an Isothermal Indoor Flow Problem . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Isothermal Indoor Flow Problem Description . . . . . . . . . . . . . . . . . . 8.3 Simulation Methodology and Boundary Conditions . . . . . . . . . . . . 8.3.1 Simulation Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Parameters to Discuss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Instantaneous and Time-Averaged Scalar Velocities . . . . . . 8.4.2 Effects According to Grid Resolution . . . . . . . . . . . . . . . . . . 8.4.3 Effects According to Relaxation Time and Discrete Velocity Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Effects According to the Discrete Time Interval . . . . . . . . . 8.4.5 Discussion on Compressibility Errors . . . . . . . . . . . . . . . . . . 8.4.6 Discussion on Oscillations Caused by Over-Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Comparison Between LBM-LES and FVM-LES . . . . . . . . . . . . . . . 8.6 Discussion on Computational Performance . . . . . . . . . . . . . . . . . . . . 8.6.1 Computational Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Parallel Computational Performance . . . . . . . . . . . . . . . . . . . 8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 146 148 148 149 150 150 151

9 LBM-LES in the Outdoor Wind Environment Problem Around a Single Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction to the Outdoor Wind Environment Problem . . . . . . . . 9.2 Problem Description of Flow Around a Single Building . . . . . . . . . 9.2.1 Simulation Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Wind Tunnel Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Simulation Methodology and Boundary Conditions . . . . . . . . . . . . 9.3.1 Simulation Conditions and Parameter Settings . . . . . . . . . . 9.3.2 Inlet Boundary Data and Approaching Flow . . . . . . . . . . . .

141 142 143

155 156 160 163 165 166 167 168 169 170 173 173 174 174 175 175 176 180

Contents

9.3.3 Sampling Time Convergence Criteria . . . . . . . . . . . . . . . . . . 9.3.4 Simulation Accuracy Evaluation Index . . . . . . . . . . . . . . . . . 9.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Instantaneous Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Time-Averaged Velocity and Flow Structure . . . . . . . . . . . . 9.4.3 Effect According to Grid Resolution . . . . . . . . . . . . . . . . . . . 9.4.4 Effect According to SGS Model . . . . . . . . . . . . . . . . . . . . . . . 9.4.5 Effect According to the Relaxation Time and Discrete Velocity Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.6 Effect According to the Solid Wall Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Comparison Between LBM-LES with FVM-LES in Terms of Predicted Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Discussion on Computational Time and Efficiency . . . . . . . . . . . . . 9.6.1 Computational Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Discussion on CTR and PCE . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

181 185 186 186 188 190 194 194 199 202 206 206 208 209 209

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

Nomenclature, Symbols, and Abbreviations

A+ b C CSM C DSM Ce Cm Cw Cs DTo e es ea (ea )i F fa f a∗ f a+ f a− fa eq

fa eq+ fa eq− fa f a(n) neq fa fs F CS F(e2 ) H k

A constant in the damping function for the Smagorinsky-Lilly SGS model (–) Building model width (m) Constant in the coherent structure SGS model (–) Constant in the dynamic Smagorinsky SGS model (–) Constant E in the WALE SGS model (–) Constant in the Smagorinsky-type SGS model (–) Constant W in the WALE SGS model (–) Smagorinsky constant (–) Tolerance deviation for hit rate (%) Particle speed magnitude in the LBM (–) Lattice sound speed in the LBM (–) Discrete velocity vector of the virtual particle in the a-direction (–) i-direction component of ea (–) External force (kg·m2 ·s–2 ) The distribution function of the virtual particle in the a-direction (–) Distribution function after collision step of f a (–) Symmetric part of f a in the TRT collision function (–) Antisymmetric part of f a in the TRT collision function (–) Corresponding symmetric distribution function of f a in a discrete velocity scheme, utilized in the TRT collision function (–) Local equilibrium function of f a (–) eq Symmetric part of f a in the TRT collision function (–) eq Antisymmetric part of f a in the TRT collision function (–) nth-order non-equilibrium part of f a (–) Non-equilibrium component of f a (–) Damping function for the Smagorinsky SGS model (–) Coherent structure function in the coherent structure SGS model (–) Maxwell-Boltzmann equilibrium distribution (s·m−1 ) Room or building model height (m) Arbitrary small quantity with the same order as the Knudsen number (–) xv

xvi

Nomenclature, Symbols, and Abbreviations

kB lt M

Boltzmann constant, = 1.38064852 × 10–23 (J·K–1 ) Turbulent length scale (m) Matrix structure used to transform the distribution functions into moments (–) Mass of a fluid particle, molecule, atom, ion, and so forth (kg) nth-order moment (–) Moment corresponding to quantity Φ (–) Number of measurement points (–) N-index proposed by Zuo et al. (–) Neglected high-order small quantities of nth-order on Φ (–) Spatial operator in the Smagorinsky SGS model (s–1 ) Local fluid pressure (Pa) Local momentum, P = mu (kg·m·s–2 ) Second invariant quantity (s–2 ) Number of discrete particle velocities (–) Wall surface distance coefficient in the BFL scheme and Guo scheme, i.e., the ratio of the wall surface distance from the fluid grid point to the grid length (–) Position vector of the distribution function (–) Strain-rate tensor (s–1 ) Relaxation coefficient diagonal matrix corresponding to M (–) Traceless symmetric part of the squared velocity gradient tensor (s–2 ) Absolute temperature (K) Specific time (s) Turbulent integral time scale (s) Sampling time (s) Physical time of flow motion (s) Time cost for simulation (s) Local fluid velocity vector (m·s–1 ) Magnitude of u, = |u| (m·s–1 ) Dimensionless velocity, = u/u τ (–) x-direction component of instantaneous velocities (m·s–1 ) x-direction component of time-averaged velocities (m·s–1 )

m mn mΦ N Nt O(Φ n ) OP p P Q q qw

r S S Sd T t t int t smp t flow t sim u u u+ ux ⟨u x ⟩ / ⟨ , ⟩ u x2 uτ ⟨uy ⟩ u /⟨y ⟩ u ,y2 uz ⟨u z ⟩

x-direction component of the standard deviation of fluctuating velocities (m·s–1 ) √ Friction velocity, = τw /ρ (m·s–1 ) y-direction component of instantaneous velocities (m·s–1 ) y-direction component of time-averaged velocities (m·s–1 ) y-direction component of the standard deviation of fluctuating velocities (m·s–1 ) z-direction component of instantaneous velocities (m·s–1 ) z-direction component of time-averaged velocities (m·s–1 )

Nomenclature, Symbols, and Abbreviations

/⟨

u ,z2

⟩

xvii

z-direction component of the standard deviation of fluctuating velocities

(m·s–1 ) Uunc (α) Statistical uncertainty of a% for the time-averaged velocity (m·s–1 ) W Vorticity tensor (s–1 ) wa Weight of ea (–) x Streamwise direction component of the spatial coordinate (m) XF Reattachment length of the flow behind the building (m) XR Reattachment length of the flow on the roof of the building (m) y Spanwise direction component of the spatial coordinate (m) y+ Dimensionless distance from the wall in the y-direction, = yu τ /ν (–) z Vertical direction component of the spatial coordinate (m) z+ Dimensionless distance from the wall in the z-direction, = zu τ /ν (–)

Greek Symbols α Δ δi j , δi jk δL δt Ω ρ τ τ+ τ− τ τw μ ν Λ ω κ σ Q

Power exponent in urban inlet boundary flow, indicating the degree of velocity reduction near the ground (–) Filter width in the LES SGS  model (m)  1, i = j 1, i = j = k Kronecker symbol, δi j = , δi jk = (–) 0, i /= j 0, else Discrete lattice length (m) Discrete time step (s) Collision function (–) Local fluid density (kg·m–3 ) Relaxation time in the Bhatnagar-Gross-Krook scheme (–) Relaxation time corresponding to the shear velocity in the TRT collision function (–) Free relaxation time in the TRT collision function (–) Stress tensor (kg·m–1 ·s–2 ) Shear drag on the wall (kg·m–1 ·s–2 ) Molecular dynamic viscosity (kg·m–1 ·s–1 ) Molecular kinematic viscosity, = μ/ρ (m2 ·s–1 ) Magic parameter in the TRT collision function (–) Relaxation coefficient, = 1/τ (–) Von Karman constant (–) Variance for statistical uncertainty Uunc (α) (–) Discrete density (kg·m–3 )

xviii

Nomenclature, Symbols, and Abbreviations

Superscript Φ LBU Φ PHU Φ bc Φt

Quantity Φ Quantity Φ Quantity Φ Quantity Φ

in the lattice Boltzmann unit (–) in the physical unit (–) on the boundaries (–) at time step t (–)

Subscript PEXP(i) PFVM(i) PLBM(i) Φ ij , Φ ijk Φ SGS Φ tot Φ ref

Quantities obtained experimentally at point i (–) Predicted quantities obtained via FVM-LES at point i (–) Predicted quantities obtained via LBM-LES at point i (–) ij-component or ijk-component of quantity Φ (–) SGS component of quantity Φ (–) Total component of quantity Φ (–) Quantity Φ for reference or normalization (–)

Acronyms, Abbreviations 1D, 2D, 3D BGK CC CFD Co CPU CRT CSM DSM EOS FAC2 FB FDM FVM FVM-LES GS HR Kn LB LBE LBM

One-, two-, and three-dimensional (–) Bhatnagar-Gross-Krook model (–) Correlation coefficient (–) Computational fluid dynamics (–) Courant number (–) Central processing unit (–) Computational time ratio (–) Coherent structure SGS model (–) Dynamic Smagorinsky SGS model (–) Equation of state (–) Fraction of data within a factor of two of observations (–) Fractional bias (–) Finite difference method (–) Finite volume method (–) Finite volume method-based large-eddy simulation (–) Grid scale (–) Hit rate (–) Knudsen number (–) Lattice Boltzmann (–) Lattice Boltzmann equation (–) Lattice Boltzmann method (–)

Nomenclature, Symbols, and Abbreviations

LBM-LES LES LGA Ma MB Mbps MD ME MFP MRT NMSE NSE PCE PCT PSD(s) q.v. RANS Re SCT SGS SI SRT TKE TRT URANS WALE WFB

Lattice Boltzmann method-based large-eddy simulation (–) Large-eddy simulation (–) Lattice gas automata (–) Mach number (–) Mean bias (–) Million bits per second (–) Mean difference (–) Mean error (–) Mean free path of molecules (m) Multi-relaxation time scheme (–) Normalized mean square error (–) Navier–Stokes equations (–) Parallel computational efficiency (–) Parallel computational time, computational time of a multi-core simulation (s) Power spectral density(s) (–) Refer to (from Latin quod vide) (–) Reynolds-averaged Navier–Stokes (–) Reynolds number (–) Serial computational time, computational time of a one-core simulation (s) Subgrid scale (–) International system of units (–) Single-relaxation time scheme (–) Turbulence kinetic energy (m2 ·s–2 ) Two-relaxation time scheme (–) Unsteady Reynolds-averaged Navier–Stokes (–) Wall-adapting local eddy-viscosity SGS model (–) Wall-function bounce boundary condition (–)

Symbols, Operators exp() M –1 ∇ ∇1 ()T · |Φ| ⟨Φ⟩ Φ˙ [[Φ]]

xix

Exponential function (–) Inverse of matrix M (–) ∂ Nabla operator,∇ = ∂∂x i + ∂∂y j + ∂z k (–) ∂ ∂ Nabla operator,∇1 = ∂ x r1 i + ∂ yr1 j + ∂ z∂r1 k (–) Transpose of a vector or matrix (–) Dot product of tensors or vectors (–) Magnitude of vector or tensor Φ (–) Time-averaged component of Φ (–) Dimensionless value of Φ (–) Spatial-averaged value of Φ (–)

xx

Nomenclature, Symbols, and Abbreviations

Φ GS component of Φ (–) Φ˜ Quantity Φ updated via model functions (e.g., the SGS model) (–) var(Φ) Variance of Φ (–) ※ Bold characters represent a vector or matrix form. Italicized characters represent variables.

Part I

Fundamental Theory and Implementation of the Lattice Boltzmann Method

Chapter 1

Introduction

1.1 Introduction Accompanied by industrialization and urbanization, human populations are suffering from increasingly severe urban environmental problems, such as urban heat islands and air pollution. These problems can manifest themselves on various scales, from urban to human scales (Murakami et al. 1999). In recent decades, with significant improvements in computer performance, computational fluid dynamics (CFD) analysis (Anderson 1995), partially replacing wind tunnel experiments (WTEs), has been employed in the prediction of urban environmental problems and assessment of their environmental impact (Mochida and Lun 2008; Tominaga et al. 2008a; Tominaga and Stathopoulos 2013). Many CFD simulation methods have been developed, aided by the continuous evolution of mathematical and physical tools, which apply different physical and mathematical models to handle various temporal-spatial scales. Figure 1.1 illustrates the various fluid scales and their corresponding CFD methods. At the macroscopic scale, the fluid particles (e.g., molecules, atoms, and ions) fill the entire vacuum so that the fluid is assumed to be a continuous medium. Fluid motion satisfies mass, momentum, and energy conservation and is described by equations such as Euler’s equation and the Navier–Stokes equations (NSE, including the mass, momentum, and energy conservation equations), among others. The CFD methods at this scale take a group of nonlinear partial differential Euler or NavierStokes equations and discretize them into various algebraic equations via discrete methods, and then solve them. Most of the existing simulation methods belong to this category, including the finite volume method (FVM), finite element method (FEM), and finite difference method (FDM). These methods are relatively mature and can be used not only to study the mechanism of physical problems but also to discover new physical phenomena. In the CFD analysis of the built environment, the primary approaches are NSE-based methods, with the FVM-based large-eddy simulation (FVM-LES) and FVM-based Reynolds-averaged Navier–Stokes (FVMRANS) methods being the most widely used (Tominaga et al. 2008b). In general, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Han and R. Ooka, Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems, https://doi.org/10.1007/978-981-99-1264-3_1

3

4

1 Introduction

Scale macroscopic

mesoscopic

Navier-Stokes equations: finite difference method, finite volume method, finite element method, etc.

Boltzmann equation, lattice Boltzmann equation: lattice Boltzmann method, etc.

Hamilton s Equation: microscopic

molecular dynamics, etc.

Fig. 1.1 Various fluid scales and the corresponding simulation methods

the FVM-LES model performs a relatively high-accuracy analysis compared with the FVM-RANS model and is becoming increasingly popular. However, its computational load is enormous (Ferziger et al. 2020), causing few reports on the implementation of FVM-LES to predict actual built environment engineering problems at this stage, and most of them are applied for research alone. At the mesoscopic and microscopic levels, fluids are no longer assumed to be a continuum. At the microscopic level, the fluid is composed of a massive number of discrete molecules whose motion characteristics are affected by intermolecular interaction forces and external forces. The macroscopic properties and laws of motion appear as microscopic irregular thermal motions. Therefore, a straightforward idea is to obtain the macroscopic law of fluid flow by tracing the motion of every molecule and then performing statistical averaging based on different laws. A typical method for this is molecular dynamics simulation (Alder and Wainwright 1957). However, molecular dynamics simulation is complex, computationally intensive, and has high memory requirements. Due to computational and memory limitations, the simulation spatial and time scales are limited. Therefore, this method is currently challenging to apply to the built environment because of its relatively large scale. At the mesoscopic level, the fluid is dispersed into a series of virtual fluid particles (parcels). These virtual particles are larger than at the molecular level but are macroscopically infinitely small, and their mass is much smaller than that in the FVM. Considering that the motion details of a single molecule do not affect the macroscopic fluid properties, we can let these fluid parcels undergo evolutionary calculations by constructing an evolutionary mechanism that satisfies specific physical laws. The lattice gas automata (LGA), lattice Boltzmann method (LBM), and Monte Carlo simulation are methods usually employed at the mesoscopic level. Among them, the LBM, which is derived from the LGA, has gathered increased attention over the past two decades. In the LBM, in addition to the fluid being dispersed into virtual

1.2 Review of Navier–Stokes Equations (NSE)-Based Computational Fluid …

5

fluid particles, the physical regions are also separated into a series of lattices, and the time is dispersed into a series of time steps. The equation describing the motion of a fluid particle in the LBM is the lattice Boltzmann equation (LBE). Compared with macroscopic and microscopic methods, the LBM is advantageous owing to its clear physical meaning, simple boundary condition processing, easy program implementation, and good parallel computing performance. Accordingly, it is considered a good prospect to address built environment problems.

1.2 Review of Navier–Stokes Equations (NSE)-Based Computational Fluid Dynamics (CFD) in the Built Environment We believe that the readers of this have general CFD and fluid mechanics experience and knowledge; nevertheless, in this section, we would like to briefly review the fundamental ideas of fluid mechanics and classic CFD methods (i.e., NSE-based methods), which will aid readers understanding the LBM by analogy. Given the purpose and length of this book, the scope of this review is limited to topics related to the built environment, particularly the isothermal wind environment, which is consistent with the topics covered in subsequent chapters.

1.2.1 Continuity Assumption of Fluid Fluid molecules are essentially free to move, and there are gaps between molecules. Nevertheless, from a macroscopic perspective (i.e., the spatial scale is much larger than the molecular diameter and intermolecular distances), one can view the fluid as a continuous substance (i.e., a continuum). The fluid moves under various external forces, such as pressure, gravity, and shear forces. Although the motion of molecules is complex and varied, these complexities manifest in some common features at the macroscopic level. It is necessary to understand these properties, including the fluid density, viscosity, momentum, and Reynolds number (q.v. Sect. 1.2.3), regardless of the perspective and method used to study the fluid. Some properties are also reflected in specific problems, such as the specific heat, Prandtl number, and surface tension. When a flow field is simulated using CFD, the temporal-spatial distribution of these physical properties in fluid motion is solved using their conservation laws (q.v. Sect. 1.2.2).

6

1 Introduction

1.2.2 Governing Equations In classic CFD methods, the governing equations of fluid flow consist of the laws of mass, momentum, and energy conservation. This book does not cover variations in temperature and heat; therefore, next, we only review the laws of mass (1.1) and momentum conservation (1.2). ∂ρ + ∇ · (ρu) = 0, ∂t

(1.1)

∂ρu + ∇ · (ρuu) = −∇ p + μ∇ 2 u + F. ∂t

(1.2)

Here, ρ is the density (or mass) of the fluid molecule, and ρu is its momentum. μ is the fluid dynamic viscosity, which we consider invariable during the simulation (or, at least, generally invariable in built environment issues). F is the external force. The so-called Navier–Stokes equation is represented by (1.2). Some scholars also refer to the three conservation equations collectively as the Navier–Stokes equations (NSE). In this book, the CFD method for solving the flow field using these three conservation equations is referred to as the NSE-based method. In the built environment, fluids usually move slowly (relative to the speed of sound); thus, fluids in built environment simulations are generally considered incompressible (i.e., ρ is constant). Therefore, the mass and momentum conservation equations can be rewritten as: ∇ · u = 0,

(1.3)

∂u F ∇p + ∇ · (uu) = − + ν∇ 2 u + . ∂t ρ ρ

(1.4)

The NSE-based method uses (1.3) and (1.4) to solve fluid issues in the built environment. However, one will find that the LBM does not actually achieve the form of (1.3) and (1.4) (at least not the standard LBM), but is equivalent to (1.1) and (1.2) to some extent, as shown to a certain extent in Chaps. 2 and 4. Therefore, the LBM actually solves incompressible fluid issues in the built environment with equations in compressible form, which we call a pseudo-compressible method.

1.2.3 Important Physical Quantities in CFD (1) Momentum Momentum is a fundamental concept in fluid mechanics. Momentum P is defined as P = mu, where m is the fluid mass and u is the fluid velocity. In conventional NSE-based methods, the focus is mainly on the macroscopic

1.2 Review of Navier–Stokes Equations (NSE)-Based Computational Fluid …

7

fluid velocity u, while the macroscopic fluid velocity u should be distinguished from the mesoscopic or microscopic particle velocity e in LBM. This issue is covered in more detail in Chap. 2. Mass and momentum conservations are central to both NSE-based methods and the LBM. Newton’s second law relates force F to momentum P as:

F = ma = m

dP du = . dt dt

(1.5)

(2) Viscosity Viscosity μ is a physicochemical fluid property that resists fluid movement due to resistance, such as internal friction. For example, the viscosity of water is smaller than that of oil. The viscosity of air, which is the fluid commonly found in built environments, obeys Newton’s law of friction, which relates shear stress τ to the velocity gradient as: τ =μ

du du dP = νρ =ν . dx dx dx

(1.6)

Here, μ is the dynamic viscosity, and ν is the kinematic viscosity, which is μ divided by the fluid density (i.e., ν = μ/ρ). Kinematic viscosity is more common in built environments because the air velocities are generally low and the air is treated as an incompressible fluid, the[ density ]of which can be eliminated. The dimension of kinematic viscosity is m2 · s−1 . (3) Reynolds number (Re) There are many critical dimensionless numbers in CFD that characterize essential flow properties. The most fundamental dimensionless number is the Reynolds number (Re), which physically expresses the ratio of the inertial force level to the viscous force level when the fluid moves. Re is defined as: Re =

Lu . ν

(1.7)

Here, L is the characteristic length in the flow field (e.g., the length of the building), and u is the characteristic velocity of the flow (e.g., the inflow velocity at the room window or the time-averaged velocity at a specific height in the urban and standard atmospheric pressure environment). ν of air at room [ temperature ] is approximately 1.5 × 10−5 m2 · s−1 .

8

1 Introduction

1.2.4 Dimensional Analysis and Dimensionless Form of the NSE Dimensional analysis is a useful tool in fluid mechanics. It is a “compression technique” for reducing the number and complexity of experimental variables that affect a given physical phenomenon. In other words, dimensional analysis eliminates physical variables that depend on other variables, leaving as few independent variables as possible to simplify the problem. Similar to the unit system of physics, all physical quantities units can be reduced to a combination of seven fundamental physical units. Although its main purpose is to reduce the number and complexity of variables, dimensional analysis has several other benefits. First, it can help save time and reduce costs associated with wind tunnel experiments (WTE) and CFD calculations. More importantly, dimensional analysis provides similarity laws (q.v. Sect. 1.2.5), which can transform data from inexpensive small models to real information for expensive large prototypes. It can even make the comparison of specific problems possible when a direct comparison is difficult or impossible. More information on dimensional analysis can be found in related works on fluid mechanics (White 2015). In fluid mechanics, only four fundamental dimensions are needed to make most problems dimensionless (non-dimensionalization), namely mass M, length L, time T, and temperature Q, which form the MLTQ system for short. Alternatively, the FLTQ system is used with force F instead of M (White 2015). The laws of mass, momentum, and energy conservation can be dimensionless through the MLTQ system. For example, the common (1.3) and (1.4) in non-external force form in the built environment can be dimensionless as follows: ∇˙ · u˙ = 0, ( ) ∇˙ p˙ νTref ∂ u˙ ˙ + ∇˙ · u˙ u˙ = − + 2 ∇˙ 2 u, ∂ t˙ ρ˙ L ref with Tref t , t˙ = , ∇˙ = L ref ∇, u˙ = u L ref Tref p˙ = p

2 L ref Tref L3 , ρ˙ = ρ ref . Mref Mref

(1.8)

Here, Mref , L ref , and Tref are the referenced mass, length, and time for nondimensionalization, respectively. It is not difficult to see that (1.8) and the original NSE are almost identical in form. The only difference is that ν in the original NSE ref , which is just the reciprocal of Re. Therefore, when Re is consistent, becomes νT L 2ref (1.8) and the original NSE can obtain similar results, which is a part of the similarity (q.v. Sect. 1.2.5).

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9

1.2.5 Similarity Similarity is a fundamental principle in fluid mechanics. It is often challenging to ensure that the research object is precisely the same as the prototype in WTE and CFD. For example, we cannot build an identical high-rise building to conduct surface wind pressure measurements in a wind tunnel. Sometimes, it is also not feasible to perform a full-scale wind environment simulation of a large city. Accordingly, researchers usually scale the research object to a certain extent based on the similarity principle, which is convenient for research while ensuring that the fluid motion properties remain unchanged. Similarity mainly includes geometric and physical similarity. Geometric similarity is the premise of ensuring that the flow field is similar. This similarity indicates that the flow space is geometrically similar; that is, the intersection angle of any corresponding line segment of the geometries (e.g., flow fields and buildings) is the same, and the length of any corresponding line segment maintains a certain proportion. It is also important to understand that the geometric model must be proportionally scaled based on the prototype. Physical similarity is manifested as physical field similarity, which is composed of a series of physical quantities. It generally includes kinematic and dynamic similarity. Kinematic similarity means that the velocities and accelerations at all corresponding points in the two flow fields are in the same direction and have the same ratio. That is, two flows with similar motions have geometrically similar streamlines and flow spectra. Dynamic similarity means that the various forces acting at the corresponding fluid positions in the two flow fields, such as gravity, pressure, and viscous force, have the same direction and magnitude ratios. This implies that the force polygons composed of the forces acting at the corresponding fluid positions are geometrically similar if the two flow fields are dynamically similar.

1.2.6 NSE-Based CFD Simulation Methods The NSEs are partial differential equations that need to be discretized first when used to solve the flow field. According to different discrete methods, NSE-based simulation methods can be roughly divided into three categories: FDM, FVM, and FEM. FDM is the longest existing and simplest differential method. FDM covers the flow field with a grid system and then approximates the differential equations using the function nodal values at each grid point. FDM transforms continuous partial differential equations into algebraic equations for each grid node, in which the variable values at the local grid and several adjacent grids are unknown (Ferziger et al. 2020). FDM is very effective on structured grids, where it is easy to obtain highorder accuracy. However, it suffers some shortcomings, including that extra attention

10

1 Introduction

should be paid to the conservation of physical quantities and has certain limitations for complex geometries. FVM is based on the integral form of the conservation equations. FVM divides the flow field into a finite number of control volumes (CVs), applying the conservation equations to each CV. FVM computes the variable value at the centroid of each CV and represents the flux at the interface of the CVs using the centroid variables of several adjacent CVs. FVM is suitable for grids with complex geometries and strictly guarantees conservation. Therefore, it is prevalent in CFD at present; however, its main disadvantage is that the high-order accuracy scheme is complex. FEM is somewhat similar to FVM in that it divides the flow field into discrete volumes or finite elements, which are usually unstructured. FEM multiplies the discretized algebraic equation by a weight function before applying it to each finite element, which is one of its distinguishing features. In the simplest FEM, the solution is approximated by a linear shape function within each element to ensure solution continuity across element boundaries (Ferziger et al. 2020). FEM can handle arbitrary geometries. It is relatively easy to analyze mathematically and can prove optimal for certain types of equations. Nevertheless, FEM also suffers from the disadvantages of using the unstructured grid method, namely the structure of the linearized equation matrix is not as good as that of structured grids, making finding an efficient solution difficult.

1.3 Development of the Lattice Boltzmann Method (LBM) 1.3.1 Lattice Gas Automata (LGA) The LBM originates from the lattice gas automata (LGA), which is a broader application of cellular automata (CA) in fluid mechanics. LGA is a mathematical model that discretizes time and space and then simulates this evolution process based on several simple local rules to solve specific problems, such as cell growth and fractal structures. LGA is a fluid simulation method based on 0- and 1-bit calculations. It treats fluids as hypothetical particles on a regular lattice system. Particles collide and migrate on the grid according to specific rules, and the macroscopic quantity is obtained via statistical averaging (Wolfram 1986). In the 1970s, Hardy et al. (Hardy et al. 1973, 1976) proposed the first LGA model, the Hardy-Pomeau-de Pazzis (HPP) model, which was the first fully discrete model. It discretizes the fluid into a series of particles and space-time onto a two-dimensional (2D) square grid system. These hypothetical particles continuously stream and collide in the square grid system to reproduce the fluid motion. However, the HPP model does not satisfy the isotropy condition and cannot recover the NSE due to the lack of symmetry of the lattice system. As a result, the HPP model lacked practicality for a long time.

1.3 Development of the Lattice Boltzmann Method (LBM)

11

Fig. 1.2 Grid systems of FHP and FCHC LGA model

In the 1980s, Frisch et al. (1986) proposed the Frisch-Hasslacher-Pomeau (FHP) model using a 2D regular hexagonal lattice system (Fig. 1.2a), which ensures sufficient symmetry. At the same time, D’Humières et al. (1986) proposed the facecentered hyper-cubic (FCHC) model using an FCHC lattice system (Fig. 1.2b), which also ensures sufficient symmetry. These two models can recover 2D and 3D incompressible NSE, respectively. The FHP and FCHC models have been the most widely used LGA models for a long time. In LGA, fluid particles exist on discrete grid points and migrate along prescribed routes. All particles collide and stream synchronously according to specific rules. Particle evolution only involves adjacent grids, leading to LGA’s good parallel computing ability. The LGA boundary processing is simple and can be applied to the simulation of complex geometric regions. Furthermore, LGA can be unconditionally stable because it utilizes 0- and 1-bit calculations. LGA has the advantages of clear physical meanings and calculation stability. Nonetheless, it also faces some shortcomings, such as statistical noise and unsatisfying Galilean invariance. Furthermore, the computational complexity is exponential. These LGA shortcomings greatly limit its application. It is not difficult to see that LGA contains the rudimentary idea of the LBM. To retain the advantages of the LGA method and overcome its shortcomings, the LBM was developed.

1.3.2 From LGA to LBM Inspired by LGA, the underlying ideology of the LBM was proposed at the end of the 1980s. In 1988, to eliminate the statistical noise of LGA, McNamara and Zanetti (McNamara and Zanetti 1988) first proposed to replace the Boolean variable (0 or 1) in LGA with the particle distribution function (continuous real number)

12

1 Introduction

after local statistical averaging. That is, utilizing the Boltzmann1 equation instead of LGA for evolution in the simulation. This is the earliest LBM model and has since opened up the LBM field. The LBM removes most of the noise while retaining many of the advantages of LGA. However, it uses multi-particle collisions, and the collision model is very complex when the number of particles increases, and it still has exponential complexity. In 1989, Higuera and Jimenez (1989) introduced the equilibrium distribution function, which reduces the collision function to a linear matrix. The model satisfies the conservation of mass and momentum and is easy to construct. This matrix model dramatically reduces the computational and storage requirements of the simulation, but other disadvantages of LGA still remain. From 1991 to 1992, Chen et al. (1991, 1992) and Qian et al. (1991, 1992) independently proposed the single-relaxation time (SRT) model, which used one relaxation time coefficient to control the speed of various particles approaching their respective equilibrium states, further simplifying the collision model. This model is also called the BGK model, derived from the model proposed by Bhatnagar, Gross, and Krook (BGK) (Bhatnagar et al. 1954) to simplify the collision integral term in the Boltzmann equation. SRT (or BGK) completely abandons the Fermi–Dirac equilibrium distribution and adopts the Maxwell–Boltzmann distribution for gas. The specific form of the equilibrium distribution is related to the dimension of the flow and the discrete velocity model. One can derive the NSE that satisfies Galilean invariance by choosing an appropriate equilibrium distribution. To date, the LBM has completely overcome the shortcomings of the LGA method and has gradually become more mature, becoming an independent and widely used simulation method.

1.3.3 LBM Development Since the 1990s, LBM has made significant progress, in theory, modeling, and engineering applications. Numerous studies have shown that LBE can be rigorously derived from the continuous Boltzmann equation with appropriate discretization (Abe 1997; He and Luo 1997). Regarding the collision function, which is the core of LBM, the two-relaxation time (TRT) model (Ginzburg 2005) and the multi-relaxation time (MRT) model (D’Humières et al. 2002) have been successively established successively in addition to SRT (BGK) model. These models have shown great vitality in numerous fields, including the built environment. In addition, other advanced collision models, such as the cumulant LBM model (Geier et al. 2015), have been developed for very high-Re flow fields such as urban turbulence (Lenz et al. 2019). 1

Ludwig Edward Boltzmann (1844–1906) was an Austrian physicist and philosopher. He was one of the founders of thermodynamics and statistical physics. He developed statistical mechanics to explain and predict the physical properties of matter (such as viscosity, heat conduction, and diffusion) through the properties of atoms (such as atomic weight, electric charge, and structure) and explained the second law of thermodynamics from a statistical perspective.

