1 Introduction Denition the physical hydraulic model A physical model is a scaled representation of a hydraulic 57519ow situation Both the boundary conditions eg channel bed sidewalls the upstream 57519ow conditions and the 57519ow eld m ID: 26452
Download Pdf The PPT/PDF document "Physical modelling of hydraulics" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Physicalmodellingof14.1IntroductionDe®nition:thephysicalhydraulicmodelAphysicalmodelisascaledrepresentationofahydraulic¯owsituation.Boththeboundaryconditions(e.g.channelbed,sidewalls),theupstream¯owconditionsandthe¯ow®eldmustbescaledinanappropriatemanner(Fig.14.1).Physicalhydraulicmodelsarecommonlyusedduringdesignstagestooptimizestructureandtoensureasafeoperationofthestructure.Theyhaveanimportantfurtherroletoassistnon-engineeringpeopleduringthe`decision-making'process.Ahydraulicmodelmayhelpthedecision-makerstovisualizeandtopicturethe¯ow®eld,beforeselectinga`suitable'design.Incivilengineeringapplications,aphysicalhydraulicmodelisusuallyasmaller-sizerepresentationoftheprototype(i.e.thefull-scalestructure)(e.g.Fig.14.2).Otherapplicationsofmodelstudies(e.g.watertreatmentplant,¯otationcolumn)mayrequiretheuseofmodelslargerthantheprototype.Inanycasethemodelisinvestigatedinalaboratoryundercontrolledconditions.DiscussionHydraulicmodellingcannotbedisassociatedfromthebasictheoryof¯uidmechanics.Tobeecientanduseful,experimentalinvestigationsrequiretheoreticalguidancewhichderivesprimarilyfromthebasicprinciples(seeChapter13)andthetheoryofsimilarity(seethenextsubsection).Inthepresentsection,wewillconsiderthephysicalmodellingofhydraulic¯ows:i.e.theuseoflaboratorymodels(withcontrolled¯owconditions)topredictthebehaviourofprototype¯owsituations.14.2BasicprinciplesInaphysicalmodel,the¯owconditionsaresaidtobesimilartothoseintheprototypeifthemodeldisplayssimilarityofform(geometricsimilarity),similarityofmotionkinematicsimilarity)andsimilarityofforces(dynamicsimilarityTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. BasicscaleratiosGeometricsimilarityimpliesthattheratiosofprototypecharacteristiclengthstomodellengthsareequal: lplmˆ dpdmˆ wherethesubscriptspandmrefertoprototype(full-scale)andmodelparametersrespectively,andthesubscriptrindicatestheratioofprototype-to-modelquantity.Length,areaandvolumearetheparametersinvolvedingeometricsimilitude.Kinematicsimilarityimpliesthattheratiosofprototypecharacteristicvelocitiestomodelvelocitiesarethesame: VpVmˆ …V1†p…V1†mˆ VelocityDynamicsimilarityimpliesthattheratiosofprototypeforcestomodelforcesare F1pF1mˆ ForceWorkandpowerareotherparametersinvolvedindynamicsimilitude.1.Geometricsimilarityisnotenoughtoensurethatthe¯owpatternsaresimilarinbothmodelandprototype(i.e.kinematicsimilarity).2.Thecombinedgeometricandkinematicsimilaritiesgivetheprototype-to-modelratiosoftime,acceleration,discharge,angularvelocity. Fig.14.1Basic¯owparameters.PhysicalmodellingofhydraulicsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. Fig.14.2Exampleofphysicalmodel:breakwaterjettyfortheChanghuaReclamationarea,alongthenorth-westcoastlineofTaiwan,RepublicofChina(January1994).(a)Prototypebreakwaterjetty.(b)Modelbreak-waterjettyinawave¯ume(TainanHydraulicLaboratory).14.2BasicprinciplesTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. SubsequentscaleratiosThequantities(de®nedinequations(14.1)to(14.3))arethebasicscaleratios.Severalscaleratioscanbededucedfromequations(14.1)to(14.3):Mass LrVrTime…14:5†QrˆVrL2rDischarge…14:6†Prˆ Pressureisthe¯uiddensity.Furtherscaleratiosmaybededucedinparticular¯owsituations.ApplicationInopenchannel¯ows,thepresenceofthefree-surfacemeansthatgravitye€ectsareimportant.TheFroudenumberisalwayssigni®cant.Second-aryscaleratioscanbederivedfromtheconstancyoftheFroudenumberwhichimplies:VelocityOtherscaleratiosarederivedfromtheFroudesimilarity(e.g.Henderson1966): MrLrT2rˆrL3rForcePrˆ Pressure14.3Dimensionalanalysis14.3.1BasicparametersThebasicrelevantparametersneededforanydimensionalanalysis(Fig.14.1)maybegroupedintothefollowinggroups.(a)Fluidpropertiesandphysicalconstants(seeAppendixA1.1).Theseconsistofthedensityofwater),thedynamicviscosityofwater(Ns/m),thesurfacetensionofairandwater(N/m),thebulkmodulusofelasticityofwaterandtheaccelerationofgravityItisassumedthatthegravityaccelerationisidenticalinboththemodelandtheprototype.PhysicalmodellingofhydraulicsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. (b)Channel(or¯ow)geometry.Thesemayconsistofthecharacteristiclength(s)(m).