Table Of Content

Michel O. Deville An Introduction to the Mechanics of Incompressible Fluids An Introduction to the Mechanics of Incompressible Fluids Michel O. Deville An Introduction to the Mechanics of Incompressible Fluids MichelO.Deville SchoolofEngineering SwissFederalInstituteofTechnology ÉcolePolytechniqueFédéraledeLausanne Lausanne,Vaud,Switzerland ISBN 978-3-031-04682-7 ISBN 978-3-031-04683-4 (eBook) https://doi.org/10.1007/978-3-031-04683-4 ©TheEditor(s)(ifapplicable)andTheAuthor(s)2022.Thisbookisanopenaccesspublication. OpenAccessThisbookislicensedunderthetermsoftheCreativeCommonsAttribution4.0International License(http://creativecommons.org/licenses/by/4.0/),whichpermitsuse,sharing,adaptation,distribu- tionandreproductioninanymediumorformat,aslongasyougiveappropriatecredittotheoriginal author(s)andthesource,providealinktotheCreativeCommonslicenseandindicateifchangeswere made. Theimagesorotherthirdpartymaterialinthisbookareincludedinthebook’sCreativeCommonslicense, unlessindicatedotherwiseinacreditlinetothematerial.Ifmaterialisnotincludedinthebook’sCreative Commonslicenseandyourintendeduseisnotpermittedbystatutoryregulationorexceedsthepermitted use,youwillneedtoobtainpermissiondirectlyfromthecopyrightholder. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Notallequationstellthewholestory ToChristinawithloveandpassion Foreword Thebook:AnIntroductiontotheMechanicsofIncompressibleFluidsbyProf.Michel Deville is another feather in the cap of the author, who is extremely prolific in writingbooksonmultipleaspectsofcontinuummechanics(withemphasisonfluid mechanics), high accuracy scientific computing by high order methods and topics ofappliedmathematics.Thepresentbookisafineadditiontothislist.Eventhough thebookistitledasanintroductiontothesubjectofincompressibleflow,theauthor has covered extensively from topics on potential flow theory to direct numerical simulationoffluidflow.Apartfromteachingthesubjecttohisstudentsforalong time,hehasalsoperformedextensiveresearchonmanytopicsinthebook.Hehas broughtagoodbalancewithearlyexpositiononthetopics,whilecoveringthemost recentdevelopmentsinsomeothertopics.Oneofthemostdistinguishingfeatures of the book is that the author has introduced the essentials of the topics, while he has cited enough references to augment the details which the reader can consult. This has kept the book quite concise, while the topics covered are quite wide. I am very happy to see that the book covers many useful topics of classical field like conformal mapping that influenced early developments in aerodynamics; to applicationsinappliedmathematics;tocontemporarytopicsinbio-fluidmechanics. Aspecialfeatureistheintroductiontodirectnumericalsimulationandlargeeddy simulation of incompressible flows. The book is very well organized, written in a lucid style, which the students will find very easy to follow. This will be an ideal bookwrittenwithinformationandreferenceveryup-to-date.Ifindthethirdchapter tobeveryinterestingcontainingmanyequilibriumunsteadyflows,whichcanbeof immensevaluetoresearcherswhowouldliketoreadthereceptivityandinstabilityof suchflows.TheinstabilityofflowhasbeenintroducedwiththeTaylor-Couetteflow asanaptexample,whichisknowntosuffermodalinstability.Inclusionofthetopic ofReynoldsaveragedNavier-Stokesequationtoexplainturbulentflowisperhapsthe easyandcorrectapproachtointroducereaderstothistopic.Theauthorisawidely acclaimed expert in the theoretical and computational aspects of incompressible fluid flow, specifically on spectral and higher order methods. He and his group of researchers are well-known for scholarly contributions which have been pursued vii viii Foreword overmorethanfivedecades.Hehasauthoredmanybooksonvariousaspectsoffluid flowanditscomputingandthepresentbookwillbeaniceadditiontothat. October2021 TapanK.Sengupta IITDhanbad Jharkhand,India Preface FluidMechanicsisafascinatingsubjectinscienceandengineeringbecausefluids are everywhere in nature, technology and in facts of every day life. The water for the shower, the air we breathe, the blood in our cardiovascular