Think Before You Compute: A Prelude to Computational Fluid Dynamics

243 Pages·2020·3.068 MB·English

by  E. J. Hinch

ThinkBeforeYouCompute
Everyfluiddynamicistwillatsomepointneedtousecomputation.Thinkingaboutthe
physics,constraintsandtherequirementsearlyonwillberewardedwithbenefitsin
time,effort,accuracyandexpense.Howthesebenefitscanberealisedisillustratedin
thisguideforwould-beresearchersandbeginninggraduatestudentstosomeofthe
standardmethodsandcommonpitfallsofcomputationalfluidmechanics.Basedona
lecturecoursethattheauthorhasdevelopedover20years,thetextissplitintothree
parts.Thequickintroductionenablesstudentstosolvenumericallyabasicnonlinear
problembyasimplemethodinjustthreehours.Thefollow-uppartexpandsonallthe
keyessentials,includingdiscretisation(finitedifferences,finiteelementsandspectral
methods),time-steppingandlinearalgebra.Thefinalpartisaselectionofoptional
advancedtopics,includinghyperbolicequations,therepresentationofsurfaces,the
boundaryintegralmethod,themultigridmethod,domaindecomposition,thefast
multipolemethod,particlemethodsandwavelets.
e. j. hinchhasbeenateacherandresearcherinfluidmechanicsandapplied
mathematicsattheUniversityofCambridgeforover45years.Heistheauthorof
PerturbationMethods(CambridgeUniversityPress,1991)andhasbeenawardedthe
FluidDynamicsprizesoftheEuropeanMechanicsSocietyandtheAmericanPhysical
SocietyDivisionofFluidDynamics.
CambridgeTextsinAppliedMathematics
AlltitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversity
Press.Foracompleteserieslisting,visitwww.cambridge.org/mathematics.
Flow,DeformationandFracture
G.I.BARENBLATT
GeometricandTopologicalInference
JEAN-DANIELBOISSONNAT,FRE´DE´RICCHAZAL&MARIETTEYVINEC
IntroductiontoMagnetohydrodynamics(2ndEdition)
P.A.DAVIDSON
AnIntroductiontoStochasticDynamics
JINQIAODUAN
Singularities:Formation,StructureandPropagation
J.EGGERS&M.A.FONTELOS
StochasticModellingofReaction-DiffusionProcesses
RADEKERBAN&S.JONATHANCHAPMAN
Microhydrodynamics,BrownianMotionandComplexFluids
MICHAELD.GRAHAM
DiscreteSystemsandIntegrability
J.HIETARINTA,N.JOSHI&F.W.NIJHOFF
AnIntroductiontoPolynomialandSemi-AlgebraicOptimization
JEANBERNARDLASSERRE
AnIntroductiontoComputationalStochasticPDEs
GABRIELJ.LORD,CATHERINEE.POWELL&TONYSHARDLOW
Self-ExcitingFluidDynamos
KEITHMOFFATT&EMMANUELDORMY
NumericalLinearAlgebra
HOLGERWENDLAND
Think Before You Compute
A Prelude to Computational Fluid Dynamics
E. J. HINCH
UniversityofCambridge
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Informationonthistitle:www.cambridge.org/9781108479547
DOI:10.1017/9781108855297
©E.J.Hinch2020
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
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Firstpublished2020
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Contents
Preface pagexi
PARTI AFIRSTPROBLEM 1
1 Thedrivencavity 3
1.1 Theproblem 3
1.2 Knowyourphysics 3
1.3 KnowyourPDEs 5
1.4 Specialphysicsofthecorner 6
1.5 Nondimensionalisation 7
1.6 Steadyvstransientcalculations 7
1.7 Pressure! 8
2 Streamfunction-vorticityformulation 9
2.1 Formulation 9
2.2 Finitedifferences(simple) 10
2.3 Poissonproblem 13
2.4 Testthecode 16
2.5 Codequality 19
2.6 Simplegraphs 20
2.7 Vorticityevolution 21
2.8 Time-stepinstability 22
2.9 Accuracyconsistency 24
2.10 Results 27
3 Primitivevariableformulation 35
3.1 Formulation 35
3.2 Pressureequation 35
3.3 Algorithm1withpressureequation 36
3.4 Incompressibilityasaconstraint:splittime-step 38
v
vi Contents
3.5 Algorithm2byprojection–spuriouspressuremodes 40
3.6 Algorithm3withastaggeredgrid 42
3.7 Resultsfromalgorithm3 46
PARTII GENERALITIES 51
4 Finitedifferences 53
4.1 Higherorders 53
4.1.1 Centraldifferencing 53
4.1.2 One-sideddifferencing 54
4.1.3 Nonequispacedpoints 55
4.2 Compactfourth-orderPoissonsolver 56
4.2.1 One-dimensionalversion 56