(c)Flowproperties.Theseconsistofthevelocity(ies)(m/s)andthepressuredi€erence(s)14.3.2DimensionalanalysisTakingintoaccountallbasicparameters,dimensionalanalysisyields:;;;Thereareeightbasicparametersandthedimensionsofthesecanbegroupedintothreecategories:mass(M),length(L)andtime(T).TheBuckingham(Buckingham1915)impliesthatthequantitiescanbegroupedinto®ve(5independentdimensionlessparameters: V2P; VL; Lr; The®rstratioistheFroudenumber,characterizingtheratiooftheinertialforcetogravityforce.istheEulernumber,proportionaltotheratioofinertialforcetopressureforce.ThethirddimensionlessparameteristheReynoldsnumberwhichcharacterizestheratioofinertialforcetoviscousforce.TheWebernumberisproportionaltotheratioofinertialforcetocapillaryforce(i.e.surfacetension).ThelastparameteristheSarrau±Machnumber,characterizingtheratioofinertialforcetoelasticityforce.1.TheFroudenumberisusedgenerallyforscalingfreesurface¯ows,openchannelsandhydraulicstructures.AlthoughthedimensionlessnumberwasnamedafterWilliamFroude(1810±1879),severalFrenchresearchersuseditbefore:e.g.Belanger(1828),Dupuit(1848),Bresse(1860),Bazin(1865a).FerdinandReech(1805±1880)introducedthedimensionlessnumberfortest-ingshipsandpropellersin1852,andthenumbershouldreallybecalledtheReech±Froudenumber.2.LeonhardEuler(1707±1783)wasaSwissmathematicianandphysicist,andaclosefriendofDanielBernoulli.3.OsborneReynolds(1842±1912)wasaBritishphysicistandmathematicianwhoexpressed®rstthe`Reynoldsnumber'(Reynolds1883).4.TheWebernumbercharacterizingtheratioofinertialforceoversurfacetensionforcewasnamedafterMoritzWeber(1871±1951),GermanProfessoratthePolytechnicInstituteofBerlin.5.TheSarrau±MachnumberisnamedafterProfessorSarrauwho®rsthigh-lightedthesigni®canceofthenumber(Sarrau1884)andE.Machwho14.3DimensionalanalysisTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. introduceditin1887.TheSarrau±MachnumberwasoncecalledtheCauchynumberasatributetoCauchy'scontributiontowavemotionanalysis.DiscussionAnycombinationofthedimensionlessnumbersinvolvedinequation(14.9)isalsodimensionlessandmaybeusedtoreplaceoneofthecombinations.ItcanbeshownthatoneparametercanbereplacedbytheMortonnumber,alsocalledtheliquidparameter,since: TheMortonnumberisafunctiononlyof¯uidpropertiesandthegravityconstant.Forthesame¯uids(airandwater)inbothmodelandprototype,isaconstant(i.e.14.3.3DynamicsimilarityTraditionallymodelstudiesareperformedusinggeometricallysimilarmodels.Inageometricallysimilarmodel,truedynamicsimilarityisachievedifandonlyifeachdimensionlessparameter(or-terms)hasthesamevalueinbothmodelandproto-Scalee€ectswillexistwhenoneormore-termshavedi€erentvaluesinthemodelandprototype.PracticalconsiderationsInpractice,hydraulicmodeltestsareperformedundercontrolled¯owconditions.Thepressuredi€erencemayusuallybecontrolled.Thisenablestobetreatedasadependentparameter.Furthercompressibilitye€ectsaresmallinclear-waterandtheSarrau±Machnumberisusuallyverysmallinbothmodelandproto-type.Hence,dynamicsimilarityinmosthydraulicmodelsisgovernedby: PV2ˆF3 VL; HydraulicmodeltestsThereareamultitudeofphenomenathatmightbeimportantinhydraulic¯owsituations:e.g.viscouse€ects,surfacetension,gravitye€ect.Theuseofthesame¯uidonbothprototypeandmodelprohibitssimultaneouslysatisfyingtheFroude,ReynoldsandWebernumberscalingcriteria(equation(14.12))becausetheFroudeThisstatementisnottrueinair±water¯ows(e.g.free-surfaceaerated¯ows)asthesoundceleritymaydecreasetoabout20m/sfor50%volumeaircontent(e.g.Cain1978,Chanson1997).PhysicalmodellingofhydraulicsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. numbersimilarityrequires,theReynoldsnumberscalingimpliesthatandtheWebernumbersimilarityrequires:Inmostcases,onlythemostdominantmechanismismodelled.Hydraulicmodelscommonlyusewaterand/orairas¯owing¯uid(s).Infully-enclosed¯ows(e.g.pipe¯ows),thepressurelossesarebasicallyrelatedtotheReynoldsnumber.Hence,aReynoldsnumberscalingisused:i.e.theReynoldsnumberisthesameinbothmodelandprototype.Infree-surface¯ows(i.e.¯owswithafreesurface),gravitye€ectsarealwaysimportantandaFroudenumbermodellingisused(i.e.)