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Home , Fluid dynamics, Heat transfer

The in Transfer and Dynamics Third Edition

J. N. Reddy Department of Mechanical Texas A&M University College Station, Texas, USA 77843—3123 D. K. Gartling Engineering Sciences Center Sandia National Laboratories Albuquerque, New Mexico, USA 87185

CRC Press Boca Raton London New York Washington, D.C. • • • Library of Congress Cataloging-in-Publication Data To our wives Aruna and Laura

vii

Contents

Preface to the Third Edition ...... xvii Preface to the Second Edition ...... xix Preface to the First Edition...... xxi About the Authors ...... xxiii 1. of and ...... 1 1.1 Introduction...... 1 1.1.1 Heat Transfer ...... 1 1.1.2 Fluid Mechanics ...... 2 1.2 Present Study ...... 3 1.3 Mathematical Preliminaries...... 3 1.3.1 Vectors and Tensors...... 3 1.3.2 Index Notation and Summation Convention ...... 5 1.3.3 The Operator and Calculus of Vectors and Tensors ...... 7 1.4 Governing Equations of a Continuum ...... 10 1.4.1 Introduction ...... 10 1.4.2 ; the Continuity ...... 10 1.4.3 Conservation of Momenta ...... 11 1.4.4 Conservation of ...... 12 1.4.5 ...... 13 1.4.6 Constitutive Equations...... 14 1.4.7 Divergence and Forms ...... 15 1.5 Governing Equations in Terms of Primitive Variables ...... 16 1.5.1 Vector Form ...... 16 1.5.2 Cartesian Component Form ...... 17 1.5.3 Cylindrical Component Form ...... 17 1.5.4 Closure ...... 19 1.6 Porous Flow Equations ...... 19 1.7 Low-Speed Equations ...... 20 1.8 Auxiliary Transport Equations ...... 22 1.9 Chemically Reacting Systems ...... 23 1.10 Boundary Conditions...... 26 1.10.1 Viscous Flow Boundary Conditions ...... 26 1.10.2 Porous Flow Boundary Conditions ...... 29 1.10.3 Thermal and Transport Boundary Conditions ...... 30 1.10.4 Initial Conditions ...... 31 viii CONTENTS

1.11 Change of Phase ...... 32 1.12 Enclosure ...... 34 1.13 Summary of Equations ...... 36 Problems ...... 37 References for Additional Reading...... 40 2. The Finite Element Method ...... 43 2.1 Introduction ...... 43 2.2 Model Differential Equation ...... 44 2.3 Finite Element Approximation...... 45 2.4 Weighted- Statements and Weak Forms ...... 47 2.4.1 Preliminary Comments...... 47 2.4.2 Weak Form Development ...... 47 2.5 Finite Element Model ...... 50 2.6 Interpolation Functions...... 52 2.6.1 Properties of Approximation Functions ...... 52 2.6.2 Linear Triangular Element ...... 52 2.6.3 Linear Rectangular Element...... 54 2.6.4 Evaluation of Boundary ...... 55 2.7 Assembly of Elements ...... 55 2.8 Time-Dependent Problems ...... 58 2.8.1 Introduction ...... 58 2.8.2 Semidiscretization ...... 58 2.8.3 Temporal Approximation ...... 60 2.9 Axisymmetric Problems ...... 61 2.10 Library of Finite Elements ...... 63 2.10.1 Introduction ...... 63 2.10.2 Triangular Elements ...... 63 2.10.3 Rectangular Elements...... 65 2.11 Numerical Integration ...... 66 2.11.1 Preliminary Comments ...... 66 2.11.2 Coordinate Transformations ...... 68 2.11.3 Integration over a Master Rectangular Element...... 70 2.11.4 Integration over a Master Triangular Element ...... 71 2.12 Modeling Considerations ...... 72 2.12.1 Mesh Generation ...... 72 2.12.2 Representation of Boundary ...... 74 2.12.3 Imposition of Boundary Conditions ...... 74 2.13 Illustrative Examples ...... 75 2.13.1 Example 1 ...... 76 2.13.1.1 Problem description ...... 76 2.13.1.2 Solution by linear triangular elements...... 77 2.13.1.3 Solution by linear rectangular elements ...... 79 2.13.1.4 Discussion of the results ...... 79 CONTENTS ix

