I 1 ) ) I \ J I MeGRA W-HItr,CIVILENGLNEERINGSERIES HARMERE.DAVIS,ConsultingEditor BABBI'IT '.Engineering.in PublicHealth BENJAMIN'Statically Indeterminate Cnow .Open,-cha,nnelHyqraulics DAVIS,TROXELL,'ANDWrsKoCIL.Tl1eTesting andInspection of' EngineeringMaterials DUNl'iAM.FoundationsofStructures DUNHAM'The Theory and Practice ofReinforcedConcrete DUNHAMANDYOUNG.'Contracts,Specifications,andLaw forEngineers GAYLORDANDGAYLORD'StructuralDesign HALLERT'Photogrammetry HENNESANDEKSE.Fundamentals ofTransportationEngineering KRYNINEANDJUDD'Principles of Engineering GeologyandGeot.echnics LINSLEYANDFRANZINI.Elements ofHydraulic Engineering LmsLIDY,KOHLER,ANDI'A ULHUB'AppliedHydrology LINSLEY,KOHLER,ANDPAULHUS'Hydrologyf9rEngineers LU:8DER. AerialPhotographic Interpretation MA'l'SON,SMITH,ANDHURD'TrafficEngineering MEAD,MEAD,ANDAKERMAN'Contracts,Specifications,and Engineering Relations NORRIS,HANSEN,HOLLEY,BIGGS,NAMYET,AND1fINAMI. :Structural Design forDyiramicLoads PEURIFOY'ConstructionPlanning,Equipment,andMethods' PEURIFOY'gstimatingConstructi()uCosts TROXELLANDDAVIS'Composition andPropertiesof Concrete TSCHEBOTARIOFF.SoilMechanics,Foundations,andEarthStructures URQUHART,O'ROURllU"'" Emphasis is given to the qualities of "teachability" and" practicability," andt1tt,emptsweremadeinpreaentingthematerialtobridgethegap which is generallyto exist between the theory and the practice. Inordertoachievetheseobjectives,the useof .advanced mathematics is deliberately avoidedas much as possible, and theof hydraulic 1Such. as:EtienneOrausse,"Hydrauliquc des decouvelts enregimeperma-nent"(" HydrauliosofOpenChanneldwithSteadyFlow"),EditionsEyrolles, Paris,1951;R.Silber,"EtudeattracedesecouleroentspermanentsencanaUl(et rivieres"("StudyandSket !

