Essentials of computational fluid dynamics

236 Pages·2016·17.263 MB·English

by  Müller, Jens-Dominik

E S S E N T I A L S O F
COMPUTATIONAL
FLUID DYNAMICS
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
E S S E N T I A L S O F
COMPUTATIONAL
FLUID DYNAMICS
Jens-Dominik Mu¨ller
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
Lotus Formula 1 race car, with streamlines coloured by pressure coefficient.
Simulation performed with STAR-CCM+ of CD-Adapco.
(Image courtesy of Dr. N. Forsythe, Lotus F1).
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2016 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20151013
International Standard Book Number-13: 978-1-4822-2731-4 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use. The authors and
publishers have attempted to trace the copyright holders of all material reproduced in this publication
and apologize to copyright holders if permission to publish in this form has not been obtained. If any
copyright material has not been acknowledged please write and let us know so we may rectify in any
future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information stor-
age or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copy-
right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222
Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro-
vides licenses and registration for a variety of users. For organizations that have been granted a photo-
copy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Foreword xi
1 Introduction 1
1.1 CFD, the virtual wind tunnel . . . . . . . . . . . . . . . . . . 1
1.2 Examples of CFD applications . . . . . . . . . . . . . . . . . 2
1.3 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Organisation of the chapters . . . . . . . . . . . . . . . . . . 13
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Physical and mathematical principles of modern CFD 15
2.1 The physical model . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Continuum assumption . . . . . . . . . . . . . . . . . 15
2.1.2 Lagrangian vs. Eulerian description . . . . . . . . . . 15
2.1.3 Conservation principles . . . . . . . . . . . . . . . . . 16
2.2 The mathematical model: the equations of fluid flow . . . . . 16
2.2.1 Mass conservation in 1-D . . . . . . . . . . . . . . . . 17
2.2.2 Mass conservation in 3-D . . . . . . . . . . . . . . . . 19
2.2.3 Divergence and gradient operators, total derivative . . 21
2.2.4 The total or material derivative . . . . . . . . . . . . . 21
2.2.5 The divergence form of the total derivative . . . . . . 23
2.2.6 Reynolds’ transport theorem . . . . . . . . . . . . . . 23
2.2.7 Transport of a passive scalar . . . . . . . . . . . . . . 25
2.3 The momentum equations . . . . . . . . . . . . . . . . . . . 25
2.3.1 Examples of momentum balance . . . . . . . . . . . . 26
2.3.2 The inviscid momentum equation — the Euler equation 26
2.3.3 The viscous momentum equations — Navier-Stokes. . 29
2.3.4 The incompressible Navier-Stokes equations . . . . . . 31
2.3.5 Energy balance . . . . . . . . . . . . . . . . . . . . . . 32
2.3.6 Summary of properties for the Navier-Stokes equations 33
2.4 Simplified model equations . . . . . . . . . . . . . . . . . . . 34
2.4.1 Linear advection equation . . . . . . . . . . . . . . . . 34
2.4.2 Inviscid Burgers’ equation . . . . . . . . . . . . . . . . 35
2.4.3 Heat equation . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Excercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
v
vi Contents
3 Discretisation of the equations 41
3.1 Discretisation of the linear advection equation . . . . . . . . 42
3.1.1 Finite difference discretisation of linear advection . . . 42
3.1.2 Solving the finite difference approximation. . . . . . . 44
3.1.3 Mesh refinement . . . . . . . . . . . . . . . . . . . . . 46
3.1.4 Finite volume discretisation of the 1-D advection . . . 47
3.1.5 Solving the finite volume approximation . . . . . . . . 51