2.13.2 Example 2 ...... 81 2.13.3 Example 3 ...... 82 Problems ...... 83 References for Additional Reading...... 86 3. Conduction Heat Transfer ...... 87 3.1 Introduction ...... 87 3.2 Semidiscrete Finite Element Model ...... 88 3.3 Interpolation Functions...... 90 3.3.1 Preliminary Comments...... 90 3.3.2 Hexahedral (Brick) Elements ...... 91 3.3.3 Tetrahedral Elements ...... 92 3.3.4 Prism Elements ...... 93 3.3.5 Pyramid Elements ...... 94 3.4 Numerical Integration ...... 95 3.5 Computation of Flux ...... 96 3.6 Semidiscrete Finite Element Model ...... 99 3.7 Solution of Nonlinear Equations ...... 100 3.7.1 Preliminary Comments ...... 100 3.7.2 Steady-State Problems ...... 100 3.7.3 Transient Problems ...... 102 3.7.3.1 General formulation ...... 102 3.7.3.2 Predictor-corrector methods ...... 105 3.7.3.3 Time step control ...... 106 3.7.3.4 Initialization ...... 107 3.7.3.5 Linear multi-step methods ...... 107 3.7.3.6 Convergence and stability ...... 108 3.7.3.7 Mode superposition methods ...... 111 3.8 Radiation Solution Algorithms ...... 113 3.9 Variable Properties...... 118 3.9.1 -Dependent Properties ...... 118 3.9.2 Phase Change Properties ...... 119 3.9.3 Anisotropic Properties ...... 121 3.10 Post-Processing Operations...... 122 3.10.1 ...... 122 3.10.2 Heat Flow Function ...... 124 3.11 Advanced Topics in Conduction ...... 125 3.11.1 Introduction ...... 125 3.11.2 Specialty Elements ...... 126 3.11.3 Computational Boundary Conditions...... 129 3.11.3.1 Contact boundary conditions...... 129 3.11.3.2 Multipoint constraints ...... 132 3.11.3.3 Partially covered ...... 133 3.11.4 Bulk Nodes...... 134 3.11.5 Reactive Materials...... 136 x CONTENTS

3.11.6 Material ...... 138 3.12 Example Problems ...... 139 3.12.1 Introduction ...... 139 3.12.2 Element Convergence ...... 139 3.12.3 Conduction/Radiation Solution ...... 141 3.12.4 Temperature-Dependent Conductivity...... 143 3.12.5 Anisotropic Conductivity ...... 143 3.12.6 One-Dimensional ...... 145 3.12.7 Bit Analysis ...... 147 3.12.8 Brazing and Welding Analysis ...... 149 3.12.9 Investment Casting ...... 152 Problems ...... 154 References for Additional Reading...... 155 4. Flows of Viscous Incompressible ...... 161 4.1 Introduction ...... 161 4.1.1 Background...... 161 4.1.2 Governing Equations...... 161 4.2 Mixed Finite Element Model ...... 164 4.2.1 Weak Form ...... 164 4.2.2 Finite Element Model...... 165 4.3 Penalty Finite Element Models ...... 167 4.3.1 Introduction ...... 167 4.3.2 Penalty Function Method ...... 168 4.3.3 Reduced Integration Penalty Model ...... 171 4.3.4 Consistent Penalty Model ...... 171 4.4 Finite Element Models of Porous Flow...... 172 4.5 Computational Considerations...... 174 4.5.1 Properties of the Matrix Equations...... 174 4.5.2 Choice of Interpolation Functions ...... 175 4.5.2.1 Quadrilateral elements (2-D) ...... 176 4.5.2.2 Triangular elements (2-D) ...... 179 4.5.2.3 Hexahedral elements (3-D)...... 179 4.5.2.4 Tetrahedral elements (3-D) ...... 180 4.5.3 Evaluation of Element Matrices in Penalty Models ...... 180 4.5.4 Calculation ...... 181 4.5.5 Traction Boundary Conditions ...... 184 4.6 Solution of Nonlinear Equations ...... 186 4.6.1 General Discussion...... 186 4.6.2 Fully Coupled Solution Methods ...... 189 4.6.2.1 Picard method ...... 189 4.6.2.2 Newton’s method...... 190 4.6.2.3 ModiÞed and quasi—Newton methods ...... 192 4.6.2.4 Continuation methods ...... 192 4.6.3 Pressure Correction/Projection Methods ...... 194 CONTENTS xi