-"1"'-I "., CHAPTER,1 FLOWANDITSCLASSIFICATIONS 1-1.Description.The flowofwater inaconduit may be either open-channel flotuorflow.The two kinds of floware similar in many ways butdifferinoheimportantrespect.Open-channelflowmusthavea freesurface,whereaspipeflowhasnone,sincethewatermustfillthe wholeconduit.A freesurface issubject to atmosphericpressure.,flow,beingconnnedina.closedirectatl!lo,heri pressurebut hydraulic pressureonly.., , "--,,,,-Thet\VOkindsofflowarecomparedinFig.1-1.Shownontheleft sideispipeflow.Twopiezometertubesareinstalledonthepipeat' sections1and2.,Thewaterlevelsinthe tubesaremaintainedbythe pressureinthepipeat elevationsrepresentedby g.radeline.Thepressureexertedbythewaterineachsectionofthe . pipeisindicatedinthe correspondingtube by theheight yofthe watet c;olumnabovethecenter lineofthepipe.ThetIJt.alenergyinthe flow of the section with referenceto adatum line is the sum ofthe elevation z of thepipe-centerline,thepiezoine'tricheighty,andthevelocityhead V2/2g,whereVisthe mean velocity offlow.l The energy isrepresented inthefigureby what iscalledtheenergygradelineorsimplytheenergy line.Thelossof energythatresultswhenw:aterflowsfromseqtion1 tosectiori2is . representedbyhi.Asimilardiagramforopen-channel flowisshownonthe right sideofFig.1-1.For simplicity,it is assumed . that the flowisparallel and hasa uniformvelocity distribu,tionandthat the slopeofthechannelis,small.lP,.. i,hisc!1;,e,the is ,the .hldraulicandpiezometric'.,.. Despitethe similaritybetween the twokindsofflow,it ismuchmore cl.ifficultto solve problems of flow inopen channels than.in pressure pipes. conditionsinopen arecomplicatedPythe ',factthatthe :1It ishereassumedthatthevelocityisuniformlydistrib\.itedacrosstheconduit othenvise acorrec,tion ,,",ouldhave tobemade, such Ilf is desaribed in Art. 2-7 fbropenchanrrels.!, :2If the flowwere curvilinear or if the slope of the channellarge, the piezometric height would be appreciably different fromthe depth of flow(lILandless in the latter.possibilitY_2I'il!!PQ.sJ:- sin4.j.] YI=ifcos10(2-13) (2-14) where :1'1 and YI,respectively,aretheordinateandabscissameasuredfromt,h.emid-pointofthe freesurface;k=sin(110/2);'"=sin-1 f [sin(1/>/2)l!kl;and11 isthe slope angle at the point(XI,l!I),varying from 0at the bottom ofthe curve to9,at the ends. Theaboveequationswilldefine' the. crosssectionwhentheflowisatitsfulldept.h. Theslopeangleattheendsofahydrostaticcatenaryofbesthydraulicefficiency is foundmathema.ticallytobeII,=35'37'7".(a)Plotthissectionwith' adepth y=10ft,and(bldeterminethevalues ofA, R,D,andZ atthefulldepth. 2-7.Estimatethe Yllluesofmomentum coefficient (jfor., the- given values of energy c(lefficientex =1.00,1.50,and 2.op., 2-8.Compute the energy andcoefficientsof the cross st'ction shown in Fig.2-3(a)by Eqs.(2-4)and(2-5),and(b)by (2-6)and(2-7).The crosssec-tionand the curvesofequalvelocitycan betransferredto apieceofdrawingpaper andenlargedfordeSired ll.ccuracy. 2-9.Indesigningsidewallsofsteepchutes andoverflowspillways,provethat the overturning moment due tothe pressure of theflowingwater isequaltoYswy'cos'9, wherew.is the unit weight of water,y isthe vertical depth of the flowingwater,and9 ist,heslopeangle ofthechannel. 2 .. 10.Prove Eq.(2-10). 2-11.Ahigh-headoverflowspillway(Fig.2-10)hasa60-ft-radiusflipbucketu.t itsdownstream end.The bucket isnot submerged,but acts tochange the direction of theflowfromthe slope ofthe lipillwayfacetothehorizontalandtodischargethe flov1intothe air' betweenverticaltraining walls soft apart.,At: adischarge of 55,100 ds, ;the water surface at the vertical sectionOB is at El. 8.52.'Verifyt.hecurvethat represents the computedhydraulic ,pressure acting onthe training wallat section DB. ThecomputatiQnisbailedanEq,' (2-9)andon'thefollowingassumptions:(1)the velqcityisuniformly acrossthe section;(2)the vo.lu,eusedforr,fQrpres- valuesnearthewallbase,is'equaltotheradiusofthebucketbut,forother pre;isurevalues,isequaltothe radius ofthe concentric flowlines;and(3)theflowis entto.inedwithair,andthe density ,oftheair-waterbe estimatedby the j \\ j \ .I l I ( ", ,i j )) . I .1! ) i. /,i \ I , . 36BASICPRINCIPLES Douma.formula,' that is, 'U - 10 - 1 .gR (2-15) whereuisthepercentageofentrained .airbyvoiume,Visthe velocityofflow,and RIsthe hydraulicradIus. . 2-12.ComputethewallpressureonthesectionOA(Fig.2-10)ofthespillway describedin Prob.2-11.It isassumedthat the depth oftlowsection isthe same!l.S that at sectionDB. ;:; C .2 ::l :-OJt;j 2 / 4,a Un;lpressare, II01 woler \$PilIW1!Y Irainingwall, eo IIcpO!1 FIG,2-10.Side-wall pressuresonthe flipbucketof aspillwa.y. 2-13.ComputethewallpressureonthesectionOA(Fig.2-10)ofthespillway descrIbedinProb.2-11ifthe bucketissubmergedwithatailwater levelatEL75.0. It is!l.SSulnedthat thepressure resultbg fromthecentritugalforceorthe submerged jet neednotbeconsidered beca.usethe submergencewillreaultinaseverereduction invelocity. REFERENCES 1.S.F.Averillnov:0gidravlicheskomraschete ruselkrivolineinoI formypoperech, nogosecheniia(Hydraulicdesignofchannelswithcurvilinear formoithe crosS section),lzvestiiaAkademiiNaukS.S.S.R.,Otdelenie Nauk" Moscow,no.1,pp.54-58,1956. 2.LeonardMetcalf and H.P. Eddy: "American SeweragePra.ctice,"McGraw-Hm BookCompany,1M.,New York,3ded., ,1935,vo!. 1. 3.Harold E. Babbitt: "Sewerage and Sewage Treatment," JohnWiley &:Sons, Inc., .NewYork, 7th ed.,1952,pp. 60-:.66. 4.H.M.Gibb:Curvesforsolvingthehydrostaticoatenary,EngineeringNews, vol.73,no .. 14,pp.668-670,Apr,8,1915. IThisiormull!.[26Jisbasedonda.taobtainedfromactual conoreteandwooden chutes,involving errOnlof10%.' OPENCHANNELS.ANDTHEIRPRO'PERT1ES37 5.George Higgins: "Water Channels," Crosby, Lockwood&:Son Ltd., London, 1927, pp.15-36... 6.AhmedShukry:Flowaroundbendsinanopenflume,Transactions,AmericilTl Societyof CivilEngineers,vol.115,pp.751-779,1950.' 7.A.II. Gibson: "Hydraulics and Its Applications,'"Constable &:Co., Ltd., London, 4th ed.,1934,p .. 332., 8.J. R. Freeman: "Hydra.ulic Laboratory Practice," Amedcan Society of Mecha.nical Engineers,NewYork,1929,p. 70:', 9.DonM.Corbettandot.hers:8trealn-ga.gingprocedure,U.S.GeologicnlSlI1vey, WaterSupply Paper888,1943. 10.N.C.GroverandA.W.Harri'ngtoo.:"S.ream FlOW,"JohnWiley&;80ns,Inc.) NewYork,Hl43. 11.Standards formethodsandrecordsofmeasurements,UnitedNatio7ls EconomicComm.isslcnforAsia:andtheFa:rFloodControlSeries,No.6, Ba.ngkok,1954,pp.26-30., 12.G.CorioUs: Sur.l'etablissemellt de Ill. formulequi donnela figuredes remons, et .sIU 12. ilorrectiontiu'ondoHyint,roduirePOllrtenircomptedesdiffel'encesdevitesse dans lesdiVerspoints d'unemarne sectiond'un COUl'ant(Ontheba.ckwater-curve equation a.tidthecorrections to be introduced to!lccount forthe difference ofthe velocitie$atdifferentpointsonthesamecrosssection),IvnmoireNo.268, ..,l,n'nalcadu punts etchaw;sees,vol.11,ser.1,pp.314-335,1836. 13.J.Boussinesq:Esg's'isurla theoriedeseauxcourantes(Onthetheoryofflowing waters), ]fr/;sentespardivensavantsril'AcademiedesSciences,Paris, 1877... 14.ErikG.W.Lindquist:DiscussionunPrecise.weirmeasurements,byErnesf W. Schader andT(ennethB. Turner, 1"7'(tllaac:l.ions,American Society of Civil Engineers, vol.93,pp.1163-1176,1929. 15.N. M. Shcha.pov: H Gidrometriia Gidrotelchnicheskikh SoorllllheniI i Gicir,omashin" (" HydrometryofHydrv.lllicStructuresandMacJ:Jnery ") IGosenel'goizciat, Moscow,1957,p.88... 16.StcponasKolupaila:Methodsofdetermin!l.tionofthekineticenergy facto!',The PortEngineer;Calcutta,India.,vol.5,no.I,pp.12-18,Januo.ry,1956. 17.M.P.O'BrienandG: H.Hickox:"AppliedFluidMechallics,"McGraw-Hill BookCompany,Inc.,NewYork,1st ed.,1937,p'.272.' 18,HoraceWilliamKing;i'Handbook ofHydraulics,"4thed.,l'evisedbyErnestF. Brater,McGraw-Hill Book Company, Inc., New York, 1954, p. '7-12. 19.MorroughP.O'Brien and Joe W. Johnson: Velocity-head correction for hydrau1ia flow,EngineeringNews-Record,vol.113,0.0.7,pp. 214-216,Aug.16,1934.. 20.Th.P..ehbQck':Die Bestimmung der I,agederEnergieliniebeiftiessendenGewful-sern mit HilIe des GeschwindigkeitshOhen-Ausgleichwertes(The determina.tion of thepositionoftheenergylineinflowingwaterwiththeo.idofvelocity-head a.djustment),DerBau.ingenieuT, Berlin, vol.. 3, no.15,pp. 453-455, Aug.15,11122. 21.Boris A.Bak&meteff: CorioIis and the energy principle in hydraulics, in "Theodore von !Urman Anniversary Volume," California. Institute of Teohnology, Pasadena, 1941,pp.59-65. 22.W. S.Eisenlohr: Coefficient's forvelocity distribution in open-Channelflow,Tra.ns-ac:I.ior/.$,American.Socie4/ofCivilEnginee7's,voL11:0,pp.633-644,1945.Dis-cussions,pp. 645-668. 23.J. B.Bela.nger:"Essai sur la solutionnumeriqnedeql,lelquesproblemesrelatifs aumou.-ementpermanentdeseaux courantes"("Essa.yontIleNumerica.l Solu-tionofSomeProblemsRelativetoSteadyFlowofWa.ter"),Carilian-Goeury, Paris,1828,pp.10-24. \ , 38BASICPRINCIPI..ES 24.It Ehrenberger: Versuche Iiber dieVerteilung der Drucke anWehrriicken infolge desI1bsturzcnden'.Vassers(Experiments on the distribution 'of pressuresa\ong the of w(d ..;;resulting from the impact of the fa.llingwater),Die W IMJserwirtschaft, Vienna,vol.22,no.5,pp.65-72,1929. 25.'H&raldLauffer:Druck,EnergieundFliesszustandinGerinnenmitgrossem Gefiille(Pressure,energy,andflowtype inchannels withhighgradients),Was-serkrafl, und Wasserwirtschaft,Munich,vol.30,no.7,pp. 78--82,1935. 26.J.H.Douma: Discussion on Open channel flowat highvelocities,by L.Standish Hall,in Entra.inment of atr in flowing water: a symposium,T1'ansactions, American Societyof CivilEngineers,vol.108,pp.1462-1473,1943. LeI!.'foI.l- Pit'67/\1c: ,Il.1 ,f;.