Mathematical Modeling of Unsteady Inviscid Flows PDF
Preview Mathematical Modeling of Unsteady Inviscid Flows
Interdisciplinary Applied Mathematics 50 Jeff D. Eldredge Mathematical Modeling of Unsteady Inviscid Flows Interdisciplinary Applied Mathematics Volume 50 Editors AnthonyBloch,UniversityofMichigan,AnnArbor,MI,USA CharlesL.Epstein,UniversityofPennsylvania,Philadelphia,PA,USA AlainGoriely,UniversityofOxford,Oxford,UK L.Greengard,NewYorkUniversity,NewYork,USA Advisors L.Glass,McGillUniversity,Montreal,QC,Canada R.Kohn,NewYorkUniversity,NewYork,NY,USA P.S.Krishnaprasad,UniversityofMaryland,CollegePark,MD,USA AndrewFowler,UniversityofOxford,Oxford,UK C.Peskin,NewYorkUniversity,NewYork,NY,USA S.S.Sastry,UniversityofCaliforniaBerkeley,CA,USA J.Sneyd,UniversityofAuckland,Auckland,NewZealand RickDurrett,DukeUniversity,Durham,NC,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/1390 Jeff D. Eldredge Mathematical Modeling of Unsteady Inviscid Flows 123 JeffD.Eldredge MechanicalandAerospaceEngineering UniversityofCalifornia,LosAngeles LosAngeles,CA,USA ISSN0939-6047 ISSN2196-9973 (electronic) InterdisciplinaryAppliedMathematics ISBN978-3-030-18318-9 ISBN978-3-030-18319-6 (eBook) https://doi.org/10.1007/978-3-030-18319-6 MathematicsSubjectClassification:76G25,76B99 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland ForMina,Alex,andDaphne. Preface Inthismoderneraofreadyaccesstocomputationalresources,bothforresearchand forinstruction,potentialflowtheoryhasbecomesomewhatofanendangeredspecies. Though the subject persists in most undergraduate- and graduate-level aerospace and mechanical engineering curricula, a typical course covers only the most basic aspects and generally culminates in steady flows past streamlined planar bodies, settingthefoundationforclassical(steady)aerodynamicsinstruction.Theassociated courseworktendstoconsistofcontrivedproblemsadaptedfrommagicallyattached flowspastcircularcylindersthateventheinstructorisatalosstoexplaintherelevance of.Itisnosurprise,then,thatastudentoftenleaves such aclasswondering about the modern relevance of potential flow. Indeed, even experienced researchers tend todismissitasahistoricallegacy.However,itispreciselybecauseofitstruncated treatment that even well-respected researchers and practitioners of fluid dynamics haveadoptedaratherdimviewofthesubjectofinviscidflows. But the true utility of inviscid fluid dynamics—both to illuminate fundamental concepts and toserveas thefoundation forpractical modeling—only emerges just pastthispointoftruncation,whenthesubjectisextendedtounsteadyflows.Inthat context,theflowpastacircularcylinderfinallydisplaysoneofitsvirtues:outfitted withconformaltransformations,wecandevelopthisbenignflowintotheflowpast any closed planar body. By adding a few simple tools in the circle plane and the transportrulesforvortexelements,wecanconstructanendlessvarietyofphysically plausible unsteady models. Even the potential flow generated by the cylinder has animportantphysicalrole,forifthecylinderisacceleratedfromrestinaquiescent fluid, the flow that it develops at its first instant is entirely potential. Furthermore, in unsteady inviscid flows, we have a powerful milieu from which to understand and distinguish a variety of fundamental concepts in flow physics: flows induced by moving bodies, the influence of added mass, vortex–vortex and vortex–body dynamics,andtheassociatedforces,moments,andenergetics. It must be emphasized that the subject of unsteady inviscid flows is as rele- vant as ever, providing a foundation from which to analyze the dynamics of agile aerial and aquatic vehicles, the performance of wind turbines and other devices thatharvestenergyfromfluidflows,andthelocomotionoflargecreaturesinfluid vii viii Preface media.Furthermore,thetoolswedevelopinthissubjectenableuniqueinsightinto interactinggroupsofsuchsystems—schools,swarms,andwindfarms. Themannerinwhichinviscidflowmodelsareusedhasevolvedinrecentyears. Inthe1970s–1990s,numericalcodesbasedoninviscidmodelswereincommonuse in the design of aircraft, based on complex panelizations of the fuselage and wing system. This is still the case, but there is now a heavier reliance on higher-fidelity analysisfromcomputationalfluiddynamics(CFD).Incontrast,inviscidflowmodels are increasingly used in a data-driven capacity, wherein the flow model serves as a dynamical template: the model describes the underlying dynamical evolution of the flow, but unspecified parameters of the model are obtained from experiments orCFDbymeansofdataassimilationormachinelearning.Theresultingreduced- ordermodelisthereforeadistillationoftherealflowphysicsandmightserveasthe backboneofaflowestimationand/orcontrolstrategy,forexample. Therefore,thepurposeofthisbookistoextendtheinstructionofpotentialflow theorytothisrichermoderncontext.Myobjectiveinwritingthebookhasbeento make this extended treatment accessible to as broad an audience as possible and tohighlight thepowerfulsetoftoolsthatitprovides forexploring fluiddynamics. The book consolidates various mathematical tools and theoretical frameworks for describing the physics of unsteady inviscid flows, particularly in the context of lifting and propulsive surfaces. Though many of these tools are classical, others are relatively recent. It is important