4.7 Time-Approximation Schemes ...... 196 4.7.1 Preliminary Comments ...... 196 4.7.2 Forward/Backward Euler Schemes ...... 197 4.7.3 Adams—Bashforth/Trapezoid Rule ...... 198 4.7.4 Implicit Integration and Time Step Control ...... 198 4.7.5 Explicit Integration ...... 199 4.8 Stabilized Methods ...... 200 4.8.1 Preliminary Comments ...... 200 4.8.2 Galerkin/Least-Squares Formulation ...... 202 4.8.3 Polynomial Pressure Projection ...... 204 4.8.4 Variational Multiscale Methods ...... 205 4.9 Least-Squares Finite Element Models ...... 212 4.9.1 Introduction ...... 212 4.9.2 Governing Equations...... 215 4.9.3 Least-Squares Formulation ...... 216 4.9.4 Finite Element Model...... 217 4.9.5 Computational Aspects ...... 218 4.10 Post-Processing ...... 219 4.10.1 Computation...... 219 4.10.2 Computation ...... 221 4.10.3 Particle Tracking ...... 223 4.11 Flows ...... 224 4.11.1 Preliminary Comments ...... 224 4.11.2 Time-Independent Free Surfaces ...... 224 4.11.3 Time-Dependent Free Surfaces ...... 229 4.12 ...... 235 4.12.1 Preliminary Comments ...... 235 4.12.2 Governing Equations ...... 236 4.12.3 General Turbulence Models ...... 237 4.12.3.1 Correlations ...... 238 4.12.3.2 Integral methods...... 238 4.12.3.3 One-point closure ...... 238 4.12.3.4 Two-point closure...... 238 4.12.3.5 Large ...... 238 4.12.3.6 Direct numerical (DNS) ...... 239 4.12.4 One-Point Closure Turbulence Models ...... 239 4.12.4.1 Zero-equation model ...... 240 4.12.4.2 One-equation model ...... 240 4.12.4.3 Two-equation model ...... 241 4.12.5 Finite Element Modeling of Turbulence ...... 242 4.12.5.1 Zero-equation model ...... 242 4.12.5.2 Two-equation model ...... 243 4.12.6 Variational Multiscale (VMS) ...... 244 4.13 Numerical Examples ...... 247 4.13.1 Preliminary Comments ...... 247 4.13.2 Fluid Squeezed between Parallel Plates ...... 248 xii CONTENTS

4.13.3 Flow of a Viscous in a Slider Bearing ...... 250 4.13.4 Wall-Driven 2-D Cavity Flow ...... 252 4.13.5 Wall-Driven 3-D Cavity Flow ...... 254 4.13.6 Evaluation of the EBE Iterative Solvers ...... 256 4.13.7 Backward Facing Step ...... 258 4.13.8 Flow Past a Submarine ...... 260 4.13.9 Growth from the Melt ...... 262 4.13.10 Mold Filling ...... 263 4.13.11 Examples Using Least-Squares Finite Element Models...... 267 4.13.11.1 Kovasznay ßow ...... 267 4.13.11.2 Flow over a backward-facing step...... 269 4.13.11.3 Flow past a cylinder at low ...... 270 Problems ...... 273 References for Additional Reading...... 275 5. Coupled Fluid Flow and Heat Transfer ...... 285 5.1 Introduction ...... 285 5.2 Nonisothermal Incompressible Flows...... 286 5.2.1 Governing Equations...... 286 5.2.2 Boundary Conditions ...... 288 5.3 Mixed Finite Element Model ...... 289 5.4 Penalty Finite Element Model ...... 293 5.4.1 Preliminary Comments ...... 293 5.4.2 Reduced Integration Penalty Model ...... 294 5.4.3 Consistent Penalty Model ...... 295 5.5 Finite Element Models of Porous Flow...... 295 5.6 Nonisothermal, Low-Speed, Compressible Flows ...... 297 5.6.1 Governing Equations...... 297 5.6.2 Boundary Conditions ...... 299 5.6.3 Mixed Finite Element Model ...... 299 5.7 Solution Methods ...... 302 5.7.1 General Discussion...... 302 5.7.2 Newton’s Method ...... 303 5.7.3 Segregated Equation Methods...... 304 5.8 with Change of Phase ...... 306 5.9 Convection with Enclosure Radiation ...... 308 5.10 Post-Computation of Heat Flux ...... 308 5.11 Turbulent Heat Transfer ...... 310 5.12 Chemically Reacting Systems...... 311 5.12.1 Preliminary Comments ...... 311 5.12.2 Finite Element Modeling of Chemical Reactions ...... 311 5.13 Numerical Examples ...... 312 5.13.1 Preliminary Comments ...... 312 5.13.2 Concentric Tube Flow ...... 312 CONTENTS xiii