(e.g.Fig.14.2).Wheninertialandsurfacetensionforcesaredominant,aWebernumbersimilar-itymustbeselected.Studiesinvolvingairentrainmentin¯owingwaters(i.e.whitewaters),de-aerationinshaftorbubbleplumesareoftenbaseduponaWebernumberscaling.TheEulernumberisusedinpracticeforthescalingofmodelsusingairratherthanwater:e.g.hydraulicmodelsinwindtunnels,oramanifoldsystemwithwater¯owwhichisscaledatasmallersizewithanair¯owsystem.14.3.4ScaleeffectsScalee€ectsmaybede®nedasthedistortionsintroducedbye€ects(e.g.viscosity,surfacetension)otherthanthedominantone(e.g.gravityinfree-surface¯ows).Theytakeplacewhenoneormoredimensionlessparameters(seeSection14.3.3)di€erbetweenmodelandprototype.Scalee€ectsareoftensmallbuttheyarenotalwaysnegligiblealtogether.Consid-eringanover¯owaboveaweir,the¯uidissubjectedtosomeviscousresistancealongtheinvert.Howeverthe¯owabovethecrestisnotsigni®cantlya€ectedbyresistance,theviscouse€ectsaresmallandthedischarge±headrelationshipcanbededucedasforideal-¯uid¯ow.Infree-surface¯ows,thegravitye€ectisdominant.Ifthesame¯uid(i.e.water)isusedinboththemodelandtheprototype,itisimpossibletokeepboththeFroudeandReynoldsnumbersinthemodelandfull-scale.IndeeditiselementarytoshowthataFroudesimilitudeimplies,andtheReynoldsnumberbecomesmuchsmallerinthemodelthanintheprototype(ifNotethatdi€erent¯uidsmaybeusedtohavethesameReynoldsandFroudenumbersinboththemodelandprototype,butthisexpedientisoftennotpracticalnoreconomical.SomeexamplesofscaleeffectsExampleNo.1Consideringthedragexertedontwo-dimensionalbodies,Fig.14.3showsthee€ectsoftheReynoldsnumberonthedragcoecient.Dynamicsimilarity14.3DimensionalanalysisTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. Fig.14.3Dragcoef®cientontwo-dimensionalbodies.(equation(14.12))requiresthedragcoecienttobethesameinthemodelandprototype.IftheReynoldsnumberissmallerinthemodelthanatfull-scale(inmostpracticalcases),Fig.14.3suggeststhatthemodeldragcoecientwouldbelargerthanthatoftheprototypeanddynamicsimilaritycouldnotbeachieved.Moreover,thedragforcecomprisestheformdragandthesurfacedrag(i.e.skinfriction).Insmall-sizemodels,thesurfacedragmightbecomepredominant,particularlyifthemodel¯owisnotfully-roughturbulentorthegeometricalscalingofroughnessheightisnotachievable.Inpractice,animportantrule,inmodelstudiesisthatthemodelReynoldsnumbershouldbekeptaslargeaspossible,sothatthemodel¯owisfully-roughturbulent(ifprototype¯owconditionsarefully-roughturbulent).ExampleNo.2Anotherexampleisthee€ectofthecornerradiusonthedragforceontwo-dimensionalbodies(Fig.14.4).Figure14.4showssigni®cantdi€erencesintheReynoldsnumber±dragcoecientrelationshipsdependingupontherelative.Whenthecornerradiusontheprototypeissmallandlarge,itisimpossibletohavethesameratioofcornerradiustobodysizeinthemodelandprototypebecausethemodelcannotbemanufacturedwiththerequiredaccuracy.Insuchcases,thedragforceisnotscaledadequately.PhysicalmodellingofhydraulicsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. Fig.14.4Effectofcornerradiusandsurfaceroughnessonthedragcoef®cientoftwo-dimensionalExampleNo.3Adi€erentexampleisthe¯owresistanceofbridgepiers.Henderson(1966)showedthattheresistanceto¯owofnormalbridgepiershapesissuchthatthedragcoecientisaboutoroverunity,implyingthattheformdragisasigni®cantcomponentofthetotaldrag.Insuchacase,theviscouse€ectsarerelativelysmall,anddynamicsimilarityisachievable,providedthatmodelviscouse€ectsremainnegligible.14.3DimensionalanalysisTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. Ifscalee€ectswouldbecomesigni®cantinamodel,asmallerprototype-to-modelscaleratioshouldbeconsideredtominimizethescalee€ects.Forexample,ina100:1scalemodelofanopenchannel,thegravitye€ectispredominantbutviscouse€ectsmightbesigni®cant.Ageometricscaleratioof50:1or25:1maybeconsideredtoreduceoreliminateviscousscalee€ects.Anotherexampleistheentrainmentofairbubblesinfree-surface¯ows.Gravitye€ectsarepredominantbutitisrecognizedthatsurfacetensionscalee€ectscantakeplacefor10to20(or05to0.1)(e.g.Wood1991,Chanson1997).Atthelimit,noscalee€ectisobservedatfull-scale(i.e.1)asallthe(equation(14.11))havethesamevaluesintheprototypeandmodelwhen14.4Modellingfully-enclosed¯ows14.4.1ReynoldsmodelsFully-enclosed¯owsituationsincludepipe¯ows,turbomachinesandvalves.Forsuch¯owsituations,viscositye€ectsonthesolidboundariesareimportant.PhysicalmodellingisusuallyperformedwithaReynoldssimilitude:i.e.theReynoldsnumberiskeptidenticalinboththemodelandprototype:Ifthesame¯uidisusedinboththemodelandprototype,equation(14.13)implies:(Reynoldssimilitude)1,themodelvelocitymustbelargerthanthatintheprototype.DiscussionForexample,ifthemodelscaleis10:1(i.e.10),thevelocityinthemodelmustbetentimesthatintheprototype.Byusingadi€erent¯uidinthemodel,theratiobecomesdi€erentfromunityandcanbereduced.14.4.2DiscussionFlowresistanceinpipe¯owsForpipe¯ows,theDarcyequationrelatesthepressurelossestothepipegeometry,length)andtothe¯owvelocity