Intermittency as a transition to turbulence in pipes: A long tradition from Reynolds to the 21st century

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A century of ﬂuid mechanics: 1870–1970 / Un siècle de mécanique des ﬂuides : 1870–1970

Intermittency as a transition to turbulence in pipes: A long tradition from Reynolds to the 21st century

Les intermittencies comme transition vers la turbulence dans des tuyaux : Une longue tradition, de Reynolds au XXI

^e

siècle

Christophe Letellier

NormandieUniversité,CORIA,avenuedel’Université,76800Saint-Étienne-du-Rouvray,France

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received21October2016 Accepted3April2017 Availableonline3July2017 PresentedbyFrançoisCharru Keywords:

Pipeﬂows Laminarregime Turbulence Intermittency Frictioncoeﬃcient Mots-clés :

Écoulementdanslesconduitescylindriques Régimelaminaire

Turbulence Intermittences Coeﬃcientdefrottement

Intermittencies are commonly observed in fluid mechanics, and particularly, in pipe flows. Initially observed by Reynolds (1883), it took one century for reaching a rather fullunderstandingofthisphenomenonwhoseirregulardynamics(apparentlystochastic) puzzledhydrodynamicistsfordecades.Inthisbrief(non-exhaustive)review,mostlyfocused on the experimental characterization of this transition between laminar and turbulent regimes, we present some key contributions for evidencing the two concomittant and antagonistprocessesthatare involvedinthiscomplextransitionand weresuggestedby Reynolds. It is also shown that a clear explicative model was provided, based on the nonlineardynamicalsystemstheory,theexperimentalobservationsinfluidmechanicsonly providinganappliedexample,duetoitsobviousgenericnature.

©²⁰¹⁷Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

ré s u m é

Les intermittencies sont communément observées en mécanique des ﬂuides et, plus particulièrement, dans les écoulements dans des conduites cylindriques. Initialement obervéespar Reynoldsen 1883, il afallu unsièclepour parveniràunecompréhension plutôtcomplètedecephénomènedontladynamiqueirrégulière(apparemmentstochas- tique) déconcertaleshydrodynamiciensdurant plusieurs décades. Parcette brèverevue (nonexhaustive), essentiellementfocalisée surlacaractérisation expérimentale de cette transition entrerégimes laminaire etturbulent,nous présentonsquelques contributions clésayantconduitàmettreenévidencelesdeuxprocessusconcomittantsetantagonistes impliquésetquiavaientdéjà étésuggérés par Reynolds. Ilestégalement montréqu’un modèleexplicatifclairfutproposé,surlabasedelathéoriedessystèmesdynamiquesnon linéaires,lesobservationsexpérimentalesenmécaniquedesﬂuidesayantserviuniquement d’exemple,etceenraisondesoncaractèregénériqueévident.

©²⁰¹⁷Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

E-mailaddress:Christophe.Letellier@coria.fr.

http://dx.doi.org/10.1016/j.crme.2017.06.004

1631-0721/©²⁰¹⁷Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccessârticleûnder^the^CC^BY-NC-ND^license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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In thispaper,I willshow that the transitionto turbulencevia intermittencyfollows moreor lessthisexample.First, thereisalongtraditionininvestigatingtheroutetoturbulence,andintermittencieswerealreadyobservedintheearlyex- perimentsmadebyReynolds.Roughlyonemaysaythatacenturywasneededforclarifyingthistransition(see,forinstance, Barkley’sreview[16]).Second,inthe1980s,anotherapproachtothisproblemcamefromnumericalexperimentsandpro- videdsimpleandgeneric modelstodescribe such transition.Ifit contributedto popularizedtheidea ofintermittencyas a route toturbulencein ﬂuid mechanics,thistheoretical explanationhad amuch moresigniﬁcant impact inthenonlin- ear dynamical systemstheory andopened ourminds to thepossibility tohave intermittentbehaviorsinwell-conducted experiments,thatis,withoutanyexternalperturbations.