5.13.3 Tube Flow with Change of Phase ...... 313 5.13.4 Heated Cavity Boussinesq Model ...... 314 5.13.5 Heated Cavity − Acoustically Filtered Model ...... 316 5.13.6 Solar Receiver ...... − 317 5.13.7 Tube Bundle ...... 320 5.13.8 Volumetrically Heated Fluid ...... 322 5.13.9 Porous/Fluid Layer ...... 322 5.13.10 Curing of an Epoxy...... 326 5.13.11 Heated Channel ...... 329 5.13.12 Closure...... 331 References for Additional Reading...... 331 6. Flows of Non-Newtonian Fluids ...... 335 6.1 Introduction ...... 335 6.2 Governing Equations of Inelastic Fluids...... 336 6.2.1 Conservation Equations ...... 336 6.2.2 Boundary Conditions ...... 337 6.2.3 Constitutive Equations ...... 338 6.2.3.1 Power-law model ...... 339 6.2.3.2 Carreau model ...... 340 6.2.3.3 Bingham model ...... 340 6.3 Finite Element Models of Inelastic Fluids ...... 341 6.3.1 Introduction ...... 341 6.3.2 Mixed Model ...... 341 6.3.3 Penalty Model ...... 343 6.3.4 Matrix Evaluations ...... 344 6.4 Solution Methods for Inelastic Fluids ...... 346 6.5 Governing Equations of Viscoelastic Fluids ...... 350 6.5.1 Conservation Equations ...... 350 6.5.2 Constitutive Equations ...... 351 6.5.2.1 Differential models ...... 352 6.5.2.2 Integral models...... 355 6.5.3 Boundary Conditions ...... 356 6.6 Finite Element Model of Differential Form ...... 357 6.6.1 Preliminary Comments ...... 357 6.6.2 Summary of Governing Equations ...... 357 6.6.3 Finite Element Model...... 358 6.6.4 Solution Methods ...... 362 6.7 Additional Models of Differential Form ...... 363 6.7.1 Explicitly Elliptic Equation Method...... 364 6.7.2 Elastic Viscous Stress Splitting Method ...... 365 6.8 Finite Element Model of Integral Form ...... 367 6.9 Unresolved Problems ...... 368 6.9.1 General Comments ...... 368 6.9.2 Choice of ...... 369 xiv CONTENTS

6.9.3 Uniqueness and Existence of Solutions ...... 370 6.9.4 Numerical Algorithm Problems ...... 370 6.9.5 Equation Change of Type ...... 371 6.9.6 Closure ...... 372 6.10 Numerical Examples ...... 372 6.10.1 Preliminary Comments ...... 372 6.10.2 Driven Flow in a Cavity ...... 372 6.10.3 Driven Cavity Flow ...... 374 6.10.4 Squeeze Film Flow...... 374 6.10.5 Time-Dependent Poiseuille Flow ...... 377 6.10.6 Four-to-One Contraction Problem...... 380 Problems ...... 381 References for Additional Reading...... 382 7. Multiphysics Problems ...... 387 7.1 Introduction ...... 387 7.2 Coupled Boundary Value Problems ...... 387 7.3 Fluid Mechanics and Heat Transfer ...... 388 7.3.1 Introduction ...... 388 7.3.2 Continuum Equations...... 388 7.3.3 Finite Element Models ...... 390 7.4 Mechanics ...... 390 7.4.1 Introduction ...... 390 7.4.2 Kinematics of ...... 391 7.4.2.1 Descriptions of motion ...... 391 7.4.2.2 Displacement vector ...... 393 7.4.2.3 Deformation tensor ...... 393 7.4.2.4 Green strain tensor ...... 394 7.4.3 Kinetics ...... 395 7.4.3.1 Stress measures...... 395 7.4.3.2 Equilibrium statements ...... 395 7.4.4 Constitutive Relations ...... 396 7.4.5 Boundary Conditions ...... 397 7.4.6 Finite Element Models ...... 397 7.4.7 Solution Methods ...... 399 7.5 Electromagnetics ...... 399 7.5.1 Introduction ...... 399 7.5.2 Maxwell’s Equations ...... 400 7.5.2.1 Constitutive relations ...... 400 7.5.2.2 Electromagnetic and heating ...... 402 7.5.2.3 Quasi-static approximation ...... 402 7.5.3 Electromagnetic Potentials ...... 403 7.5.4 Boundary and Interface Conditions ...... 405 7.5.5 Gauge Conditions...... 407 CONTENTS xv