3.1.6 Finite difference vs. finite volume formulations . . . . 51
3.2 Burgers’ equation: non-linear advection and conservation . . 54
3.3 Heat equation in 1-D . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1 Discretising second derivatives . . . . . . . . . . . . . 55
3.3.2 1-D Heat equation, differential form . . . . . . . . . . 56
3.3.3 Solving the 1-D heat equation . . . . . . . . . . . . . . 56
3.4 Advection equation in 2-D . . . . . . . . . . . . . . . . . . . 57
3.4.1 Discretisation on a structured grid . . . . . . . . . . . 58
3.5 Solving the Navier-Stokes equations . . . . . . . . . . . . . . 62
3.6 The main steps in the finite volume method . . . . . . . . . 62
3.6.1 Discretisation on arbitrary grids . . . . . . . . . . . . 63
3.6.2 Transport through an arbitrary face . . . . . . . . . . 64
3.6.3 The concept of pseudotime-stepping . . . . . . . . . . 65
3.6.4 Time-stepping for compressible flows . . . . . . . . . . 67
3.6.5 Iterative methods for incompressible flows . . . . . . . 68
3.6.6 The SIMPLE scheme. . . . . . . . . . . . . . . . . . . 70
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Analysis of discretisations 73
4.1 Forward, backward and central differences . . . . . . . . . . 73
4.2 Taylor analysis: consistency, first- and second-order accuracy 74
4.2.1 Round-off errors . . . . . . . . . . . . . . . . . . . . . 76
4.2.2 Order of accuracy and mesh refinement . . . . . . . . 77
4.3 Stability, artificial viscosity and second-order accuracy . . . . 78
4.3.1 Artificial viscosity . . . . . . . . . . . . . . . . . . . . 80
4.3.2 Artificial viscosity and finite volume methods . . . . . 81
4.3.3 Stable second-order accurate discretisations for CFD . 84
4.3.4 Monotonicity and second-order accuracy: limiters . . . 86
4.4 Summary of spatial discretisation approaches . . . . . . . . . 89
4.5 Convergence of the time-stepping iterations . . . . . . . . . . 91
4.5.1 Explicit methods . . . . . . . . . . . . . . . . . . . . . 92
4.5.2 Implicit methods . . . . . . . . . . . . . . . . . . . . . 93
4.5.3 Increasing mesh resolution . . . . . . . . . . . . . . . . 95
4.5.4 Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.6 Excercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Contents vii
5 Boundary conditions and flow physics 99
5.1 Selection of boundary conditions . . . . . . . . . . . . . . . . 99
5.1.1 Some simple examples . . . . . . . . . . . . . . . . . . 99
5.1.2 Selecting boundary conditions to satisfy the equations 101
5.2 Characterisation of partial differential equations . . . . . . . 102
5.2.1 Wave-like solutions: hyperbolic equations . . . . . . . 103
5.2.2 Smoothing-type solutions: elliptic equations . . . . . . 105
5.2.3 The borderline case — parabolic equations . . . . . . 105
5.2.4 The domain of dependence, the domain of influence . 106
5.2.5 Example of characterisation: surface waves . . . . . . 107
5.2.6 Compressible and incompressible flows . . . . . . . . . 109
5.2.7 Characterisation of the Navier-Stokes equations . . . . 110
5.3 Choice of boundary conditions . . . . . . . . . . . . . . . . . 111
5.3.1 Boundary conditions for incompressible flow . . . . . . 111
5.3.2 Boundary conditions for hyperbolic equations . . . . . 113
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6 Turbulence modelling 115
6.1 The challenges of turbulent flow for CFD . . . . . . . . . . . 115
6.2 Description of turbulent flow . . . . . . . . . . . . . . . . . . 117
6.3 Self-similar profiles through scaling . . . . . . . . . . . . . . 120
6.3.1 Laminar velocity profiles . . . . . . . . . . . . . . . . . 120
6.3.2 Turbulent velocity profile . . . . . . . . . . . . . . . . 120
6.4 Velocity profiles of turbulent boundary layers . . . . . . . . . 121
6.4.1 Outer scaling: friction velocity . . . . . . . . . . . . . 122
6.4.2 Inner scaling: non-dimensional wall distance y+ . . . 124
