Open Channel Hydraulics

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OpenChannelHydraulics i .l I . l

TerrvW. Sturm Georgia Ins'tut. oI Te.hnology

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lA Madison,Wl NewYork Boston BunRidge,lL Dubuqu_e, SanFranciscoSt.Louis BanEROkBogote Caracas KualaLumpur Lisbon LondonMaddd MexicoCity Milan Montreal NewDelhi SantiagoSeoul SingaporeSydney Taipei Toronto

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McGraw-Hill Seriesin Water Resourcesand Environmental Engineering CoNsuLTtNcEDrroR George Tchobanoglous, Univenity

of Caldornia,

Davis

Bailey and Ollist Biochemical Engineering Fundamentals Bedsmin: WaterChemistry Bishopt Pollution Prevention: Fundamentals ond practice Canlert Environmcnral Impact Assessment Cha etli Envirc^rnantal Protection Chapran SurfaceWarer-Quality Modeting Choq Maldment, ard May$ Applied Hydrology Crites and Tchobanog)otrsi Snwll and Decentrclized WastewoterMonogerrun, Sysrems Davis and Cornwelft Inrruduction to Errvimnnqatal Engineering deNevers: Air PollutioaCon,rol Engineering Eckenfefder: Industrial WaterPollurion Contml Ewels, Ergas, Chang, and Schroeder: Bioretwdiation principles LaGrega, Buckingham, and Evans: Hdzardous Wd$teManogenan, Linsley, Franzini, Frtyberg, and Tchobanogtous: Water Resourcel. and Engineeing M&het WaterSupplyand Sewage Metceff & Fddy,Inc.i WastewaterEngineering: Collection and pumping olWastewater M€tcalf & Fddy, Inc-i WastewoterEngineering: Treatment, Disposal, Reusc Peary, Rowe, and Tchobanoglous: Environmental Engineering Rittmann snd McCarTli Environmentol Biotechnology: principles and Applicotions Rubin: Introduction to Engineering and the Environment Sswyer, McCarty, and Partkin: Chemistry for Environmental Engineering Slurmr Open Charnel Hydraulics Tchobanoglous,.Thies€n, and Vigit: Integrated Solid WasteManagement:Engineering Principlesand ManagementIssues

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OPEN CHANNEL TTDRAIJLICS Intematioml Edition200I This Exclusiverights by McGraw'Hill Book Co Singapore,tbr manufactureandexPort The by *{rich is sold McGraw-Hill it to counlry from the roexported .aruroibe book InternationatEditionis not availablein Norlh Amenca' Publislredby McGraw-Hitl,an imprint of The Mccnv-Hill Companies'lnc-, l22l Avenue of tire Americas,New Yodq NY l@20. CopyriSbtO 2001,by TheMcGraw-Hill or iomoanies, hc. ell rights reservedNo pan oftbis publicationaraybe reproducei system, ol retrieval a databsse in or s1orcd distriiured in anyfonn or by anymeans, without tlle Drio;written consentofthe Mcclaw-Hill ComPanics,Inc , iocluding,bl,ttoot or hoadcastfor ti-lt"a to, ii -v ""t""rk or oths electooic storageor u'ansrnission' distanc€teaming. iorte ancil"riei inctudingelectronicarrdprint compooents,maytrot be svailableto customersoutsidcthe United Slet€s

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CONTENTS

Preface

xl

BasicPrinciples l.l 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 LlO

lntroduction Characteristics of OpenChannelFlow SolutionofOpenChannelFlowProblems Purpose HistoricalBackground Definitions BasicEquations Surfacevs. FormResistance DimensionalAnalysis ComputerPrograms

I I

I ) J

) 6 lt IJ

l7

Specific Energr

2l

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

2l 23 26 28

Definitionof SpecificEnergy SpecificEnergyDiagram Choke DischargeDiagram ContractionsandExpansionswith Headloss Critical Depthin Nonrectangular Sections OverbankFlow Wein

JI

34 39 48

Sharp-Crcsted Recmngulor Notch Weir / Sharp-Crested Triangulu NoEh Weir / Brcad-Crested Weir

2.9 EneryyEquationin a StratifiedFlow

))

Momentum

6l

3.1 3.2 3.3 3.4 3.5 3.6

6l 6l 74 78 8l 84

Introduction HydraulicJump StillingBasins Surges BridgePiers SupercriticalTransitions Design of Supercritical Con rcction / Designol Supercritical &pansion

Conlents

vut

4

Uniform Flow 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Introduction AnalYsis Dimensional MomentumAnalysis of theChezyandManningFormulas Background LogarithmicFormulafrom ModernFluid Mechanics Discussionof FactorsAffecting/and n

Selectionof Manning'sn in NaturalChannels Channelswith CompositeRoughness Uniform Flow ComPutations 4.10 Pattly Full Flow in Smooth,CircularConduits 4.1I GravitySewerDesign 4.12 ComPoundChannels 4.13 Riprap-LinedChannels 4.14 Grass-LinedChannels 4.15 SlopeClassihcation 4.16 Best HydraulicSection Manning'sFormula 4.17 DimensionallyHomogeneous

97 98 99 l0O 102 109 I 14 ll4 I 19 l2l 122 126 129 132 137 l4l

4.18 ChannelPhotograPhs

142 142

Gradually Varied Flow

t59

5.1 5.2 5.3

Introduction Equationof GraduallyVariedFlow Classificationof WaterSurfaceProfiles

5.4 5.5

Lake DischargeProblem WaterSurfaceProfile Computation DistanceDeterminedfrom DepthChanges DirectStepMelhod / DirectNumeicalIntegration

5.6 5.7 5.8

DepthComputedfrom DistanceChanges NaturalChannels

5.9 FloodwayEncroachmentAnalysis 5.10 Bress€Soludon 5.1I SpatiallyVariedFlow

159 159 l6l 165 t67 168 174 l8l 189 190

r92

Hydraulic Structurts

201

6.1 Introduction 6.2 Spillways 6.3 SpillwayAeration

zol 202 2to

lx

Contcnts, 6.4 SteppedSpillways 6.5 Culvens hlet Control / Ou,lerContml / Road Otenopping / Improvedlnlets

2t3 215

6.6 Bridges HEC-2 and HEC-MS / HDS-I / USGSVtidh Contraction Method / WSPROModel / IISPRO Input Data / WSPROOutput Data

233

Governing Equations of UnsteadyFlow

26'l

7.1 Introduction ?.2 Derivationof Saint-Venant Equations

267 269

Conlinuity Equatibn / Momentun Equatiorl

7.3 7.4 7.5 7.6

Transformation to Characteristic Form Mathematical Interpretation of Characteristics Initial andBoundary Conditions SimpleWave

274 277 219 282

Dom-Break Problcm

NumericalSolutionof the UnsteadyFlow Equations 295 8.1 Introduction 295 8.2 Methodof Characteristics 8.3 BoundaryConditions 8.4 ExplicitFiniteDifference Methods

297 301 305

Lal Difasive Scheme/ kapfrog Scheme / Lax.Wendrof Scheme / Predictor-Corrector Methods / Flux-Splining Schemes/ Stability 8.5

Implicit Finite Difference Method

8.6 E.7 8.8 8.9

Comparison of NumericalMethods Shocks Dam-BreakProblem PracticalAspectsof RiverComputations

Simplified Methods of Flow Routlng 9.1 lntroduction 9.2 HydrologicRouting

313 319

3m 324 326 JJJ JJJ

334

Resemoir Routing / River Routing

9.3 KinematicWaveRouting 9.4 Diftrsion Routing 9.5 Muskingum-CungeMethod

345 352 356

Contents

l0

Flow in Alluvial Channels l0.l Introduction Properties 10.2 Sediment Pa icle Size/ PorticleShape/ PanicleSpecifc Gravity/ BulkSpecifcWeight/ Fall Veloci4/ Arain SizeDistribution 10.3 Initiationof Motion 10.4 Applicationto StableChannelDesign 10.5 Bed Forms Relationships 10.6 Stage-Discharge Method/ VanRijn'sMethod/ Karin' Engelund's KennedyMethod Discharge 10.7 Sediment Sediignl Bed-loadDischorge/ Suspended Discharge Discherge/ TotalSediment 10.8 StreambedAdjustmentsand Scour AggradationandDegradation/ BridgeContraction Scour / Incal Scour/ ToralScour

AppendixA NumerlcalMethods A.l Introduction A.2 NonlinearAlgebraicEquations Intenal HolvingMethod/ SecontMethod / Netton' RaphsonMethod A.3 Finite DifferenceApproximations

AppendixB Examplesof ComputerProgramsln VisualBASIC B.l YOYC Programfor Calculationof Normal andCritical Depth in a TrapezoidalChannel Y|YCFormCode/ Y|YCModuleCode B.2 Ycomp Programfor FindingMultiple Critical Depths in a CompoundChannel YcompFormCode/ YconpModuleCode B.3 WSP Programfor Water SurfaceProfrle Computation WSPFormCode/ WSPModuleCode Index

311

3'tI 372

380 388 389 396

404

457 457 458

463

467 467

469 476

483

PREFACE

The studyof open channelhydraulicsis a challengingand excitingendeavor becai.rse of the influenceof gravity on free surfaceflows. The position of the free surfaceis not known a priori, andcounterintuitivephenomenacan occur from the liewpoint of the first-time sludentof open channelflow. This book offers a study of gravity flows staning from a firm foundarionin modern fluid mechanicsthat includesboth experimentalresultsand numericalcomputationtechniques. The developmentof the subject matter proceedsfrom basic fundamentalsto selected applicationswith numerousworked-outexamples.Experimentalresultsand their comparisonwith theoryare usedthroughoutthe book to developan understanding flow phenomena.Computationaltools range from spreadsheets of free-surface to computerprogramsto solve moredifficult problems.Somecomputerprogramsafe providedin Vsual BASIC, both as leamingtools and asexamplesto encouragethe use of computationalmethods regardlessof the platform available in a very dynamicenvironment.In addition, severalwell-known computerpackagesavailable in the public domain are demonstratedand discussedto inform userswith respectto lhe methodologiesemployedand their limitations. The basicequationsofcontinuity, energy,and momentumare derivedfor open cbannelflow in the first chapter,from the viewpoint of both a finite control volume and an infinitesimalcontrol volume,althoughthe completederivationof the general unsteadyform of the differential momentumequationis savedfor Chapter7. Dimensionalanalysisis introducedin somedetail in the hrst chapterberauseof its use throughout the book. This is followed by Chapters 2 and 3 on the specific energyconceptand the momentumfunction.respectively,and their applicationsto open channelflow problems.Designof open channelsfor uniform flow is examined in Chapter4 with a detailedconsiderationof the estimationof flow rcsistance. Applicationsincludethe designofchannelswith vegetativeandrock riprap linings, and the design of storm and sanita4r sewers.Chapter 5, on gradually varied flow, emphasizesmodem numerical solution techniques. The methodology for watersurface profile computation used in current computer prcgrams promulgated by federal agenciesis discussed,and example problems are given. The design of hydraulic structures,including spillways,culverts, and bridges,is the subjectof Chapter 6. Accepted computer programs used in such design are introduced and their methodologiesthoroughly explored.Chapters7, 8, and 9 developcunent techniquesfor the solution of the one-dimensionalSaint-Venantequationsof unsteadyflow and their simplifications. In Chapter 7, the Saint-Venantequations are derived, and the method of characteristics is introduced for the simple wave problem as a meansof understandingthe matbematical transformation of the governing equationsinto characteristic form. The numerical techniquesof explicit and implicit finite differences and the numerical method of characteristicsare given in Chapter 8, with applications to hydroelectric transientsin headracesand tailraces,

xl

Prcface

xii

co'rerssimplified problem,and flood routingin rivers Chapter9 the dam-break nrethod' diffusion method' wave of flow routingincludingrhe kin-ematic methods chanalluvial of subjecr complcx ;;;';; lt;;;ki"s";-b,i,lg" nt"tttod Finrllv' the explored is surface free adjusiable nel flows thathavea movablebed as rvell asan links amongsedimentdisinponant the emphasizcs chapter rrtls r" ti "pli ioof essentialto an.u.nderstanding .n-gaiU"O forms,and flow "ti'tunt" that are adjustchannel alluvial are l0 op.n'.itunn"tnonuin rivers.Also coveredin Chapter flow blockase and shapeiand bed scouiin responseto the J;;;';iilLrm, causedbY bridgefoundations' to supPlementthe text material The first is The book includestwo appendices techniquesthat can be used , ".n"iui discussionof somi selectednumerical someexamplecomputerprocontains ,rrt?r"rr"*-t.trebook. The secondappendix ;i noIilal and critical depth in prismaticchannels' ;l#ffi;';;;;;";;,t* of waler-surfaceproirles These channels'and computation. i".i"oi"g .t.p"""h aids for. more exrensrveprc. as leaming ;" written in visual BASIC ;;;;r".: chapters'6n a website for the book' addiu,',i. "na-or*tal !;ffi;;;;t;i;;; exerciseson unsteadyflow comadvinced ionJ proir^*. for solutionof the more if necessaryin a dynamic updated be can ""o,iJ", i"" be found, where they