system. From the technological point of view, water is a source of energy in turbines, pumps, and hydraulicsystems.Withoutairwecouldhardlyflyoruseaboomerangforfun.In summary,fluidsoccupyacentralplaceinhumanityreflectedinthepopularsaying “Water is life”. Therefore, the investigation of fluid behavior has been a central theme in research for many centuries. Recall Archimedes running in the streets of Syracuse with joy “Eureka”. In the following centuries, many great minds tackled thestudyofthefluidflows.TheSwissmathematicianLeonhardEuler(1707–1783) broughtsignificantdevelopmentsandconceivedthe“Euler”equationsthatdescribe themechanicsofinviscid(non-viscous)fluids.Thiswasamajorbreakthroughthat impactedthefluiddynamicsscience.Inthefollowingcentury,ClaudeLouisMarie HenriNavier(1785–1836)andGeorgeGabrielStokes(1819–1903)wrotethecele- brated“Navier–Stokes”equationsthatformthebasisforviscousfluidmechanics.In thismonograph,wewillrestrictourattentiontoincompressiblefluids.Therefore,we willavoidallphysicalphenomenaassociatedwithcompressibilitylikesoundwaves orshocksthatappearassoonaswedealwithairflowsatfinitevaluesoftheMach number. Writing a book on fluid mechanics constitutes a real challenge. The topic is broadandincorporatesalotofexperimentalresults,theoreticalconsiderationsand modelingaspects.Themathematicalcoreoffluidmechanicsrestsuponappliedmath- ematicstoolslikethesolutionofordinaryandpartialdifferentialequationsmaking it less tasty in an era where everything is available through the Internet. However, myteachingpracticetomechanicalengineeringandphysicsstudentsattheCatholic UniversityofLouvaininBelgiumandattheSwissFederalInstituteofTechnology inLausanneovermorethanthreedecadesledmetothreechoiceswhilewritingthese lecturenotes.Firstofall,thecontentsofthisbookiswhatIconsiderasunmissable knowledgetostartafreshwithfluidmechanicsproblems.Secondly,withtheinvasion ofcomputersandomnipresentsoftwarestofacilitatemathematicaldevelopments,I havedecidedtogothrougheverystepindetailtoshowthatassoonwemasterthe ix x Preface main mathematical tools, we are able to carry through any major chunk of fluid mechanicsproblem.Thirdly,sometopicsliketheflowaroundanairfoilinChap.6 orthestabilityofthecircularCouetteflowinChap.8comefrommyprofessional lifeasaconsultantatONERA(OfficeNationald’ÉtudesetdeRecherchesAérospa- tiales,theFrenchAerospaceLab)andmypostdocyearsattheMITDepartmentof MathematicswithSteveA.Orszag.Stevehadadeepinfluenceonmycareerandmy researchasheeducatedmewithspectralmethodswhichbecamethecentralthemeof mywork.TheapplicationoftheFourier–ChebyshevspectralmethodtotheCouette flowstabilityledtoajointIUTAMsymposiumcommunicationthatwaspublished later. ThecourseItaughttothePhysicsstudentswasnamedHydrodynamics,adecision that pertained to the EPFL (École Polytechnique Fédérale de Lausanne) Physics Department.Therefore,Iwastemptedtotitlethemonographwiththesameheadline. However, comparing my contribution to the renowned book by Sir Horace Lamb refrained my enthusiasm and this is the reason of the present title. As I am not an experimentalisttheflavorofthebookcomesfrommathematicaldevelopmentsand theconclusionswecandrawfromthem. Let us consider the red thread that goes through the text. We start focusing on basicfluidmechanicsfromthegeneralprinciplesofcontinuummechanicsinChap.1. Governingequationsarepresentedtogetherwiththeboundaryandinitialconditions. Specialattentionisdevotedtothemeaninganddifferencesbetweenincompressible andcompressiblefluids.Thischapterisaveryabridgedpresentationofthenecessary conceptsofcontinuummechanics.Thereaderwhowantstogetalongeranddeeper understandingofthesetopicsisreferredtothemonographthatmyEPFLcolleague JohnBotsisandmyselfpublishedwiththeEPFLPress[16]. Chapter2developstheprinciplesofdimensionalanalysisallowingthedefinition oftheReynoldsnumber,adimensionlessnumbercharacterizingtheflowproperties. TheapplicationoftheVaschy-Buckinghamtheoremshedssomelightonthebenefits ofdealingwithdimensionlessequations.TheanalysisofthecompressibleNavier– Stokes equations shows howtheincompressible equations arerecovered whenthe