LD istheDarcy±Weisbachfrictionfactor.Aftertransformationandcombiningwithequation(14.10),itleads: PhysicalmodellingofhydraulicsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. Inpipe¯ows,gravityandsurfacetensionhavenoe€ectonthepressurelosses.Forsteadyliquid¯ows,thecompressibilitye€ectsarenegligible.Theroughnessheightis,however,anadditionalcharacteristiclength.Forauniformlydistribu-ted-roughness,equation(14.15)becomes: fL2DˆF4Re; Equation(14.16)expressesthedimensionlessrelationshipbetweenfrictionlossesinpipes,theReynoldsnumberandrelativeroughness.AnillustrationistheMoodydiagram(Fig.14.5).SkinfrictionandformdragConsideringthedragonabody(e.g.Figs14.3and14.4),thepressurelossesassociatedwiththemodi®cationofthe¯ow®eldcausedbythepresenceofthebodyareusuallyexpressedintermsofthedragforceonthebody.TheEulernumberisrewrittenas:,whereistheprojectionofthebodyintheplanenormaltothe¯owdirection.isequivalenttothepressureEquations(14.10)and(14.15)maybecombinedtorelatethedragcoecient Eu2ˆ Fd Inequation(14.17),theReynoldsnumberisrelatedtotheskinfrictiondragduetoviscousshearaswellastoformdragresultingfromtheseparationofthe¯owstream-linesfromthebody. Fig.14.5FrictionfactorversusReynoldsnumberinpipe¯ows.14.4Modellingfully-enclosed¯owsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. 14.4.3Practicalconsiderationsinmodellingfully-enclosed¯owsThe¯owregimeinpipesiseitherlaminarorturbulent.Inindustrialapplications,itiscommonlyacceptedthatthe¯owbecomesturbulentforReynoldsnumberslargerthan2000to3000,theReynoldsnumberbeingde®nedintermsoftheequivalentpipediameterandofthemean¯owvelocityForturbulent¯ows,the¯owregimecanbesub-dividedintothreecategories:smooth,transitionandfullyrough.EachcategorycanbedistinguishedasafunctionofashearReynoldsnumberde®nedas: istheshearvelocity.Thetransitionbetweensmoothturbulenceandfully-roughturbulenceisapproximatelyde®nedas:FlowsituationOpenchannel¯owPipe¯owRef.(Henderson1966)(Schlichting1979)SmoothturbulentTransition41005Fullyroughturbulent100Dynamicsimilarityoffully-enclosed¯owsimpliesthesameresistancecoecientinboththemodelandtheprototype.ThiscanbeachievedwiththeReynoldsnumberbeingthesameinthemodelandprototype,orwithboth¯owsinthemodelandprototypebeingfully-roughturbulent(Fig.14.5).Ifthefull-scale¯owisturbulent,itisextremelyimportanttoensurethatthemodel¯owisalsoturbulent.Laminarandturbulent¯owsexhibitveryimportantbasicdi€erences.Inmostcases,turbulent¯owsshouldnotbescaledwithlaminar¯owTheReynoldsnumbercanbekeptconstantbychangingthe¯owing¯uid.Forexampletheatmosphericwind¯owpastatallbuildingcouldbemodelledinasmall-sizewatertunnelatthesameReynoldsnumber.14.5Modellingfree-surface¯ows14.5.1PresentationInfree-surface¯ows(e.g.rivers,wavemotion),gravitye€ectsarepredominant.Model-prototypesimilarityisperformedusuallywithaFroudesimilitude:PhysicalmodellingofhydraulicsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. Ifthegravityaccelerationisthesameinboththemodelandprototype,aFroudenumbermodellingimplies:(Froudesimilitude)Notethatthemodelvelocityislessthanthatintheprototypefor1andthetimescaleequalsFroudenumbermodellingistypicallyusedwhenfrictionlossesaresmallandthe¯owishighlyturbulent:e.g.spillways,over¯owweirs,¯owpastbridgepiers.Itisalsousedinstudiesinvolvinglargewaves:e.g.breakwaterorshipmodels.Amainconcernisthepotentialforscalee€ectsinducedbyviscousforces.Scalee€ectscausedbysurfacetensione€ectsareanotherconcern,inparticularwhenfree-surfaceaeration(i.e.airentrainment)takesplace.14.5.2ModellinghydraulicstructuresandwavemotionInhydraulicstructuresandforwavemotionstudies(Fig.14.2),thegravitye€ectisusuallypredominantintheprototype.The¯owisturbulent,andhenceviscousandsurfacetensione€ectsarenegligibleintheprototypeifthe¯owvelocityisreasonablysmall.InsuchcasesaFroudesimilitudemustbeselected.Themosteconomicalstrategyis:1.tochooseageometricscaleratiosuchastokeepthemodeldimensionssmall,2.toensurethatthemodelReynoldsnumberislargeenoughtomakethe¯owturbulentatthesmallesttest¯ows.14.5.3Modellingriversand¯oodplainsInrivermodelling,gravitye€ectsandviscouse€ectsarebasicallyofthesameorderofmagnitude.Forexample,inuniformequilibrium¯ows(i.e.normal¯ows),thegravityforcecomponentcounterbalancesexactlythe¯owresistanceandthe¯owconditionsarededucedfromthecontinuityandmomentumequations