1.3 Development of the Lattice Boltzmann Method (LBM)

13

For convective heat transfer and chemical reaction systems, advection-diffusion (Guo et al. 2002; Peng et al. 2003), reaction-diffusion (Ponce Dawson et al. 2002), and combustion (Succi et al. 1997) models were established. For the non-standard regular lattice systems, non-standard LBE models have been developed, such as interpolation complement (Nannelli and Succi 1992), local mesh refinement (Filippova and Hänel 1998), finite difference (Mei and Shyy 1998), and finite volume (Chen 1998). In terms of boundary condition development, in addition to the traditional Dirichlet and Neumann boundaries, LBM has also developed rich boundary conditions such as symmetry, cycle, and free slip (Sukop and Thorne 2006; Krüger et al. 2017). Especially for solid walls, in addition to the standard bounce-back (Humieres and Lallemand 1987; Ziegler 1993; Ginzbourg and Adler 1994), boundaries such as interpolation, extrapolation, and immersed are developed for complex traits. Similar to traditional NSE-based methods, LBM-based LES (LBM-LES) (Dong and Sagaut 2008) and LBM-based RANS (LBM-RANS) (Sajjadi et al. 2017a) have also been developed. In the recent decade, LBM-LES was applied to a variety of complex turbulent problems, particularly in the built environment (Sajjadi et al. 2017a, 2017b, 2016; Crouse et al. 2002; Kuznik et al. 2010; Aidun et al. 2010; Habilomatis and Chaloulakou 2013; Salimi et al. 2015; Chang et al. 2013; Han et al. 2019a, 2020, 2021a, 2021b). After two or three decades of development, LBM has become a complete and independent simulation method that can be applied to solve many complex fluid problems, and it is still in progress.

1.3.4 LBM Application in the Built Environment Because of its excellent potential, LBM has increasingly attracted the attention of scholars in the built environment. Many related academic reports and applications have been reported in recent years. Indoor airflow is an important built environment field. Indoor airflow produces rich flow dynamics, such as separation, recirculation, and reattachment phenomena. These physical behaviors are very suitable for testing the accuracy of CFD methods in simulating ventilation systems. Over the past few decades, researchers have utilized the LBM to simulate indoor turbulent flow. As early as 2002, Crouse et al. (2002) simulated the thermal-flow field of an actual office building by implementing the LBM on a computer-aided design platform. Employing the 2D indoor turbulent flow model of IEA Annex 20 (Lemaire et al. 1993), Elhadidi and Khalifa (2013) compared the accuracy of LBM and FVM at various discrete time steps. Ito (2012) discussed the averaged velocity of indoor flow obtained via LBM-RANS with that obtained by integrating FVM-LES and argued that the results achieved via 3D LBM are consistent with the experimental results, while 2D LBM produces relative errors. Sajjadi et al. (2017a) introduced the LBM-RANS with the k-ε model to simulate indoor ventilation and particle transportation, demonstrating satisfactory accuracy. They also employed the LBM-LES coupled with the MRT scheme to simulate flow, particle diffusion, and

14

1 Introduction

accumulation; the results were in good agreement with experimental results (Sajjadi et al. 2016). Recently, Han et al. discussed the influence of various LBM parameters, such as grid resolution, collision function, discrete velocity model (Han et al. 2019a, 2018a), solid-wall boundary conditions (Han et al. 2021b), and compressible errors (Han 2022), on accuracy when LBM-LES simulates the indoor airflow environment. Outdoor and urban turbulent flow is another important built environment field. Currently, NSE-based methods are widely used to simulate flows in outdoor wind environment, especially the FVM-LES. Due to the complexity and large scale of the outdoor environment (from tens of meters to tens of kilometers), the FMV-LES grids are usually quite massive, leading to high simulation hardware and computational time costs. Therefore, the LBM-LES was considered an attractive alternative to simulate outdoor flows and began to gain attention in the 2010s. Using a self-developed LBM-LES code, Onodera et al. (2013) produced a large-scale LES instantaneous velocity field solution in a 10 × 10 km2 metropolitan area in Tokyo, Japan. Andre et al. (2014) compared the performance of FEM-LES and LBM-LES for simulating bluff bodies in the atmospheric boundary layer, evaluating metrics such as the averaged velocity, fluctuating velocity, aerodynamic drag, and lift coefficient. They argue that the time-averaged velocities obtained from the LBM are in better agreement with experimental data than those from FEM; however, the fluctuating velocity obtained from FEM is better than that from the LBM. Obrecht et al. (2015) used LBM-LES to simulate flows around one or several wall-mounted cubes using a multi-core graphics processing unit (GPU) platform, showing good agreement with experimental data within a reasonable computational time range. King et al. (2017) simulated the airflow in and around a cubic building in isolation and within an array of similar buildings using LBM-LES and FVM-LES. Their results show that LBM-LES can significantly reduce the computation time while maintaining acceptable accuracy levels for pressure, velocity, and ventilation rates. Ahmad et al. (2017) used LBMLES to simulate a pedestrian-level unsteady wind environment in built-up areas to investigate the intensity of wind gusts using a large computational domain to allow sufficient boundary layer development. Lenz et al. (2019) simulated the wind flow in a complex neighborhood-scale urban canopy in Basel, Switzerland, using cumulant LBM-LES on a GPU platform. They compared three different grid resolutions and indicated that the averaged velocity and turbulent kinetic energy profiles satisfy the acceptance criteria. Han et al. (2020, 2021b, 2018b, 2019b) systematically studied the average velocity, turbulent fluctuating velocity, turbulent structure, turbulent spectrum, and other turbulent information after using LBM-LES to simulate the wind environment around a single building. They verified the robustness of LBM-LES in simulating a high-Re turbulent environment. In addition, many achievements have been made using LBM-LES in indoor and outdoor thermal convection, pollutant diffusion, and other fields. These achievements are not listed here due to space limitations and because they are out of the scope of this book.

1.4 Purpose and Outline of This Book

15

1.4 Purpose and Outline of This Book This book presents a supplementary solution to the LBM-LES applied to the built environment for scientific and engineering personnel. The authors hope this book will provide relevant research and engineering personnel with (i) the fundamental theoretical knowledge and (ii) specific implementation methods for LBM-LES in the built environment. We mainly focus on airflow and wind in the built environment and do not address issues such as temperature variations, diffusion of substances and particles, and chemical changes. This book consists of two parts. The first part summarizes the basic LBM theory, while the second part introduces several practical LBM-LES applications in typical built environment problems. As mentioned, Part I introduces the fundamental LBM theory. The authors do not wish to discuss the underlying LBM theory from a physical or mathematical perspective. To enhance the guidance for engineering personnel related to the built environment, in Part I, the authors mainly introduce the basic principles from the perspective of engineering personnel, summarize the basic LBM theory in engineering simulations, and present some related basic LBM knowledge related to solving practical engineering problems. To facilitate practical application, the authors discuss the workflow and error analysis when using the LBM for actual engineering simulations. Part I consists of six chapters. Chapter 1, i.e., this chapter, briefly reviews traditional building environment simulation methods, the development process of LBM, and its achievements. Chapter 2 systematically introduces the system theory of LBM, from the most basic LBM ideas, to equations, to related discrete methods and models. Chapter 3 summarizes various boundary conditions used in LBM simulations, including their basic theoretical ideas and implementation methods. Chapter 4 mathematically derives the relationship between the LBE and NSE, further deepening the reader’s understanding of the LBM. Readers who are not interested in the derivation process or in the mathematical details can skip this chapter. Chapter 5 introduces the ideas and implementation method of LBM-LES. Finally, Chap. 6 introduces the process and precautions of applying the LBM to actual simulations and discusses the possible errors when conducting LBM-LES. Part II, which comprises three chapters (Chaps. 7, 8 and 9), presents three typical built environment cases: a simple cavity flow, indoor airflow, and an outdoor wind environment. These three cases cover essential problem prototypes in the built environment and serve as a simplified basis for many other issues. Each case explains how to use LBM-LES to simulate each particular problem. The boundary conditions and parameter settings, the influence of various parameter conditions on the simulation results, and a comparison with the traditional FVM-LES simulation results are discussed in detail. Furthermore, the simulation time and computational efficiency are also discussed. These cases serve as a simulation guide for built environment issues, providing readers with deep insight into using the LBM to simulate the built environment field.

16

1 Introduction

The potential readers of this book are researchers and engineering technicians in the fields of architecture, building technology science, urban planning, HVAC, built environment engineering, and civil engineering, among other related fields. This book hopes to serve as a theoretical reference and technical guide for their research and engineering practice using the LBM for simulating and solving built environment problems. It can also be used as a reference textbook for teachers, students, and engineering technicians to study the LBM and conduct architecture and urban wind environment simulations.

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Hertwig D, Efthimiou GC, Bartzis JG, Leitl B (2012) CFD-RANS model validation of turbulent flow in a semi-idealized urban canopy. J Wind Eng Ind Aerodyn 111:61–72. https://doi.org/10. 1016/j.jweia.2012.09.003 Higuera FJ, Jiménez J (1989) Boltzmann approach to lattice gas simulations. Europhysics Letters (EPL) 9:663–668. https://doi.org/10.1209/0295-5075/9/7/009 Humieres D, Lallemand P (1987) Numerical simulations of hydrodynamics with lattice gas automata in two dimensions. Complex Syst 1(1):599–632 Ito K (2012) 41300 Analysis of airflow and particle dispersion in indoor environment by the lattice Boltzmann method. Summ Tech Papers Annual Meet 2012:607–608 King M-F, Khan A, Delbosc N, Gough HL, Halios C, Barlow JF, Noakes CJ (2017) Modelling urban airflow and natural ventilation using a GPU-based lattice-Boltzmann method. Build Environ 125:273–284. https://doi.org/10.1016/j.buildenv.2017.08.048 Krüger T, Kusumaatmaja H, Kuzmin A, Shardt O, Silva G, Viggen EM (2017) The lattice Boltzmann method. Springer International Publishing, Cham Kuznik F, Obrecht C, Rusaouen G, Roux J-J (2010) LBM based flow simulation using GPU computing processor. Comput Math Appl 59:2380–2392. https://doi.org/10.1016/j.camwa. 2009.08.052 Lemaire AD, Chen Q, Ewert M, Heikkinen J, Inard C, Moser A, Nielsen PV, Whittle G (1993) Room air and contaminant flow, evaluation of computational methods, subtask-1 summary report. IEA Annex 20: air flow patterns within buildings 82 Lenz S, Schönherr M, Geier M, Krafczyk M, Pasquali A, Christen A, Giometto M (2019) Towards real-time simulation of turbulent air flow over a resolved urban canopy using the cumulant lattice Boltzmann method on a GPGPU. J Wind Eng Ind Aerodyn 189:151–162. https://doi.org/ 10.1016/j.jweia.2019.03.012 McNamara GR, Zanetti G (1988) Use of the boltzmann equation to simulate lattice-gas automata. Phys Rev Lett 61:2332–2335. https://doi.org/10.1103/PhysRevLett.61.2332 Mei R, Shyy W (1998) On the finite difference-based lattice Boltzmann method in curvilinear coordinates. J Comput Phys 143:426–448. https://doi.org/10.1006/jcph.1998.5984 Mochida A, Lun IYF (2008) Prediction of wind environment and thermal comfort at pedestrian level in urban area. J Wind Eng Ind Aerodyn 96:1498–1527. https://doi.org/10.1016/J.JWEIA. 2008.02.033 Murakami S, Ooka R, Mochida A, Yoshida S, Kim S (1999) CFD analysis of wind climate from human scale to urban scale. J Wind Eng Ind Aerodyn 81:57–81. https://doi.org/10.1016/S01676105(99)00009-4 Nannelli F, Succi S (1992) The lattice Boltzmann equation on irregular lattices. J Stat Phys 68:401– 407. https://doi.org/10.1007/BF01341755 Obrecht C, Kuznik F, Merlier L, Roux J-J, Tourancheau B (2015) Towards aeraulic simulations at urban scale using the lattice Boltzmann method. Environ Fluid Mech 15:753–770. https://doi. org/10.1007/s10652-014-9381-0 Onodera N, Aoki T, Shimokawabe T, Kobayashi H (2013) Large-scale LES wind simulation using lattice Boltzmann method for a 10 km× 10 km area in metropolitan Tokyo. TSUBAME e-Sci J Global Sci Inf Comput Center 9:1–8 Peng Y, Shu C, Chew YT (2003) Simplified thermal lattice Boltzmann model for incompressible thermal flows. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 68:8. https://doi. org/10.1103/PhysRevE.68.026701 Ponce Dawson S, Chen S, Doolen GD (2002) Lattice Boltzmann computations for reaction-diffusion equations. J Chem Phys 98:1514–1523. https://doi.org/10.1063/1.464316 Qian YH, D’Humières D, Lallemand P (1992) Lattice bgk models for Navier-Stokes equation. EPL 17:479–484. https://doi.org/10.1209/0295-5075/17/6/001 Richards PJ, Mallinson GD, McMillan D, Li YF (2002) Pedestrian level wind speeds in downtown Auckland. Int J Wind Struct 5:151–164. https://doi.org/10.12989/was.2002.5.2/3/4.151

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Chapter 2

Fundamental Theory of the Lattice Boltzmann Method

2.1 Introduction In this chapter, we introduce fundamental lattice Boltzmann method (LBM) concepts and its equation, the lattice Boltzmann equation (LBE). LBM theory is rich and constantly evolving; accordingly, it cannot be covered in its entirety in this book, as an in-depth explanation would require extensive and complex mathematical derivations. Given that most of the readers of this book come from an engineering background, especially researchers and practitioners in the built environment sciences, complicated mathematical derivations are not included in this chapter and are presented in Chap. 4. For a more reader-friendly discussion, this chapter focuses on introducing the LBM knowledge closely related to the simulation process from an engineering and physics perspective, excluding its mathematical logic and aspects less related to the simulation implementation.

2.2 Fluid From a Mesoscopic Perspective 2.2.1 Distribution Functions From a microscopic perspective, a continuous fluid consists of a massive number of discrete particles, e.g., atoms, molecules, and ions. Under standard conditions, even a volume as small as a cubic centimeter of air contains 2.7 × 1019 molecules. These particles are under continuous Brownian motion, and there is a complete vacuum between molecules.1 Theoretically, if we can track the motion of each particle, we 1

In classical physics, the space between molecules is a vacuum. However, in modern quantum mechanics, quantum fluctuations constantly exist in a vacuum, and an absolute vacuum does not

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Han and R. Ooka, Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems, https://doi.org/10.1007/978-981-99-1264-3_2

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22

2 Fundamental Theory of the Lattice Boltzmann Method

Approximation as virtual particles

Macroscopic continuum of fluid molecules, governed by the Navier-Stokes equation

Discretization

Mesoscopic fluid molecules within a grid are approximated as fluid particles, governed by the Boltzmann equation

Mesoscopic discrete virtual particles into distribution functions in various directions, governed by the lattice Boltzmann equation

Fig. 2.1 Evolution process from macroscopic continuous fluid molecules to mesoscopic distribution functions

can know exactly any fluid property and state. Unfortunately, because of the massive number of particles, tracing them one-by-one is practically impossible. Let us make a compromise: the number of fluid molecules within a certain space (e.g., a grid or cell) is approximated into a virtual particle from a mesoscopic perspective (Fig. 2.1). In this perspective, we reduce a countless number of fluid molecules in the entire space to countable virtual particles, each containing the information of a group of molecules. Then, we can replace the motion of all fluid molecules with this set of particles. This field is governed by the Boltzmann equation. Herein, we make the massive number of fluid molecules countable and traceable by introducing virtual particles. Nevertheless, the motion patterns of these particles are still varied: they can move in any direction and at any speed. Let us go a step further. We artificially prescribe several motion patterns (including direction and speed) and discretize all virtual particles into these motion patterns (Fig. 2.1). This operation reduces the motion patterns of particles to a level that can be easily handled via computational methods. Accordingly, virtual particles are discretized into the number of particles (or a proportion) in each motion pattern. Thus, we touch on the well-known distribution function concept. A distribution function is essentially the percentage of molecules whose velocity is within a specific range in the total number of molecules of the entire system at a given time according to the above discretization and, therefore, is a continuous real number. Many researchers also call them populations. At this point, we enter the field governed by the LBE. The distribution function concept appears in many disciplines with various interpretations. In LBM theory, it is created to make the particles and their motion patterns countable. The distribution function f (r, t). represents a set of fluid particles on a specific temporal-spatial scale, which is the subject operated by the LBE. f (r, t) can be regarded as a generalization of the particle count or probability at a specific position r at time t. exist. Nevertheless, this is not the focus of this book and has no impact on the LBM. Interested readers can refer to books and publications on modern quantum mechanics.

2.2 Fluid From a Mesoscopic Perspective

23

As previously stated, to integrate the countless motion patterns of fluid particles into a finite number of patterns, the particle velocities are also reduced to finite discrete velocities so that f (r, t) is substituted by f a (r, t). Here, a represents the finite discrete velocityirection. Subsequently, by assuming a fundamental virtual particle velocity ea , we can employ one fundamental variable, f a (r, t), to describe the information of both the particle amount and the velocity vectors. Once the distribution functions are determined, the macroscopic local fluid density ρ and velocity u can be obtained by simply aggregating all the discrete velocity directions of the local point. This is demonstrated in (2.1), which bridges the macroscopic physical quantities and mesoscopic virtual particles. ρ=

Σ

f a (r, t), ρu =

a

Σ

ea f a (r, t).

(2.1)

a

Here, ea is the fundamental virtual particle velocity. The local pressure p is determined based on the gas molecular kinetic theory for the LBM, as expressed in (2.2). p = ρes2 .

(2.2)

Here, es represents the speed of sound in the lattice Boltzmann unit (LBU). It should be noted that this pressure calculation is based on the equation of state (EOS) of the fluid and is completely independent of the velocity. This is a distinct advantage of the LBM, whereas the pressure solution in FVM is usually coupled with the velocity, such as in the popular SIMPLE and PISO algorithms. For further information, readers can refer to some classic CFD works (Versteeg and Malalasekera 2007; Ferziger et al. 2020).

2.2.2 Equilibrium Distributions and Maxwell–Boltzmann Distributions James Maxwell2 established the concept of statistical average because it was difficult to track the trajectory of each molecule in a macroscopic system. Maxwell believed that the velocity and position of a single fluid particle (i.e., molecule and atom) at a specific time was unimportant. Meanwhile, the distribution function is an essential parameter to describe molecular effects. When a fluid is in thermodynamic equilibrium, its molecules are distributed uniformly. As distribution functions are sets of 2

James Clerk Maxwell (1831–1879) was a British physicist and mathematician. He was the founder of classical electrodynamics and one of the founders of statistical physics. He was mainly engaged in the research of electromagnetic theory, molecular physics, statistical physics, optics, mechanics, and elasticity theory. In particular, his electromagnetic field theory, which unifies electricity, magnetism, and optics, is considered as the most brilliant achievement in physics in the nineteenth century and one of the greatest syntheses in the history of science.

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2 Fundamental Theory of the Lattice Boltzmann Method

fluid particles statistically based on their positions and velocities, the local equilibeq rium distribution functions f a should obey the thermodynamic equilibrium state established by Maxwell (Barrow 1996). This is the so-called Maxwell–Boltzmann equilibrium distribution, which is expressed as: ( ) F e2 =

(

m 2π kB T

) 23

) ( me2 . exp − 2kB T

(2.3)

Here, m and e (represent the molecular mass and speed (i.e., velocity magnitude), ) respectively. F e2 represents the distribution function of a molecule with speed e, and its units are the reverse of the velocity units, i.e., [s · m−1 ]. T represents the absolute temperature [ ] of the fluid, and k B is the Boltzmann constant, which is 1.38 × 10−23 J · K−1 . The Maxwell–Boltzmann equilibrium distribution indicates the proportion of fluid particles within a specified speed range in the equilibrium system. Some premise assumptions are necessary for this equilibrium distribution. One such assumption is that the three directional components of the molecule’s velocity are independent of each other, meaning that the probability of one component having a specific value is the same regardless of the value of the other components. Another assumption is that the fluid is isotropic, meaning its properties do not change according to direction. These assumptions are reasonable and generally adopted in CFD simulations, particularly in architectural wind engineering. Figure 2.2 illustrates the equilibrium distribution functions of oxygen, nitrogen, and hydrogen at T = 300 K. The Maxwell–Boltzmann distribution is a Gaussian distribution. Taking oxygen as an example, its distribution function is symmetric about the zero-speed line, regardless of the type of molecules. This means that at any speed level, in the same speed interval, the number of molecules with opposite speeds is the same, resulting in the average molecule speed being zero. A peak is observed in the middle of the curve, indicating that the speed of most molecules is close to zero, while fewer molecules have a higher speed. In the temperature toward certainty, the higher the molecular mass, the higher the peak value, indicating that the motion of molecules with more mass is less severe under similar temperature conditions. Similarly, when the gas temperature is constant, the higher the molecular mass is, the higher the peak speed value, and molecules with more mass move less violently. According to the Maxwell–Boltzmann distribution, we can obtain the three most common molecular speeds in statistical mechanics. The most probable speed e p is the molecular speed corresponding to the maximum distribution function value (not the maximum speed value). The arithmetic mean of the molecular speed is called square root of the average squared molecular speed is the the average speed e. The √ root-mean-square speed e2 . The positional relationship of these three molecular speeds on the distribution function curve is shown in Fig. 2.3.

2.2 Fluid From a Mesoscopic Perspective 0.0016 0.0014 0.0012

distribution function

Fig. 2.2 Maxwell–Boltzmann equilibrium distributions of oxygen, nitrogen, and hydrogen at T = 300 K

25 oxygen nitrogen hydrogen

0.001 0.0008 0.0006 0.0004 0.0002 0 -1500 -1000 -500 0 500 1000 1500 molecular speed [m·s-1]

Fig. 2.3 Positional relationship on the Maxwell–Boltzmann distribution curve of three commonly used molecular speeds

According to statistical mechanics theory and applying some mathematical skills, it is not difficult to derive the expressions for these three molecular speeds from the Maxwell-Boltzmann distribution, as shown in (2.4). Showing the full derivation process is not within the scope of this book, and readers can refer to works on statistical mechanics. / / /⟨ ⟩ / 3kB T 2kB T 8kB T . (2.4) , ⟨e⟩ = , e2 = ep = m m mπ From (2.4), we can obtain the following relationship: e p < ⟨e⟩
1/2, the solid barrier is closer to S, the time taken for the particle to reach W is greater than δt /2 and eventually bounces back to a position between A and S at δt . It is easy to see that the BFL boundary is identical to halfway bounce-back when qw = 1/2. When qw /= 1/2, the particles do not lie on the grid point after bounceback. In the BFL scheme, the unknown particle is solved via interpolation based on the distribution function at A and B and the particle’s position in relation to A and B. The detailed BFL formulation is as follows:

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3 Implementation of the Boundary Conditions

f i (r A , t + δt ) =

2qw f i∗ (r A , t) + (1 − 2qw ) f i∗ (r B , t) q < 1/2 . 1 f ∗ , t) + 2qqww−1 f i∗ (r A , t) q ≥ 1/2 2qw i (r A

(3.24)

The formulation in (3.24) uses linear interpolation, i.e., interpolation based on information from an adjacent grid point on either side of the unknown particle. The BFL scheme can also use quadratic interpolation, which requires information from two grid points on either side of the unknown particle (Bouzidi et al. 2001). It should be noted that the BFL scheme uses interpolation, leading to slight mass conservation errors near the boundary. Enhancement methods have been proposed to address the shortcomings of the original BFL scheme. However, most of these methods exhibit a higher computational complexity, especially considering the nature of solid barriers and extensive barrier areas.

3.10 Extrapolation Method for Curved-Wall BC: The Guo Scheme In addition to the BFL scheme based on a combination of bounce-back and 1D interpolation, Guo et al. (2002) proposed another curved-wall BC based on a nonequilibrium extrapolation method. As shown in Fig. 3.17, the curved-wall boundary node W is normal to solid grid S and fluid grid A. For this motion path, we need f i (r A , t + δt ) for point A, while all other directional distribution functions are independent of S and W. Additionally, based on the stream pattern, we have f 5 (r A , t + δt ) = f 5∗ (r S , t), where f 5∗ (r S , t) is the collision distribution function at solid grid S. Therefore, we need to solve for f 5∗ (r S , t). Similar to the BFL scheme, the distance factor qw = AW/AS is also defined in the Guo scheme, which plays an essential role in the extrapolation. The basic idea behind the Guo scheme is to decompose the unknown distribution function f 5∗ (r S , t) into equilibrium and non-equilibrium parts, as: f 5∗ (r S , t) = f 5 (r S , t) + f 5 *eq

*neq

(r S , t).

(3.25)

The equilibrium part can be determined from the equilibrium part of the adjacent fluid grids, and the non-equilibrium part can be extrapolated via the 1st-order accuracy method. For the equilibrium part, we actually lack the local density. In the Guo scheme, the density of the neighboring fluid grid A is borrowed, as:   e5 · u S u 2S (e5 · uS )2 *eq . f 5 (r S , t) = ρ(r A , t)ω5 1 + + − es2 2es4 2es2

(3.26)

The equilibrium distribution calculation can be found in Sect. 2.4.4. Here, uS is determined by the local velocity at W, A, and B as:

3.11 Other Solid-Wall BCs

75

Fig. 3.17 Schematic of the Guo scheme boundary. A and B are fluid grids, S is a solid grid, and W is the wall position. W can be located anywhere between A and S

B

wall BC

5

A

2

6

S 3

W

5 1

0 7

uS =

4

8

fluid grid

solid wall BC

solid grid

solid wall node

uS1 qw ≥ 0.75 , with qw uS1 + (1 − qw )uS2 qw < 0.75

1 [u(r W , t) + (qw − 1)u(r A , t)], qw 1 = [2u(r W , t) + (qw − 1)u(r B , t)]. 1 + qw

uS1 = uS2

(3.27)

For the non-equilibrium distribution, a similar definition can be made: *neq

f5

(r S , t) =

*neq f 5 (r A , t) *neq qw f 5 (r A , t)

qw ≥ 0.75 + (1 −

*neq qw ) f 5 (r B , t)

(3.28) qw < 0.75

Guo et al. (2002) reported that the above extrapolation scheme for the nonequilibrium part is 1st-order accurate; however, the overall accuracy of the distribution function is 2nd-order and satisfies numerical stability.

3.11 Other Solid-Wall BCs Among the various BC types for the LBM, solid-wall BCs are always a hot research and engineering topic. In addition to the bounce-back and extrapolation schemes introduced in this chapter, other BC schemes have been developed for the LBM,

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3 Implementation of the Boundary Conditions

including the virtual equilibrium scheme proposed by Filippova and Hänel in 1998 (1998) and recently the more popular immersed boundary method (IBM) scheme (Kang and Hassan 2011; Cheng et al. 2014). Due to space constraints, it is not possible to expound on all of them in this book, and the reader is encouraged to refer to relevant literature for further information.

3.12 Summary This chapter introduces the most commonly used BCs in the LBM. First, we describe the most basic and general first-type (the Dirichlet BC, Sect. 3.2) and second-type (the Newman BC, Sect. 3.3) BCs. All other BCs are variations or combinations of these two low-level BC types. Then, we introduce the most commonly used BC types in the building environment. These BCs can be roughly divided into two categories: non-solid-wall BCs and solid-wall BCs. For the former, we introduce the periodic (Sect. 3.4), symmetric (Sect. 3.5), and free-slip (Sect. 3.6) BCs. These are common flow-field BCs in the built environment, especially in urban turbulent environment simulations. Solid-wall BCs are more complex and are mainly divided into straight solid walls and curved solid boundaries. For straight solid-wall boundaries, bounce-back (Sect. 3.7) is among the most commonly used and successful BCs. There are many ways to implement bounce-back, and the implementation method and simulation accuracy differ slightly among them (Sects. 3.7.1, 3.7.2 and 3.7.3). We should note that bounce-back corresponds to the no-slip BC, so the near-wall accuracy may be poor when simulating a high-turbulence built environment. Therefore, we introduce an improvement of the bounce-back condition, namely the WFB model (Sect. 3.8), which absorbs the wall function and can achieve higher accuracy. For curved solid boundaries, bounce-back has an extended use method, namely the BFL scheme (Sect. 3.9), which combines bounce-back and 1D interpolation to calculate the distribution function according to the relative position of boundary grid points. In addition to interpolation, we also introduced the Guo scheme (Sect. 3.10) for calculating solid-wall surface distribution functions via extrapolation.

References

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Vasaturo R, Kalkman I, Blocken B, van Wesemael PJV (2018) Large eddy simulation of the neutral atmospheric boundary layer: performance evaluation of three inflow methods for terrains with different roughness. J Wind Eng Ind Aerodyn 173:241–261. https://doi.org/10.1016/j.jweia. 2017.11.025 Versteeg H, Malalasekera W (2007) An introduction to computational fluid dynamics—the finite volume method, 2nd Second. Pearson Education Limited, Harlow Werner H, Wengle H (1993) Large-Eddy simulation of turbulent flow over and around a cube in a plate channel. Turbulent shear flows 8. Springer, Berlin Heidelberg, Berlin, Heidelberg, pp 155–168 Wolf-Gladrow DA (2004) Lattice-gas cellular automata and lattice Boltzmann models: an introduction. Springer Yang XIA, Meneveau C (2016) Recycling inflow method for simulations of spatially evolving turbulent boundary layers over rough surfaces. J Turbul 17:75–93. https://doi.org/10.1080/146 85248.2015.1090575 Ziegler DP (1993) Boundary conditions for lattice Boltzmann simulations. J Stat Phys 71:1171– 1177. https://doi.org/10.1007/BF01049965 Zou Q, He X (1997) On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys Fluids 9:1591–1598. https://doi.org/10.1063/1.869307

Chapter 4

From the Lattice Boltzmann Equation to Fluid Governing Equations

4.1 Introduction Conventional CFD (e.g., FVM) solves the fluid governing equations, including the continuity equation, NSE, and energy equation. These equations essentially describe one thing: the conservation and transport of physical quantities, including mass, momentum, and energy (White 2015). The basic LBM equation is the LBE, which also represents the conservation and transport of a physical quantity; in this case, the physical quantity is the distribution function. Compared with the fluid mass, momentum, and energy, the distribution function is nothing but the information of the fluid molecules on a smaller scale. Therefore, the mesoscopic LBE and the macroscopic fluid governing equations describe the same phenomenon on various temporal-spatial scales. In this chapter, we recover the fluid governing equations from the non-externalforce LBE with the BGK model (Bhatnagar et al. 1954), as represented in (4.1), based on statistical mechanics, which will help readers gain a more in-depth understanding of the lattice Boltzmann method. As mentioned in Chap. 2, the LBE describes both the stream and collision processes. In the stream step, the distribution functions move directly to the neighboring lattice points or rest in the current points in the next step without variation. While in the collision step, the distribution functions approach the local equilibrium state based on the BGK model. f a (r + δt ea , t + δt ) − f a (r, t) = −

 1 f a (r, t) − f aeq (r, t) τ

(4.1)

In the derivation process, we first expand the LBE and relate it with the physical quantities via Chapman-Enskog multi-scale analysis (Wolf-Gladrow 2004) in Sects. 4.2 and 4.3. Then, we establish the connection between the mesoscopic temporal-spatial scale and the macroscopic scale (Sect. 4.4). Subsequently, by adding the macroscopic quantities and equilibrium distributions (Sect. 4.5), we finally derive

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Han and R. Ooka, Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems, https://doi.org/10.1007/978-981-99-1264-3_4

81

82

4 From the Lattice Boltzmann Equation to Fluid Governing Equations

the continuity (Sect. 4.6) and NSE (Sect. 4.7) equations. For the derivation, several mathematical manipulations are required. To ensure the smoothness of the derivation process, we detail these mathematical manipulations separately in Sect. 4.8. This will help the reader understand the physical logic of the derivation process without tripping over some subtle mathematical pitfall. The derivation does not cover the energy equation because it is slightly more complicated, and this book mainly addresses isothermal built environment problems. A similar derivation process can be found in the literature (Chen et al. 1992; Han et al. 2018a, b), which can be referred to by readers.