;'f"':l"J#ilj?i,),.Ti;n, r.*?;l;;;;;"i"

andnrsrundersraduales roradvanced is inrended environmental and i" trt" g"**r fieldsof waterresources sufficient provide 9. ? throush Chapters and i'il"dsh 5

ch;ffi coveringbothsteadyand hydraulics courserr openchannei materialfor a semester courseor a sengraduate u*i*Oy no". fn" bookalsocanbe usedfor a first-year andriverhydraulics'utilizingChapters i", .i.tii"i-.."*" on hydraulicstructures andexample whichincludesseveralapplications +,'s,'is, "iJ 10.n"s materia.l, for responsibilitv ; ur.rur ,o the fractitionerchargedwith the ilii #il;r, culvert spillwaydesignfor smallteseryoirs' i""ft'lrks "t nt aplainmanagement' of alluoi stabilityandflow resistance investigaion -J."*", a"tig" r,ir orainage, applied this of "stimatioriofbridgeiackwaterandstour.Because il ;;;il;J library consultingengineer's i*"r J,ii"'m"r. it shouldbe a uJefuladditionto a flow' ofopenchannel ' u, " po.ti"al textbookon *re fundamentals -ur-*"fi to ard.intheunderstandproblems example worked-out ;;;h'rp";"ontains form Wherepostibt"'tolution'aregiven.indimensionless ins of thetextmaterial. and problem the of iniuiJu'"una.''ondingof thephvsics llt#n"il ;;il;; chapeach of the end of its solutionovera widerangeof variablesAt thebehavior in thechapteras of the.material thatinvolveapplication ;" ptesented ;;; ;;;;;i; In some material' the text ""'U "t ti"J* "'-pf"tation of furtherrarnificationsof by presentation and i"do.utoryresultsaregivenfor datareduction ;;;#;;;i verify text material' studenb "'""d; to experimentally of it"ttctional andresearchmaterialsdevelopedover out bJ;;*to*n in openchannelflow andsedcout'esequence *J "i"A in a g'adoate **r"iy"-. thatI havetaughtat the eduiation.course i**io't .pon ^t *ell asii a continuing of its uniquefocus on fundamentalsas 6'""r *iJi i*" of Technology'Becau-se resultsas well as numericalanalvsis'this ffiiuuJ *fi;;il'-*J"*pJtl-""ttl

ffitffi;;:

book shouldfill a niche b€tweenexhausiivehandbooksand purely academicuea_ tiseson ihe subjectof open channelhydraulics. I am indebtedto more peoplethan t can enumerateherefor the completionof .. this project.My initial motivationfor preparingfor an academiccareerin hydraul_ ics datesbackto a keynoteaddressthatI hearddeliveredby HunterRouse,who was an accomplishedorator as well as writ€r, at a conferenceheld at the Universitv of Iowa. The subjectwas the careersof famoushydrauliciansincluding rheir foiLles as well as achievements.I later graduatedfrom the Universityof Iowa under the lateJackKennedy,who was a continuinginspirationto a strugglingph.D. student. I am much indebtedto the continuingencouragement given by BJn C. yen at the Universityof Illinois, where I receivedmy B.S. and M.S. degreesin Civil Engi_ neering,and Fiward R. Holley at the Universityof Texasat Austin over the course of my careerC. SamuelMartin hasservedas mentorandcolleaguefor many years at GeorgiaTech.The encouragement and researchcollaborationof my coileague Amit Aminharajahhas been invaluible. I owe much to the previous treatiseson openchannelhydraulicsby Ven Te Chow and F. M. Henderson,as do many olher authorsaswell as practirioners.Reviewcommentsby JohnnyMorris, Larry Mays, and Ben C. Yen, and suggestionsby EdwardR. Holley have led to an improved manuscript,although I bear the responsibility for any errors or shortcomings that remain.I expressmy gratitudeto Mark Landersofthe USGSfor locating and providing copiesof the river slidesby Bames. My students have been a continuing source of motivation for me to try rc explain complex aspectsof open channelhydraulicswirh clarity. I have leamed much from their curiosity and probing questionsabout the details of o;rn channel flow phenomena. Finally, I am forever indebtedto my wife, Candy,whosepadence,love, and s-upportbrought me through this project, and to my grown children, Geofrrey, Sarah,and Christy, through whoseeyesI continually seethe wodd anew.

Torhememoryol my brorherTim ( 1949-1998), entlangdem sichewig windendenStrontdesI'ebens' Reisegenosse

C]IIAPTI]R I

BasicPrinciples

t.l INTRODUCT'ION O p e nc h a n n ehl y d r a u l i ci s r h es r u d yo f r h ep h y s i c o s f f l u i df l o w i n c o n re ya n c e si n whjch the flowingfluid formsa freesurfaceand is drivenby gravity.The primary cascof inrerestin thisbookis waterastheflowing fluid ha!ing an inlerfaceor free surfaceformcd*,ith theantbientatmosphcre, but lhe basicprinciplesalsoapplyto othercasessuchasdensity-stratified flows.Naturalopenchannels includebrooks, streams,rivers,and estuaries. Anificial openchannelsare exemplifiedby storm sewers,sanitarysewers,and culvens flowing partly full, as well as drainage ditches,irrjgationcrnals,aqueducts, andflooddiversionchannels. Applications of opcnchanneihydraulics rangefrom the designof anificial channels for beneficial purposes suchasirrigation,drainage, watersupply,and wastewater conveyance to the analysisof floodingin naturalwaterwaysto delineatefloodplainsand assess flood damagesfor a flood of spccifiedfrequency.Principlesof open channel hydraulicsalsoareulilizedto dcscribethetransporland faleof environrnental contan'linanls, includingthosecarriedby sedinrcnts in morion,as *ell as to predict flood surgescausedby dambreaksor hurricanes.

1.2 CIIARACTERISTICS OF OPEN CHANNEL FLOW Althoughthe basicprinciplesof fluid mechanicsare applicableto openchannel flow, suchflow is considerably morecomplexthanclosedconduitflow dueto the frce surface.The relcvantforcescausingandresistingmotionandthe inefliamust form a balancesuchthatthe frcc surfaccis a strca',rlinealongwhichthe pressure is coDsl.lr)t andequalto almospheric pressure. Tltis extradegrceof freedomin open

2

: B a s i cP r i n c i p l e s C B P T T - :l R

chanacl flow means rhat the flow boundariesno longer are fixed by the conduit geometry, as in closed conduit flow, but rather the free surfacead.justsitself to .rccomnrodatethe gi\ en flow conditionr. Another importantcharacteristicof open channel florv is the extremevariability encounteredin cross-scctionalshapeand roughness.Conditions range from a circular gravity scwerflowing partly full to a natural river channelwith a floodplain subject to overbank flow. Roughnessheights in the gravity sewer correspondto those encounteredin closedconduit flow, while roughnesselementssuch as brush, vegetation,and deadfallsin natural open channels make the roughnessextremely difficult to quantify. Even in the case of the circular gravity sewer,resistlnce to flou is complicatedby the changein cross-secrionalshapeas the depthchanges.In allur ial channels,the boundaryitsclf is movable, giving rise to bed forms that pror ide a funher conlributiunto florr re.istrnce. Becauseof the free surface,gravity is the driving force in open channeltlow. The ratio of inertial to gravity forces in open channel flow is the most irnportanr governing dinrensionlessparanreter.It is called the Froude number, deltned Lty

F: - Y* (sD)'-

(l.l)

in which V is the meanvelocity,D is a lengthscalerelatedto depth.andg is gravIn someinstances the Re1'nolds numberalsois imponant,as itationalacceleration. in closedconduitflow, butoneof the few simplifications in naturalopenchannels is the existence of a largeReynoldsnumberso that viscouseffectsassumeless imponance.Flow resistance in this casecan be dominatedby form resistance, pressure which is associated with asymmetric disfibutionsresulting fromflow separation.The success Manning's equation in characterizing of openchannelflow resistance in factdepends on theexistence of a Reynoldsnumberlargeenoughthat the Manning'sresistance factoris invariantu irh Reynoldsnumber.

1.3 SOLUTION OF OPEN CHANNEL FLO\Y PROBLEMS The complexitiesofferedby open channelflow often can be dealtwith througha combination of theoryandexperiment, asin otier branches of fluidmechanics. The basicprinciplesof continuity, energyconsenation,andforce-momentum flux balancemustbe satisfied, but we often mustreson to experiments to completethe solutionof theproblem.Theresultingrelationships canbe quitecomplicated, espegeometryis considered. cially whenthe variabilityof thecross-sectional past,the designof openchannels ln the not-too-distant wasachieved with the aid of numerous nomographs andgraphicalrelationships because of thenonlinearity of thc govemingequations combinedwith complexgeometry. More extensive analysisof unsteadyflow problemsor gradually variedflow problemsassociated with river floodplainsrequiredmainframecomputers.Presently, the proliferationof personalcomputersandengineeringworkstationshasprovidedmuchgreateraccessibility and flexibilityfor simpleas well as complexproblemsin openchannel

I 1 B a s i cP r i n c i p l c s l CtlAPrF-R \ \ ' i t hi m m c d i a l ef c c d b a c ko f r c s u l t s h l t l r a u l i c s .P r o g r a m sl h a t a r e l r u l ) i n l c r a c t i ! e T h e h 1 d r a u l i ec n p i n c ccr a n i n t h e f o r n l o fs c r c c n g r a P h l c s r ' t n t ' \ - * r i r t c n u i t h e a s e 3 n d t h e i r i n l p l i c a t i o n si n a c o m p l c l e l ) i " r a . u i g " , "a * i d e a r r - a yo f d c s t g ns o l t t l i o n s O n t h e . o t h e rh a n d 's u c h r n o d ci n t h c n t o d c r nc n g i n c c r i n gr r o r t s t a t i o n in(cractive o a p p l i c a t i o n sf a c c c p t e dp r o g r a r l st h a t . l " r a o i u r a , o t n . , i t n e sI c r d st o n r i s i n f . r n t e d 10 flcrsonalcomptrlcrs havc bccn lran\Poned from lltc tnainfrlmc

1.4

PURPOSE n u n l c r i c atl c c h n i q u c sf o r t h e s o l u t i o n T h c t h ! ' m eo f t h i s b o o k t s t o p r c s c n ln r c d c t n

n":^o:::l:',::.i:,ii",...,lli"l,ir'l'r"il'*r:::i:i;l:::'JT "i'"p."'ii-".r exPcnmclr well as to en'lphaslze transponin ailub1'sedinrent ;;;t. Tt; proii.. or u variablebed surfacccaused of aPplication the placcdon .'ii .f,r"".it is (reatedas well ln addition'focusis probflow oinu,o -"tr"nitt ro the formulationof .pen channel ;;'';;;i:';i;: modelsnow widely ano limitationsof the nurnerical lems.so thatthe assumptlons andnumerexperimenta!' of theoretical' ;;;il;;i" "r" rnadecleai The cornbination become has that to openchannelflow providesa synthesis ;;;h-",q;;plied thehallmarkof modernfluid mechanics'

1.5 HIS'[ORICAL BACKGROUND of hydraulics relieson the excellenthisloricaltreatment The followingdiscussjon f u r t l r e dr e t a i l s ' f o r i s r e f c r r c d r e a d e r t h e "'v R o u s ea n dI n c e( 1 9 5 7 )t,o w h i c h b has channels opcn in \tater of cotrvcyance the ;ro; irr" Jt*i "i.i"irization' andMesopolamiasinigationfor thellSyptians becnusedto mcetbasicneeds,sLrch in theMiddle U'1ry:lt for disposal *istt and ans,watrr supplyfor theRonlans' ln transmission somecases' discase ,og;t,;irh ,ttl'ail"strousresultsof waterborne while in otbersnaturalriver channels artificialopen channelswcre ton'trutttd' \""'e r eu r i l i / e dt o c o n \ c y$ a t c rJ n d \ \ a s t e s ' throughcanals ur.d u du* for waterdiversionandgravityflow d;;;;pii^"t developedcanalsto Mesoporami.ans to ,rirtnuu['iut", from the Nile River,andthe rivers'but thereis no recordedevi. transferwater from the Euphrat€sto the figris principJesinvolved'The Chinese i""i. "f""y ""a"tstanding of the theoreticalflow for prorectionfrom.flooding several are known to havedeviseda systemof dikes of watersupplvpipesandbrickconduitsfor drainage Evidence ,h;;; ;;;;;g" of valley The success y;rs B.c.hasbcenfoundin int indut River ;;;;i000 onlv' likelv the resultof experience ;;;; ;;tiy,;;;""tive hvdraulicworkswas water from springsto distribution transport to were used Romanaqueducts by masonry supported masonry.canals rvcrereclangular, The aqueducts reservoirs. in longitudinal slope.The "r.rr.r, ""0 they conforned to the n,iural topography areaof flow is the cross-sectional waterdischargein the aqueducts*ut 'neu'u"d

4

C H A p T T Rl : B a s i cP r i n c i p l e s

u,ith no regard for the r elocity or slopc producingrhc velocity.Alrhough rhe exis_ t e n c eo f a c o n s e r v a t i opnr i n c i p l ew a sr c c o g n i z e dt .h c c o n s c r v e d q u a n t i t ro f v o l u m e flux was misundcrstood.Yct, these itqueductsserved their .ngin".r,,,g purpore, albeit inefficiently and uneconomicallyin nrotlernlerms. The philosophicalapproachof the Grceks toward physical phenomcnawas rcvived by the Scholasticismof the N4iddle.{ges, and it renrainedior L_eonardo da Vinci to introducethe exprcrimcnlal rnethodin open channelflow during the Renais_ sance.konardo's prolific writings includedobservations of the velocitv distribution in riversand a correctunderstanding ol the continuityprinciptein srcams with nar_ rowing width. Someearlt experimental resultson pipe andchannelflo*, $ ere rcponcd by Du Buat in 1816,but rhe cxperintcntalwork on canalsbegunby Darcl,ard com_ pletedby Bazinin the late l9th centurv,and Bazin'sexperimentson weirs.\\ cre unsur_ passedat the timc and remainan enduringlegacyto the experimentalapproach. The problcm of o;xn channclflow resistancewas recognizedas imponant by many engineersin the I Sth and l9rh centuries.The work of Chezy on fl oq, resisrrnce beganin 1768,originaringfrom an engineeringproblemof sizing a canal to deliver water from the Yvctte River to paris. The resistancecoefficieni attributed to him, however,was introducedmuch laterbecausehis work dealt only with ratios of the independentvariablesof slopeand hydraulicradiusto the l/2 power in a relationship for velcrity ratiosin difrerentstreams.His work was not publisheduntil the lgth century.The Manning equationfor openchrnnel flow reiistarrce, aboutwhich much will be said in this book, has a complex historical development but was based on field observations.Thc Irish engineerRoben Manning actually discarded the formula becauseof its nonhomogeneityin fayor of a more complex one in l gg9. and Gauck_ ler in 1868preccdedManning in introducinga formulaof the type that no$. bears the nameof Manning. The theoreticalapproach to open channel flow rcsts on the lirm foundation built by Newton, Leibniz, Bernoulli. and Euler, as in other branches of fluid mechanics;but one of its early fruits was the analyticalsolution of the equationof graduallyvariedflow by Bressein I860 and the conect lormulation of thi momentum equationfor the hydraulic jump, which he attributedro the lg3g lecturenotes of Belanger.In addition, Julius Weisbachextendedthe sharp_crested u eir equation in l84l to a form similar to that usedtoday.By the end oflhe lgth centurv.manv of the elementsof the modem approachto open channelflow, which inclujes borir theory and experiment,had beenestablished. The work of Bakhmeteff, a Russianemigre to the United States.had oerhaos the most imponant influence on the developmentof open channel hl,draulics in the early 20th century. Of course, the foundations of modern fluid mechanics (boundarylayer theory, turbulentvelocity and resistancelaws) were beins laid bv Prandtl and his students, including Blasius and von Kiirmiin, but Bakh-meteffis contributionsdealt specifically with open channelflow. ln 1932, his book on the subject was published, based on his earlier l9l2 notes developed in Russia (Bakhmereff 1932). His book concentraredon .'varied flow" and introduced the notion of specific energy, still an imponant tool for the analysisof open channel flow problems. In Germany at this time, rhe conributions of Rehbock to weir flow also were proceeding,providing the basisfor many further weir expenmenrs and weir formulas.