.Inpractice,rivermodelsarescaledwithaFroudesimilitude(equation(14.19))andviscousscalee€ectsmustbeminimized.Themodel¯owmustbeturbulent,andpossiblyfully-roughturbulentwiththesamerelativeroughnessasfortheprototype:Thetestingofshipmodelsisveryspecialized.Interestingly,F.ReechandW.Froudewereamongthe®rsttousetheFroudesimilitudeforshipmodelling.14.5Modellingfree-surface¯owsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. wheretheReynoldsnumberisde®nedintermsofthehydraulicdiameter(i.e.DistortedmodelsAdistortedmodelisaphysicalmodelinwhichthegeometricscaleisdi€erentbetweeneachmaindirection.Forexample,rivermodelsareusuallydesignedwithalargerscalingratiointhehorizontaldirectionsthanintheverticaldirection:.Thescaledistortiondoesnotdistortseriouslythe¯owpatternanditusuallygivesgoodAclassicalexampleofadistortedmodelisthatoftheMississippiriver,builtbytheUSArmyCorpsofEngineers.TheMississippibasinisabout3100000kmandtheriverisnearly3800kmlong.Anoutdoormodelwasbuiltwithascaleof2000:1.Ifthesamescalingratiowasappliedtoboththeverticalandhorizontaldimensions,prototypedepthsofabout6mwouldimplymodeldepthsofabout3mm.Withsuchsmall¯owdepths,surfacetensionandviscouse€ectswouldbesigni®cant.TheMississippimodelwasbuilt,infact,withadistortedscale:100and2000.Altogetherthemodelsizeisabout1.5kmper2km!AdistortedmodelofriversisdesignedwithaFroudesimilitude:wheretheFroudenumberscalingratioisrelatedtotheverticalscaleratio: Asforanundistortedmodel,thedistortedmodel¯owmustbeturbulent(equation(14.20)),andpreferablyfully-roughturbulentwiththesamerelativeroughnessasfortheprototype:TheFroudesimilitude(equation(14.22))implies:VelocityDischarge XrVrˆ XrZrpTime…14:26†…tan†rˆ Longitudinalbedslopeistheanglebetweenthechannelbedandthehorizontal.Withadistortedscalemodel,itispossibletoselectsmallphysicalmodels(i.e.large).Inadditiontotheeconomicalandpracticalbene®ts,distortedmodelsalsohavethefollowingadvantagescomparedwithnon-distortedmodels:the¯owvelocitiesandturbulenceinthemodelarelarger(equation(14.24)),thetimescaleisreduced(equation(14.26)),themodelReynoldsnumberislarger,improvingtheprototype-to-modeldynamicsimilarity,andPhysicalmodellingofhydraulicsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. thelargerverticalscale(i.e.)allowsagreateraccuracyonthe¯owdepthmeasurements.Practicallyitisrecommendedthatthemodeldistortion(i.e.theratioshouldbelessthan5to10.Somedisadvantagesofdistortedmodelsmaybementionedforcompleteness:thevelocitydirectionsarenotalwaysreproducedcorrectly,andsomeobserversmightbedistractedunfavourablybythemodeldistortionleadingtoinaccurateorincorrectjudgements.Movable-bedmodelsMovable-bedhydraulicmodelsaresomeofthemostdiculttypesofmodelsandtheyoftengiveunsatisfactoryresults.Theprimarydicultyistoscaleboththesedimentmovementandthe¯uidmotion.Furthermore,thebedroughnessbecomesafunctionofthebedgeometryandofthesedimenttransport.EarlymovablebedmodelstudiesontheRiverMersey(England)andSeineRiver(France)inthe1880sshowedthatthetimescalegoverningthe¯uid¯owdi€ersfromthetimescalegoverningsedimentmotion(seeAppendixA3.1).AdetailedanalysisofsedimenttransportmodellingisdevelopedinAppendixA3.1.Severalauthors(e.g.Henderson1996,pp.497±508,Graf1971,pp.392±398)alsodiscussedvariousmethodsfor`designing'amovable-bedmodel.Themostimportantpointistheneedtoverifyandtocalibrateamovable-bedmodelbeforeusingitasapredictiontool.14.5.4ResistancescalingThemodellingof¯owresistanceisnotasimplematter.Oftenthegeometricsimilarityofroughnessheightandspacingisnotenough.Forexample,itisobservedsometimesthatthemodeldoesnotreproducethe¯owpatternsintheprototypebecausethemodelistoo`smooth'ortoo`rough'.Insomecases(particularlywithalargescaleratio),themodel¯owisnotasturbulentastheprototype¯ow.Asolutionistouseroughnesselements(e.g.mesh,wire,verticalrods)toenhancethemodel¯owturbulence,hencetosimulatemoresatisfactorilytheprototype¯owpattern.Anotheraspectisthescalingoftheresistancecoecient.The¯owresistancecanbedescribedintermsoftheDarcyfrictionfactororanempiricalresistancecoecient(e.g.ChezyorGauckler±Manningcoecients).Inuniformequilibrium¯ows,themomentumequationimplies: Foranundistortedmodel,aFroudesimilitude(equation(14.19)and(14.28))impliesthatthemodel¯owresistancewillbesimilartothatintheprototype:14.5Modellingfree-surface¯owsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. Mostprototype¯owsarefully-roughturbulentandtheDarcyfrictionfactorisprimarilyafunctionoftherelativeroughness.AnotherapproachisbasedupontheGauckler±Manningcoecient.TheCheequationimpliesthat,ingradually-variedanduniformequilibrium¯ows,thefollow-ingscalingrelationshipholds: Foranundistortedscalemodel,equation(14.30)becomes:Equation(14.31)indicatesthatthenotionofcompletesimilarityisappliedbothtothetextureofthesurfaceandtotheshapeofitsgeneraloutline(Henderson1966).Inpractice,thelowestachievablevalueofisabout0.009to0.010s/mforglass).Withsuchavalue,theprototyperesistancecoecienttheGauckler±Manningcoecientsimilaritycouldlimitthemaximumgeometricalsimilarityratio.Ifistoosmall(typicallylessthan40),thephysicalmodelmightnotbeeconomicalnorconvenient.Insummary,aphysicalmodel(baseduponaFroudesimilitude)hasproportion-allymoreresistancethantheprototype.Iftheresistancelossesaresmall(e.g.ataweircrest),theresistancescalee€ectsarenotconsidered.Inthecasesofriverandharbourmodelling,resistanceissigni®cant.Themattermaybesolvedusingdistortedmodels.DistortedmodelsWithadistortedscalemodel,equations(14.28)and(14.30)becomerespectively: …DH†r…sin†rfrs…14:32†VrˆZrpˆ Forawidechannel(i.e.)anda¯atslope(i.e.),thescalingof¯owresistanceindistortedmodelsimplies: widechannelandflatslope widechannelandflatslopeDiscussionInpractice1andequation(14.34)wouldpredictamodelfrictionfactorlowerthanthatintheprototype.Butequation(14.35)couldimplyamodelPhysicalmodellingofhydraulicsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. resistancecoecientlargerorsmallerthanthatintheprototypedependingupontheratio14.6Designofphysicalmodels14.6.1IntroductionBeforebuildingaphysicalmodel,engineersmusthavetheappropriatetopographicandhydrological®eldinformation.Thetypeofmodelmustthenbeselected,andaquestionarises:Whichisthedominanteffect:e.g.viscosity,gravityorsurfacetension?14.6.2GeneralcaseInthegeneralcase,theengineermustchooseapropergeometricscale.Theselectionprocedureisaniterativeprocess.Step1.Selectthesmallestgeometricscaleratioto®twithintheconstraintsofthelaboratory.Step2.,andforthesimilitudecriterion(e.g.FroudeorReynolds),checkifthedischargecanbescaledproperlyinthemodel,baseduponthemaximummodeldischargeandthesimilitudecriterion,isthemaximummodeldischargelargeenoughtomodeltheprototype¯owconditions?Step3.Checkifthe¯owresistancescalingisachievableinthemodel.Isitpossibletoachievetherequiredinthemodel?Step4.CheckthemodelReynoldsnumberforthesmallesttest¯owrate.,whatarethe¯owconditionsinthemodel:e.g.laminarorturbulent,smooth-turbulentorfully-rough-turbulent?Iftheprototype¯owisturbulent,model¯owconditionsmustbeturbulent(i.e.typically14.6DesignofphysicalmodelsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. Step5.Choosetheconvenientscale.Whenasimplephysicalmodelisnotfeasible,moreadvancedmodellingtechniquescanbeused:e.g.atwo-dimensionalmodel(e.g.spillway¯ow),adistortedscalemodel(e.g.river¯ow).14.6.3DistortedscalemodelsForadistortedscalemodel,theengineermustselecttwo(orthree)geometricscales.Themodeldesignprocedureisagainaniterativeprocess:Step1.Selectthesmallesthorizontalscaleratioto®twithintheconstraintsofthelaboratory.Step2.Determinethepossiblerangeofverticalscalesuchas:thesmallestscaleisthatwhichgivesthelimitofthedischargescalingratio,baseduponthemaximummodeldischargethelargestscaleisthatwhichgivesthefeasible¯owresistancecoe-cient(i.e.feasible,andcheckthedistortionratioshouldbelessthan5to10.)Step3.CheckthemodelReynoldsnumberforthesmallesttest¯owrate.ThismightprovideanewlargestverticalscaleratiocheckthedistortionratioStep4.Selectaverticalscaleratiowhichsatis®es:.Ifthisconditioncannotbesatis®ed,asmallerhorizontalscaleratiomustbechosen.checkthedistortionratioInpracticeitisrecommendedthatshouldbelessthan5to10.Step5.Choosetheconvenientscales(14.7SummaryPhysicalhydraulicmodellingisadesigntechniqueusedbyengineerstooptimizethestructuredesign,toensurethesafeoperationofthestructureand/ortofacilitatethedecision-makingprocess.Inpractice,mosthydraulicmodelsarescaledwitheitheraFroudeoraReynoldssimilitude:i.e.theselecteddimensionlessnumberisthesameinthemodelandintheprototype(i.e.full-scale).Themostcommon¯uidsareairandwater.Free-surface¯owmodellingismostoftenperformedwiththesame¯uid(e.g.water)infull-scaleandthemodel.Fully-enclosed¯owmodellingmightbeperformedwithwaterintheprototypeandairinthemodel.Theselectionof¯uidinthemodelandtheprototype®xesthedensityscaleratioTable14.1summarizesthescalingratiosfortheFroudeandReynoldssimilitudes.PhysicalmodellingofhydraulicsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. 