reynold number.docx

reynold number.docx

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reynoldnumber

Reynoldsnumber

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Avortexstreetaroundacylinder.Thisoccursaroundcylinders,independentlyofthefluid,thecylindersizeandthefluidspeed,providedthatthereisaReynoldsnumberofbetween250and200,000.Picturecourtesy,CesareodeLaRosaSiqueira.

Influidmechanics,theReynoldsnumberReisadimensionlessnumberthatgivesameasureoftheratioofinertialforcestoviscousforcesandconsequentlyquantifiestherelativeimportanceofthesetwotypesofforcesforgivenflowconditions.

TheconceptwasintroducedbyGeorgeGabrielStokesin1851,[1]buttheReynoldsnumberisnamedafterOsborneReynolds（1842–1912）,whopopularizeditsusein1883.[2][3]

Reynoldsnumbersfrequentlyarisewhenperformingdimensionalanalysisoffluiddynamicsproblems,andassuchcanbeusedtodeterminedynamicsimilitudebetweendifferentexperimentalcases.

Theyarealsousedtocharacterizedifferentflowregimes,suchaslaminarorturbulentflow:

laminarflowoccursatlowReynoldsnumbers,whereviscousforcesaredominant,andischaracterizedbysmooth,constantfluidmotion;turbulentflowoccursathighReynoldsnumbersandisdominatedbyinertialforces,whichtendtoproducechaoticeddies,vorticesandotherflowinstabilities.

Contents

[hide]

∙1Definition

o1.1Significance

o1.2FlowinPipe

o1.3Flowinanon-circularduct（annulus）

o1.4FlowinaWideDuct

o1.5FlowinanOpenChannel

o1.6Objectinafluid

▪1.6.1Sphereinafluid

▪1.6.2Oblongobjectinafluid

▪1.6.3Fallvelocity

o1.7PackedBed

o1.8StirredVessel

∙2TransitionReynoldsnumber

∙3Reynoldsnumberinpipefriction

∙4Thesimilarityofflows

∙5Reynoldsnumbersetsthesmallestscalesofturbulentmotion

∙6ExampleoftheimportanceoftheReynoldsnumber

∙7Reynoldsnumberinphysiology

∙8Reynoldsnumberinviscousfluids

∙9Derivation

∙10Seealso

∙11Referencesandnotes

o11.1Furtherreading

∙12Externallinks

[edit]Definition

Reynoldsnumbercanbedefinedforanumberofdifferentsituationswhereafluidisinrelativemotiontoasurface（thedefinitionoftheReynoldsnumberisnottobeconfusedwiththeReynoldsEquationorlubricationequation）.Thesedefinitionsgenerallyincludethefluidpropertiesofdensityandviscosity,plusavelocityandacharacteristiclengthorcharacteristicdimension.Thisdimensionisamatterofconvention–forexamplearadiusordiameterareequallyvalidforspheresorcircles,butoneischosenbyconvention.Foraircraftorships,thelengthorwidthcanbeused.Forflowinapipeoraspheremovinginafluidtheinternaldiameterisgenerallyusedtoday.Othershapes（suchasrectangularpipesornon-sphericalobjects）haveanequivalentdiameterdefined.Forfluidsofvariabledensity（e.g.compressiblegases）orvariableviscosity（non-Newtonianfluids）specialrulesapply.Thevelocitymayalsobeamatterofconventioninsomecircumstances,notablystirredvessels.

[4]

where:

∙

isthemeanvelocity,

oftheobjectrelativetothefluid（SIunits:

m/s）

∙

isacharacteristiclineardimension（travelledlengthofthefluid;hydraulicdiameterwhendealingwithriversystems）（m）

∙

isthedynamicviscosityofthefluid（Pa·sorN·s/m²orkg/（m·s））

∙

isthekinematicviscosity（ν=μ/ρ）（m²/s）

∙

isthedensityofthefluid（kg/m³）

NotethatmultiplyingtheReynoldsnumber,

by

yields

whichistheratio,

.[5]

[edit]Significance

[edit]FlowinPipe

Forflowinapipeortube,theReynoldsnumberisgenerallydefinedas:

[6]

where:

∙

isthehydraulicdiameterofthepipe;itscharacteristiclength,

（m）.

∙

isthevolumetricflowrate（m³/s）.

∙

isthepipecross-sectionalarea（m²）.

∙

isthemeanvelocity,

oftheobjectrelativetothefluid（m/s）

∙

isthedynamicviscosityofthefluid（Pa·sorN·s/m²orkg/（m·s））.

∙

isthekinematicviscosity（ν=μ/ρ）（m²/s）.

∙

isthedensityofthefluid（kg/m³）.

[edit]Flowinanon-circularduct（annulus）

Forshapessuchassquares,rectangularorannularducts（wheretheheightandwidtharecomparable）thecharacteristicdimensionforinternalflowsituationsistakentobethehydraulicdiameter,DH,definedas4timesthecross-sectionalarea（ofthefluid）,dividedbythewettedperimeter.Thewettedperimeterforachannelisthetotalperimeterofallchannelwallsthatareincontactwiththeflow.[7]ThismeansthelengthofthewaterexposedtoairisNOTincludedinthewettedperimeter

Foracircularpipe,thehydraulicdiameterisexactlyequaltotheinsidepipediameter,ascanbeshownmathematically.

