Mathematical and physical theory of turbulence

197 Pages·2006·3.147 MB·English

by  John Cannon, Bhimsen Shivamoggi

Mathematical and
Physical Theory of
Turbulence
Edited by
John Cannon
University of Central Florida
Orlando, U.S.A.
Bhimsen Shivamoggi
University of Central Florida
Orlando, U.S.A.
Boca Raton London New York
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Library of Congress Cataloging-in-Publication Data
Mathematical and physical theory of turbulence / edited by John Cannon, Bhimsen Shivamoggi.
p. cm. -- (Lecture notes in pure and applied mathematics ; 250)
Includes bibliographical references.
ISBN 0-8247-2323-6 (alk. paper)
1. Turbulence. I. Cannon, John. II. Shivamoggi, Bhimsen K. III. Series.
QA913.M38 2006
532’.0527--dc22 2006040268
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Preface
Turbulencearisesinpracticallyallflowsituationsthatoccursnaturallyorinmoderntechnological
systems.Theturbulenceproblemposesmanyformidableintellectualchallengesandoccupiesacen-
tral place in modern nonlinear mathematics and statistical physics. As the great physicist Richard
Feynmanmentioned,itisthelastgreatunsolvedproblemofclassicalphysics.TheAmericanMath-
ematical Society has listed the turbulence problem as one of the four top unsolved problems of
mathematics.ItisalsooneofthesevenproblemsforwhichtheClayInstitutehasrecentlyannounced
a$1millionprize.
The powerful notions of scaling and universality, which matured when renormalization group
theorywasappliedtocriticalphenomena,hadbeenmanifestedinturbulencepreviously.Therecent
dynamical system approach has provided several important insights into the turbulence problem.
However,deepproblemsstillremaintochallengeconventionalmethodologiesandconcepts.There
is considerable need and opportunity to advance and apply new physical concepts as well as new
mathematical modeling and analysis techniques. There is also an ongoing need to bridge the gap
betweenthegrandtheoriesofidealizedturbulenceandtheharshrealitiesofpracticalapplications.
Theturbulenceproblemcontinuestocommandtheattentionofphysicists,appliedmathematicians,
andengineers.
Several turbulence research groups in Florida collaborated to hold an international turbulence
workshop at the University of Central Florida, May 19–23, 2003. The sponsors of this workshop
were: University of Central Florida (Department of Mathematics, College of Arts and Sciences,
andOfficeofResearch),FloridaStateUniversity,FloridaA&MUniversity,UniversityofFlorida,
Embry Riddle Aeronautical University, and area industrialist Dr. Nelson Ying. The idea was to
bringtogetherexpertsfromphysics,appliedmathematics,andengineeringworkingonthiscommon
problemandpromotetheinfluxofexpertiseintothesubjectfromallthesegroupstothebenefitof
allinunderstandingcomplexissuesofthisproblem.
Thisworkshopaimedatadiscussionofrecentprogressandsomemajorunresolvedbasicissues
inthree-andtwo-dimensionalturbulenceandscalarcompressibleturbulence:
• Three-dimensional turbulence: theory, experiments, computational and mathematical
aspectsofNavier–Stokesturbulence
• Two-dimensionalturbulence:geophysicalflowsandlaboratoryexperiments
• Scalarturbulence:theory,modeling,andlaboratoryexperiments
• Compressible/magnetohydrodynamicseffects
Therewasenormousinterestinthisworkshopworldwide.Thefollowingleadingexpertswere
invitedspeakers(*overviewspeakers):
EberhardBodenschatz(Cornell) JohnKrommes(Princeton)
GeorgeCarnevale(ScrippsInstitution)* JacquesLewalle(Syracuse)
PeterConstantin(UniversityofChicago)* TasosLyrintzis(Purdue)
GregoryFalkovich(WeizmannInstitute)* DavidMontgomery(Dartmouth)
ThomasGatski(NASALangley) SutanuSarkar
SharathGirimaji(TexasA&M) (UniversityofCalifornia,SanDiego)
MarvinGoldstein(NASAGlenn) EranSharon(UniversityofTexas,Austin)
JerryGollub(UniversityofPennsylvania)* SivaThangam(StevensInstitute)
JackHerring(NCAR) EdrissTiti(UniversityofCalifornia,Irvine)
YukioKaneda(Nagoya) ZelmanWarhaft(Cornell)
ShigeoKida(Nagoya) VictorYakhot(BostonUniversity)*
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ProfessorPeterHilton,DistinguishedProfessorofMathematics,UniversityofCentralFlorida,was
thebanquetspeaker.
