finite elemente method for flow problem

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Finite Element M ethods for F low Problems

Finite Element Methods for Flow Problems. Jean Donea and Antonio HuertaCopyright 2003 John Wiley & Sons, Ltd.

ISBN: 0-471-49666-9



Jean Donea and Antonio Huerta

WILEY

Finite ElementMethods for Flow Probelms



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Preface xi

1 Introductiona nd preliminaries 1

1.1 Finite elementsin fluiddynamics 11.2 Subjects covered 21.3 Kinema tica l descriptionsof the flow field 4

1.3.1 Lagrangiana nd Eulerian desc riptions 51.3.2 ALE descriptionof motion 81.3.3 Thefundamental ALE equation 111.3.4 Time derivativeof integrals over moving

volumes 121.4 The ba sic c onservation equa tions 131.4.1 Mass equation 131.4.2 Momentum equation 131.4.3 Internalenergy equation 151.4.4 Total energy equation 171.4.5 ALEform of the conservation equations 18

1.4.6 Closureof the initia l boundary va lue problem 191.5 Basic ingredientsof the finiteelement method 19

1.5.1 M a thema tica l preliminaries 19

Contents



Vi CONTENTS

1.5.2 Trial solutionsa nd weighting functions 211.5.3 Compactintegralforms 231.5.4 Strong a nd wea k formsof a boundary value

problem 231.5.5 Finite element spa tia l discretization 2 7

2 Steady tra nsportproblems 332 .1 Problem statement 33

2.1.1 Strong form 332.1.2 Weakform 35

2.2 Ga lerkin a pproximation 362.2.1 Piecewise l inear a pproxima tionin 1D 372.2.2 Analysisof the discrete equation 402.2.3 Piecewise qua dratic ap proxima tionin ID 452.2.4 Analysisof the discrete equations 47

2.3 Ea rly Petrov Galerkin m ethods 502.3.1 Upwind approximationof the convective term 502.3.2 First finite elementsof upwind type 512.3.3 The conceptof balancingdiffusion 53

2.4 Stabilization techniques 592.4.1 The SU PG method 602.4.2 The Ga lerkin/Lea st-squa res m ethod 632.4.3 The stabilizationparameter 64

2.5 Other stabilization technique s a nd new trends 652.5.1 Finite increment ca lculus 662.5.2 Bubble Junctions a nd wa velet a pproximations 672.5.3 The variational multisca le method 682.5.4 Complements 70

2 .6 Applicationsa nd solved exercises 702.6.1 Constructionof a bubble function m ethod 702.6.2 One-dimensional tran sport 722.6.3 Convection diffusion across a source term 742.6.4 Convection diffusion skew to the mesh 752.6.5 Convection diffusion-reaction in 2D 762.6.6 The Hem ker problem 78

3 Unsteady convec tive tra nsport 793.1 Introduction 793.2 Problem sta tement 81



CONTENTS vii

3.3 Themethodof characteristics 323.3.1 The conceptof characteristic lines 823.3.2 Propertiesof the linear convection equation 84

3.3.3 Methods ba sed on the cha rac teristics 873.4 Classica l timeand space discretization techniques 913.4.1 Time discretization 923.4.2 Ga lerkin spa tia l discretization 94

3.5 Stabilityand accuracy analysis 983.5.1 Analysisof stabilityby Fourier techniques 993.5.2 Analysisof classical time-stepping schemes 101

3.5.3 The modified equation method 1053.6 Taylor Galerkin Methods 1073.6.1 The needfor higher-order time sc hemes 1073.6.2 Third-order explic it Taylor Galerkin method 1083.6.3 Fourth-order explicitleap-frog method 1133.6.4 Two-step explicitTaylor Galerkin methods 114

3.7 Anintroductionto monotonicity-preserving schemes 117

3.8 Least-squares-based spatial discretization 1203.8.1 Least-squaresapproach for the 6family ofmethods 121

3.8.2 Taylor least-squa res method 1223.9 Thediscontinuous Galerkin method 1243.10 Space timeformulations 126

3.10.1 Time-discontinuous Ga lerkin formulation 1263.10.2 Time-discontinuous least-squa res formulation 1283.10.3 Spa ce-timeGa lerkin/Lea st-squa resformula tion 128

3.11 Applicationsand solved exercises 1293.11.1 Propagationof a cosineprofile 1293.11.2 Travelling wa ve pa cka ge 1333.11.3 The rotating coneproblem 1353.11.4 Propagationof a steep front 137

CompressibleFlow Problems 1474.1 Introduction 1474.2 Nonlinear hyperbolic equations 149

4.2.1 Sca lar equations 1494.2.2 Weaksolutionsand entropy c ondition 1524.2.3 Time and space discretization 156

4.3 TheEuler equations 159



Viii CONTENTS

4.3.1 Strong formof the conservation equa tions 1594.3.2 The qua si-linea rform of the Euler equations 1614.3.3 Ba sic propertiesof the Euler equa tions 163

4.3.4 Bounda ry conditions 1654.4 Spatial discretization techn iques 1664.4.1 Ga lerkin formula tion 1674.4.2 Upwind-type discretizations 168