4.2 Taylor Expansion of the LBE First, let us implement the 2nd-order Taylor expansion on the first term on the lefthand side of (4.1) in position vector r and time t to obtain 

∂ ∂ + δt f a (r + δt ea , t + δt ) = f a + ea · δt ∂r ∂t   + O δt3 .



  ∂ ∂ 2 1 + δt fa fa + ea · δt 2 ∂r ∂t (4.2)

By substituting (4.2) into (4.1) and rearranging it, (4.3) is obtained, which holds a 2nd-order accuracy.        ∂ ∂ 2 δ2 ∂ 1 ∂ + f a + t ea · + f a − f aeq + O δt3 = 0 fa + δt e a · ∂r ∂t 2 ∂r ∂t τ (4.3) The LBE indicates the temporal-spatial transportation of the distribution functions of the virtual particles. Hence, the macroscopic fluid governing equations are obtained once the distribution functions are given. What we need is the so-called ChapmanEnskog multi-scale analysis.

4.3 Chapman-Enskog Multi-scale Analysis The mesoscopic LBE and the macroscopic fluid governing equations can be connected via the Chapman-Enskog multi-scale analysis (Wolf-Gladrow 2004). In this analysis, we employ three types of temporal and two types of spatial scales, as shown in Table 4.1. Here, k is an arbitrary amount holding the same order as the Knudsen number (Kn = MFP/L, where MFP is the mean free path of molecules and L is the representative macroscopic length).

4.3 Chapman-Enskog Multi-scale Analysis Table 4.1 Temporal and spatial scales employed in Chapman-Enskog multi-scale analysis

83

Temporal scales

Spatial scales

(a) Collision temporal scale of particles k 0

(a) Discrete spatial scale of particles k 0

(b) Convection temporal scale k −1

(b) Continuous spatial scales k −1

(c) Diffusion temporal scale k −2

For example, if k = 0.001, the collision temporal scale is k 0 = 1, the convection temporal scale is k −1 = 103 , and the diffusion temporal scale k −2 = 106 . This implies that the convection temporal scale is larger than the collision temporal scale, while the diffusion temporal scale is even larger. By assuming a collision time t, we can represent it using the convection temporal scale as t1 = kt and using the diffusion temporal scale as t2 = k 2 t. Therefore, ∂t∂ can be represented using t1 and t2 as ∂ ∂t1 ∂ ∂t2 ∂ ∂ ∂ = + =k · · + k2 . ∂t ∂t1 ∂t ∂t2 ∂t ∂t1 ∂t2

(4.4)

Similarly, by assuming a particle discrete spatial vector r, we represent the discrete spatial scale using the continuous spatial scale as r 1 = k r. Likewise, the nabla operator ∇ can be rewritten using ∇ 1 as ∇=

∂ ∂ r1 ∂ ∂ = =k · = k∇ 1 . ∂r ∂ r1 ∂ r ∂ r1

(4.5)

The LBE assumes that the unsolved flow field is in a non-equilibrium state, and the solving process involves approaching a local equilibrium state. Therefore, the distrieq bution function is considered to consist of an equilibrium component f a and a nonneq neq equilibrium component f a . As it is difficult to determine f a , we have to employ the perturbation expansion to f a in the Chapman-Enskog analysis using factor k eq around the equilibrium component f a . The perturbation expansion to f a (r, t) with factor k is expressed as:   f a = f aeq + k f a(1) + k 2 f a(2) + O k 3 .

(4.6)

The expansion in (4.6) demonstrates that f a is constituted by the equilibrium and multi-order non-equilibrium parts. The idea behind perturbation expansion is that the order of each k term is semi-independent, so the higher-order terms can be regarded as a correction term (Krüger et al. 2017). In order to retain the 2nd-order accuracy ofthe derivation, we keep the 2nd-order k term and ignore the higher-order terms O k3 .

84

4 From the Lattice Boltzmann Equation to Fluid Governing Equations

Substituting (4.4)–(4.6) into (4.3), we obtain      ∂ 2 ∂ δt k +k + kea · ∇ 1 f aeq + k f a(1) + k 2 f a(2) + O k 3 ∂t1 ∂t2 2   eq   ∂ ∂ δ2 f a + k f a(1) + k 2 f a(2) + O k 3 + k2 + kea · ∇ 1 + t k 2 ∂t1 ∂t2     1  (1) + k f a + k 2 f a(2) + O k 3 + O δt3 = 0. (4.7) τ Next, we arrange (4.7) by the orders of k and obtain (4.8).       ∂ ∂ ∂ eq 1 (1) eq δt + e a · ∇ 1 f a + f a k + δt f + δt + ea · ∇ 1 f a(1) ∂t1 τ ∂t2 a ∂t1

2  δt2 ∂ 1 + + ea · ∇ 1 f aeq + f a(2) k 2 2 ∂t1 τ  3  4  3 + O k + O k + O δt = 0. (4.8) To satisfy (4.8), every term of different k order is necessarily zero as k is an arbitrary amount. That is, for k and k 2 , the coefficients should be zero, as shown in (4.9) and (4.10), respectively. 

 ∂ 1 + ea · ∇ 1 f aeq + f a(1) = 0. (4.9) ∂t1 τ   2  ∂ ∂ eq δ2 ∂ 1 δt f a + δt + ea · ∇ 1 f a(1) + t + ea · ∇ 1 f aeq + f a(2) = 0. ∂t2 ∂t1 2 ∂t1 τ (4.10) δt

We arrange (4.9) and (4.10) into the forms of (4.9) and (4.10), respectively. 

 ∂ 1 (1) f =0 + ea · ∇ 1 f aeq + (4.11) ∂t1 δt τ a  2   ∂ eq ∂ δt ∂ 1 (2) (1) f = 0 (4.12) fa + + ea · ∇ 1 f a + + ea · ∇ 1 f aeq + ∂t2 ∂t1 2 ∂t1 δt τ a Substituting (4.11) into (4.12) to eliminate the 2nd-order differential term, we obtain    1 ∂ eq ∂ 1 (2) f a(1) + f = 0. fa + + ea · ∇ 1 1 − (4.13) ∂t2 ∂t1 2τ δτ a

4.5 Definition of Macroscopic Quantities and the Equilibrium Distribution …

85

4.4 From Mesoscopic Temporal-Spatial Scale to Macroscopic Scale This section utilizes the results of the Chapman-Enskog analysis to connect the mesoscopic temporal-spatial scale with the macroscopic scale. By implementing (4.11) · k + (4.13) · k 2 , we obtain  ∂ eq k (1) ∂ fa + k 2 + ea · ∇ 1 f aeq + f ∂t1 δt τ ∂t2 a    1 ∂ k 2 (2) f a(1) + f = 0. + ea · ∇ 1 1 − + k2 ∂t1 2τ δt τ a

 k

(4.14)

Rearranging (4.14) into the form as:      (1) ∂ 1 ∂ fa ∂ k f aeq + ea · k∇ 1 f aeq + k 2 1 − + k2 + ea · ∇ 1 f a(1) ∂t1 ∂t2 2τ ∂t1 k (1) k 2 (2) f + f = 0. + (4.15) δt τ a δt τ a Based on (4.4) and (4.5), we rewrite (4.15) into the following form. This equation bridges the LBE and the fluid governing equations.    (1) eq 1 k (1) ∂ fa ∂ fa k 2 (2) eq 2 (1) + + ea · ∇ f a + k 1 − fa + f = 0. + ea · ∇ 1 f a ∂t 2τ ∂t1 δt τ δt τ a (4.16)

4.5 Definition of Macroscopic Quantities and the Equilibrium Distribution Function After converting the mesoscopic temporal-spatial scale to macroscopic scale, we need to perform a similar conversion for physical quantities. This section converts the mesoscopic distribution function into macroscopic mass (density) and momentum (velocity). Let us review how the macroscopic quantities in the LBE are defined, which is as ρ=

 a

fa , u =

1 ( f a ea ), ρ a

p = ρes2 .

(4.17)

86

4 From the Lattice Boltzmann Equation to Fluid Governing Equations

As mentioned previously, the equilibrium distribution function in the DdQq scheme (Qian et al. 1992) (q.v. Sect. 2.4.4) can be formulated as (Krüger et al. 2017) f aeq

 ea · u (ea · u)2 u2 = ρwa 1 + 2 + − 2 . es 2es4 2es

(4.18)

The equilibrium distribution function also satisfies the following relation mass: ρ =



f aeq

a

momentum: ρu =



 f aeq ea .

(4.19)

a

With (4.6), (4.17), and (4.19), we can naturally determine the following constraint condition for the non-equilibrium part. (n) fa =  0 a (n) n = 1, 2, 3, 4 . . . a f a ea = 0

(4.20)

Then, (4.21) and (4.22) are obtained from (4.18) and (4.20), which are the key points connecting the mesoscopic f a and ea with the macroscopic ρ and u. The process of deriving these two equations is somewhat complicated and requires more space. We temporarily omit it here to ensure a better flow for reading. The detailed derivation is presented in Sect. 4.8 for interested readers. 

 ea ea · f aeq = ρuu + pδi j .

(4.21)

a

      1  1 2  (1) 1− es δt ∇ 1 u + (∇ 1 u)T ea ea · f a = −ρ τ − 2τ 2 a   1 ∇ 1 ρuuu. + δt τ − 2

(4.22)

4.6 Derivation of the Continuity Equation (Mass Conservation) After converting both the temporal-spatial scales and physical quantities to macroscopic scales, we can finally derive the fluid governing equations. First, we derive the continuity equation in this section, while the NES is derived in the next section.

4.7 Derivation of the NSE (Momentum Conservation)

87

Let us calculate the 0th-order moment of (4.16). As a result, the following equation is obtained.      ∂ f aeq   1  ∂ f a(1) + ea · ∇ f aeq + k 2 1 − + ea · ∇ 1 f a(1) ∂t 2τ ∂t1 a a a +

k  (1) k 2  (2) fa + f = 0. δt τ a δt τ a a

(4.23)

Each term in (4.23) can be transformed into the forms of (4.24)–(4.27). Here, (4.19) and (4.20) are employed.  ∂ f aeq 

a

∂t

=

∂  eq ∂ f a = ρ, ∂t a ∂t

   ea · ∇ f aeq = ∇ · f aeq ea = ∇ · (ρu),

a

(4.24) (4.25)

a

 ∂ f (1) a

 a

a

∂t1

=

∂  (1) f = 0, ∂t1 a a

   ea · ∇ 1 f a(1) = ∇ 1 · f a(1) ea = 0.

(4.26) (4.27)

a

Finally, by substituting (4.24)–(4.27) into (4.23), we transform (4.23) to (4.28), which is the continuity equation (mass conservation). ∂ ρ + ∇ · (ρu) = 0. ∂t

(4.28)

4.7 Derivation of the NSE (Momentum Conservation) Now, let us derivate the NSE. First, we calculate the 1st-order moment of (4.16), as 

    eq    1  ∂ fa ∂ f a(1) eq 2 (1) ea ea + e a ea · ∇ f a + k 1 − + ea ea · ∇ 1 f a ∂t 2τ ∂t1 a a a   1  1  ea f a(1) + k 2 ea f a(2) = 0. (4.29) +k δt τ a δt τ a

Each term in (4.29) can be transformed into (4.30)–(4.33). For this, (4.19) and (4.20) are utilized again.

88

4 From the Lattice Boltzmann Equation to Fluid Governing Equations

 a

 a



eq 

∂ fa ea ∂t

=

∂  eq  ∂ f ea = (ρu), ∂t a a ∂t

m    eq    ea ea · ∇ f aeq = ∇ · f a ea ea = ∇ · ρuu + pδi j ,

(4.30)

(4.31)

a

     1  1 ∂  (1)  ∂ f a(1) ea = 1− f ea = 0, (4.32) k 1− 2τ ∂t1 2τ ∂t1 a a a   

   1  1 2  2 (1) es δt ∇u + (∇u)T ea ea · ∇ 1 f a = ∇ · −ρ τ − k 1− 2τ 2 a    1 ∇ρuuu . (4.33) +δt τ − 2 2

Combining (4.30)–(4.33), we can transform (4.29) to (4.34), as 

   ∂ 1 δt es2 ∇u + (∇u)T (ρu) + ∇ · (ρuu) = −∇ p + ∇ · ρ τ − ∂t 2    1 −δt τ − ∇ρuuu . (4.34) 2 Suppose the macroscopic fluid groups move much slower than the virtual particles (i.e., the fluid velocity is much slower than the speed of sound). In that case, the last term on the right-hand side of (4.34) is small enough and can be omitted.   1 ∇ρuuu ≈ 0. δt τ − 2

(4.35)

Finally, (4.34) becomes (4.34), which is the derived NSE.     ∂ 1 δt es2 ρ ∇u + (∇u)T . (ρu) + ∇ · (ρuu) = −∇ p + ∇ · τ − ∂t 2

(4.36)

In (4.36), the form of the kinematic viscosity in the viscous term is different from the standard form of the NSE shown in (4.37).   ∂ (ρu) + ∇ · (ρuu) = −∇ p + ∇ · νρ ∇u + (∇u)T . ∂t

(4.37)

Comparing the two forms of kinematic viscosity, we obtain the relationship between fluid viscosity and the relaxation time in the collision function, as shown in (4.38) (Krüger et al. 2017). Note that (4.38) contains δt because we did not apply non-dimensionalization during the derivation, and all the quantities involve their units. δt will disappear if the equation is dimensionless, as described in Chap. 2.

4.8 Detailed Mathematical Operations in the Derivation Process

  1 δt . ν = es2 τ − 2

89

(4.38)

4.8 Detailed Mathematical Operations in the Derivation Process In Sect. 4.5, we have obtained (4.21) and (4.22), the fundamental equations connecting the mesoscopic with macroscopic scales. Nonetheless, we did not explain the detailed derivation process because tedious mathematical operations are involved. In this section, we will address this issue. Readers who are not interested in these mathematical details can skip this section entirely, which will not affect their general LBM knowledge.

4.8.1 Derivation of (4.21) First, we focus on (4.21). A specific discrete velocity scheme is involved in the derivation process. We take the D3Q19 scheme, which is the most commonly used in built environment problems, as an example to illustrate the derivation. As mentioned in Chap. 2, the equilibrium distribution function of the D3Q19 discrete velocity scheme is demonstrated as follows, where the lattice sound speed √ is es = 1/ 3.  ea · u (ea · u)2 u2 f aeq = ρwa 1 + 2 + − es 2es4 2es2 = wa ρ + 3wa (ea · ρu) +

3wa ρu 2 9wa (ea · u)2 − . 2 2

(4.39)

eq

Therefore, we supersede f a in (4.21) and obtain (4.40). It should be noted that here the quantity of a becomes specific. 18   a=0

18 18    ea ea wa ρ + ea ea [3wa (ea · ρu)] ea ea f aeq = a=0

+

18  a=0

a=0

    18  ea ea 3wa u 2 ea ea 9wa (ea · u)2 + . 2 2 a=0

(4.40)

90

4 From the Lattice Boltzmann Equation to Fluid Governing Equations

Each term on the right-hand side of (4.40) is solved following (4.41)–(4.44). Recall that according to the EOS, the pressure is acquired as p = ρes2 = ρ/3. For the detailed calculation process, please refer to Sect. 4.8.2. 18 

ρ δi j = pδi j , 3

(4.41)

ea ea [3wa (ea · ρu)] = 0,

(4.42)

ea ea wa ρ =

a=0 18  a=0

  18   ea ea 9wa (ea · u)2 ρu i u j  δi j δkl + δik δ jl + δil δ jk , = 2 2 a=0

(4.43)

  18  ea ea 3wa u 2 ρu i u j = δi j δkl . 2 2 a=0

(4.44)

Therefore, the following equation is obtained, which is the same as (4.21). 18   a=0

  ρu i u j ρu i u j  δi j δkl + δik δ jl + δil δ jk − δi j δkl ea ea f aeq = pδi j + 2 2  ρu i u j  δik δ jl + δil δ jk = pδi j + 2  ρ u k u j δ jl + u l u j δ jk = 2 ρ = (u k u l + u l u k ) = ρuu + pδi j . (4.45) 2

4.8.2 Derivation of (4.41)–(4.44) First, let us derive (4.41). We consider a = 0, a = 1−6 , and a = 7−18 separately. (e0 )i (e0 ) j = 0, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 6 100 100 000    (ea )i (ea ) j = ⎝ 0 0 0 ⎠ + ⎝ 0 0 0 ⎠ + ⎝ 0 1 0 ⎠ a=1 000 000 000 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 000 000 000 + ⎝0 1 0⎠ + ⎝0 0 0⎠ + ⎝0 0 0⎠ 000 001 001

(4.46)

4.8 Detailed Mathematical Operations in the Derivation Process

91

⎛

⎞ 200 = ⎝ 0 2 0 ⎠ = 2δi j , 002 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 18 110 110 1 −1 0    (ea )i (ea ) j = ⎝ 1 1 0 ⎠ + ⎝ 1 1 0 ⎠ + ⎝ −1 1 0 ⎠ a=7 000 000 0 0 0 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 −1 0 101 101 + ⎝ −1 1 0 ⎠ + ⎝ 0 0 0 ⎠ + ⎝ 0 0 0 ⎠ 0 0 0 101 101 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 −1 1 0 −1 000 + ⎝ 0 0 0 ⎠ + ⎝ 0 0 0 ⎠ + ⎝0 1 1⎠ −1 0 1 −1 0 1 011 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 0 0 0 000 0 0 0 + ⎝ 0 1 1 ⎠ + ⎝ 0 1 −1 ⎠ + ⎝ 0 1 −1 ⎠

(4.47)

0 −1 1

011 0 −1 1 ⎞ 800 = ⎝ 0 8 0 ⎠ = 8δi j . 008 ⎛

(4.48)

Using (4.46)–(4.48), we obtain the following relation, which is the same as (4.41). 18 

ea ea wa ρ = ρ

a=0

18 

ea ea wa =

a=0

ρ (e0 )i (e0 ) j 3

  ρ  ρ  (ea )i (ea ) j + (ea )i (ea ) j 18 a=1 36 a=7 6

+ =

18

ρ δi j = pδi j . 3

(4.49)

Similarly, for (4.42), we also consider a = 0, a = 1−6 , and a = 7−18 separately. The results are as in (4.50)–(4.52).   (e0 )i (e0 ) j (e0 )k ρu k = 0, ⎛ ⎛ ⎞ ⎞ 6 100 100    (ea )i (ea ) j (ea )k ρu k = ⎝ 0 0 0 ⎠(ρu 1 ) + ⎝ 0 0 0 ⎠(−ρu 1 ) a=1 000 000 ⎛ ⎛ ⎞ ⎞ 000 000 + ⎝ 0 1 0 ⎠(ρu 2 ) + ⎝ 0 1 0 ⎠(−ρu 2 ) 000 000

(4.50)

92

4 From the Lattice Boltzmann Equation to Fluid Governing Equations

⎛

⎛ ⎞ ⎞ 000 000 + ⎝ 0 0 0 ⎠(ρu 3 ) + ⎝ 0 0 0 ⎠(−ρu 3 ) = 0, (4.51) 001 001 ⎛ ⎞ 18 110    (ea )i (ea ) j (ea )k ρu k = ⎝ 1 1 0 ⎠(ρu 1 + ρu 2 ) a=7 000 ⎛ ⎛ ⎞ ⎞ 110 1 −1 0 + ⎝ 1 1 0 ⎠(−ρu 1 − ρu 2 ) + ⎝ −1 1 0 ⎠(ρu 1 − ρu 2 ) 000 0 0 0 ⎛ ⎛ ⎞ ⎞ 1 −1 0 101 + ⎝ −1 1 0 ⎠(−ρu 1 + ρu 2 ) + ⎝ 0 0 0 ⎠(ρu 1 + ρu 3 ) 0 0 0 101 ⎛ ⎛ ⎞ ⎞ 101 1 0 −1 + ⎝ 0 0 0 ⎠(−ρu 1 − ρu 3 ) + ⎝ 0 0 0 ⎠(ρu 1 − ρu 3 ) 101 −1 0 1 ⎛ ⎛ ⎞ ⎞ 1 0 −1 000 + ⎝ 0 0 0 ⎠(−ρu 1 + ρu 3 ) + ⎝ 0 1 1 ⎠(ρu 2 + ρu 3 ) −1 0 1 011 ⎛ ⎛ ⎞ ⎞ 000 0 0 0 + ⎝ 0 1 1 ⎠(−ρu 2 − ρu 3 ) + ⎝ 0 1 −1 ⎠(ρu 2 − ρu 3 ) 011 0 −1 1 ⎛ ⎞ 0 0 0 + ⎝ 0 1 −1 ⎠(−ρu 2 + ρu 3 ) = 0. (4.52) 0 −1 1 Using (4.50)–(4.52), we obtain the following relation, which is the same as (4.42). 18 

ea ea [3wa (ea · ρu)] =

a=0

18 

  (ea )i (ea ) j 3wa (ea )k ρu k

a=0

  1 (e0 )i (e0 ) j 3wa ρ(e0 )k u k 3 6   3wa  + (ea )i (ea ) j ρ(ea )k u k 18 a=1

=

18   3wa  + (ea )i (ea ) j ρ(ea )k u k = 0. 36 a=7

Next, we prove (4.43).

(4.53)

4.8 Detailed Mathematical Operations in the Derivation Process

  (e0 )i (e0 ) j ρu k u l (e0 )k (e0 )l = 0,

93

(4.54)

⎛ ⎛ ⎞ ⎞ 6 100  100       (ea )i (ea ) j ρu k u l (ea )k (ea )l = ⎝ 0 0 0 ⎠ ρu 21 + ⎝ 0 0 0 ⎠ ρu 21 a=1 000 000 ⎛ ⎛ ⎞ ⎞ 000  000    + ⎝ 0 1 0 ⎠ ρu 22 + ⎝ 0 1 0 ⎠ ρu 22 000 000 ⎛ ⎛ ⎞ ⎞ 000  000    + ⎝ 0 0 0 ⎠ ρu 23 + ⎝ 0 0 0 ⎠ ρu 23 001 001 ⎞ ⎛ 2 2u x 0 0 ⎝ (4.55) =ρ 0 2u 2y 0 ⎠, 0 0 2u 2z 18  a=7

  (ea )i (ea ) j ρu k u l (ea )k (ea )l ⎛

⎛ ⎞ ⎞ 110 110 = ⎝ 1 1 0 ⎠(u 1 + u 2 )2 + ⎝ 1 1 0 ⎠(−u 1 − u 2 )2 000 000 ⎛ ⎛ ⎞ ⎞ 1 −1 0 1 −1 0 + ⎝ −1 1 0 ⎠(u 1 − u 2 )2 + ⎝ −1 1 0 ⎠(−u 1 + u 2 )2 0 0 0 0 0 0 ⎛ ⎛ ⎞ ⎞ 101 101 + ⎝ 0 0 0 ⎠(u 1 + u 3 )2 + ⎝ 0 0 0 ⎠(−u 1 − u 3 )2 101 101 ⎛ ⎛ ⎞ ⎞ 1 0 −1 1 0 −1 + ⎝ 0 0 0 ⎠(u 1 − u 3 )2 + ⎝ 0 0 0 ⎠(−u 1 + u 3 )2 −1 0 1 −1 0 1 ⎛ ⎛ ⎞ ⎞ 000 000 + ⎝ 0 1 1 ⎠(u 2 + u 3 )2 + ⎝ 0 1 1 ⎠(−u 2 − u 3 )2 011 011 ⎛ ⎛ ⎞ ⎞ 0 0 0 0 0 0 + ⎝ 0 1 −1 ⎠(u 2 − u 3 )2 + ⎝ 0 1 −1 ⎠(−u 2 + u 3 )2 0 −1 1 0 −1 1 ⎛ 2 ⎞ 2 2 2 u 1 + u 2 + 2u 1 u 2 u 1 + u 2 + 2u 1 u 2 0 = 2⎝ u 21 + u 22 + 2u 1 u 2 u 21 + u 22 + 2u 1 u 2 0 ⎠ 0 0 0

94

4 From the Lattice Boltzmann Equation to Fluid Governing Equations

⎛

⎞ u 21 + u 22 − 2u 1 u 2 −u 21 − u 22 + 2u 1 u 2 0 + 2⎝ −u 21 − u 22 + 2u 1 u 2 u 21 + u 22 − 2u 1 u 2 0 ⎠ 0 0 0 ⎛ 2 ⎞ 2 2 2 u 1 + u 3 + 2u 1 u 3 0 u 1 + u 3 + 2u 1 u 3 ⎠ + 2⎝ 0 0 0 2 2 2 2 u 1 + u 3 + 2u 1 u 3 0 u 1 + u 3 + 2u 1 u 3 ⎛ 2 ⎞ u 1 + u 23 − 2u 1 u 3 0 −u 21 − u 23 + 2u 1 u 3 ⎠ + 2⎝ 0 0 0 2 2 2 2 −u 1 − u 3 + 2u 1 u 3 0 u 1 + u 3 − 2u 1 u 3 ⎛ ⎞ 0 0 0 + 2⎝ 0 u 2 + u 2 + 2u 2 u 3 u 2 + u 2 + 2u 2 u 3 ⎠ 2

⎛

3

2

3

0 u 22 + u 23 + 2u 2 u 3 u 22 + u 23 + 2u 2 u 3

⎞ 0 0 0 + 2⎝ 0 u 22 + u 23 − 2u 2 u 3 −u 22 − u 23 + 2u 2 u 3 ⎠ 0 −u 22 − u 23 + 2u 2 u 3 u 22 + u 23 − 2u 2 u 3 ⎛ 2 ⎞ 4u 1 + 2u 22 + 2u 23 4u 1 u 2 4u 1 u 3 ⎠. = 2⎝ 4u 1 u 2 2u 21 + 4u 22 + 2u 23 4u 2 u 3 2 2 2 4u 1 u 3 4u 2 u 3 2u 1 + 2u 2 + 4u 3

(4.56)

Then, we combine (4.54)–(4.56) and have   18  ea ea 9wa (ea · u)2 2 a=0   1 (ea )i (ea ) j 9wa ρu k u l (ea )k (ea )l 2 a=0 ⎞ ⎛ 2 2u 0 0 9ρ ⎝ 1 2 =0+ 0 2u 2 0 ⎠ 36 0 0 2u 23 ⎞ ⎛ 2 4u 1 u 2 4u 1 u 3 4u 1 + 2u 22 + 2u 2z 18ρ ⎝ ⎠ + 4u 1 u 2 2u 21 + 4u 22 + 2u 2z 4u 2 u 3 72 2 2 2 4u 1 u 3 4u 2 u 3 2u 1 + 2u 2 + 4u 3 ⎞ ⎛ 2 2 2 2u 1 u 2 2u x u 3 3u + u 2 + u 3 ρ⎝ 1 ⎠. = (4.57) 2u 1 u 2 u 21 + 3u 22 + u 23 2u 2 u 3 2 2u x u 3 2u 2 u 3 u 21 + u 22 + 3u 23 18

=

Meanwhile, let us construct a term following relations:

ρu i u j 2

(δi j δkl + δik δ jl + δil δ jk ). We have the

4.8 Detailed Mathematical Operations in the Derivation Process

⎛ ⎞ 100  ρu i u j ρu i u i ρ 2 δi j δkl = δkl = u + u 22 + u 23 ⎝ 0 1 0 ⎠ 2 2 2 1 001 ⎞ ⎛ 2 2 2 0 0 u + u2 + u3 ρ⎝ 1 ⎠, = 0 0 u 21 + u 22 + u 23 2 2 2 2 0 0 u1 + u2 + u3 ⎛ ⎞  u1 ρu i u j ρ ρu k u l δik δ jl = = u1 u2 u3 ⎝ u2 ⎠ 2 2 2 u3 ⎞ ⎛ u1u1 u1u2 u1u3 ρ = ⎝ u 1 u 2 u 2 u 2 u 2 u 3 ⎠, 2 u1u3 u2u3 u3u3 ⎛ ⎞  u1 ρu i u j ρ ρu l u k δil δ jk = = u1 u2 u3 ⎝ u2 ⎠ 2 2 2 u3 ⎞ ⎛ u1u1 u1u2 u1u3 ρ = ⎝ u 1 u 2 u 2 u 2 u 2 u 3 ⎠. 2 u1u3 u2u3 u3u3 Therefore, we can reshape

ρu i u j 2

95

(4.58)

(4.59)

(4.60)

  δi j δkl + δik δ jl + δil δ jk as

 ρu i u j  δi j δkl + δik δ jl + δil δ jk 2 ⎞ ⎛ 2 2u 1 u 2 2u 1 u 3 3u + u 22 + u 23 ρ⎝ 1 ⎠. = 2u 1 u 2 u 21 + 3u 22 + u 23 2u 2 u 3 2 2 2 2 2u 1 u 3 2u 2 u 3 u 1 + u 2 + 3u 3

(4.61)

We find that (4.61) is equal to (4.57), thus providing the proof for (4.43). Finally, we derive (4.44) in a similar manner as (4.41) and (4.42). (e0 )i (e0 ) j (ρu k u l δkl ) = 0,

(4.62)

⎛ ⎛ ⎞ ⎞ 6 100 100  (ea )i (ea ) j (ρu k u l δkl ) = ⎝ 0 0 0 ⎠(ρu k u l δkl ) + ⎝ 0 0 0 ⎠(ρu k u l δkl ) a=1 000 000 ⎛ ⎛ ⎞ ⎞ 000 000 + ⎝ 0 1 0 ⎠(ρu k u l δkl ) + ⎝ 0 1 0 ⎠(ρu k u l δkl ) 000 000

96

4 From the Lattice Boltzmann Equation to Fluid Governing Equations

⎛

⎛ ⎞ ⎞ 000 000 + ⎝ 0 0 0 ⎠(ρu k u l δkl ) + ⎝ 0 0 0 ⎠(ρu k u l δkl ) 001 001 ⎛ ⎞ 200 = ⎝ 0 2 0 ⎠u k u l δkl = 2ρu k u l δkl δi j , (4.63) 002 ⎛ ⎛ ⎞ ⎞ 18 110 110  (ea )i (ea ) j (ρu k u l δkl ) = ⎝ 1 1 0 ⎠(ρu k u l δkl ) + ⎝ 1 1 0 ⎠(ρu k u l δkl ) a=7 000 000 ⎛ ⎛ ⎞ ⎞ 1 −1 0 1 −1 0 + ⎝ −1 1 0 ⎠(ρu k u l δkl ) + ⎝ −1 1 0 ⎠(ρu k u l δkl ) 0 0 0 0 0 0 ⎛ ⎛ ⎞ ⎞ 101 101 + ⎝ 0 0 0 ⎠(ρu k u l δkl ) + ⎝ 0 0 0 ⎠(ρu k u l δkl ) 101 101 ⎛ ⎛ ⎞ ⎞ 1 0 −1 1 0 −1 + ⎝ 0 0 0 ⎠(ρu k u l δkl ) + ⎝ 0 0 0 ⎠(ρu k u l δkl ) −1 0 1 −1 0 1 ⎛ ⎛ ⎞ ⎞ 000 000 + ⎝ 0 1 1 ⎠(ρu k u l δkl ) + ⎝ 0 1 1 ⎠(ρu k u l δkl ) 011 011 ⎛ ⎛ ⎞ ⎞ 0 0 0 0 0 0 + ⎝ 0 1 −1 ⎠(ρu k u l δkl ) + ⎝ 0 1 −1 ⎠(ρu k u l δkl ) 0 −1 1 0 −1 1 ⎛ ⎞ 800 = ⎝ 0 8 0 ⎠(ρu k u l δkl ) = 8ρu k u l δkl δi j . (4.64) 008 Using (4.62)–(4.64), we obtain the following relation, which is the same as (4.44).   18 18  ea ea 3wa u 2 1 = (ea )i (ea ) j (3wa ρu k u l δkl ) 2 2 a=0 a=0

  1 3 = (e0 )i (e0 ) j ρ(e0 )k u k 2 3   3  (ea )i (ea ) j ρ(ea )k u k 18 a=1 6

+

4.8 Detailed Mathematical Operations in the Derivation Process

 18   3  + (ea )i (ea ) j ρ(ea )k u k 36 a=7 ρu i u j δi j δkl . = 2

97

(4.65)

4.8.3 Derivation of (4.22) (1) For deriving (4.22), let us first consider  more. If we supersede f a by f a  (4.9) once (1) utilizing (4.9), the left-side term a ea ea · f a in (4.22) can be reshaped as

eq

 a

ea ea ·

f a(1)



     ∂ eai ea j −τ δ = + ea · ∇ 1 f aeq ∂t 1 a 

   ∂  eq eq = −τ δ eai ea j f a + ∇ 1 eai ea j eak f a ∂t1 a a

   ∂  ∂  ρu i u j + pδi j + eai ea j eak f aeq = −τ δ ∂t1 ∂r1k a   

 ∂  ∂  2  ∂  eq = −τ δ ρu i u j + ρes δi j + eai ea j eak f a ∂t1 ∂t1 ∂r1k a   

 ∂  ∂  2 eq = −τ δ ρu i u j − es ∇ 1 · (ρu)δi j + eai ea j eak f a . ∂t1 ∂r1k a (4.66)

  Then, ∂t∂1 ρu i u j and and (4.68), respectively.

