Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com An Introduction to Modeling and Simulation of Particulate Flows cs04_ZohdiFM-a.qxp 5/17/2007 1:44 PM Page 1 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Computational Science and Engineering (CS&E) is widely accepted, along with theory and experiment, as a crucial third mode of scientific investigation and engineering design. This series publishes research monographs, advanced undergraduate- and graduate-level textbooks, and other volumes of interest to a wide segment of the community of computational scientists and engineers. The series also includes volumes addressed to users of CS&E methods by targeting specific groups of professionals whose work relies extensively on computational science and engineering. Editor-in-Chief Omar Ghattas University of Texas at Austin Editorial Board COMPUTATIONAL SCIENCE & ENGINEERING David Keyes, Associate Editor Columbia University Ted Belytschko Northwestern University Clint Dawson University of Texas at Austin Lori Freitag Diachin Lawrence Livermore National Laboratory Charbel Farhat Stanford University James Glimm Stony Brook University Teresa Head-Gordon University of California–Berkeley and Lawrence Berkeley National Laboratory Rolf Jeltsch ETH Zurich Chris Johnson University of Utah Laxmikant Kale University of Illinois Efthimios Kaxiras Harvard University Jelena Kovacevic Carnegie Mellon University Habib Najm Sandia National Laboratory Alex Pothen Old Dominion University Series Volumes Zohdi,T. I., An Introduction to Modeling and Simulation of Particulate Flows Biegler, Lorenz T., Omar Ghattas, Matthias Heinkenschloss, David Keyes, and Bart van Bloemen Waanders, Editors, Real-Time PDE-Constrained Optimization Chen, Zhangxin, Guanren Huan, and Yuanle Ma, Computational Methods for Multiphase Flows in Porous Media Shapira,Yair, Solving PDEs in C++: Numerical Methods in a Unified Object-Oriented Approach cs04_ZohdiFM-a.qxp 5/17/2007 1:44 PM Page 2 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com An Introduction to Modeling and Simulation of Particulate Flows T. I. Zohdi University of California–Berkeley Berkeley, California Society for Industrial and Applied Mathematics Philadelphia cs04_ZohdiFM-a.qxp 5/17/2007 1:44 PM Page 3 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Copyright © 2007 by the Society for Industrial and Applied Mathematics. 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Trademarked names may be used in this book without the inclusion of a trademark symbol.These names are used in an editorial context only; no infringement of trademark is intended. Figures 2.1–2.4, 4.1–4.3, 5.2, and 5.3 are reprinted with permission from Zohdi,T.I., 2004, Modeling and direct simulation of near-field granular flows, The International Journal of Solids and Structures,Vol. 42, issue 2, pp. 539–564. Copyright © 2004 by Elsevier Ltd. Figures 6.1–6.6 are reprinted with permission from Zohdi,T.I., 2003, Computational design of swarms, The International Journal of Numerical Methods in Engineering,Vol. 57, pp. 2205–2219. Copyright © 2003 John Wiley & Sons Ltd. Figures 7.1, 7.2, and 7.4–7.11 are reprinted with permission from Zohdi,T.I., 2005, Charge-induced clustering in multifield granular flow, The International Journal of Numerical Methods in Engineering,Vol. 62, issue 7, pp. 870–898. Copyright © 2004 John Wiley & Sons Ltd. Figures 8.1–8.6 are reprinted with permission from Zohdi,T.I., in press, Computation of strongly coupled multifield interaction in particle-fluid systems, Computer Methods in Applied Mechanics and Engineering. Copyright © Elsevier Ltd. Figures 9.1, 9.2, 9.4, 9.7, and 9.11–9.18 are reprinted with permission from Zohdi,T.I., 2006, Computation of the coupled thermo-optical scattering properties of random particulate systems, Computer Methods in Applied Mechanics and Engineering,Vol. 195, issues 41–43, pp. 5813–5830. Copyright © 2005 Elsevier Ltd. Figures 9.5, 9.6, 9.8–9.10, B.3, and B.4 are reprinted with permission from Zohdi,T.I., 2006, On the optical thickness of disordered particulate media, Mechanics of Materials,Vol. 38, pp. 969–981. Copyright © 2005 Elsevier Ltd. Figures B.5–B.9 are reprinted with permission from Zohdi,T.I. and Kuypers, F.A., 2006, Modeling and rapid simulation of multiple red blood cell light scattering, Journal of the Royal Society Interface,Vol. 3, no. 11, pp. 823–831. Copyright © 2006 The Royal Society of London. The cover was produced from images created by and used with permission of the Scientific Computing and Imaging (SCI) Institute, University of Utah; J. Bielak, D. O’Hallaron, L. Ramirez-Guzman, and T.Tu, Carnegie Mellon University; O. Ghattas, University of Texas at Austin; K. Ma and H.Yu, University of