6.5 Levels of turbulence modelling . . . . . . . . . . . . . . . . . 127
6.5.1 Direct Numerical Simulation (DNS) . . . . . . . . . . 127
6.5.2 Reynolds-Averaged Navier-Stokes (RANS). . . . . . . 128
6.5.3 Large Eddy (LES) & Detached Eddy Simulation (DES) 130
6.5.4 Summary of approaches to turbulence modelling . . . 132
6.6 Eddy viscosity models . . . . . . . . . . . . . . . . . . . . . . 133
6.6.1 Mixing length model . . . . . . . . . . . . . . . . . . . 133
6.6.2 The Spalart-Allmaras model . . . . . . . . . . . . . . 134
6.6.3 The k ε model . . . . . . . . . . . . . . . . . . . . . . 135
−
6.7 Near-wall mesh requirements . . . . . . . . . . . . . . . . . . 137
6.7.1 Estimating the wall distance of the first point . . . . . 138
6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7 Mesh quality and grid generation 143
7.1 Influence of mesh quality on the accuracy . . . . . . . . . . . 143
7.1.1 Maximum angle condition . . . . . . . . . . . . . . . . 143
7.1.2 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.1.3 Size variation . . . . . . . . . . . . . . . . . . . . . . . 147
7.2 Requirements for the ideal mesh generator . . . . . . . . . . 147
viii Contents
7.3 Structured grids . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.3.1 Algebraic grids using transfinite interpolation . . . . . 149
7.4 Unstructured grids . . . . . . . . . . . . . . . . . . . . . . . . 153
7.4.1 The Advancing Front Method . . . . . . . . . . . . . . 155
7.4.2 Delaunay triangulation. . . . . . . . . . . . . . . . . . 156
7.4.3 Hierarchical grid Methods . . . . . . . . . . . . . . . . 157
7.4.4 Hexahedral unstructured mesh generation . . . . . . . 159
7.4.5 Hybrid mesh generation for viscous flow . . . . . . . . 161
7.5 Mesh adaptation . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.5.1 Mesh movement: r-refinement . . . . . . . . . . . . . . 164
7.5.2 Mesh refinement: h-refinement . . . . . . . . . . . . . 165
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8 Analysis of the results 169
8.1 Types of errors . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.1.1 Incorrect choice of boundary conditions . . . . . . . . 169
8.1.2 Insufficient convergence . . . . . . . . . . . . . . . . . 171
8.1.3 Artificial viscosity . . . . . . . . . . . . . . . . . . . . 173
8.1.4 Modelling errors . . . . . . . . . . . . . . . . . . . . . 174
8.2 Mesh convergence . . . . . . . . . . . . . . . . . . . . . . . . 175
8.2.1 Cost of error reduction . . . . . . . . . . . . . . . . . . 175
8.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
9 Case studies 179
9.1 Aerofoil in 2-D, inviscid flow . . . . . . . . . . . . . . . . . . 179
9.1.1 Case description . . . . . . . . . . . . . . . . . . . . . 179
9.1.2 Flow physics . . . . . . . . . . . . . . . . . . . . . . . 180
9.1.3 Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.1.4 Simulation results for the C-mesh. . . . . . . . . . . . 182
9.1.5 Comparison of C- vs O-mesh . . . . . . . . . . . . . . 189
9.1.6 Analysis of lift and drag values . . . . . . . . . . . . . 192
9.2 Blood vessel bifurcation in 2-D . . . . . . . . . . . . . . . . . 194
9.2.1 Geometry and flow parameters . . . . . . . . . . . . . 194
9.2.2 Flow physics and boundary conditions . . . . . . . . . 196
9.2.3 Velocity and pressure fields . . . . . . . . . . . . . . . 198
9.2.4 Velocity profile in the neck . . . . . . . . . . . . . . . 200
9.2.5 Effect of outlet boundary condition . . . . . . . . . . . 200
9.3 Aerofoil in 2-D, viscous flow . . . . . . . . . . . . . . . . . . 202
9.3.1 Flow physics . . . . . . . . . . . . . . . . . . . . . . . 203
9.3.2 Turbulence modelling . . . . . . . . . . . . . . . . . . 205
9.3.3 Flow results . . . . . . . . . . . . . . . . . . . . . . . . 206
9.3.4 Lift and drag . . . . . . . . . . . . . . . . . . . . . . . 206
Contents ix
10 Appendix 211
10.1 Finite-volume implementation of 2-D advection . . . . . . . . 211
Bibliography 221
Index 223