∂ ∂r1k



eq  a eai ea j eak f a

in (4.66) can be reshaped as (4.67)

 ∂ρu j ∂  ∂ρu i ∂ρ + uj − ui u j ρu i u j = u i ∂t1 ∂t ∂t1 ∂t1  1    ∂  ∂  2 2 ρu j u k + ρes δ jk + u j − ρu i u k + ρes δik = ui − ∂r1k ∂r1k ∂ρ − ui u j ∂t1  ∂ρ ∂ ∂ρ ∂  ρu j u k − u i es2 − uj = −u i (ρu i u k ) − u j es2 ∂r1k ∂r1 j ∂r1k ∂r1i ∂(ρu k ) + ui u j ∂r1k

98

4 From the Lattice Boltzmann Equation to Fluid Governing Equations

 ∂ρ ∂ρ ∂u i ∂u i  ρu j u k − u j es2 − ρu j u k − ui ∂r1 j ∂r1i ∂r1k ∂r1k   ∂ρ ∂ρ ∂u i ρu i u j u k , − u j es2 − = −u i es2 ∂r1 j ∂r1i ∂r1k    ∂  ∂  2 eq ρes u i δ jk + u j δik + u k δi j eai ea j eak f a = ∂r1k a ∂r1k = −u i es2

= es2 ∇ 1 · (ρu)δi j + ρes2 + ρes2

(4.67)

∂u j ∂ρ + es2 u j ∂r1i ∂r1i

∂u i ∂ρ + es2 u i = 0. ∂r1 j ∂r1 j

(4.68)

Therefore, combining (4.66)–(4.68), we have  a

  ∂ρ ∂ρ ea ea · f a(1) = −τ δ −u i es2 − u j es2 ∂r1 j ∂r1i  ∂u i  ρu i u j u k − es2 ∇ 1 · (ρu)δi j ∂r1k ∂u j ∂ρ + es2 ∇ 1 · (ρu)δi j + ρes2 + es2 u j ∂r ∂r1i 1i ∂u ∂ρ i + es2 u i +ρes2 ∂r1 j ∂r1 j     ∂u j ∂u ∂  i − = −τ δ ρes2 + ρu i u j u k ∂r1 j ∂r1i ∂r1k    2 T = −τ δ ρes ∇ 1 u + (∇ 1 u) + ∇ 1 ρuuu . −

(4.69)

Finally, we obtain the following relationship, which is the same as (4.22).          1  1 1− δt ρes2 ∇ 1 u + (∇ 1 u)T + ∇ 1 ρuuu ea ea · f a(1) = −τ 1 − 2τ 2τ a    1 2  es δt ∇ 1 u + (∇ 1 u)T = −ρ τ − 2   1 ∇ 1 ρuuu. (4.70) + δt τ − 2

References

99

4.9 Summary This chapter successfully recovers the macroscopic continuity equation and NSE based on the mesoscopic LBE. In the recovery process, the cornerstone is the Chapman-Enskog multi-scale analysis, which links the microscopic and macroscopic temporal-spatial scales. Furthermore, the perturbation expansion assumes how the equilibrium and non-equilibrium components constitute the distribution functions. The effect of the non-equilibrium component corresponds to the viscosity term of the NSE. During the recovery, 2nd-order accuracy is retained in all of the expansion operations, including the Taylor and perturbation expansions. From this perspective, we conclude that the macroscopic fluid governing equations can be considered a 2ndorder approximate solution of the LBE. It should be noted again that the energy equation is not discussed in this book; interested readers can refer to relevant literature for further information (Krüger et al. 2017; Guo and Shu 2013; Mohamad 2011).

References Bhatnagar PL, Gross EP, Krook M (1954) A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys Rev 94:511–525. https://doi.org/10.1103/PhysRev.94.511 Chen H, Chen S, Matthaeus WH (1992) Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. Phys Rev A (Coll Park) 45:5339–5342. https://doi.org/10.1103/PhysRevA. 45.R5339 Guo Z, Shu C (2013) Lattice Boltzmann method and its applications in engineering. World Scientific, Singapore Han M, Ooka R, Kikumoto H (2018a) Derivation of fluid governing equations from the lattice Boltzmann equation Part 1 Chapman-Enskog expansion of the lattice Boltzmann equation (in Japanese). In: Proceeding of the architectural research meetings, Kanto chapter of Architectural Institute of Japan. Architectural Institute of Japan, Tokyo, pp 199–202 Han M, Ooka R, Kikumoto H (2018b) Derivation of fluid governing equations from the lattice Boltzmann equation Part 2 Derivation of the continuity equation and Navier–Stokes equation (in Japanese). In: Proceeding of the architectural research meetings, Kanto chapter of Architectural Institute of Japan. Architectural Institute of Japan, Tokyo, pp 203–206 Krüger T, Kusumaatmaja H, Kuzmin A, Shardt O, Silva G, Viggen EM (2017) The lattice Boltzmann method. Springer International Publishing, Cham Mohamad AA (2011) Lattice Boltzmann method: fundamentals and engineering applications with computer codes. Springer Science & Business Media, Berlin Qian YH, D’Humières D, Lallemand P (1992) Lattice BGK models for Navier-Stokes equation. EPL 17:479–484. https://doi.org/10.1209/0295-5075/17/6/001 White FM (2015) Fluid mechanics, 8th edn. McGraw-Hill Education, New York Wolf-Gladrow DA (2004) Lattice-gas cellular automata and lattice Boltzmann models: an introduction. Springer, Berlin

Chapter 5

Turbulence Models and LBM-Based Large-Eddy Simulation (LBM-LES)

5.1 Introduction Turbulent flow dominates in the built environment, including both indoor and outdoor environments. One of its most striking features is the eddies at various scales. In particular, small-scale wind velocities continuously vary if Re is large. In turbulent flow, turbulence kinetic energy (TKE) is sequentially transferred from large eddies (long-wavelength wind fluctuations) to small eddies (short-wavelength wind fluctuations), which is called a cascade. The viscous force has little effect on largescale eddies but gradually dissipates TKE to thermal energy in small-scale eddies. This dissipation process and the short-wavelength wind variation are very important in turbulence and must be accurately captured in CFD simulations; otherwise, the simulation cannot fully grasp the nature of turbulence. In NSE-based methods, solving the fluid fields using the NSE directly is called direct numerical simulation (DNS). DNS obtains the most accurate solutions and, therefore, is able to solve any fluid problem theoretically. It is well known that DNS solves the smallest-sized eddy in the fluid field (i.e., the Kolmogorov scale), which generally corresponds to Re−3/4 . The number of meshes required for a DNS of a 3D problem can be roughly estimated to be approximately proportional to Re9/4 . Thus, the number of meshes would be staggering if solving high-Re flow problems, such as those encountered in the built environment (where Re can be tens of thousands or up to millions), using DNS, which is beyond the capabilities of most computer systems. Kim et al. (1987) showed that the number of meshes could reach 2 × 106 even when solving a low-Re channel flow (Re = 3300) via DNS. Therefore, researchers usually utilize turbulence models for high-Re flow fields to obtain approximate solutions while minimizing the computational cost. The most widely used methods are RANS and LES. RANS applies statistical turbulence theory and performs the Reynolds-average operation (usually time average) for the unsteady NSE, obtaining time-averaged flow field results. Meanwhile, LES starts from the various scales of eddies in turbulent flows. It performs some kind of filtering on turbulence and decomposes turbulence into large-scale translation and small-scale © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Han and R. Ooka, Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems, https://doi.org/10.1007/978-981-99-1264-3_5

101

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5 Turbulence Models and LBM-Based Large-Eddy Simulation (LBM-LES)

Table 5.1 Correspondence between LBM and FVM methods

Simulation idea

FVM

LBM

DNS

FVM-DNS (FVM for short)

LBM-DNS (LBM for short)

RANS

FVM-RANS

LBM-RANS

LES

FVM-LES

LBM-LES

pulsation. LES solves the former directly and models the latter using subgrid-scale (SGS) models. It should be noted that DNS, RANS, and LES are fluid simulation ideas that need to be implemented using specific fluid simulation methods. In the field of CFD, the most commonly used simulation implementation methods are NSE-based methods, such as the finite volume method (FVM)-based LES, which we call FVM-LES1 here. In FVM-LES, the time-dependent 3D momentum and continuity equations are solved, which enables analysts to gain a deeper insight into turbulent fluid motions, including the statistical properties of the flow, compared with conventional RANS methods. Accordingly, FVM-LES has been widely used to simulate turbulent flows in the built environment (Murakami 1993; Shah and Ferziger 1997; Krajnovic and Davidson 2002; Tominaga et al. 2008; Ikegaya et al. 2017; Kato et al. 2003; Nielsen 1998; Béghein et al. 2005). Even though FVM-LES is particularly advantageous for improving the richness of the results, this is somewhat offset by its significantly higher computational time and cost (Krajnovic and Davidson 2002; Sagaut 1998). The parallel computation capability of the LBM is considered superior to that of FVM because its algorithm is significantly simpler and easier to implement with boundary conditions. These advantages make the LBM an effective tool for turbulent simulation. Similar to FVM, the LBM can also perform DNS, LES, and RANS simulations. The correspondence between LBM and NSE-based methods (taking FVM as an example) is shown in Table 5.1. LBM-based LES (LBM-LES) overcomes the complexity of high-RE turbulent flows in the built environment. Furthermore, as the pressure Poisson equation must be solved, FVM-LES has the apparent disadvantage of high computational and time costs. In contrast, the LBM has tremendous potential for performing high-speed LES for massively parallel simulations. As with NSE-based simulations, solving the fluid directly with the LBE is DNS (i.e., LBM-DNS). It is also impractical to solve built environment problems using LBM-DNS. Therefore, turbulent flow methods such as RANS and LES are also required in the LBM. To date, there are some reports in the literature on using LBMRANS (Sajjadi et al. 2016, 2017a) and LBM-LES (Sajjadi et al. 2017b; Ahmad et al. 2017; Onodera et al. 2013; Han et al. 2019, 2020, 2021) to solve built environment problems.

1

Currently, NSE-based fluid simulation methods are widely used, among which FVM is dominant. Therefore, when researchers and scholars say DNS, RANS, and LES in the field of CFD, they usually refer to FVM-DNS, FVM-RANS, and FVM-LES. Some researchers also use FEM, namely FEM-DNS, FEM-RANS, and FEM-LES.

5.2 LES Implementation

103

Essentially, LES is still simulating unsteady turbulence, only relaxing the spatial scale of turbulent eddies. By contrast, RANS changes the problem, abandoning the unsteady turbulence information and only seeking averaged flow results. Therefore, LES can better describe the unsteady characteristics of turbulent flow, thus becoming increasingly popular in the built environment modeling community. In summary, this chapter provides a theoretical introduction to LBM-LES, which is the main approach adopted in this book. Section 5.2 describes how to implement LES in the LBM. Section 5.3 introduces the most basic SGS model in LES, the Smagorinsky model, and its improved version. Section 5.4 presents two advanced SGS models currently used for built environment simulations. In Sect. 5.5, we summarize the main simulation workflow of LBM-LES, hoping to aid readers who write LBM-LES codes.

5.2 LES Implementation LES has been widely used for solving turbulent flows over the past decade. The core idea behind LES is to filter the turbulent eddies of various scales in space. The larger eddies are calculated directly, while the smaller ones are modeled. Generally, low-frequency large-scale eddies vary according to specific problems. Conversely, high-frequency small-scale eddies hold universality independent of the specific flow field. Additionally, high-frequency eddies are isotropic, and their kinetic energy will eventually dissipate into heat. Therefore, small eddies can be modeled in a generic form. In LES, a fluid quantity Φ is decomposed into a grid-scale (GS) component Φ and a subgrid-scale (SGS) component Φ SGS . Taking the velocity u as an example, we have u = u + uSGS .

(5.1)

In LES, we aim to solve u directly and approximate uSGS using certain models (e.g., SGS models).

5.2.1 LES for the NSE-Based Method By defining the GS and SGS components of the physical quantities, the NSE can be filtered according to GS components with a residual SGS stress term, as (Davidson 2015):   ∂ (ρu) + ∇ · (ρuu) = −∇ p + ∇ · νρ ∇u + (∇u)T + ∇τ SGS . ∂t

(5.2)

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5 Turbulence Models and LBM-Based Large-Eddy Simulation (LBM-LES)

The most popular method to model τ sgs in LES is the eddy viscosity model. That is:  1  τ SGS = νSGS ρ ∇u + (∇u)T + δi j (τ SGS )kk . 3

(5.3)

Hence, we can yield:   ∂ (ρu) + ∇ · (ρuu) = −∇ p + ∇ · (ν + νSGS )ρ ∇u + (∇u)T + ∇τ SGS . (5.4) ∂t

5.2.2 LES for LBM Similar to LES in the NSE-based method, the total kinematic viscosity in the LBM consists of the molecular kinematic viscosity ν and the SGS kinematic viscosity νSGS according to the eddy viscosity model (Dong and Sagaut 2008), which is νtot = ν + νSGS .

(5.5)

Meanwhile, if we follow the relationship between viscosity and relaxation time in the LBM as introduced in (2.19), we obtain the following relation between total viscosity and total relaxation time. νtot = es2 (τtot − 0.5).

(5.6)

Here, τtot represents the total relaxation time. Hence, according to (5.6), τtot is obtained from νtot and then substitutes the original τ in the collision function (2.18) to implement LES for the BGK model. For the MRT scheme, τtot substitutes the relaxation coefficient corresponding to the shear viscosity in matrix S in (2.35) to implement LES. At this point, the only remaining problem is how to calculate νSGS . In the following two sections, we introduce several popular SGS models used in the built environment to calculate νSGS .

5.3 Smagorinsky SGS Model and Its Development The oldest and most straightforward SGS model is the Smagorinsky SGS model, developed by Smagorinsky (1963). It is so successful that it remains the most commonly applied SGS model in LES, despite being developed over half a century

5.3 Smagorinsky SGS Model and Its Development

105

ago. Nevertheless, its shortcomings are also apparent; accordingly, researchers have developed several improved versions. The classical Smagorinsky model and its improved versions can be expressed in general form as: νSGS = Cm Δ2 O P.

(5.7)

Here, Cm is a constant, Δ is the filter width (usually the grid size), and O P is a spatial operator in the GS component. The classical Smagorinsky model and its improved versions almost all follow this form.

5.3.1 Smagorinsky-Lilly SGS Model The most primitive and straightforward SGS model is the so-called SmagorinskyLilly SGS model. Smagorinsky proposed this model in 1956, and Lilly formally derived it using the evolution equation for stress tensors (Wyngaard 2010). In this model, νSGS is calculated as follows: | | νSGS = Cs2 Δ2 | S |.

(5.8)

| | Here, Cs is the Smagorinsky constant, Δ is the grid width (Δ > 0), and | S | is the magnitude of the strain rate tensor in the GS component and is expressed as:   | | / | S | = 2S i j :S i j , with S i j = 1 ∂u j + ∂u i . 2 ∂ xi ∂x j

(5.9)

The Smagorinsky-Lilly model follows the form of (5.7). Here, Cm = Cs2 , and the spatial operator of this model is the strain rate S i j . Note that the same form, i.e., as in (5.8) and (5.9), is applied in NSE-based methods. In the LBM, we can also employ this form to calculate νSGS and then obtain τtot based on (5.5) and (5.6). However, this process has more mutual conversion processes between distribution functions and macroscopic physical quantities, and its logic is still based on the NSE on the macoscopic level. Nevertheless, we can implement this model in the form | | of a mesoscopic distribution function. The core idea is to express the strain rate | S | using the distribution functions (Hou et al. 1996), such as: / 2 2 −τ e τ es2 δt + 2Cs Δ2 ⊓/ρ δ + | | s t |S| = , with 2Cs Δ2 / Σ  neq neq neq ⊓ = π i j π i j and π i j = (5.10) eai ea j f a − f aeq . a

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5 Turbulence Models and LBM-Based Large-Eddy Simulation (LBM-LES)

We can combine (5.5), (5.6), (5.8), and (5.10) to reshape the form of the Smagorinsky SGS model using the LBM objects, reducing the interconversion step with macroscopic variables, which in turn could reduce computational errors and increase speed. Despite its many advantages, the Smagorinsky-Lilly model has several disadvantages. First, it is highly dependent on Cs ; however, Cs varies according to specific problems and cases. Unfortunately, there are no well-defined rules to determine Cs . It is insufficient to dampen the fluctuations in small high-frequency scales if Cs is too small, which may cause numerical instability. Conversely, if Cs is too large, it may lead to significant viscosity-dependent errors (Chen et al. 2008). Some studies have determined that Cs = 0.1 is an optimal value for balancing numerical stability and accuracy (Fernandino et al. 2009). However, Murakami (1993) proposed that Cs = 0.12 is suitable for large-scale outdoor wind simulations. Some works have employed this value for both indoor and outdoor flows (Han et al. 2019, 2018). Another disadvantage is that the near-wall νSGS is obviously overestimated in the standard Smagorinsky model. This is especially evident in built environment problems, especially in outdoor turbulence problems. Boundary layer theory indicates that near walls, ν, dominate compared to νSGS due to the small velocity. However, as Cs is a constant, the near-wall νSGS will be overestimated by the Smagorinsky model because of the apparent velocity gradient in that region. In the FVM-LES, a straightforward way to compensate for this overestimation is to introduce a damping function in the Smagorinsky model. In the FVM-LES, a van Driest-style (Driest 1956) damping function is widely employed, which is expressed as: f s = 1 − exp

−y + . A+

(5.11)

Here, A+ is a constant with a value of ~ 25. The damping function f s is introduced in νSGS as shown in (5.12), which makes νSGS approach zero near the wall. Thus, the standard Smagorinsky form with the van Driest-style damping function is: | | νSGS = (Cs f s Δ)2 | S |.

(5.12)

However, the damping function is an ad hoc (or empirical formulation) modification based on the distance to the wall without a distinct physical meaning (Nicoud and Ducros 1999). Furthermore, it is difficult to implement in the general case for complex geometries and it utilizes the near-wall friction velocity to non-dimensionalize the local velocity. Therefore, it destroys the locality of the simulation, which is the core of the parallel LBM simulation.

5.3 Smagorinsky SGS Model and Its Development

107

5.3.2 Dynamic Smagorinsky SGS Model (DSM) To overcome the limitations of the Smagorinsky-Lilly model, Germano et al. (1991) proposed the dynamic Smagorinsky SGS model (DSM). This model is based on the idea that the smallest resolvable scales in the simulation are similar to the modeled smaller scales. The concept of a scale similarity model implies that the smallest resolvable scale eddies can provide information for the smaller eddies that need to be modeled by the SGS model. The DSM defines two filter scales: a regular grid-scale Δ2 and a test-scale Δ1 , which is larger than Δ2 (e.g., two grids). The SGS constant in the DSM is CDSM , replacing Cs2 in the Smagorinsky-Lilly model. The DSM assumes that CDSM is identical in transferring the kinetic energy from the test scale to the grid scale and from the grid scale to the subgrid scale. Therefore, the spatially dependent CDSM and the viscosity can be calculated adopting these assumptions as:   | | 1 1 L i j − L kk δi j , νSGS = CDSM Δ22 | S |, with CDSM = Mi j 3 | | | | | | L i j = u i j u i j − u i j u i j , and Mi j = 2Δ21 | S | S i j − 2Δ22 | S | S i j .

(5.13)

However, Germano et al. (1991) pointed out that the DSM could produce a region with negative values (CDSM < 0), causing solution numerical instability. This problem is usually solved by employing a volume average, as in (5.14), in which [[ ]] means the volume averaging operation. | | L i j − 13 L kk δi j | S¯ | | | Δ22 . | | = 2 |¯|¯ 2 | | 2Δ1 S S i j − 2Δ2 |S |S i j 

νSGS

(5.14)

Although the DSM achieves the dynamic computation of spatially dependent CDSM , its volume averaging operation greatly destroys the simulation’s locality, which can be fatal for the LBM. The damping function in the Smagorinsky-Lilly model also faces this problem to some extent. Another disadvantage of the DSM is that a larger test scale is defined; therefore, its application to unstructured grid systems is difficult. Furthermore, the standard Smagorinsky model and its successor (DSM) mainly focus on the correction of constant Cm . In these models, the spatial operator O P is based on the strain rate tensor, so these SGS models cannot detect the vorticities in turbulence (Nicoud and Ducros 1999). This is an inherent flaw of the Smagorinsky model, whether in the FVM-LES or LBM-LES. These disadvantages make the Smagorinsky model relatively less accurate, although its implementation is simple.

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5 Turbulence Models and LBM-Based Large-Eddy Simulation (LBM-LES)

5.4 Advanced SGS Models in LBM-LES for Built Environment Simulations In recent years, increasingly advanced SGS models have been developed for LES for larger spatial scale issues, especially in the built environment. This section introduces two popular models: the wall-adapting local eddy-viscosity (WALE) SGS model and the coherent-structure SGS model (CSM).

5.4.1 Wall-Adapting Local Eddy-Viscosity (WALE) SGS Model In recent years, a new SGS model, the so-called WALE model, has become popular for NSE-based LES (Nicoud and Ducros 1999). In the WALE model, the focus is no longer on the correction of a constant but on the spatial operator O P. Therefore, it models the asymptotic decay of the eddy viscosity in the near-wall regions without relying on a dynamic procedure for a constant Cm . It is also suitable for complex geometries, such as unstructured grid systems, because it only requires the local velocity gradient instead of an explicit filter to calculate νSGS . Furthermore, it is sensitive to both the strain and rotation rate of small turbulent structures. Therefore, this model overcomes some of the disadvantages of the Smagorinsky model and exhibits higher accuracy in complex LES. In the WALE model, a traceless symmetric part of the square of the GS velocity gradient tensor Sidj is defined as: Sidj

  1 ∂u k ∂u l ∂u k ∂u i 1 ∂u k ∂u j − δi j + . = 2 ∂ xi ∂ xk ∂ x j ∂ xk 3 ∂ xl ∂ x k

(5.15)

Here, δi j is the Kronecker symbol and u i is the velocity of the GS component in the i-direction. Then, Sidj Sidj , the square of Sidj , possesses the following relationship (Nicoud and Ducros 1999). Sidj Sidj =

2 1 2 2 S S + W 2 W 2 + S 2 W 2 + 2I VSW . 6 3

(5.16)

In (5.16), S and W are the magnitudes of the strain rate tensor S i j and the rotation rate tensor W i j , respectively, as expressed in (5.17). In other words, Sidj Sidj is related to the behavior of both S i j and W i j , and S i j and W i j are the velocity-strain and vorticity tensors of the GS component in the turbulent structure, respectively. S2 = Si j Si j , W 2 = W i j W i j ,

I VSW = S ik S k j W ik W k j ,

5.4 Advanced SGS Models in LBM-LES for Built Environment Simulations

with S i j =

    ∂u j ∂u j 1 ∂u i 1 ∂u i and W i j = . + − 2 ∂x j ∂ xi 2 ∂x j ∂ xi

109

(5.17)

In the WALE model, a new spatial operator O P, as shown in (5.18), is constructed based on Sidj Sidj .

OP = 

Si j Si j

Sidj Sidj

5/2

3/2

5/4 .

+ Sidj Sidj

(5.18)

Then, we can yield νSGS as in (5.19), which is the main WALE model equation. Here, Cw = 0.325.

νSGS = (Cw Δ)2 

Si j Si j

Sidj Sidj

5/2

3/2

5/4 .

+ Sidj Sidj

(5.19)

In a pure shear flow, Sidj Sidj = 0, meaning that νSGS naturally becomes zero in the near-wall regions, which is attractive. The SGS turbulence kinetic energy kSGS and its dissipation rate εSGS in the WALE model can be calculated using (5.20), in which Ck = 0.094 and Ce = 1.048.  kSGS =

εSGS =

2 Cw2 Δ Ck





Si j Si j

Sidj Sidj

5/2

3

5/4 2 ,

+ Sidj Sidj

Ce (kSGS )3/2 . Δ

(5.20)

5.4.2 Coherent Structure SGS Model (CSM) Another recently popular SGS model in built environment LES is the coherent structure SGS model (CSM) proposed by Kobayashi (2005), which determines the SGS stress based on turbulent flow structures. The CSM imitates the idea of extracting coherent fine-scale eddies from the full second invariant in DNS and extracts a similar second invariant Q (Q-criterion) in the LES, which is given by Q=

1 ∂u i ∂u j 1 W i j W i j − Si j Si j = − , 2 2 ∂ x j ∂ xi

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5 Turbulence Models and LBM-Based Large-Eddy Simulation (LBM-LES)

with S i j =

    ∂u j ∂u j 1 ∂u i 1 ∂u i and W i j = . + − 2 ∂x j ∂ xi 2 ∂x j ∂ xi

(5.21)

Similar to the WALE model, here, S i j and W i j are the velocity-strain and vorticity tensors of the GS component, respectively. After that, the CSM constructs a coherent structure function FCS , which is expressed as: FCS =

Q E

, with E =

  1 ∂u j 2 1 . W i j W i j + Si j Si j = 2 2 ∂ xi

(5.22)

Here, Q and E are the second invariant and the squared magnitude of the velocity gradient tensor in the GS component, respectively. The CSM defines the SGS constant as CCSM , which is related to FCS . Therefore, the viscosity in the CSM is calculated using (5.23). It should be noted that the coherent structure function FCS is close to zero in the near-wall regions. This model can be applied to turbulent flow around objects with complex geometries (Kajishima and Taira 2018). Kobayashi (2005) also suggested that it can be extended to turbulent flow under the influence of rotation. | | 1 3 |FCS | 2 (1 − FCS ). νSGS = CCSM Δ2 | S |, with CCSM = 22

(5.23)

5.5 LBM-LES Workflow Figure 5.1 illustrates the primary workflow of the SGS model in the LBM-LES. This workflow will aid readers in understanding at which step of the LBM simulation loop the SGS model comes into play and provides a reference for readers who wish to write their own LBM-LES code. Here, Φ˜ indicates the physical property Φ updated by the SGS model. The SGS model mainly acts before the collision step. The main LBM-LES steps are as follows: (1) Implement an LBM collision-stream loop between f a (r, t) and f a (r + δt ea , t + δt ); (2) After a complete collision-stream loop, calculate the necessary physical properties for the SGS model: • If the SGS model is in LB form, calculate τtot , f aeq , f aneq , and other LB properties directly from the LBE; • If the SGS model is in NSE form, calculate τtot and f aeq and convert them into νSGS , u, and other macroscopic properties.

References

111

Fig. 5.1 Principal workflow for implementing the SGS model in the LBM-LES. The SGS model modifies τtot before every collision step. The SGS model in LB form directly updates τtot using distribution functions, while the SGS model in NSE form calculates νSGS using macroscopic quantities and then updates τtot

(3) Calculate the new total relaxation coefficient τ˜tot according to the SGS model: • Update τtot to τ˜tot directly according to the SGS model if it is in LB form; • Calculate the new ν˜ SGS according to the SGS model if it is in NSE form and then calculate τ˜tot using ν˜ SGS . (4) The former relaxation time τtot is updated to the new value τ˜tot , which is introduced into the LBE for the subsequent collision step, and then returns to step 1 for the next time step.

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Germano M, Piomelli U, Moin P, Cabot WH (1991) A dynamic subgrid-scale eddy viscosity model. Phys Fluids A 3:1760–1765. https://doi.org/10.1063/1.857955 Han M, Ooka R, Kikumoto H (2018) Comparison between lattice Boltzmann method and finite volume method for LES in the built environment. In: The 7th international symposium on computational wind engineering 2018, Seoul, pp 2–5 Han M, Ooka R, Kikumoto H (2019) Lattice Boltzmann method-based large-eddy simulation of indoor isothermal airflow. Int J Heat Mass Transf 130:700–709. https://doi.org/10.1016/j.ijheat masstransfer.2018.10.137 Han M, Ooka R, Kikumoto H (2020) Validation of lattice Boltzmann method-based large-eddy simulation applied to wind flow around single 1:1:2 building model. J Wind Eng Ind Aerodyn 206:104277. https://doi.org/10.1016/j.jweia.2020.104277 Han M, Ooka R, Kikumoto H (2021) Effects of wall function model in lattice Boltzmann methodbased large-eddy simulation on built environment flows. Build Environ 195:107764. https://doi. org/10.1016/j.buildenv.2021.107764 Hou S, Sterling J, Chen S, Doolen GD (1996) A lattice Boltzmann subgrid model for high Reynolds number flows. Fields Inst Commun 151–166. http://doi.org/10.48550/arXiv.comp-gas/9401004 Ikegaya N, Ikeda Y, Hagishima A, Tanimoto J (2017) Evaluation of rare velocity at a pedestrian level due to turbulence in a neutrally stable shear flow over simplified urban arrays. J Wind Eng Ind Aerodyn 171:137–147. https://doi.org/10.1016/J.JWEIA.2017.10.002 Kajishima T, Taira K (2018) Computational fluid dynamics: incompressible turbulent flows. Springer International Publishing, Cham Kato S, Ito K, Murakami S (2003) Analysis of visitation frequency through particle tracking method based on LES and model experiment. Indoor Air 13:182–193. https://doi.org/10.1034/j.16000668.2003.00173.x Kim J, Moin P, Moser R (1987) Turbulence statistics in fully developed channel flow at low Reynolds number. J Fluid Mech 177:133. https://doi.org/10.1017/S0022112087000892 Kobayashi H (2005) The subgrid-scale models based on coherent structures for rotating homogeneous turbulence and turbulent channel flow. Phys Fluids 17:045104. https://doi.org/10.1063/1. 1874212 Krajnovic S, Davidson L (2002) Large-eddy simulation of the flow around a bluff body. AIAA J 40:927–936. https://doi.org/10.2514/2.1729 Murakami S (1993) Comparison of various turbulence models applied to a bluff body. J Wind Eng Ind Aerodyn 46–47:21–36. https://doi.org/10.1016/0167-6105(93)90112-2 Nicoud F, Ducros F (1999) Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow Turbul Combust 62:183–200. https://doi.org/10.1023/A:1009995426001 Nielsen PV (1998) Selection of turbulence models for prediction of room airflow. ASHRAE Trans 104:1119–1127 Onodera N, Aoki T, Shimokawabe T, Kobayashi H (2013) Large-scale LES wind simulation using lattice Boltzmann method for a 10 km × 10 km area in metropolitan Tokyo. TSUBAME e-Sci J Global Sci Inf Comput Cent 9:1–8 Sagaut P (1998) Large eddy simulation for incompressible flows: an introduction. Springer Science & Business Media, Berlin Sajjadi H, Salmanzadeh M, Ahmadi G, Jafari S (2016) Simulations of indoor airflow and particle dispersion and deposition by the lattice Boltzmann method using LES and RANS approaches. Build Environ 102:1–12. https://doi.org/10.1016/j.buildenv.2016.03.006 Sajjadi H, Salmanzadeh M, Ahmadi G, Jafari S (2017a) Lattice Boltzmann method and RANS approach for simulation of turbulent flows and particle transport and deposition. Particuology 30:62–72. https://doi.org/10.1016/j.partic.2016.02.004 Sajjadi H, Salmanzadeh M, Ahmadi G, Jafari S (2017b) Turbulent indoor airflow simulation using hybrid LES/RANS model utilizing lattice Boltzmann method. Comput Fluids 150:66–73. https:// doi.org/10.1016/j.compfluid.2017.03.028

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Chapter 6

From LBE to LBM: Using the LBM to Solve Built Environment Problems

6.1 Introduction In the previous chapters, we presented the preliminary information needed for solving environmental problems using the LBM, including the fundamental equation and its discretization method (Chap. 2), boundary conditions (Chap. 3), and the necessary turbulence model (Chap. 5). To finalize the LBM theory, this chapter comprehensively introduces how to utilize the LBE for LBM simulations. As with general CFD simulations, the first step is to discretize and normalize the simulation problem. The idea of discretization and normalization in the LBM differs from conventional CFD methods, such as FVM, which we explain in detail in Sect. 6.2. Then, Sect. 6.3 introduces the process framework for LBM simulations. This is a general process framework; thus, built environment problems naturally apply. Finally, the simulation errors that may be encountered in LBM simulations are discussed in Sect. 6.4. Computational simulations can produce a variety of errors. We will not cover the LBM algorithm or turbulence model errors in this chapter, which should be left to works devoted to LBM theory and development. This book mainly deals with numerical and user errors, particularly those specific to LBM simulations. We discuss the discrete errors that are encountered in general CFD (Sect. 6.4.1), the unique compressibility errors in the LBM (Sect. 6.4.2), and the over-relaxation numerical oscillation that may occur when solving high-Re problems, such as built environment problems (Sect. 6.4.3). In this chapter, only the causes and possibilities of these errors are discussed. In Part II, we present quantitative discussions on these errors for several case studies in Chaps. 8 and 9. We hope that the discussion presented herein will inspire our readers to avoid or reduce such simulation errors.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Han and R. Ooka, Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems, https://doi.org/10.1007/978-981-99-1264-3_6

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6.2 Discretization and Normalization 6.2.1 Spatial Discretization As with all CFD methods, a time and space discretization of the solving domain is required (currently, at least, there is no way to solve for continuous time and space). In conventional CFD, spatial discretization is known as the meshing process. Taking FVM as an example, one can divide the solving domain into multiple mesh spaces (Versteeg and Malalasekera 2007; Ferziger et al. 2020). These discrete mesh spaces can be of various shapes, such as hexahedrons, tetrahedrons, honeycombs, or any other shape, and their size can also be adjusted according to specific needs. In the LBM, the fluid molecules on a continuous temporal-spatial scale are discretized into mesoscopic distribution functions of finite virtual particles. This process is the discrete velocity scheme described in Sect. 2.4, which is how the LBM spatially discretizes the solving domain. Taking the widely used DdQq scheme as an example (Qian et al. 1992), the space is discretized into cubic meshes (or squared meshes for 2D). One can determine the grid resolution δ L according to the solution accuracy, and δ L remains constant in the entire space. It should be noted that the LBM mesh is uniform and cannot be locally refined or coarsened. This is indeed a unique feature of the LBM because the discrete velocity scheme fixes the velocity of the particles and the distance they can travel in each time step (one mesh length). The advantage of this approach is that its theory and practical programming are elementary; however, a problem arises: the increase in the number of meshes. To this end, researchers have studied many local grid refinement technologies to develop the currently used uniform mesh systems, which are the quadtree (2D) or octree (3D) grid refinement technologies. In places where the local mesh is refined, physical quantities such as particle velocity require special treatment. Figure 6.1 shows an example of spatial discretization in FVM and LBM. This book does not discuss such mesh refinement techniques, and interested readers can refer to relevant literature for further information (Filippova and Hänel 1998; Lagrava et al. 2012; Dorschner et al. 2016; Gendre et al. 2017).