-NeN .2.. h}j c/rct,;.d.i :/'t'--10c: c ,'" Y.5.,.(, < W/ibn.ftu;o 4,4ffluitf&Yl7 tUi,rv:5(id.. t f?. dtU. ,lillJ-' foe-tY-'.c,tt/).Je1' ..k.":Ina 4t- I_P J:. ..c."e J'.,t:;'I tt'l"1::>..;t;.1".!/f"'''V"'r1.tl.-7VW'-t&.4i-:5 p,ttt'a . e>Jrret,/1/0 fall. .. ,he.ellA-of ;$dtre,r/6..,;tLul vV(I!'n'YII!'.Vr';,t.. cd cbf #..is/l. J'Vi4 v..u ,pver.( e@'we - According to Newton's second law ofmotion,the change ofmomentum perunit oftimeinthebodyofwaterinaflowingchannel isequaltothe . 0( . resultant ofall the external forces that are acting on the body,Applying this principletO achannel of large slope (Fig.3-7),theexpl:ession . forthemomentumchangeperunittirnein;thebodyofwaterenclosed betv.:eensections1 and2 maybe written: J(..1 .;..(3-14) - cit!,., whereQ,w,and1:::' are :as .. previouslydefined,withsubscl'ipts refe1'l'in'gto' )>1'C( _,Yn 2; P! andP2arethe resultantsof pressuresactingonthe two sections; W isthe weight ofwaterenclosed between the sections; and F! isthetotalexternalforceQL friE,tionandthe lLUliaceof conta,Qtbet\V'een.thewater and the cha;nl)el.The aboveequa-' tion .is knownasthe m;omentumequation,l .I, 1The applicationofthe inomentum principlewasfirstsuggestedbyBelanger[5J. , r-j: 50 BASICPRINCIPLES For aparallei or gradually variedflow,the valuesofPI andP2in the momentumequationmaybecomputedbyassumingahYdrosta.tic distributionofpressure.For acur:-vilinea.rorrapidlyvariedflo"l,' how-ever,the pressure distribution isnolongerhydrostatic; hencethe of PI and Pz cannot be socomputed but must be cOl'rected forthe curva-ture effect of the streamlines of the flow.For simplicity,P1and P2 may bereplaced,respectively,by{)!'P1andwhere{:Jt'and{)z'arethe correctioncoefficientsat thetwosections.The coefficientsarereferred Fro.3-7.Application ofthe momentum.principle. toaspressure-dislributioncoefficients,SjnceandP2areforces,the. coeffi(Jientsmay be (JaIledforcecoefficients.It canbeshown thu,tthe forcecoefficientisexpressedby {:J' =1-::fAhdA=1 + fAcdA(3-15) AZ)owherez isthedepthofthecentroidofthewa.terareaAbelowthefree surface,histhepressureheadontheelementaryareadA,andc isthe pressure-headcorrection[Eq.(2-9)1,Itcaneasilybeseenth'1tpIis !.Treaterthan 1.0 forooncave flow,less than 1.0 forand equai to1.0f '1[1""< a-"1,..""; 7 ,rgf1lJ,o/,Jedinthe specific-energy . curve(Fig.Asthe curve isalmost verticalliear the critical depth, aslightinenergywouldchangethedepth toamuchsmalleror muchgreateralternatedepthcorrespondingtothespecificenergyafter lFor a accountof thetheory ofcritical low,see[1J. 63 ) ., I . ('. I ,\ I 64BASIC.PRINCIPLES thechange.It canbeobservedalsothat,whentheflowisnearthe criticalstate,thewatersUl'faceappearsunstableandwavy.Such phenomena aregenerallycausedbytheminorchnngesinenergy dueto variationsinchannelroughness,crosssection,slope,01'depositsof sedimentor debris.In thedesignofachannel,if thedepthisfoundat or neart,he criticaldepth fOl"agreat lengthofthechinnel,theshapeor slopeof the channelshouldbealtered,if practicable,inordertosecure greater stability. The criterionforacriticalstateofflow3-:3)isthebasisforthe computationofcriticalflow/whichwillbeexplainedinsubsequent articles.Twomajor applications of critiClll-fiow theory nre flowcontrol and flowmeasurement,whichwillalsobediscussedinthis chapter. 4-2.TheSectionFactor forCritical-flowComputation.Substituting VQ/ AinEq.(3-10)and Simplifying, .Q z=-v'g Whenthe energy coefficientisnO&assumedtobe unity, .Q z=--v'g!cY. (4-1) ('1-2) Intheaboveequations,ZA.v'D;whichisthet;ectionfactorJor critical-flowcompu.tation[Eq.(2--3)].Equation(4-2)statesthatthe section facto!:Zfora channel sectionat the critical state of flow equal tothedischargedividedbythe rootofU/Ol.Sincethesection factorZ is!l.functionofthe depth,theequationindicatesthatthereis onlyonepossiblecriticaldepth formaintainingthegivendischargeina. channelandsimilarlythat,whenthedepthisfixed, canbe'only onedischargethat maintains acritical flowand makes the depthcritical inthe g'ivenchannel section. Equation(4-1)or(4-2)isll.veryusefultoolforthecomputationand analysisof critical'flowin1111 openchannel.Whenthedischargeis given,theequationgivesthecriticalsectionfactorZcand,hence,the criticaldepthYo_Ontheotherhand"whenthedepthand,hepce,the sectionfactw'aregiven,thecritical!dishargecanbe by Eq.(4-1) 'inthe followingform: Q=Z Vg I orbyEq.(4-2)inthe followingform: . Q=z:(2 N;' (4-3) (4-4) ....o ci OpI puo' q/,( lOSGnlOI\ 65 tJ11"'1 .II . .., c 0,-, '" !" ...Q:5 . .., ... 'w .....':!.., "'0 '0 "-N1> '"OJa dt5 (5 g ;;:...lregiona,gainstoverflow;anduponthe deepening. ofthemOllths;baseduponsurveysandinvestigations... ,"J.E, LippincottCompany,Philadelphia,1861; reprinted inWashington, D.C., in18157, and asU.B.Army Corps(JfEllllineel's,PtofC8sionaiPap.,.No.13,1875. 12.H.DI.\!'cya.ndH.Ballin:"Rechercheshydra.uliques,'lrepartie,Recherches experimenta.lessurl'ecoulementdeFeaudansiescanauxdecouverts;2epartie, Recherchesexperiment&lesrel().tivesauxramousetil.In.propaga.tiondes,andes ("HydralllicResea.rches,"pt.1,'Experimentalresearchonflmvof water inopen channels;2,Experimenta.lresearchon 'bMkwaterandthepropagationof 'wave:;),ACMiemiedes Sciences,Paris,1865. 13.H.Bnzin: li:tuded'one nouvellefonnulspourca.lculerIedebit descanaux deooll-verts(AllllWformula.' forthe calcula.tion ofdischargeinopen channels),Mtmoire No.41, deBpontivol.14,seT.7,4metrimestrs,pp.20-70, 1897.. 14.RalphW.Powell: toflowinl'oughcha.nnels,'l',u,naadion'5,American GeophysicalUnion,vol.31,[lO,4,pp.575-582,August,1950. 1.5.HobertManning: Onthe flowofwaterinopenchannelsand pipes,Transactions, ofCivilEllgine(Jl'8'ofIreland,vol.20,pp.161-207,Dublin,lSIH; supplement, voL24,pp.179-207,1895, 15.VenTeChow:AnoteontheManningformula,Transadidns,AmericanGeo-physicalvol.36,no.4,p.688,August,1955. 17.AllenJ. C.OUlmingham:Recentbydrl'luIicexperiments,Proceedings,Institution .of CivilEngineel's,London,voL7l, pp.1-36,1883.. 18.Ph.Gauckler:Du1l10UVeme[lt l'eauda.::lsles()onduites(Theflowof water conduits),Annates .1chriu.ssees,vol.15,ser.4,pro229-281,1868. 10.A.St.ridder:J3eitriigezurFragederGeschwindigkeitsformr.iundderRauhi,g-keitsia.hlenfurStrome,!{a,nii.leundge.schbsseneLeitungen(Somecontributions tothe problem of velocity fonnul" and roughness c(lI;lfficientforrivers,canals,and dosed conduits). Jfillcil!l.nge'l deseidgen6:isischenAmteofiir }Vasserwirtachajt,Bern, no.16,1923. 20.ThomasBlench:Anewtheoryofturbulentfto,vinliquidsofsmallviscosity, Journal,b'stittlt'ion of einilEngineer.,Lolkion,vol.11,no.6,pp. 611-612,April, . 21.N. N.Pavlov'skiI:"GidravlkheskiI Spravochnik"("Handbook of Hydraulics"). This bookmany editions: (1)"Giclrl\vlicheskiI Spravochnilc," Put, Letlingrad, 1924,192 pp.; (2)"Uchebnyl GidravlicheskiI Spravoclwik'"(for 'Leningrad,lng, 100 pp.; 2ded,W3l,168 pp.;(3)"Gidrnvlicheskii Spravochnik," Onti,LeningradandMoacow,H}37,890pp;!lnd(4)"Kratki!GidravlicheskiI. Sprflvochnik,"(conciseversion),Gosstrolizd'at,LeningradandMoscow,1940,. 314pp., 22.GeorgeW; Pickets:Run-off investigat.ions in central IllinQis,University of Illinois, EngineeringExperim.entSta.iion,Bulletin232, ,vol.29,no.3,September,1931. 23.Frederick C. Scobey: The flowofwater influmes,U.S.Departm.ent of ;Agriculture, TechnicalBulletin No.393,December,1933. 24.Methodologyforcmpandpastureinundationdamageappraisal:"Training manual forhydrologistsonwatershedprotectionand floodpreventiQn work pIau DEVELOPMENTOFUNIFOIUI{FLOWITSFORM (JLAS 127 parties,"U.S. SoilConservation Sel'vice,Milwaukee, Wis., 1954. 25.E.W.Lane:DISCUSSIononSlopedischargeformulaeforalluvialstreamsa.nd byE. C. Schnackenberg,New Zealandof Enqillee!'s vol.37,pp.4.35-438,Wellington,1951.'.) 25.J .S.:'I,ndE.A.Schultz:Canal:Thesea-levelpI'oject,inA symposIUm.Tlda.lcurrents, AmeticanSocietyofCivilEngineers voL114,pp.668-571,' 27.Thomas R.Camp: Designof sewerstofacilitateflow,Sewage'Work.:;Joumal,voL 18,pp.I-HI, .January-December,194.6. 28.1i"lomasIt edlater,inArt.17-6.. TheSlope-areaMethod.The following inform:ttionisnecessary forthe slope-areamethqd:thedeterminationoftheenergyslopeinthechannel reach; the measurement ofthe a vel'agecross-sectionalarea,and the le.ngth ofthereach;andtheest.imationoftheroughnesscoefficientapplicable tothechannelreach,sothatfrictionallossescanbecalcu!il.ted.When this information is obtained, the discharge can be computed by auniform-flowformula,suchasl'vIa.nlling's.Theprocedureofcomputationisas follows: 1.FromtheknownvaluesofA,R,andn,computethe conveyances Ie, andKd,respeetively,ofthe Epstl'eam and dO\\'llstrealllof the reach. 2.ComputetheaverageconveyanceKofthereachasthegeometric meanof1(uandF:.o,' or (6-37) 3.Assumingzerovelocityhead,theenergy slopeisequal tothe fallF ofwatersurfaceinthereachdividedby the length Lofthe reach,or F' S =(6-38) The corresponding 'discharge may,therefore,be computed by Eq.(6-3), or Q=KvB. (6-3) whichgivesthefirstapproximationofthe discluirge. 4.Assumingdisclul.rgep.qltaltothefil'stapproximation,compute the velocity heads at theupstream [1,nddownstream sections, or au V,,2j2g andad V dZj2g.The energy slopeis,therefore,equalto s=!!:!. L (6-39) where (6-40) andkisafactor.'Whenthereachis(;ontracting(Vu discharge,slope,viscosity,anddegreeofsurfaceroughness.If velocitiesanddepths of floware relatively small,theviscositybecomesa domina.tiQgfactorandtheflowislaminar.InthiscasetheNewton's lD.wofviscosityapplies.Thislawexpressesthe the ......... 'r- .'\\ f'"'"(i"', r.' 1'.. ., (_'flop'li' ....\.., "Ii!.;.:.; fr. .(._,(.J,;.iii:. i n7 ....,)' ...' .,/,., AifC

