Brief History of the Early Development of Theoretical and Experimental Fluid Dynamics John D. Anderson Jr. AeronauticsDivision,NationalAirandSpaceMuseum,SmithsonianInstitution,Washington,DC,USA 1 INTRODUCTION 1 Introduction 1 Asyoureadthesewords,therearemillionsofmodernengi- 2 EarlyGreekScience:AristotleandArchimedes 2 neeringdevicesinoperationthatdependinpart,orintotal, 3 DAVinci’sFluidDynamics 2 ontheunderstandingoffluiddynamics–airplanesinflight, 4 TheVelocity-SquaredLaw 3 ships at sea, automobiles on the road, mechanical biomedi- 5 NewtonandtheSine-SquaredLaw 5 caldevices,andsoon.Inthemodernworld,wesometimes take these devices for granted. However, it is important to 6 DanielBernoulliandthePressure-Velocity pauseforamomentandrealizethateachofthesemachines Concept 7 is a miracle in modern engineering fluid dynamics wherein 7 HenriPitotandtheInventionofthePitotTube 9 manydiversefundamentallawsofnatureareharnessedand 8 TheHighNoonofEighteenthCenturyFluid combinedinausefulfashionsoastoproduceasafe,efficient, Dynamics–LeonhardEulerandtheGoverning andeffectivemachine.Indeed,thesightofanairplaneflying EquationsofInviscidFluidMotion 10 overheadtypifiesthelawsofaerodynamicsinaction,andit iseasytoforgetthatjusttwocenturiesago,theselawswere 9 InclusionofFrictioninTheoreticalFluid so mysterious, unknown or misunderstood as to preclude a Dynamics:theWorksofNavierandStokes 11 flying machine from even lifting off the ground; let alone 10 OsborneReynolds:UnderstandingTurbulent successfullyflyingthroughtheair. Flow 14 Inturn,thisraisesthequestionastojusthowdidourin- 11 TheCirculationTheoryofLift:Kuttaand tellectual understanding of fluid dynamics evolve? To find Joukowski 17 the answer, we have to reach back over millennia of intel- 12 LudwigPrandtlandHisBoundary-LayerTheory 19 lectual thought, all the way back to ancient Greek science. However,properlyaddressingthehistoryoffluiddynamics 13 Summary 21 inacompletefashionrequiresmanymorepagesthanavail- References 22 ableinthepresentchapter.Severalbookshavebeenwritten onthesubject,notablythosebyRouseandInce(1957)and Tokaty(1971).Aninclusivestudyofthehistoryofbothfluid dynamicsandaerodynamicscanbefoundintherecentbook byAnderson(1997). Instead,wewillfocusonafewthemesandcasehistories thatexemplifythehistoricalevolutionoffluiddynamicsand provide a flavor of the intellectual thought and the human EEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinngg,,OOnnlliinnee©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. 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DDOOII:: 1100..11000022//99778800447700668866665522..eeaaee000011 2 FundamentalsofFluidFlows dynamicsthathaveledtothestate-of-the-artoffluiddynam- drawinggeometricfiguresinSyracusesand.Archimedesis icsasweknowittoday.Wewillchooseachronologicalap- usually known for his concepts in fluid statics, and particu- proachtothesubject,andwillmarbletogetheradvancements larlyforhisvagueconceptofpressureinafluid.Hesensed inboththeoreticalandexperimentalfluiddynamics.Muchof thateverypointofthewettedsurfaceareaofabodyinafluid thefollowingmaterialisexcerptedfromtheauthor’sbroader wasundersomeforceduetothefluidalthoughtheconcept studyofthesubjectinAnderson(1997). of “force” was not quantified during the age of Greek sci- ence.However,therewassomevague,intuitivefeelingabout what we today technically label as force, and Archimedes 2 EARLY GREEK SCIENCE: ARISTOTLE realizedthatsuchforceisdistributedoverthebodysurface. AND ARCHIMEDES Archimedesstatedthat,inafluid,“eachpartisalwayspressed bythewholeweightofthecolumnperpendicularlyaboveit.” Thescienceoffluiddynamicscantraceitsrootstoamanborn Thiswasthefirststatementoftheprinciplethat,inmodern in384B.C.intheIoniancolonyofStagiraontheAegeanSea, terms,thepressureatapointinastationaryfluidisduetothe andeducatedatPlato’sAcademyinAthens.Aristotle(384– weightofthefluidaboveit,andhenceislinearlyproportional 322B.C.)livedatthemostintellectuallyfruitfultimeinGreek tothedepthofthefluid.Thisisatruestatement,aslongas history,wenttothebestschool,andassociatedwithsomeof thefluidisnotinmotion,thatis,forfluidstatics. themostinfluentialpeople.Throughoutallofthis,Aristotle However,Archimedesmadeacontributiontofundamental developed a corpus of philosophy, science, ethics, and law fluiddynamicsasfollows.Today,wefullyunderstandthat,in thatinfluencedtheworldforthefollowing2000years. ordertosetastagnantfluidintomotion,adifferenceinpres- Aristotle’s scientific thoughts established two concepts suremustbeexertedacrossthefluid.Wecallthispressuredif- that bear on the development of fluid dynamics. The first ferenceoveraunitlengththepressuregradient.Archimedes istheconceptofcontinuum.Hewrotethat had a vague understanding of this point when he wrote “if fluidpartsarecontinuousanduniformlydistributed,thenthat ofthemwhichistheleastcompressedisdrivenalongbythat Thecontinuousmaybedefinedasthatwhichisdivisibleinto whichismorecompressed.”Liberallyinterpreted,thismeans parts which are themselves divisible to infinity, as a body that when a pressure gradient is imposed across a stagnant which is divisible in all ways. Magnitude divisible in one fluid,thefluidwillstarttomoveinthedirectionofdecreas- directionisaline,inthreedirectionsabody.Andmagnitudes ingpressure.TheabovestatementbyArchimedesisaclear whicharedivisibleinthisfashionarecontinuous. contributionofGreeksciencetofluiddynamics. Itisnotwidelyappreciatedthatthefundamentalconcept of a continuum, upon which most fluid dynamic theory is based, is one of Aristotle’s contributions to the science of 3 DA VINCI’S FLUID DYNAMICS fluiddynamics. The second of Aristotle’s contributions to aerodynamics Thetime-spanfromthedeathofArchimedestothetimeof was the idea that a moving body passing through the air or Leonardo da Vinci covers the zenith of the Roman Empire, anotherfluidencounterssomeaerodynamic“resistance,”He itsfall,thedearthofintellectualactivityinWesternEurope wrote: duringtheDarkAges,andthesurgeofnewthoughtthatchar- acterizedtheRenaissance.Intermsofthescienceofaerody- “Itisimpossibletosaywhyabodythathasbeensetinmotion