6.2.2 Temporal Discretization In addition to spatial discretization, temporal discretization also needs consideration in unsteady fluid simulations. In NSE-based methods, such as LES and unsteady RANS (URANS), the primary basis of temporal discretization is that the Courant number (Co) in the mesh should not exceed unity, that is Co =

u PHU δt ≤ 1. δL

(6.1)

6.2 Discretization and Normalization Fig. 6.1 Spatial discretization in FVM and LBM. FVM utilizes meshing technology to divide the space into non-uniform polyhedral discrete meshes. LBM divides the space into uniform cubic meshes via the DdQq scheme, and the local meshes are refined via local grid refinement technology using quadtree and octree meshes

117

solving domain

spatial discretization in LBM (DdQq scheme)

spatial discretization in FVM (meshing)

local grid refinement technology (quadtree & octree )

The superscript PHU implies that u uses a physical unit, e.g., SI units. Co physically represents the ratio of the fluid movement distance in a time step δt to the unit length of the mesh δ L . Indeed, the definition of Co in complex 3D polyhedral meshes is more complicated. However, we can simply understand that Co ≤ 1 implies that the fluid flow should not exceed one mesh within one δt (i.e., limitation of the information dissemination speed). This is the main principle of δt in FVM, particularly in built environment simulations dominated by incompressible fluid problems. In other words, as long as the discrete time step satisfies (6.1), any value is acceptable in FVM. Of course, one should consider the computational cost. In the LBM (or LBM-LES), determining δt is slightly different. First, Co ≤ 1 is permanently established in the LBM, which is the premise of a stable simulation. This is further discussed in Sect. 6.4.2. Second, one usually does not directly set δt in the LBM, and it is set indirectly by defining the lattice speed u LBU [the superscript LBU implies that u uses the lattice Boltzmann unit (LBU)], which satisfies u LBU = u PHU

δt . δL

(6.2)

The relationship in (6.2) is discussed in detail in Sect. 6.2.3. Here, we note that δt is generally set indirectly by defining the representative physical speed u PHU and its corresponding lattice speed u LBU . Furthermore, u LBU should also satisfy a series of conditions, including that it should be smaller than 0.3 times the lattice Mach number. These complex conditions, which are described in Sect. 6.4.2, jointly

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constrain the value of δt . One can see that the temporal discretization in the LBM is a relatively complex issue. Improperly setting δt may cause simulation errors, which are discussed in Sect. 6.4.

6.2.3 Normalization of Physical Quantities After implementing the temporal-spatial discretization, the normalization (or nondimensionalization) of the LBE is a top priority. Similar to NSE-based methods, the LBM also satisfies the law of similarity. In FVM, one obtains results that are in agreement with experimental data or the natural phenomenon if a set of appropriate dimensionless parameters are selected and the characteristic dimensionless numbers are equivalent (e.g., Re and Pr). Still, many engineers tend to select dimensional parameters similar to the natural quantities (e.g., physical quantities in SI units) or do not conduct the normalization but simulation in original units for convenience. However, normalization is indispensable in the LBM. Regardless of the dimension set being set up, the LBM will normalize the parameters using its internal logic. This is done to make the normalized physical quantities fit the distribution function f a and its particle velocities ea . Next, we discuss how to conduct the normalization in the LBM. First, geometric similarity is extremely important. Then, all simulation parameters are converted to the LBU systems. In isothermal turbulent problems, the characteristic quantities are length L, density ρ, time t, velocity u, and kinematic viscosity ν. In most LBM cases, only the discrete density  (or mass), discrete length (i.e., grid resolution) δ L , and discrete time interval δt are essential. All quantities in physical units (indicated by the superscript PHU ) can be transformed to dimensionless parameters or in LBUs (indicated by the superscript LBU ). It is worth noting that the , δ L , and δt used for the transformation are in physical units. Then, we have 1 1 1 ρ LBU = ρ PHU , L LBU = L PHU , t LBU = t PHU ,  δL δt δ δ t t u LBU = u PHU , ν LBU = ν PHU 2 . δL δL

(6.3)

Let us continue with the 2D solving domain in Sect. 6.2.1 as an example. Here, we go a little deeper and give this domain some specific meaning. As illustrated in Fig. 6.2, the domain size is 1.0 × 1.0 [m]. The lid-driven velocity is 1.0 [m s−1 ], and the fluid kinematic viscosity is 1 × 10−3 [m2 · s−1 ]. The turbulent dimensionless number is Re = 103 . We want to simulate a flow period of 10 [s]. Next, suppose that we choose δ L = 0.2 [m] for the spatial discretization and the edge length of the domain in the LBU is 5, which means that we divide the edge into five grids. Following the temporal discretization, let us set the lattice velocity as u LBU = 0.1. Then, we can calculate the discrete time step as:

6.3 LBM Simulation Workflow

119

Fig. 6.2 Example of the transformation from physical to LB units

δt =

0.1 [−] u LBU · 0.2 [m] = 0.02 [s]. δL = u PHU 1.0 [m · s−1 ]

(6.4)

In the LBM, the length of one grid in the LBU is always unity. Via normalization, the inlet velocity and fluid kinematic viscosity in LBUs are converted to 0.1 [–] and 5×10−4 [−], respectively. The number of simulation time steps is 500. In the physical system, 1.0 s is required for the fluid with a velocity of 1.0 [m · s−1 ] to traverse a 1.0-m-long edge. Similarly, the LB system requires 50 iterations for the fluid with a dimensionless velocity of 0.1 [–] to traverse an edge with a dimensionless length of five. Note that, in the LB system, Re is equitant to that in the physical system, as: u PHU δδLt LδL u LBU L LBU = = ν LBU ν PHU δδ2t

PHU

Re

LBU

= RePHU .

(6.5)

L

The above relationship is the paramount similarity in the LBM, which means that we can obtain the correct results if the LB and physical systems have the same Re. Indeed, geometric similarity cannot be neglected.

6.3 LBM Simulation Workflow So far, we know that the cornerstone of the LBM is the LBE and its collision functions. We also know how to implement the boundary conditions. We know we should discretize the velocity and space using the DdQq scheme and normalize the parameters by choosing an appropriate δ L and δt on the premise of geometric and Re similarity. It seems that we know many details needed for a simulation; however, how should we implement the LBM for simulation? Figure 6.3 summarizes the primary LBM simulation process. Here, a general framework for how the LBM algorithm solves the problem is presented, which naturally applies to the built environment as well. Similar frameworks are also presented elsewhere (Guo and Shu 2013; Krüger

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Fig. 6.3 Overview of the standard LBM simulation process

et al. 2017). These frameworks differ in detail but are consistent in the LBM core: the collision-stream loop. An LBM simulation task can be generally divided into three parts:

6.3 LBM Simulation Workflow

121

1. Simulation initialization and boundary settings (a) Identify the characteristic parameters and geometry, and then choose appropriate discrete parameters. For non-external-force isothermal fluid problems, , δ L , and δt are the essential parameters. (b) Temporal-spatially discretize the solving domain and convert the physical properties from physical to LB units. Geometric similarity and characteristic dimensionless parameter consistency (e.g., Re) should be ensured. (c) Set all boundary conditions (e.g., inlet, outlet, walls, and others) and map the properties to distribution functions. 2. Main LBM iteration loops (a) Implement local collision at all grids according to the relaxation time scheme at the current time step t: • If SRT(BGK) is used, directly relax f a to f a∗ using the relaxation parameter τ ; • If MRT is used, first, transform f a to m using M, then relax m to m∗ using a set of relaxation parameters S. Finally, convert m∗ back to f a∗ . (b) Implement the stream step in a straightforward manner at the current time step t. (c) Treat the boundary grids if f a∗ is propagated there. Note that the stream process at the boundaries is still at the current time step t. (d) Recover the physical quantities in the LBU from f a and then calculate the local equilibrium distribution functions using them. At this point, a complete time step ends. (e) Evaluate whether the simulation convergence condition1 is satisfied; otherwise, move to the next time step t → t + δt . (f) Implement the next collision and stream steps. 3. Results output (a) If necessary, recover the macroscopic quantities from the LB equilibrium distribution functions to physical units and output them at any time step. 4. This series of steps represents a standard simulation workflow. Among the three parts, Part 2, “Main LBM iteration loops,” is the core algorithm of the LBM. 1

The built environment is usually a time-varying turbulent environment, and thus, there is generally no so-called convergence state that the flow field reaches a complete steady state similar to laminar flow, unless utilizing RANS to simulate the time-averaged flow filed. The LES used in this book is to simulate the temporal-spatial eddy variation at a certain spatial scale, and thus, the flow field will never reach the completely stable state. Therefore, the “convergence” in the built environment LES simulations is more complicated. What’s more, if we want to simulate a dynamically varying process using LES (e.g., dispersion process of pollutants starting from the source), there is no convergence state even during the whole simulation. Therefore, this book does not intend to deal with the “convergence” concept theoretically. We will deal with one of the common “convergence” states in built environment LES simulations in Sect. 9.3.3, which usually corresponds to the statistical averaged flow field no longer changing drastically.

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This part may append some additional actions, such as the calculation of νSGS in the LBM-LES or the implementation of wall functions. Generally, most of the additional actions are appended between the stream and collision steps.

6.4 Common User-Induced Simulation Errors in the LBM We should always be aware of the various simulation errors when performing simulations. A high-quality simulation should already avoid errors as much as possible. Here, we do not attempt to present systematic errors due to LBM model and algorithm limitations or truncation errors due to computational bit limitations, which are beyond the scope of this book. As researchers and engineering practitioners, our main concern is determining if the errors encountered in the simulations are due to an improper simulation setup. In this book, this is referred to as user-induced simulation errors. This section will briefly introduce some common user-induced simulation errors to provide a reference for readers to reduce such simulation errors. Due to the complexity of the simulation setup and the variety of error sources, it is not possible to conduct a comprehensive analysis in this book. Accordingly, only the more relevant or easily overlooked errors in LBM simulations are presented here.

6.4.1 Grid Discretization Errors Grids or meshes are the basis for spatial discretization and the vehicle for CFD simulations. In CFD simulations, meshing the simulation object geometry is a timeconsuming task and is the first threshold that determines the simulation accuracy. The influence of the grid system on the simulation accuracy is mainly manifested in two aspects: grid quality and grid size. In classic FVM, meshes come in various forms, the most basic of which can be divided into structured and unstructured meshes, as shown in Fig. 6.4. Structured meshes are usually orthogonal (e.g., rectangles or hexahedra), while unstructured meshes are mostly non-orthogonal (e.g., triangles or tetrahedra). Different mesh types lead to different degrees of errors in the simulation results (e.g., orthogonal meshes outperform non-orthogonal meshes in terms of accuracy in many cases), thus giving rise to the concept of “mesh quality.” A high-quality mesh is beneficial in terms of accuracy and fast convergence. Researchers have also invented a variety of indicators to describe mesh (or) grid quality. In the LBM, however, the DdQq grid system is widely used, which is a typical orthogonal grid (square or cubic grids) and therefore does not have grid quality in the usual sense. The grid size (i.e., grid resolution) is the primary source of error and affects the simulation accuracy. The grid resolution fundamentally determines the accuracy of the spatial discretization and has a direct impact on the simulation accuracy, especially in LES, where the grid sensitivity is even higher. Therefore, most

6.4 Common User-Induced Simulation Errors in the LBM

123

Fig. 6.4 Examples of 2D structured and unstructured meshes in FVM

CFD simulations are subjected to “grid independence” tests to assess the influence of the grid on the simulation accuracy; the LBM-LES is no exception. A common and easy way to determine grid independence in built environment simulations is to select several representative grid sizes (usually reduces exponentially) and perform the same simulation (i.e., with all other parameters fixed). Then, the overall error or the error at some representative points is assessed based on the difference between the simulation results and experimental or other standard values to determine grid independence. Figure 6.5 shows the grid independence test results for the LBM-LES simulation of a built environment case. Four grid sizes were selected for this test, namely, b/4, b/8, b/16, and b/32 (b is the characteristic length). This test compares the error between the simulation results and experimental values and obtains a curve of the simulation error according to grid size. The results show that the simulation error decreases as the grid resolution increases. When the grid resolution reaches b/16, the error essentially ceases to change. Therefore, in this case, with a grid resolution of b/16, the simulation results can be considered to have become “grid independent.” It is worth noting that the grid resolution test results should be considered on a case-by-case basis and that there are no universal results or conclusions for this test. Fig. 6.5 Grid independence test for an LBM-LES case. This test compares the differences between simulated and experimental results for four different grid sizes

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6 From LBE to LBM: Using the LBM to Solve Built Environment Problems

6.4.2 Compressibility Errors In Chap. 4, we derived the NSE from the LBE with the BGK scheme in the form of (4.34). We can rewrite (4.34) in the following form (Reider and Sterling 1995; Han et al. 2018a, b):       ∂(ρu) + ∇ · (ρuu) = −∇ p + ∇ · νρ ∇u + (∇u)T + O Kn2 + O Ma3 . ∂t (6.6) Here, Ma is the Mach number in LBU. Kn is the Knudsen which has been   number,  proven to be proportional to Ma such that O Kn2 ∼ O Ma2 (Reider and Sterling 1995). Thus, additional terms in the recovered equation relate to the lattice Ma, which is defined as:  LBU   PHU  u   u  δt Ma = = . (6.7) es es δ L   Here, uPHU  is the magnitude of the local macroscopic flow velocity uPHU . According to (6.6), it is clear that the recovered NSE are in compressible form. This indicates that the LBM is a pseudo-compressible model when addressing incompressible fluid problems and generates so-called compressibility errors (Reider and Sterling 1995; He and Luo 1997; Ponce Dawson et al. 2002). According to previous studies, compressibility errors include the deviations associated with density gradients and those associated with Ma via the abovementioned additional terms and velocity field divergence (Martínez et al. 1994; Klainerman and Majda 1982). This implies that compressibility errors may appear when using LBM-LES to solve incompressible turbulent problems compared to FVM-LES. Furthermore, it is observed that Ma ∼ δt in (6.7) indicates that variations in δt result in different Ma values in LBMLES and cause obvious compressibility errors, despite the prototype flow and mesh being the same. It is worth noting that the compressibility errors in the BGK model differ from the errors caused by an improper Co, although they are both caused by an inappropriate δt . In FVM-LES, when addressing low-Ma incompressible fluid problems, a proper discrete time step δt should be selected to control Co to less than unity. This prevents the fluid from flowing over more than one mesh in one step; otherwise, a visible error will occur in some implicit solutions. However, LBM-LES, one requirement   in uLBU  < es  1. Subsequently, Ma < 1, implying that for BGK model stability is   Co = uLBU δtLBU /δ LLBU < 1 is always tenable (because δtLBU = 1 and δ LLBU = 1 are prescribed in the LBM), and the simulation can proceed stably. Skordos (1993) simulated both a 2D Taylor vortex flow and a shear flow and found that in the Taylor vortex simulation, the velocity errors between the simulation and analytical results decreased with decreasing Ma until finally achieving stability. In the shear flow simulation, although the error decreased initially with decreasing Ma, it subsequently increased with a further decrease in Ma. Reider and Sterling

6.4 Common User-Induced Simulation Errors in the LBM

125

(Reider and Sterling 1995) theoretically derived compressibility errors and tested them using a decaying Taylor vortex in a 2π periodic domain with Re = 100 while considering three types of grid resolutions and Ma values. Their results confirmed that the simulation accuracy improved as Ma decreased. Other studies indicate that Ma < 0.3 should be satisfied when performing simulations in order to prevent obvious compressibility errors (Krüger et al. 2017; Han 2022). We will quantitatively discuss the compressibility errors using an isothermal indoor flow simulation in Part II: Chap. 8.

6.4.3 Over-Relaxation and Numerical Oscillations According to the LBE with the BGK model, it is evident that the purpose of the eq BGK model is to evolve f a (r, t) toward the equilibrium f a (r, t), which is also known as the relaxation process. However, as the LBE can be rewritten in the form eq eq evolve toward f a (r, t), or even exceed f a (r, t), of (6.8), f a (r, t) can immediately   1 contingent on the value of 1 − τ . Thus, the collision in the BGK model can be divided into three patterns.   1 1 f a (r, t) + f aeq (r, t). f a (r + δt ea , t + δt ) = 1 − τ τ

(6.8)

eq

1. If 1 − τ1 > 0, or τ > 1, f a gradually evolves toward f a stepwise, which is called under-relaxation. eq 2. If 1 − τ1 = 0, or τ = 1, f a directly evolves toward f a at one step, which is called full-relaxation. eq 3. If 1 − τ1 < 0, or 21 ≤ τ < 1, f a goes over f a , which is called over-relaxation. It should be noted that τ cannot be less than 21 , as ν cannot be negative. Kruger et al. (2017) investigated these three collision patterns for the spatially homogeneous eq lattice BGK equation with an initial condition of f 0 (0)/ f a = 1.1 and a constant eq f a . Their results are shown in Fig. 6.6. Therefore, the collision pattern is determined by the relaxation time parameter τ . According to (2.19) and (6.3), τ is determined using (6.9). This indicates that τ is sensitive to δt if the grid resolution δ L is determined. τ = 3ν LBU +

δt 1 1 = 3ν PHU 2 + . 2 2 δL

(6.9)

The ideal collision patterns are the full- or under-relaxation patterns, which result eq in f a evolving directly or smoothly, respectively, toward f a . In the over-relaxation eq pattern, f a oscillates around f a with an exponentially damping amplitude. Therefore, the ideal τ is not less than unity, and according to (6.9), requires δ2 LBU ν ≥ 16 or ν PHU ≥ 6δLt . It should also be noted that in LBM-LES, ν is substituted

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6 From LBE to LBM: Using the LBM to Solve Built Environment Problems

Fig. 6.6 Examples of under-, full-, and over-relaxation for the BGK model [Reproduced from Krüger et al. (2017)]

by νtot . Unfortunately, this is extremely difficult to comply with in the built environment. In most built environment cases, the fluid is air, which viscosity is in the order of 10−5 [m2 · s−1 ]. It is challenging to make the order of air viscosity larger δ2 than δLt in this situation, although νSGS can partially increase the value of νtot . For example, typically, it is difficult to set the grid size to 1 m with a discrete step of 10−5 [s] or utilize 0.001 m to match 10−1 [s] because of limitations related to both Co and computational resources. Therefore, in wind engineering, LBM-LES often exhibits an over-relaxation pattern. Usually, this over-relaxation pattern corresponds to normal turbulence fluctuations. However, extreme fluctuations in LBM-LES may cause numerical oscillation problems and will affect the simulation accuracy. Under the same grid resolution, a smaller δt results in a smaller τ , which may incur more serious oscillations. Han (2022) discussed the result accuracy of LBM-LES with the BGK model based on variations in δt in an indoor turbulent flow simulation and reported that very small δt could cause apparent oscillations in both the time-averaged and fluctuating velocities. We will further discuss these oscillations in Chap. 8.

6.5 Summary This chapter, as the last chapter of Part I, involves the basic LBM-LES workflow and the main problems that may be encountered when using it for specific simulations. We focus on LBM discretization and normalization, which is different from classical FVM. Finally, some common user-induced errors in LBM simulations are described and discussed. Particularly, compressibility errors and numerical oscillations caused by setting an improper discrete time interval are unique problems to the

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127

LBM, as opposed to FVM, and should be taken into consideration when performing simulations.

References Dorschner B, Frapolli N, Chikatamarla SS, Karlin IV (2016) Grid refinement for entropic lattice Boltzmann models. Phys Rev E 94:053311. https://doi.org/10.1103/PhysRevE.94.053311 Ferziger JH, Peri´c M, Street RL (2020) Computational methods for fluid dynamics, 4th edn. Springer International Publishing, Cham Filippova O, Hänel D (1998) Grid refinement for lattice-BGK models. J Comput Phys 147:219–228. https://doi.org/10.1006/jcph.1998.6089 Gendre F, Ricot D, Fritz G, Sagaut P (2017) Grid refinement for aeroacoustics in the lattice Boltzmann method: a directional splitting approach. Phys Rev E 96:023311. https://doi.org/10.1103/ PhysRevE.96.023311 Guo Z, Shu C (2013) Lattice Boltzmann method and its applications in engineering. World Scientific, Singapore Han M (2022) Effect of time steps on accuracy of indoor airflow simulation using lattice Boltzmann method. Tongji Daxue Xuebao/J Tongji Univ(Nat Sci) 50:793–801. http://doi.org/10.11908/j. issn.0253-374x.21486 (in Chinese) Han M, Ooka R, Kikumoto H (2018a) Derivation of fluid governing equations from the lattice Boltzmann equation Part 1 Chapman-Enskog expansion of the lattice Boltzmann equation. In: Proceeding of the architectural research meetings, Kanto chapter of Architectural Institute of Japan. Architectural Institute of Japan, Tokyo, pp 199–202 (in Japanese) Han M, Ooka R, Kikumoto H (2018b) Derivation of fluid governing equations from the lattice Boltzmann equation Part 2 Derivation of the continuity equation and Navier-Stokes equation. In: Proceeding of the architectural research meetings, Kanto chapter of Architectural Institute of Japan. Architectural Institute of Japan, Tokyo, pp 203–206 (in Japanese) He X, Luo LS (1997) Lattice Boltzmann model for the incompressible Navier-Stokes equation. J Stat Phys 88:927–944. https://doi.org/10.1023/b:joss.0000015179.12689.e4 Klainerman S, Majda A (1982) Compressible and incompressible fluids. Commun Pure Appl Math 35:629–651. https://doi.org/10.1002/cpa.3160350503 Krüger T, Kusumaatmaja H, Kuzmin A, Shardt O, Silva G, Viggen EM (2017) The lattice Boltzmann method. Springer International Publishing, Cham Lagrava D, Malaspinas O, Latt J, Chopard B (2012) Advances in multi-domain lattice Boltzmann grid refinement. J Comput Phys 231:4808–4822. https://doi.org/10.1016/j.jcp.2012.03.015 Martínez DO, Matthaeus WH, Chen S, Montgomery DC (1994) Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics. Phys Fluids 6:1285–1298. https://doi.org/10.1063/1.868296 Ponce Dawson S, Chen S, Doolen GD (2002) Lattice Boltzmann computations for reaction-diffusion equations. J Chem Phys 98:1514–1523. https://doi.org/10.1063/1.464316 Qian YH, D’Humières D, Lallemand P (1992) Lattice BGK models for Navier-Stokes equation. EPL 17:479–484. https://doi.org/10.1209/0295-5075/17/6/001 Reider MB, Sterling JD (1995) Accuracy of discrete-velocity BGK models for the simulation of the incompressible Navier-Stokes equations. Comput Fluids 24:459–467. https://doi.org/10.1016/ 0045-7930(94)00037-Y Skordos PA (1993) Initial and boundary conditions for the lattice Boltzmann method. Phys Rev E 48:4823–4842. https://doi.org/10.1103/PhysRevE.48.4823 Versteeg H, Malalasekera W (2007) An introduction to computational fluid dynamics—the finite volume method, 2nd edn. Pearson Education Limited, Harlow

Part II

Practice of LBM-LES in Built Environment

Chapter 7

LBM-LES in Ideal 3D Lid-Driven Cavity Flow Problems

7.1 Introduction In Part II of this book, starting with this chapter, we introduce three benchmark cases for built environment problems. For each case, we provide an in-depth description of the primary LBM-LES method and parameters to aid readers replicating the simulation. As the first benchmark case, a lid-driven 3D square cavity flow is presented in this chapter. This square cavity flow represents a simple and ideal flow; however, it is a prototype problem for many built environment flows (Fig. 7.1), such as indoor flows and street canyon flow (flow between urban buildings). This is a good start to gain the necessary knowledge for more advanced cases. Although built environment flows are often in a high-Re turbulent state, let us first exclude the complex influence of turbulence, simplify the problem, and mainly focus on low-Re flow simulation in this chapter. Herein, we set multiple flow states of various Re to explore the LBMLES performance when the flow gradually transitions from fully laminar to smallscale turbulent flow. For comparison purposes, the FVM-LES simulation results are also presented. Section 7.2 describes the specific lid-driven 3D square cavity flow problem, while Sect. 7.3 presents the specific simulation methodology and parameters. In Sect. 7.4, the simulation results are analyzed in terms of various aspects and their agreement with the FVM-LES results is discussed. Finally, the computational time and parallel computational efficiency of both models are discussed in Sect. 7.5.

7.2 Description of the Ideal 3D Lid-Driven Cavity Flow A typical 3D lid-driven flow is shown in Fig. 7.2, using a cubic cavity with a length of H = 1.0 [m] on all sides. A fixed velocity Ulid = 1.0 [m s−1 ] was implemented on the lid with all other sides having solid fixed walls. Although the fluid moving

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Han and R. Ooka, Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems, https://doi.org/10.1007/978-981-99-1264-3_7

131

132

7 LBM-LES in Ideal 3D Lid-Driven Cavity Flow Problems

Fig. 7.1 Ideal 3D lid-driven cavity flow as the prototype for many built environment problems, including indoor airflow and urban canyon turbulent flow

Fig. 7.2 Schematic of a typical 3D lid-driven cavity flow

horizontally at the top lid generally does not enter the cavity, it will drive the adjacent fluid layers in the cavity to move under the influence of viscosity, which eventually causes the overall movement of the fluid in the cavity. In addition to being a prototype for many built environment flow problems, there are numerous other benefits to using the lid-driven cavity flow as the first test case. First, it has a simple flow pattern; second, it has simple BCs, using mainly boundaries with a given velocity value and flat wall boundaries, which are the most basic and vital BCs in LBM-LES.

7.4 Results and Discussion

133

Table 7.1 General simulation conditions of the 3D lid-driven cavity flow Item

FVM-LES

Simulation domain

21.5b(x) × 13.75b(y) × 11.25b(z)

LBM-LES

Basic grid size

Width: 1/100H; Total: approximately 1 million

Time step and marching

0.01 [s], PISO

0.005 [s]

Time discretization

Euler-implicit

–

Space discretization

2nd-order central difference

–

Simulation period

Preparatory: 0–60 [s]; Averaging: 60–120 [s]

Lid BC

Uniform velocity boundary, Ulid = 1.0 [m s−1 ] (no fluctuations)

Other BCs

Wall function (Spalding’s law)

Bounce-back condition

For this test case, we varied Re (Re = 500, 1000, 5000, and 10,000); however, it is basically in a laminar or transition state. Re is defined by H , Ulid , and the kinematic viscosities.

7.3 Simulation Methodology and Boundary Conditions Both LBM-LES and FVM-LES were conducted for the first test case, adopting the simplest Smagorinsky-Lilly SGS model. Table 7.1 lists the simulation and boundary conditions used in this case. The Smagorinsky constant Cs was set to 0.12, as proposed by Murakami (1993) for the built environment. As the Smagorinsky-Lilly model can overestimate νSGS near the walls, a damping function is usually utilized for νSGS correction (e.g., the van-Driest damping function) (van Driest 1956). LBU The grid resolution was fixed as δ L = H/100 and the lattice speed to Uref = 0.1. LBU are determined based on The discrete time interval δt and the viscosity in LBU ν these parameters, and the results are shown in Table 7.2. We normalized the results with H and Ulid .