FHl.fr9.Uniform laminaropen-channelflow. dynamicviscm:ityi"andtheshearstressratadist.anceyfromthe bOlmdarysurface(Fig.6-9)"asfollows: du r=I> dy (6-41) Foruniformlaminarflow,thecomponentofthegravitationalforce paralleltotheflowinanylaminarlayerisbalancedbythefrictional force.In otherwords,thcshearstress Tperunitarel;\ofthe flowalong 'the laminar layer PP (Fig.6-9)is equal to the effectivecomponent ofthe gravitationalforce,thatis,r=tV(Ym- y),c;.Sincetheunitweight tV=pgand1>/ P =v(Art.1-3),T=gl>(Ym- y)S/v.Thus,fromEq. (6-41), dv=g8(Ym- y)dy .J' Integrating andnotingthat v=0w;henY=0, v=g8(YYm_t) )I .2 (8-42) Thisisaquadraticequationindicatingthat' thevelocityofuniform, laminar flowinawideopenchannelhasaparabolicdistribution.Inte-150 UNIFORMFLOW grateEq.(6-42)fromY=O. toY =Ymanddividetheresultby Yrn;the averagevelocityis 11011' V=- vdy Y0 (6-43) andthe di:;;chargeperunit width is (6-44) whereCL= gSj3v,acoefficientinvolving slopeand viscosity. Uniformsurfaceflowbecomesturbulentifthesurfaceisroughandif thedepthofftmvissufficientlylargetoproducepersisting'eddies.In thiscasethesui'faceroughnessisadominatingfactor,andthevelocity canreadilybeexpressedbytheManning formula.Thus,the discharge perunit width is (6-45) wherey",istheaveragedepthofflowandwhereCT=a coefficientinvolvingslopeand roughness, Thechangeofstateofsheetflowfromlaminartoturbulenthn.sbeen studiedbyffi8.nyhydraulicians.Thetransitionalregionwasfound variously at R=310 by JeffreYmputethe corresponding disch'al'ge. 6-26.Showthat thecritic!!.lslope atn givennorma.ldepth !i4maybe expressed by (6-52) a.ndthatslope forawidecluumelis S=14.5n> crt. lIH (6-53) 6-27.Deterrnhiethelimit slopeof thechanneldescribed inExa.mple6-4, 6-28.Constructthecritical-slopecurvesofthecha.nneldescribedinExampie6-5 forbottom widths b== 1 ft,4It,2()ft,and"'. 6-29.Determinethecritical-slope.curvl',soi thechanneldesi::ibedinExample6-4 forsideslopeszI, 0.2,0.5,1,2,5,and"'. 6-30.Acha.nnelreac)11,OOOftloug afallof0.35ftinsurfa.ceduringIt flood.Computeflooddischargethroughthisreach,usingthe followingdata: SubsectionA,ft'11. Of Upstream: Main channel. ,. .......4,2500.0381.101.0'1 Sidechannel .... .. ,....... .25,6202,0500.0381.20L08 Downstream: channel ... , . ...... ..5,7603200.0421.101 ..04 SidechanneL........ ...25,610l,gOo0.038 i 1.181.06 6-:11.ProveEq.(6-46), 6-32.UsingEqs.(1-5)and(6-43),determinethevalueof]( inEq.(1-8}. 6-33.thedischargesperunitwidthofaghectflowona. surflLcwith 11 0.01andAS = 0.035when thedepthofflowis(a)0.01ftand(b)0.004ft.The temperature of water is6soF...'6-34.CompareHorton's criteria. forsheet flowinProb.6-33withthoseshOW\lby . cha'ttof Fig. 1-3.".... . 6-35.Showthatthevelocity-distributioncoeffici(mtsforla.minaruniformflowin wideopenchannelsarea""1,.54and1.20.. 6-36.UsingtheBlasiusequation(1-6)forturbulentflowinopencha.nnels, ;ahowthat the cOI'nsponding exponent, in Eq.(6-47)is:c= 177. I ,. i I ! I I I" . .'!"> ,,/.-:' COMPUTA'l'lONOFUNIFOR1I1FLOW155 REFERENCES 1.YenTeChow:Integratingtheeqltationofgraduallyvariedflol\',paper838, ProceediniJs,American,"!ocietl!of CiviiEngineers,voL81,pp.1-32,November, . 1955. 2.R.R. Chugaev: NekotorY(lvoprosy neravnomemogo dvizheniia vody votkrytykh prizmaticheskikhruslakh(Abollt somequestionsconcerningnonuniformflowof w!).ter in open channels), /zv6stiia.l'Besoiuznogo Nauchno-Issierlov(Ltel'skoIlO [nstitllta. aidrotekhniki(Tri.lnsa:clioM,AU-UnionScientific institl,/eofHydraulic Enyincerinp),1,pp. 157-227,1931. 3.PhillipZ.Kirpich:Dimeru;ionlessforhydr>l.ulicelementsofopell-cha.nnelCivilEngineering,vol.18,no,10,p.47,October,1948.. 4.N.N.PavlovskiI:"Gidra.vlicheskii Spravochnik"("Handbook ofHydmulics"), Onti,Leningra.dandMoscow,1937,p.515. 5.A.N.Rakhmanoff:0post.roeniikrivykhsvobodnolpoverlthnostiV.prir-mat-icheskikhi ruslakhpriusta.novivshemsia dyil!henii (Onthecon-structionofcurves01freesurfacesinprismaticnndcylindricalchannelswith establishedflow),hvestiiaV sesoiuznoyoNaudmo-Is:sledouaf.eI'.kO'lobl.lititula Gidrotekhllim(1'ranS(lctions,All-UnionScienli.fi.cResC(LrcflInstituteDfHydraulic Engineering),Leningrad,vol.3,pp. 75-114,1ll31. 6.Robert E.Horton; Separate roughnesscoefficientsforchannelbottom and sides, Jj7'dl""'.""diNews-ltccord,vol.111,no.22,pp.052-653,Nov.30,1933. 7,H.A.Einstein;Del'hydraulische odsrProJil-Radius(Thehydraulic(11'cross sec-tion radius), Ba.uzeilv,1!g,ZUrich,vo!'103,no.8,pp.89-91,Feb. 24, 1934. 8.AhmedM,Yassin:Meanroughnesscoefficientinopencho.nnelswithdifferent, roughnessofbedandsidewalls,technischeH oCMchllleZiirich, MitieilungenrutsderVcnu.chsa.nst(lltflITTfr(lsse!'oauunaErd:bau,No.27,Verlag LeemannJZUrich,1954. 9.N.N ..Pavlovskii:KvoprOStl0raschetnoI formuledlia ravnomernogodvizheniia yyociotoka.hk sllcodnorodllymi stenkami (On adesign formula. for uniform move inchannelswithnonhomogeneouswalls),Izul1siiaVsesoiUZllogoNau.cir.lIo-I 8sledovatel' skolJoInstiluta.Gidratekhnikt(T"(LMrtClions,,111-Scienlifu; Inslill1leof Hyd1"altiic Leningrad, vol.3,pp. 157-164,1931. 10.L.MUhlhofer:Rauhigiteitsuntersuclmngen in einem Stollen mit betonierter Soble undWand en(R(Jughne5Sinvestigationsin ashaft withconcrete bOUomandunlinedwo.lis),Wass8rlcra!tu.naWasserwir/,sdtafl,Munich,vol.28, no.8,pp. 85-83,1933.. ll.H. A.Einstein andR.B.Banks: Fluidresistanceof composite roughness,Trans- . actions,Am.erican GeophysicalUnion,vol.31,no.4,pp. 603-6jO,August,1950, 12.G.l{.Lott.er:SOQbrazheniiakgidravlicheokomuraschetucuselsl'!>tlichnoI sherokhovatosliiustenoI!:(Collsidera.tionsonhydraulicdesignofchannelsdifferentroughnessof\ir8.1I8),/zu.estiia. Nauchno-!ssledvvateJ,'skof/Q Instiluta.Gid1'otekhniki(Transa.ctiqna,.till-UnionSde1\tijicReseprchIn.stilutcof En(Jineering),Leningrad,voL9,pp.238-241,1933.. 13.G.It Lotter: Vliianieuslovii ledoobrazovaniia. itolshchinY l'da naraschct deriva-tsionnykh ka.nruov(Influenceof ofice formationandthicknessonthe designofderivationca-nlLls),IZTleiltiia.Vsesoiu.znogo Ins/itu/aGidrolekhltiki(Tra.nsactions,All-UnionScimtiji.cReseO;rch17Ultituteof Engineering),Leningrad,vol.7,pp.5&-80,1932. 14.G.Ie LoLter;Metod akademilca N. N. Pavlovskogo dUo.ojJcedeleniia koeflitsienta I i I i 156. UNIFORMFLOW sherokhovl1tostlrusel,pokrytyl,hI'dam(MethodbyAcademyMemberN.N. Pavlovskilforder.crminationDfroughnesscoefficientsofice-{loveredchannels), I TllVest{iaV seso;wm30 BHigh 11-24 CModerate 6-10 DLow 2-8 DLow Yn>y"the sUDcritlcalflowmust occur inamild channl!l(i.e.,achannel of subcritical slope).Onthe other hand,if y>y,>Yn,thesubcriticalflowmustoceurinasteepchannel (i.e.,achannelofsupercriticalslope).Similarly,thesecondcaseindi-cates Y< Ynand y< yo.Thecorresponding flowmust be supercritical; anditoccursi:namildchannelifYn>y,> !Iandinasteep channelif y,>Yn>y. Foradrawdowllcurve,dy/d:!:isnegativeaildEq.(9-13)givestwo possiblecases: 1.1- (Kn /K)2>0and1 2.1.- (Kn/K)"< 0and1 (Zc/Z)2< 0 (Z./Z)">0 The firstcaseindicates that Yo> Y>Yn. and, thus,that the flowissuper-Ci:iticalinasteepchanneL'Slffiiiarly,thesecondcaseindici,testhat V>Yo,9E.. that the flowissubcritical inarild channel. .! vVhenthewatersurfaceisparalleltothebottomofthechannel, dy/dx=0,and Eq.(9-13)gives1- (1(',/ K)2=0,or Y== yn,which indi-catesauniformnow.Tbe flowisaunjformcritical flowify=Yn=Yo, 8,Ulliform subcritjcal flowify=y ... >Yetand auniform sllpercritieaIflow ifYo> Yn=y.l70-r purposes of discussion,cha.nnel slope may be classified as sv.stair.ing, andnonsusta.ining.Asustainingslopeisachai1l1elslopethatfallsin thedirectionofflow.,Hence,asustainingslopeisalwayspositiveand' may alsobe ,called apositive slope.'A sustaining orpositive slopemay be critical,mild(sllbcritical),orsteep(supel'critical).Anonsustaining slopemaybeeitherhorizontaloradverse.Ahorizontalslopeisazero slope.. Anadverseslopeisanegativeslopethat risesinthedirectionof flow.. In achannelofhorizontalslope,Qt" So=(9-11)givesKn=00 . or K ... =Q,Eq.(9-13)hOrizo\ltal channels may bewritten Considering Un= dy_ dx- 1- (Zo/Z)2 (9-19) 00,thisequation indic,1testwopossibleconditions: -y>Yo flowwith r dy/ilxiss0\ve.esecondcaserepresentsasupercritical aackwatercurve, shownaspositive. ... 0:-----. II I I j j 224Gfu\DUALLYVARIEDFLOW In a channel of adverseor SD< 0,Eq:(9-11)indicatesthat,for negativevaluesofSD,K" :mustbeimaginaryor[(Himustbenegative [8].Consequently,Eq.(9-13)givestwopossiblecases: 1.A8ubcl'itical flowinwhichy> 2.A supercriticaJflowin' which Yyn > y.I B ...kwnt;,r! SubcrlU",,1 Mild -"12 lin l)u>Drnwdo\itnSuberitir.t'l NJy""> );.>!IBllckwn.terSupererltjoal 1.'1i 11>y, -I/n Sub.ducal j ParalleltoCritical C2 1),:; :=11-channelen tlcnl 8,-8,>0 bQttom C3II, -y. >11 Bll.ckwJlterSuper-critical 81 I '11> 1I.>Yn BackwaterSubcrit.icul Steep 1-]Sup.relitlcal 8. > 8, > 0 82y,>!I >y. .-133 . 11>'lin> !II"81-lpexcri,ti.en.l None J. 1i>(lIA)*>II' None AdverA2!(lin)' >!I >Y,8ubcriticnl S, y"> y,(Fig.9-4.Classification' ofFlowProfiles. 3 Forthegivendischargeand channelconditionst.henormal-depthandcriticahiepthlinesdividethe space inaehannel into threeLlones:. Zone1.The space ,,;bovetheupperline Zone2.The spacebetweenthe two lines Zone3.Th;:;spacebelowthe lowel'line Thus,theflowprofilesmaybeclassifiedintothirteendifferent;types accordingto t,henatureofthechannelslopeandthezoneinwhichthe flowsurfacelies.Thesetypesa.redesignatedasH2, H3;MI, M2,1113; CI,02,C3; 81,S2, S3;and A2, A3;rvhere the letter isdescriptive oithe slope: Hforh/)l'izontal,11-/ formild(sub-critical),0forcritical,S forsteep (supercritiCal), and A foradverse slope; and where the numeral represents thezonenumber.Ofthethirteen flowprofiles,twelveare forgraduJ1lly Bowand one,02,isforlltl.ieffll:fiaw It shouldbenotedthat a continuousflowprofileusuallyoccursonlyinoneLlone.Thegeneral characteristics of profilesare given in Table 9-1,and the shapesshown in9-2 and 9-4.Sincethe profiles near the critical depth and thechannel bottom co.nnotbe definedby the theoryofgradu-1It isbelievedthatthequestionofpointsofinflectionwasfirstdiscussedby Merten(13). This point ofinflectionoccurs becausetheprofile must have!J. horizont.al slope in crossingthetra.nsitionalprofileandthenbendforwardtangenttothe downstream pool level(Art.9-6).. , It isbeliP.ivedthat acomprehensive descriptiona.ndc!llSsification of tflow profileswerefirstgivenby Boudin [14]. ! 1 228GRADUALLYVARIEDFLOW allyvariedflow,theyareshownwithshortdashedordottedlines, Various flowprofilesare discussedbelow. A,MP1'ofiles.SQ< Seand Yll>Y. The -"11pt'ofilethemostwell.. knownbackwatercurve; it is themostimportantofall.Howprofilesfromthepracticalpointofview. ThiSprofileoeem'swhenthedownstreamendofalongmildchannelis submergedina l'eseryoir toa; greater depth than the normal depthofthe flowinthe channeLThis flowprofileliesillzoneLThe upstream end ofthecurveistangenttothenormal-depthline,sincedyldx=0as y=y,,;and the downstream end is tangent to the horizontal pool surface, Rincedy/dx=Snasyc:Q.Typieal exampleaof the 1111profile arethe profilebehindadam inanaturalriver(Fig,9-4a)andtheprofileilla canaljoining two reservoirs(Fig,9-4b).. AnMZprofileoccmswhenthebottomofthechannelatthedown-streamundissubmergedinareservoirtoa dept.hlessthanthe normal depth.Theupstreamendofthe flowprofileistangenttothenormal-depthline,sinaedyld:r:=0asYYn.If theamountofsubmergence at the downstreamend is lessthan the critical depth, the flowprofilewill terminate abruptly, with its end tangent to a:vertical line at a depth equaJ tothe .sincedyJdx=S. andy"< Yo_The Sl profilebegins with ajump' at the upstream and becomes tangent tothehorizontalpoollevelatthedownstreamend.Examplesarethe r