namics,theseventeencenturiesthatseparateArchimedesand in a vacuum should ever come to rest. Why, indeed, should Leonardoresultedinnoworthwhilecontributions.Although itcometorestatoneplaceratherthananother.Asaconse- theRomansexcelledinhighlyorganizedcivil,military,and quence,itwilleithernecessarilystayatrest,orifinmotion, politicalactivities,aswellasinlargeengineeringfeatswith willmoveindefinitelyunlesssomeobstaclecomesintocol- buildingconstructionandthewidedistributionofwaterfrom lision.” reservoirstocitiesviaaqueducts,theycontributednothingof substancetoanyscientifictheory.Moreover,althoughthean- A conclusion from this reasoning is that, since bodies cientGreekscienceandphilosophywaskeptaliveforfuture eventuallycometorestinafluid,theremustbearesistance generations by eastern Arabian cultures through the Dark actingonthebody.Today,wecallthisfluiddynamicdrag. Ages,nonewcontributionsweremadeduringthisperiod. The other Ancient Greek scientist to contribute to fluid This changed with the work of Leonardo da Vinci. Born dynamics was born in 287 bc in Syracuse, and was killed in1452inthesmallTuscanvillageofVinci,nearFlorence, unceremoniouslyin212bcbyRomansoldierswhilehewas LeonardodaVinciwentontorevolutionizetheworldsofart, EEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinngg,,OOnnlliinnee©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleeiiss©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleewwaassppuubblliisshheeddiinntthheeEEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinnggiinn22001100bbyyJJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. DDOOII:: 1100..11000022//99778800447700668866665522..eeaaee000011 BriefHistoryoftheEarlyDevelopmentofTheoreticalandExperimentalFluidDynamics 3 science,andtechnology.Heisrecognizedtodayasbeingin theforefrontoftheworld’sgreatestintelligences. Pertinent to this article, Leonardo had an interest in the characteristics of basic fluid flow. For example, one of the fundamentalprinciplesofmodernfluidmechanicsisthefact that mass is conserved; in terms of a fluid moving steadily inatube,thismeansthatthemassflow(e.g.,thenumberof poundspersecond)passingthroughanycrosssectionofthe tubeisthesame.Foranincompressibleflow(flowofafluid, or low-speed flow of a gas), this principle leads to a basic relationthat AV =constant (1) where A is the cross-sectional area of the duct at any lo- cation, and V is the velocity of the fluid at that same loca- tion. This relation is called the continuity equation, and it states that in moving from one location in the duct to an- otherwheretheareaissmaller,thevelocitybecomeslarger in just the right amount that the product of A times V re- Figure1. SketchesbydaVincishowingcomplexflowfieldsover objectsinaflowingstream. mainsthesame.Leonardoobservedandrecordedthiseffect inregardtotheflowofwaterinrivers,where,inthoseloca- that can be taken in any modern fluid dynamic laboratory, tionswheretheriverbecomesconstricted,thewatervelocity andtheydemonstratethedetailtowhichLeonardoobserved increases.Moreover,hequantifiedthisobservationinthefol- variousflowpatterns. lowingstatementthatreferstowaterflowthroughapassage Inmodernfluiddynamicsandaerodynamics,thewindtun- wherethedepthmnchangestoasmallerdepth,ab,smaller nelisanabsolutelyessentiallaboratorydevice.Althoughwe byfactorof4. takeforgrantedtodaythattherelativeflowoverastationary body mounted in a wind tunnel is the same as the relative “Each movement of water of equal surface width will run flowoverthesamebodymovingthroughastationaryfluid, the swifter the smaller of the depth...and this motion will we have Leonardo to thank for being the first to state this beofthisquality:Isaythatinmnthewaterhasmorerapid fact.Hisstatementofwhatwecancalltodaythe“windtun- movementthaninab,andasmanytimesmoreasmnenters nelprinciple”canbefoundintwodifferentpartsoftheCodex intoab;itenters4times,themotionwillthereforebe4times Atlanticus.Leonardomadethefollowingstatements:“Asitis asrapidinmnasinab.” tomovetheobjectagainstthemotionlessairsoitistomove the air against the motionless object,” and “The same force Here we have, for the first time in history, a quantitative asismadebythethingagainstair,ismadebyairagainstthe statementofthespecialformofthecontinuityequationthat thing.”Therefore,thebasicprinciplethatallowsustomake holdsforlow-speedflow. wind tunnel measurements and apply them to atmospheric Inadditiontothisquantitativecontribution,Leonardo,be- flightwasfirstconceivedbyLeonardo380yearsbeforethe ingaconsummateobserverofnature,mademanysketchesof inventionofthefirstwindtunnel. variousflowfields.Aparticularlygraphicexampleisshown inFigure1,foundintheCodexAtlanticus.Hereweseethe vortexstructureoftheflowaroundaflatplate.Atthetop,the 4 THE VELOCITY-SQUARED LAW plate is perpendicular to the flow, and Leonardo accurately sketches the recirculating, separated flow at the back of the We now address what is perhaps the most important break- plate,alongwiththeextensivewakethattrailsdownstream. throughinexperimentalfluiddynamicsinthe17thcentury. Atthebottom,theplateisalignedwiththeflow,andwesee Putyourselfintheshoesofaself-stylednaturalphilosopher thevortexthatiscreatedatthejunctionoftheplatesurface in the Middle Ages. In thinking about the question of how andthewatersurface,aswellasthebowwavethatpropagates theforceonanobjectimmersedinamovingfluidvarieswith at an angle away from the plate surface. These sketches by the velocity of the fluid, intuition is most likely to tell you Leonardoarevirtuallyidenticaltophotographsofsuchflows that, when the velocity doubles, the force doubles. That is, EEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinngg,,OOnnlliinnee©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleeiiss©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleewwaassppuubblliisshheeddiinntthheeEEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinnggiinn22001100bbyyJJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. DDOOII:: 1100..11000022//99778800447700668866665522..eeaaee000011 4 FundamentalsofFluidFlows youareinclinedtofeelthatforceisdirectlyproportionalto