7.4 Results and Discussion 7.4.1 Instantaneous and Time-Averaged Velocities The instantaneous velocity field at the central vertical cross-section (y/H = 0) at t = 120 [s] is shown in Fig. 7.3. In the relatively low-Re flow case (i.e., Re = 500 and 1000), there are almost no fluctuating velocities, a noticeable circulation flow formed in the entire cavity, and the LBM-LES and FVM-LES results agree well. In contrast, when Re becomes larger (i.e., Re = 5000 and 10,000), turbulence occurs gradually. This is one of the main characteristics of turbulence: the instantaneous

134

7 LBM-LES in Ideal 3D Lid-Driven Cavity Flow Problems

Table 7.2 LBM parameters for flow under various Re Re

500

L PHU ref [m]

1.0

PHU [m s−1 ] Uref

1.0

ν PHU [m2 s−1 ]

0.002

δ L [m]

0.01

δt [s]

0.001

L LBU ref [–]

1

1000

5000

10,000

0.001

0.0002

0.0001

LBU [–] Uref

0.1

ν LBU [–]

0.02

0.01

0.002

0.001

τ [–]

0.56

0.53

0.506

0.503

velocity is chaotic and irregular. As Re increases, the turbulence effect intensifies and the chaotic nature of the instantaneous velocity intensifies as well. In this case, for Re = 10,000, it is difficult to have agreement between the two methods in terms of the instantaneous velocity as strongly irregular fluctuations appear. Turbulence would be much more intense in a real building environment. To investigate fluid motion patterns, researchers usually take a statistical sampling approach to obtain time-averaged physical quantities of the flow field and their standard deviations. Furthermore, turbulence in the built environment is usually unsteady, i.e., the flow field varies over time. However, in many cases, the unsteady turbulent environment can be treated as steady state over a large temporal scale; that is, the time-averaged statistics of the flow field reach stability. For example, in a ventilated room where the wind varies over time in all parts of the room, the time-averaged velocity over a sufficiently long period of time can reach a steady state. As described in Sect. 5.8, the statistical averaging method based on time sampling is used when performing such unsteady turbulence simulations with LES, as shown in (7.1). 1 Σ t Φ, N t ⎡ ⎡ | | N N |1 Σ |1 Σ √ 2 √ ,t ,2 ⟨Φ ⟩ = (Φ ) = √ (Φ t − ⟨Φ⟩)2 . N t N t N

⟨Φ⟩ =

(7.1)

Here, Φ t is the instantaneous √ value of physical quantity Φ at time t, ⟨Φ⟩ is the time-averaged value of Φ, and ⟨Φ ,2 ⟩ is the standard deviation of Φ after the timeaveraging operation, which reflects its temporal fluctuation intensity. N is the total number of samples. It is worth noting that N should be large enough to ensure that the samples cover the entire temporal variation of the turbulence structure. Meanwhile, an excessive sampling frequency should be avoided because the results are meaningless

7.4 Results and Discussion Fig. 7.3 Instantaneous velocity field at the central vertical cross-section (y/H = 0) at t = 120 s for Re = 500 (a) and 10,000 (b)

135

136

7 LBM-LES in Ideal 3D Lid-Driven Cavity Flow Problems

if the sampling frequency is higher than the turbulence energy frequency of the LES simulation. We recommend a sampling √ frequency of 10–500 Hz for typical built environment turbulence simulations. ⟨Φ ,2 ⟩ in (7.1) is actually its defining equation; however, in practice, we usually calculate it as: √ √ ⟨Φ ,2 ⟩ = ⟨Φ ,t Φ ,t ⟩ − ⟨Φ ,t ⟩⟨Φ ,t ⟩.

(7.2)

Figure 7.4 shows the time-averaged velocity at the central vertical cross-section (y/H = 0) for all Re cases. For the low-Re cases (i.e., Re = 500 and 1000), the fluid in the cavity moves as a steady laminar flow with almost no turbulent effects. The circulating flow in the entire cavity is legible, and the time-averaged velocity is almost identical to the instantaneous velocity. Figure 7.5a, b show the velocity profiles at the vertical and horizontal centerlines for Re = 500 and 1000, respectively, indicating that LBM-LES and FVM-LES are in excellent agreement. As Re increases (Re = 5000 and 10,000), the time-averaged circulating flow within the cubic cavity becomes less pronounced. As the degree of turbulence increases, the instantaneous velocities become progressively more chaotic, and the advective energy is transferred to the fluctuations and eventually dissipated as heat. The advective energy for maintaining the circulating flow is much smaller compared to the low-Re cases. For the high-Re cases, as shown in Fig. 7.5c, d, the LBM-LES and FVM-LES results disagree slightly, and deservedly so. Turbulence prediction is extraordinarily complex, and no current method can perfectly simulate turbulent motion. All CFD methods, including FVM-LES and LBM-LES, are gradually approaching the turbulence “truth” based on their respective physical perspectives and theories. Accordingly, LBM-LES cannot perfectly match the experimental or measured values, or even the classical FVM-LES results in the turbulent built environment. We should establish an appropriate simulation result acceptance level according to our purpose and the specific simulation case. The reader should accept this fact, but it does not mean that the simulations are baseless or that the results are useless. Indeed, there are many techniques and settings to improve the simulation accuracy and obtain results closer to the real situation. In the following chapters, we will discuss some simple techniques for specific built environment problems. However, CFD, including the LBM, is a complex subject and cannot be exhaustively discussed in this book.

7.4.2 Comparison Between LBM-LES and FVM-LES The time-averaged velocity profiles visually show the difference in velocity between LBM-LES and FVM-LES but cannot quantify such difference. Therefore, we can evaluate the agreement in time-averaged velocity between both methods at all grid points in the central vertical cross-section using the mean difference (MD) and mean bias (MB) indices, as defined in (7.3), and the correlation coefficient (CC).

7.4 Results and Discussion

137

Fig. 7.4 Normalized time-averaged scalar velocity at the central vertical cross-section. The gray scale indicates the scalar magnitude of the time-averaged velocity and the white arrows indicate the direction of the time-averaged scalar velocity at that location N | 1 Σ|| PLBM(i ) − PFVM(i) |, MD = N i=1

MB =

N 1 Σ PLBM(i) − PFVM(i ) . N i=1

(7.3)

Here, PFVM(i) and PLBM(i ) are the time-averaged velocities of the FVM-LES and LBM-LES at grid point I, respectively. N represents the total number of grid points. Figure 7.6 shows the comparison results for all Re cases. For Re = 500 and 1000, most points are located on the 45° line, implying a “perfect” agreement in velocity between LBM-LES and FVM-LES. Conversely, for Re = 5000 and 10,000, more points deviate from the 45° line, implying a decrease in agreement. The CC is 0.99

138 Fig. 7.5 Time-averaged velocity profile comparison for all Re cases (Left: ⟨u⟩ = vertical profile; Right: ⟨w⟩ = horizontal profile)

7 LBM-LES in Ideal 3D Lid-Driven Cavity Flow Problems

7.4 Results and Discussion

139

or larger for Re = 500 and 1000 but decreases to 0.91 for Re = 5000 and 10,000. As Re increases, the turbulence influence increases, and the agreement between LBMLES and FVM-LES decreases; nevertheless, the results are still within an acceptable range (> 0.9). The mean difference also shows the same trend, with a mean difference of approximately 0.002–0.003 for the low-Re cases, increasing almost tenfold to 0.02–0.03 for the high-Re cases. Meanwhile, the mean bias results show that LBM-LES overestimated the time-averaged velocity compared with FVM-LES; however, the results differ by only ~ 0.1%, which is negligible.

Fig. 7.6 Time-averaged velocity comparison at all grid points on the central vertical cross-section. The velocity ratio, correlation coefficient (CC), mean difference (MD), and mean bias (MB) are utilized to evaluate the agreement between methods

140

7 LBM-LES in Ideal 3D Lid-Driven Cavity Flow Problems

Fig. 7.7 An example of vortex structures at the central vertical section of the cavity (Re = 1000)

7.4.3 Comparison Between Vortex Structures A classic way to estimate the flow structure for a cavity flow is to compare the positions of the vortices the simulation reproduces (Erturk et al. 2005). In general, a clockwise large primary vortex is generated in the cavity. As Re increases, small vortices (bottom-right and bottom-left vortices) tend to occur on both sides near the bottom of the cavity (Fig. 7.7). Table 7.3 lists the positions of the central points of these vortices in the central vertical cross-section. Both LBM-LES and FVM-LES reproduce these vortices successfully. The center positions of the primary vortices obtained by both methods are in agreement; however, for Re = 10,000, the position difference is at its maximum (11%). The difference between the center positions of the bottom-right and bottomleft vortices is up to 6%. It is worth noting that the bottom-left vortex is not generated at Re = 500 because of the larger viscosity. The vortex structure also demonstrates that the difference trend in the central position of the primary and bottom vortices is larger as the turbulence effect becomes more significant.

7.5 Discussion on Computational Time and Parallel Computational Efficiency

141

Table 7.3 Vortex center position in the cavity flow Vortex position Simulation method

Re = 500

Re = 1000

Re = 5000

Re = 10,000

Primary vortex LBM-LES

0.605, 0.540

0.608, 0.468

0.610, 0.432

0.675, 0.443

FVM-LES

0.612, 0.568

0.608, 0.466

0.575, 0.465

0.576, 0.500

Distance

0.029

0.002

0.048

0.114

LBM-LES

0.911, 0.152

0.900, 0.120

0.950, 0.061

0.935, 0.055

FVM-LES

0.921, 0.150

0.905, 0.121

0.892, 0.061

0.955, 0.044

Distance

0.010

0.005

0.058

0.023

LBM-LES

–

0.089, 0.069

0.080, 0.070

0.075, 0.075

FVM-LES

–

0.086, 0.059

0.070, 0.110

0.072, 0.046

Distance

–

0.013

0.041

0.029

Bottom-right vortex

Bottom-left vortex

Note The number pairs represent the x- and z-coordinates of the points where the vortex centers are located, “-” indicates that the vortex did not appear in the simulation

7.5 Discussion on Computational Time and Parallel Computational Efficiency One of the main advantages of the LBM is its algorithmic simplicity and reliable parallelism. To compare the computational time and parallel computational efficiency (PCE) between FVM-LES and LBM-LES, we use the Re = 10,000 case results at different grid resolutions. The simulations were carried out using one CPU core (Intel (R) Xeon (R) E5-2667 v4 @ 3.20 GHz) in series and two CPUs with 32 cores in parallel. We call the simulation time cost for the former the serial computational time (SCT), and that for the latter the parallel computational time (PCT). Several parameters have been proposed to describe the computational efficiency of CFD methods. Among them, we select the N-index Nt proposed by Zuo et al. (2009), the computational time ratio (CTR), and PCE for the following discussion. N-index Nt is defined as (7.4), where t flow is the physical time of the flow motion (120 [s] in this case) and t sim is the elapsed computational time required for FVM-LES or LBM-LES. Nt =

t flow . t sim

(7.4)

CTR is defined as the ratio of the computational time of FVM-LES to that of LBM-LES under the same number of cores, as in (7.5). A smaller CTR indicates that LBM-LES is faster than FVM-LES in the calculation. CTR =

SCT(or PCT)of FVM-LES . SCT(or PCT)of LBM-LES

(7.5)

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7 LBM-LES in Ideal 3D Lid-Driven Cavity Flow Problems

Table 7.4 Computational time and PCE comparison Grid resolution Method [m] (mesh size [million])

Discretization Nt (SCT) [–] time step δt [s]

Nt (PCT) [–]

H/50 (0.125)

LBM-LES

1/100

9.700 × 10−2

3.085 × 10−1

3.18

FVM-LES

1/50

5.568 × 10−2

4.286 × 10−1

7.70

LBM-LES

1/200

2.031 × 10−2

2.105 × 10−1

10.36

FVM-LES

1/100

2.516 × 10−3

1.998 × 10−2

7.94

LBM-LES

1/800

1.200 × 10−3

1.330 × 10−2

11.09

1/200

1.168 × 10−4

9.559 × 10−4

8.18

H/100 (1.0)

H/200 (8.0)

FVM-LES

PCE [–]

CTR

1.74 (SCT) 0.722 (PCT) 8.07 (SCT) 10.54 (PCT) 10.27 (SCT) 13.91 (PCT)

Here, PCE is employed to evaluate how the computational speed could be improved when utilizing more CPU cores. PCE is defined as the ratio of SCT to PCT, as in (7.6). PCE =

SCT . PCT

(7.6)

Table 7.4 shows the above indices for LBM-LES and FVM-LES. In the simulations, Nt is less than unity in all cases, indicating that the computational time is larger than the physical time for flow field development. When the number of grids is small, the SCT of LBM-LES is almost half that of FVM-LES. Conversely, the PCT of LBM-LES is larger than that of FVM-LES. As the number of grids increases, the computational time of LBM-LES is increasingly smaller than that of FVM-LES, reaching a minimum close to 1/14 of that of FVM-LES. The maximum FVM-LES PCE of this simulation case approaches approximately 7–8, independent of the number of grids. Conversely, that of LBM-LES improves significantly as the number of grids increases. This could be attributed to the local LBM-LES algorithm. Therefore, the LBM-LES PCE is higher compared to that of FVM-LES in the simulations, and, in particular, its PCE is more strongly dependent on the number of grids.

7.6 Summary In this chapter, a simple test case of an ideal 3D lid-driven cavity flow is presented and discussed. We conducted LBM-LES simulations for low-Re cases. We initially show that LBM-LES is able to correctly represent the physical flow structure, similar

References

143

to FVM-LES. We also explored the level of agreement between the LBM-LES and FVM-LES results and compared their computational efficiency. As a first case study, the 3D lid-driven cavity flow was investigated as an ideal problem with almost no turbulence. In the following chapters, we will discuss LBM-LES simulations for practical, more complex indoor and outdoor turbulent flows in the built environment.

References Erturk E, Corke TC, Gökçöl C (2005) Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. Int J Numer Meth Fluids 48:747–774. https://doi.org/10. 1002/fld.953 Murakami S (1993) Comparison of various turbulence models applied to a bluff body. J Wind Eng Ind Aerodyn 46–47:21–36. https://doi.org/10.1016/0167-6105(93)90112-2 van Driest ER (1956) On turbulent flow near a wall. J Aeronaut Sci 23:1007–1011. https://doi.org/ 10.2514/8.3713 Zuo W, Chen Q (2009) Real time or faster-than-real-time simulation of airflow in buildings. Indoor Air 19:33–44

Chapter 8

LBM-LES in an Isothermal Indoor Flow Problem

8.1 Introduction The indoor flow environment is a very important component of the built environment. Modern humans spend 60–90% of their time indoors in buildings. Indoor thermal comfort and air quality have an important impact on people’s health and quality of life. Among the various indoor environmental factors that affect our health, airflow plays a fundamental role. Indoor airflow produces rich flow dynamics, including phenomena such as separation, recirculation, and reattachment. These physical phenomena and behaviors make indoor airflow suitable for testing the accuracy of CFD methods in reproducing and predicting ventilation systems. In recent decades, numerous researchers have utilized the LBM to simulate indoor turbulent flow. By implementing the LBM on a CAD platform, Crouse et al. (2002) simulated the thermal-flow field inside an actual building. Elhadidi et al. (2013) compared the accuracy of LBM and FVM over various discrete time intervals by employing the 2D indoor turbulent flow model of the IEA Annex 20 (Lemaire et al. 1993) their results showed that an FVM with a coarse grid system exhibited a higher accuracy and was less computationally expensive than the LBM. ITO (2012) compared the mean velocity field of an indoor flow obtained via LBM-RANS to that obtained via integrated FVM-LES. In his research, the 3D LBM implementation results agreed with the experimental results, whereas the 2D LBM yielded a relative error. Sajjadi et al. (2017) introduced LBM-RANS with the k-ε model to simulate indoor ventilation and particle transport, and their results exhibited satisfactory accuracy. Then, they employed the LES-LES coupled with the MRT scheme to simulate indoor flow and particle diffusion and accumulation; the simulation results were found to agree well with the experimental results (Sajjadi et al. 2016). Han et al. (2019), Han (2022) provided an in-depth discussion of the effects of various simulation parameters on indoor airflow simulations via LBM-LES. They further adopted wall functions to improve the accuracy of the bounce-back BCs for indoor airflow simulation (Han et al. 2021a).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Han and R. Ooka, Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems, https://doi.org/10.1007/978-981-99-1264-3_8

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8 LBM-LES in an Isothermal Indoor Flow Problem

In this chapter, the standard indoor airflow case from Annex 20 published by the International Energy Agency (IEA Annex 20 (Lemaire et al. 1993)) in 1990 is adopted for LBM-LES. Nielsen et al. (2010) originally implemented this case for indoor airflow CFD simulation. In 1998, Nielsen (1998) provided a general guide for the turbulence model choice for this case. Later, Sørensen and Nielsen (2003) presented guidelines specifically for indoor air environments and made recommendations to improve the quality of the CFD simulation results. By then, this case had become the standard system for CFD testing. Section 8.2 describes the IEA Annex 20 indoor airflow case, while Sect. 8.3 presents the specific simulation parameters, boundary conditions, and the parameters that will be the focus of the subsequent discussion. In Sect. 8.4, the simulation results are analyzed according to various aspects and the effect of each parameter on the simulation results is discussed. As with the 3D lid-driven cavity flow case presented in Chap. 7, a comparison of the simulation results between LBM-LES and FVMLES is discussed in Sect. 8.5. Finally, the LBM-LES computational time and parallel computational efficiency are discussed in Sect. 8.6.

8.2 Isothermal Indoor Flow Problem Description The room model for the isothermal indoor airflow benchmark case of the IEA Annex 20 (Lemaire et al. 1993) is presented in Fig. 8.1. The room has a rectangular shape, with air flowing in through a narrow inlet at the top of one side and out through a narrow outlet at the bottom of the opposite side. The room specifications are as follows: L/H = 3.0, h/H = 0.056, and r/H = 0.16, where L , H, h, and r are the room length, room height, inlet slot height, and outlet slot height, respectively. Here, L = 9.0[m], meaning that a full-scale room model is  employed  to simulate the actual indoor flow. The inflow air velocity is Uin = 0.455 m · s−1 . According to the IEA report, the inlet turbulence intensity during the experiment is only 4%, which can be considered to have a negligible impact on the velocity. Therefore, the inlet velocity fluctuations are ignored in our simulations, as assumed in previous studies (Emmerrich and McGrattan 1998; Zhang and Chen 2000). Fig. 8.1 Schematic of the isothermal indoor flow benchmark case (from IEA Annex 20 (Lemaire et al. 1993))

8.2 Isothermal Indoor Flow Problem Description

147

According to the IEA Annex report (Lemaire et al. 1993), the Reh defined by the inlet height and velocity is Reh =

hUin = 5000. ν

(8.1)

Meanwhile, as the full-scale room model was employed, the Re H defined by the room height and inlet velocity is Re H =

HUin = 91,000. ν

(8.2)

The turbulent length scale lt is estimated as lt =

h = 0.0168 [m]. 10

(8.3)

Nielsen et al. (2010) conducted scaled-down wind tunnel experiments for this flow case. In their experiments, ⟨u√ x ⟩ (i.e., the mainstream directional component of the time-averaged velocity) and u ,2 x (i.e., the root-mean-square of the velocity fluctuations in the x-direction) were measured via laser Doppler anemometry (LDA). All measurement points were located in the room’s central vertical cross-section, parallel to the airflow streamwise direction. Considering the airflow symmetry, the spanwise component of the airflow in this section can be considered zero (at least its time-averaged value). Two vertical measurement point sets were located at x = H and 2H , respectively, and two horizontal measurement point sets were located at y = 1/2h and H − 1/2h, respectively, as shown in Fig. 8.2. We employed the experimental data from Nielsen et al. (2010) as a basis for the simulation.

Fig. 8.2 Schematic of the measurement point locations in the Nielsen et al. (2010) experiment. All measurement points are located in the central vertical cross-section of the room (z/H = 0)

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8 LBM-LES in an Isothermal Indoor Flow Problem

8.3 Simulation Methodology and Boundary Conditions 8.3.1 Simulation Conditions Similar to the previous case, both LBM-LES and FVM-LES were conducted for the isothermal indoor flow case. The difference is that we no longer use FVM-LES as a benchmark to assess the simulation results, instead, the comparison is made with Nielsen et al.’s experimental results. FVM-LES is mainly used to evaluate the consistency and computational time of the LBM-LES results compared with popular methods. Table 8.1 lists the primary simulation parameters and BCs used in this case. LES employs the standard Smagorinsky SGS model for both the LBM-LES and FVM-LES models. The Smagorinsky constant Cs is set to 0.12, as proposed by Murakami et al. (Murakami 1993) for the built environment. As the standard Smagorinsky model can overestimate νsgs near the walls, a damping function is usually utilized for νsgs corrections in FVM-LES, e.g., the van Driest-style damping function (Driest 1956). However, the standard bounce-back BC in the LBM does not contain a damping function. Therefore, as in comparable cases, the FVM-LES in this case also avoids using a damping function. No-slip BCs are employed for the walls in both LBM-LES and FVM-LES to ensure similar simulation conditions. Notably, the standard bounce-back condition is implemented in LBM-LES, which is the most popular no-slip wall BC in LBM-LES largely due to its simplicity. The main principle of the bounce-back condition is to describe the particles, which hit the wall and bounce back rather than move forward, implying that there is no flux Table 8.1 Simulation conditions for the isothermal indoor flow case

Item

LBM-LES

FVM-LES

Simulation domain 9.0 (L) × 3.0 (W) × 3.0 (H) [m] SGS model

Standard Smagorinsky model, Cs = 0.12 (Murakami 1993)

Time step and marching

0.01 s, PISO

0.005 s

Time discretization –

Euler-implicit

Space discretization

–

2nd-order central difference

Simulation period

Preparatory: 18 [min]; Sampling: 6 [min] (air change rate: 0.172 [min−1 ])

Inlet BC

Uniform velocity boundary,   Uin = 0.455 m · s−1 , , (no fluctuations), h = 0.056 H

Outlet BC

Gradient-zero, r = 0.16 H

Other BCs

Bounce-back condition

No-slip BC

8.3 Simulation Methodology and Boundary Conditions

149

across the wall and no relative transverse motion between the fluid and wall (Krüger et al. 2017). As LES is sensitive to the sampling period, a period of 6 min is utilized, similar to the room air exchange time after a sufficient preparatory flow. According to our advance test, this sampling period is adequate. For the FVM-LES, the discrete time interval is set such that the maximum Courant number (Co) remains less than unity; for the LBM-LES, the time interval is set to the maximum value required to ensure simulation stability (q.v. Sects. 6.4.2 and 6.4.3). One computational node with two CPUs, each with an Intel (R) Xeon (R) E5-2667v4 3.20-GHz eight-core processor, was used for this case. Additionally, to evaluate the LBM-LES and FVM-LES PCE, 1-, 2-, 4-, 8-, and 16-core simulations were performed.

8.3.2 Parameters to Discuss In addition to the main simulation parameters described in Sect. 8.3.1, in this chapter we also discuss the parameters that may affect the simulation results, particularly the unique parameters that distinguish the LBM from the classical FVM. First, different uniform grid resolutions are used to verify the grid dependence of LBMLES (Sect. 8.4.2). Then, the effects of different discrete-velocity and relaxation time schemes are examined (Sect. 8.4.3). In particular, as discussed in Sect. 6.4, the simulation result accuracy is affected by compressibility errors and over-relaxation oscillations if an improper discrete time interval δt is selected, for which LBMLES and FVM-LES differ in an important aspect. Therefore, the influence of δt is also discussed in this chapter (Sect. 8.4.4), as well as the compressibility errors (Sect. 8.4.5) and over-relaxation oscillations (Sect. 8.4.6) derived therefrom. The selection of these parameters is listed in Table 8.2. Subsequently, based on the above parameters, we summarize a set of LBM-LES parameters suitable for indoor environments according to the simulation accuracy, and discuss its agreement with FVM-LES (Sect. 8.5), as well as their computational performance (Sect. 8.6). Table 8.2 Various LBM parameters of isothermal indoor airflow from IEA Annex 20 Parameter

Parameter settings

Grid resolution [m]

H/75; H/150; H/300

Discrete time interval [s]

1/50; 1/100; 1/200; 1/400; 1/800; 1/1600;1/3200

Relaxation time schemes

SRT(BGK); MRT

Discrete velocity schemes

D3Q19; D3Q27

150

8 LBM-LES in an Isothermal Indoor Flow Problem

8.4 Results and Discussion 8.4.1 Instantaneous and Time-Averaged Scalar Velocities Indoor airflow is typically a turbulent environment with Re in the tens of thousands, and its instantaneous wind velocity is often chaotic (Fig. 8.3), making it difficult to understand the macroscopic information on turbulence by analyzing the specific instantaneous velocity alone. Accordingly, we adopted a statistical approach and implemented the time-average operation to obtain the time-averaged velocity ⟨u x ⟩ √ and turbulent fluctuating u ,2 x components. Figure 8.4 shows the time-averaged scalar velocity at the central vertical crosssection. A large flow circulates throughout the entire room, with region-stagnated airflow occurring in the center of the room. LBM-LES can reflect the flow features, similar to FVM-LES. This is a preliminary and intuitive demonstration of the LBMLES simulation results. Next, we will discuss the quantitative impact of various simulation parameters on the LBM-LES simulation accuracy.

Fig. 8.3 Raw instantaneous scalar velocity at the central vertical cross-section

Fig. 8.4 Normalized timed-averaged scalar velocity at the central vertical cross-section

8.4 Results and Discussion

151

8.4.2 Effects According to Grid Resolution In Sect. 6.4.1, we explained how the size of the grid system may impact the simulation accuracy. Here, we examine the influence of grid resolution on the simulation accuracy; note that the SRT and D3Q19 schemes are implemented in all simulation cases presented in this section. As illustrated in Fig. 8.5, except for δ L = H/75 [m], which exhibits the lowest accuracy, LBM-LES is generally able to simulate the ⟨u x ⟩ profiles with an accuracy that is comparable to that of FVM-LES. Additionally, at the inlet height where x = H and 2H , it can be clearly observed that the velocity accuracy increased as the grid resolution increased. This is also the case for y = 1/2h. For y = H − 1/2h, all LBM-LES cases yielded results that were in agreement with the experimental results,√except for δ L = H/75 [m]. ,2 Figure 8.6 shows the √ u x distributions. Near the ceiling where x = H , the FVMLES overestimates u ,2 x ,√resulting in higher LBM-LES accuracies. Conversely, LBM-LES overestimates u ,2 x near the inlet √ where y = 1/2h. It is notable that both LBM-LES and FVM-LES underestimate u ,2 x values at a lower area of x = 2H and along y = H − 1/2h. This underestimation has been previously reported as a result of implementing FVM-LES (Voigt 2007; Davidson et al. 1996). Therefore, it may not be an inherent problem of LBM-LES but may be due to conditional differences between the simulations and the experiments. In Chap. 7, we examined the differences between LBM-LES and FVM-LES based on the mean difference (MD) and mean bias (MB). In this chapter, we use the mean error (ME) to quantitatively assess the LBM-LES simulation accuracy compared with experimental values. ME is defined as in (8.4); smaller ME values indicate a better agreement between the simulated and experimental results, which implies a higher simulation accuracy. Attentive readers may have noticed that the definition of ME is almost identical to the form of MD, except that the FVM results in MD are replaced by the experimental values. Despite their definitions being similar, ME has a different meaning than MD and MB. We cannot arbitrarily assume that LBM-LES is “more correct” or that FVM-LES is “more accurate”; accordingly, in Chap. 7, we only discussed their “differences” and “biases”. On the contrary, the experimental values are “correct” (this is common practice in the built environment field, where experimental data are generally considered reliable despite the errors and limitations of the experiment), we consider the difference between the LBM-LES results and the experimental values to be an “error”. ME =

N | 1 Σ|| PLBM or FVM(i ) − PEXP(i) |. N i=1

(8.4)

√ Here,PLBM or FVM (i) is ⟨u x ⟩ or u ,2 x predicted via LBM-LES or FVM-LES at point i, PEXP(i) is the experimental value for the corresponding point, and N is the number of data points used in the evaluation. Note that all measurement points in Nielsen et al.’s experiment are considered here.

152

8 LBM-LES in an Isothermal Indoor Flow Problem

Fig. 8.5 ⟨u x ⟩ profile according to grid resolution (Han et al. 2019)

√ Figure 8.7 shows the ME for ⟨u x ⟩ and u ,2 x at all simulated points (All Data), as well as at the points near the ceiling (y = 1/2h), near the ground (y = H − 1/2h), and away from the walls (x = H and 2H ). The subregional ME values for ⟨u x ⟩ reveal that the maximum MEs are located and FVM-LES, regardless of grid near the ceiling (y = 1/2h) for both LBM-LES √ resolution. However, the differences in u ,2 among various subregions are small. x Among all LBM-LES cases, the ⟨u x ⟩ accuracy tends to increase markedly when the grid resolution increases from H/75 to H/150 [m], and most of the increase is attributed to that at y = 1/2h. With an increase in resolution from H/150 to

8.4 Results and Discussion

Fig. 8.6

153

√ u ,2 x profile according to grid resolution (Han et al. 2019)

H/300 [m], the ⟨u x ⟩ accuracy only improves slightly. The FVM-LES accuracy √ is equivalent to that of LBM-LES between H/150 and H/300 [m]. Regarding u ,2 x, the LBM-LES accuracy is not significantly affected by grid resolution and is slightly higher than that of FVM-LES. Thus, the LBM-LES accuracy is highly dependent on grid resolution, similar to classic NSE-based LES. δ L = H/300 [m] produces the highest accuracy in this simulation. However, considering both accuracy and grid cost, the accuracy for H/150 is adequate, producing an accuracy comparable to that of FVM-LES. It should be noted that the standard halfway bounce-back wall condition in the LBM is not as accurate near the wall as FVM-LES. A finer near-wall grid is required

154

8 LBM-LES in an Isothermal Indoor Flow Problem

Fig. 8.7 ME according to grid resolution for the indoor airflow LBM-LES simulation results (Han et al. 2019)

to produce a similar accuracy as FVM-LES. This is because bounce-back corresponds to a no-slip BC that cannot completely capture the correct shear drag at the wall when the grids are coarse in high-Re flows (q.v. Sect. 3.7). Elhadidi and Khalifa (2013) and Han et al. (2019) also reported this issue. Accordingly, an improved bounce-back condition, such as the WFB proposed by Han et al. (2021a, b), can be used to obtain higher near-wall accuracy (q.v. Sect. 3.8).

8.4 Results and Discussion

155

8.4.3 Effects According to Relaxation Time and Discrete Velocity Scheme The grid dependency discussed in the previous section is mandatory for most CFD methods; however, the relaxation time and discrete velocity schemes are parameters unique to the LBM. In this section, the effects of different relaxation time (SRT or MRT) and discrete velocity (D3Q19 or D3Q27) schemes on the simulation accuracy are examined, as shown in Figs. 8.8 and 8.9. Grid independence tests are conducted in advance to essentially rule out the influence on grid systems. As illustrated in Fig. 8.8, for x = H , D3Q27 yields u x values that are in good agreement with the experimental values at y > 0.6H ; other schemes slightly underestimate u x in this area. In the other areas, the D3Q27 and MRT results are very similar

Fig. 8.8 ⟨u x ⟩ profiles for various relaxation time and discrete velocity schemes (Han et al. 2019)

156

Fig. 8.9

8 LBM-LES in an Isothermal Indoor Flow Problem

√ u ,2 x profiles for various relaxation time and discrete velocity schemes (Han et al. 2019)

√ to those of the SRT + D3Q19 scheme. From the u ,x2 profiles shown in Fig. 8.9, the MRT and D3Q27 results are also comparable to those of SRT + D3Q19. Figure 8.10 shows the ME√ values according to the relaxation time and discrete velocity schemes. The u x and u ,2 x accuracies under the conditions of the MRT and D3Q27 schemes are almost the same. Thus, in this isothermal indoor flow problem, the MRT and D3Q27 schemes have little influence on the simulation accuracy.

