Vorticity and incompressible flow

558 Pages·2002·2.69 MB·English

by  Andrew J. Majda, Andrea L. Bertozzi

VorticityandIncompressibleFlow
This book is a comprehensive introduction to the mathematical theory of vorticity
and incompressible flow ranging from elementary introductory material to current
researchtopics. Althoughthecontentscenteronmathematicaltheory,manypartsof
thebookshowcasetheinteractionsamongrigorousmathematicaltheory,numerical,
asymptotic,andqualitativesimplifiedmodeling,andphysicalphenomena. Thefirst
halfformsanintroductorygraduatecourseonvorticityandincompressibleflow. The
second half comprises a modern applied mathematics graduate course on the weak
solutiontheoryforincompressibleflow.
AndrewJ.MajdaistheSamuelMorseProfessorofArtsandSciencesattheCourant
InstituteofMathematicalSciencesofNewYorkUniversity. Heisamemberofthe
NationalAcademyofSciencesandhasreceivednumeroushonorsandawardsinclud-
ing the National Academy of Science Prize in Applied Mathematics, the John von
NeumannPrizeoftheAmericanMathematicalSocietyandanhonoraryPh.D.degree
fromPurdueUniversity. Majdaiswellknownforbothhistheoreticalcontributionsto
partialdifferentialequationsandhisappliedcontributionstodiverseareasbesidesin-
compressibleflowsuchasscatteringtheory,shockwaves,combustion,vortexmotion
and turbulent diffusion. His current applied research interests are centered around
Atmosphere/Oceanscience.
Andrea L. Bertozzi is Professor of Mathematics and Physics at Duke University.
ShehasreceivedseveralhonorsincludingaSloanResearchFellowship(1995)and
the Presidential Early Career Award for Scientists and Engineers (PECASE). Her
researchaccomplishmentsinadditiontoincompressibleflowincludeboththeoretical
andappliedcontributionstotheunderstandingofthinliquidfilmsandmovingcontact
lines.
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Cambridge Texts in Applied Mathematics
MaximumandMinimumPrinciples
M.J.SEWELL
Solitons
P.G.DRAZINANDR.S.JOHNSON
TheKinematicsofMixing
J.M.OTTINO
IntroductiontoNumericalLinearAlgebraandOptimisation
PHILIPPEG.CIARLET
IntegralEquations
DAVIDPORTERANDDAVIDS.G.STIRLING
PerturbationMethods
E.J.HINCH
TheThermomechanicsofPlasticityandFracture
GERARDA.MAUGIN
BoundaryIntegralandSingularityMethodsforLinearizedViscousFlow
C.POZRIKIDIS
NonlinearWaveProcessesinAcoustics
K.NAUGOLNYKHANDL.OSTROVSKY
NonlinearSystems
P.G.DRAZIN
Stability,InstabilityandChaos
PAULGLENDINNING
AppliedAnalysisoftheNavier–StokesEquations
C.R.DOERINGANDJ.D.GIBBON
ViscousFlow
H.OCKENDONANDJ.R.OCKENDON
Scaling,Self-Similarity,andIntermediateAsymptotics
G.I.BARENBLATT
AFirstCourseintheNumericalAnalysisofDifferentialEquations
ARIEHISERLES
ComplexVariables:IntroductionandApplications
MARKJ.ABLOWITZANDATHANASSIOSS.FOKAS
MathematicalModelsintheAppliedSciences
A.C.FOWLER
ThinkingAboutOrdinaryDifferentialEquations
ROBERTE.O’MALLEY
AModernIntroductiontotheMathematicalTheoryofWaterWaves
R.S.JOHNSON
RarefiedGasDynamics
CARLOCERCIGNANI
SymmetryMethodsforDifferentialEquations
PETERE.HYDON
HighSpeedFlow
C.J.CHAPMAN
WaveMotion
J.BILLINGHAMANDA.C.KING
AnIntroductiontoMagnetohydrodynamics
P.A.DAVIDSON
LinearElasticWaves
JOHNG.HARRIS
IntroductiontoSymmetryAnalysis
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Vorticity and Incompressible Flow
ANDREWJ.MAJDA
NewYorkUniversity
ANDREAL.BERTOZZI
DukeUniversity
PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING)
FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF
CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge CB2 IRP
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia

http://www.cambridge.org

© Cambridge University Press 2002
This edition © Cambridge University Press (Virtual Publishing) 2003

First published in printed format 2002

A catalogue record for the original printed book is available
from the British Library and from the Library of Congress
Original ISBN 0 521 63057 6 hardback
Original ISBN 0 521 63948 4 paperback