C l l A p r F RI : B a \ i cP r i n c r p l e s5 B y t h c n r i d - 2 0 t hc c n t u r yn. l i l n ) o l t h c g r i n s i n I n o w l e d g ei n o p c nc h l n n c l f l o w h a d b c c n c o n s o l i d a t c da n d c r t c n t j c Liln t h c b o o k sb y R o u s e( 1 9 5 0 ) C , how ( 1959), a n d l J c n c j e r s o(n1 9 6 6 ) .i n r v h i c l cr r t r ' n s i v cr c f c r c n c ccsa n b c f o u n d .T h e s eb u l k s \ e t t h e s t a g cf o r r p p l i c a t i o n so f n t o c l c nnr u n t c r i c aal n a l y s i st e c h n i q u cas n dc x p c r i n r c n I a l i n s t n r n r c n t a t i ol no p r o b l c o r so f r ' f . ' n c h l n n e l f l o * .

1.6 DI.]FIN IT'IONS I n a s t c a d yo p en c h a n n c l l o w . t h c d c 1 l l ha n dr c l t , , - i t ya t a p x ) i n dl o n o t c h a n g ca s a f u n c t i o no f t i m e . I n t b c n r o r eg c n e r i rcl a s co f u n s t c a d yf l o q . b o t h v c l m i t y a n dd c p l h v a r y u i t h t i l r e . a s i n t h e c a s eo f t h e p r s s a g co f a f l o o d $ a r c i n a r i v e ra s s h o w ni n Figure L la relative to a fired obscrrer on lhc rivcrbank.Thc changein rc'lrn--it;and

(a)

(c) Rainfall

TITTTTTTTT|ITTT|N Q

(d)

r

I

(e)

F I G U R I :T . T T)pesof L,penchannelflow: (a) unstcidy;(b) stcady, riniform:(c) steady. gradutllyvaried (GVF).tn,lsleady,rapidly!aried(RVF)i(d) uns{ead}'. rapidll rrried: (e).paliall}varied.

6

C H A P T E Rl : B a s i cP r i n c i p l e s

deplh in a large river nlay occur so gradually and over such long distancesthat the obsen er can seeonly a gradualrise rnd flll of rir er stlge. If rhe flood wave results from a dam break,on the other hand,an abrupr changein depth and velocity and a distinct u'avefront or surgemay be observed.In the former case,only ncarthe peak of the flood wave could the florv be consideredapproxirlately steaoy,or quasl_ stead). allowing steadyflow analyses. Spatialvariationsin velocityand dcpth in the flow directionare disringuishedbv the tcrms uniform nnd nonunifornt. In a uniform flow. the mean crosi-sectional velmity and dcpth are constantin the flow direction. as shown in Figure l.lb. This flou conditionis difficult to createin lhe laboraroryamdrarcly occursin the field, but oflen js usedas the basisfor opcn channcldesign.It requircsthe existenceof a chan_ nel of unifornrgcometryand slopein the flow direcrion;that is. a prismaricchannel. The nonunifornrflow conditioncan be divided inro tuo rypes:gridLrallyvariecland rapidly varied.Gradually varied flow is nonuniform flow. but the curvatureof the free surfaceand ofthe accotnpanyingstreamlinesis so slightthatthe transvcrsepres, sure distributionat any stationalong the flou can bc approximatedlr hl,tlrosiarrc. This assumptionallows the flow to be treated\{ith one-dimensionalfonns of the gor eming diffcrentialequationsin which we are concemedwith variationof the flow lariables in the flow directiononly. Fortunatel\'.most river flows can be feated in this ntanner.Rapidly variedflow, on the other hand, is not amenableto this approach and often requircsapplicationof the monentum equationin control volume ftrm as in the hydraulicjurnp or a two dimensionalformulationof the gor eming differential equationsas in the highly cunilinear flow over a spillway crest.Examplesof gradu_ allv raried and rapidly variedflow are shown in Figures l.lc and I.ld. Spatiallyvaricd flow really is a classof nonuniform flow but oues irs nonuni_ formiry to Iariation in the flow dischargein the directionof motion as well as to an inrbalanceof gravity and resistingforces.Examplesof spatiallyvariedflow include side channelspillwaysard continuousrainfall additionsto gutter flow, as snown rn Figure 1.1e.

1.7 BASIC EQUATIONS Thebasicequations of fluidmechanics areapplied to openchannel flowwrrnsome modificationsdue to the f'reesurface.These equationsare the continuity.momen_ tum. and energyequations,which can be derived directly from the Rcynoldstrans_ pon theorem applied to a fixed control volume as shown in Figure 1.2a. The Retnolds transporttheorem is derived in mrnr elementaryfluid mccnanrcsrextbooks (Robersonand Crowe 1997:White 1999) and is civen bv d

dr

B

d

I

dr

t

b p d Y + I b p ( V . nd )a I

(1 . 2 )

inuhich8:systemproperty:/:time; b : r h e i n l e n s i v ev a l u eo f B p e r u n i t r n a s s nr, dB/dm. p = fluid density;V : volume of rhe control volume (ca); V : veloc,

C l l ^ f l t - R l : B : l s i cP r i n c l p l c \

Control (b)

(a)

F - o il

r

t

i

sEc.c-c

c (FpQv,),"

(1.10)

If the monrentumequationis applied to a differential control rolurne aJonga slreamline,as in Figure L2d, and only pressureand gravity forces are considered, the result is Euler'scquationfor an incompressible.frictionlessfluid:

au, - = p - ' t p L " ou, " _ 0 p - p 8 a; ( t . t dt dt as

( r . ll )

i n u h i c h p = p r e s s u r e : :- e l e v a t i o nui , = s t r e a m l i n ev e l o c i t y ;t = t i m e : a n d s = coordinatein the streamlinedirection.lf only steadyflow is consideredand Euler's equation is intcgratedalong a streamline,the resuh is the familiar Bemoulli equation \\ ritten here in terms of head betweenany t\\'o points along the streamline: ui rP jr + - ' .r + : : = i + P- . - + " )o v v

Ui :

0 .12)

in \\ hich y is the specificweighr of watcr = pg. In this form. the Bernoulli equation rcrms have dimensionsof energyor work per unit weight of lluid. and so it is trul) a work-energyequation derived fiom. but independentof. rhe momentum equation.The terms are scalarsand reprcsentpressurework, potentialenergy,and kinetic energy in that order. For applicationsto open channel llou, we need to expand the equationfrom a strearrlineto a streamtubeand include the energyhead loss term due to friction, /rr,for a real fluid, which resultsin

- .+ a . $ + r , 1f + .:, + "g 5 =f 4 + ' 2 8

(l.r..)

This expansionof the Bemoulli equationto a srreamtuberi'ith headloss includedis cafled the extendedBentoulli equotiotr or the €/tergr.equation. ll requires the assumptionof a hydrostaticpressuredistriburion at points I and 2, bccausethis meansthat the piezometrichead(p/7 + a) is a constantacrossthe crosssecrion.The use of the meanvelocity in the velocity headterm necessitates a kinetic encrgy flux correcticn coefficientdefined bv

i,: o.r *

vlA

(l.ll)

C H A p T F -IR : B a s rP crinciplcs ll I o a c c o u n tf o r a n o n u n i f o r m\ c l { r ' r t \ d i \ ( n h u t i o n A . s llc shrll scc in succecding c h a p l c r st,h e v a l u co f r r c a n b c . , , - i r r l i c l n t l vl a r g c ' trh r n u n i t y r n r i v c r sw i t h o v c r blnk flow and thcre'forecilnnol b. n.!lr'aled. T o c n r p h a s i ztch e i n d c - p c ' n d cenr., f r h c c x t c n t l c dI l c r n o u l l io r e n c r g yc q u l t i o n f r o m l h c I n o n r c n t u rcn( l i l l t i o n ,i t ' h , r u l dh c p o i n t e do u t t h a tl h c c n c r g yc q l r r l t o l lc a n bc dcrivcd in a nrorc gcncral*l; Ironr tltc Rcynoldstransponthcorernand the first lar of thcrmodynamics:

dE

dQ^

;,=a,

dlv,

i-

dll;

d I

t

+ I e p ( V . n )d A

d , = * 1 , , " ' o oI

(1.15)

in which B has bcen replaccdbl the total cncrgy E; 0^ - lhc hcat transferlo rhe tluid; lti ,. thc shaft work donc by thc fluid on hl,ilraulicmachincs;ly. = lhc work dcrreby the fluid prcssureforccs;and e is dE/dn = the intemal cncrgy plus kinctic e n c r g yp l u sp o t c n t i acl n c r g yp c r u n i t n t a s sF . o r s t c a d yo, n e , d i n t c n s i o n fal lo w o f a n i n c o m p r e s s i b fl 1 e u i d .t h c c n c r g yb a l a n c eg i v c nb y E q u a r i o nL l 5 r c d u c e st o E q u a t i o n 1 . 1 3 .i n w h i c h t h e h e a dl o s st e r m r e p r e s c n ttsh e i r r e v e r s i b lceh a n g ei n i n r e r n a l energyand the energyconvertedinro heatdue to viscousdissipation(White 1999). The continuityequationis a statementof (heconservation of mass.Likewise,the energyequationcxpressesconservationof energy.It is a scalarequationand in the form of work/energybecauseof the spatialinregrarion of IF = nra.The momentum equationalsocomesfrom Newton'ssecondlaw appliedto a fluid but is a vectorequation thatstatesthatthe sum of forcesin anycoordinatedirectionis equalto tie change in momentumflux in thatdirection.In tie controlvolumeform, the momentumequation can be appliedto quitecomplicatedflow siluations.as long as the extcmal forces on the conrol volume can be quantified.The energyequation,on the other hand, requiresthe capabilityof quantifyingcnergydissipationinsiderhe control volume. Often, all three fundamentalequationsare applied simultancouslyto solve what otherwisewould be intractableproblems.The hydraulicjump is an example in which the momentum and continuity equationsare applied first to obtain the sequcntdepth (depth afler the jump). and lhen the energyequarionis employedto solve for the unknown energyloss. Even experienced hydraulicians sometirncsmisapply the nromentum and energyequations.The cardinalrule is that the cncrgyequarionnruslinclude all sign i f i c a n te n e r g yl o s s c sa n d t h e n r o m e n t u mc q u a t i o nm u s t i n c l u d ca l l s i g n i f i c a n t forces.Breakingthis rule sometimesleadsto conflicting rcsults from the momentum and energyequationsbecauseof misapplicationrarherthan a breakdownof the fundamentalphysicallaws.

r.8 SURFACEVS.FORM RESISTANCE F"lowresistrn,e in fluid flow can rcsult fron two fundarnentallydiffercnr physical p r ( ) c e s s cu s ,l , i c h t a k e o n s | c c i a l m e a n i n g' , , h c nw e d i s c u s so p e n c h a r r n e fl l o w resistancecoclficients. Surfirce rcsistanceis the lrilditiorral form of r sislance

t2

C H A P T E R l : B a s i cP r i n c i p l e s

Separation Separatjon

lu t"\

Broad r,vake

- 1 -

Narrow wake

(a)

(b)

1.0

J,

0.0 Turbulent

\.

!

dl*E I l>t - 1 . 0 oo

\ / i fl""irt

-2.O

\-

-3.0 0"

,/..