8.4.4 Effects According to the Discrete Time Interval In Chap. 6, we discussed how the selection of discrete time interval δt may cause compressibility errors and affect the LBM-LES results, which is an important feature of the LBM as opposed to the classical incompressible NSE-based

8.4 Results and Discussion

157

Fig. 8.10 ME according to the relaxation time and discrete velocity schemes for the indoor airflow LBM-LES simulation results (Han et al. 2019)

method. Next, to discuss the effect of δt on the indoor turbulent environment simulation results, a total of 11 cases are configured as follows: δt = 1/50, 1/100, 1/200, 1/400, 1/800, and 1/1600 [s] for a grid resolution of δ L = H/75 [m] and δt = 1/200, 1/400, 1/800, 1/1600, and 1/3200 [s] for a grid resolution of δ L = H/150 [m]. As this is an important issue in LBM simulations that is often overlooked by engineers and practitioners, we break it down into three sections for a detailed discussion. In this section, the simulation results for the abovementioned 11 cases are presented and discussed. In Sect. 8.4.5, the compressibility errors due to δt are discussed for some cases. Finally, the over-relaxation numerical oscillations that occur in the remaining cases are discussed in√ Sect. 8.4.6. Figures 8.11 and 8.12 show the u x and u ,2 x results for the δ L = H/75 and H/150 [m] case groups, respectively. In the δ L = H/75 [m] group, nearly all cases reproduce the correct u x , except for δt = 1/50 [s], which exhibits the worst accuracy. In general, the LBM-LES

158

Fig. 8.11 u x (left) and

8 LBM-LES in an Isothermal Indoor Flow Problem

/ , u x2 (right) profiles for the δ L = H/75 [m] case group

√ accuracy improves as δt decreases, with u ,2 x showing a similar trend. √ It is worth noting that slight spatial numerical oscillations occur in both u x and u ,2 x for δt = 1/800 and 1/1600 [s]. In the δ L = H/150 √ [m] group, the best accuracy is achieved for δt = 1/200 [s] for both u x and u ,x2 . As δt decreases, no obvious√improvement in accuracy is observed. By contrast, evident oscillations occurred in u ,2 x , which negatively affects the accuracy. In addition to the ME, we introduce another common index used for evaluating simulation errors in fluid mechanics, the L2 error norm ε P (Krüger et al. 2017), which is defined as ⎡ |Σ

2 | i PLBM(i ) − PEXP(i) √ Σ 2 (8.5) εP = i PEXP(i )

8.4 Results and Discussion

Fig. 8.12 u x (left) and

159

/ , u x2 (right) profiles for the δ L = H/150 [m] case group

Here, PEXP(i) and PLBM(i ) represent physical property P at position i obtained experimentally and via LBM-LES, respectively. All experimental data are considered. A smaller ε P indicates a smaller error between the simulation and experimental results, in turn indicating a higher accuracy. The ε P results for all cases are shown in Fig. 8.13. We speculate that the increase in accuracy with a change in δt from 1/50 to 1/200 [s] for the δ L = H/75 [m] case group may be due to the corresponding compressibility error reduction, which is discussed in Sect. 8.4.5. The subsequent decrease in accuracy may √ be due to over-relaxation, as apparent oscillations are observed in both u and u ,2 x . Meanwhile, for the δ L = H/150 [m] case group, the decrease in accuracy with a decrease in δt should also be due to the oscillation-induced √ over-relaxation, especially in u ,x2 . We will discuss this assumption in Sect. 8.4.6.

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8 LBM-LES in an Isothermal Indoor Flow Problem

Fig. 8.13 L2 error norm according to δ t for all cases

8.4.5 Discussion on Compressibility Errors Next, let us discuss the compressibility errors using three cases, namely, δt = 1/50, 1/100, and 1/200 [s], under a grid resolution of δ L = H/75 [m]. These cases were selected for this purpose because an apparent difference in their results was observed with no oscillations, suggesting that compressibility errors may have caused the difference. Figure 8.14 shows the streamwise component distributions of normalized time-averaged velocity u x in the central vertical plane for these three cases. For δt = 1/50[s], the airflow from the inlet tended to move downward and away from the ceiling, rendering the least accurate results. As δt decreases, the airflow tends to be more horizontal. Figure 8.15 shows the normalized time-averaged density ρ for the three cases. As the LBM is a pseudo-compressible simulation method, the results indicate that ρ varies significantly according to δt , even though the prototype is an incompressible flow problem and should remain constant. A larger δt generates a change in ρ from the initial value for δt = 1/50 [s], especially in the area near the airflow inlet. This significant density change near the inlet corresponds to the airflow deviation

8.4 Results and Discussion

161

Fig. 8.14 Streamwise component distributions of normalized time-averaged velocity u x in the central vertical plane for three δt cases and δ L = H/75[m]

shown in Fig. 8.14a, implying that the larger density gradient resulted in significant compressibility errors. By obtaining the spatial average of ρ for the entire simulation domain, one can intuitively see the relationship between the change in [[⟨ρ⟩]] from ρ0 according to δt (Fig. 8.16). Here, [[·]] means the spatial average operation. The change rate decays almost as a power function as δt decreases by half. When the deviation is below 0.50%, the change rate decay is not as apparent, even though δt continues to decrease. At that time, the distribution of ρ approaches the initial value, and thus, the effect of compressibility can be ignored. Figure 8.17 shows the distribution of Ma (in LBU, defined by u LBU , q.v. Sect. 6.4.2) on the central vertical section. Ma near the inlet for δt = 1/50 [s] is larger than that in other areas or that for other cases, with values exceeding 0.3. This is the area with a significant ρ change and downward airflow. This finding accords with those of previous studies (Krüger et al. 2017), which suggested that Ma in the LBM-LES field should be small (Ma < 0.3) and a larger Ma would yield more significant compressibility errors. The airflow tends to be horizontal with decreasing Ma in other areas for this case, indicating that the compressibility errors were compensated by decreasing Ma. In other cases, Ma < 0.3 is tenable such that the compressibility errors are neglected.

162

8 LBM-LES in an Isothermal Indoor Flow Problem

Fig. 8.15 Normalized time-averaged density ρ distributions in the central vertical plane for the three test cases

Fig. 8.16 Spatially averaged density [[⟨ρ⟩]] change according to δt for all cases

These findings confirm that apparent compressibility errors may occur when solving turbulent flows via LBM-LES due to large Ma. By adjusting δt , Ma can decrease and compensate for such errors. It is noteworthy that Ma is an LB property defined by u LBU , and can vary by utilizing different parameters despite the simulation prototype being the same; this distinguishes it from the Ma defined by the actual velocity in the physical field. Therefore, to avoid apparent compressibility errors, we

8.4 Results and Discussion

163

Fig. 8.17 Ma distribution in the central vertical plane for the three test cases

have to adjust δt to satisfy Ma < 0.3 where the local velocity reaches the maximum, as 0.3es δ L δt < | PHU | . |u |

(8.6)

max

| | Here, |uPHU |max is the magnitude of the maximum local scalar velocity in physical units.

8.4.6 Discussion on Oscillations Caused by Over-Relaxation As discussed in Sect. 8.4.4, significant oscillations can be observed when δ L = H/75 [m] and δt < 1/200 [s]. Furthermore, the oscillations are more obvious when δ L = H/150 [m] and δt < 1/400 [s]. This is most likely due to the over-relaxation collision pattern. Figure 8.18 illustrates the relaxation situation of distribution function f 0 in a 1-s period (1080–1081 [s]) at point (x, y, z) = (H, H/2, H/2). This point

Fig. 8.18 Over-relaxation of f 0 in a 1-s period for all cases at point (x, y, z) = (H, H/2, H/2)

164 8 LBM-LES in an Isothermal Indoor Flow Problem

8.5 Comparison Between LBM-LES and FVM-LES

165

is interesting because it shows a critical state in which the simulation performs well when δt > 1/200 [s] but significant numerical oscillation occurs when δt < 1/200 [s] under a grid resolution of δ L = H/75 [m]. eq f 0 shuttles back and forth over f 0 in all cases, indicating that all the collision steps in the simulations are of the over-relaxation type and not the ideal under-relaxation eq pattern (q.v. Sect. 6.4.3), as f 0 does not always decay exponentially toward f 0 . This result verifies that in turbulent flows, over-relaxation is more typical than underrelaxation for the collision step. In the δ L = H/150 [m] case group, the “frequency” for f 0 shuttling increases by decreasing δt . Meanwhile, the “amplitude” for f 0 shuttling decreases with decreasing δt . However, from δt = 1/50 to 1/200 [s] under δ L = H/75 [m], the shuttles do not transform into oscillations; in contrast, one can consider them to be formed by fluctuations in turbulence. Conversely, the fluctuations transform into visible oscillations when the δt reaches 1/400 [s] or smaller, concurring with cases in which velocity oscillations occur. Analogously, in the δ L = H/150 [m] case group, the turbulence fluctuations transform into apparent numerical oscillations when δt decreases to a certain degree; ultimately, oscillations in the fluctuating velocities occur, especially when δt = 1/1600 and 1/3200 [s]. Accompanied by time series oscillations in the undersize of δt , the standard deviations of the fluctuating velocities become larger, as shown in Fig. 8.12. In summary, the relaxation time parameter τ decreases accompanied by a decrease in δt , which also means that the viscosity in LBU νLBU decreases. When decreasing to a certain degree, the viscosity becomes insufficient, and the over-relaxation collision pattern results in distribution function oscillations, ultimately causing abnormal macroscopic velocity oscillations. However, it is difficult to determine an acceptable minimum δt to avoid oscillations because it is affected by νSGS , which in turn varies in a case-by-case basis. Furthermore, it is also challenging to quantitatively determine the relationship between the “amplitude” of the distribution function shuttling and the macroscopic velocity oscillations.

8.5 Comparison Between LBM-LES and FVM-LES According to the discussion in Sect. 8.4, we can roughly determine a set of recommended LBM simulation parameters for turbulent indoor environments, as shown in Table 8.3. It is worth noting that these parameters are currently only suitable for the flow problem presented in this chapter but can act as a reference for other indoor flow problems. In this section, we present a comparison between LBM-LES and FVM-LES based on the LBM parameters in Table 8.3. To evaluate the consistency between the two methods, we use the hit rate (HR), as defined in (8.7). HR is among the evaluation metrics recommended by the popular European COST 732 project (Schatzmann et al. 2010) and the best practice guideline for the CFD simulation of flows in the urban environment (Franke et al. 2007). It is commonly used to evaluate the accuracy of CFD simulation results in the built

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8 LBM-LES in an Isothermal Indoor Flow Problem

Table 8.3 Recommended LBM parameters for the isothermal indoor flow problem in IEA Annex 20

Parameter

Settings

Grid resolution

H/150 [m]

Discrete time interval

1/200 [s]

Relaxation time schemes

SRT(BGK) or MRT

Discrete-velocity scheme

D3Q19

environment, especially for outdoor built environments. We will also employ it in the outdoor flow problem presented in the next chapter. HR indicates the rate of points (or possibility) at which the LBM-LES deviations relative to FVM-LES are not larger than the allowed tolerance. 1 Σ n i , with n i N i=1 | |  | | | P ) −PFVM(i ) | 1 for | LBM(iPFVM(i | ≤ DTo or| PLBM(i) − PFVM(i) | ≤ WTo ) N

HR = =

0 else

.

(8.7)

Here,PLBM(i) and PFVM(i ) are the LBM-LES and FVM-LES predicted values (scalar velocity or k) at point i, respectively. N is the total number of points, and DTo accounts for the relative uncertainty of the comparison data. WTo describes the repeatability of the comparison data. The predicted values for all measurement points are taken into account in HR. Fig. 8.19 shows the consistency between the LBM-LES and FVM-LES results for all experimental measurement points; the HR √ results when the consistency is 70, 90, and 95% are also shown. For both ⟨u x ⟩ and u ,2 x , over 90% of the LBM-LES results satisfy the 70%-consistency with FVM-LES, and approximately half of the results satisfy the 95%-consistency, both for the time-averaged and fluctuating velocity. This implies that similar simulation results can be obtained via LBM-LES and FVM-LES. Regarding the subregions, the ⟨u x ⟩ consistency results showed that for x = 2H and y = H − 1/2h, most points are concentrated within the 90%-consistency range. Conversely,√a few points for x = H and y = 1/2h are outside the 70%-consistency range. The u ,2 x consistency results show that nearly all points for x = H , x = 2H , and y = H − 1/2h are concentrated within the 70%-consistency range, with only a few outliers for y = 1/2h.

8.6 Discussion on Computational Performance Similar to the previous chapter, we want to assess the LBM-LES and FVM-LES performance in terms of computational time and PCE in the indoor turbulent environment simulation. The simulations for this problem were carried out using one

8.6 Discussion on Computational Performance

Fig. 8.19 HR of the consistency between LBM-LES and FVM-LES for u x (left) and Consistencies of 70, 90, and 95% are evaluated

167

/ , u x2 (right).

CPU (Intel (R) Xeon (R) E5-2667 v4 @ 3.20 GHz) with one core (serial simulation) and two CPUs with up to a maximum of 32 cores (parallel simulation). The computational time is discussed in Sect. 8.6.1, including the SCT and PCT (q.v. Sect. 7.5), while the parallel computational efficiency (PCE) is discussed in Sect. 8.6.2.

8.6.1 Computational Time We normalize the LBM-LES and FVM-LES computational time based on the physical flow time (24 min) for the entire simulation period, and obtain the relationship between computational time and the number of CPU cores, as shown in Fig. 8.20. As the MRT and D3Q27 schemes do not affect the accuracy, the discussion is based on the SRT and D3Q19 schemes. In fact, using the D3Q27 scheme will significantly increase the computational time, which we will discuss in the next chapter. The computational time increases by approximately 16 times when the grid width is halved for both the LBM-LES and FVM-LES. For all cases, the computational times of both the LBM-LES and FVM-LES exhibit an approximately linear decrease with an increase in the number of cores. Furthermore, the LBM-LES computational time for δ L = H/150 [m] is comparable to that of FVM-LES for H/75 [m], while the LBM-LES computational time for H/300 [m] is comparable to that of FVMLES for H/150 [m]. Although the LBM-LES SCT for H/150 [m] is significantly longer than that of FVM-LES for H/75 [m], its PCT approaches that of FVM-LES as the number of cores increases, even falling below that of the FVM-LES when 16 cores are employed. One can observe a similar trend in the relationship between the computational times of LBM-LES for H/300 [m] and FVM-LES for H/150 [m].

168

8 LBM-LES in an Isothermal Indoor Flow Problem

Fig. 8.20 Normalized LBM-LES and FVM-LES computational time under various grid resolutions. The solid and dashed lines represent the LBM-LES and FVM-LES results, respectively

As the number of CPU cores approaches 32, we can observe an increase in the LBM-LES computational time. This is likely due to the communication bandwidth limitation between computation nodes. For the 32-core simulations, two computational nodes (each node contains 16 cores) are utilized with a hub for intercommunication. In these cases, the communication bandwidth limitation for all hub ports is such that the data exchange becomes a bottleneck in terms of computational speed. We will discuss the hub performance bottleneck in detail in Sect. 9.6.1. These results indicate that the LBM-LES computational speed is significantly greater than that of FVM-LES for the same grid resolution. In particular, the LBMLES is faster than FVM-LES when more cores are utilized (16 cores). As the communication bandwidth between ports becomes a limitation, the computational time will increase even when using more cores.

8.6.2 Parallel Computational Performance In addition to computational time, we also discuss the parallel computational performance of LBM-LES and FVM-LES. We describe their computational performance by employing the computational time ratio (CTR), defined by (7.5), and the parallel computational efficiency (PCE), defined by (7.6). CTR indicates how fast the LBMLES is compared to FVM-LES when using the same number of cores. PCE indicates how the LBM-LES or FVM-LES computational speed increases when using multiple cores compared to a single core. Figure 8.21 shows the CTR and PCE results for δ L = H/75 and H/150 [m]. In general, the CTR results are far greater than unity, indicating that the LBM-LES computational speed is indeed greater than that of FVM-LES, and this trend is more evident as the number of grids increases. The computational speed of LBM-LES in

8.7 Summary

169

Fig. 8.21 CTR and PCE for all cases

this simulation can reach a maximum of approximately 27 times that of FVM-LES. Meanwhile, CTR does not increase when the number of cores reaches 32, implying that there exists an optimal number of CPU cores. As stated previously, the PCE indicates the reduction in PCT for FVM-LES or LBM-LES relative to their SCT. The LBM-LES and FVM-LES PCE are similar for the 2- and 4-core simulations; however, the LBM-LES PCE increases significantly faster than that of FVM-LES as the number of cores increases. This trend also becomes more pronounced as the number of grids increases. In these simulations, the maximum parallel speedup value (i.e., PCE) for FVM-LES is only 12, while that for LBM-LES is ~ 24. This confirms that LBM-LES outperforms FVM-LES in terms of PCE, and this advantage becomes more pronounced as the number of cores increases. Thus, when more cores are used, LBM-LES exhibits a higher PCE than FVM-LES, and if more cores are utilized for parallel simulation, LBM-LES is likely to outperform FVM-LES more significantly. At the same time, the PCE of both methods diminishes somewhat when 32 cores are used in the simulation. As shown in the previous section, when the number of CPU cores is too large, the data exchange bottleneck in the hub outpaces the CPU computational speed. Therefore, as with FVM, it is unwise to attempt to make the LBM achieve a faster computational efficiency by indiscriminately increasing the number of CPU cores. Instead, an optimal solution should be determined by weighing the trade-off between computational volume and hardware performance.

8.7 Summary In this chapter, the first real LBM-LES case for the built environment of this book, an isothermal indoor flow case, is implemented. We discussed the LBM-LES accuracy for different grid resolutions, relaxation time schemes, discrete velocity schemes,

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and discrete time intervals for the isothermal indoor airflow problem. Additionally, we also investigated the consistency between the LBM-LES and FVM-LES results. Finally, the computational speed and PCE of both simulation methods were discussed. In this chapter, a set of simulation parameters was proposed, which can be used as a reference to carry out indoor turbulent environment simulations.

References Crouse B, Krafczyk M, Kühner S, Rank E, Van Treeck C (2002) Indoor air flow analysis based on lattice Boltzmann methods. Energy Build 34:941–949. https://doi.org/10.1016/S0378-778 8(02)00069-5 Davidson L, Nielsen PV (1996) Large eddy simulations of the flow in a three-dimensional ventilated room. In: 5th International conference on air distributions in rooms roomvent, vol 96, no 2, pp 161–168 Elhadidi B, Khalifa HE (2013) Comparison of coarse grid lattice Boltzmann and Navier Stokes for real time flow simulations in rooms. Build Simul 6:183–194. https://doi.org/10.1007/s12273013-0107-x Emmerrich SJ, McGrattan KB (1998) Application of a large eddy simulation model to study room airflow. ASHRAE Trans 104:1128–1137 Franke J, Hellsten A, Schlünzen H, Carissimo B (2007) Best practice guideline for the CFD simulation of flows in the urban environment. Brussels Han M, Ooka R, Kikumoto H (2019) Lattice Boltzmann method-based large-eddy simulation of indoor isothermal airflow. Int J Heat Mass Transf 130:700–709. https://doi.org/10.1016/j.ijheat masstransfer.2018.10.137 Han M, Ooka R, Kikumoto H (2021b) A wall function approach in lattice Boltzmann method: algorithm and validation using turbulent channel flow. Fluid Dyn Res 53:045506. https://doi. org/10.1088/1873-7005/ac1782 Han M (2022) Effect of time steps on accuracy of indoor airflow simulation using lattice boltzmann method (in Chinese). Tongji Daxue Xuebao/J Tongji Univ (Nat Sci) 50:793–801. https://doi. org/10.11908/j.issn.0253-374x.21486 Han M, Ooka R, Kikumoto H (2021a) Effects of wall function model in lattice Boltzmann methodbased large-eddy simulation on built environment flows. Build Environ 195:107764. https://doi. org/10.1016/j.buildenv.2021.107764 ITO K (2012) 41300 Analysis of airflow and particle dispersion in indoor environment by the lattice Boltzmann method. Summ Tech Pap Annual Meet 2012:607–608 Krüger T, Kusumaatmaja H, Kuzmin A, Shardt O, Silva G, Viggen EM (2017) The lattice Boltzmann method. Springer International Publishing, Cham Lemaire AD, Chen Q, Ewert M, Heikkinen J, Inard C, Moser A, Nielsen PV, Whittle G (1993) Room air and contaminant flow, evaluation of computational methods, subtask-1 summary report. IEA Annex 20: Air Flow Patterns Build 82 Murakami S (1993) Comparison of various turbulence models applied to a bluff body. J Wind Eng Ind Aerodyn 46–47:21–36. https://doi.org/10.1016/0167-6105(93)90112-2 Nielsen PV (1998) Selection of turbulence models for prediction of room airflow. ASHRAE Trans 104:1119–1127 Nielsen PV, Rong L, Olmedo I (2010) The IEA annex 20 two-dimensional benchmark test for CFD predictions. In: Clima 2010, 10th REHVA world congress 978-975-6907-14–6 Sajjadi H, Salmanzadeh M, Ahmadi G, Jafari S (2016) Simulations of indoor airflow and particle dispersion and deposition by the lattice Boltzmann method using LES and RANS approaches. Build Environ 102:1–12. https://doi.org/10.1016/j.buildenv.2016.03.006

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Chapter 9

LBM-LES in the Outdoor Wind Environment Problem Around a Single Building

9.1 Introduction to the Outdoor Wind Environment Problem Currently, the most widely used method to simulate outdoor wind environments is the NSE-based method, especially FVM-LES (Stathopoulos and Baskaran 1996; Richards et al. 2002; Tominaga et al. 2004, 2008a; Hertwig et al. 2012). With FVMLES, the time-dependent 3D momentum and continuity equations are solved, as opposed to traditional RANS methods. This enables analysts to gain a deeper insight into the complex turbulent temporal-spatial features in urban wind environments, which is extremely helpful for problems such as the wind resistance of building structures and urban wind hazards. For this reason, FVM-LES has been widely used to simulate turbulent wind environments (Tominaga et al. 2008a; Murakami 1993; Rodi 1997; Shah and Ferziger 1997; Krajnovic and Davidson 2002; Ikegaya et al. 2017). Even though FVM-LES is particularly advantageous in providing rich and accurate results, it is somewhat hampered by its significantly higher computational time and cost (Krajnovic and Davidson 2002; Sagaut 1998). Due to the complexity and large scale of the built outdoor environment (from tens of meters to tens of kilometers), the FMV-LES grids are usually extensive, leading to high simulation hardware and computational costs. Due to the parallel computing advantages of LBM-LES, researchers have attempted to apply it to outdoor and urban-scale wind environment simulations since the 2010s. Onodera et al. (2013) performed large-scale LES of 10 km × 10 km metropolitan areas in Tokyo, Japan, using LBM-LES to represent the instantaneous velocity field. Andre et al. (2014) compared the performance of the finite element method (FEM) and LBM-LES for simulations of bluff bodies in an atmospheric boundary layer using LES. Obrecht et al. (2015) used LBM-LES to simulate flows around one or several wall-mounted cubes using a multi-core graphics processing unit (GPU) platform. Their simulation results agreed well with the experimental © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Han and R. Ooka, Large-Eddy Simulation Based on the Lattice Boltzmann Method for Built Environment Problems, https://doi.org/10.1007/978-981-99-1264-3_9

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data in the literature. King et al. (2017) simulated the airflow in and around a cubic building in isolation and within an array of similar buildings. They demonstrated that the pressures, velocities, and ventilation rates obtained via LBM-LES were comparable to experimental data. Ahmad et al. (2017) used LBM-LES to simulate a pedestrian-level unsteady wind environment in built-up areas. They investigated the intensity of wind gusts using a sizeable computational domain to allow sufficient boundary layer development and fine grid spacing for explicit LBM-LES. Lenz et al. (2019) simulated the wind flow in a complex neighborhood-scale urban canopy in Basel, Belgium, using cumulant LBM-LES on a GPU platform. Han et al. (2020) systematically assessed a number of LBM-LES parameters by simulating the wind distribution around a single building and discussed the applicability of LBM-LES in outdoor wind environments. In this chapter, we will discuss the performance of LBM-LES in outdoor turbulent wind environments. For this, we select the most fundamental problem, the turbulent wind flow surrounding a single high-rise building, and verify the simulation accuracy of LBM-LES for turbulent outdoor flows under different simulation conditions, as described in Sect. 9.2. This problem is taken from the CFD guidebook published by the Architectural Institute of Japan (AIJ) (2016), Tominaga et al. (2008b), a popular benchmark case for outdoor wind simulations. In Sect. 9.3, the case settings and simulation method are introduced in detail as a simulation reference for the reader. Many simulation techniques and parameter settings presented in this book are integrated into this outdoor turbulent wind problem. The simulation results, including the effect of different parameters on the simulation accuracy and the LBM-LES consistency with FVM-LES are discussed in Sects. 9.4 and 9.5. Finally, the computational time and parallel computational efficiency are discussed in Sect. 9.6. Here, we take a step further, as we discuss the hardware limitations on the LBM-LES computational speed, which is usually a major bottleneck in applying LBM-LES for large-scale outdoor environment simulations.

9.2 Problem Description of Flow Around a Single Building 9.2.1 Simulation Target The simulation target is a building with a scale ratio of H: b: b (height:length:width), and the building is immersed in a turbulent wind environment. Figure 9.1 shows a schematic of the computational domain. Considering the self-similarly at a high Re, the building model has an equal length and width of 0.16 m and a height of 0.32 m (H = 2b), which is sufficient to obtain fine results. Re is approximately 40,800, which was determined based on the building width b and a time-averaged inflow velocity in the x-direction of Uref = 3.824 m/s at a height of b. This simulation case satisfies the self-similarity high-Re condition with turbulent structure and velocity distribution proportional to the actual full-size outdoor wind environment. The Re

9.3 Simulation Methodology and Boundary Conditions Fig. 9.1 Schematic of the computational domain for turbulent flow around a single building

175

9.25b

2b

10b

b

14.5b

6.375b b 6.375b

of this case is of the scale model. In fact, the Re of the real scale can reach tens of millions. But considering the self-similarity of the high-Re flow, the current Re is high enough to express the turbulence characteristics. The coordinate origin is set at the center point of the building on the ground surface.

9.2.2 Wind Tunnel Experiment The AIJ conducted a detailed wind tunnel experiment (WTE) for this case and obtained accurate results, which will be the basis of our simulation. Using a split-fiber probe, they measured the 3D time-averaged flow velocities and standard deviations of the fluctuating velocities. The measurement points were located in the vertical plane (y = 0), near-ground horizontal plane (z/b = 0.125), and far-from-ground horizontal plane (z/b = 1.25). The measurement point distribution is shown in Fig. 9.2. Detailed measurement methods and results can be found in Meng and Hibi (1998).

9.3 Simulation Methodology and Boundary Conditions The simulation of turbulent outdoor environments is one of the most complex problems in built environment simulation. Here, we comprehensively employ the various simulation techniques presented in this book, such as the parameter settings and boundary conditions (BCs), among many others. First, we describe the general simulation conditions and discuss the parameters that may affect the simulation results (i.e., case settings). Then, we show the inflow data and convergence criteria for the outdoor wind simulation. Additionally, we introduce the indices used to evaluate the simulation accuracy.

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Fig. 9.2 Measurement point distribution in the vertical and horizontal planes (Meng and Hibi 1998). The vertical plane is located in the centerline of the wind tunnel. The measurement points in the horizontal planes are only arranged on one side of the y = 0 axis because this problem is symmetric about this axis

9.3.1 Simulation Conditions and Parameter Settings 9.3.1.1

General Simulation Conditions

Table 9.1 lists the general simulation conditions for LBM-LES. These parameters are not unique to the LBM but are general parameters that must be considered for any simulation method. Commonly used FVM-LES parameters are also listed for

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177

Table 9.1 General simulation conditions for flow around a single building Item

LBM-LES

Simulation domain

21.5b(x) × 13.75b(y) × 11.25b(z)[m]21.5b(x) × 13.75b(y) × 11.25b(z)[m]

FVM-LES

SGS model

Standard Smagorinsky model (Smagorinsky 1963), C s = 0.12 (Murakami 1993) / WALE model (Nicoud and Ducros 1999)

Time – discretization

Euler-implicit

Space – discretization

2nd-order central difference

Simulation period

2 [min] (Preliminary-simulation preparation: 1 [min]; Sampling period 1 [min]; sampling frequency: 250 [Hz])

comparison. For the SGS model, in addition to the simplest standard Smagorinsky model, we also utilize the WALE model, which has recently become increasingly popular in the architectural field for comparison. The time and space discretization for FVM-LES adopts the popular second-order difference schemes to ensure accuracy. By contrast, no differential schemes are required for LBM-LES because it is an explicit solution. Nevertheless, the LBM requires a unique velocity discretization scheme, which we will discuss later. For the turbulent wind environment, we generally need to perform a preliminary simulation (e.g., a 1-min simulation) to ensure that the turbulence in the computational domain is fully developed. Then, the time-averaged sampling can start. The sampling period also needs to be set carefully to ensure a sufficient number of samples (or what we can call “computational convergence”), which is discussed in Sect. 9.3.3.

9.3.1.2

Boundary Conditions

Table 9.2 lists the BCs for each simulation element in this case. LBM-LES generally utilizes the same BCs as FVM-LES except for the solid wall boundary. The solid wall boundary in LBM-LES is distinctive. Hitherto, bounce-back is the most popular boundary for solid walls and has been utilized in several studies to successfully solve turbulent outdoor flow problems (Onodera et al. 2013; King et al. 2017; Ahmad et al. 2017; Habilomatis and Chaloulakou 2013). It is worth noting that bounce-back was adopted in the first two flow problems presented in this book (q.v. Chaps. 7 and 8), achieving good results. However, as stated in Chap. 3, because bounce-back is a no-slip boundary, if near-wall grids are too coarse, bounce-back may mispredict the shear drag. In our simulations, the first near-wall grids are not in the viscous sublayer, which leads to the above problem (q.v. Sect. 9.3.1.3). In addition to the popular bounce-back boundary (Cornubert et al. 1991; Ziegler 1993), we also utilize the wall-function bounce (WFB) boundary proposed by Han et al. (2021b). Readers can refer to Chap. 3 to explore the

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9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

Table 9.2 Boundary conditions for each simulation element Item

LBM-LES

Inlet BC

Dirichlet velocity, using the approaching flow velocity boundary data (q.v. Sect. 9.3.2)

FVM-LES

Outlet BC

Zero-pressure fixed value

Left- and right-side BCs

No-slip

Top side BC

Free-slip

Ground and building BCs

Bounce-back/WFB (Han et al. 2021b)

Wall function using Spalding’s law (Spalding 1961)

implications of these boundaries. FVM-LES employs a Spalding’s law wall-function boundary, which is widely applied in outdoor wind environment simulations. Remarkably, the inlet boundary uses the approaching flow data for turbulent urban wind, a particular setting for outdoor environment simulation that will be discussed in Sect. 9.3.1.3.

9.3.1.3

Grid Resolution

Due to the greater turbulence intensity of the outdoor wind environment, the grid requirements for LES are higher than those of the indoor environment. Therefore, a grid accuracy test is essential. We chose three grid sizes for testing; Table 9.3 lists detailed information for each grid resolution. The z + of the first layer grids around the building and ground are approximately 18, 10, and 6, leading to the first grids being located in the logarithmic layer, transition layer, and viscous sublayer, respectively. This allows us to better evaluate the solid wall BCs. As the LBM utilizes a uniform cubic grid system, we slightly modified some of the non-orthogonal bluff bodies to conform to the grid system, as shown in Fig. 9.3. This is the most straightforward method for the LBM to deal with non-orthogonal Table 9.3 Mesh information for three grid test cases Grid resolution

Edge length of a cubic cell [m]

z + of the first layer grids

Cells on each edge

Number of cells (×106 )

Building (x × y × z)

Target domain (x × y × z)

Target domain

b/8

0.02

∼ 18

8 × 8 × 16

172 × 110 × 90

1.7

6.0

7.7

b/16

0.01

∼ 10

16 × 16 × 32

344 × 220 × 180

13.6

48.1

61.7

b/32

0.005

∼6

32 × 32 × 64

688 × 440 × 360

108.8

384.8

493.6

Preparator domain

Total

9.3 Simulation Methodology and Boundary Conditions

179

Fig. 9.3 Schematic of the uniform cubic grid system. Non-orthogonal-shaped boundaries are modified in a stepped manner to fit the grid system

geometries. The local mesh refinement technique is not employed in this book. Interested readers are referred to the relevant literature (Lagrava et al. 2012; Dorschner et al. 2016; Gendre et al. 2017).

9.3.1.4

Parameters to Discuss

In addition to the general simulation conditions and BCs, we selected various simulation parameters to discuss their effect on simulation accuracy in depth. The specific settings of these parameters are listed in Table 9.4. Each variable parameter setting becomes an individual simulation case so that the effects of each one can be discussed separately. The case name is in the following format: “LBM_Param.” The first string field is “LBM” or “FVM,” representing the simulation method utilized in each case. The subsequent string, i.e., “Param,” represents the variable parameter value in the simulation, such as the grid resolution and relaxation time scheme. Please note that “Param” is not fixed but depends on the discussion’s objective in Sect. 9.4. Table 9.4 Simulation parameter settings

Parameter

Parameter settings

SGS models

Smagorinsky model; WALE model

Grid resolution

b/8; b/16; b/32

Relaxation time schemes

SRT(BGK); MRT

Discrete-velocity schemes D3Q19; D3Q27 Solid wall BCs

Bounce-back; WFB

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9.3.2 Inlet Boundary Data and Approaching Flow As opposed to the indoor environment, the outdoor wind environment is complex and highly turbulent, and is affected by various factors, including atmospheric and oceanic currents, solar radiation, temperature, and terrain. In addition, the nearground wind velocity is exponentially distributed in the vertical direction due to the friction force of the ground surface (AIJ 2016), such as: ( u z = Uref

z z ref

)α

.