Machnumbergoestozero.Thenatureofpressureisalsodiscussed. Chapter3coversextensivelyvariousexactsolutionsoftheNavier–Stokesequa- tionsforsteady-stateandtransientcases.Ofparticularinterestarethepulsatingflows inachannelandinacircularpipeasthesesolutionsarerelevantforbloodflowanal- ysis,afieldofbioengineeringthathasgrownatanextremelyrapidpaceoverthelast decade. Chapter4introducestheconceptsofvorticityandcirculation.Thisleadstothe famousBernoulliequationthatistaughtineveryphysicscourse.Adetailedstudyof thevorticityproductiononasolidwallisundertaken.Theflowbehindagridsolved byKovasznayprovidesabenchmarksolutionfortheeagernumericist.Thechapter ends with the Taylor-Green vortex where the Clebsch potentials are introduced to computethevorticitylines. Then in Chap. 5 Stokes flows, also called creeping flows because the viscous effectsaredominant,areconsidered.TheMoffatteddiesinacorneraredescribed.The flowaroundasphereisdetailedandleadstotheStokesformula.Stokeseigenmodes Preface xi are analyzed and a three-dimensional Stokes solution is given. The flow around a circularcylinderleadstotheStokes’paradox. Chapter6describesthemechanicsofinviscidfluidsthroughtheuseofcomplex variablesandpotentials.Theflowaroundacircularcylinderisdetailed.Thenusing conformal mapping and especially the Joukowski transformation, it is possible to consideranaerodynamicsapplication,namelytheflowaroundanairfoil.TheBlasius theorem allows for the computation of the forces and moment generated by flow around an immersed body. It is applied to the case of the cylinder and Joukowski profile. TheboundarylayertheoryisthesubjectofChap.7.ThePrandtl’sequationsare obtainedviadimensionalanalysisconsiderations.Thecaseoftheflatplateistreated asasuitableexampleforthedevelopmentoftheboundarylayeronsimplegeometry. The von Kármán integral equation allows the elaboration of the approximate von Kármán–Pohlausenmethodwherethevelocityprofileisgivenasapolynomial.This leadstothecalculationofthevariousthicknesses. Chapter8treatsflowinstabilities.Thestabilityofplaneparallelchannelflowleads tothewell-knownOrr-SommerfeldequationwhichissolvedbyaChebyshevspectral method.TheassociatedFortranprogramisgivenintheappendix.Thenthestability ofthecircularCouetteflowbetweentwoconcentriccylindersiscarriedoutfirstby an inviscid approach that yields the Rayleigh stability criterion. The incorporation oftheviscousandpressuretermsgeneratesthroughalinearizationprocessasetof differential equations again solved by high-order discretization methods through a generalized eigenvalue problem. The chapter ends with the case of the non-linear axisymmetricTaylorvortices. Thepenultimatechapterdealswithturbulencewhichisaphysicalphenomenon presentinnaturelikeriversandoceancurrents,wavesonabeach,storms,tsunamis, etc.,andintechnologicalapplicationsliketheairturbulenceinaroughflight.Turbu- lenceisstillaphenomenonthatisfarfrombeingcompletelyunderstood.Reynolds averagedNavier–Stokes(RANS)equationsareobtained.Severallinearturbulence modelsarepresentedintheRANSframework:K−ε,K−ω.Non-linearmodelsare builtontheanisotropytensorandtheincorporationoftheconceptofintegritybases. The chapter ends with the theory of large eddy simulations with a few up-to-date models:dynamicmodel,approximatedeconvolutionmethod. Eachchapterproposesafewexercisesthatcomefromthelargesetusedforthe EPFLrecitationclasses.ThesolutionsaregiveninChap.10. Chapter6onthetheoryofcomplexvariablesandpotentialsisbasedonmathe- matics that goes back mainly to the first half of the twentieth century. I have long debatedtodecideifIwouldskipthismatterorleaveitasanicepieceoftheorythat generatesdeepinsightintoimportantfluidflows.Thisisespeciallycrucialwhenwe all know that it is so easy to compute the flow around an airfoil using numerical methodslikefinitevolumesorfiniteelements.Thefinaldecisionwasbasedonthe factthatthistheoryisacornerstoneoffluidmechanicswhichnobodyescapes. Themonographcomesfrommyownlecturenotes.Initspresentformitisintended tobethesupportforaone-semestercoursewiththreelecturesof1houraweek.The teachershouldchoosethetopicsathis/herowntaste.