LaminarandTurbulentFlowObservationshowsthattwoentirelydifferenttypesoffluidflowexist.Thiswasdemon-stratedbyOsborneReynoldsin1883throughanexperimentinwhichwaterwasdischargedfromatankthroughaglasstube.Therateofflowcouldbecontrolledbyavalveattheoutlet,andafinefilamentofdyeinjectedattheentrancetothetube.Atlowvelocities,itwasfoundthatthedyefilamentremainedintactthroughoutthelengthofthetube,showingthattheparticlesofwatermovedinparallellines.Thistypeofflowisknownaslaminar,viscousorstreamline,theparticlesoffluidmovinginanorderlymannerandretainingthesamerelativepositionsinsuccessivecross-sections.Asthevelocityinthetubewasincreasedbyopeningtheoutletvalve,apointwaseventuallyreachedatwhichthedyefilamentatfirstbegantooscillateandthenbrokeupsothatthecolourwasdiffusedoverthewholecross-section,showingthattheparticlesoffluidnolongermovedinanorderlymannerbutoccupieddifferentrelativepositioninsuccessivecross-sections.Thistypeofflowisknownasturbulentandischaracterizedbycontinuoussmallfluctuationsinthemagnitudeanddirectionofthevelocityofthefluidparticles,whichareaccompaniedbycorrespondingsmallfluctuationsofpressure.Whenthemotionofafluidparticleinastreamisdisturbed,itsinertiawilltendtocarryitoninthenewdirection,buttheviscousforcesduetothesurroundingfluidwilltendtomakeitconformtothemotionoftherestofthestream.Inviscousflow,theviscousshearstressesaresufficienttoeliminatetheeffectsofanydeviation,butinturbulentflowtheyareinadequate.Thecriterionwhichdetermineswhetherflowwillbeviscousofturbulentisthereforetheratiooftheinertialforcetotheviscousforceactingontheparticle.TheratiovlconstforceViscousforceInertialThus,thecriterionwhichdetermineswhetherflowisviscousorturbulentisthequantityρvl/μ,knownastheReynoldsnumber.Itisaratioofforcesand,therefore,apurenumberandmayalsobewrittenasul/vwhereisthekinematicviscosity(v=μ/ρ).ExperimentscarriedoutwithanumberofdifferentfluidsinstraightpipesofdifferentdiametershaveestablishedthatiftheReynoldsnumberiscalculatedbymaking1equaltothepipediameterandusingthemeanvelocityv,then,belowacriticalvalueofρvd/μ=2000,flowwillnormallybelaminar(viscous),anytendencytoturbulencebeingdampedoutbyviscousfriction.ThisvalueoftheReynoldsnumberappliesonlytoflowinpipes,butcriticalvaluesoftheReynoldsnumbercanbeestablishedforothertypesofflow,choosingasuitablecharacteristiclengthsuchasthechordofanaerofoilinplaceofthepipediameter.Foragivenfluidflowinginapipeofagivendiameter,therewillbeacriticalvelocityofflowcorrespondingtothecriticalvalueoftheReynoldsnumber,belowwhichflowwillbeviscous.Inpipes,atvaluesoftheReynoldsnumber2000,flowwillnotnecessarilybeturbulent.LaminarflowhasbeenmaintaineduptoRe=50,000,butconditionsareunstableandanydisturbancewillcausereversiontonormalturbulentflow.Instraightpipesofconstantdiameter,flowcanbeassumedtobeturbulentiftheReynoldsnumberexceeds4000.PipeNetworksAnextensionofcompoundpipesinparallelisacasefrequentlyencounteredinmunicipaldistributionsystem,inwhichthepipesareinterconnectedsothattheflowtoagivenoutletmaycomebyseveraldifferentpaths.Indeed,itisfrequentlyimpossibletotellbyinspectionwhichwaytheflowtravels.Nevertheless,theflowinanynetworks,howevercomplicated,mustsatisfythebasicrelationsofcontinuityandenergyasfollows:1.Theflowintoanyjunctionmustequaltheflowoutofit.2.Theflowineachpipemustsatisfythepipe-frictionlawsforflowinasinglepipe.3.Thealgebraicsumoftheheadlossesaroundanyclosedcircuitmustbezero.Pipenetworksaregenerallytoocomplicatedtosolveanalytically,aswaspossibleinthesimplercasesofparallelpipes.Apracticalprocedureisthemethodofsuccessiveapproximations,introducedbyCross.Itconsistsofthefollowingelements,inorder:1.Bycarefulinspectionassumethemostreasonabledistributionofflowsthatsatisfiescondition1.2.Writecondition2foreachpipeintheformhL=KQn(7.5)whereKisaconstantforeachpipe.Forexample,thestandardpipe-frictionequationwouldyieldK=1/C2andn=2forconstantf.Minorlosseswithinanycircuitmaybeincluded,butminorlossesatthejunctionpointsareneglected.3.Toinvestigatecondition3,computethealgebraicsumoftheheadlossesaroundeachelementarycircuit.∑hL=∑KQn.Considerlossesfromclockwiseflowsaspositive,counterclockwisenegative.Onlybygoodluckwilltheseaddtozeroonthefirsttrial.4.Adjusttheflowineachcircuitbyacorrection,ΔQ,tobalancetheheadinthatcircuitandgive∑KQn=0.TheheartofthismethodliesinthedeterminationofΔQ.ForanypipewemaywriteQ=Q0+ΔQwhereQisthecorrectdischargeandQ0istheassumeddischarge.Then,foracircuit0100/QhnhQKnQKQLLnn(7.6)ItmustbeemphasizedagainthatthenumeratorofEq.(7.6)istobesummedalgebraically,withdueaccountofsign,whilethedenominatorissummedarithmetically.ThenegativesigninEq.(7.6)indicatesthatwhenthereisanexcessofheadlossaroundaloopintheclockwisedirection,theΔQmustbesubtractedfromclockwiseQ0’sandaddedtocounterclockwiseones.Thereverseistrueifthereisadeficiencyofheadlossaroundaloopintheclockwisedirection.5.Aftereachcircuitisgivenafirstcorrection,thelosseswillstillnotbalancebecauseoftheinteractionofonecircuituponanother(pipeswhicharecommontotwocircuitsreceivetwoindependentcorrections,oneforeachcircuit).Theprocedureis