(9.1)

Here, u z represents the velocity at height z and Uref is a representative velocity at representative height z ref . α represents the degree of velocity reduction near the ground, and its value increases as the surface roughness increases. The AIJ recommends α = 0.1 for flat surfaces, such as the sea or a lake; α = 0.2 for urban areas dominated by low-rise buildings; α = 0.27 for urban areas dominated by mid-rise buildings; and α = 0.35 for cities with high-rise buildings. For LES, using (9.1) for the inflow BC is not sufficient. LES needs to read the turbulent wind inflow at every moment, whose instantaneous value is chaotic and its time-averaged value satisfies (9.1). Figure 9.4 shows an example of the turbulent inflow velocity. Several technologies have been developed to obtain the time series of these turbulent inflow data. Among them, the simplest method is to simulate the entire turbulent wind generation process via a WTE. Figure 9.5 shows a schematic of the entire WTE setup for LES simulation. Triangular spires are set for turbulent wind generation, while the cubic block array is designed to mimic the roughness of the ground, such as buildings and other obstacles, so that the turbulent wind satisfies (9.1). At the tail of the wind tunnel is the simulation target building. Fully reproducing the entire WTE is the simplest, most straightforward, and most accurate method to obtain the turbulent inlet data, as it can generally reproduce the complete temporal-spatial turbulent structure. The obtained boundary turbulent flow data are the so-called approaching flow. Figure 9.6 shows a comparison between the approaching flow data obtained via LBM-LES and the WTE. The streamwise direction component of the time-averaged /⟨ ⟩ velocity ⟨u x ⟩ and three directional components of the fluctuating velocity u ,2 x , /⟨ ⟩ /⟨ ⟩ u ,2y , and u ,2 z are shown. The power law exponent is ~ 0.27 for the vertical profile of ⟨u x ⟩. It can be seen from the figure that the approaching flow characteristics in the experiments are accurately reproduced. Some approaching flow databases that are openly accessible are also provided by other organizations (AIJ 2020). Although reproducing the WTE is simple and accurate, the simulation time and hardware cost are noteworthy. Recently, several mathematical methods have been developed to minimize the computation while generating turbulence data. Among them, synthetic methods, such as the vortex (Sergent 2002), weighted amplitude wave superposition (Klein et al. 2003), and spectral (Smirnov et al. 2001) methods, have

9.3 Simulation Methodology and Boundary Conditions

181

Fig. 9.4 Normal-direction cross-section of the turbulent inlet velocity magnitude for a typical outdoor environment simulation

become popular. Each of these methods has its own merits. Their specific working principle and algorithms are not within the scope of this book and interested readers can refer to related literature.

9.3.3 Sampling Time Convergence Criteria Due to the turbulent features of outdoor wind, the velocity varies at every moment. Meanwhile, LES is an instantaneous simulation method and does not have a similar convergence state as the RANS method. Furthermore, there is no “air change rate” concept as in the indoor airflow simulation (q.v. Sect. 8.3.1) to estimate the effective

Fig. 9.5 Generation of the approaching flow from an additional LES using the boundary layer wind tunnel experiment. The uniform inlet wind passes through the spires and roughness blocks to generate turbulence and mime the roughness of the ground in the built environment. Then, the turbulent wind is obtained similar to the actual urban wind to flow around the target building. The velocity shown in Fig. 9.4 is obtained from the approaching flow gathering plane

182 9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

9.3 Simulation Methodology and Boundary Conditions

183

Fig. 9.6 Time-averaged and fluctuating velocity profiles at the central vertical line of the approaching flow obtained via LBM-LES. The simulation results agree with the experimental results of Meng and Hibi (1998)

turbulence development period and the period required to reach the “steady state” of the outdoor airflow. Therefore, judging whether the simulation has “converged” during the time-averaging operation is not easy. Here, we introduce two methods for determining the sampling time convergence criteria.

9.3.3.1

Time-Averaged Value Stability

The simplest and most direct method is to assess whether the time-averaged velocity has reached stability. We can select several representative measuring points and observe whether the statistical average remains unchanged as the sampling proceeds. The selection of such representative measurement points plays a critical role. Generally, points with severe turbulence changes or those upon which we focus are good candidates. Figure 9.7 shows the ⟨u x ⟩ variation at two representative points as the sampling proceeds for different grid resolutions. The two points are located at the windward corner above the building roof (x = −0.5b, y = 0b, z = 2.125b) and within the wake area behind the building (x = 2.00b, y = 0b, z = b). These regions, in which the turbulence is intense, are representative in this problem. Figure 9.7 clearly shows that at the beginning of the sampling process (within 10 [s] or less, 2500 samples), the ⟨u x ⟩ variation is very drastic due to the insufficient number of samples. As the number of samples increases, the ⟨u x ⟩ variation gradually stabilizes regionally. The magnitude of ⟨u x ⟩ variation varies at different points, and the sampling time required to reach stability is also different. When the sampling reaches 60 [s], i.e., the number of samples reaches 15,000, almost all curves exhibit no change, and the sampling can be considered stable.

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9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

Fig. 9.7 Relationship between u x at two representative points and sampling time under different LBM-LES grid resolutions (Han et al. 2020)

9.3.3.2

Time-Averaged Value Statistical Uncertainty

Determining the stability of the time-averaged velocity as the number of samples increases is a simple but crude method. Here, we introduce another quantitative verification method for the sampling time convergence criteria. The convergence criteria are evaluated using the statistical uncertainty Uunc (α) of the time-averaged wind velocity (Kikumoto et al. 2018). Uunc (α) is defined as Uunc (α) = 2z α/2 · var(⟨u⟩)0.5 .

(9.2)

Here, z α/2 is a percentage point corresponding to the significance level α of the random velocity following a standard normal distribution. var(⟨u⟩) is the variance of the sample average, which is calculated as var(⟨u⟩) = 2σ 0.5

t int . t smp

(9.3)

Here, σ is the sample variance, t int is the integral time scale, and t smp is the sampling time. Figure 9.8 shows the Uunc (α) curves for different LBM-LES cases for the outdoor wind around a single building. In this simulation problem, the sampling frequency is set to 250 Hz and the 95% confidence interval (α = 0.05, z 0.025 = 1.96) is employed for the uncertainty estimation. We choose two representative points to observe the 5% statistical uncertainty Uunc (5%) when the sampling time is 10, 20, 30, 40, 50, and 60 s. As the sampling time increases, Uunc (5%) decreases in all cases; when the sampling time approaches 60 s (up to 60,000 samples), the 5% uncertainty reduces to 10% or less of the Uref in all cases. Therefore, under 60 s, the 95% reliability in all cases can reach over 90%. We can consider that a sampling time of 60 s (15,000 samples in total) is sufficient as the time-averaged results generally cease to vary.

9.3 Simulation Methodology and Boundary Conditions

185

Fig. 9.8 Statistical uncertainty Uunc (5%) according to sampling time. The legend shows the simulation method utilized for each case and its main simulation conditions, including grid resolution, relaxation time scheme, and discrete velocity scheme

9.3.4 Simulation Accuracy Evaluation Index In Chap. 8, the ME and HR are utilized to evaluate the overall simulation accuracy in the isothermal indoor flow problem. However, the turbulent outdoor flow problem is more complex and is not enough to describe the accuracy using only two indices. In addition to the ME and HR, a group of evaluation metrics (Schatzmann et al. 2010), widely used in outdoor wind environment simulation assessment, is also employed in this chapter, including FAC2, FB, and NMSE. HR and FAC2, the fraction of data points in which the predictions are within a factor of two of the are used ⟨ observations, ⟩ to assess the accuracy of the time-averaged velocities ⟨u x ⟩, u y , and ⟨z x ⟩ as well as (⟨ ⟩ ⟨ ,2 ⟩ ⟨ ,2 ⟩) the turbulence kinetic energy (TKE = 21 u ,2 x + u y + u z ). The fractional bias (FB) and normalized mean square error (NMSE) are also used to assess the TKE accuracy because these metrics cannot be used to evaluate the accuracy of variables that can take both positive and negative values. HR is defined as in (8.7). The other

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9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

evaluation metrics are defined by (9.4)–(9.6). N 1 Σ n i with n i = FAC2 = N i=1

⎧

1, for 0.5 ≤ 0, else

PLBM or FVM(i ) PEXP(i )

≤2

,

⟨PEXP ⟩ − ⟨PLBM or FVM ⟩ with 0.5(⟨PEXP ⟩ + ⟨PLBM or FVM ⟩) N N 1 Σ 1 Σ ⟨PEXP ⟩ = PEXP(i ) , and ⟨PLBM or FVM ⟩ = PLBM or FVM(i) , N i=1 N i=1 ⟩ ⟨ (PEXP − PLBM or FVM )2 NMSE = with ⟨PEXP ⟩⟨PLBM or FVM ⟩ N 1 Σ ⟨PEXP ⟩ = PEXP(i) , N i=1

(9.4)

FB =

⟨PLBM or FVM ⟩ =

(9.5)

N 1 Σ PLBM or FVM(i) , N i=1

N ⟨ ⟩ )2 1 Σ( and (PEXP − PLBM or FVM )2 = PEXP(i) − PLBM or FVM(i) N i=1

(9.6)

In (9.4)–(9.6), PLBM or FVM(i) is the predicted value at point i obtained via LBMLES or FVM-LES, PEXP(i ) is the experimental value for the corresponding point, and N is the number of data points used for the evaluation. The evaluation considers all measurement points in Meng and Hibi (Meng and Hibi 1998).

9.4 Results and Discussion In this section, the LBM-LES results are presented, primarily focusing on the effect of various LBM-LES parameters on the simulation accuracy, based on Meng and Hibi (1998). In this section, we will not deal with FVM-LES; a comparison between LBM-LES and FVM-LES is presented and discussed in the next section. All results are normalized by the characteristic length b and reference flow velocity Uref unless otherwise stated.

9.4.1 Instantaneous Velocity LES solves for the instantaneous velocity at each discrete time step, rather than for the ensemble-averaged velocity as in RANS simulation. Figure 9.9 presents the

Fig. 9.9 Instantaneous scalar velocity distribution for the entire WTE at a specific time frame. The turbulent wind gradient generation can be clearly observed

9.4 Results and Discussion 187

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9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

instantaneous scalar velocity distribution at the vertical section for the entire WTE along the streamwise direction as simulated via/LBM-LES. The scalar velocity is the Euclidean norm of the velocity vector (= u 2x + u 2y + u 2z ), which ignores the velocity direction and only expresses the magnitude distribution in space. The inlet uniform wind passes through the spires and generates chaotic turbulence. The turbulent wind passes over the roughness blocks of different sizes. Due to the frictional resistance of the roughness blocks, a turbulent wind gradient similar to the urban wind is formed. Finally, the resulting approaching wind reaches the target building to form the surrounding turbulent wind environment.

9.4.2 Time-Averaged Velocity and Flow Structure The instantaneous turbulent wind is chaotic and random. Accordingly, it is challenging to obtain an overall perspective of turbulence properties by analyzing the instantaneous wind velocity distribution at a specific time. Therefore, similar to Chap. 8, researchers and scholars often sample the turbulent wind to obtain the time-averaged velocity and its fluctuation to investigate its properties in a statistical sense. Figure 9.10 shows the normalized time-averaged scalar velocity contours at the central vertical plane for some simulation cases. In general, all LBM-LES cases reproduce similar flow structures. The incoming wind produces an impingement phenomenon when it comes in contact with the building’s windward surface and sweeps over its top, where a stripping effect occurs, creating a small eddy above the roof. The wind finally reattaches to the ground far from the building, forming a large wake area in the leeward area at the rear of the building. Although the flow structures are broadly similar for all cases, there are some subtle differences among them. For example, b/8_MRT_D3Q19 (Fig. 9.10a) does not form the eddy at the top of the building; b/8_SRT_D3Q19 (Fig. 9.10c) exhibits numerical oscillations in the windward region; and b/8_SRT_D3Q27 (Fig. 9.10d) forms a complex multi-eddy structure above the building and numerical oscillations occur in the windward region. In addition, the size and shape of the wake area behind the building vary from case to case. These differences are related to the effects of different LBM simulation parameters, which we will discuss in subsequent sections. The recirculation zones over the roof and in the wake area of the building are essential to examine the flow structure around the building. To evaluate the size of the recirculation zones, the flow reattachment length on the roof X R and in the wake area of the building X F (Adams and Johnston 1988), as shown in Fig. 9.11, are commonly used. The reattachment point is actually the dividing point between areas where ⟨u x ⟩ < 0 and ⟨u x ⟩ > 0. The normalized reattachment lengths for some simulation cases are listed in Table 9.5. For comparison, the results obtained by Gousseau et al. (2013) using FVM-LES are also listed in the table.

9.4 Results and Discussion

189

Fig. 9.10 Normalized time-averaged scalar velocity contours at the central vertical plane for some LBM-LES cases

Fig. 9.11 Schematic of the recirculation zones over the roof and in the wake behind the building. The size of the recirculation zones is usually measured in terms of reattachment lengths X R and XF

Recirculation zone over the roof Inlet flow

Building Recirculation zone in the wake area of the building

LBM_b/8_SRT_D3Q19 does not reproduce the recirculation zone on the roof because the grid is not sufficiently fine to capture the full turbulent characteristics of the flow. All other cases are able to reproduce this recirculation zone even though their X R values are larger than those in the experiment. The recirculation zone on the roof changes in the LBM_Grid16_SRT_D3Q27 test case, as shown in Fig. 9.10d. The reattachment lengths of the flow in the wake of the building X F

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9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

Table 9.5 Normalized reattachment lengths in the roof X R /b and wake X F /b recirculation zones

Case

X R /b

X F /b

LBM_b/8_SRT_D3Q19

–

2.94

LBM_ b/16_SRT_D3Q19

0.69

1.84

LBM_ b/16_MRT_D3Q19

0.47

1.85

LBM_ b/32_MRT_D3Q19

0.64

1.79

LBM_ b/16_SRT_D3Q27

0.17

2.32

0.59

1.66

0.52

1.42

FVM-LES by Gousseau et al.

(2013)a

Experiment (Meng and HIBI 1998) a The

results of Gousseau et al. (2013) were obtained from the LES20-2 test case (b = 20; Cs = 0.15) via FVM-LES

are larger than those in the experiment for all test cases. X F is greatly overestimated in the LBM_b/8_SRT_D3Q19 and LBM_b/16_SRT_D3Q27 test cases. In general, LBM_b/16_MRT_D3Q19 and LBM_b/32_MRT_D3Q19 most accurately reproduce the X F value from the experiment and are the closest to the FVM-LES result of Gousseau et al. (2013).

9.4.3 Effect According to Grid Resolution In the previous two chapters, we learned that a grid independence analysis is an essential preliminary task in LBM-LES. In this section, we use three grid resolutions to examine the grid dependence of the LBM-LES results. The standard halfway bounce-back BC (q.v. Sect. 3.7.2) is employed for all solid wall boundaries. The MRT and D3Q19 schemes and the WALE SGS model are utilized in the simulations presented in this section. For the collision function, we use MRT instead of the SRT used in Chap. 8 as SRT generates numerical oscillations in this problem, which interfere with the simulation accuracy. This issue is discussed in Sect. 9.4.5. Figure 9.12 shows the velocity and TKE vertical profiles around the building for the three grid resolution cases. Here, we will not discuss the velocity fluctuations of each component separately but integrate them and discuss the TKE. For the flow separation region over the roof (−0.5 < x/b < 0.5; 2 < z/b 0.5,z/b < 2), whereas the b/16 and b/32 simulation results agree well with the experimental data. Figure 9.13 shows the time-averaged velocity and fluctuating velocity profiles of the turbulent flow at the near-ground horizontal plane (z/b = 0.125). This plane lies at a pedestrian-level height in the full-scale model; therefore, care should be taken when interpreting the simulation results. In the b/8 case, this plane lies one cell above the ground level. In the b/16 and b/32 cases, this plane lies ~ 2–4 cells above

9.4 Results and Discussion

191

Fig. 9.12 Time-averaged velocity and TKE profiles of the turbulent flow at the central vertical plane (y = 0) for three LBM-LES grid resolution cases. All cases utilize the WALE SGS model, MRT and D3Q19 schemes, and halfway bounce-back BC

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9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

Fig. 9.13 Time-averaged velocity and TKE profiles of the turbulent flow at the near-ground horizontal plane (z/b = 0.125) for three LBM-LES grid resolution cases. All cases utilize the WALE SGS model, MRT and D3Q19 schemes, and halfway bounce-back BC

the ground level. It is apparent that ⟨u x ⟩ is underestimated at almost all points in b/8. The same trend is observed for the other time-averaged velocity component and

9.4 Results and Discussion

193

TKE, although the underestimation of ⟨these ⟩ components is smaller than that of the mainstream component. The absolute u y and ⟨u z ⟩ values are also underestimated in b/8 because the time-averaged velocities are in the negative direction, indicating that the flow is in a region where it impinges on the windward walls. The source of simulation error in the near-wall region is complex when the grids near the solid wall are coarse. It includes excessive SGS viscosity in LES due to coarse grids, improper near-wall SGS viscosity evaluation by some specific SGS models, and improper near-wall shear drag evaluation by the standard bounce-back BC in high-Re flow problems. The former is an issue faced by all LES methods, and the reader can refer to LES textbooks for more information (Sagaut 2004; Kajishima and Taira 2018; Versteeg and Malalasekera 2007). The following two points are discussed in Sects. 9.4.4 and 9.4.6, respectively. We can compare the simulated and experimental results at all measurement points and visualize the relationship between overall simulation accuracy and grid resolution using a uniform error metric (e.g., ME, which we have previously adopted, q.v. Sects. 8.4.2 and 8.4.3), as shown in Fig. 9.14. The simulation error decreases as the grid resolution increases. In general, the accuracy of the LBM-LES results depends on the grid resolution. In the present problem, it can be seen that the accuracy of each physical property improves significantly as the grid resolution increases from b/8 to b/16, while the accuracy increase is limited when the grid resolution increases to b/32. One can expect that when the grid resolution continues to increase (e.g., b/64), the simulation accuracy will continue to improve, but only slightly. Thus, we consider that a grid resolution of b/32 is sufficient to reach a stable level for this problem, i.e., the simulation accuracy is independent of grid resolution. Further considering the simulation cost in engineering or industrial applications (for every doubling of the grid resolution, the computational cost increases by approximately 16 times), b/16 can also be accepted to some extent.

Fig. 9.14 Average ME according to grid resolution. All cases utilize the WALE SGS model, MRT and D3Q19 schemes, and halfway bounce-back BC. It can be considered that grid independency is reached at a grid resolution of b/32

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9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

Fig. 9.15 TKE comparison between the Smagorinsky and WALE SGS models at the near-ground region (z/b = 0.125). Other simulation conditions are b/16_MRT_D3Q19_bounce-back. The WALE SGS model improves the simulation accuracy in high-turbulence regions such as the impinging and stripping regions on the side of the building

9.4.4 Effect According to SGS Model In addition to grid resolution, the choice of the SGS model is another fundamental issue in LBM-LES. In high-Re flow cases, the Smagorinsky model may produce large SGS viscosity near the wall, as shown in Sect. 5.3, which is non-physical and leads to accuracy errors in the simulation results in the near-wall regions. In this section, we take b/16_MRT_D3Q19 as an example to explore the influence of the Smagorinsky and WALE models on simulation accuracy. Recently, the WALE model has become increasingly popular in the built environment field. The simulation results show that for the time-averaged velocity, the difference between the two SGS models is small. For TKE, the WALE model improves the simulation accuracy to a certain extent in the areas with higher turbulence intensity near the ground, such as the impingement and stripping region on the side of the building, as shown in Fig. 9.15. This suggests that when the fluid turbulence in the near-wall region is high, we should employ an advanced SGS model to ensure that the simulations reproduce the correct complex turbulent processes and obtain satisfactory results.

9.4.5 Effect According to the Relaxation Time and Discrete Velocity Schemes In this section, the effects of the relaxation time scheme (SRT or MRT) and discrete velocity scheme (D3Q19 or D3Q27) on LBM-LES accuracy are discussed. The Smagorinsky SGS model and a grid resolution of b/16 are used in the simulations presented in this section.

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195

Figure 9.16 shows the time-averaged and fluctuating velocity profiles at the y = 0 vertical plane. Based on the results for different relaxation time schemes, it is evident that SRT causes pronounced spatial ⟨u z ⟩ oscillations over the roof of the building, while MRT diminishes such oscillations. Nevertheless, the MRT scheme only provides a minor improvement in accuracy compared to the SRT scheme. Regarding the discrete velocity schemes, D3Q27 does not reduce the ⟨u z ⟩ oscillations. Instead, the time-averaged and fluctuating velocities in the proximity of the roof surface are overestimated. In particular, TKE near the roof was abnormally overestimated, indicating that the LBM-LES result accuracy decreases when using D3Q27 compared with D3Q19. Figure 9.17 shows the time-averaged and fluctuating velocity profiles of the turbulent flow at the near-ground horizontal ⟨ ⟩ plane (z/b = 0.125). Similar to the vertical section results, there are apparent u y oscillations in all cases, irrespective of grid resolution. These are regions where velocity oscillations significantly overlap with those where the flow impinges on the windward face, where the three components of the time-averaged velocity are large. There are no oscillations in regions where the time-averaged velocity components are small. It can be deduced that the numerical oscillations occur in high-Re regions. According to previous studies (Elhadidi and Khalifa 2013; Han et al. 2018; Sajjadi et al. 2017), these oscillations do not occur in low-Re flows, such as laminar and turbulent indoor flows. This numerical oscillation problem ⟨ ⟩ arises from the SRT in high-Re flows. MRT significantly reduces the ⟨u x ⟩ and u y oscillations on the side of the building, whereas they are significant when using SRT. Significant velocity oscillations appeared in some regions in several cases in Figs. 9.16 and 9.17. To assess the presence of noise, one can examine the power spectral densities (PSDs) of the instantaneous velocity at a representative point in these regions, such as (x/b = 0.5, y/b = 0, z/b = 2.125), as shown in Fig. 9.18. Several non-physical peaks occurred at ∼ 3.5 f H/U f , ∼ 7.8 f H/U f , ∼ 11.4 f H/U f , and ∼ 13.1 f H/U f for all SRT cases, indicating that noise occurred in the highfrequency regions in these cases. In addition, D3Q27 starts to decay from a lower frequency compared to other cases. By contrast, noise is not observed in MRT. The PSD results indicate that noticeable noise appeared in the high-frequency regions in SRT regardless of the discrete velocity scheme, resulting in velocity oscillations in these cases. Meanwhile, the PSDs of the MRT case decayed more smoothly in the high-frequency region. The velocity distribution profiles and PSDs show that SRT leads to simulation instability and generates numerical oscillations at high frequencies in a high-Re flow field such as the turbulent outdoor environment. This phenomenon does not occur in the indoor flow field in Chap. 8 as the Re in that case is not sufficiently large. MRT can effectively mitigate this oscillation and improve the simulation accuracy. It should be noted that MRT is not the ultimate solution. Researchers have found that MRT also loses its usefulness when Re is much higher, producing numerical oscillations. As a result, more sophisticated collision functions for higher Re fluid environments have been proposed, such as the cumulant LBM mentioned in Chap. 2 (Geier et al. 2015, 2017). Meanwhile, D3Q19 is sufficiently accurate for simulating

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9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

Fig. 9.16 Time-averaged and fluctuating velocity profiles on the y = 0 vertical plane for different relaxation time and discrete velocity schemes. A grid resolution of b/16 is utilized in all cases

9.4 Results and Discussion

197

Fig. 9.17 Time-averaged and fluctuating velocity profiles on the near-ground horizontal plane (z/b = 0.125) for different relaxation time and discrete velocity schemes. A grid resolution of b/16 is utilized in all cases

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9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

Fig. 9.18 Normalized PSDs for different relaxation time and discrete velocity schemes at (x/b = 0.5, y/b = 0, z/b = 2.125)

9.4 Results and Discussion

199

the outdoor wind environment. Although D3Q27 increases the degree of freedom of particles, it does not effectively improve the simulation accuracy for outdoor wind environments and increases the simulation hardware cost.

9.4.6 Effect According to the Solid Wall Boundary Condition In simulations presented in this chapter to this point, LBM-LES yielded relatively accurate results. However, when the near-wall grids are coarse, large simulation errors are produced in the near-ground region (z/b = 0.125, q.v. Sect. 9.4.4). This is partly due to the limitations of the standard halfway bounce-back BC. As mentioned in Sect. 3.8, there is an extremely thin viscous sublayer in the nearwall regions where the inertial force is almost the same as or only slightly larger than the viscous force, leading to Re = ~ 1. In this sublayer, the viscosity affects the fluid to the extent that the motion does not follow a free-flow pattern. When simulating very high Re problems, such as the outdoor flow problem, the near-wall grids are usually much coarser than the viscous sublayer, leading to unsatisfactory results in the near-wall region for no-slip BCs (e.g., the bounce-back BC). Many schemes for bounce-back improvement have been proposed to overcome this situation. In this section, the WFB scheme (q.v. Sect. 3.8.1) is compared with the bounce-back scheme to investigate the effect of different solid BCs on simulation accuracy. The core difference between the bounce-back and WFB schemes is whether the near-wall velocity is calculated directly by the no-slip BC or adjusted by the wall function. The main purpose of the wall function model is to correct the shear drag on the wall. However, it is difficult to determine which shear drag is more precise by directly comparing the bounce-back and WFB results due ⟨ to ⟩ the lack of experimental ⟨ ⟩ data. Alternatively, we compared the distribution of u + according to z + using Spalding’s ⟨ ⟩ law⟨ to ⟩determine the shear drag that is more consistent with Spalding’s law. u + and z + represent the dimensionless velocity and distance at the grids normalized by the friction velocity on the wall, respectively. We utilize Spalding’s law to calculate the non-dimensional velocity u + Spalding , + which we consider the “correct” value, based on the corresponding z LBM . Although u+ LBM is affected by the prediction accuracy of the local velocity, the deviation between + u+ LBM and u Spalding can indicate the precision of the shear drag to a certain degree, + i.e., a smaller the deviation between u + LBM and u Spalding implies that the calculated shear drag is more precise. ⟩ ⟨ u+

−u +

Figure 9.19 shows the distribution of this deviation ( LBMu + Spalding ) at the first off⟨ u + −u + ⟩ Spalding LBM Spalding wall grids near the ground (z/b = 0.125). is the time-averaged value u+ Spalding

of the normalized deviation of u + obtained via the LBM and Spalding’s law based on the dimensionless distance of the first off-wall grids. Deviations of all bounceback cases are generally within − 10 and − 40%, and the deviation increases as

200

9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

⟨ Fig. 9.19 Relationship between

+ u+ LBM −u Spalding

u+ Spalding

⟩

+ and z LBM at the first off-ground grids for all cases.

BB indicates that the standard halfway bounce-back BC is adopted (Han et al. 2021a)

Re increases. WFB reduces most of the deviations to under 10% in high-Re regions + + >∼ 100). In addition, in low-Re regions (z LBM 0.66 > 0.5

1

1

1

> 0.66 > 0.5

1

1

> 0.66 > 0.5

NMSE

0

0

< 0.3

0.66 and FAC2 > 0.5; while Lenz

204

9 LBM-LES in the Outdoor Wind Environment Problem Around a Single …

Table 9.7 Recommended simulation parameters for the isothermal outdoor wind environment around a single building Item

Recommended value

SGS model

WALE SGS model or other advanced SGS model

Grid resolution

Not coarser than b/16 if WFB is utilized; otherwise, b/32 for halfway bounce-back

Relaxation time scheme

MRT

Discrete velocity scheme D3Q19 Solid wall BC

WFB or other bounce-back BC that includes wall-function

Simulation period

Sampling period: 1 [min]; Sampling frequency: 250 [Hz]

et al. (2019) suggest FB < 0.3 and NMSE < 4. These acceptable ranges are also listed in the table. The simulation results can be considered satisfactory if these metrics fall within their corresponding acceptance range. First, the grid resolution is fundamental. A too coarse grid resolution (e.g., b/8) will cause unacceptable simulation results in most cases. In the process of improving the grid resolution, a more accurate shear drag can be calculated using the WFB solid wall boundary, which can accommodate coarser near-wall grids than the standard bounce-back condition. Second, MRT and D3Q19 are more accurate than SRT and D3Q27. The evaluation metric results also show that the simulation accuracy of LBM_b/16_MRT_WFB_D3Q19 is comparable to that of FVM-LES and that it can obtain satisfactory results. In general, among all the LBM-LES cases investigated in this chapter, LBM_b/16_MRT_WFB_D3Q19 yields optimized accurate results and numerical stability, with simulation results that are in agreement with the experimental results. The consistency between the LBM-LES and FVM-LES results was assessed based on the results for the LBM_b/16_MRT_D3Q19 and FVM-LES test cases. Figure 9.22 shows the consistency between LBM-LES and FVM-LES for the time-averaged scalar velocity and TKE. The HR metric (Schatzmann et al. 2010) was used once more to evaluate the consistency, as defined by (8.7). The allowable relative deviation tolerance (DTo ) used in this study was 10% and 30%, corresponding to HR values of 90% and 70%, respectively. HR indicates the fraction of data points in which the relative deviation between LBM-LES and FVM-LES is less than DTo . The 70%—HR values for the time-averaged scalar velocity and TKE are found to be 0.892 and 0.441, respectively. The 90%—HR values for the time-averaged scalar velocity and TKE are lower than the 70%—HR values (0.634 and 0.231, respectively). Overall, the consistency between LBM-LES and FVM-LES is lower for TKE than for the time-averaged scalar velocity.

0.92

0.81

LBM _ b/16_MRT_WFB_D3Q19

0.79

0.57

0.69

LBM_b/16_SRT_BB_D3Q27

FVM-LES

0.76

0.13

0.75

0.72

0.73

0.74

0.52

0.33

0.13

0.13

> 0.66

1

HR

⟨ ⟩ uy

0.73

0.33

0.96

0.72

0.98

0.70

0.87

0.58

0.37

0.32

> 0.5

1

FAC2

⟨u z ⟩

0.71

0.18

0.69

0.71

0.80

0.69

0.59

0.59

0.22

0.11

> 0.66

1

HR

0.79

0.31

0.92

0.82

0.93

0.78

0.89

0.74

0.32

0.26

> 0.5

1

FAC2

TKE

0.80 0.82

0.70

1.00

0.96

0.97

0.98

0.87

0.68

0.76

0.13

> 0.5

1

FAC2

0.56

0.52

0.72

0.43

0.69

0.41

0.47

0.53

0.05

> 0.66

1

HR

Note Bold values indicate values that did not satisfy the acceptance range. BB stands for halfway bounce-back BC

0.88

0.81

0.84 0.93

0.66

0.79

LBM_b/32_MRT_BB _D3Q19

LBM_b/32_MRT_WFB_D3Q19

0.81

0.61

0.67

LBM _ b/8_MRT_WFB_ D3Q19

0.67

LBM _ b/16_MRT_BB _ D3Q19

0.44

LBM _ b/8_MRT_BB_D3Q19

0.78

0.45

0.23

0.52

> 0.5

> 0.66

LBM_ b/8_SRT_BB_D3Q19

1

1

Ideal value

Acceptance range

LBM_ b/16_SRT_BB _D3Q19

FAC2

⟨u x ⟩

HR

Test case

Table 9.8 Evaluation metrics for selected cases

0.28

0.18

0.17

0.63

0.02 0.05

0.05

0.11

− 0.20 − 0.16

0.17

0.35

0.76

0.37

2.32