2. Thehistoricalapproach 2.1. Earlyexperimentalevidences

HenryDarcy(1803–1858),preoccupiedbytheengineeringofwatersupply,investigatedindetailsthelawprovidingthe headloss hper unit oflength ina tubeof diameterd.Darcystartedfroma generallawforthehead lossh per unit of lengthmadeoftheﬁrsttwopowerofthemeanvelocity V,thatis,from

dh

4 =A V +B V² (1)

Heclearlyunderstoodthatthesecondterm,B V²,wassufficienttoexplaintheflowwithlargevelocities(thuscorresponding to the turbulentregime inour presentterminology). This“turbulent” component was forhim themost importantsince directlyrelatedtotheroughnessofpipes.DarcyunderconsideredthecaseswherethetermA V wasnegligible,thatis,when thevelocity was small(lessthan 0.10 m s⁻¹) orwhenthe roughnessofpipeswasimportantasmostly encountered ina workingsystemforwatersupply, pipesare “very quicklycoveredby slits, tubersor limescale”[17,p. 91].Consequently, Darcy mostly focused his attention on the turbulent regime, arguing that “the experiments for which the slope or the velocitywas smallare ingeneralthelessaccurate onesand, contrarytothis, thoseforwhichslopeandvelocityarelarge providemorereliableresults.”Nevertheless,Darcyremarkedthatthereisathresholdvelocitybeyondwhich“removingthe firstterm(A V)doesnotaffectthevalueoftheflow”[17,p. 120].Heevenaddedthat“thefirsttermwasthereforerelevant onlyforvelocities(V<0.10 m s⁻¹)andforpipeswithsufficientlysmoothwalls”[17,p. 120]:inthiscase,thesecondterm canberemoved.Darcymentionedthatforintermediatevelocities,thefrictionofwateragainstwallsbecomesproportional to a binomial made ofthe first and the second power of the velocity” [17, p. 121], thus suggesting that there are two concomittantunderlying phenomena. He was convinced that beyond a velocity equal to 0.10 m s⁻¹,“a new phenomena occurs”[17,p. 214].Whenhekeptthetwoterms,Darcyrecommendedtouse

⎧⎪

⎨

⎪⎩

A=0.000 031 655+⁰.000 000 007 5112 D

B=⁰.000 442 939+⁰.000 012 402 D

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In1854,GotthilfHagen(1797–1884)investigatedtheinfluenceoftemperatureonwaterflowinpipes[18].Inorderto dothis,Hagenuseddifferentpipesandvariedthetemperatureofwater:indoingso,hewasinfactvaryingveryslowlythe propertiesofthewaterflow.Hethusremarkedthat,forcertainconditions(pressureanddiameterofthepipe,forinstance), thewaterflow, whenplottedversus thetemperature,was presentingamaximumfollowedbya minimum(Fig. 1).Hagen observed the evolution ofthe waterflow by looking atthe jet at the end of the tube. He remarked that betweenthe maximumandtheminimumvelocity(17<T<30^◦C)inthe caseofFig. 1,thejet wasjerking. Initially, hebelievedthat sucha featurewasduetoabadlyconductedexperiment, butfinally convincedhimselfthatthisphenomenonwasinfact

“normal”. He also understood that this was a transition from one regime to another one. Since he developed a theory

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Fig. 1.Watervelocityinapipe(d=⁰.28 cm,l=⁴⁷.2 cm)withaloadof11.08waterinch(2934.7 PawhentheoldFrenchinchistakenequalto27 mm) whenthetemperatureisvaried.ThevaluesforthevelocitywhereassessedwiththehelpofPoiseuille’slaw.RedrawnfromHagen,1854.

Fig. 2.“Theﬂasheswouldoftencommencesuccessivelyatonepointinthepipe.Theappearancewhentheﬂashessucceededeachotherrapidly[ishere]

shown.”FromReynolds,Plate72,Fig. 16,1883.

basedonthehypothesisthatthevelocitywasproportionaltothedistancefromthewall,hewas notabletoprovideafull agreementbetweenhistheoreticalrulesandobservationaldata.