14.8ExercisesNumericalsolutionstosomeoftheseexercisesareavailablefromtheWebatwww.arnoldpublishers.com/support/chansonAbutter¯yvalveistobetestedinalaboratorytodeterminethedischargecoecientforvariousopeningsofthedisc.Theprototypesizewillbe2.2mindiameteranditwillbemanufacturedfromcaststeelwithmachinedinsidesurfaces(roughnessheightestimatedtobeabout0.5mm).Themaximumdischargetobecontrolledbythevalveis15m/s.Thelaboratorymodelisa5:1scalemodel.(a)Whatsurfaceconditionisrequiredinthemodel?Whatmodeldischargeisrequiredtoachievecompletesimilaritywiththeprototype,ifwaterisusedinboth?(b)Cantheseconditionsbeachieved?(c)Ifthemaximum¯owavailableformodeltestsis200L/s,couldyouaccuratelypredictprototypedischargecoecientsfromtheresultsofthemodeltests?Summarysheet(b)Yes/NoReasons:TheinletofaFrancisturbineistobetestedinalaboratorytodeterminetheper-formancesforvariousdischarges.Theprototypesizeoftheradial¯owrotorwillbe: Table14.1ScalingratiosforFroudeandReynoldssimilitudes(undistortedmodel)ParameterUnitScaleratiowithFroudelawFroudelaw(distortedmodel)Reynoldslaw(1)(2)(3)(4)(5)GeometricpropertiesLengthmAreamKinematicpropertiesVelocitym/sDischargeperunitwidthmDischargemTimesDynamicpropertiesForceNPressurePaDensitykg/mDynamicviscosityPasSurfacetensionN/mNote:assumingidenticalgravityaccelerationinmodelandprototype.14.8ExercisesTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. inletdiameter0.6m,width0.08m,inletcross¯owareadiameterwidth.Itwillbemanufacturedfromcaststeelwithmachinedinsidesurfaces(roughnessheightestimatedtobeabout0.3mm).Themaximumdischargetobeturbined(bytheFranciswheel)is1.4m/s.Thelaboratorymodelisa5:1scalemodel.(a)Whatsurfaceconditionisrequiredinthemodel?Whatmodeldischargeisrequiredtoachievecompletesimilaritywiththeprototype,ifwaterisusedinboth?(b)Cantheseconditionsbeachieved?(Computetheminimumrequiredmodeltotalheadand¯owrate.Comparethesewiththepumpperformancesofatypicalhydrauliclaboratory:10m,100L/s.)(c)Ifthemaximum¯owavailableformodeltestsis150L/s,wouldyoubeabletopredictaccuratelyprototypeperformancesfromtheresultsofthemodeltests?(Justifyyouranswer.)Summarysheet(b)Yes/NoReasons:Anover¯owspillwayistobedesignedwithanuncontrolledcrestfollowedbyasteppedchuteandahydraulicjumpdissipator.Themaximumspillwaycapacitywillbe4300m/s.Thewidthofthecrest,chuteanddissipationbasinwillbe55m.A50:1scalemodelofthespillwayistobebuilt.Dischargesrangingbetweenthemaximum¯owrateand10%ofthemaximum¯owratearetobereproducedinthemodel.(a)Determinethemaximummodeldischargerequired.(b)Determinethemini-mumprototypedischargeforwhichnegligiblescalee€ectsoccurinthemodel.(Com-mentonyourresult.)(c)Whatwillbethescalefortheforceratio?Laboratorytestsindicatethatoperationofthebasinmayresultinunsteadywavepropagationdownstreamofthestillingbasinwithamodelwaveamplitudeofabout0.05mandmodelwaveperiodof47seconds.Calculate:(d)theprototypewaveamplitudeand(e)theprototypewaveperiod.Summarysheet(a)Maximum(b)Minimum(c)ForceA35.5:1scalemodelofaconcreteoverfallspillwayandstillingbasinistobebuilt.Theprototypedischargewillbe200m/sandthespillwaycrestlengthis62m.(a)DeterminethemaximummodeldischargerequiredandtheminimumprototypedischargeforPhysicalmodellingofhydraulicsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. whichnegligiblescalee€ectsoccurinthemodel.(b)Intestsinvolvingba‚eblocksforstabilizingthehydraulicjumpinthestillingbasin,theforcemeasuredoneachblockwas9.3N.Whatisthecorrespondingprototypeforce?(c)Thechanneldownstreamofthestillingbasinistobelinedwithrip-rap(angularblocksofrock)approximately650mminsize.Thevelocitymeasuredneartherip-rapisaslowas0.2m/s.CheckwhetherthemodelReynoldsnumberislargeenoughforthedragcoecientofthemodelrockstobethesameasintheprototype.Whatwillbethescalefortheforceratio?Summarysheet(a)Maximum(b)ForceForceratioAsluicegatewillbebuiltacrossa25mwiderectangularchannel.Themaximumprototypedischargewillbe275m/sandthechannelbedwillbehorizontalandcon-crete-lined(i.e.smooth).A35:1scalemodelofthegateistobebuiltforlaboratorytests.(a)Whatsimilitudeshouldbeused?Calculate:(b)modelwidthand(c)maximummodel¯owrate.Foroneparticulargateopeningand¯owrate,thelaboratory¯owconditionsare:upstream¯owdepthof0.2856m,downstream¯owdepthof0.0233m.(d)Computethemodeldischarge.Statethebasicprinciple(s)involved.(e)Computethemodelforceactingonthesluicegate.Statethebasicprinciple(s)involved.(f)Whatwillbethecorrespondingprototypedischargeandforceonthegate?