4.5 Numerical treatmentof shocks 1764.5.1 Introduction 1764.5.2 Earlyartificial diffusion methods 178

4.5.3 High-resolution m ethods 1804.6 Nearly incompressible flows 1864.7 Fluid-structure interaction 187

4.7.1 Acoustic approximation 1894.7.2 Nonlinear tra nsient dyna micproblems 1914.7.3 Illustrative exa m ples 196

4.8 Solved exercises 199

4.8.1 One-stepTaylor Galerkin solutionof Burgers'equation 1994.8.2 The shock tubeproblem 202

5 Unsteady convection diffusion problems 2095.1 Introduction 2095.2 Problem statement 2105.3 Time discretization proc edures 21 1

5.3.1 Classica l methods 2115.3.2 Fra ctional-step methods 2135.3.3 High-order tim e-stepping schem es 215

5.4 Spatial discretization procedu res 2225.4.1 Ga lerkin formula tionof the semi-discrete

scheme 2225.4.2 Ga lerkin formulationof 0 family methods 2245.4.3 Ga lerkin formulationof explicitPade schemes 2285.4.4 Ga lerkin formula tionof implicit mu ltista ge

schemes 2295.4.5 Stabilizationof the semi-discrete scheme 2315.4.6 Stabilizationof mu ltistage sc hemes 233

5.5 Stabilizedspace-time formulations 2415.6 Solved exerc ises 24 3



CONTENTS IX

5.6.1 Convection-diffusion of a Ga ussian hill 2435.6.2 Tra nsient rotating pulse 2465.6.3 Stea dy rotating pulse problem 250

5.6.4 Nonlinear propag a tionof a step 2505.6.5 Bu rgers' equa tionin 1D 2515.6.6 Tw o-dimensiona l Burg ers' equa tion 252

Appendix Lea st-squa res in transient/relaxa tion problems 254

Viscous incompressible flows 2656.1 Introduc tion 26 5

6.2 Ba sic concepts 2676.2.1 Strain rate a nd spin tensors 2676.2.2 The stress tensor in a Ne wtonia n fluid 2686.2.3 The Navier Stokes equations 269

6.3 M a in issues in incom pressible flow problems 27 26.4 Trial solutionsa nd weighting functions 2736.5 Stationa ry S tokes problem 275

6.5.1 Formulationin terms of Ca uchy stress 2756.5.2 Formulationin terms of velocitya nd pressure 2786.5.3 Ga lerkin formula tion 27 96.5.4 M a trix problem 2816.5.5 Solva bility co ndition a nd solution proc edure 28 36.5.6 The LBBcompatibility condition 2846.5.7 Som e popularvelocity pressure couples 2856.5.8 Stabilizationof the Stokes problem 2876.5.9 Pena lty method 288

6.6 SteadyNavier Stokesproblem 2936.6.1 Weakform an d Ga lerkin formulation 2936.6.2 M a trix problem 293

6.7 Unsteady Navier Stokesequations 2946.7.1 Weakformulation a nd spa tia l discretiza tion 2956.7.2 Sta bilized finite element formula tion 29 66.7.3 Time discretizationbyfractional-step methods 297

6.8 Applicationsa nd solved exercices 3066.8.1 Stokes flow with a na lytica l solution 3066.8.2 Cavityflow problem 3076.8.3 Planejet simulation 3136.8.4 Na tural convec tionin a square ca vity 317



Preface

This book is based on lectures regularly taught in the fifth year of engineering diplomasor basic graduate courses at the University of Liege and at the Universitat Politecnicade Catalunya. The goal of the book is not to provide an exhaustive account of finiteelements in fluids, which is an extremely active area of research. The objective isto present the fundamentals of stabilized finite element methods for the analysis ofsteady and time-dependent convection diffusion and fluid dynamics problems with anengineering rather than a mathematical bias. Organized into six chapters, it combines

theoretical aspects and practical applications and covers some of the recent researchin several areas of computational fluid dynamics.

The authors wish to thank their colleagues and friends at the Joint Research Cen-tre of the European Commission, Ispra, Alenia, Torino, ENEL Research Hydraulicsand Structures, Milano, Politecnico di Milano, Universite de Liege and Universi-tat Politecnica de Catalunya, who contributed to the development of several of theconcepts presented in this book and provided worked examples. We are especiallygrateful to Folco Casadei, Luigi Quartapelle and Vittorio Selmin who reviewed parts

of this book, provided numerous suggestions and corrections, and contributed withseveral illustrative examples. We are particularly in debt to our colleagues at theLaboratori de Calcul Numeric: Sonia Fernandez-Mendez, Esther Sala-Lardies, An-tonio Rodriguez-Ferran, Pedro Diez, Josep Sarrate, Agusti Perez-Foguet and XeviRoca. Their reviews, suggestions and contributions have made this book a reality.The second author must also thank Gilles Pijaudier-Cabot and the Ecole Centrale deNantes for giving him sanctuary during the last stages of this project.

Above all, however, our thanks go to our wives, Marie-Paule and Sara, for their

continuous and unfailing support and patience for many years and particularly duringthe writing of this book.

J . DONEA A N D A . H U E RTA

XI

finite elemente method for flow problem

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