Osborne Reynolds(1842–1912) ismostly knownforhaving evidencedthat there aretwo very differenttypesofﬂow, namelythelaminarandturbulentﬂows[19,p. 935].Heunderstoodthat

thegeneralcharacterofthemotionoffluidsincontactwithsolidsurfacesdependsontherelationbetweenaphysicalconstantof thefluidandtheproductofthelineardimensionsofthespaceoccupiedbythefluidandthevelocity.

In ordertodistinguishlaminarfromturbulent flows,hethusderived a dimensionlessnumber,the“Reynoldsnumber”as namedbyArnoldSommerfeld[20].ItismorerarelyknownthatReynoldsalreadydescribed“theintermittentcharacterofthe disturbance[...]givingtheappearanceofflashes”(Fig. 2).ForReynolds,theconceptofthecriticalReynoldsnumberwasclear, inspiteoftheexperimentaldifficultiesencounteredformanagingtheinitialconditionsoftheflow[19,p. 955]:

TheonlyideathatIhadformedbeforecommencingtheexperimentswasthatatsomecriticalvelocitythemotionmustbecome unstable,sothatanydisturbancefromperfectlysteadymotionwouldresultineddies.

[...]Ihadnotbeenabletoformanyideaastoanyparticularformofdisturbancebeingnecessary.Butexperiencehavingshown theimpossibilityofobtainingabsolutelysteadymotion,Ihadnotdoubtedbutthatappearanceofeddieswouldbealmost simultaneouswiththeconditionofinstability. Ihadnot,therefore,consideredthedisturbancesexcepttotryanddiminish themasmuchaspossible.Ihadexpectedtoseetheeddiesmaketheirappearanceasthevelocityincreased,atﬁrstinaslowor feeblemanner,indicatingthatthewaterwasbutslightlyunstable.Anditwasamatterofsurprisetometoseethesuddenforce withwhichtheeddiessprangintoexistence,showingahighlyunstableconditiontohaveexistedatthetimethesteadymotion brokedown.

Thus, in1895,heproposed thattheﬂow inround tubeswas stablefor ρV d

μ <1900 and unstablewhenitisgreater than 2000 [21].But theresearch fora “criticalReynoldsnumber” characterizingthe transitionfromlaminarto turbulentﬂow wasmadediﬃcultbythedependencesofthetransitiononinitialdisturbances.

Maurice Couette(1858–1943) investigatedwhetherthe twotermsinDarcy’s lawwere duetodifferentphenomena or not.IfhediscoveredReynolds’resultsjustbeforethepublicationofhispaper(hethankBoussinesqforhavingshowedhim Reynolds’results),thereisnomentionofHagen’sworks.Couetteusedthreedifferentexperimentstocheckthatthereisa discontinuitybetweenthetwo(laminarandturbulent)regimes[22]:i)hemeasured thefrictionappliedtothewallsofa cylinder bya fluidentrained by therotationofa secondcoaxialcylinder, ii)hemeasured theheadloss producedby the flow ofa liquidthrough a pipeandiii) heobserved thejet comingout froma long tube.He showedthat foraflow Q lessthanathresholdflow Q0,thejet wasalwayssmooth;contrarytothis,whentheflowwasgreater than Q1>Q0,the jet was alwaysrough. Whenthe flowwas such Q0<Q <Q1,heobservedajet with“suddenvariations,withoutapparent regularity,initsshapeaswellasinitsamplitude,thelatterbeingalwayslongerforthesmoothjetthanfortheroughjet.”

Inhisbookpublishedin1907,MarcelBrillouin(1854–1948)madeaquitecomprehensivedescriptionofHagen’sobser- vations [23]: whenthe temperatureisincreased, theviscosity decreases,andirregularmotions areno longer“suﬃciently damped”forallowingwatermotionstoremainalmostrectilinear.Brillouinused“Poiseuilleregime”fordesignatingthelam- inar regime and“hydraulicregime”fortheturbulent regime.We systematicallyreplaced histerminologyby the“modern”

one betweenbrackets.Brillouinalsodistinguished “ondulatorymotion”fromturbulentmotion,adistinctionwhichwasnot later followed: wewill thereforeuseturbulent motion.So, forBrillouin,turbulent motion“appearintheliquid,amplifythe

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Fig. 3.AcapillarytubeBCisadaptedhorizontallytoameasuringtubeAﬁlledwithmercury.TheendBofthecapillarytubehasafunnelshapetoobtain moreeasilyalaminarregime.ThejetﬂowingoutinCisobservedwhiletheheightHofmercuryisdecreasing.