J dy!d + C'_ --- THEORYANDANALYSIS' (0) 229 / /"/ ;1, . d 230GRADUALLYVARIEDFLOW profiles of flowbehind adamina steep channel (Fig.9:-4g)and ina steep canal emptyingintoapoolofhighelevation(Fig.9-4h). The 82 profile isa drawdown curve.It is usually very short and rather likeatransitionbetweenahydraulicdropanduniformflow,sinceit starts upstieo.mwithavertical slopeat the criticaldepthandistangent tothenormal-depthlineatthedownstreamend.Examplesarethe profilesformedonthedownstreamsideofanenlargementofchannel section. (Fig.9-41.)andonthesteep-slopesideasthechannelslope changes from steepto(Fig. TheS3profileisalsoofthetra!Jsitionaltype,formedbetweenan issuing supercriticoJ flow .and the normal-depth line to which the profileis tangent.Examples aretheprofifeonthe steep-slope side as the channel slopechangesfromsteeptomildersteep(Fig.9-4k),andthatbelowa with the depth ofthe entering flowlessthan the normal depth ona steep slope9-4l). C.C P1ofilcs.Soandy ... =y . Theseprofilesthetransitionconditioi15betweenAiandS profiles.Assumingawiderectangularchannel,Eq.(9-17)showsthat C1andC3pl'ofilesIwecurved andthattheClprofile isasymptotic to a horizon talline(Fig.9-4mandn).WhenhhaGhezyfot'mula used JEq.(9-18)willshowthatthetwoprofilesarehorizontallines.TheC2 p7'ofilethecaseofuniformcriticalflow. D.HP1ofili3s.800 andy"=00 . Thesearethelimit.ingcasesof]..fprofileswhenthechannelbottom becomeshorizontal.The H2o,ndH3profile3correspondtothe1lf2.and 11{3profiles,but noH1profilecanactuallybeest,ablished,since y ... is infinite.Examples of Hprofilesare showni!l Fig.9-40' andp. E.AProfiles. < o..'The A 1 profileisimpossible,since the valueofy. isnot reaLThe/12 andA3profilesaresimilartotheH2andH3profiles,respectively.In general,Aprofilesoccurinfrequently.ExamplesareshowninFig. 9-4qand r. F.Profiles in Conduits witha GraduallyClosingTflp..For nnyconduit withagraduallyclosingtop,thenormaldischargewillincreaseasthe depth of flowincreases.It will increase firstto the value of fulldischarge atadepthyo'less'thanthe fulldepthyo.Thereafter,thedischarO'e willreacham!)'ximumvalueQ,matadepthYn *.Furtherincreasein depthof flowwilldecreasethedischargeeventuallytothe fulldischarge atthemomentwhentheflowsurfacetouchesthetopoftheconduit. 9-5a .showsthe variationofnormaldischargeinsuchaconduit. fllthe case of a circular conduit, the depth Yo'""O.82yo and y" * =O.938yo whereyoisthe diumeter ofthe conduit(Art.6-4).Withinthe regionof IJ ya'and y= Yo,there a.retwopossiblenormaldepths foragiven dis- . charge,namely,thelowernormaldepthy"andtheupperorconjugate itorrnaldepthy ,.'.. THEORYANALYSm 231 Followingtheprincipleusedintheprecedingparagraphs,itcanbe demonstrated that four types of Howprofiles are possible fora given slope [15-19J.Figure9-5b,c,anddshowsthesefol'mildandst.eep 8lopes.The positionsofthe depthsy"andy,,.'areassumed constant in FIG.9-5.Flowprofilesin aclosedthesefigures .. It shuuldbenotedthatthecriticaldepthinFig.9-5dis . greaterthan.thenormal. depthsy,.'andY.. ,butthatitscorresponding lowernormaldepthislessthanYfo'andYn.Consequently,thecorre-,spondingcriticalslopeshouldbelessth!l.l1thenormalslope,andthe channel slopeisconsideredmild. \ I ( I II 232GltADUALLYVARIEDFLOW