7.5.6 Static Problems ...... 408 7.5.6.1 Electrostatics ...... 408 7.5.6.2 Steady current ßow ...... 409 7.5.6.3 Magnetostatics ...... 409 7.5.7 Finite Element Models for EM Fields ...... 409 7.5.7.1 Quasi-static potential equations ...... 409 7.5.7.2 Gauge condition ...... 412 7.5.7.3 Static Þeld equations ...... 413 7.5.8 Solution Methods — EM Fields ...... 414 7.6 Coupled Problems in Mechanics ...... 415 7.6.1 Introduction ...... 415 7.6.2 Heat Conduction Viscous Fluid Interactions 1&2...... 416 7.6.3 Heat Conduction − Quasi-Static Solid Interactions 1&3 ...... 416 7.6.4 Heat Conduction − Electric Field Interactions 1&4 ...... 418 7.6.5 Heat Conduction − Electromagnetic Field Interactions 1&4&5...418 7.6.6 Viscous Flow Quasi-Static− Solid Interactions 2&3 ...... 420 7.6.7 Viscous Flow − Electric Field Interactions 2&4...... 421 7.6.8 Viscous Flow − Electromagnetic Field Interactions 2&4&5 ...... 422 7.6.9 Quasi-Static Solid− — Electromagnetic Field Interactions 3&4&5 ..423 7.7 Implementation of Coupled Algorithms ...... 424 7.8 Numerical Examples ...... 426 7.8.1 Introduction ...... 426 7.8.2 Thermal-Stress Example ...... 426 7.8.3 Thermal-Electromagnetic Example ...... 428 7.8.4 Fluid-Solid Interaction Example...... 431 7.8.5 Fluid-Electromagnetic Example ...... 432 References for Additional Reading...... 436 8. Parallel Processing ...... 439 8.1 Introduction ...... 439 8.2 Parallel Systems ...... 440 8.2.1 ClassiÞcation ...... 440 8.2.1.1 Granularity of the processing elements ...... 440 8.2.1.2 Topology of interconnections ...... 440 8.2.1.3 Distribution of control across the processing elements ....441 8.2.1.4 Memory access ...... 442 8.2.2 Languages and Communication Utilities...... 442 8.2.3 Performance ...... 443 8.2.3.1 Algorithmic efficiency ...... 443 8.2.3.2 Actual/BeneÞcial efficiency ...... 443 8.2.3.3 Scalability ...... 444 8.3 FEM and Parallel Processing ...... 444 8.3.1 Preliminary Comments ...... 444 8.3.2 Generic FEM Steps ...... 445 8.3.3 External Preprocessing ...... 445 8.3.4 Internal Preprocessing ...... 447 xvi CONTENTS

8.3.5 Solution Processing ...... 447 8.3.5.1 Element matrices...... 447 8.3.5.2 Matrix solvers...... 448 8.3.5.3 Solution control ...... 450 8.3.6 Internal Postprocessing ...... 451 8.3.7 External Postprocessing...... 451 8.3.8 Other Parallel Issues ...... 451 8.3.8.1 Nonlocal data ...... 452 8.3.8.2 Multiphysics simulations ...... 452 8.4 Summary ...... 454 References for Additional Reading...... 454 Appendix A: Computer Program FEM2DHT ...... 457 A.1 Introduction ...... 457 A.2 Heat Transfer and Related Problems ...... 457 A.3 Flows of Viscous Incompressible Fluids...... 458 A.4 Description of the Input Data...... 458 A.5 Source Listings of Selective Subroutines ...... 469 Reference for Additional Reading ...... 470 Appendix B: Solution of Linear Equations ...... 477 B.1 Introduction ...... 477 B.2 Direct Methods ...... 478 B.3 Iterative Methods...... 479 B.3.1 General Comments ...... 479 B.3.2 Solution Algorithms ...... 479 References for Additional Reading...... 483 Appendix C: Fixed Point Methods and Contraction Mappings 485 C.1 Fixed Point Theorem ...... 485 C.2 Method...... 486 C.3 Newton’s Method...... 487 C.4 The Newton—Raphson Method ...... 488 C.5 Descent Methods ...... 488 References for Additional Reading...... 489 Subject Index ...... 491 xvii

Preface to the Third Edition

Computational ßuid mechanics (CFD) and computational heat transfer (CHT) continue to evolve at a signiÞcant pace and have become an ever increasing presence in standard engineering design and analysis practice. The seemingly endless increase in computing power in both single processor and parallel environments has allowed realistic problems of signiÞcant complexity and Þdelity to be routinely solved and utilized in technological advances. Commercial software has made rapid progress in providing a broad spectrum of analysis capabilities to a variety of industries. Though software is increasingly robust, accurate CFD and CHT simulations still require a knowledgeable user, with a background in both mechanics and numerical methods. The present edition of this book remains focused on providing the information required by an individual who is interested in good numerical methods for the study and understanding of ßuid mechanics and heat transfer phenomena. This book remains practical in scope and content with an emphasis on computational procedures that we have found effective on a wide spectrum of applications. Little, if any, material has been deleted from the second edition. New material has been added primarily in the Þrst Þve chapters and reßects the research thrusts over the last eight years. Chapter 1 contains the general description of the boundary value problems of interest and has been expanded with a section on mathematical preliminaries and a section on low-speed compressible ßows. Chapter 2 continues with the introduction of the Þnite element method and is essentially unchanged. The and radiation problem is discussed in great detail in Chapter 3, and has a new discussion of mode superposition methods and a more detailed account of radiation solution methods. The isothermal, viscous ßow problem is the topic of Chapter 4. The section on stabilized methods has been expanded with more discussion of variational multiscale methods (VMM), and a new section on least-squares Þnite element models (LSFEM) has been added. Chapter 5 extends the Þnite element method to non-isothermal ßows and now includes a section on the formulation of low-speed, compressible ßows. Non-Newtonian ßow problems, both inelastic and viscoelastic, are included in Chapter 6, which is largely unchanged. Chapter 7 remains focused on the formulations and algorithms for multidisciplinary problems involving ßuid mechanics, heat transfer, , and electromagnetics; the last chapter is on parallel computing, including a general discussion of the parallel architecture and the implementation of Þnite element models. Organization of the text, equation numbering, references, and symbols retain the same style as used in the previous editions. References have been added as needed without the removal of older citations. We believe it is important to retain the historical record. xviii PREFACE