,r'"0'=,,-0",n', 45"

90' B

135"

180"

(c) FIGURE 13 Separation andformresistalce in realfluid flowarounda circularcylinder:(a) larninarsep(c)realandidealfluidpressure aration;(b) turbulent separation; (Whire1999). distributions (Source: F White,FluidMechanics, 4e,@ 1999, McGraw-Hill. Reproduced teithpelmission of TheMcCraw-HillCompanies.) resultingfrom surfacefriction or shearstressat a solid boundary.Integrationof the shearstressover the surfaceareaof the circularcylinder in Figure 1.3, for example,wouldresultin surfacedrag. Surfaceresistancealonecannotaccountfor the measuredflow resistanceof a bluntobject,suchasa circularcylinder.Because of thephenomenon of flow separationof a real fluid, an asymmetricpressuredistributionoccursaroundthe circular cylinder,leadingto form dragas shownin Figure1.3with higherpressure on the upstreamfaceof the cylinder thanon the downstreamfacein the zoneof sepa-

C l l A p r rR l : B a r r cP r r n c i p l c r l . l r a l i o n .I n c o n l r i l s t .i n \ i s c r dl l r \ r t h c o r vp r c d i c t sa s l n r n c l r i c p r c s s u r cd i \ t r j b u l i o n a n d n o f o r m d r a g ( a s * c l l a r n o r L r r f r t cdcr a g ) o n t h c c \ l i n d c r . a s s h o r + ni n F i r u r c 1 . 3 .I f l h c c o n ' r p o n c notf t h r 'f , r c s . u r lt i r r c cr n t h c f l o * d j r r - ' c t i oins o b l l i r r c db t r n r r ' , g r l t i n g l h e r c a l f l u i d p r c ' s u r cJ i \ t n b u l i o na r o u n dt h c \ p h c r c . t h c r c s u l ti s a f r r r m dr;rgor form rc\islanrc {h.rtrsutrnrIlr't.'lyscparllc frorn surfaccdrag.The totaldrlg 'fhc thcn is thc sunr of thc rurfrrccdra-guni! lornr cirag. nragnitudcof the'fornr drag d c p c n d sh i g h l y ( ) nl h c p o i n lo f \ c p . r . r t i o nw. h i c h i s d i l f c l c n t i n t l r el a r n i n aar n dl u r h , r i l c ncl a s c s .a s s h o * n b y F i g u r cI 3 . I n o p c n c h r n n c l l l o \ \ . t h c r c s i \ t i l n c o c ff('rcd h r l a r g c r o u r h n c s sc l c n t . ' n t so r l l l u r i r l b t ' d f o r n t s n r a r b e d u c l a r g c l yt o f o r m r c \ i s t a n c cT. h i s p o i n t * i l l b e d i s c u s s cidn m o r c d c t a i l i n C h r p t c r s . la n d 1 0 .

1.9 DI}1I'NSIO\A

L A\ALYSIS

Thc purposeof dinrt'nsionalanall,sisis to reducethe number of indcpcndentvari, ablcs in an opcn channelflow problemor any olher fluid mechanicsproblernby transforming the depcndentvariableand severalindependentvariablesthat form a functional relation:,hipinto a snaller nunrberof dimcnsionlessratios.This reduces the number of experimentsinvolvcd in devcloping an experinrentalrelationship. paramctersneed to be variedratherthan since only the independcntdimcnsionless each individual indcpendentvarjable.Ratherthan varying the ve)ocity,depth,and gravitationalaccelcrationindcpendently in a hydraulicjump experiment,for example, it is necessaryto vary only thc Froudenurnber,rr hich is a dimensionless combinalion of thesevariables.and presenlthe resultsfor the ratio of depthsbcforeand after the jump in terns of thc Froudenumber.In addition. the dimensionlcssvariablesoften representratiosof forces,suchas inertia and gravity, so that the magnitude of a panicular dimensionlessvariableand its variation in a given experiment rclate to an understandingof the physicsof lhe flow siluation. Iirdbennore, presentationof experimentalresultsin terms of dimensionlessvariablesgeneralizes the resultsto a wider rangcof applicationsand confirms the validity of the dimensionlessratios chosento model a panicularflow phenomenon. lf the goreming equationscan be completelyformulated for a given problem, the equationscan be nondimensionalized to deduce the embeddeddimensionless paramelersof imponance.For example,spplicationof the rnomentumequationto a hydraulicjunrp and nondimensionalization of the resulting equationfor the depth after the junp resultsdirectly in the appearance of the Froude numberas the only independentdimensionlessparameterfor this problem.The necessary conditionfor nondirnersionalizationof an equationis dimensionalhomogenei{y,whicb simply requiresevery term to have the samedimensionsin any propcrly posedequation describinga physical phcnomcnon.Once the govcnring equationsare transformed into dimensionlessform, the solution can be obtaincd in terms of the resulting or numerically,for a conrplctclygcneral dimcnsionlessvariables,eitheranalyti,-.rlly solution. This solution can be appli,d to similaf flo\\ siturtions under conditions different from those for which the rcsultsrvcreobtained, so long as the rangesof the dimensionlessvariablesare the same.

1 . 1 C H A P - r F .l R : B i r \ i cP r i n c i p l e s I n s o r n ec a s c se, q u a t i o nos f o p e n c h a n n e fl l o u s u c ha s t h c M l n n i n g , sc q u a t i o n or thc head dischargeequationfbr llow over a \\r'ir:lt first may not appcar to be dimensionally hontogencous.In thesc cases,son'te"constant" nrust have dimensions for lhc equationto be dinrcnsionallyhomogeneous.lf the equation for discharge Q over a sharp-crested weir. for example. is wrilten as a constantCr times lH"r. where l, is the crest lcngth and H is the head on the crest,it is clear that the e q u a t i o ni s n o t d i t n e n s i o n a l lhyo m o g c n e o uusn l e s sC , h a sd i m e n s i o n os f l e r g t h t o the l/2 power dividcd by time. Thesein fact arc rhe dimensionsof the squareroor of the gravitationalaccelerarion. e, * hich has bcen incorporaredimpliciriy into the value of C,. This practicerequiresthat rhe coefficient Cr take on a differentnumerical r,alue for different systemsof units, which is less desirablerhan leaving the original equationin tenns of the grar itational acccleration. As an exantplcof nondintcnsionalization of rhe govcming equations,the inviscid flow solution shown in Figure 1.3 can be obtained fronr an application of Bemoulli's equationbetweenthc approachflo\ (variablcswith a subscriptof -) and any point on the circumfcrenceof the cylinder:

p , , p ' 2 1o , r :

(l.16)

If the equationis nondimensionalized, thereresults p - P ,

vi

,o 2

/ r \ t

\ v-l

( 1 .1 7 )

in u hich Cois definedasa dimensionless pressure coefficient. Thesolutionfor the pressure coefficient is obtained by substituting the inviscidflow solutionfor thecircumferential velocityu : 2y_ sin d into Equationl.i7 with theresult C p= I - , 1 s i n r a

( 1 .l 8 )

EquationLl8 givesthetheoretical distribution of thedimensionless pressure coefficientCoshownin Figure1.3.Thus,if the govemingequationof a fluid mechanics problemis klown, then the equarionitself can be madedimensionless, as in EquationL17, andtheresultingsolutionalso*ill be dimensionless. In many problemsof open channelflou,, the theoreticalsolutionis not directly applicablewithout the addition of experimentalresultsto evaluate unkrown parameters, or it may not be possibleto formulateand solvethe goveming equations in verycomplicated flows.This requiresa differentapproachfor obtainingthe importantdimensionless parameters of the problem.In the caseof dragon a circularbridgepier,for example,specification of theexperimental drag coefficientis necessary to calculatethe drag force,which includesboth surface and fonn drag,the latterof which is not easily calculatedfrom the goveming equations.Presentation of the exp€rimental resultsfor the drag forcein dimensionlessform requiresa generaltechniquesuchas thataffordedby the Buckingham fl theorem(see,for example,White 1999).The BuckinghamfI theoremcan be statedas follows:

C l t \ r t t , R I : R . r : i cp r r n c i p l c s I 5 l f a p h l r i c a l p r \ \ e s sr n \ o l \ e s r f u n c t i t r n arlr l : r r r o n . h i rpm o n g , r \ t n a b l c \ , $ h i c h c a n b c e r p r e ' r s c idn t e r m r o [ , r b a s i cd i n t e n s i o n sr .t a u n b c r c J L ] a c tdo a r c l a l r o nb c l t r c c n , r ) d i r r r a n \ i o n l c sr sa r r r b l c s o . r f l t c r n r . .t ' \ i h o o \ i n g r r r . p a , r t t n 'ga r i n b l e s c. i t c h {n o f \ r h r c hi r r o n r b r n r tiln t u r n \ \ i l h ( h c r t ' n r a r n r n\rr r r , l h l c \l o f L r r r rt h r c I l I c r m sa s p r M 'fhe u ( r \ ( ) f t h r ' \ r r j J b [ - 5l l l c n t o r h e i ] n p r o p r i r r cp r * . - r r / , r c l a . l r r n -rca r j a b l e sm u s r a ( r r r l . i nr r r r ( ) n 8t h t n r a l l [ r : r s i cd i m e n s i o n sf o u n d | t l ] l I h c r a r r r h l c sb u l c l r n n o t n e m s c l r ' c sf o n r ta I I t c r m . (rr -

I n n r ; i ( h c r r u ( i c al lc r n t s . i f a d c p c n c l c n tv a r i l h l c , 1 , c a n b c c ' r f r c s s c d i n t c r n r s o f l ) i n d c p c n d c ' nvl a r i a b l c ' sa s

A , = J ( 4 2A. . .. . . . , \ " )

0.19)

t h e nl h e B u c k i n g h aIm ] t h e o r c ranl l o u ' st h er r r a r i a b l etso b c c x p r e ' s sacsda f u n c , ( i o n arl c l a t i o n a n o n g( r r a r )I l g r o u p s :

d ( n r . I t : . . . . f. I , . ) : 0

(i .20)

T h c b a s i cd i n r c n s i o n us s u a l l ya r e t a k e n a s n r a s s( M ) , I c n g r h( / _ ) .a n d r i m e ( f ) , allhoughforcc (F), Iength,and tirne are an equally valid choice.The force dimension is uniquely relatedto the rcmaining dimensionsby Newron'ssecondlaw; that i s . F = M L T r . I n c e n a i ni n s t a n c e st h, e f u n d a m e n r ad li m c n s i o n m s a y b e f e w e rt h a n three;for exantple,only lcngth and time may be involred. Whcn choosingrepeating variables,it is importantto recognizethal ir is b€tternot to choosethe dependent variableas a rep€atingvariable,so that it \r,ill appearin only one fl term. If, for example,n = 5 and rn - 3 with M, L and fas rhe basicdimcnsions.the two fI terms can be found from

It7it = uatro = [A,]''1.,r,], Ie.],,[a,]

(1.21)

Lll.l = untoro = [A,]',lArl,,[A4],,lA5l

(t .22)

in which the squarebrackctsdcnote"dimensionsof'the enclosedvariables;and A" A.. andAn havebcen chosenas repcatingvariables.By substirutingthe dinrensionl of the variablesinto the right hand sidesof EquationsI .2 | and t .22 and equating the exponerltson M, L, and f on both sides of the equalions,rhe rcsultingalgebraic equalionscan be solvedfor the unknorvnexponentsand the resultinglf tcrms. Now considerthe drag prob)em for a conrpletelyimnrersedcylinder in whicb the drag force, D, can be cxprcssedin termsof tre cylinder diameter,d; the cylinder length, /.; the approachvelocity, V-; rhe fluid density,p; and rhe fluid viscosrty, 1t:

D : f, (d,r,,v-,p,t")

(t.23)

A loral ci six variableswith all rhree basicdimensions(M, I- Tl arerepresent€d, so therc will be three II terms.The rcpeatingvariablesare chosento be the density, velocity, and cylindcr diamelcr, which contain among rhem M, I-, and T as basicdinrensionsbul do not lhenrscivesform a dinrcnsionlessgroup.The cylinder diarnt'terand lcngth could not bc ( itoscn togethcr-,s rcp,:atingvariablesbecause lhey would form a lI gloup. Irirst, thc (lrag force is couibilcd with powers of the

l6

CTAPTER l : B a s i cP r i n c i p l c s

repeatingvariablcs,either algcbraicallyor by inspection.ro vield rhe firsr Il tcrnt; then the same processis rcpcatedfbr the cylinder length and rhe fluid viscosity. T h e r e s u l ti s g i v e n b y

D

trvi

- / . \l ;t R e\ /

(l.24)

which gives the dintensionlessdrag ralio in ternrsof the Rel nolds number, Re = pV*dlp and the rario of cylinder length to diametcr,/./d. Tradirionall),,rhe drag ratio is redefinedas a more generaldrag coefficicnt,applicableto other shapesof immersedobjects as D/(pAV:12r. with A in rhe coefficienrof drag defined as the frontal area of thc inrmersed object projectcdonto a plane perpendjcularto rhe oncoming flow (1,.X d). Also. a factor of 2 is added ro the dcfinirion of the drag coefficientas a natter of tradition. For an infinitely long $ Iinder,the ratio {./d no longer has an influence becausethereare no end effects.so the experimentalcoefficicnt ofdrag is determinedfronr the Reynoldsnumberalone rnd ;sed ro calculate the drag force. The choice of the repeatingvariablesis not unique, so rhereare equally valid altemativefomrs of the fl groups.If, for example,the repeatingvariableswerc chosen to be p. V-. and d in the cylinder drag problem, the resulrwould be

1 \ D =n(ne / --.; ,.lv-a'

(r .25)

l{owever,the alternatedependentll group in ( 1.25)could be deducedfrom taking the productof the drag ratio and Reynoldsnumber in ( 1.2.1).In the same manner. the justification for replacing d I in the denominatorof the drag ntio in ( I .24) with the frontal area is that the drag ratio in (1.24) can be divided b1 {/d and rep)acedby the result.ln general,it is possibleto statethat a new II group can be formcd as

n; : ili n!r:

(l .26)

and usedto replaceone of the original fl groups. In the more generalcaseof severalbridgepiers,eachwirh diameterd and spacing s betweenpiers and in open channelflow with a finite depthof water I0. the formation of gravity surface waves around the piers may give rise to additional flow resistanceso that the drag force can be written as

D - [email protected]'0, V-, p, p. g)

(t 27)

in which the gravitationalaccelerationhasbeenaddedto the list of variables.Alternatively,the specific weight 7 could be addedto the lisr insteadof g, but rhe ratio 7/p, which is equal to g, then would appearin the dimensionlessgroup relatedto the gravity force. Now, rhere are eight variablesand still rhreebasic dimensions resultingin five ll groups that can be expressedas