(a) ____ _ M;ldslOn/. , Milder slope Dependinc;;on M Idownstream (f) ---'---:::::.-__----:.---- Depending'lIl downstrecm t2.(tJ

.. ---- Jump Sfse':3 P./oPe,se s\O?e . FIG.9-6.Profiles of gradually variedflowin along prismatic channelwith 11 hreak in' bottomslope. 9":5.AnalysisofFlowProfile.Flow'"profileanalysisisaproQedure usedtopredictthegeneralshapeoftheflowprofile.It enablesthe engineertoleatnbeforehandthe possibleflowprofilesthatmay occur in agivenla.yout.Thisconstitutesaverysignificant part ofallproblemsinchanneldesign forgraduallyvariedflow. A.Pi'ismatiaChannelwithConstantSlope.Theflowprofileinlong prismaticchannelwithaconstant slope:hasbeendescribedin 94. [

, I I THEORYANDA.NALYSIS233 em)-"....CI; ...

---Cr'-"-. JlicG/ Slo,oe'! Mild.slope .. (q}------- ....... --..-. .:':'.:::.:..-::

(n) (s) Dependingon -.:.... C l'aownst ream (P) ___ Critic.C"\01/.'\:\...OPe IAdverse.slope "'-" - '0 I" 1/1171777Tmn,.,!._ LEGEND,I Thlcklinesindicate water surJoce FIG.96(Continued). Typicalexamples shown in Fig.9-4shduldbehelpful indetermining the typeofflowprofilein agivenproblem. B.PrismaticCha.nnelwithIi Change in Slope.This channel isequiva-lent to apair of connected prismatic channels of the same cross section but withslopes.Twentytypica.lflowprofilesinalongprismatic channelwith abreakin'slope are shown inFig.9-(3,Theseprofilesare ' self-explanatory.Howevel;,somespecial featuresshould bementioned; 1.' The profile near or at the critical depth caml0t be predicted precisely i:lythe theory of varied flow,sincetheflow isgenerally rapidly varied.., 2.In passing acritical line,the flow profile should, theoretically, have 0.234 GRADUALLYVARIEDFLOW vertical slope.Sincethe flowisusuallyrapidlyvariedwhenthe criticallinejtheactualslopeoftheprofilecannotbepredicted precisely bythetheory.Forthesamereason,thecriticaldepthmaynotoccur exactlya.bove breakoftheehannelbottomandmaybedifferent fromthedepth shown in the figure. 3.In some cases (Fig. 9-6gand l),the hydraulic jump may occur either intheupatl'eamchannelorinthe downstreamchannel,upon the relative steepness of the two slopes. 1 In case {I,for instance, the jump willoccur in tHedowllstream channel ifthe normal depth thischannel iscomparativelysmalLWhentheslopeofthedownstreamchannel decreasesand,.accordingly,thenormaldepthincreases,thejumpwill moveupstream,eventuallyintotheupstreamchannel.Theexact locationofthejump willbe disclissedinArt.15-7. 4.If theupstreamchannelhasanadverseslope(Fig.9-Bqtot),the dischargeisfixednotbyupst.reamchannelconditionsbut by theeleva-tionofthe upstreampoollevel,whichis the horizontal asymptote ofthe )12profile.Theprocedureofanalysisis .toassumeadischargeandto determine to whichcase q to t the profile shouldbelong.Then,compute theflowprofileintheupstreamdirectionanddeterminethepoollevel. H the computed level does not agree with the given pool level,then repeat the computation with another assumed discharge until the computed level agrees with thegivenleveL 5.Typical profiles(Fig.!H3)are illustratedforlongchannelsin,vhich auniformflowcanbe established farupstreamand downstream. C.Prismatic Chann,el with SeueralChanges in Slope.;For such channels the generalprocedure ofanalysis isas follows: 1.Plot the channel profilewith allexaggerated vertical scale. 2.Computey.,.foreachreach,and plot thenormal-depthline,shown bydashedlines,throughouttheentirechanneL' 3.Compute y. for each reach, and plot the critical-depth line, shown by dottedlines,throughout tilechanneL .4.Locateallpossiblecontrolsections.Atthecontrolsection,2:flow mustpassthroughacontrol depthwhichmaybethecritica,ldepth; thenormaldepth,or anyotherknowndepth.There arethreetypes of cont.rolsection: a.UPSTREAM CONTROL SIWTION.This occurs in any steep reach at the upstl'eamend,sincethe flowin asteepchannelhasto passthroughth$ critical sectionatthe upstreamendandthenfollow.either the Sl orS2 profile.Thecriticaldepththerefore,thecontrol depth(seealsoArt. 1 Alsodependingonrelativeroughnessandshapeofthe twoconnectingchan-' nels.Inthisdiscussionfactorsarcassumedconstant. Theterm"controlsection"usedherehasa. broadmeaning.It referstoany' sectionatwhichthe depth of flowis knownorcanbecontrolledto!l.requiredsta.ge. f THEORYANDANALYSIS235 4-5).If the downstream water surface isvery high, it may raisethe flow surfaceatthe upstreamcontrol.Whenseveralst.eepreachesoccurin succession,thecontrol section isattheupstreamendof theuppermost reach.Upstreamcontrolalsooccursinlongmildreaches,becausethe lVIlorlIlZcurveswillapproachthe normaldepth at theupstream end. b.DOWNSTREAMCONTROLSECTION.Thisoccursatthedownstream endinanylongsteepreach,becausetheflowwillapproachth.enormal at the downstream end.If the downstream end of a mild channel terminatesatafreeoverfall,the control Section 'maybe assumed at the brink wherethe depthiscrit.icaL r C.ARTIFICIALCONTROLSECTION.Thisoccmsatacontrolstructure, sur.hasav;eil',dam,orsluicegate,at whichthe control deptheitheris knownorcanbedetermined. 5.Starting D,tthecontroldepthateachcontrolsection,traceineach reachacontinuousprofile.The positionofthe profile in each reachcan belocated withrespecttothe andcritical-depth lines. For thispurpose,typicalprofilesdescribedpreviously(Art.94)should befounduseful. B.Whenflo''''issuper criticalintheupsi:.reamportionofareachbut subcriticaIinthec1ownst,reamportion,theflowprofilehastopassthe critical depth soniewhere in the reach.In crossing the critical-depth line, ahydraulic jump isusually crsl:'.ted in raising the wat,et surfaf -.8 I tangle SeboldiaehIIras.d IgnoredCBm8' .tangle MononobeAU.hape.1 COlloli'idered P1UI;OD"\ A.!c tttfM:t Le.AilConoideredManningK: tt AI cc lICOI.l,at VQn 6egll'Olrn AllConsideredManningK.:ctZ21% 1JJlKeifur-Clm butCon.ideredM .. nningNone .he method mAY beex tandedt.oother 3hB.pea253 ..Refer--'cnce tiJ i2J [31 lSI (5] Ii,S) [IOJ1111(12]lIS1 [14.15] [16] (17) theintegrationiscarriedoutby short-rangestepsandwiththe aidofa varied-flowfunction. In an attempt toimprove Bakhmeteff's method,Mononobe [13]intro-duced twoasllumptionsforhydraulic exponents.Bytheseassumptions the effects9fvelocitychangeandfrictionheadaretakenintoaccount' integrally without the necessityofdividingthechannellength 'into short reaches.Thus, the Mononobe method affords a more direct and accurate computation procedure wherebyresuH.s, canbeobtained without recourse. in kinetic energy tothe friction slope,or rinEq.(9-14),i3a.ssUI;nedcOMbnt in each rea.ch.Sinceanincrea.seordecreaseindepthwillcha.ngeboththese fa.ctorsinthe sa.me direction,their ratiois relativelystable a.ndcan-be IloBsumedconstantforprall-tical purpose!!. 254GRADUALLYVARIEDFLO"," tosuccessivest.eps.Inapplyingthismethodtopracticalproblems,it has been found!.ha,t the firstassumption (see Table 10-2)is not very satis-factoryinmany cases.Another drawback ofthismethodperhaps lies in the difficulty of using the accompanying. charts, ."hich arenot sufficiently accurate forpl'acticalpurposes. Later,Lee Tl4]and Von Seggern[16]suggested newassumptions which resultinmoresatisfactorysolutions.VonSeggernintroducedanew varied-flowhmctionin[t,dclitiontothefUllctionusedbyBakhmeteff j hence,aDadditional table for the newfuncticn isnecessary in hismethod. In Lee'smethod,however,nonew.functionisrequired. Themethod[18]describedhereIStheoutcomeofastudyofmany existing methods.'By this method,the hydraulic exponents arE;expressed in terms ofthe depth of flow.From Eqs.(6-10)and (4-6),/{,,2=Gly"N, /(2=G,yN,Z.2=C2y,M,and Z2=C2yM, where G, and C2 are coefficient,s. If the;3eexpressionsaresubstituted inEq.(9-13),thegradually-v.aried- .' flow. equationbecomes (10-2) Letu=yly,,;the above'equationmaybeexpressedfOl'dxas dx [1- 1ltN+1 ] dtt(10-3) This equation can be integrated for the length ,;ofthe fio,vprofile.Since thechangeindepthofa.graduallyvariedflowisgeneraIJysmall,the hydraulicexponentsmaybeassumedconstantwithintherangeofthe limitsofintegration.II:_...U!.e _eXQonentL1l y"the channelsiope ismild.As thedepthofflowissUingfromthe slUIce!!;atels.1essthan the criticaldepth,theflowprofileisoftheM3type..T'Considering aflilverage depth of1.61 Ct,the hydrauhc exponents areI\0=3.43and M=3.17.Thus,J.. 2.72,N IJ=1.26,lindIN=0,442,.':. Table10-3 'showst.hecomputationof theflowprofile.ForconvenUl[lcemmter-- -..:..-. -----..: ___

---- ... - .... -------258GRADUALLYVARIEDFLOW polating values af F(u,N) fromthe varieti-flOlv,function.tablervalues of uare assigned atregularinter',:[LI:0c.o 1""""1 C"J O. ': Xh' "> ... p;j'" -0-'" '": "" '"--;0-'".,; '"...!::-d ;: I'" j 0 r ., ., --.-.... -0 ;e ,,; o co ;:; '" 0 '" 0 --Oil .....0 0 0 '" 0 --0-'" '" '"-'-_.... -j 0 ... 0 '" t";'".. ... iii'"--",-.;::'" " 0

gj

"xx X:;; 1 (10-59) hdhdtback"'atereffectarealso wherethevelocity- eacn.nges'ue0n neglected.FromEqs.(10-58)u.nd(10-59) I '1." L'r=Q/v'F (10-60) ,Q/ y'F iscalledthedischargefor1-ftfal!.1!his equationC8,l:be usedilltheflow-profileeomputntionifthestage-fall-dischargerelatlOn-shtpforuniformflowinthe reachis known. The stLlge-fall-dischargerelationship f.or11 selected reach may bedeter-mined froni.records ofobserved stages and dischargesCfable 10-9).The stagesorwater-surfaceelevationsat thebeginningsection _ofthereach areplotted asol'dinates,andcorrespondingvaluesofQ/y'F areplotted as resultingina C'urve(Fig.10-13).When any water-surface elevation the_beginning sectionofthe reach fSgiven, thecorresPQndingvalueofQ/ v'Fcanbereadfromthecurve,andthe fallforadischrLrgeQ.becomputed by Eq.(10-60).Thecomputedfall, whenadded tothe water-surface elevation at the beginning section ofthe -reachgivesthewater-surfaceelevationattheendsectionofthereac.h, isalsothewater-surfaeeelevationatthebeginningsectionofthe nextreach.The procedureisrepeated foreach reachuntilthecomplete ,requiredflowprofileisob,tained. :Thestage-versus-Q/v'Fcurveisgenerally asanaverage 'curveforvaryingriverconditions;suchasrisingandfallingofstage, ,lIn asimilarmethodbyRakhmanoff[341atermF /Q'is. us7dinlieuof Q/ This termhasthenatureof/J. resistance factor and therdore ISgiven!I IIame ofresistance'modulusbyPavlovskiI [21,p.1151. e.;.-, 282GRADUALLYVARIEDFLOW fluctuating stream bed,and effects of wind,aquatic growth, ice, .and over-bankflow .. Owingtothesevaryingconditions,theplottedpoints20re oftenscattered;andasmoothline,givingconsiderationtothevarying conditions,shouldbedrawn through the points,representing the average conditionofthe-channel.Wheresufficientmeasurementsareavailable, dataofdoubtfulaccuracyshouldberejected.Ingeneral,themore recentmeaSurementsshould begivengreate: weight,asreflectingrecent channelchanges.Other factorsthat shouldbeconsideredinconstruct-ingthecurvearethe relativeaccuracyofindividualdischargemeasure-ments;the,flowconditionduringthemeasurements,whetherrising, falling,or stationary; conditions affecting the stage-fall-discharge relation-ship,such as the changes in channel roughness,levee bteaks,and shifts of cha,nnel controls; and the existence of substantial local inflow between the stations. The stage-yersus-Q/VF curve may qe extrapolated !].boveor below t.he rangeoftheobserveddata by extendingthecurve at its endsinaccord-ancewiththegeneraltrendofthecurvature.Howe-.:er,anyabrupt changeinhydraulicelementsofthechannelsectionwillproducea,eor-respondingchange inthecurvature ofthecurve.In thiscase,acorrec.-tion forthe change,if known,shouldbemade in extrapolating the curve. This method is used most advantageously when a number ofdischarges correspondingtoknownstages,orviceversa,a.redesiredinastream. By makingproperallowanceforvariableconditions,satisfactoryresults ca.nbe obtained for reaches of large rivers 50to100 miles from the measur-ingstation.Thedata requiredbythemethOdareoften .les;:!expensive the,nthose requiredby the standard step method.However, this advan-tage.is usually.offset by the inaccuracy ofthe-results, because the effecl of thechange in velocityhead 1s ignoredinthepresentmethod.Forthis' reason,the stage-falldischargemethod ismore satisfactoryforproblems . iriwhichthevelocityiswellbelowcritical enddecreasesinthedown-stream direction. E:x:amplelO-ll.Computethe water-sudace elevation at section1 of theMissouri Riverproblem in Example 10-10 by the sto.ge-fall-discharga method.The reach from section1tosection5istakea asthefirstreach.Thewater-surfaceelevationsare availablefromstagerecordsforgageslocatedatsections1and5.Thedischarges havebeen observed at the A.S.B.Bridge located about 3,000 ft downstream fromsec-tion1.These data are tabulated in Table10-9.'. Solution.The data aridcornputa tionsforthe sta.geversusdischarge foraI-it-fall! curve aregivenin Table10-9,which contains the followingheadings: Col.1.Recorded water-surfaceelevations at section1 Col.2.Recorded wo.ter-surfaceelevationsat section 5 Col.3-. Fa.llinft,whichisequal tothe betweenelevationsenteredin cols.2and1 1Thisexample istakenfrom[291withmodifications. METHODSOFCOMPUTATION 283 Col.4.0bserveddischarges,at.the, A ..8J3.Btidgeji!lds Col.5.DischargeperI-ft faIl,orQ/ y'F, whereQ:is,th'edischa.rgein col.4and F istilE;fa.llincol.3. Usingwater-surfa.cecle\'ationsat section1aslistedincol.1oftheta.bleandthe CQl:respondingvaluesofQ/ ..\/ifiincol.5,construct0. sta.ge-versus-Q; vFcurve 10-13). 755752,25\1- ' 750 crt Ii :745, !III 740r--r- I '0 .. ., '0735 c .120; > '" 730 '" -[- w-" "": .. g .. 725 0 oI II 3;

