Encyclopedia of Aerospace Engineering

5810 Pages·2010·4.74 MB·English

by  Richard Blockley

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
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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,
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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,
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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
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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
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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
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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,
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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.
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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
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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
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