workofLeonardoandGalileo.Indeed,Mariottewasagifted velocity. This seems “logical,” although there is (up to the experimenterwhotookpainstotrytolinkexistingtheoryto 17thcentury)noproperexperimentalevidenceortheoretical experiment–anovelthoughtinthatday.TheAcademywas analysistosayonewayoranother.Likesomuchofancient essentiallyMariotte’slaterlife;heremainedinParisuntilhis science,thisfeelingwasbasedsimplyontheimageofgeo- deathon12May121684. metricperfectioninnature,andwhatcouldbemore“perfect” TheparticularworkofMariotteofinteresttoourdiscus- than the force doubling when the velocity doubles. Indeed, sion was conducted in the period before 1673. He was par- bothLeonardoandGalileo–twoofthegreatestmindsinhis- ticularlyinterestedintheforcesproducedbyvariousbodies tory–heldthisbelief.Uptothemiddleofthe17thcentury, impacting on other bodies or surfaces. One of these “bod- theprevailingthoughtwastheincorrectnotionthatforcewas ies”wasafluid;Mariotteexaminedandmeasuredtheforce directlyproportionaltotheflowvelocity. createdbyamovingfluidimpactingonaflatsurface.Thede- However, within the space of 17 years at the end of the viceheusedfortheseexperimentswasabeamdynamometer seventeenthcentury,thissituationchangeddramatically.Be- whereinastreamofwaterimpingesononeendofthebeam, tween1673and1690,twoindependentsetsofexperiments andtheforceexertedbythisstreamisbalancedandmeasured duetoEdmeMariotte(1620–1684)inFranceandChristian byaweightontheotherendofthebeam.Thewaterjetem- Huygens (1629–1695) in Holland, along with the theoreti- anatesfromthebottomofafilledverticaltube,anditsvelocity calfundamentalspublishedbyIsaacNewton(1642–1727)in isknownfromTorricelli’slawasafunctionoftheheightof England,clearlyestablishedthattheforceonanobjectvaries the column of water in the tube. From the results obtained asthesquareoftheflowvelocity,thatis,ifthevelocitydou- withthisexperimentalapparatus,Mariottewasabletoprove bles,theforcegoesupbyafactoroffour.Incomparisontothe that the force of impact of the water on the beam varied as previous centuries of halting, minimal progress in fluid dy- thesquareoftheflowvelocity.Hepresentedtheseresultsina namics,therathersuddenrealizationofthevelocity-squared paperreadtotheParisAcademyofSciencein1673,entitled lawforaerodynamicforcesrepresentsthefirstmajorscien- “Traite´ delaPercussionouChocdesCorps,”–thefirsttime tific breakthrough in the historical evolution of the subject. in history that the velocity-squared law was published. For Letusexaminethisbreakthroughmoreclosely,aswellasthe thiswork,EdmeMariottedeservesthecreditforthefirstma- menwhomadeitpossible. joradvancementtowardtheunderstandingofvelocityeffects Creditfortheoriginofthevelocity-squaredlawrestswith onaerodynamicforce. Edme Mariotte, who first published it in the year 1673. To As a final note on Mariotte, the esteem in which he was gain an appreciation for the circumstances surrounding this held by some of his colleagues is reflected by the words of development,letusconsiderMariotte’sbackground.Helived J.B.duHamel,whosaidafterMariotte’sdeathin1864, inabsoluteobscurityforaboutthefirst40yrofhislife.There isevencontroversyastowhereandwhenhewasborn.There is a claim that he was born in Dijon, France, in 1620, but “Themindofthismanwashighlycapableofalllearning,and there are no documents to verify this, let alone to pinpoint theworkspublishedbyhimattesttothehighesterudition.In anexactbirthdate.Wehavenoevidenceconcerninghisper- 1667,onthestrengthofasingulardoctrine,hewaselectedto sonal life, his education, or his vocation until 1666, when theAcademy.Inhim,sharpinventivenessalwaysshoneforth very suddenly he was made a charter member of the newly combined with the industry to carry through, as the works formedParisAcademyofSciences.Mostlikely,Mariottewas referredtointhecourseofthistreatisewilltestify.Hisclev- self-taught in the sciences. He came to the attention of the erness in the design of experiments was almost incredible, Academy through his pioneering theory that sap circulated andhecarriedthemoutwithminimalexpense.” through plants in a manner analogous to blood circulating through animals. Controversial at that time, his theory was However, there was at least one colleague who was not confirmedwithinfouryearsbynumerousexperimentalinves- sohappywithMariotte,andwhorepresentsanothersideof tigators.ItisknownthathewasresidinginDijonatthetime thehistoricalproprietorshipofthevelocity-squaredlaw.This ofhisappointmenttotheAcademy.Mariottequicklyproved man was Christian Huygens (1629–1695). Huygens’ back- tobeanactivememberandcontributortotheAcademy.His groundisbetterknownthanthatofMariotte.ChristianHuy- areasofworkwerediverse;hewasinterestedinexperimen- genswasbornonApril14,1629,intheHague,TheNether- tal physics, hydraulics, optics, plant physiology, surveying, lands,toafamilyprominentinDutchsociety.Hisgrandfather andgeneralscientificandmathematicalmethodology.Mari- servedWilliamtheSilentandPrinceMauriceassecretary.His otteiscreditedasthefirstinFrancetodevelopexperimental father,Constantine,wassecretarytoPrinceFrederickHenry. science,transferringtothatcountrythesameinterestinex- Indeed,severalmembersofthefamilywerediplomatsunder periments that grew during the Italian renaissance with the thereignoftheOrangefamilyinHolland.Christianwaswell EEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinngg,,OOnnlliinnee©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleeiiss©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. 