III 7201 III 715 50100ISO200250 0 ValuesOfQ/IFinl.o00 I 1 h __I I 300.,,350 "., "., FIG.10-13.The stage-vs.-Q(V'F curveforExample10-11. For awe.ter-surfaceelevationof752.25,avalue ofQ/ -../F i5obtained by extrapolation.By Eq.(10-60),the fall between sections 1 ancH isequal to(431,000/ 355,000)'=1.65ft.Ad'ding- thisva.lue. totheelevationatsection1,required water-suffaceelevationat section5 is753.90.Thisisabouthalfafootlowerthan theelevation.by the standa.rd step method; the differenceresultsprimarily fromthe neglectofvelocity-headchanges inthepresent-method. Thecomputationmaybecontinuedforsubsequent reaches.Atabulation,.as 5hownin Table10-10,issuggestedforthecomputationifa. completeflowprofileis requiled.. ,I 1 "\ 284 GRADUALLYVARIEDFLOW Ifthe,,:,ater-surface at the intermediate sections2to4rnabe by breakmg upthe reach1-5 Into four short reachesThe profile1atth.td'.e eva,.lons eIIIermela/,,",sectlOnsmaybeobtainedbyin'erpel-t'"Th Q/, 1- ,,}B Ion..estage-versus-.vFcurves can bedrawn foreach sectionand the cbmputat'ob.d fol'the subdividedreaches.'.Incane carneout TAllLE10-9.,DATA,ANDCOMPUTATIONSFORSrAGE-vs.-Q/YFCURVEUSED IN 10-11 (MissouriRiveratE:ansasCity,Mo.,sections1to5) Water-surfaceelevation,m,s.l.11 ,F11D'hII, ,ISCal'ge, Section1Section5 ftcfs VF (1) (2) I (3) I (4)

Q 724.8725,7 0.933,600 35,40',) 725',3 726.20.036,100 38,000 729.13 I 730.2O.G66,100 85,30U 727.4 728.30.969,500 7'6,200 727.8728.81.076,000 76,UOO 730.2 731.21.097,200 97,200 730.8 731.70.9105,000 111,000 731.3732.31.0113,000 113,000 734.6 735.610141,000141,000 735.8 736.70,9 157,000165,000 736,6 737.71.1104,000156,000 745,0 746.61.6326,000'258,000 722.2 723.10.822,90025,600 724.6 72561.045,400 45,100 725.0726.01.049,900 49,9UO 725.3 I 726.41.152,300 '19,800 TA.BLE10-10. OFTHEFLOWP'ROFILEFOil.EXAMPLE10_11BY THEST,lGE-FA.LL-nrSCHARGEII-fETHOD (MissouriRiver at KansasCityMo'1 to5f) .'".','.,=431,000ds) I Sea.RiverLengthof I no.milereachelevation Q/-./FF. (Q/ -./F) 2 i 1377.58 752.25..... r 335,000 1.65 1.65 I 5378.655,655753,90 .... '.. . "....... ....To becontinuedifdesired'... 10.,8.The Method forNaturalChannels.'If flowprofilesfora numbel'ofdischargesorstagesaredesired,thestage-fall.discharge can be use.d most advantageously for a simple and economical but approXlrnate solutlOn.However)ifa precisecomputation,including the METHODSOFCOMPUTATION285 effectsofvelocity-headchangesandeddylosses,isrequired,theEzra methoddescribedin Art.10-5shouldprovidemoresatisfactoryresults. Example 1012.Determinethew!),ter-surfaceelevationsa.tsections1to5ofthe MissouriRiveratlCa-nsasCity,M.a.,asdescribedinExample10-10.Theda.ta requiredforthe computation bythe Ezra method aregiveninTable10-7.The dis-chargeis431,000cis.The initia.lwater-surfaceelevationatsection1is752,25.It isassumed that eddy lossesare included in the frictionlosses. S:)/u.t-ior..ThefirststepistocomputevalueofZ+ F(Z)fromthegivendata. The computation'istabulatedinTable10-11withthe followingcolumn headings: Col.1.'Channel-section number Col.2.Rivermileage CoL3.Lengthofreachinft.Theuppervaluef>:Cdisthelengthofthedown-streamreachfromthe selected section,andthelower va.lueLlx.isthelengthofthe upstream reach.,' CoL4.'\Vater-surfa.ceelevations.Threeelevationsaregivenforeachsection. Generally,atleast,threeelevationsareseiectedforeaahsectiontoprovideatleast three points forplottingeachZ+ F(Z)curve. Cols.5to14.ThesecolumnscorrespondexactlytothoseinTable10-8forthe standard step method.The values inthe top rowforeachelevlltional'efor the mllio cha!lnel,and thoseinthebottomrowarc fortheleft-overbank section. CoL,15.}'rictioii slope,whichisequalto where Q=431,000 cisand lCis fromcol.10 Col.16.Valueof.-8,ClXd/2,whereS.isthe 'valuefromcol.15andClXd is the uppervalueincol.3 Col.17.ValueofS, t..x./2,where S. isthe value fromcol.15and C.x"isthe,lower value incol.3' Col.18.ValueofF(Z,),whichisequaltothe sumofthe value incoL14.and the valueincol.16_ ('..01.19.of F(Z2),whichisequaltothe sumofthe va.hl"- incol.14 and the value in col.17 Col.20.Sum ofthe valuesofZincol.1and F(Z,)incoL18 Col.21.Sumoft.hevaluesofZincol.4and F(Z,)in col.Ii) The second stP.pis to plot curves ofZ+ F (Z)against Zforeach oross section,using, valuesfromcols.4,20,and21ofTl1blelCl-11.Theresultiugcurvesareshownin Fig.10-14.. Thethirdstepistodeterminethewater-surfaceelevationsfromtheZ+ F(Z) curves.AtsectionI,fora.ninitialwater-surfaaeelevationof752:25,thevalueof + isfoundfromt.heappropriatecurve(Fig.10-14)tobe754.14.Taking thisvaluetotheZ,+ F(Z ,)curve forthe next upstream section2,the corresponding water-slUiaceelevationisfoundtobe752.72.Contiuuingtheprocedureforother sections,the valuesaretrD.Cedinthe direction shown by thcdashed line inFig,1014. The rcsults of the IV ater-surfac,e-elevation determinat.ionare tabulated in Table10-12. Theyareinverycloseagreementwith those obtainedbythe standard stepmethod, Example 10-13..Solvetheproblent in Example10-12 fora dischargeof500,000 ds, The correspondinginitialwuter-surfaceelevatior.&tsection1 wasestimated fromthe ratingCUrvetobe752.30, Solution.Thevaluesof F(Z ,)and F(Z2)forQ=500,000cfsmaybeobtailledby mUltip1yingthecorrespondingvaluesinTable10-11by(500,000/431,000)'=1.34.' valuesthus aretabulatedin.cols.3and 4ofTable10-13,respectivel}', and the values ofZI+ F(Z.)and Z,+ F(Z.)in cols.5and6,t:,espectively. ;;< .t co 0> 8ee. no, . "" co -l i TA.BLE10-11.COMPllT.AT10NOFZl+ F(Z!)ANnZ2+F(Z,)FORE:u.MPLE10-12 (MissouriRiverat Kl\llsasCity.Mo.;Q- 431,000cfs) ApR I R% ".K Ie' A' . I.f J -0.151 0.12 .... 11,&3 0.11.... 11.80 I' 0.10... 1.71 0.131.461.70 1.391.65 1.3111.55 TAllLE10-11.COMPUTATIONOFZl+ F(Z,)ANDz. + jt'(Z.)FORl:i:::CHtI'LR10-12(conli'n"lJ.ed) 0.24 1.1111,45\ 0.21l.oJ 1.341 -0.080..1.001.27 -O.Hi0.5J0;90 -0.150.120.5,0.83 -0.130.11 0.78 0.62 H'OUI,,u;onr'"x I ... 1 7,9007411,10.64.B3,O.Q5011 0.58 0.54 ,.....---. +F(Z.) 753.93 754.80 7.55.71 753.46753.7il 154..39754.66 755.31755.M + F(Z.l 753.11753."'5 754.04754.34 75S.00755: 27 M2.61752 .. 90' 753.5u753.g3 751.54754.78 752,62 753.58 288 GRADUALLYVARIEDFLOW TABLE10-12.COMPUTATIOl\OFTHEFLOWPROFILEEXAMPLE10-12DY THEEZRAMETHOD ;:: .: o :;::

OJ0; ., '-' 754 " '" , OJ753 (MissouriRiveratCity,Mo.,Q = 431,000cfs) I Sec.River no.mile 37758 2377.78 3377.9,1 4378.33 5378.65 Z,+ f(Z.) Iz, + F(Z,) 'ffater-surface elevationZ, It . . . .. .754..[4 752.25 754.14754.41.75272 754.41754.68753.38 754.G8754.93. 7154.15 754.93 764.43 754.43

75ZL-_________L__ 756 755754753.754755 756 Zl+F(Zll,ftZ2+F{Z21.ft FIG.10-14. 'CurvesofZ+ F(Z)forExample10-12. 755756. FIG.10-15.CurvesofZ+ F(Z)forExample10-13; METHODSOFCOMPUTATION289 TABLE10-13.COMPUTATIONOFZ+ F(Z) EXAMli'LE10-13 . (Missouri Riverat.KansasCity,Mo.,Q =500,000cfs)

