Mathematical and physical theory of turbulence PDF
Preview Mathematical and physical theory of turbulence
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 Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business © 2006 by Taylor & Francis Group, LLC DK3004_Discl.fm Page 1 Tuesday, March 28, 2006 11:08 AM Published in 2006 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2323-6 (Hardcover) International Standard Book Number-13: 978-0-8247-2323-1 (Hardcover) Library of Congress Card Number 2006040268 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. 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 storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.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 provides licenses and registration for a variety of users. For organizations that have been granted a photocopy 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. 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 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group and the CRC Press Web site at is the Academic Division of Informa plc. http://www.crcpress.com © 2006 by Taylor & Francis Group, LLC P1:Binaya May9,2006 17:10 3004 DK3004˙C000 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)* vii © 2006 by Taylor & Francis Group, LLC P1:Binaya May9,2006 17:10 3004 DK3004˙C000 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 viii © 2006 by Taylor & Francis Group, LLC P1:Binaya May9,2006 17:10 3004 DK3004˙C000 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). ix © 2006 by Taylor & Francis Group, LLC P1:Binaya May12,2006 12:41 3004 DK3004˙C000 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 xi © 2006 by Taylor & Francis Group, LLC P1:Binaya May9,2006 17:10 3004 DK3004˙C000 MikhailM.Shvartsman SivaThangum DepartmentofMathematics DepartmentofMechanicalEngineering UniversityofSt.Thomas StevensInstituteofTechnology St.Paul,Minnesota Hoboken,NewJersey StephenL.Woodruff CenterforAdvancedPowerSystems FloridaStateUniversity Tallahassee,Florida xii © 2006 by Taylor & Francis Group, LLC P1:Binaya May9,2006 17:10 3004 DK3004˙C000 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 xiii © 2006 by Taylor & Francis Group, LLC P1:Binaya May9,2006 17:10 3004 DK3004˙C000 Chapter12 TheRoleofAngularMomentumInvariantsinHomogeneousTurbulence...................................165 P.A.Davidson Chapter13 OntheNewConceptofTurbulenceModelinginFullyDevelopedTurbulentChannelFlowand BoundaryLayer.............................................................................................................................183 EkachaiJuntasaroandVarangratJuntasaro xiv © 2006 by Taylor & Francis Group, LLC P1:Binaya April24,2006 15:2 3004 DK3004˙C001 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 © 2006 by Taylor & Francis Group, LLC
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