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DDOOII:: 1100..11000022//99778800447700668866665522..eeaaee000011 BriefHistoryoftheEarlyDevelopmentofTheoreticalandExperimentalFluidDynamics 5 educated;hewastutoredbyhisfatheruntiltheageof16,after yearslatertomakesuchcharges?Thisauthorhasnodefinite which he studied law and mathematics at the University of answertothisquestion.However,usingthewrittenscientific Leiden.Devotinghimselftophysicsandmathematics,Huy- literatureasthemeasureofproprietorship,Mariotteisclearly gensmadesubstantialcontributions,includingimprovements thefirstpersontopublishthevelocity-squaredlaw.Takenin in existing methodology, developing new techniques in op- conjunctionwithHuygens’silenceatthetimeofthispubli- tics,andinventingthependulumclock.Eventoday,alltext- cation,wehavetoconcludethatMariottedeservesfirstcredit booksonbasicphysicsdiscussHuygens’lawofoptics.For forthislaw.However,itisquiteclearthatHuygens’experi- hisaccomplishments,Huygenswasmadeachartermember ments,whichwerecarriedoutbeforeMariotte’spublication, oftheParisAcademyofSciencein1666–thesameyearas alsoprovedthevelocity-squaredlaw.Ofcourse,ofgreatim- Mariotte. Huygens moved to Paris in order to more closely portancetothedevelopmentoffluiddynamicsissimplythe participateintheactivitiesoftheAcademy;helivedinParis factthat,bytheendofthe17thcentury,wehavedirectexper- until1681.Duringthislife,HuygenswasrecognizedasEu- imentalprooffromtwoindependentinvestigationsthatfluid rope’sgreatestmathematician.However,hewasasomewhat dynamic force varies as the square of the velocity. Of even solitarypersonwhodidnotattractafollowingofyoungstu- greaterimportanceisthat,atthesametime,thesamelawwas dents.Moreover,hewasreluctanttopublish,mainlybecause derivedtheoreticallyonthebasisoftherational,mathemati- ofhisinordinatelyhighpersonalstandards.Forthesetworea- callawsofmechanicsadvancedbyNewtoninhisPrincipia, sons,Huygens’workdidnotgreatlyinfluencethescientists publishedin1687. of the next century; indeed, he became relatively unknown duringthe18thcentury. 5 NEWTON AND THE SINE-SQUARED In1668,Huygensbegantostudythefallofprojectilesin resistingmedia.FollowingLeonardoandGalileo,hestarted LAW outwiththebeliefthatresistance(drag)wasproportionalto velocity.However,withinoneyearhisanalysisoftheexper- It is fitting that the end of the 17th century saw the natural imentaldataconvincedhimthatresistancewasproportional fruition of experimental work such as that by Mariotte and tothesquareofthevelocity.ThiswasfouryearsbeforeMar- Huygensinthedevelopmentofarationalmathematicalthe- iotte published the same result in 1673; however, Huygens ory by Isaac Newton (1642–1727). Newton’s contributions delayed until 1690 in publishing his data and conclusions. tophysicsandmathematicswerepivotal.Thepublicationof Thissomewhatcomplicatesthequestionastowhomshould his“PhilosophiaeNaturalisPrincipiaMathematica”–widely thevelocity-squaredlawbeattributed.Thepictureisfurther knownas“thePrincipia”in1687representedthefirstcom- blurred by Huygens himself, who accused Mariotte of pla- plete,rational,theoreticalapproachtothestudyofmechani- giarism;however,HuygensleviedthischargeafterMariotte’s calphenomena. deathin1684.Huygensstatedthat“Mariottetookeverything Newton was born on 25 December 1642, in the hamlet from me.” In regard to Mariotte’s paper in 1673, Huygens of Woolsthorpe-by-Colsterworth near the English town of complains that “he should have mentioned me. I told him Grantham.Hewasraisedbyhismother;hisfatherhaddied thatoneday,andhecouldnotrespond.” fivemonthsbeforehisbirth.Heshowedaninterestinmechan- Inthepresentauthor’sopinion,hereisaclassicsituation icaldiagrams,whichhescratchedonthewallsandwindow that frequently occurs in scientific and engineering circles edgesofhishouseinWoolsthorpe.Withtheencouragement eveninmoderntimes.Wehavealearnedsociety–theParis ofanuncle,NewtonenteredTrinityCollegeatCambridgein Academy of Sciences – the members of which frequently 1661andreceivedhisB.A.degreein1665.Forthenexttwo gathered to discuss their experiments, theories, and general yearsheretreatedtothecountryinLincolnshiretoavoidthe feelingsaboutthenaturalworld.Ideasandpreliminaryresults plague that was running rampant in Europe and had closed weresharedandcritiquedinacollegialatmosphere.Mariotte theUniversity.Itwasduringthattwo-yearperiodthathecon- andHuygenswerecolleagues,andfromHuygensownwords ceived many of his basic ideas on mathematics, optics, and above, they clearly discussed and shared thoughts. In such mechanicsthatwerelatertoappearinprint.Newtonsaidof an atmosphere, the exact credit for the origin of new ideas those two years that “I was in the prime of my age of in- is sometimes not clear; ideas frequently evolve as a result ventionandmindedmathematicsandphilosophymorethan of discussion among groups. What is clear is this. Mariotte atanytimesince.”In1667,NewtonreturnedtoCambridge published the velocity-squared law in a paper given to the andbecameaminorfellowatTrinity.HeearnedanM.A.de- Academy in 1673; Huygens published the same conclusion greein1668andwasappointedLucasianprofessorin1669. 17 years later. Moreover, in 1673 Huygens critiqued Mari- NewtonremainedatCambridgeforthenext27years. otte’spaper,andsaidnothingaboutplagiarismornotbeing Newton’scontributionstofluiddynamicsappearinBook referenced. Why did he wait until after Mariotte’s death 11 II of the Principia, subtitled “The Motions of Bodies (in EEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinngg,,OOnnlliinnee©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleeiiss©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleewwaassppuubblliisshheeddiinntthheeEEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinnggiinn22001100bbyyJJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. DDOOII:: 1100..11000022//99778800447700668866665522..eeaaee000011 6 FundamentalsofFluidFlows Resisting Mediums).” Book II deals exclusively with fluid of any corporeal fluid, excepting, perhaps, some extremely dynamicsandhydrostatics.Duringthelastpartoftheseven- rarevaporsandtheraysoflight.”ForNewton,thatwasthe teenthcentury,practicalinterestinfluiddynamicswasdriven crowningaccomplishmentfromhisstudyoffluiddynamics. by problems in naval architecture, particularly the need to In regard to aerodynamics, Newton’s work in Book II understand and predict the drag on a ship’s hull, an impor- of the Principia contributed a second fundamental finding, tant concern for a country that was ruling large portions of namely, a relationship for the shear stress at any point in a the world through the superior performance of its powerful fluid in terms of the velocity gradient existing at that same navy.Newton’sinterestinfluidmechanicsmayhavederived