D p d:'nvl

"(d d Jr r' \

\

./ .nen

( |.28)

C r t A r ' 1R t I

R J \ ' (P r l n ( r l l c \ l 1

'fhc

a r l c l i t i o n a! cl o t n c l r i cr a r i a t r l cr c s r t l l si n l n a d d i t i o n : rSl e o n ) c ( r i rca t i o ,a r l dt h e ' r l \ r l l t l i o n l lf o r c cn c c c s s a r i lbyr i n g si n t o P l l y t h e I ' r o u ( j cn u r n of lhL8 inrroduction (1 28) t . l r r ,! - . T h e r c l r i t r r ci t t r l o n r i n c e ' otl h c l l g r o u p so n t h c r i g h t h l r n ds i d c o f r , ' o u l cbl c d c t c tt t r r r r r db ) c \ i r r ' r l l n c l l t s 'Ilrc cxistcnccof thc' frce sltrfaceitt o1^-nchanncl florr incritabll ir\ol\cs thc gravity forcc. r'ithcrthroughthc folllllllion of srtrfaccualcs. lllc cxislcncclrf a componcntof the body forcr'in thc llo* dircction,or a tliffercntill pr!-ssurcf(irct'due to changesin dcpth.'lhercforc.a ilitncnsionalanalysisof an opcn channclflo* problcm includcsthc gravitalionalaccclctllionin thc list of variablcs.and thc Froudcnunrbcr parametcr'asdiscussedpreviously' ncccssarilycrtrcrgcsrs an intponantditlrensionlcss v a r i a b l e si s c r u c i a lt o t h c s u L c e sos f d c p c n d e n t o f i r r c l e p e n d cannt d Thc c-hoice o n c ' d c p c n d e nr ta r i l l b l c .l n d t h e i n d e p c n o n l y c a n b c r . l i n r c n s i o naanl a l y s i sT. h c r e i s o n c o f t h e i n d e p c n d c nvt a r i a b l e cs a n h a t r c d u n d a n t ; d c n t v a r i a b l c sn r u \ t n o t b c -l-he inclusionof extra inde{he others. of combination not bc obtrincd front sonrc the expcrintental is not fatal because indcPcnd!'nl atc truly ocndent rariablcs that g r o u p s i s u n i n P o ( a n t 'b u t d i n l c n s i o n l e s s r e s u l l i n g o f t h e r e s u l t s* ' i l l s h o w w h i c h a n i n complete xPerv a r i a b l e c a n i n d c p c n d c n t Sive f a i l i n gt o i n c l u d ea s i g n i f i c a n t of rescarch course are nrade in lhe decisions such Ultirnately, imental rclationship. the final set of at error to arrive trial and involve and may problcrn on a panicular r a t i o s . importand t imensionless

1.10 CO]\'IPUTERPROGRAMS B in \lsual BASICcode,whichis programs aregivenin Appendix Someconrputer

aoolicable to ttre Microsoft Windows environment.The BASIC languagehas evolved frorn a DOS-basedlangua8eto *re presentform that utilizes the graphicaluser interface of \Mndou s. It is an event-drivenlanguagecomposedof both form nrodules,which contain the graphical user interface,and standardmodules. whjch contain the computational code. Th. progt"., in the appcndix include standard modules that consist of (o numerical pro,:eduresor subprograms.They can be convened easily otier languages such as Fortran or C, contbincd $'iti fomr nrodulesin Msual BASIC for input and outusing\4sualBASIC for ApplicationsThe put,or jnco+rcralcdinto Ixccl sprcadshects pu.pose here is to doelop thc core methtxlology for the use of numerical analysisto iolve opcn channelflow problcms.To this end.AppendixA containssomebasicmaterial on numcrical methodsthat will be used throughoutthe text. Appendix B includes someexample programsthat arc jntendedto serveas leaming tools to explorethe apPlication of numerical techniquesto open channelflow problems.

RE[-EREr.'CES B. A. ll\Llrdrticsof OpcnChunnelFlott Neu York: McGraw'Hill,1932' Rakhnrcteff. NewYork:McCra*'Hill, I959' Chow.V T. OpcnChantelHttlrtLr,lics New York:lr'latmillan.1966 f'ltw. L M. OpenChonnel llenderson.

l8

C H , \ P r E R l : B a s i cP r i n c i p l e s

Robe.son. J. A.. anclCro$e. C.'f. E gitvering FlLti(r,ilri) "

Kr lA2

(l.ll)

.10

C B A P T E R2 : S p e c i f iE cnergy

in which A, : the conveyanceofthe ith subsection:a, : the areaofthe lth subsec_ tion: and K : It, - the conveyance of the total crossscction.The conveyance of the ith subsectionis calculatedfrom a unifomt florv fomtula such as N,tanning's cquarion. Differcntiating the kinetic encrgycorection coefficient as definedbv Ecuarion 2.31 and subsritutinginto Equation 2.28 leadsto a working definition ui th. .orn_ pound channel Froudenurnber:

".-l#(+-,)i'

in which

-,,f - f il)] ?[(f)'(,,,

, rq) ,

\4;

(2.32)

(2.33a)

(2 . 3 1 b )

/

? [ ( f ) ( ,"' * - . ; : ? ) ]

( 2 . 33 c )

in which a,. p,,.r,, t,, ni, and l, representthe flow area. \\,ettedpedmeter,hydraulic radius, top width, roughnesscoefficient. and convel.anceoi the ith subscction. respectively,and K = total conveyance.AII the terms on the right hand side are cvaluatedio the courseof water surfaceprofile computarion, .*aapt dp,/dy,,which can be evaluatedas shown in Figure 2.15 becausethe cross sectionis composetlof a seriesof ground pointsconnectedby straightlines.At any given \r.atersurfaceelevation, only those portionsof the boundary that intersectthe free surfaceare con_ sidcredto contributeto dp,/d-rAt the point of minimum specificenergy,F. can be cxpectedto have a value of unity so that Equarions2.12 and 2.3j can be used to solve for critical depth in a compoundchannel. For r spceific range of dischargein some comJnund chanrrelcross rections, multiple valuesof critical depthcan exist with one minimum in the specificenergy occumng in the overbankflow case and (he other occuning in the caseof main c h a n n e lf l o w a l o n e .B l a i o c k a n d S r u r m 1 1 9 8 1 Id e m o n s r r a t ; dr h e v a l i d i t vo f t h e compound channelFroudenurnberin conectly predicting multiple point, tf mini_

FIGURE 2.I5 Evalualionof dpldy at the water surfaceintersection!\ ith the channelbank (BIalock and Sturm, l98l ). (Soune: M. E. Blalockand T. W. Stunn. "Minimun SpecifcE .r|y in Conpowtd Open Channel," J. H\d. Dir'., A 1981, ASCE. Reproduc.edbt pernission of ASC')

C l l ^ p r t R 2 r S r ) ( , c l fEi cn c r g y . 1 1 r r r u ms J r c c i f iccn c r g yb t i n v e s t i g a t i ntgh c h \ p o r h c r i c a l c r o l s s c c t i 0 nA , a s ! h o \ \ n i n I : i g u r c2 . 1 6 f o r a f i \ c d d i . , c h a i g e o ,f S O O O c t s f t f : n r i l s r .f , r ' r t , i r , t , r . r , , , r g " r t " c r o s ss ( ' c t i o nh a s t w o p o i n t so f m r r i n r u n tr p . , c r f i c c n c r g St C t u n J C : t . a s c a n b c

],,11^fl:..:l,n,l,"d chann-cr l:roudc .,,.,ii",,, "q""r,. Lrnrry rr thc :::l.T ::i,[ !',"\r' u ( l ' r ' r \ .L o r r c \ p o n d t nt op D o n t l so f m i n i m u m

s p c c i f i ce n c r g ya. s s h o w nj n F r g u r cI I 8 I n r d d r l r o nI.r g u r c ' 2 . 1 g s h o * sr h a r . . r " ' . " " r f " i i . , r i r ' , i " , i u i r i n n , of Fr,ru,-ic' nrrrrbcrgire 'n.,.rr.-ctraruesof thc criricarocpri. Thc iLrr,ra uuu,r.)"., n", i s d e f i n c db y E q u i l ( i o2n. 1 9 a , si s F * i r h r r = 1 . 0 .

r ! 72 tl

T'ICL]RE 2.I6 Hlpothcticalcorrpoundchannelcross,section A.

Cross sectionA O = 5000 cts

z0

4.O 6.0 Specific Energy, ft

8.0

1O.O

F I G t ' R u2 . l 7 Specificenergydiagram for cross seclion (Blalock A and Srumt. lggl). (.\our..e;M. E. Blalock and T. W Srunt, l,lininun Speci/ic Energt in Co^pn,na'Op",) ino,,n"t,- l. HtLl. Dir,.,A 1981,ASCE. Reprtttlucctl bl.pe rntission ttf ASCE.)

C H A P T E R2 : S p e c i f i cE n e r g y

10.0 8.0

t

-e 6.0 E 6 4.0 o

S=

Crosssection A O = 5000cts -c1

S:-:---&-___-

Topof bank \

/c2

2.0 0.0

1.0 Froude Number

3.0

T T G U R2E. I 8 Froudc numbers forcross,secrion A (Blalock andStu.m,lggl).(Sourcer M. E. Btatock

dnd T. W. Sturm, " Milimum Specijic Energt in Conpound Open Channel,', J. H1d. Div., A 1981,ASCE.Reproducedby pennission ofASCE.1

FIGURE 2.19 Experimental conrpound (BIalockandSturm.lggl). (Source;M. E. channelcross-section Blalockartd T. ll. Sturm, "Mininum SpecificEnergyin ConpoutttlOpenChannel,,,J. Htd. Div., @ 1981,ASCE.Reprodutedbt permissionof ASCE.) The conceptof two points of mininlum specificenergy,as illustratedby crosssectionA in Figure 2.17, was investigatedexperimentallyby Blalock and Sturm ( 1981)in a tilting flume wirb the crosssectionshown in Figure 2. 19. Uniform flow was estat'lishedin the flume for variousslopesat an averageconstantdischargeof 1.69cfs (ftr^). Detailedvelocity distributionst"ere meorure,lto computea and the specificenergyat eachmeasureddepthof flow. The experirnentrlresultsare shown in Table 2-2, in which two poinrs of minimum specifii energy lRuns 2 and g) are predictedby a value of unitv for the compoundchannelFroude number u ithin the

ClrA|TLR 2: Spcific l-ncrgy

4l

T A T I , E2 . 2 Fl)ipcrinlental ralucs ol conrporrndchanncl Froudc nunrber for \!rious deplhs of flow i n t h c c r o s s - s e c l i oonf F i g u r c 2 . 1 9x i t h a n a r e r a g ed i s c h a r g eo [ 1 . 6 9 2c f s 1 0 . 0 { 7 9m r / s )

Run

E, lr

), ft

F

1

0 6,i0

tl92

07t8

070

.t

0 6t_5

tt98

0.70:

082

2

0.600

1.221

0.700

0.97

3

0 561

I :18

0.701

I t5

l0

0.5t1

t09.1

0 ?0.1

082

1

0 5(n

| 087

0.700

090

b

0..167

r.0:6

0.690

r.00

I

0 Jll

I 100

070t

I Il

. S . ! r . c D a r . f u o mE l i i r { l u d S l u n n l 9 8 l

cxperi,nenta)uncertainty.The two valuesof critical depth also correspondto mini n r u m v a l u e so f t h e m o n r c n t u nfru n c t i o n( B l a l o c k a n d S t u r m , 1 9 8 3 ) .a s e x p l a i n e d i n C h a p t e r3 . The compound channelFroude number also can be derived by setting V - c, rvherec is the wave celrrity in a compoundsection,in the equationsof the characteristicsof the generalunsteadyform of either the energy or momenturnequation ( B l a l o c ka n d S t u r m 1 9 8 3 ;C h a u d h r ya n d B h a l l a m u d i1 9 8 8 ) .O n c e a n e x p r e s s i ofno r the wave celerity c is devclopedfrom the characteristicsof the unsteadyenergyor momentumequation(seeChaptcr7), the compoundchannelFroude numbercan be defined as V/c uith a result identjcalto lhat of mininizing the specificenergyor rnonrentumfunctions.Kcinemann(1982) also sugges(san expressionfor the compound channel F'roudenurrrberby minintizing the expressionfor spccific cncrgy, cxcept that the tcrms involving the rate ofchange of wetl!-d p€rimcterwith respect to depth of flow, dp/d\', Ne ncglccted.Inlerp.ctationof the flou' rcgimc of the separatcfloodplain and nraincltannclsubscctionshasbecn proposedby Schoellhamer, Pcters.rnd l-artxk ( 1985) using a subdivisionFroude numberi however,{he compound channel Froudc nunrbergiven herein applies lo the entire cross scctionfor the purposeof watcr surlaceprofile computation,as discussedin Chapter5. For a particularcompound channcl geometry and roughness,it is possibleto establisha range of valucs of the discharge(if any) over which multiple critical depths can be expected(Stun.nand Sadiq 1996).The key to such a determination is to recognizethat curvesof depth versuscompound channel Froude number can be made dinrensionlessand independentof discharge Q. The bank-full Froude nurnberfor the main channelis definedby

o B"t'

F, = j?r

(2 . 3)4

C H A P T E R2 : S p e c i f i cE n e r g y

2.O CrossseclionA (Fc /Fr )mar

0.0

1.0

2.O

3.0

F cl F 1 FIGLRE 2.20 Dimensionless contpound channelFroudenumberfor cross,section A.