ISBN 0 511 01917 3 virtual (netLibrary Edition)
Contents
Preface page xi
1 AnIntroductiontoVortexDynamicsforIncompressible
FluidFlows 1
1.1 TheEulerandtheNavier–StokesEquations 2
1.2 SymmetryGroupsfortheEulerandtheNavier–StokesEquations 3
1.3 ParticleTrajectories 4
1.4 TheVorticity,aDeformationMatrix,andSomeElementary
ExactSolutions 6
1.5 SimpleExactSolutionswithConvection,VortexStretching,
andDiffusion 13
1.6 SomeRemarkablePropertiesoftheVorticityinIdealFluidFlows 20
1.7 ConservedQuantitiesinIdealandViscousFluidFlows 24
1.8 Leray’sFormulationofIncompressibleFlowsand
Hodge’sDecompositionofVectorFields 30
1.9 Appendix 35
NotesforChapter1 41
ReferencesforChapter1 42
2 TheVorticity-StreamFormulationoftheEulerand
theNavier-StokesEquations 43
2.1 TheVorticity-StreamFormulationfor2DFlows 44
2.2 AGeneralMethodforConstructingExactSteadySolutions
tothe2DEulerEquations 46
2.3 SomeSpecial3DFlowswithNontrivialVortexDynamics 54
2.4 TheVorticity-StreamFormulationfor3DFlows 70
2.5 FormulationoftheEulerEquationasanIntegrodifferentialEquation
fortheParticleTrajectories 81
NotesforChapter2 84
ReferencesforChapter2 84
3 EnergyMethodsfortheEulerandtheNavier–StokesEquations 86
3.1 EnergyMethods: ElementaryConcepts 87
vii
viii Contents
3.2 Local-in-TimeExistenceofSolutionsbyMeansofEnergyMethods 96
3.3 AccumulationofVorticityandtheExistenceofSmoothSolutions
GloballyinTime 114
3.4 Viscous-SplittingAlgorithmsfortheNavier–StokesEquation 119
3.5 AppendixforChapter3 129
NotesforChapter3 133
ReferencesforChapter3 134
4 TheParticle-TrajectoryMethodforExistenceandUniqueness
ofSolutionstotheEulerEquation 136
4.1 TheLocal-in-TimeExistenceofInviscidSolutions 138
4.2 LinkbetweenGlobal-in-TimeExistenceofSmoothSolutions
andtheAccumulationofVorticitythroughStretching 146
4.3 GlobalExistenceof3DAxisymmetricFlowswithoutSwirl 152
4.4 HigherRegularity 155
4.5 AppendixesforChapter4 158
NotesforChapter4 166
ReferencesforChapter4 167
5 TheSearchforSingularSolutionstothe3DEulerEquations 168
5.1 TheInterplaybetweenMathematicalTheoryandNumerical
ComputationsintheSearchforSingularSolutions 170
5.2 ASimple1DModelforthe3DVorticityEquation 173
5.3 A2DModelforPotentialSingularityFormationin3DEulerEquations 177
5.4 PotentialSingularitiesin3DAxisymmetricFlowswithSwirl 185
5.5 Dothe3DEulerSolutionsBecomeSingularinFiniteTimes? 187
NotesforChapter5 188
ReferencesforChapter5 188
6 ComputationalVortexMethods 190
6.1 TheRandom-VortexMethodforViscousStrainedShearLayers 192
6.2 2DInviscidVortexMethods 208
6.3 3DInviscid-VortexMethods 211
6.4 ConvergenceofInviscid-VortexMethods 216
6.5 ComputationalPerformanceofthe2DInviscid-VortexMethod
onaSimpleModelProblem 227
6.6 TheRandom-VortexMethodinTwoDimensions 232
6.7 AppendixforChapter6 247
NotesforChapter6 253
ReferencesforChapter6 254
7 SimplifiedAsymptoticEquationsforSlenderVortexFilaments 256
7.1 TheSelf-InductionApproximation,Hasimoto’sTransform,
andtheNonlinearSchro¨dingerEquation 257
Contents ix
7.2 SimplifiedAsymptoticEquationswithSelf-Stretch
foraSingleVortexFilament 262
7.3 InteractingParallelVortexFilaments–PointVorticesinthePlane 278
7.4 AsymptoticEquationsfortheInteractionofNearlyParallel
VortexFilaments 281
7.5 MathematicalandAppliedMathematicalProblemsRegarding
AsymptoticVortexFilaments 300
NotesforChapter7 301
ReferencesforChapter7 301
8 WeakSolutionstothe2DEulerEquationswithInitialVorticity
∞
inL 303
8.1 EllipticalVorticies 304
8.2 Weak L∞SolutionstotheVorticityEquation 309
8.3 VortexPatches 329
8.4 AppendixforChapter8 354
NotesforChapter8 356
ReferencesforChapter8 356
9 IntroductiontoVortexSheets,WeakSolutions,
andApproximate-SolutionSequencesfortheEulerEquation 359
9.1 WeakFormulationoftheEulerEquationinPrimitive-VariableForm 361
9.2 ClassicalVortexSheetsandtheBirkhoff–RottEquation 363
9.3 TheKelvin–HelmholtzInstability 367
9.4 ComputingVortexSheets 370
9.5 TheDevelopmentofOscillationsandConcentrations 375
NotesforChapter9 380
ReferencesforChapter9 380
10 WeakSolutionsandSolutionSequencesinTwoDimensions 383
10.1 Approximate-SolutionSequencesfortheEulerand
theNavier–StokesEquations 385
10.2 ConvergenceResultsfor2DSequenceswith L1and Lp
VorticityControl 396
NotesforChapter10 403
ReferencesforChapter10 403
11 The2DEulerEquation: ConcentrationsandWeakSolutions
withVortex-SheetInitialData 405
11.1 Weak-*andReducedDefectMeasures 409
11.2 ExampleswithConcentration 411
11.3 TheVorticityMaximalFunction: DecayRatesandStrongConvergence 421
11.4 ExistenceofWeakSolutionswithVortex-SheetInitialData
ofDistinguishedSign 432
NotesforChapter11 448
ReferencesforChapter11 448