point.Newtonadvancedthefollowinghypothesis:“There- partly from such a practical problem, but he had a much sistance arising from the want of lubricity in the part of a more compelling reason for calculating the resistance of a fluidis,otherthingsbeingequal,proportionaltothevelocity bodymovingthroughafluid.Therewasaprevailingtheory, withwhichthepartsofthefluidareseparatedfromonean- advanced by Rene Descartes, that interplanetary space was other.”Inmodernterms,the“wantoflubricity”istheaction filledwithmatterthatmovedinvortex-likemotionsaround offrictioninthefluid,namely,theshearstressτ.The“veloc- theplanets.However,astronomicalobservations,suchasthe ity with which the parts of the fluid are separated from one definitiveworkofJohannesKeplerinhisRudolphineTables, another”istherateofstrainexperiencedbyafluidelement published in 1627, indicated that the motions of the heav- intheflow,whichinturncanbemathematicallyrepresented enlybodiesthroughspacewerenotdissipated,butratherthat bythevelocitygradient,dV/dn.Amathematicalstatementof thosebodiesexecutedregular,repeatablepatterns.Theonly Newton’shypothesisissimply explanationthatthebodiesweremovingthroughspacefilled withacontinuousmediumasDescarteshadtheorizedwould τ ∝dV/dn (3) be for the aerodynamic drag on each body to be zero. The centralpurposeofNewton’sstudiesinfluidmechanicswas Withtheproportionalityconstantdefinedatthecoefficientof toprovethattherewasafinitedragonabody(includingthe viscosity,µ,thatbecomes heavenly bodies) moving through a continuous medium. If thatcouldbeshowntobetrue,thenthetheoryofDescartes τ =µ(dV/dn) (4) would be disproved. Indeed, in Proposition 23 of the Prin- cipia,Newtoncalculatedfiniteresistanceonbodiesmoving ThisequationiscalledtheNewtonianshear-stresslaw,andall through a fluid and showed that such resistances were “in fluidsthatobeythelawarecalled“Newtonianfluids.”Virtu- a ratio compounded of the squared ratio of their velocities, allyallgases,includingair,areNewtonianfluids.Hencethe andthesquaredratiooftheirdiameters,andthesimpleratio Newtonian stress law, as first hypothesized in thePrincipia of the density of the parts of the system.” That is, Newton representedamajorcontributiontothestateoftheartoffluid discussed the velocity-squared law, while at the same time dynamicsattheendoftheseventeenthcentury. showing that resistance varies with the cross-sectional area In an indirect sense, Isaac Newton was responsible for of the body (the “squared ratio of their diameters”) and the thefirsttechnicalcontributiontowardtheanalysisofangle- firstpowerofthedensity(the“simpleratioofthedensity”). of-incidence(angleofattack)effectsonaerodynamicforce. Insodoing,Newtonpresentedthefirsttheoreticalderivation Proposition34inBookIIofthePrincipiaisaproofthatthe oftheessenceofthedragequation resistanceofaspheremovingthroughafluidishalfthatof a circular cylinder of equal radius with its axis oriented in D∝ρSV2 (2) thedirectionofitsmotion.Thefluiditselfispostulatedasa collectionofindividualparticlesinrectilinearmotionthatim- However,inNewton’smind,hiscontributionwassimplyto pactdirectlyonthesurfaceofthebody,subsequentlygiving refutethetheoryofDescartes.Thatwasstatedspecificallyby uptheircomponentsofmomentumnormaltothesurface,and NewtoninthescholiumaccompanyingProposition40,deal- then traveling downstream tangentially along the body sur- ing with experimental measurements of the resistance of a face.Thatfluidmodelwassimplyahypothesisonthepartof spheremovingthroughacontinuousmedium.Becausesuch Newton;itdidnotaccuratelymodeltheactionofarealfluid, spheres had been shown both theoretically and experimen- asNewtonreadilyacknowledged.However,consistentwith tallytoexhibitfiniteresistanceswhilemovingthroughafluid, thatmathematicalmodel,burieddeepintheproofofProposi- Newton reasoned that “the celestial spaces, through which tion34istheresultthattheimpactforceexertedbythefluid theglobesoftheplanetsandcometsarecontinuallypassing on a segment of a curved surface is proportional to sin2θ, towards all parts, with the utmost freedom, and without the where θ is the angle between a local tangent to the surface leastsensiblediminutionoftheirmotion,mustbeutterlyvoid andthefree-streamdirection.Thatresult,whenappliedtoa EEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinngg,,OOnnlliinnee©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleeiiss©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleewwaassppuubblliisshheeddiinntthheeEEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinnggiinn22001100bbyyJJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. DDOOII:: 1100..11000022//99778800447700668866665522..eeaaee000011 BriefHistoryoftheEarlyDevelopmentofTheoreticalandExperimentalFluidDynamics 7 L thesmallvalueofsin2αandtoincreasetheliftsothatitwould equaltheweightoftheflyingmachine: R 1. Increase the wing area S. That would lead to enormous α wingareas,whichwouldmaketheflyingmachinetotally impractical. D 2. Increasetheangleofattackα.Unfortunatelythatwould α lead to greater drag because D varies as sin3α, that is, Flow the drag increases faster than the lift as α is increased, puttingagreaterdemandonthepowerplant.Thelift-to- Figure2. Aerodynamicforceonaflatplateatanangleofattacka. dragratiowoulddecreasedramatically.BecauseL/Disa measureofaerodynamicefficiency,flyingatlargeangles flatsurface(e.g.,aflatplate)orientedatanangleofattackα ofattackwouldbeundesirable,tosaytheleast. tothefreestream(Figure2),givestheresultantaerodynamic forceRontheplate: Indeed,Anderson(2002)showsthatiftheWrightbroth- ers had used the Newtonian sine-squared law to design the R=ρV2Ssin2α (5) 1903WrightFlyer(theydidnot),thewingareawouldhave beenawhopping23448ft2 –animpossiblylargeareafora where S is the planform area of the plate. This equation is flyingmachineatthattime–incomparisontotheactualwing calledtheNewtoniansine-squaredlaw. areaof510ft2.IftheWrightshadbasedtheirdesignonthe TheNewtoniansine-squaredlawisasimplerelationship sine-squared law, they would have quit their efforts imme- for the calculation of the fluid dynamic force on a