in *hich the subscript I refers to bank-full values of the geometricparameters. Dir iding either(2.28)or (2.32)by F, effectivelyremovesrhe influenceof discharse. so thar the curve for F.,/F, can be plotted as a function of ry'_r,, alone. as sho* n in Fig_ ure 2.20 for cross-section A. Tb find critical dcpth,r...F, is setto a yalueof unity, so thar it is obviousfrom Figure 2.20 that therc is a range of valuesof l/F, and, there_ fore. a range of discharges,ovcr which two values of crilical depth extst, one in overbankflow and the other in main channelflow alone. (The interrnediate depth is a local maximumin specificenergyratherthan a point of rninimumspecificenergy..y Becausel/F, decreases with increasingdischarge.\r.ecan seefrom Figure 2.20 that an upper limit is placedon the dischargep, beyond which on)y one critical depth exists for the caseof overbank flow The limit Qu tx'curs when F, = F, and for F- = l; hence,Q.. can be calculatedfrom the condilion F, : I as

\ / . ,^ \ : VB'

(2 . 3 5)

The lower limiting dischargeQ. for the dischargerange of multiple critical depths occurs when F./[', lakeson a maximum value as shown in Figure 2.20. In this cise, Fl for 0 - Q. can be expressed, as Qr/Q, from (2.35) and combinedwith the con_ dition F. : l. We have /F,\

\t/,..._ Q , Qt

(2.-r6)

C \ , I r . ' rl : S 1 " - t r fli:cn . , r g r 1 5 'I

h e v r rl t t co i ( F , . / Fr ) , , , .c, .a n r o u: t an ! ' r ; i t ( ' ldl t , r n u : r r r cs o I r l l r r c so l t J cp th I l 9 r 'a n ) t f j s t h r r r . l ca.l r h t r u g h i t i s c o n r c n i c n tt o u s c a d i : e h , r r g r( '( ) r r c \ f r ) n ( J i n6g e = e t [ ] q u i t t i ( r r1\ . . 1 5l n t l 1 . . 1 6p r o r i d c t h c r r r c u ofsr t r i , . o l u r i n t! h r r { ) ( )\t( i 1 r c hf o r c r i t i c a l d c p t l tr r l t c n n r u l t i p l cc r i t i c l r lt J c I t h st ' r r s t A l o l l r n c l r r : r l t c b r : r i cc r l L r i r l j o 'no l r c r , \ u ( ' hi \ l h c i n t c r r r r l - h l r l r i tncge h n i q u cc.l n h c l r j r l l l i c tdo r o l t c l , I uhcn (hc l r o u o d so l l h c r o o t s c u r c ha r t p r o | c r l r c l c l i n c d. \ I t cr nt rt i r c l r ' . ( ' h u I L l l t r vl n d I l h a l l i r n r u ( l1i 1 9 8 8 ) l r o l o s ca n i t c r : r l r r cn u - r n c r r cpurlo e r . d t r rt co s o l r e t h r c ( l L r i i t i r )gni \ c n b ; l ' , . l . r r r r r h r t h l - , i s d c f i n t ' d l r { ) n tt h c l J l r r n t c n l u lcl q l t l i o n . u n d p r L r r i d ca ( i ( l l t i l f d p f o ( c d u r cf o r a s l l t n t c t r i c l l . r ! ' . t i l l l l ] u l aero r ] ] l t c r i t nt hdu n n c l . 'l-hc e o n r l r i r t u t i oonf c r j t i c a l d c p t hu i t h t h c c o n t l ) o i r n d . h i r n n cFlr ( ) u d cn u n l b e r r i c l r D c cbll F . q u l t i o n2 . 3 2r c q U i r t ' rl h c d c r c r r n i n ; r l i ,o. rf nt h c t c o l t . ' l r i c p r o p c r t i c so f t h c n l r l u r l l c r o s ss L ' c l i o nA. o a l r o r i l l l r ) tl o l c c o n t l ) l i s hl h i s l i t \ k i r s h o r , . ni n l h e V i s u u ll l i \ S I C p r o c r ' d u rY c c o n t p i n A p p c n d i xB . T h e l l g o r i r h n tr c q u i r c s. r n i l J ' u t d l t a f i l e ' o f d i s t a n c cc l c r a r i o n p ! i r s . b c t \ \ c e ns h i c h a . , t r l i g h t l i n c r . r r r . r t r c rins l \ \ u r ) 1 ! ' dI.n a r l d i t i o nt.l r cd i s l a n c c s. r t\ \ h i eh s u b s er i o n h ( , u n d i r n eJ\r c l L x - - a t cadh d r h c \ i r l u c \o l N l l n n i n g ' sr r i n e a c h s u b s c c l i o n t u s tb c s p c ' c i f i c -Tdh. c r . r r i o u sq u a n { i t i c s n c c c s s a r fyo r t h c c v a l u a t i o no f t h e c o n t p o u n dc h a n n e lF r o u d c n u n t b c r b y E q r r r t i o n s2 3 2 a n d 2 . 1 3a r e c o m p u t c d T . h e p r o c e d u r c ' c abnc u s c d1 ( )e v a l l r a t et h e c r i t i c a ld e p t hi n a c o n r p o u n dc h a n n e lo r a s i m p l en a t u r a lc h a n n e cl r o s ss e L l i r ) na, s illustratcb dy t h ef o l l o * i n g e x a n r p l e . I,rxA\tpI-E:.3. Forcross-secrionA.prcriouslvdefinedinFigure2.l6,findrhedis(hecriricaldcprhfor dischargerangeof multiplecriricaldeplhs.if any.anddelerrnine charges of .1000. 5000.and 6500cfs ( l 13.1.12. and 184ml/s). So/trlrbn. Fir\1.thevaluesof thecross,scclional areaandtop widthfor bank-firllflow aredetcrmined to beAr = .16Efrr {.{1.5m:) and8r = 8.1.0fr (25.6m). Tbcn.rhe upper p1. is calculared linrilinSdischarge. as

" - VIt: rrt,8,' - 6 : 6 8c l \ ( l / ?6 m ' \ ) O, v 8,1 lle\'alueof (F,,/Fr)...: LjJ6 is calculatcdfrorn a sr.rjc's of incrcasingIalues of _rA,, as shown prc!iously in Figure 2 f0. The lo\ler Iimiting dischargeis given by, ^ Qt

6168 ";'

J r r 5 r f \ ( l - \ \ 8 r n\ )

Thercfore.two valuesof critical depth should be expectedin the range from ,1335to 6268 cfs ( 122.8to 177.6mr/s) for c.oss-secrion A. The equation lo b€ solr ed for critical dcptlr is given by setting the comground channel Froudenumber,F., in F4uation 2.32equalto unity and defininga new function given by F1r) = F. - I = 0. The only difficulty is in conrpuringrhe geomerricproperties rcq':iredfor thc evaluationof F.. This can be accomllished by assumingstraight lines bctwccn su^'eyedground points and computingthe leoDtet.ic propcrtiesas a sunlmatron of thosefor regulargeornetricfiguresfrom one ground poinl lo rhe ncxt. This has b€cn done in the funcliotl subpro )l anopb > po.

56

2 : S P e c i f iEcn e r g Y CHAPTER

a point dre overlying stagnantlayer' The cnergy equation from thc approach flow to ovcr the obstacleis

v1, + pr,g(t:+ J:) + pn-

(2.50)

Collectingtermsanddividingby p, resultsin ap P}r p " '

t

+ = + s ( r : + a : ) + T( 1 . 5 1 )

for in which Ap/p : (pa p,)/pa.This equationis identicalto thePrcviousrcsults gravithe reduced by replaced is gravirarional acceleration sinsle-lareiflow if ihe (,\t/p)B = 8'. The specificenergythenis writtenasE' = -r * acceleration rari"onal = 0 is V2/2p' andthe Froudenumberfrom taking dt'ld)

v

F o = , , . , r1 8l )

rr 5?l

is call ed the tlensinetric Froude nLtntber'Note that the Froude number in which Fo "defined for a single-layerflow of water really is just a specialcaseof the previously - 1' two-layer flow of water under air, in which l-plp The densimetricFroudenumberrepresentsthe ratioof inenial forceto buoyancy force, which is just anothcrmanifestationof the influenceof gravity ln movabledensimetric bed channe|s,which are treatedin Chapter l0' yet anotherfbrm of the the scdiment lt uses Froude number,called the sedimentnunber' is encountered force to the inenial of grain diameteras the length scaleand sl mbolizes the ratio throughnumber Froude iubmerged weight of a sedimentgrain. \'e encounterthe jumps' uniform flow gradout the iemaindir of the text; for example' in hydraulic ually varied flow, and unsteady flow.

REFERENCES of OpenChantel Flttw'NewYork;McGraw-Hill'l()]2 B. A. H,-droulics Bakhmeteff, ' "MininrumSpecificEnergyin Compound OpenChannel Blalock.M. E..andT. W Sturm. pp 699-'711' J. Htd. Div.,ASCE,107,no.6 (1981)."\4inimum Open SpecificEnergyin Compound to Closure Blalock. l\f. 8., andT. w Sturm. (1983)' pp 483-8?' no. I 109, ASCE' Dirr, J. Hrrl. Channel." 20' 3rd RevisedEdition' ILRI Publication Bos.M. G. DischdryeMeasurenentStructures' (1988) the Netherlands Wageningen. icr, 6th ed' NewYork:McGraw-Hill' 1976' Brater,E: F..a;d H. w .King Hamlbookof Hldraai "Computation of CriticalDepth in Symnrelrical Bhallamudi S M H., and M. Chaudhry. " ChannelsJ. Hydr. Res..26.no.4 (1988),pp 311-96 Compound

C r l A P T l , RI

S p e c i f i cE n c . g y

51

IIEC.RAS ttrclroulit Rt[trtnce 'Vonual' version2 2 Davis. CA L S Arnr! Corps of I-n8inccrs, HydrotogrcEngtnecrrngCcntcr. 1998 llendcrson,F. )l Optn Chonnel Flctu Nc\r York: Macmillan, 1966 Kinds\rrcr,C.8..rndR.\\'.CCartcr"DischargcCharritcristicsofRectan8ularThinPlalc \ \ ' c r r s . -" , rl l r d . D r r . A S C F . 8 1 . n o t l Y 6 ( 1 9 5 7 ) P p l J 5 3 - l t o 1 5 KonerDrnn,N. Discussionof NlinirrlunlSpccific Encrgl in CornryrundOpcn Channel J H r d D i r , A S C E . 1 0 8 .n o t i Y 3 ( 1 9 8 2 ) f. ' p { 6 l S Schffllharner,D. H.. J. C. Pctcrs.and B E. Iaro':k Subdi\ision FroudeNunrber"J //rr/r E r r 3 r gA . ,S C E .I l l . n o 7 ( 1 9 8 5 ) . p p 1 0 9 9 l l ( l l Sh"a.,rrin.J. O.. w. tl. Kirbi, V. R. Schncidcr'and H N Flippo BndSe\\'aleruays Analysis Model: ResearchRcpon.' Fcdcral lligh*'ay Adlllini\tr;llion. R'Por1 No lltVtA/ W. a s h i n g r o nD. C . 1 9 8 6 R D - 8 6 / 1 0 8U . . S . D c P l .o f T r a n s p o n a t i o n 'User's\lanual for WSPRO A Compuler Model for Water SurfaccPro' Shearnran,J. O. file Cor)rpulalions.Fcrleral tligh*aS Adnrinislrrlion Rtport I lt\lA 1P'89 027 U 5 . a s h i n S l o nD. . C . I 9 9 0 D e p t .o f T r a n s p o n i l i o n W Srunn, i W. and Af(ab Sadiq. Waler SurfaceProfiles in ConrpoundChanoeI \rith \1ulti' p l c C r i r i c a lD c p t h s . ' - 1H. r d r E n g r g A S C t ' l '1 2 2 .n o l 2 ( 1 9 9 6 ) p, p 7 0 3 - 1 0 '

EXERCISES 2.1. \\'areris flo*ing at a depri of lO fl with a velxity of l0 ft/sin a channelof rectangular section.Find the depthand changein watersurfaceelcvationcausedby a \ rhalis themaximumallowable inroothup*ard slePin the channelbottomof I ft: 0) (Use loss coefficient a head prevcnted? is that choking so srepsize 2 l wilh a smoothconfactionin arethesameasin Exercise conditions 2.2. The upsrream {hedepthof flow andchange Find bot{om a horizontal 9 ft and widthfrom l0 ft to What is the grealestallowable sectjon conltacted the in elevation water surface in (lleadlosscoefficicnt= 0 ) in *idth so thalchokingis preventedl contraction elein q alersurface andthechange 2.3, Detcnninerhedo\\nslreamdepthin thetransition conditions area vcloclty vationif thechannelbottomrises0 15m andthe upstream o f 4 5 n / . a n da d e p t ho f 0 6 m ransitiontf Q - 262 cfs and the depthin a sub'crilical 2.4. Dctcrminethedo*'nstream circularchannello a downin goingfrcm an upslrcanl channelbolromriscs3 279ft'fhe circularchannelhasa diameterof 9 18 ft upstream channel slreamrectanguliu channelhasa width of tectangulat and a deplhof flotl of 7.34ft. The downstream loss. the head 6.56ft. Neglect 2,5. DererminetheupstreamdePthof flow in a subcriticaltransitionfrom an upstreamrecchannelwith a widti of traPezoidal tangularflume that is 49 fi wide to a downstream from tbe upstreamflume drops I ft botlom transition 2: l. The of slopes 75 it and side ro the downstreamtrapezoidalchannelThe flow rate is 12,600cfs. and t-hedepti in of 0 5 channelis 22 ft. Usea hr:adlossco€fficjent trapczoidal thedownstream 2 . 6 . I n a h o r i z o n trael c l t n g u l a r f l u m e , s r r p p o s e t h a t a s m c x r t h ' b u m p " * i 0 r a h e i g h t o f 0 3 3 f t P€runitwidtl in thc flrlne is hasbccnolacedon the channelbottont.Thedischargc

58

C H A P T E R 2 : S p e c i f i cE n e r g y 0.,1cfs/fl- Delerminc the depth ar rhe obslructionfor a tail\\arer dcpth oi 1.0 it and neSligiblehcad losses.Sketch thc rcsultson a specific enereYdiagram.