4.2.2 Twodimensions 56
4.3 Upwinding 58
4.4 Othergrids 59
4.5 Conservativeschemes 60
5 Finiteelements 64
5.1 Thetwoideas 64
5.2 Representationsinonedimension 65
5.2.1 Constantelements 65
5.2.2 Linearelements 65
5.2.3 Quadraticelements 65
5.2.4 Cubicelements 66
5.2.5 Basisfunctions 67
5.3 Representationsintwodimensions 67
5.3.1 Constantelements 68
5.3.2 Linearelements 68
5.3.3 Quadraticelements 69
5.3.4 Cubicelements 70
5.3.5 Basisfunctions 71
5.3.6 Rectangles 71
5.4 VariationalstatementofthePoissonproblem 72
5.5 Detailsinonedimension 73
5.6 Detailsintwodimensions 74
5.7 Galerkinformulation 78
5.8 Diffusionequation 80
5.8.1 Weakformulation 80
5.8.2 Inonedimension 80
5.8.3 Intwodimensions 81
Contents vii
5.9 Navier–Stokesequation 81
5.9.1 Weakformulation 81
5.9.2 Timeintegration 82
5.9.3 Pressureproblem–locking 82
5.9.4 Pressureproblem–spuriousmodes 84
6 Spectralmethods 86
6.1 Thetwoideas 86
6.2 Globalvslocal 87
6.3 Choiceofspectralbasisfunctions 89
6.4 Chebyshevpolynomials 90
6.5 Ratesofconvergence 90
6.6 Gibbsphenomenon 91
6.7 DiscreteFourierTransform 93
6.8 Aliasing 94
6.9 FastFourierTransform(FFT) 96
6.10 Differentialmatrix 97
6.11 Navier–Stokes 98
6.12 Bridgingthegap 99
7 Timeintegration 100
7.1 Stability 100
7.2 ForwardEuler 102
7.3 BackwardEuler 103
7.4 MidpointEuler 104
7.5 Crank–Nicolson 104
7.6 Leapfrog 105
7.7 Runge–Kutta 105
7.8 Multistepmethods 106
7.9 Symplecticintegrators 107
7.10 Navier–Stokes 108
8 Linearalgebra 110
8.1 Lapack 111
8.2 Gaussianelimination 112
8.2.1 Pivoting 113
8.2.2 LUdecomposition 114
8.2.3 Errors 115
8.3 QRdecomposition 115
8.3.1 QRbyGram–Schmidt 116
8.3.2 QRbyGivensrotations 117
8.3.3 QRbyHouseholderreflections 118
viii Contents
8.4 Sparsematrices 119
8.5 Conjugategradients 119
8.6 Eigenproblems 121
8.7 Poweriteration 122
8.8 Jacobi 122
8.9 Mainmethod 122
PARTIII SPECIALTOPICS 125
9 SoftwarepackagesandFreeFem++ 127
9.1 Poissonproblem 128
9.2 Drivencavity 131
10 Hyperbolicequations 136
10.1 Simplest,butunstable 137
10.2 Lax–Friedricks,toostable 139
10.3 Upwinding 142
10.4 Crank–Nicolson 142
10.5 Lax–Wendroff 144
10.6 AngledDerivative 145
10.7 Propagationofdiscontinuities 148
10.8 Fluxlimiters 149
10.9 Nonlinearadvection 151
10.10 Godunovmethod 153
11 Representationofsurfaces 156
11.1 Curvesintwodimensions 157
11.1.1 Splines 157
11.2 Surfacesinthreedimensions 158
11.2.1 Redistributingpoints 158
11.2.2 Curvature 159
11.3 VolumeofFluid(VoF)method 162
11.4 Diffuseinterfacemethod 163
11.5 Levelsets 165
11.5.1 FastMarchingMethod 165
12 Boundaryintegralmethod 167
12.1 IntegralequationforLaplaceequation 167
12.1.1 Greensfunctions 168
12.1.2 Eigensolutions 169
12.1.3 Singularintegrals 169
12.2 Discretisation 169
Contents ix
12.2.1 Evaluationofthematrixelements 170
12.2.2 Avoidingtheeigensolution 171
12.2.3 Tests 171
12.2.4 Costs 172
12.3 Free-surfacepotentialflows 173
12.4 Stokesflows 173
13 FastPoissonsolvers 175
13.1 Multigridmethod 175
13.1.1 AV-cycle 176
13.1.2 Accuracyandcosts 177
13.2 FastFourierTransforms 178
13.3 Domaindecomposition 180
13.3.1 Costs 182
14 FastMultipoleMethod 183
14.1 Trees,rootsandleaves 184
14.2 Barnes–Hutalgorithm 184
14.3 FastMultipolealgorithm 186
14.3.1 Upwardpass 186
14.3.2 Downwardpass 187
14.3.3 Errors 187
14.3.4 Costs 188
15 Nonlinearconsiderations 190
15.1 Findingsteadystates 190
15.1.1 FindingtheJacobian 191
15.1.2 ExampleofthelimitcycleoftheVanderPol
oscillator 192
15.2 Parametercontinuation 193
15.3 Searchingforsingularitiesofphysicalproblems 193
15.3.1 Useofcomputeralgebra 195
16 Particlemethods 197
16.1 Moleculardynamics 197
16.2 LatticeGas 199
16.3 LatticeBoltzmann 201
16.4 Dissipativeparticledynamics 204
16.5 Stokesiandynamics 205
16.5.1 Hydrodynamicinteractions 205
16.5.2 Brownianmotion 207
16.6 ForceCouplingMethod 209