surface, diately.Indeed,throughoutthenineteenthcentury,theNew- andforthatreasonotherexperimenterswerequicktouseit. toniansine-squaredlawwasmisusedbymanynaysayersto However, the accuracy of the Newtonian sine-squared law “prove”thatheavier-than-airpoweredflightwasnotpossible. soon came into question in the 18th century. For example, Ironically, the Newtonian sine-squared law has had a re- in 1777 the great French scientist and mathematician Jean birthinmodernaerodynamics,namely,forthepredictionof leRondd’Alembertparticipatedinaseriesofexperimental pressuredistributionsonthesurfacesofhypersonicvehicles. measurementsofthedragonships’hulls.Forthemostpart, Thephysicalnatureofhypersonicflow,wherethebowshock calculationsfromthesine-squaredlawdidnotagreewiththe wave lies very close to the vehicle surface, closely approx- experimental data. However, other researchers continued to imates the fluid model used by Newton – a stream of parti- usethesine-squaredlawforanothercentury. clesinrectilinearmotioncollidingwiththesurfaceandthen The Newtonian sine-squared law was used by various movingtangentiallyoverthesurface.Hence,thesine-squared would-be flying machine inventors in the 19th century for lawleadstoreasonablepredictionsforthepressuredistribu- thepredictionofliftLanddragD.InFigure2,theresultant tions over blunt-nosed hypersonic vehicles, an application force R is resolved into two mutually perpendicular forces: thatNewtoncouldnothaveforeseen. liftLperpendiculartoVanddragDparalleltoV.Hence, L=Rcosα=ρV2Ssin2αcosα (6) 6 DANIEL BERNOULLI AND THE D=Rsinα=ρV2Ssin3α PRESSURE-VELOCITY CONCEPT and,thelift-to-dragratioL/Dis The fundamental advances in fluid dynamics that occurred L inthe18thcenturybeganwiththeworkofDanielBernoulli =cotα (7) (1700–1782). Newtonian mechanics had unlocked the door D to modern hydrodynamics, but the door was still closed at Examiningtheforegoingequationforlift,wenotethatfora thebeginningofthecentury.DanielBernoulliwasthefirstto flyingmachineatagivenvelocitywithagivenwingareaS, open this door, albeit just by a crack; Euler and others who thesine-squaredlawpredictsverysmallliftatsmallanglesof followedflungthedoorwideopen. attack.However,forsteadylevelflight,theliftmustequalthe DanielBernoulliwasborninGroningen,TheNetherlands weight.IfweweretoaccepttheNewtoniansine-squaredlaw on February 8, 1700. His father, Johann, was a professor ascorrect,thenwewouldhaveonlytwooptionstocounter at Groningen but returned to Basel, Switzerland, in 1705, EEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinngg,,OOnnlliinnee©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleeiiss©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleewwaassppuubblliisshheeddiinntthheeEEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinnggiinn22001100bbyyJJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. DDOOII:: 1100..11000022//99778800447700668866665522..eeaaee000011 8 FundamentalsofFluidFlows to occupy the Chair of Mathematics that had been vacated decreases.Thislowerpressureonthetopsurfaceincombi- bythedeathofJacobBernoulli.AttheUniversityofBasel, nation with a higher pressure on the bottom surface gener- Danielobtainedamaster’sdegreein1716inphilosophyand ates lift. A quantitative statement of Bernoulli principle is logic. He went on to study medicine in Basel, Heidelberg, Bernoulli’sequation,writtenasfollows.Ifpoints1and2are and Strasbourg, obtaining his Ph.D. in anatomy and botany twodifferentpointsinafluidflow,then in1721.Duringthesestudies,hemaintainedanactiveinterest inmathematics.Hefollowedthisinterestbymovingbrieflyto p +1/2ρV2 =p +1/2ρV2 (8) 1 1 2 2 Venice,wherehepublishedanimportantworkentitled“Ex- ercitationes quaedam Mathematicae” in 1724. This earned ThisisthefamousBernoulliequation–perhapsthemost him much attention and resulted in his winning the prize famous equation in all of fluid dynamics. Examining this awardedbytheParisAcademy–thefirstof10hewaseven- equation,clearlyifV islargerthanV ,thenp issmallerthan tually to receive. In 1725, Daniel moved to St. Petersburg, 2 1 2 p ;thatis,asVincreases,pdecreases.Question:Howmuch Russia, to join the academy. The St. Petersburg Academy 1 of this did Bernoulli ever state? The answer is, not much. hadgainedasubstantialreputationforscholarshipandintel- In his book Hydrodynamica, which is the central reference lectual accomplishment at that time. During the next eight used by all subsequent investigators for his contributions, years,Bernoulliexperiencedhismostcreativeperiod.While Bernoullididattempttoderivetherelationbetweenpressure atSt.Petersburg,hewrotehisfamousbookHydrodynamica, and velocity. Using the concept of “vis viva,” Bernoulli ap- completed in 1734, but not published until 1738. In 1733, plied an energy conservation principle to the sketch shown DanielreturnedtoBaseltooccupytheChairofAnatomyand inFigure3;thisisacopyofhisoriginalillustrationforHy- Botany, and in 1750 moved to the Chair of Physics created drodynamica. Here we see a large tank, ABGC, filled with exclusivelyforhim.Hecontinuedtowrite,giveverypopular water,towhichhasbeenattachedahorizontalpipe,EFDG. andwell-attendedlecturesinphysics,andmakecontributions The end of the pipe is partially closed; it contains a small tomathematicsandphysicsuntilhisdeathinBaselonMarch orificethroughwhichthewaterescapes.Statingthatthesum 17,1782. ofthepotentialandkineticenergiesofthefluidinapipeis Daniel Bernoulli was famous in his own time. He was constant (an incorrect statement, because in a flowing fluid a member of virtually all the existing learned societies and thereisworkdonebythepressureinadditiontotheexistence academies, such as Bologna, St. Petersburg, Berlin, Paris, ofkineticandpotentialenergies–such“flowwork”wasnot London,Bern,Turin,Zurich,andMannheim.Hisimportance understood by Bernoulli), he obtained the following differ- to fluid dynamics is centered on his 1738 book, Hydrody- ential equation for the change in velocity, dV, over a small namica(withthisbook,Danielintroducedtheterm“hydro- dynamics” to literature). In this book, he ranged over such topics as jet propulsion, manometers, and