) 1

A rcctaneularchannel3.6 m !\'idecontractsto a 1.8-m \r ide rectangularchanneland then expands back to the 3.6 m width. The contractionis gradual enough that head losses can be neglected,but rhe expansion loss coefficienl is 0.5. The discharge through t})c transitionis l0 mr/s. lf the do*'nstreamdepth at the recxpandedsection is 2..1m. calculatethe depthsal the approachscctionand rhe contractedsection.Show the positions of the depth and spccific energy for all three sectionson a specific energy dragram.

tJl

Dctcrminc the dischargein a circulaJculven on a steepslop€ if the diameteris 1.0 nl and the upstreanrhcad is L3 m rvith an unsubnlergedentrance.Aiso calculaterhc cntical dcpth. Neglect entrancclosses.Repeatfor a box culren that is 1.0 m square.

2.9. An openchannelhasa senlicircular botlomandvertical.parallel\{alls.lf thediametcr.d, is 3 ft. calculate thecriticaldepthandthenrinimumspecificenergyfor two discharges.l0 cfs and30 cfs. 2.10. Derivean exactsolutionfor criticaidepthin a parabolic channelandplaceit in dimensionlessform. Repeatfic procedure for a triangularchannel. 2.11. A parabolic-shaped irrigation canalhasa top$idrh of l0 m ar a bank-fulldepth of 2 m. p. (i.e.,thedischarge Calculaterhecriticaldischarge, for q hichrhedepthof uniform flow is equalto criticaldepth)for a uniformflow depthof 1.0m. If Q < O, for the uniformflow depthof 1.0m. $ ill the uniformflow be supercritical or subcritical? 2.12, A USCSstudyof naturalchannel shapes in thewcstemUniredStatesreponsan averageratio of maximumdepth1(]hydraulicdepthin the main channel(with no overflow) of t,zD= 1.55for 761 measuremenls. (d) Calculalethe ratioof marinrumdepthro hydraulicdeprhfor a ( I) triangula. (3) reclangular channcl.(2) parabolicchannel. channel.Whardo you conclude? (r) Calculatethedischarge for a bank-fullFroudenumberof Fr = 1.0if r/D = L55 and 8, : 100ft for r., : l0 fr. Wharis rhesignificance of rhisdischarge? 2.13. A naturalchannelcrosssectionhasa bank full cross-sectional areaof ,15ml anda top widthof 37.5m. The maximumvalueof F./F, hasbeencalculated ro be 1.236.Find thedischarge range.if any.$ilhin whichmultiplec.iticaldcprhscouldbe expected. 2,14. Designa broad-crested *eir for a laboratoryflume with a *idth of l5 in. The dischargerangeis 0.1{o 1.0cfs.Themaximumapproach flou depthis l8 in. Determine theherghtof theweirandthe* eir lengrhin rhef'lowdirecrion. Plotrheexpecred headdischarge relationship. 2.15. Plot and comparethe head-discharge relationships for a reclangular. sharp-crested weir haling a crestlengthof 1.0fr in a 5-ft widechannel\\ irh tharfor a 90. V-notch,

C A p r l R 2 : S p c c i UEc n c r g y 5 9 rrcir if bothweir crest-s arc I li bo\c thcchunnelbottonr.Considcra shlrp,crc'sted hcadrangeof 0-{.5 ft. 2.16. Derivelhehead-discharge rclationship for a triangular. brotd-crcsted weiranda corresponding relationship for C" analoeous to Equation:.JS. 2.17. Dcrivethehead-discharge rclationship for a truncated. lriangular, sharp-cresred weir with nolchangled andvenicalwallsthatbeginar a heighrofir abovelhe triangular crest.AssumethatI/ > h,. programYoYC in AppendixB to calculare 2.18. Modify theconrputer rhecriticaldepthin a circularchlnnel. programthat compulesrhe dc|th in a widthcontraction 2.19. W.ite a computer and the upstream depthgivena subcritical rail$aterdeprhasin Figure2.1L Assumethatrhe channelis rectangular at all threesecrionsand makeprovisionfor a h€ad-losscoefllcientthatis nonzero: includea checkfor possible choking. 2.20. A laboratoryexperimenthasbeenconductedin a horizonralflume in which a shar?crestedwet plalehasbeeninstalledto determinethe head-discharge relationshipfor a rectangular, sharp-crested weir. With referenceto Figure2.23,P = 0.506 ft, L = 0.25 ft, and, = 1.25ft. The dischargewas measuredby a bendmeterfor which the calibrationis givenby Q = 0.015 Al0'r, in which @ = dischargein cubic feet per second(cfs): Al = manom€terdeflectionin inchesof warer;and the uncertaintyin the calibra(ionis 10.003 cfs, The head on the crestof the weir was measuredby a point gaugeandis givcnin the data rablerhatfollows.An upstreamview of lhe weir nappecanbe seenin Figure2.28.

FIGURE2.28 Upstreamviewof theflow overa rectangular. sharp-crested $.ek(photograph by G. Sturm).

b0

C H c P r E R l : S P e c i f i cE n e r P Y

llr, in.

H, tt

r3.2

0.,198

I1.5

0.,176

I1.2

0..{71

8.3

0.,r25

8.0

0.,{21

6.2

0.3u,l 0.386

,{.3

0._113

1.2

0.ll,l

2.1

0.212

2.0

0.25?

(.o) PIotthe headon the verticalscaleand thc discharge on the horizonlalscaleof

\b)

loglog axesandobtain a least squaresregressionfit forcing the inverseslopero be the theoreticalvalue of 3/2. What are the singlebest fit valueof C, and the "0 esti$ate" with the standardenor in Cr? Comparethe standarderror of the calibratron. uncenainty in thebend-meter relationship and then first usingthe Kindsvater-Caner Cdlculatethe discharge usingthesinglebest-fitvalueof Cr.Comparebothsetsof resultswith the measthepercentdifferences andalsoplottingthemeasureddischarges by calculating discharges. uredvs-calculated

CIIAPTF]R3

Momentum

3.1 INTRODUCTION e q u a t i o ni n c o n t r o l - r o l u l nfeo r m i s a v a l u a b l et o o l i n o p c nc h a n n e l The nromentum I t f l o w a n a l y s i s . o f t e n i s a p p l i e di n s i t u a t i o n isn v o l ! i n g c o m p l e xi n t e r n a lf l o w p a t tcms with energy lossesthat initially ate unknown. The advanlageof lhe monlentum equationis that the detailsof thc intenlal flow palternsin a control volume are immaterial.It is necessaryonly to be able to quantify the forces ard rnonlcntum fluxes at the control surfacesthal form the boundariesof thc control rolune. This propertyof the rromenlulll cqualionallows it 10 bc used in a cornplenlentatlfashion with the encrgy equation to solve for unknown energy losses in otherwis€ intractableprobicms.

3.2 IIYDRAUI,IC JUJ\IP n f t h e m o l l l c n t u me q u a t i o ni n o p e n c h a n n e fl l o w i s T h e m o s tc o m m o na p p l i c a t i o o t h e a n a l y s i so f t h c h y d r a u l i cj u m p . T h e h y d r a u l i cj u D l p .a n a b r u p tc h a n g ei n d e p t h from supcrcriticalto subcriticalflow, al*'ays is accompaniedbl a significant energy loss. A countcrclockwiseroller rides continuously up the surfaceof the jump, entrainingair and contributingto the generalconiplexity of the internalflow patternsillustraledin Figure 3.1.Turbulenceis producedat the boundary between the incorningjetand the roller.The turbulenteddiesdissipaleenerg) front the mean flo$, aithoughlhcre is a lag distancein the downstrearndirectionbet\r'eenthe point of maxintumproductionof lurbulenceand maximutn dissipationof energy(Rouse, Siao, and Nagaratnanr1959).Funh('rmorc,the kinetic encrgy of the turbulcnceis rapidly dissipatedalong with thc nleanflow cnergy in lhc downstreamdireclion,so 6l

6:

CflAPTLR .l: i\lonlcnlurn

t/-

'-)

V2

a . / ^

I

Protile

Sec.C-C FICURE 3.7 channel' to a hydraulicjump in a nonrectangular equation Applicationof thc momentum

that the turbulent kinetic energyis small at the end of the jump This complex flow situation is icleal for the appljcationof the nlotnentum equltion. becausepreclse mathematicaldescriptionof the intemal flow pattem is not possible lf any general nonrectangularcross section is consideredas shown in Figure 3.1. a control volume is chosen such that the hydraulic jump is enclosedat the upsfeam and downstreamboundaries,where the flow is nearly parallel This choice of conrrol volume boundariesallows the assumptionof a hydrostaticpressure force at the cntranceand exit of the control volume Also assumedis that the velocity proltles are nearlyuniform at the upstreamand do\\'nstreamcrossrections, = I The boundary with the result that the momentum corection coefficient F jump in comparisonto the neglected is of the length shearover the relativelyshort jump in a horizontalchanoccur to is assumed the Finally, changein pressureforce. flow direction in the equation momentum the nel. Under tiese assumptions, becomes ( 31 .) Frl - Fpt: PQ\V: - Vt)

I and force;pQV : momentumflux; andthesubscripts in whichf" - hydrostatic The hydrorespectively cross sections, 2 referto ihe upstreamand downstream staticforce is eipressedas7fto4,in which rlr"is the distancebelow the free surface to the centroid oi the areaon which the fcrce acts,as shown in Figure3 l ' and the

( - 1 1 . \ t tI , R l : M ( ) r r c r l u n ] 6.1

r u c a nr c l r x i l y , 1 1, , Q / A .f r o r : rt h c c , ' r : : n u r t r e q U l t i o n W i t h t h c s cs L r b s l i t u t i o n l nsd d i r i d i n g I ' , q u a t i o 3n l l b ) t h c s p t ' c i l : : * r ' r e h r . y . t h c r c r c ' \ u l l s

a)'

(-l2)

'1. h,: + Q

.(i:

\ \ ' c s c c f r o m ( h i s r c r r r a n t c l r c ' not f r h ! - e q u a l i o nt h a t , i f $ c d c f i n e a f u n c t i o n , l y ' , $ h r , h $ c u r l l r . r l l r h cn t , t n t t t t u n t f , ,tnt ,. . n .a , rtl

_ ;

(-1.-l)

t l r e ni t s c q u a l i n u p s t r c a ra r n d d o , , r n s t r c - aomf t h c h l d r a u l i c j u m p c a n b c u s c d t o d e t c r nirn et h c s c q u r ' ndt c p l h .$ h i c h i < r h c d cp t h ; i f t c rt h ej u r n p .i I t h c u p s t r c a ncr o n d i l i o n s a r c g i r e n , o r v i c e v c r s a .l v J c r i e p r c c i s c l l . t h e r n o r n c n t u nftr i n c t i o ni s f o r c c p l u s r n o r n c n l u mf l u x d i v i d e db y t h e s p r : c r f i cu e i g h t o f t h c f l u i d . a n d r h i s q u a n r i r y is conserved a c r o s st h e h y d r a u l i cj u m p . The distancefrom the free surface ro the centroid of thc flow section,ft., is a unique function of the depth,), and the geontetry of the cross s€ction.For example. the nronentum functionfor the trapr:zoidalscclion is given by M=

b] t:2- . - -: r ' t

2

3

o:

( 3 . 4)

+,. gr(b + lrr )

in uhich b - bottom width; r4 = sidcslope ratio: and y : flow dcpth as dcfined in Table 3- L ftre trapezoidalsectionhas been divided into a recranglcand two triangles, and the additivepropenyof the first moment of the areaabout the free surface has becn usedto obtainthe expressionfor Ah.. The momentunrfunction definitions for scveralother prismaticcrosssectionsalso are giten in Table 3- I The nromentum equationcan be placed in dimcnsionlessform and solved numericallyfor the sequentdepth.If Mr is known for the trapezoidalseclion from inconring flow conditions,for cxample. then setting Mt = Mz and nondimensiona l i z i n gr e s u l l si n

L5.\r

','i

* n^ r

lZ:

. \ r i ' ( t- . \ r i )

1.5

ri

rl

32:

. , i " i t- r ' i . ;

(3.5)

in rvhich A - ,r./.r',;,rj= n*,/b; and 2: : 91mtlgb5. Equation 3.5 can be solved directlyforZandthenplottedasl.,/r',:/(1'.2)asshowninFigure3.2whereZ: Z-,-n.Similarly, rhe solutionfor the sequcnl deprh ratio for the circular casecan be giren as shown in Figure3.3 with Z;^ : g:/gd5.lmplicil equationsfor yr,/_r,, and their graphicalsolutionsin a form similar to ttrat of Figures 3.2 and 3.3 for trapezoidal and circular channelswere proposedb1' )\4assey( I 961) and Thiruvengadam ( l 9 6 l ) , r e s p e e te i rl y . To solvc the nonlinearaJgcbraicequations for the sequentdepth ratio numerically, a fLrrrctionFlt) = Mr M. is defined and solved by interval halving or s o n c o l h e r n o n l i n c a ra l g c b r a i ce q u a t i o ns o l r e r . T h e c r i t i c a ld e p t hm u s t b c f o u n d f i r s t . h t x r e r , . ' rt.o l i r i i t l h e r o o t s e a r c hr o t h e a l p r o p r i l t c s u b c r j t i c aol r s u p e r c r i t i cal \olution.