Table 1

ObservationsmadebyBrillouinwithmercuryﬂowinginacapillarytube(l=18.9 cmandd=1.0 mm).TheReynoldsnumbers(notprovidedbyBrillouin) wereherecomputedaccordingtoPoiseuille’slaw.

Mercury height Regime Reynolds number

H>160 mm [Turbulent]regime Re>11131

H=^{136 mm} ^Smalloscillations Re=⁹⁴⁶¹

H=^{126 mm} ^Largeoscillations Re=⁸⁷⁶⁶

73.9<H<122 mm Hugeoscillations 5141<Re<8487

H<73.9 mm [Laminar]regimedominateswithfewsuddenlossinamplitude(sudden returnstoturbulentregime)

Re=⁸⁷⁶⁶ H=68.1 mm Endoftheblurredregime.Beginningofthestationarylaminarregime. Re=4737 inequalitiesinvelocityanddampproportionallymoreenergythan[laminar]regime,foragivenflow”.Brillouinaddedthat“when thetemperatureisstillincreased, [turbulence]isfirst,moreandmoreimportant,andthenbecomesstationary;theregimeismore regularandtheflowincreasesagain.”Brillouinunderstoodthatwhatisobservedatagivenpressurewhenthetemperatureis variedistheimageofwhatoccurswhenthetemperatureiskeptconstantandthepressureisprogressivelyincreased.

Inordertoevidencethisphasewherethelaminarregimebecomes turbulent,Brillouinuseda verysimpleexperiment (Fig. 3)hepresentedinhislectureon4February1899.Brillouindescribeshisobservationsasfollows[23]:

Atfirst,theflowproducedunderahighpressure,andthe[turbulent] regimeisobserved:thejetflowingoutisregularbutnot smooth.Sincemercuryisflowing,thepressuredecreases;atagiventime,the“blurredphase”starts;the[laminar]regime(smooth jet)startstooccurbyintermittencies;eachtimeitappears,thejetislonger,showingthattheflowincreasesandthattheresistance decreases;butthe[turbulent]regimerestartsquickly,thejetisoncemoretimesuddenlyblurredand,simultaneously,thejetis morecurvedanditsamplitudedecreases.

Thejetthusjerks,fromoneamplitudetotheother.First,theseoscillationsarerare,thesmallestamplitudebeingthemost observed;the[turbulent]regimedominates.Then,theheightyetdecreasing,theoscillationsaremorefrequent,beforebeingrare again,butthelargeamplitudebeingnowthemostoftenobserved;the[laminar]regimedominates.Finally,belowacertainvalue ofthedrivingpressure,thesole[laminar]regimeremains,thejetissmoothandtheoscillationsdisappeared.

Withagiventube(l=¹⁸.9 cmandd=¹.0 mm,Brillouinprovidedadetaileddescriptionofhisobservations(Table 1).

Herepeatedthisexperimentwithothertubesandwithwater,andgotsimilarobservations.Hewasthus abletoconclude that “foragivenglasstube,thetransitionfromoneregimetotheotherdoesnotoccursuddenly,atathresholdvelocity,butthere isa blurred phaseduringwhichthetworegimesarepossibleandalternatewithamoreorlesslargefrequency”.Intermittencies were thus alreadyclearly identiﬁed in1907! It seems that theseworks by Hagen,Couette andBrillouin remainedquite unknown...