(g)Whatwillbethescalefortheforceratio?Gateoperationmayresultinunsteady¯owsituations.Ifaprototypegateoperationhasthefollowingcharacteristics:gateopeningduration15minutes,initialdischarge180m/s,newdischarge275m/s,calculate:(h)gateopeningdurationand(i)dischargestobeusedinthemodeltests.Summarysheet(a)Similitude:Why?(f)(g)Force14.8ExercisesTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. Ahydraulicjumpstillingbasin,equippedwithba‚eblocks,istobetestedinalabora-torytodeterminethedissipationcharacteristicsforvarious¯owrates.Themaximumprototypedischargewillbe220m/sandtherectangularchannelwillbe10mwide.(Assumethechannelbedtobehorizontalandconcrete-lined,i.e.smooth.)A40:1scalemodelofthestillingbasinistobebuilt.Dischargesrangingbetweenthemaximum¯owrateand10%ofthemaximum¯owratearetobereproducedinthemodel.(a)Whatsimilitudeshouldbeused?(Justifyyourselection.)(b)Determinethemaximummodeldischargerequired.(c)Determinetheminimumprototypedischargeforwhichnegligiblescalee€ectsoccurinthemodel.(CommentonyourForoneparticularin¯owcondition,thelaboratory¯owconditionsare:upstream¯owdepthof0.019m,upstream¯owvelocityof2.38m/s,downstream¯owdepthof0.122m.(d)Computethemodelforceexertedontheba‚eblocks.(Statethebasicprinciple(s)involved.)(e)Whatisthedirectionofforcein(d):i.e.upstreamordownstream?(f)Whatwillbethecorrespondingprototypeforceactingontheblocks?(g)Computetheprototypeheadloss.Operationofthebasinmayresultinunsteadywavepropagationdownstreamofthestillingbasin.(h)Whatwillbethescaleforthetimeratio?Testswillbemadeonamodelseawallof1/18prototypesize.(a)Iftheprototypewaveclimateis:waveperiod12seconds,wavelength20m,waveamplitude2.1m,whatwaveperiod,wavelengthandwaveamplitudeshouldbeusedinthemodeltests?(b)Ifthemaximumforceexertedbyawaveonthemodelseawallis95N,whatcorrespondingforcewillbeexertedontheprototype?Summarysheet(b)ForceA®xedbedmodelistobebuiltofacertainsectionofariver.Themaximumfull-scaledischargeis2750m/s,theaveragewidthoftheriveris220mandthebedslopeis0.16mperkilometre.TheGauckler±Manningcoecientfortheprototypeisestimatedat0.035s/m.Laboratoryfacilitieslimitthescaleratioto200:1andmaximummodeldischargeis45L/s.NotethatthesmoothestmodelsurfacefeasiblehasaGauckler±Manningcoecientofabout0.014s/m.Dischargesrangingbetweenthemaximum¯owrateand15%ofthemaximum¯owratearetobereproducedinthemodel.Determinetheacceptablemaximumandminimumvaluesoftheverticalscaleratio.Selectasuitablescaleforpracticaluse,andcalculatethecorrespondingmodelvaluesoftheGauckler±Manningcoecient,maximumdischargeandnormaldepth(atmaximumdischarge).(Itcanbeassumedthattheriverchanneliswide)inboththemodelandprototypeforall¯ows.Assumethatuniformequilibrium¯owisachievedinthemodelandprototype.)PhysicalmodellingofhydraulicsTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved. SummarysheetAlternativeMax.AllowablerangeforYourchoiceforCorrespondingvaluesofatmaximum¯owrateAnarti®cialconcretechannelmodelistobebuilt.Laboratoryfacilitieslimitthescaleratioto50:1andthemaximummodeldischargeis50L/s.Themaximumfull-scaledischargeis150m/s,thecross-sectionofthechannelisapproximatelyrectangular(50mbottomwidth)andthebedslopeis0.14mperkilometre.(Note:Theroughnessheightoftheprototypeisestimatedas3mmwhilethesmoothestmodelsurfacefeasiblehasaDarcyfrictionfactorofabout03.)Dischargesrangingbetweenthemaximum¯owrateand10%ofthemaximum¯owratearetobereproducedinthemodel.Foranundistortedmodel:(a)whatwouldbethemodeldischargeatmaximumfull-scaledischarge?(b)whatwouldbetheDarcycoecientofthemodel¯ow?(c)whatwouldbetheDarcycoecientoftheprototypechannel?(d)commentanddiscussyour®ndings.(Assumenormal¯owconditions.)Adistortedmodelistobebuilt.(e)Determinetheacceptablemaximumandmini-mumvaluesoftheverticalscaleratio.(f)Selectasuitablescaleforpracticaluse.Calculatethecorrespondingmodelvaluesof:(g)Darcycoecient,(h)maximumdischargeand(i)normaldepth(atmaximumdischarge).A®xedbedmodelistobemadeofariverwithasurfacewidthof80m.TheGauckler±Manningcoecientfortheriverisestimatedat0.026s/m.Scaleratios150and25havebeenselected.(a)FindtherequiredmodelvaluesoftheGauckler±Manningcoecientcorrespondingtoprototypedepthsofwaterof2.0and5.0m,ifthecross-sectionalshapeisassumedtoberectangular.(b)Whatmaterialwouldyourecommendtouseinthelaboratorymodelforaprototypedepthof2.0m?14.8ExercisesTheHydraulicsofOpenChannelFlowbyHubertChanson.Publishedin1999byArnold,338EustonRoad,LondonNW13BH,UK.HubertChanson.Allrightsreserved.