Theworkshopprogramcontainedintroductoryoverviews,specialisttalks,andcontributedtalks,
as well as panel discussions (one every other day). The workshop was a highly stimulating and
enjoyable event and a great success. The purpose of these workshop proceedings is to distribute
theseoverviewsandcontributedtalksforthebenefitoftheturbulenceresearchcommunityatlarge.
The invited talks included in these proceedings are as follows: The invited talk by Gregory
Falkovich provides an overview of the Lagrangian description of turbulence in general and the
scalardiffusionprobleminparticular.DavidMontgomeryprovidesanaccountofrecentresultsin
decaying two-dimensional turbulence via the entropy maximization approach. The discussion by
JackHerringdescribesthedecaymechanisminvolvingphasemixingofgravitywavesinstratified
turbulence.JacquesLewalledescribesapplicationofwaveletscalingtoinvestigatetheregularityof
Navier–Stokesequations.TheinvitedtalkbyJohnKrommesprovidesanoverviewofsomeaspects
ofstatisticaltheoriesofturbulenceinstronglymagnetizedplasmas.Franc¸oiseBataillediscussesa
general framework to develop eddy viscosity models for turbulent flows which do not exhibit the
Kolmogorovenergyspectrum.TheinvitedtalkbySivaThangamprovidesanoverviewofthedevel-
opmentofcontinuousturbulencemodelsthataresuitableforlarge-eddysimulationandReynolds
averagedNavier–Stokesformulation.ThebanquettalkbyPeterHiltonprovidesfascinatingpersonal
reminiscences and anecdotes about three great mathematicians of the twentieth century — Alan
Turing,HenryWhitehead,andJean-PierreSerre.
Thecontributedtalksincludedintheseproceedingsareasfollows:ThecontributedtalkbyJacques
LewallediscussestheuseofwaveletstodescribeseveralaspectsofNavier–Stokesturbulencewhich
arenotwellhandledbytraditionalapproaches.MikhailShvartsmandiscussesaconnectionbetween
thegoverningequations,theconstitutivetheory,andtheclosureproblemfortheatmosphericbound-
arylayer.ThecontributedtalkbyPeterDavidsondiscussesthesignificanceofthelawofconservation
ofangularmomentumforfreelyevolving,homogeneousturbulence.EkachaiJuntasarodiscussesa
newconceptofturbulencemodelinginturbulentchannelflowandturbulentboundary-layerflow.
JohnR.Cannon
BhimsenK.Shivamoggi
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The Editors
JohnCannoncompletedhisB.A.degreeinmathematicsatLamarUniversityin1958.Heattended
graduate school at Rice University to study mathematics and completed his M.S. degree in 1960
andhisPh.D.degreein1962.Hisresearchexpertiseisinpartialdifferentialequationsandnumer-
ical analysis with applications to heat flow, fluid dynamics, chemical reactions, change of phase,
flow in porous media, and inverse problems, as well as applications to biology and medicine. He
hasheldfacultyandresearchpositionsatBrookhavenNationalLaboratory,Universita` diGenova,
PurdueUniversity,UniversityofMinnesota,TheUniversityofTexasatAustin,Universita`diFirenze,
ColoradoStateUniversity,theHahnMeitnerInstituteinBerlin,WashingtonStateUniversity,The
OfficeofNavalResearch,LamarUniversity,andtheUniversityofCentralFlorida.Heistheauthor
of2books,150+refereedjournalarticles,26proceedingsarticles,and5technicalreports.