Foranannularduct,suchastheouterchannelinatube-in-tubeheatexchanger,thehydraulicdiametercanbeshownalgebraicallytoreduceto

DH,annulus=Do−Di

where

Doistheoutsidediameteroftheoutsidepipe,and

Diistheinsidediameteroftheinsidepipe.

Forcalculationsinvolvingflowinnon-circularducts,thehydraulicdiametercanbesubstitutedforthediameterofacircularduct,withreasonableaccuracy.

[edit]FlowinaWideDuct

Forafluidmovingbetweentwoplaneparallelsurfaces（wherethewidthismuchgreaterthanthespacebetweentheplates）thenthecharacteristicdimensionistwicethedistancebetweentheplates.[8]

[edit]FlowinanOpenChannel

Forflowofliquidwithafreesurface,thehydraulicradiusmustbedetermined.Thisisthecross-sectionalareaofthechanneldividedbythewettedperimeter.Forasemi-circularchannel,itishalftheradius.Forarectangularchannel,thehydraulicradiusisthecross-sectionalareadividedbythewettedperimeter.Sometextsthenuseacharacteristicdimensionthatis4timesthehydraulicradius（chosenbecauseitgivesthesamevalueofRefortheonsetofturbulenceasinpipeflow）,[9]whileothersusethehydraulicradiusasthecharacteristiclength-scalewithconsequentlydifferentvaluesofRefortransitionandturbulentflow.

[edit]Objectinafluid

TheReynoldsnumberforanobjectinafluid,calledtheparticleReynoldsnumberandoftendenotedRep,isimportantwhenconsideringthenatureofflowaroundthatgrain,whetherornotvortexsheddingwilloccur,anditsfallvelocity.

[edit]Sphereinafluid

Forasphereinafluid,thecharacteristiclength-scaleisthediameterofthesphereandthecharacteristicvelocityisthatofthesphererelativetothefluidsomedistanceawayfromthesphere（suchthatthemotionofthespheredoesnotdisturbthatreferenceparceloffluid）.Thedensityandviscosityarethosebelongingtothefluid.[10]NotethatpurelylaminarflowonlyexistsuptoRe=0.1underthisdefinition.

UndertheconditionoflowRe,therelationshipbetweenforceandspeedofmotionisgivenbyStokes'law.[11]

[edit]Oblongobjectinafluid

Theequationforanoblongobjectisidenticaltothatofasphere,withtheobjectbeingapproximatedasanellipsoidandtheaxisoflengthbeingchosenasthecharacteristiclengthscale.Suchconsiderationsareimportantinnaturalstreams,forexample,wheretherearefewperfectlysphericalgrains.Forgrainsinwhichmeasurementofeachaxisisimpractical（e.g.,becausetheyaretoosmall）,sievediametersareusedinsteadasthecharacteristicparticlelength-scale.BothapproximationsalterthevaluesofthecriticalReynoldsnumber.

[edit]Fallvelocity

TheparticleReynoldsnumberisimportantindeterminingthefallvelocityofaparticle.WhentheparticleReynoldsnumberindicateslaminarflow,Stokes'lawcanbeusedtocalculateitsfallvelocity.WhentheparticleReynoldsnumberindicatesturbulentflow,aturbulentdraglawmustbeconstructedtomodeltheappropriatesettlingvelocity.

[edit]PackedBed

ForflowoffluidthroughabedofapproximatelysphericalparticlesofdiameterDincontact,ifthevoidage（fractionofthebednotfilledwithparticles）isεandthesuperficialvelocityV（i.e.thevelocitythroughthebedasiftheparticleswerenotthere-theactualvelocitywillbehigher）thenaReynoldsnumbercanbedefinedas:

LaminarconditionsapplyuptoRe=10,fullyturbulentfrom2000.[10]

[edit]StirredVessel

Inacylindricalvesselstirredbyacentralrotatingpaddle,turbineorpropellor,thecharacteristicdimensionisthediameteroftheagitatorD.ThevelocityisNDwhereNistherotationalspeed（revolutionspersecond）.ThentheReynoldsnumberis:

ThesystemisfullyturbulentforvaluesofReabove10000.[12]

[edit]TransitionReynoldsnumber

[citationneeded]Inboundarylayerflowoveraflatplate,experimentscanconfirmthat,afteracertainlengthofflow,alaminarboundarylayerwillbecomeunstableandbecometurbulent.Thisinstabilityoccursacrossdifferentscalesandwithdifferentfluids,usuallywhen

wherexisthedistancefromtheleadingedgeoftheflatplate,andtheflowvelocityisthefreestreamvelocityofthefluidoutsidetheboundarylayer.

ForflowinapipeofdiameterD,experimentalobservationsshowthatfor'fullydeveloped'flow（Note:

[13]）,laminarflowoccurswhenReD<2000andturbulentflowoccurswhenReD>4000.[14]Intheintervalbetween2300and4000,laminarandturbulentflowsarepossible（'transition'flows）,dependingonotherfactors,suchaspiperoughnessandflowuniformity）.Thisresultisgeneralisedtonon-circularchannelsusingthehydraulicdiameter,allowingatransitionReynoldsnumbertobecalculatedforothershapesofchannel.

ThesetransitionReynoldsnumbersarealsocalledcritical