ZF(Z,)F(Z.)z. + F(Z,)z. + F(Z.) no. (2)(3) .(4) (5)(6) 7522.59754.59 7532.41755.41 7542.2975G.20 27521. 962.36753.{16751L36 75:11. 862.22754.8G755.22 7541. 782.08755.78756.08 37521,491.94753.49753: 94 753lAO1.80754.40754.80 754.1.341. 70755.31,755.70 47520.821. 21752.8275321 7530.751.11753.75754.11 7540.731.05754.73755.05 57520.83752.83 7530.787fi3.78 7540.7375 =40ft,and L=50 miles. Allriversareassumedtohaverectallgularchannels.Determinethe junction deptl\ andthe flowprofilesill theriversafter thefloodflowapproaches a .stead);condition. REFERENCES 1,BorisA.Bakhliletefi:"HydraulicsofOpenChannels,"McGraw-HillBook Compo.ny,Inc.,NewYork,1932,pp.143-215. 326GHADUALLYVARIEDFLOW 2.JulhtrlHinds: The hydraulic designofflumeand siphon TrcT.nsadions, America" Societyof 'CivilEngineers,vol.92,pp.1423-1459, .1928.' '3."C;vilWorks:FloodintheLosAngelesArea,"TIl[)EngineerSchool, Beh'oir,Virginia,1950,E206.00(4-50)ML,pp.22-28 and plate10. 4.L.Hall:Openchannelflowat highvelocities,inEnt.rainmentofairill flowing\\'flter:asymposium,Transactions,AmeriwnSodetvojEngillec",3, vol.lOS,pp.1394-1447,1943. 5.FredC.Scobey:Theflowofwaterinflumes,U.S.DepC!1'lmenlojAg"ic7l!t1l7'e, TechnimlB1t/lelinNo.303,December,1933. G.Hrdraulic designd!\1;a,appendixI ofCanalSand related structures,U.S.Bure(1.(L ojReclamalion, andCons/rllction.liJ(l.n1Lal,DesignSu.pplwtentNo.3,1952, vol.X,pt,.2,paragro.phI-13.' 7.JulianHinds:Thehydraulicjumpanderit,icaldepthiuthedesignofhydraulic EngineeringNcws-RsClJrd,vol.85,no.22,pp.1034-1040,Nov.25, 1920. 8.WallD.ceM. La.nsfordand WilliD.1nD.Mitchell:AninvestigD.tionoft.hebackwater pro(ileforsteadyflowin channels,UniversityojIllinois,En!1ineering EXpcTl:lI1cnlStalion,Bulletin SeriesNo.381,vol.46,no.51,lViD.1'ch,1949. 1.1. Willi"'ffiD.Mitchell:Sf,age-fall-clisohargereiatioilsforsteadyflowin ehfl.Tlno\;l'j"'I':TI lJ")N....... - ....... M'1""""1000 -0 a0 -0 a .N .......I-l""""'.MtOr-:'ZO t'- ..-!t- '-di

]1--------------------, :3t:;:; e'r...: L..jc-i cO....c ,..... ............................ .,......1--'--1 J31-0--'-0-0-""-"'-,.....-00.--,.....--1

"'t-t-r-LOC' f-f-(/)z wl!! a:u (.Ju::3"-

"-u a:z wo >-of-u , Q, we:: f-f-z e>O U E5 ;r-n. II I , I PH/'H'poal{uo!sapo.uo JO O!'O!,l 375 0.562 - ,l.:i2.0.796 .... .. 1.53 o ,. 2:3 X/He (0) 3 .tv) 45 45 2.141.75 6 2.140.$55 6 ., , \ \ \ I \ I I I, I' I .. +dhd .He 0.201 2.140.513 4 -1.0 4 (d) 5 5 6 6 0.049 0.OS3 0.142 FIG. Typical pressure(dashed lines)and surface(solid lines)profiles forflowover submerged overflow dams.(Selected fromU.S.Bureau ofReclamationdata [1J.)(a)Supercriticn.[ flow;(b)flowinvolving hydraulic jump; (e)ftowwith!l. drowned hydraulicjump;(d)flowa.pproachingcompletesubmergence. 388UAPlD!wYVAlUEDFLOW prevalent onthe dowl1strel1mapron;(1) snpel'critical flow,(2)su bcritical flowinvolvinghydraulic jump,(3)flowaccompanied by a drownedjump withdivingjet,and(4)flowapproachingcompletesubmergence. Submergence of spillway orweirwillreduce the coeflicientofdischarge ofthecorrespondingullsubmergedflow.TheBUl'eA.uofReclamation's testresults onthisreduction,expressedinpercenta.geofthedischarge coefficientforunsubmergedflow(Fig.14-4),havebeenpresentedina chart forthe four types of flowmentioned abov;.This chart in a slightly modified form(Fig.14-17) was further checked against other datal by the uArmy Engineers Waterways Experiment StatiOll. 2It was found thltt the, chartisalsoapplicable tothedeterminationofcoefficientsforWES ,shapes under submerged conditions. In the chart (Fig.14-17)h,Jisthe drop fromthe upper poolto the tail water elevation, H, is the total hend above the crest,and d isthe tailwil.ter depth.Thegenera.!patternofthecurvesshowsthat,forlow'ratios (h.J+ d)/ H"the flowisof type 1,orsupercritical,and that the incoefficientisaffectedessentiallybythisratioo.ndispracticallyinde-pendent ofhd/l-I..Thecr-osssectionBEinthe upper right-hand corner ofthechartshowsthevariationof(hd+ d)/H,athdill,=0.78.For large 'm.lues oi (h,l+ d)/H., on the other hand, the reduction incoefficient is [1ffected by the ratio ad/H,.Under this condition. for values ofhatH, lessthan 0.10,the flowisoftype 4,the jet ison thesurface,and noJumpoccurs.Forvaluesofhd/H.greaterthan0.10,thoflowisof type3,oraccompaniedbyadrownedjumpwithdivingjet.Thecross sectionAA shows the variations ofha/H. at (h.+ d)/H, near 5.0.Sub-criticr.lliow,or flowof type 2,occurs in,theindicated onthecimrt. Otherregionsfortransitior1D,lflowconditiollSare alsoshown. Thetypicalpressureandsurfa.ceproiilesfor,submergedspillwayflow areshown fordifferentvalues of(hrl+ d)/li. and A4/H. forfolU'typesof flow(Fig.1. '" '"co" '" Disc horg e0 Coso.I '"m, '" "",' .ilL toilwclerrating DischargeQCose4 -," >-'C ICJUMPANDl'.rSUSEA.S ];}NERGYDISSIPA.TOR429 6.The slc;rpeof the chute upstre:1mJrom astilling basin has Uttleeffect onthejump u.slongas the distribution ofvelocityand depth of floware reasonablyuniformon entering thejump. 7.A snlallsolidtriangular sillwithaslopingupstream surface,placed at the end of theapron,isthe only appurtenanGe needed.This servesto lifttheflowasitleavestheapronandthusactstoconUolscour.Its dimensioasarenotcritical;themosteffectiveheighti"between0.05 and 0.10ofthe vertical distance Qfthe sequent tailwatel' elevation above the bottom of&he toe of the jump, and the surface slope Chllbe 3: 1 to 2: 1: Section A'A FIG.15.,22.Oblique hydf/mJicjtlmp . 15-17.TheObliqueJump.Whenasupercriticn,lflowisdeflected inward to thecourse ofthe flowbyaverticalbounchtry(Fig.15-22),the depthofflow'111 willincreaseabruptlytoadepthYzalongawavefront CDwhichextendsoutfromthepointofboundarydiscontinuityata waveangle{3 thatdepends in magnitudeontheangleofdeflection(J of theboundary.Thisphenomenonresemblesanormalhydraulicjump butwiththeohangeindepthoccurringalonganoblique front;henceit maybecalledan hydra'l.#cj1tmp.lWhen8=0.,obliquejump becomesthe familiarnydraulic jump inwhichI;hewavefront isnormal to the directionofftow,or(3 90., Referringtotherelationshipofvelocityvectorsbeforethe .iumpin Fig.15-22,the velocity normal to the wavefront if:1 V"IVI sill fJw11ere V lis the velocity of flow before the jump.The Froude number normal to .1For original infonnlltion see[781and [79J.Thooblique hydr!Lulicju'mpor oblique. sta.ndingwaveisalsoknown asthe shockwave,byanalogytotheC:1.Se in :mpersoniv flowof gases,The basic development of thi:5subject was aacDmplisherlby Rouse and White .[SDI. RAPIDLYVARIEDFLOW thewD,vefrontbefore thejumpis,therefore, F V"lY 1 sin (3 F.(3nl=--= ==-113111-(15-20) ConsideringasectionA-AnOl'lnaltbthe wavefront,it is seen that a nor-malhydraulicjumpoccursinthissectionandthatEq.(321)callbe applied.SubstitutingEq,(15-20)forFlinEq.(3-21),theratioofthe sequent toinitialdepth is =7Z( Vl+8F12 - 1) Y, ( 15-21) Thisistheeqllationtlmtrepresentstheconditionfor oblique hydraulicjumptotakeplace. IteferringtQFig.15-22,theta,ngentialvelocit.iesbeforeandafterthe jump are VII=TT"i/t::m (3 andV 12 =V,,2/tan(3 - 0).Since no Jl1ornen-tumchfl,l1getakesplacepatalleltothewavefront,thesetwovelocities shou1dbeequal,or V"Itan-f3 =tan ({3 - 0) (15-22) By the condition ofcontinuity 'VI V"I=Y! Y,,2,'the above equation 08,nbe written _ Y2tan f3Y;=tB.n(3 - 8) (15-23) Elimino,tingudUl fromEqs.(15-21)and(15-2:3),arelationshipinvolv-ingF1;0,and .13 isobtained: tan f3 + SFl ?t;;in2 (3 ....: 3) tfl,n0- - 2tan2 {3 + VI + 8Fl 2sin2 f3 -- 1 This equation should produce the value oft3ifF 1and.e are g-iven.How-ever,adirect solution oft.hisequation for{3 interms ofFl ande ispracti-callyimpossible.Ippen[78Jhaspreparedafour-quadrantgraph _(Fig. 1:'5-23)showingallrelationshipsexpressedbyEqs.(15-21),(15-23),and (15-24).This gl'aphis8elf-explanatory andcanbe used forthe solut,ion ofanobliquehydraulicjump. Sincetheobliquejumpisanormaljumpacrossthe sectionA-A, the energy10:0:0intheobliquej'umpcanbecomputedbyEq.(3-24).In practicalproblemsinvolvinganobliquejump,Y2!YI'isusuallysmall. Thus,theheadlossmayoftenbeneglectedindesign. normal jump,ifvdvi'Urface' atnpOlutequidistant.fromthe:thisequation the foil.oviringI'emarksshoulcl' benoted:. 1.For],. lessthan 40 tiInes the square root of t,he water area, no further deepeningofthechannelseemsttlresultfromtheincreasedcurvature; hence,insuchcases,theivalueofriJusedint.heequationshouldbe 40VA.Consequently,'bendsareconst,!'llctiveandstahlewhenr.is . greaterthem40sharperbendsaredestructive,te,ndingto shiftthechannel. 2.For r.gren.terthn.llabout110VA, the equationbacome.svalid. I I .r-Centerof Cur .... o1vreFtG.10-7.Empiricalchannelsectionat riverbend. 3.The equation ma.ybenpp!idtocU'rvedchannelsnot occupying the entire width of the wntenvay or to those at the river entrance created by a. singlecurvedjetty.Insuchcases,K26.28andthe'value1)fythus compltted shouldbe in