flow in pipes. Of most importance, however, he attempted to find a relation- shipbetweenthevariationofpressurewithvelocityinafluid flow.HeusedNewtonianmechanics,alongwiththeconcept of“visviva”or“livingforce”introducedbyLeibnizin1695. Thiswasactuallyanenergyconcept;“visviva”wasdefined by Leibniz as the product of mass times velocity squared, mV2;today,werecognizethisastwicethekineticenergyof amovingobjectofmassm.Also,Bernoullitreatedpressure in terms of the height of a fluid, much as Archimedes had done 20 centuries previously; the concept that pressure is a point property that can vary from one point to another in a flowcannotbefoundinBernoulli’swork. Let us critically examine Bernoulli’s contribution to fluid dynamics. In modern fluid dynamics there exists the “Bernoulli principle,” which simply states that in a flowing fluid,asthevelocityincreases,thepressuredecreases.This isanabsolutefactthatisfrequentlyusedtoexplainthegen- eration of lift on an airplane wing; as the flow speeds up Figure3. SketchfromBernoulli’sHydrodynamicashowingwater whilemovingoverthetopsurfaceofthewing,thepressure flowingfromatank. EEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinngg,,OOnnlliinnee©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleeiiss©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleewwaassppuubblliisshheeddiinntthheeEEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinnggiinn22001100bbyyJJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. DDOOII:: 1100..11000022//99778800447700668866665522..eeaaee000011 BriefHistoryoftheEarlyDevelopmentofTheoreticalandExperimentalFluidDynamics 9 distance,dx. thePitottube,thisdevicehasbecomethemostcommonplace instrument in modern 20th century fluid dynamic laborato- V dV a−V2 ries.Becauseofitsimportance,letusexaminethehistorical = (9) dx 2c detailssurroundingitsdevelopment. ForPitothimself,theinventionofthePitottubewasjust whereaistheheightofthewaterinthetank,andcisthelength one event in a reasonably productive life. Born in Aramon, ofthehorizontalpipe.Theaboveequationisafarcryfrom FranceonMay3,1695ofreasonablyeducatedparents,Pitot’s the “Bernoulli equation” we use today. However, Bernoulli youth was undistinguished; indeed, he demonstrated an in- wentontointerpretthetermVdV/dxasthepressure,which tensedislikeofacademicstudies.Whileservingabrieftime allows us to interpret the relation in Hydrodynamica as the inthemilitary,Pitotwasmotivatedbyageometrytextpub- form lishedinaGrenoblebookstore,andsubsequentlyspentthree yearsathomestudyingmathematicsandastronomy.In1718, a−V2 p= (10) PitotmovedtoParis,andby1723hadbecomeanassistantin 2c thechemistrylaboratoryoftheAcademyofSciences.Itwas tothisgroupthathedelivered,onNovember12,1732,hisan- Since a and c are constants, this relation says qualitatively nouncementofhisnewdeviceformeasuringflowvelocity– thatasvelocityincreasespressuredecreases. thePitottube. Fromthis,weareledtoconcludethefollowing: HisinventionofthePitottubewasmotivatedbyhisdis- satisfactionwiththeexistingtechniqueofmeasuringtheflow 1. The principle that pressure decreases as velocity in- velocityofwater,whichwastoobservethespeedofafloating creases is indeed presented in Bernoulli’s book, albeit object on the surface of the water. So he devised an instru- in a slightly obscure form. Hence, it is clearly justified mentconsistingoftwotubes;onewassimplyastraighttube to call this the “Bernoulli principle,” as is done today. openatoneendthatwasinsertedverticallyintothewater(to However, it is interesting to note that nowhere in his measure the static pressure) and the other was a tube with book does Bernoulli emphasize the importance of this oneendbentatrightangleswiththeopenendfacingdirectly principle, showing a certain lack of appreciation of its intotheflow(tomeasuretotalpressure)–namely,thePitot significance. tube.In1732,betweentwopiersofabridgeovertheSeine 2. Bernoulli’s equation does not appear in his book, nor River in Paris, he used this instrument to measure the flow elsewhere in his work. It is quite clear that Bernoulli velocityoftheriveratdifferentdepthswithintheriver.Inthis neverderivednorusedBernoulli’sequation. presentation to the Academy later that year, Pitot presented hisresults,whichhadimportancebeyondthePitottubeitself. This is not to diminish Bernoulli’s contributions to fluid Contemporarytheory,basedontheexperienceofsomeItal- dynamics. His work was used as a starting point by other ianengineers,heldthattheflowvelocityatagivendepthina investigatorsinthe18thcentury.Hewasthefirsttoexamine riverwasproportionaltothemassaboveit;hencethevelocity therelationbetweenpressureandvelocityinaflowusingthe was thought to increase with depth. Pitot reported the stun- new scientific principles of the 18th century. As far as this ning(andcorrect)results,measuredwithhisinstrument,that author can ascertain, he was the first to use the elements of inrealitytheflowvelocitydecreasedasthedepthincreased. calculustoanalyzeafluidflow,asillustratedinthedifferential Hence,Pitotintroducedhisnewinventionwithstyle.Later, equation shown above, obtained from his Hydrodynamica. in1740,heacceptedaninvitationfromtheEstatesGeneral Hisworkinspiredtheworkofotherinvestigators,including of Languedoc to supervise the draining of swamps in the thatofEuler,d’Alembert,andLagrange. province,whichthenledtohisbecomingdirectorofpublic worksoftheprovinceaswellassuperintendentoftheCanal duLanguedoc.Inhisoldage,Pitotretiredtohisbirthplace, 7 HENRI PITOT AND THE INVENTION anddiedatAramononDecember27,1771. OF THE PITOT TUBE ThedevelopmentofthePitottubein1732wasasubstan- tial contribution to experimental fluid dynamics. However, A major advancement in experimental fluid dynamics oc- in 1732, Henri Pitot did not have the benefit of Bernoulli’s curredonNovember12,1732,attheRoyalAcademyofSci- equation,whichwasobtainedbyEuler20yearslater.Pitot’s ences in Paris. On this day, Henri Pitot announced to the reasoningfortheoperationofthistubewaspurelyintuitive, Academy a new invention by which he could directly mea- and he was able to correlate by empirical means the flow