to add that this book focuses exclusively on bodiesimmersedinunboundedhomogeneousfluidmedia,generating(orsubjected to) incompressible flow; it does not contain detailed treatments of internal flows, compressibleflows,orflowswithfreesurfacesorfluid-fluidinterfaces,asidefrom theirbasicgoverningequations. In many university courses, three-dimensional inviscid flows, even the steady ones, are treated only briefly (e.g., to establish the basis for lifting-line theory) or skipped altogether due to time constraints. However, it is illuminating to see the relationships between two-dimensional and three-dimensional concepts, so wher- ever possible, I simultaneously present concepts in both contexts. With a slightly liberalizednotation,itisquiteeasytowriteequationsthatholdinbothdimensions. Forflowsintheplane,Ihavedevotedaparticularattentiontounifyingthevector andcomplexapproaches.Mostbookschooseoneortheotherandtherebylosethe opportunitytohighlighttheinherentconnectionsbetweentheseapproaches.Many students who feel comfortable with a vector-based presentation of potential flow tendtobeabitapprehensiveofthecomplexanalysisoftheproblem.Thus,Ihave takenspecialcarethroughoutthebooktodemonstratehowtomovefluidlyfromone perspectivetotheother.Whereverpossible,formulasarepresentedinbothforms. ExpectedLevelofPreparation Inaperfectworld,aninstructorwouldexpandthepotentialflowmodelingsection of a class to accommodate this extended material. However, such an expansion is practicallyimpossible,sinceitwouldrequiretrimmingsomeotheressentialsubject tofitwithinafixedacademicterm.Thus,myprimarypurposehasbeentocreatea Preface ix resourceforself-directedstudy,onewhichprovidesanopportunityforstudentstogo attheirownpacebeyondthetypicalgraduate-levelclass(oradvancedundergraduate class)onpotentialflow. While writing this book, I have imagined the reader to be a mature graduate student who has already completed a graduate-level course on inviscid flows. In other words, the reader is already familiar with the foundations of potential flow, buthasprobablyonlyseenitappliedtosimplesteady(two-dimensional)flowspast bodies and perhaps a few elementary unsteady motions (e.g., of point vortices). A reader with an aerodynamics background will likely be accustomed to a bit more, suchasthinairfoiltheoryandliftinglines,butagain,thisislikelytobeinasteadyflow setting.Itispossibleforareadertoproceeddirectlytothisbookfromanintermediate undergraduatefluidmechanicsclass.However,mostofthefoundationalconceptsare onlysummarizedinChap.3,withthepresumptionthatthisonlyservesasareview. Thereadershouldhavethemathematicalpreparationtypicalofanundergraduate engineeringcurriculum.Thoughthemathematicalanalysisisextensiveinthisbook, Ihavetriedtoensurethatitisaccessibletoareaderwhoisonlymodestlyinclined towardmathematics.Ihavealsostriventomakeitasself-containedaspossible,so that the reader can avoid looking up or learning about mathematical topics from othersources.Therefore,theAppendixofthebookcontainsanextensivesummary andbriefdiscussionofallofthemathematicalresultsessentialforthisbook.These includebasictheoremsfromvectorcalculusandtheirconsequences,definitions,and conceptsfromcomplexanalysisandsomeusefulintegrals.Theyalsoincludesome nonstandardfare,suchastimedifferentiationofvolume,surface,andlineintegrals. SomeNotesonNotation Because of the extensive collection of symbols used in this book, I have taken particularcaretoorganizethenotationalongafewrules.Thevariablesofrealand complex scalar quantities are denoted with italics, such as x and z. Vector- and tensor-valuedquantitiesarerepresentedwithbolditalics,likex,v,orM.Thearrays oftheircomponentsinsomecoordinatesystemare,incontrast,denotedwithbold sansserif:xandv,forexample. OtherSources There is a variety of references from which I have drawn the material presented in this book. For basic inviscid flow theory, I recommend the book of Karamcheti [38].Foramuchdeepersourceofmaterialonthesubject,particularlyinitsuseof complexanalysis,IamquitefondofthebooksbyMilne-Thomson[53,52].Lamb’s classicbook[43]isanabsolutetreasureofusefulresults,manyofwhicharesadly neglected by modern textbooks. For a thoughtful discussion of the foundations of x Preface steady aerodynamics, particularly on the suitability of the Kutta condition (or, by hisattribution,theJoukowskihypothesis),IsuggestGlauert’sbook[26].Anderson’s text [4] has become a modern classic and provides an accessible introduction to aerodynamics.Aeroelasticitytheoryhasbeen,historically,themostsignificantmo- tivationforstudyingunsteadyaerodynamics,andIrecommendtheclassicbookby Bisplinghoff, Ashley, and Halfman for this subject and its thorough treatment [8]. Onthesubjectofvortexdynamics,bothinviscidandviscous,Ihighlyrecommend thecompacttreatmentofSaffman’sbook[63].ThebookbyKatzandPlotkin[39] providesanexcellentresourceontheuseofnumericalmethodsforpotentialflows andshouldberegardedasapractitioner’sguidetocomplementthemoremathemat- icaltreatmentofthepresentbook.Vortexparticlemethodshavealsomaturedinto an attractive tool for high-fidelity numerical simulation of viscous external flows, andthebookbyCottetandKoumoutsakos[14]servesasaverygoodreferencefor theirimplementation. LosAngeles,CA,USA JeffD.Eldredge April2019
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