The authors thank the publisher for the opportunity to prepare this third edition. The Þrst author thanks colleagues Drs. Vinu Unnikrishnan and Ginu Unnikrishnan for their help with the scanning of Þgures from the Þrst two editions and preparing the subject index, and Feifei Cheng for proofreading of the manuscript. The second author again thanks his numerous present and former colleagues at Sandia National Laboratories who continue to present challenging engineering applications in applied and computational mechanics. SpeciÞc acknowledgments must go to Drs. Mike Glass, Rick Givler, Charles Hickox, Roy Hogan, Mario Martinez and Phil Sackinger for collaboration and assistance in much of the algorithm development and demonstration simulations cited in this . Portions of this book are taken from research work performed at Sandia National Laboratories under Contract No. DE—AC04—94AL85000 awarded by the U.S. Department of Energy and are used with permission. The authors dedicate this book to their wives, Aruna and Laura, who have graciously tolerated the authors’ preoccupation with the writing of the book.

J. N. Reddy College Station, Texas [email protected] D. K. Gartling Albuquerque, New Mexico [email protected] xix

Preface to the Second Edition

In the six years since the Þrst edition of this book appeared some signiÞcant changes have occurred in the area of computational mechanics in general, and in computational ßuid mechanics and heat transfer in particular. Foremost among these changes has been the extraordinary increase in performance in desktop computing platforms and the arrival in signiÞcant numbers of parallel computers. This widespread availability of capable computing hardware has predictably lead to the increased demand for of products and processes during the engineering design and manufacturing process. Our original thesis that the Þnite element method was very well suited to general purpose and commercial software continues to hold true, as numerous programs are now available for the simulation of all types of applied mechanics problems. The range of applications of Þnite element analysis in ßuid mechanics and heat transfer has become quite remarkable with complex, realistic simulations being carried out on a routine basis. The combination of hardware performance and reliable Þnite element algorithms has made these advances possible. Another signiÞcant change in computational mechanics is the increase in multidisciplinary (multiphysics) problems and their solution via Þnite element methods. Again, the increase in hardware performance has contributed to these types of computationally intensive problems. However, the inroads made by the Þnite element method in all areas of mechanics have also had a positive inßuence on coupled analysis. The commonality of Þnite element formulation, approximation and solution among the various boundary value problems in mechanics eases considerably the contemplation of multiphysics solutions and software. A Þnal change in the numerical simulation arena comes from the implementation side of the Þnite element method. The demand for software capability and reliability has increased in step with the hardware performance. The use of parallel computers has added to the complexity of the implementation. All of these attributes lead to the conclusion that Þnite element implementation, if done well, will require some signiÞcant knowledge from areas in . The time of general purpose codes being developed and maintained by one or two individuals is past and multi- talented teams now provide the most modern software. Our focus for the present edition of this book remains the same – the education of the individual who is interested in good numerical methods for the study of ßuid mechanics and heat transfer phenomena. The text remains practical in scope and content with an emphasis on computational procedures that we have found effective on a wide spectrum of applications. Little, if any, material has been deleted from the Þrst edition. New material has been added in almost all chapters along with some rearrangement of topics to improve overall clarity and maintain the step- wise addition of increasingly complex material. Chapter 1 contains the general description of the boundary value problems of interest and has been augmented with a section on chemically reactive systems and additional discussion of change of phase. Chapter 2 continues with the introduction of the Þnite element method xx PREFACE and is essentially unchanged. The thermal conduction and radiation problem is discussed in great detail in Chapter 3, and it has new sections covering specialized Þnite elements and advanced topics in thermal analysis. The advanced topics section includes descriptions of difficult boundary conditions, such as multipoint constraints, contact and bulk nodes, material motion and kinematics, and methods for chemically reactive . The isothermal, viscous ßow problem is the topic of Chapter 4. New sections in this chapter cover stabilized Þnite element methods and a general discussion of methods for free surface problems; the section on turbulence modeling has also been moved to this chapter. Chapter 5 extends the Þnite element method to non-isothermal ßows and is largely unchanged. Non-Newtonian ßow problems, both inelastic and viscoelastic, are now included in a revised and updated Chapter 6. A completely new Chapter 7 is focused on formulations and algorithms for multidisciplinary problems involving ßuid mechanics, heat transfer, solid mechanics, and electromagnetics. This chapter outlines many of the possible types of coupling, describes the Þnite element equations for each mechanics area, and presents a number of realistic numerical examples. The last chapter on advanced topics is now devoted exclusively to a discussion of parallel computing including some general discussion of the parallel architecture and sections on parallel implementation of Þnite element models. Organization of the text, equation numbering, references, and symbols retain the same style as used in the Þrst edition. The second author thanks his numerous present and former colleagues at Sandia National Laboratories who continue to provide a wealth of challenging problems in applied and computational mechanics. SpeciÞc acknowledgments must go to Drs. Mike Glass, Rick Givler, Charles Hickox, Roy Hogan and Phil Sackinger for collaboration and assistance in much of the algorithm development and demonstration simulations cited in this work. Portions of this book are adapted from work performed at Sandia National Laboratories under Contract No. DE— AC04—94AL85000 awarded by the U.S. Department of Energy and are used with permission. The authors dedicate this book to their wives, Aruna and Laura, who have graciously tolerated the authors’ preoccupation with the writing of the book. J. N. Reddy College Station, Texas D. K. Gartling Albuquerque, New Mexico xxi