sure the local flow velocity at a point in fluid. Later called velocity corresponding to the measured difference between EEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinngg,,OOnnlliinnee©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleeiiss©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleewwaassppuubblliisshheeddiinntthheeEEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinnggiinn22001100bbyyJJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. DDOOII:: 1100..11000022//99778800447700668866665522..eeaaee000011 10 FundamentalsofFluidFlows the stagnation pressure as measured by his Pitot tube, and theageof13,EulerenteredtheUniversityofBasel,whichat the flow static pressure, as measured by a straight tube in- that time had about 100 students and 19 professors. One of sertedverticallyinthefluid,withitsopenfacetubeparallel thoseprofessorswasJohannBernoulliwhotutoredEulerin totheflow.AsdiscussedbyAnderson(2008),theproperap- mathematics. Three years later, Euler received his master’s plicationofBernoulli’sequationtoextractthevelocityfrom degree in philosophy. It is interesting that three of the peo- the Pitot measurement of stagnation pressure was not pre- plemostresponsiblefortheearlydevelopmentoftheoretical senteduntil1913.Inthatyear,JohnAireyattheUniversity fluiddynamics–Johann,DanielBernoulli,andEuler–lived ofMichiganpublishedanexhaustiveexperimentalbehavior in the same town of Basel, were associated with the same of Pitot tubes, and presented a rational theory for their op- university, and were contemporaries. Indeed, Euler and the eration based on Bernoulli’s equation. Invented in the early Bernoulliswerecloseandrespectedfriends–somuchsothat partoftheeighteenthcentury,thePitottuberequiredtwocen- when Daniel Bernoulli moved to teach and study at the St. turiesbeforeitwasproperlyincorporatedintofluiddynamics Petersburg Academy in 1725, he was able to convince the asaviableexperimentaltool. academy to hire Euler as well. At this invitation, Euler left BaselforRussia,heneverreturnedtoSwitzerland,although heremainedaSwisscitizenthroughouthislife. Euler’sinteractionwithDanielBernoulliinthedevelop- 8 THE HIGH NOON OF EIGHTEENTH mentoffluidmechanicsgrewstrongduringtheseyearsatSt. CENTURY FLUID DYNAMICS – Petersburg.ItwasherethatEulerconceivedofpressureasa point property that can vary from point to point throughout LEONHARD EULER AND THE afluid,andobtainedadifferentialequationrelatingpressure GOVERNING EQUATIONS OF andvelocity.Inturn,Eulerintegratedthedifferentialequation INVISCID FLUID MOTION toobtain,forthefirsttimeinhistory,Bernoulli’sequationin theformweusetoday.HenceweseethatBernoulli’sequa- Today,inthemodernworldoftwenty-firstcenturyfluiddy- tionreallyisamisnomer;creditforitislegitimatelyshared namics,attheveryinstantthatyouarereadingthispage,there byEuler. areliterallythousandsoffluiddynamicistswhoaresolving When Daniel Bernoulli returned to Basel in 1733, Euler thegoverningequationsoffluidmotionforaninviscidflow. succeeded him at St. Petersburg as a professor of physics. Suchinviscidflows–flowswithoutfriction–adequatelyde- Eulerwasadynamicandprolificman;by1741hehadpre- scribe many aspects of practical fluid dynamic problems as pared90papersforpublicationandwrittenthetwo-volume longasfrictionisnotbeingconsidered.Thesesolutionsmay bookMechanica.TheatmospheresurroundingSt.Petersburg involveclosed-formtheoreticalmathematics,ormorelikely wasconducivetosuchachievement.Eulerwrotein1749,“I todaymayinvolvedirectnumericalsolutionsonahigh-speed andallotherswhohadthegoodfortunetobeforsometime digitalcomputer.However,thegoverningequationsthatare withtheRussianImperialAcademycannotbutacknowledge being solved in such a “high-tech” fashion are themselves thatweoweeverythingwhichweareandpossesstothefa- over200yearsold;theyarecalledtheEulerequations.The vorableconditionswhichwehadthere.” developmentoftheEulerequationsrepresentsacontribution However,in1740,politicalunrestinSt.Petersburgcaused tofluiddynamicsofamagnitudemuchgreaterthananyother EulertoleavefortheBerlinSocietyofSciences,atthattime wehavediscussedsofarinthischapter.Theyrepresent,for justformedbyFredericktheGreat.EulerlivedinBerlinfor allpracticalpurposes,thetruebeginningoftheoreticalfluid the next 25 years, where he transformed the society into a dynamics.TheseequationswerefirstdevelopedbyLeonhard majoracademy.InBerlin,Eulercontinuedhisdynamicmode Euler;forthisreasonEulerisfrequentlycreditedasbeingthe of working, preparing at least 380 papers for publication. “founderoffluidmechanics.”Thisissomewhatofanover- Hereasacompetitorwithd’Alembert,Eulerformulatedthe statement, because as is almost always the case in physical basisformathematicalphysics. science,Eulerbenefittedfromearlierwork,especiallythatof In 1766, after a major disagreement with Frederick the d’Alembert.Ontheotherhand,Eulerisagiantinthehistory Great over some financial aspects of the academy, Euler offluiddynamics,andhiscontributionsborderedontherevo- movedbacktoSt.Petersburg.Thesecondperiodofhislifein lutionaryratherthantheevolutionaryside.Forthesereasons, Russiabecameoneofphysicalsuffering.Inthatsameyear, letusfirsttakealookatEuler,theman. hebecameblindinoneeyeafterashortillness.Anoperation Leonhard Euler was born on 15 April 1707 in Basel, in1771resultedinrestorationofhissight,butonlyforafew Switzerland. His father was a Protestant minister who en- days.Hedidnottakeproperprecautionsaftertheoperation, joyedmathematicsasapastime.Therefore,Eulergrewupin and within a few days he was completely blind. However, afamilyatmospherethatencouragedintellectualactivity.At withthehelpofothers,hecontinuedhiswork.Hismindwas EEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinngg,,OOnnlliinnee©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleeiiss©©22001100JJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. TThhiissaarrttiicclleewwaassppuubblliisshheeddiinntthheeEEnnccyyccllooppeeddiiaaooffAAeerroossppaacceeEEnnggiinneeeerriinnggiinn22001100bbyyJJoohhnnWWiilleeyy&&SSoonnss,,LLttdd.. DDOOII:: 1100..11000022//99778800447700668866665522..eeaaee000011