Bhimsen Shivamoggi received his Ph.D. from the University of Colorado, Boulder. Following
postdoctoralresearchappointmentsatPrincetonUniversityandtheAustralianNationalUniversity,
Canberra,hejoinedthemathematicsfacultyattheUniversityofCentralFlorida,Orlandoin1984.He
hasworkedonproblemsofstabilityandturbulenceinfluidsandplasmas.Hehaswrittenseveralbooks
onthesetopicsandhasheldvisitingappointmentsinLosAlamosNationalLaboratory,Universityof
CaliforniaatSantaBarbara,TechnischeHochschuleDarmstadt(Germany),TechnischeUniversiteit
Eindhoven(TheNetherlands),ObservatoiredeNice(France),KyotoUniversity(Japan),International
CenterforTheoreticalPhysicsatTrieste(Italy),andUniversityofNewcastle(Australia).
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Contributors
F.Bataille M.YousuffHussaini
CentredeThermiquedeLyon DepartmentofMathematicsand
VilleurbanneCedex,France ComputationalScienceandEngineering
FloridaStateUniversity
G.Brillant Tallahassee,Florida
CentredeThermiquedeLyon
VilleurbanneCedex,France
R.James
NationalCenterforAtmospheric
GeorgeF.Carnevale
Research
ScrippsInstitutionofOceanography
Boulder,Colorado
UniversityofCaliforniaSanDiego
LaJolla,California
EkachaiJuntasaro
J.Clyne
SchoolofMechanicalEngineering
NationalCenterforAtmospheric
SuranareeUniversityofTechnology
Research
NakhonRatchasima,Thailand
Boulder,Colorado
P.A.Davidson VarangratJuntasaro
DepartmentofEngineering DepartmentofMechanicalEngineering
UniversityofCambridge KasetsartUniversity
Cambridge,UnitedKingdom Bangkok,Thailand
DouglasP.Dokken
Y.Kimura
DepartmentofMathematics
GradeSchoolofMathematics
UniversityofSt.Thomas
NagoyaUniversity
St.Paul,Minnesota
Nagoya,Japan
GregoryFalkovich
WeizmannInstituteofScience
J.A.Krommes
Rehovot,Israel
PlasmaPhysicsLaboratory
PrincetonUniversity
J.R.Herring
Princeton,NewJersey
NationalCenterforAtmospheric
Research
Boulder,Colorado
JacquesLewalle
DepartmentofMechanicalEngineering
PeterHilton SyracuseUniversity
DepartmentofMathematicalSciences Syracuse,NewYork
StateUniversityofNewYork
Binghamton,NewYork
and DavidC.Montgomery
DepartmentofMathematics DepartmentofPhysicsandAstronomy
UniversityofCentralFlorida DartmouthCollege
Orlando,Florida Hanover,NewHampshire
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MikhailM.Shvartsman SivaThangum
DepartmentofMathematics DepartmentofMechanicalEngineering
UniversityofSt.Thomas StevensInstituteofTechnology
St.Paul,Minnesota Hoboken,NewJersey
StephenL.Woodruff
CenterforAdvancedPowerSystems
FloridaStateUniversity
Tallahassee,Florida
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Table of Contents
Chapter1
AMathematicianReflects:BanquetRemarks...................................................................................1
PeterHilton
Chapter2
LagrangianDescriptionofTurbulence..............................................................................................7
GregoryFalkovich
Chapter3
Two-DimensionalTurbulence:AnOverview..................................................................................47
GeorgeF.Carnevale
Chapter4
StatisticalPlasmaPhysicsinaStrongMagneticField:ParadigmsandProblems..........................69
J.A.Krommes
Chapter5
SomeRemarksonDecayingTwo-DimensionalTurbulence...........................................................91
DavidC.Montgomery
Chapter6
StatisticalandDynamicalQuestionsinStratifiedTurbulence.......................................................101
J.R.Herring,Y.Kimura,R.James,J.Clyne,andP.A.Davidson
Chapter7
WaveletScalingandNavier–StokesRegularity.............................................................................115
JacquesLewalle
Chapter8
GeneralizationoftheEddyViscosityModel—ApplicationtoaTemperatureSpectrum............125
F.Bataille,G.Brillant,andM.YousuffHussaini
Chapter9
ContinuousModelsfortheSimulationofTurbulentFlows:AnOverviewandAnalysis.............131
M.YousuffHussaini,SivaThangam,andStephenL.Woodruff
Chapter10
AnalyticalUsesofWaveletsforNavier–StokesTurbulence.........................................................145
JacquesLewalle
Chapter11