California, Davis; and Mark R. Petersen, Los Alamos National Laboratory. More information about the images is available at http://www.siam.org/books/series/csecover/php. Library of Congress Cataloging-in-Publication Data Zohdi,Tarek I. An introduction to modeling and simulation of particulate flows / Tarek I. Zohdi. p. cm. (Computational science and engineering) ISBN 978-0-898716-27-6 (alk. paper) 1. Granular materials Fluid dynamics Mathematical models. I.Title. TA418.78.Z64 2007 620’.43 dc22 2007061728 is a registered trademark. cs04_ZohdiFM-a.qxp 5/17/2007 1:44 PM Page 4 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Dedicated to my patient wife, Britta, and my mother and father, Omnia and Magd cs04_ZohdiFM-a.qxp 5/17/2007 1:44 PM Page 5 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com cs04_ZohdiFM-a.qxp 5/17/2007 1:44 PM Page 6 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com 05 book 2007/5/15 page vii ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ Contents List of Figures xi Preface xv 1 Fundamentals 1 1.1 Notation 1 1.2 Kinematics of a single particle 2 1.3 Kinetics of a single particle 3 1.3.1 Work, energy, and power 3 1.3.2 Properties of a potential 4 1.3.3 Impulse and momentum 5 1.4 Systems of particles 6 1.4.1 Linear momentum 6 1.4.2 Energy principles 7 1.4.3 Remarks on scaling 8 2 Modeling of particulate flows 11 2.1 Particulate flow in the presence of near-fields 11 2.2 Mechanical contact with near-field interaction 12 2.3 Kinetic energy dissipation 15 2.4 Incorporating friction 17 2.4.1 Limitations on friction coefficients 18 2.4.2 Velocity-dependent coefficients of restitution 19 3 Iterative solution schemes 21 3.1 Simple temporal discretization 21 3.2 An example of stability limitations 22 3.3 Application to particulate flows 22 3.4 Algorithmic implementation 26 4 Representative numerical simulations 31 4.1 Simulation parameters 32 4.2 Results and observations 33 vii Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com 05 book 2007/5/15 page viii ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ viii Contents 5 Inverse problems/parameter identification 39 5.1 A genetic algorithm 40 5.2 A representative example 43 6 Extensions to “swarm-like” systems 47 6.1 Basic constructions 48 6.2 A model objective function 49 6.3 Numerical simulation 50 6.4 Discussion 52 7 Advanced particulate flow models 55 7.1 Introduction 55 7.2 Clustering and agglomeration via binding forces 55 7.3 Long-range instabilities and interaction truncation 56 7.4 A simple model for thermochemical coupling 58 7.4.1 Stage I: An energy balance during impact 59 7.4.2 Stage II: Postcollision thermal behavior 61 7.5 Staggering schemes 63 7.5.1 A general iterative framework 63 7.5.2 Semi-analytical examples 66 7.5.3 Numerical examples involving particulate flows 68 8 Coupled particle/fluid interaction 81 8.1 A model problem 82 8.1.1 A simple characterization of particle/fluid interaction . . . 82 8.1.2 Particle thermodynamics 84 8.2 Numerical discretization of the Navier–Stokes equations 86 8.3 Numerical discretization of the particle equations 89 8.4 An adaptive staggering solution scheme 91 8.5 A numerical example 95 8.6 Discussion of the results 98 8.7 Summary 101 9 Simple optical scattering methods for particulate media 103 9.1 Introduction 104 9.1.1 Ray theory: Scope of use 104 9.1.2 Beams composed of multiple rays 105 9.1.3 Objectives 106 9.2 Plane harmonic electromagnetic waves 107 9.2.1 Plane waves 107 9.2.2 Electromagnetic waves 107 9.2.3 Optical energy propagation 108 9.2.4 Reflection and absorption of energy 109 9.3 Multiple scatterers 113 9.3.1 Parametrization of the scatterers 114 9.3.2 Results for spherical scatterers 116 9.3.3 Shape effects: Ellipsoidal geometries 118 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com 05 book 2007/5/15 page ix ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ Contents ix 9.4 Discussion 119 9.5 Thermal coupling 120 9.6 Solution procedure 122 9.7 Inverse problems/parameter identification 124 9.8 Parametrization and a genetic algorithm 125 9.9 Summary 128 10 Closing remarks 133 A Basic (continuum) fluid mechanics 137 A.1 Deformation of line elements 137 A.2 The Jacobian of the deformation gradient 138 A.3 Equilibrium/kinetics of solid continua 138 A.4 Postulates on volume and surface quantities 139 A.5 Balance law formulations 140 A.6 Symmetry of the stress tensor 140 A.7 The first law of thermodynamics 141 A.8 Basic constitutive assumptions for fluid mechanics 142 B Scattering 145 B.1 Generalized Fresnel relations 145 B.2 Biological applications: Multiple red blood cell light scattering 145 B.2.1 Parametrization of cell configurations 148 B.2.2 Computational algorithm 148 B.2.3 A computational example 149 B.2.4 Extensions and concluding remarks 153 B.3 Acoustical scattering 155 B.3.1 Basic relations 155 B.3.2 Reflection and ray-tracing 156 Bibliography 159 Index 175 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com [...]