2.2. Thefrictioncoeﬃcients

In his book, Julius Weisbach (1806–1871) proposed a lawfor the head loss h per unit of length as [24] (quoted by Flamant[25,p. 143])

dh 4 =

a+√^b

V

V² (3)

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where d is the pipe diameterand V the mean velocity of thewater. The parameter values are a=⁰.0007336 and b= 0.0004828.Fewyearslater,inhisexperimentalresearchesonwaterﬂowinpipes,HenriDarcy(1803–1858)expressedthe headlosshperunitoflengthas[17,p. 14](aswrittenbyFlamant[25,p. 142])

dh 4 =

α+β

d

V² (4)

wheretheparametervaluesare α=⁰.000507 andβ=⁰.00001294[17].Tothisbackground,Jean–Louis–MariePoiseuille’s (1797–1869)contribution[26]shouldbeadded.Heobtainedaresistanceproportionaltothevelocity[26]undertheform

Q=^k^{P d}⁴

l (5)

wheredisthepipe’sdiameter,litslengthandk aconstantcoeﬃcientforthesametemperatureandthesamegravitational acceleration.Poiseuilledesignatedby Q whathenamedthe“productcorrespondingtopressures”andwhatisnamedtoday theﬂow.Reynoldsrewroteitas[19,p. 972]

α ^d

3

μ²^h⁼^β

d

μ^V ⁽⁶⁾

whereweexpressedthepressurehintermsofheightofwaterperunitofpipelength.Introducingexplicitlythepressure lossP andthelengthl ofthepipe,Reynolds’equation(6)canberewrittenas

P=γμl

d² V (7)

that is,undertheformproposedby EduardHagenbach-Bischoff(1833–1910)who usedthemeanvelocity V andnot the ﬂow Q [27]: the resistance to the ﬂow is thus proportional to the velocity. A law similar to Poiseuille’s one was also obtained by Hagen [28]. Hagenbach-Bischoff, who was aware of these two different contributions, choose to designate equation(7)asPoiseuille’slaw.

Reynoldswasthusawarethat[19,p. 995]

therelationsbetweentheresistanceencounteredby,andthevelocityof,asolidbodymovingsteadilythroughaﬂuidinwhich itiscompletelyimmersed,orofwatermovingthroughatube,presentthemselvesmostlyinoneorotheroftwosimpleforms.

Theresistanceisgenerallyproportionaltothesquareofthevelocity,andwhenthisisnotthecaseittakesasimplerformandis proportionaltothevelocity.

Nevertheless,hisexperimentsconvincedhimthatratherthanusingthelawh∝^V² ôr^h∝âV +^bV²âs“propoundedbyany ofthepreviousexperimenters”,nootherlawthanh∝^V¹^.⁷²³^should^beûsed^[19,^{p. 975],}â^law^that^he^designatedâs^the^“law ofpressures”.Reynoldsthusproposedalawforthepressurelossas

α^d

3

μ²^h⁼ βd V

μ 1.723

(8) which ishererewrittenusingour notations,h beingthepressure loss(expressed asaheight ofwater)betweenthetwo endsofoneunitoflengthoftheconsideredpipe, μbeingthedynamicviscosity.Formakingsimplerthecomparisonswith laterresultsthatwewilldiscussbelow,letusrewritethisexpressionas

h=⁰.00078V¹^.⁷²³

d¹^.²⁷⁷ (9)

wherethenumericalcoeﬃcientisobtainedfromthenumericalvaluesprovidedbyReynolds.

Slightlylater,AlfredFlamant(1839–1915)madeadetailedreviewofthedifferentlawsproposedfortheresistancehead inpipesandproposedtouse[25,p. 149]

dh=0.00092 ⁴

V⁷

d ⇔ h=0.00092V¹^.⁷⁵

d¹^.²⁵ (10)

whichisthereforenottoodifferentfromequation(9)proposedbyReynolds(whenrewritteninasimilarform).

In1901,AugustusV.SaphandErnestW.Schoderstartedajointdoctoralthesistoperform—betweenFebruary24,1902, andFebruary25,1903—approximately800experiments,testing23kindsofpipeandhoses,coveringa200-foldrangeof velocities[29].Theyobtainedfortheheadloss[30]

h=⁰.00054V¹^.⁷⁵

d¹^.²⁵ (11)