Time Averaging, Hierarchy of the Governing Equations, and the Balance of Turbulent Kinetic
Energy............................................................................................................................................155
DouglasP.DokkenandMikhailM.Shvartsman
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Chapter12
TheRoleofAngularMomentumInvariantsinHomogeneousTurbulence...................................165
P.A.Davidson
Chapter13
OntheNewConceptofTurbulenceModelinginFullyDevelopedTurbulentChannelFlowand
BoundaryLayer.............................................................................................................................183
EkachaiJuntasaroandVarangratJuntasaro
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1 A Mathematician Reflects:
Banquet Remarks
Peter Hilton
CONTENTS
1.1 AlanTuring...............................................................................................................................1
1.2 HenryWhitehead......................................................................................................................2
1.3 Jean-PierreSerre.......................................................................................................................3
1.4 Epilogue....................................................................................................................................4
First,letmethankyouverymuchforinvitingmetoparticipateinyourconferenceandforgiving
methisopportunitytosayafewinformalwordstoyoufollowinganexcellentdinner.
My area of research intersects yours precisely in our recognition of the importance of a good
numbersystemandagoodnotation,andouruseofthesamenumbersystem.Indeed,asIlookbackon
myowncareer,myclosestcontactwithturbulence—andthiswas,asinthecaseofthisconference,
amatterofinternationalturbulence—wasduringWorldWarII,whenIwasconscriptedtowork,
fromJanuary1942untiltheendofthewarinEuropeinMay1945atBletchleyPark,breakingthe
highest-grade German ciphers used for diplomatic and military traffic passing among the German
government, the German High Command, and their naval, air force, and army commanders and
U-boat captains. Those were indeed turbulent days, and I will say something about them in my
remarksthisevening.Moregenerally,IwillreflectonthewonderfulmathematiciansIhaveknown
during a career spanning more than 60 years, starting in British Military Intelligence, proceeding
alongamoreconventionalacademicroute,andcontinuingtoday,thoughatagentlerpaceconsistent
withmygrowingmaturity.
1.1 ALAN TURING
OurworkatBletchleyPark,crackingthehigh-gradeGermancodes,waswonderfullyexciting,stim-
ulating us to work tirelessly, over long stretches of time, buoyed by the intellectual challenge and
our awareness of the importance of what we were achieving. However, for me, as a very young
man — I started work at BP (as we called it) at the age of 18 — there was the thrill of work-
ing with some of the greatest British mathematicians of that period, getting to know them well
and enjoying their friendship and collegiality. They were absolutely free of any outward aware-
ness of their intellectual superiority and treated their younger colleagues as equals — which, as
cryptanalystsbutnotasmathematicians,ofcourse—wewere!Theyseemednottobeawarethat,
howeveradeptwemightbeatapplyingmathematicalthinkingandperhapscertainveryspecialized
linguistic skills to the job in hand, we had no mathematical knowledge and experience to match
theirs.
OnelessonIlearnedfrommyexperienceatBPthatIwouldliketosharewithyouisthis:tobe
abletoapplymathematicalreasoningtoproblemsthatarenotintrinsicallymathematical—inother
words,tobeabletoapplymathematics—theessentialprerequisitesareafirst-classmathematics
educationandstronginterestinandincentivetosolvetheproblem.Experienceandfamiliaritywith
some scientific discipline, although desirable, are not essential. It is an interesting and, I believe,
importantfactthat,inthegroupofmathematiciansandyoungwould-bemathematiciansworkingon
1
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