... works of Hutter and collaborators: Tai et al [188]–[190], Gray et al [80], Wieland et al [201], Berezin et al [28], Gray and Hutter [81], Gray [82], Hutter [96], Hutter et al [97], Hutter and Rajagopal [98], Koch et al [126], Greve and Hutter [85], and Hutter et al [99]; the works of Behringer and collaborators: Behringer [22], Behringer and Baxter [21], Behringer and Miller [23], and Behringer et... sand and gravel, associated with coastal erosion, landslides, and avalanches A concise introduction is given by Duran [61] Many manufactured materials also fall within this class of problems.1 For general overviews of granular media, we refer the reader to Jaeger and Nagel [100], [101], Nagel [151], Liu et al [139], Liu and Nagel [140], Jaeger and Nagel [102], Jaeger et al [103]–[105], and Jaeger and. .. Behringer and Baxter [21], Behringer and Miller [23], and Behringer et al [24]; the works of Jenkins and collaborators: Jenkins and Strack [107], Jenkins and La Ragione [108], Jenkins and Koenders [109], and Jenkins et al [110]; and the works of Torquato and collaborators: Torquato [194], Kansaal et al [119], and Donev et al [55]–[59] In this monograph, we focus on a subset of the very large field of granular... are bulk and relative motion for eo = 0.5, µs = 0.2, µd = 0.1: (1) no near-field interaction, (2) α 1 = 0.1 and α 2 = 0.05, (3) α 1 = 0.25 and α 2 = 0.125, and (4) α 1 = 0.5 and α 2 = 0.25 (Zohdi [212]) 33 The total kinetic energy in the system per unit mass for eo = 0.5, µs = 0.2, µd = 0.1: (1) no near-field interaction, (2) α 1 = 0.1 and α 2 = 0.05, (3) α 1 = 0.25 and α 2 =... products which originate as granulated or particulate media, and which are transported and constructed using flow processes, are outside the scope of this monograph For more details, see, for example, Aboudi [1], Hashin [90], Mura [150], Nemat-Nasser and Hori [152], Torquato [194], and Zohdi and Wriggers [216] 2 It is worth noting that fast computational methods, in particular efficient contact search... of two vectors u and v, we have in three dimensions 3 u·v = vi ui = u1 v1 + u2 v2 + u3 v3 = ||u|||v|| cos θ, (1.1) i=1 where ||u|| = u2 + u 2 + u 2 1 2 3 (1.2) represents the Euclidean norm in R 3 and θ is the angle between the two vectors We recall that a norm has three main characteristics for any two bounded vectors u and v (||u|| < ∞ and ||v|| < ∞): • ||u|| > 0, and ||u|| = 0 if and only if u =... irradiance (Zohdi and Kuypers [223]) The diameter (8000 cells) of the scatterers is given by Equation (B.2) Computational results for the propagation of the forward scatter of Ix (t)/ ||I (0)|| for increasingly larger numbers of cells in the sample (Zohdi and Kuypers [223]) A comparison between the computational predictions and laboratory... overshoots the target (10, 0, 0), and then undershoots it slightly and starts to concentrate itself (Zohdi [209]) 48 50 50 52 53 xi ✐ ✐ ✐ ✐ ✐ ✐ ✐ 05 book 2007/5/15 page xii ✐ Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com xii List of Figures 6.6 7.1 7.2 7.3 7.4 The swarm starts to oscillate slightly around the target and then begins to home in on the target and concentrate itself at... position over time for the best parameter set, and the components of the average ray velocity and the Euclidean norm over time for the best parameter set The normalized quantity ||v||/c = 1 serves as a type of computational “error check” (Zohdi [218]) 130 The components of the average ray irradiance and the Euclidean norm over time for the best parameter set, and the average temperature of the scatterers... 0.2, µd = 0.1, α 1 = 0.5, and α 2 = 0.25rm : (1) κ = 106 J/m2 , (2) κ = 2 × 106 J/m2 , (3) κ = 4 × 106 J/m2 , and (4) κ = 8 × 106 J/m2 (Zohdi [217]) Without clustering forces: the total kinetic energy in the system per unit mass with eo = 0.5, µs = 0.2, µd = 0.1, α 1 = 0.5, and α 2 = 0.25: (1) κ = 106 J/m2 , (2) κ = 2 × 106 J/m2 , (3) κ = 4 × 106 J/m2 , and (4) κ = 8 × 106 J/m2 . http://www.simpopdf.com Computational Science and Engineering (CS&E) is widely accepted, along with theory and experiment, as a crucial third mode of scientific investigation and engineering design work relies extensively on computational science and engineering. Editor-in-Chief Omar Ghattas University of Texas at Austin Editorial Board COMPUTATIONAL SCIENCE & ENGINEERING David Keyes,. and collaborators: Behringer [22], Behringer and Baxter [21], Behringer and Miller [23], and Behringer et al. [24]; the works of Jenkins and collaborators: Jenkins and Strack [107], Jenkins and