Preface to the First Edition

The numerical simulation of ßuid mechanics and heat transfer problems has become a routine part of engineering practice as well as a focus for fundamental and applied research. Though there are still various topical areas where our physical understanding and/or ineffective numerical algorithms limit the investigation, a large number of complex phenomena can now be conÞdently studied via numerical simulation. Though Þnite difference methods have and will continue to play a major role in computational ßuid dynamics (CFD) and heat transfer, Þnite element techniques have spurred the explosive development of “general purpose” methods and the growth of commercial software. The inherent strengths of the Þnite element method such as unstructured meshes, element-by-element formulation and processing, and the simplicity and rigor of boundary condition application are being coupled with modern developments in automatic mesh generation, adaptive meshing, and improved solution techniques to produce accurate and reliable simulation packages that are widely accessible. Improvements in computer hardware and system software (e.g., powerful workstations and window environments) have contributed signiÞcantly to streamlining the numerical simulation process. The Þnite element method in ßuid mechanics and heat transfer has rapidly caught up with the well-established solid mechanics community in simulation capabilities. As in any rapidly developing Þeld, the education of the non-expert user community is of primary importance. The present text is an attempt to Þll a need for those interested in using the Þnite element method in the study of ßuid mechanics and heat transfer. It is a pragmatic book that views numerical computation as a means to an end–we do not dwell on theory or proof. Other fundamental and theoretical textbooks that cover these aspects are available or anticipated. The emphasis here is on presenting a useful methodology for a limited but signiÞcant class of problems dealing with heat conduction, incompressible viscous ßows, and convection heat transfer. The text has been developed out of our experience and course notes used in teaching graduate courses and continuing education courses to a wide spectrum of students. To gain the most from the book the student should have a reasonable background in ßuid mechanics and heat transfer as would normally be found in most mechanical, aerospace, chemical or engineering mechanics curriculums. An introductory knowledge of Þnite element techniques would be very helpful but not essential; some familiarity with basic numerical analysis, linear algebra, and numerical integration would also be of assistance. Our approach to the Þnite element method for ßuid mechanics and heat transfer has been designed as a series of incremental steps of increasing complexity. In Chapter 1, the continuum boundary value problems that form the central focus of the book are described in some detail. We have tried to be as general as possible in describing the varied physical phenomena that may be encountered within the limits of non-isothermal, incompressible, viscous ßows. Chapter 2 xxii PREFACE introduces the Þnite element method by application to a simpliÞed, two-dimensional heat conduction problem. All of the necessary machinery for constructing weak forms of a partial differential equation and building a Þnite element model are introduced here and demonstrated by application. Chapter 3 recaps parts of Chapter 2 and extends the Þnite element method to three dimensions, time dependence, and practical applications in conduction heat transfer. Isothermal viscous ßuid mechanics formulations are described in Chapter 4 along with a signiÞcant section on the solution of nonlinear equations developed from the ßow problem. Chapter 5 extends the viscous ßow problem to consider convective heat transfer formulations and applications. Inelastic non-Newtonian ßows and free surface problems are covered in Chapter 6. The complex topic of viscoelastic ßow simulation is surveyed in Chapter 7. The last chapter concludes the text with a survey of several advanced topics, including turbulence modeling. The coverage of each topic is sufficient to allow the reader to understand the basic methodology, use existing simulation software with conÞdence, and allow development of some simpler, special purpose computer codes. Example problems ranging from simple benchmarks to practical engineering solutions are included with each topical area. Adequate references to the relevant literature have also been included for those desiring a more encyclopedic coverage of a speciÞctopic. The text is organized into major sections within each chapter. Equations are numbered consecutively within each major section. Within a section, reference to an equation is by its sequential number; references to equations outside the current section have a full section, equation number citation. Vectors, tensors, and matrices are denoted by boldface letters. The vectors of interpolation (shape) functions in this book are denoted by Greek symbols (Ψ, Θ, Φ). Bibliographic information for literature cited in the text is numbered sequentially within each chapter and collected at the end of the chapter. The Þrst author would like to thank M. S. Ravisankar for his help with Chapter 8 topics and Praveen Gramma for reading the manuscript. The second author would like to thank his numerous present and former colleagues at Sandia National Laboratories who have provided a seemingly endless stream of challenging problems and taught him much about the practice of computational mechanics. SpeciÞc acknowledgments must go to Drs. Charles Hickox, Rick Givler, Roy Hogan, Phil Sackinger, Randy Schunk, Rekha Rao, and Steve Rottler for their suggestions and comments on early versions of the text. Portions of this book are adapted from work performed at Sandia National Laboratories under Contract No. DE— AC04—76DP00789 awarded by the U.S. Department of Energy and are used with permission.

J. N. Reddy College Station, Texas D. K. Gartling Albuquerque, New Mexico xxiii

About the Authors

J. N. Reddy earned a Ph.D. in Engineering Mechanics from the University of Alabama in Huntsville, worked as a Postdoctoral Fellow at the University of Texas at Austin, was Research Scientist for Lockheed Missiles and Space Company during l974—75, and taught at the University of Oklahoma from 1975 to 1980 and Virginia Polytechnic Institute & State University from 1980 to 1992. Currently, he is a Distinguished Professor and the inaugural holder of the Oscar S. Wyatt Endowed Chair at Texas A&M University, College Station. Dr. Reddy has published over 400 journal papers and 16 textbooks on theoretical formulations and numerical simulations of problems in solid and structural mechanics, computational ßuid dynamics, numerical heat transfer, computational biology, geology and , mechanics of nanosystems, and applied . Dr. Reddy is the recipient of numerous honors and awards, including the 1998 Nathan M. Newmark Medal from the American Society of Civil , the 2003 Computational Solid Mechanics award from the U.S. Association of Computational Mechanics, the 2004 Distinguished Research Award from the American Society for Composites, and an honorary degree (Honoris Causa) from the Technical University of Lisbon, Portugal (2009). Dr. Reddy is a fellow of the American Academy of Mechanics, the American Institute of and Astronautics, the American Society of Civil Engineers, the American Society of Mechanical Engineers, the American Society for Composites, International Association of Computational Mechanics, U.S. Association of Computational Mechanics, the Aeronautical Society of India, and the Institution of Structural Engineers, U.K. Dr. Reddy serves on the editorial boards of two dozen journals, and as the Editor-in-Chief of Applied Mechanics Reviews, Mechanics of Advanced Materials and Structures, International Journal of Computational Methods in Engineering Science and Mechanics,andInternational Journal of Structural Stability and Dynamics. As a result of his extensive publications of archival journal papers and books, Dr. Reddy is recognized by ISI Highly Cited Researchers with over 10,000 citations and an H-index of over 40 to his credit. A more complete resume with links to journal papers can be found at http://authors.isihighlycited.com/ or http://www.tamu.edu/acml.

David K. Gartling is a Senior Scientist in the Engineering Sciences Center at Sandia National Laboratories, Albuquerque, New Mexico.HeearnedhisB.S.andM.S.inAerospace Engineering at the University of Texas at Austin and completed the diploma course at the von K´arm´an Institute for in Brussels, Belgium. After completion of his Ph.D. in at the University of Texas at Austin, he joined the technical staff at Sandia National Laboratories. Dr. Gartling was a Visiting Associate Professor in the Department at the University of Sydney, Australia, under a Fulbright Fellowship, and later he was a Supervisor in the Fluid and Thermal Sciences Department at Sandia National Laboratories. Dr. Gartling has published numerous papers dealing with Þnite element model development and Þnite element analysis of heat transfer and ßuid dynamics problems of practical importance. He is the recipient of the 2001 Computational Fluid Dynamics Award from the U.S. Association of Computational Mechanics and is a fellow of the American Society of Mechanical Engineers. Dr. Gartling is presently a member of several professional societies, serves on the editorial boards of several journals, and is the Co-Editor of International Journal for Numerical Methods in Fluids.