6-1

CH^['TER 3: Mornentum

IA IILE ].I

Nlonrentumfunctionfor channelsof differentshapeslr : flor depth) Reclangular

F-r--l | lv

b j l1 2+ Q l l k ' )

I

D

Trapezoidal

\ l ' l t

b.r 2 + rrrr/'3 + Qr.'lgrlb + /ll')l

t l ' D

Triangular r) rnyr,/: + P:7'(gmr

\ v l'v.;' l Circularr 'f'-,/..-\

1 2Q11 l ' L e d-' (soi n d )8 l 3 ( 0 i 2c) o s ( s l l ) ) d 1 +

1 3 ' , " { d , r ) s i n r ( 2d )

dT ffily

l'

{ \qu

Parabolic+

f'-81-l

Tl---'----f Jr'r \ lv

r \l--,/

'0 = 2 cos 'll t

( a / 1 s ) { _ r+5 Ir . 5 0 : / / ( 8 (r_)r ' r

./

2o/d)l

; = a'/;-t1n

For the rectangularcross section, there is an exact solution fbr the sequent depthratio that dependsonly on the uPstreamFroudenumber.Settingthe valuesof the momentumfunction per unit widti upstreamand downstreamof thejump equal and rearranging,we have

r r:cfr 2

2

I

1l

L.Y2 ,)rl

(3.6)

With some algebraic manipulation and nondimensionalization,Equation 3.6 becomesa quadraticequation:

, \ ' + , \ - 2 F i- o

Q7)

n y 1 l b = A 0 6 2 50 1 2 5 0 2 5

0.50

20

1 0

12 10 B d

6

E 4 2 0 0 01

0 1 Zt.p

t-tcURE-1.2 Scquentdeplh rario for a hydrauliclump in a rrapczoidalchannel (Zr,.p= pnr,n/lgtrtf?))

y1/d = O.1

10

I

I

{

./ /

o.2 I/

0.3

4

0.1

0.4

0.2

0.3 Z"n.

0.4

3.5

0.5

0.6

FIGURI]3.3 Scquentdepth ratio for a hydraulicjunrp in a crculiu channel(2.,^. : QIIS,erf"]).

65

C H \ t ' T E R J : i \ l o r n c n t ul

fzlYt L lyz

25 20 15 10

10 FroudeNumber,F1 , fzlh

/ LtY2

. Et-/E1

-

Theoretical

FIGURE 3.'l jurrp in a rectangular channel: 'r',/r'': for a hydraulic Compuisonof theoryandexperimenl : j umplen5.,h r.r|io' \I)atafrcnlBradle\ dcpthratio:Erl81= energvlossratio;Ur. sequent ttndPeterka)957.\

= The soluin which ;\ = -r'./r''and F, - the approachFroude number 1t1rl3ti)r" formula tion to Equation 3.7 is given by the quadratic

. r = l I r + . ' / r *s r i]

(3.8)

The unklou n energylosscanbe obtainedasthe differencebet\reenthe upstream E, ln dimensionless uoiue,of the specificenergy,E. - f and downstream form.this is F, .\ + Firt_\l _!=

Et

I _

I + Fii 2

(3.9)

Equations 3.8 antl 3.9 are shown in Figure 3'l and conrparedwith expenmental in da'raobtaineriby Bradley and Peterka( 1957)of the U S Bureauof Reclamation with flumes five in were obtained data The a comprehensirestudyof stilling basins. 2 and 20 The the upsrrsamFroudc numbershaving ralues betweenapproximately 3 8 and 3 9 Equations the theoretical and data ogr..r.nt betweenthe experimcntal analysis' momentum in the nrade assumptions is"quitcgood, confirmingihe initial also is experimentally onl) be determined junlp1, u'hich can fhe tenittr of the somewhat rhe experinents in defined jutnp was length shown i-n Figure 3.4. The qualitativell: as the distanc;fr;m the front of the jump to eitherthe point where the

f-llltltF

J

\lL,ntenturn

6'l

18 16 Parabolic 12 i

n

10 Triangula r

4

6

B

10

12

14

16

18

20

FroudeNumber.F, FIGTJRE 3.5 parabolic, andtriangular channels. of sequent depthratiosin rectangula-r. Contparison

jet lcft the floor or a point on lhe water surfacc immcdiatelydownstreamof the roller,whicheverwas la-rger.Basedon this data, the jump lengthoften is dcfined as c i ) ,t i n r c .t h e d c p t ha f t e r t h c j u n 1 p . Graphicalsolutionsfor the hydraulicjump in triangularand parabolicchannels can bc obtainedin the same manner as for thc trapezoidalchannel.In both cases, the Froude number is the only independentdinrensionlessparameter.Silvester (19&) summarizedsome experimentaldata for thc scquentdcpth ratio ard the agreementwith tie energyIossin triangularand parabolicchannels,and reasonable momentunrsolutions*,as demonstrated.The seclucntdcpth for the triangular,parabolic.and reclangularchannelscan be conrpareddirectly on the basisof the actual channel,as in Figure 3.5. We approachflow Froudc nunrber for a nonrectanguJar can seetbat the nragnitudeof the scqucnt-depthratio for the sameFroude nurnber "fuller" from triangularto pffabolic increascsas the channclcross sectionbccomcs to rectanBular. The ratio of the energy loss to the availableupstreamenergyE/8, is comparedfor the triangular, parabolic,and reclangularchannelsin Figure 3.6, arrdthey are remarkablyclose to cach other. The monrentum equation also has been applied to the circular, or radial, hydraulicjump (Koloscus and Ahmad t969). The major differenceb€tween tie jump in a prismatic,rectangularchanneland the radialjump is that ttre hydrostatic forccson the walls of the radially cxparrdingchannelhavca conrponentin tbe radial directjon.This, in tum, requiresthat the surfaceprofile of thejump be known. The simplestassunrption,which is adoptcd for Figure 3.7, is to take the effectivejump profile to be linear.Arbhabhiranraand Abella ( 197I ) assumedan ellipttc water surface profile, but Khalifa and McCorquodale (1979) showedthat air entrajnmcnt

Crr\t'f Fp .] \lonrentum

f l I l

fjt

z

4

6

8 1 0 1 2 1 4 1 6 1 8 2 0 FroudeNumber,Fl

FIGURE 3.6 Comparison of energylossesin a hvdraulicjump in reclangular. parabolic. ano rnangular cnanDels.

shiftsthe effectiveprofile as determinedby the hydraulic grade line toward rhe linear.shape.The sequentdepth rario for a radial jump (rn J r./r, = 2) is compared uith the rcctangularchrnnel jump (r./r, = l) in nigr..':.2. W. seethat the radial Jump nasa smalJer,sequent depth ratio for the sameapproachFroude nunlber but a rargerencrgy tos\. Lawson and phillips ( l9g3) as well as Kialifa and McCorquodale (1979) have dcmonstratedreasonablygood expcrimental ugre.m.nt *itn tn. *q*", deprh rario and relative energy )oss when aisuming the linear :11"1.]l'.1, Jump prollte. The appearanceof the hydraulicjump, as well as the sequentdepth ratio and , the dimensionlessenergy loss, is a function of the approaclin.ouo. nr_U"., u. shown in Figure 3.8. For Froude numbers between2'.5 and a.5, the enteringlet oscillates from the channel bottom to the free surface, creating -surfacewaves for long distancesdownstream.Jumps *'ith Froude numbers u.ti."n-+.-: and 9 are well balancedand stable,becausetie jet leavesthe channel boitorn o, uppro*r_ mately the samepoint as the end of the surfaceroll.r. For an approuchFroudenum_ ber in excessof 9, the downstreamwater surfacecan Ue ,ouit, but large energl, lossescan be expected. It is instructiveto considerthe shapeof the momenfum function,slncelt obvi_ ously is a function of depth1,alone foi a given ,nd g.o*.try in rnuch the sane e fashionas.thespecificenergyfunction. If-we coisioer irre ,..iinguiu, .r,rnn"r, to. example,rhe momentumfunction per unit of channelwidth is giien by M

2 -'-

(t2

;o: z ^ g+ - l

(3l0)

CIArrr'R

. 1 : l ! ' l c , m c n t u m 69

{

ro = rz/r1 (a) De{lnilionSketch

1.0 .,ra = 2

20 18

- ' 0.8'o

16 14

0.6

12

IJJ

10

Lll

8 6 4 2 0

6

8 1 0

1 2 1 4 1 6 18

20

FroudeNumber,F1

DePthandEnergyLoss (b)Sequenl F I G U R E3 . 7 d!:plhandenergyIossratiosfor a radialhydrauliclump Sequent

which has two branches and a minlmum. As -l approaches7-ero,the molnentum the parabolaf/2 function per unit of rvid1happroachesinfinity' while it approaches frrnction is the momenlum value of minintum as ,r' bec,onresvery large The z e ro: r e s u l t t o l h e s e t t i n g a n d t o w i l t ) , c s P e c t ) o b r a i n e db y d i f f c r e n t i a t i n g o2 ' 8')

( 3l l )

CHAPTFR -li l\lorrentunr

F1Between1.7 and 2.5 FormA-Prejump stage

Fl Betlveen2.5 and 4.5 FormB Transition stage

F1Beiween4.5 and 9.0 jumps FormC-Range of well-balanced

,

i

l

F1Greaterthan 9.0 Form D-Effective jump but rough sudacedownslream FIGURE 3.8 jump for diffcrentFroudenurnberranges(U.S.Bureauof ReclaAppearance of a hydraulic mation1987).

If we solve for.r', we obtain the expressionfor critical depth for a rectangularchannel derived from a considerationof minimum specific energy.Therefore.critical depth occurs not only at the minimum value of specific energy for a given discharge,O, but also at the minimum value of the momenrum function. The correspondencebetweenthe specificenergy and momentuntfunctionsis illustratedin Figure 3.9 for a hydraulicjunp in a reclangularchannelwith thc functions given in dimensionlessform. Clearly,conservationof the momentumfunction as required by the hydraulicjump analysisrequiresan enr'rgy loss.Also note that the sequentdepth ratio,_t,r/,r',, and the encrgy loss increasefor smallervaluesof the approachdepth. As the approachdepth decreases,the velocity head increases;and

Cli4Prt

R -l: \'lorientum

'71

2

uy" Mylo -

Specific energy

Momenlumfunclion

F'IGURE3.9 diagrams nondimensiondepthson specrficenergyandn]ontentum Hl draulicjunrpsequent alizedby criticaldepth.

specific so the Froude nurnb€rmust incrcase.ln othcr words, the dinrensionless energy and nronrentumdiagrams confirm the incrcasein sequentdepth ratio and energy loss with Froudc numbcr found previouslyfrom the solutionsof the energy e q u a l i o n sa n d s h o r v ni n F i g u r e3 . . 1 . and monrelrtum The gencralclse for the mininrurn value of the mootcntutnfunction can be derivcd for any nonrectangularsection for u'hich B : I = cons(ant.Setting the derivativeof the momentum function with respcctto -r lo zeroyields

!4 = !,o0., ' dy

dr

9-14 : o gA'

(3.12)

in which dA/d,r,hasbeen replacedby the top width, L Using the definition of the first momentof the areaand thc I-cibniz ntle, it can be shownthat the first term of the derivativeis cqual to the flow arta, A, frorn \rhich it is obviouslhat thc mininrurn value rf thc rnonrcnturnfuoction occurs*hen the Froudcnunlbersquarcdfor t h ( n , ' n r e 1( .r r g r r l a, r. l r r r r nlt i . . q r , , l t o u n i t l : t h r t i r . B - B l g A ' ' I 0 . bctweenexpcrinlcntalresultsard thc nl(rlllanlum Although generalagrr-cr:rcot rheory lor the hydraulicjunp has bcen dcmonslrirted,it is usefulto considcr lhe

'72

CH,rprr:t J: Mo rcntum

cffectsof the assunrptionsmade in the analysis.llarlcnran( 1959)concludedfrorn a unit h c d a t ao f R o u s e ,S i a o .a n d N a g a r a t n a n( 1 9 5 8 )t h a tt h e c f l c c t o f a s s u n r i n g a t t h e t $ o c n d s e c t i o n so f a n d n c g l c c t i n gt h e t u r b u l e n c e l o r m v e l o c i t yd i s t r i b u t i o n the hydraulicjump indeedis small. Rajaralnanr(1965),horvever.shorvedfrom his analysisof thejump as a wall jet that the intcgratedboundaryshearstresscan affect dn d e x t e n d e d t h e s e q u e n dt e p t hr a t i o .L c u t h e u s s earn d K a n h a ( 1 9 7 2 )g c n e r a l i z e a this conclusionby conductingexperimentson jumps uith fully developedinflo*s and undevelopcdinflous. From the tr,'o dirnensionalReynolds cquations, they derived an intcgratedforn of thc hl draulic jump cquationthat elinrinatesthe convcntional assumptionsby lunrping them into a singleiactor. e: ,r\,r

(1.t3)

-)tt-

and F' = approachFroude number.For c = 0, we recoverthe in which '\ = _r'./-r', result given in Equation-1.7.On tlre other hand. if rvc considerthe inlluence of the mean shearstressover the length,Z, of the jump, e is given by Ct L

-\l

2 _r2A

I

( l. I -t.)

in which Cr: overall skin friction coefficient. Leuthcusscrand Kanha (1972) showed from their experimentalresults that e has essentiallyno influence on the sequelt dcpth ratio for approachFroude numbersless than 10. For greatervalues of the Froude numbcr,however.the de\eloped-inflowjurnp had a smaller sequent depth ratio than prcdictedby Equation 3.8 due to the influence of the boundarl shcarforce. Furtlrerrlore,the developedinflow junrp *'as )ongcr and lower than in the undcveloped-inflowcase,which Lcutheusserard Kartha suggestis due to the tendencyfor thc undevelopedinflou to scparate,thus reducingthe boundary shear. lt must also be pointed out that the jump length in Equation3. l'1 is defined as the point at which no further changesare observedin the centerlinevelocity distribution in the downstreamdirection. The dimensionlesslength. L/r'.. has a value of approximately l6 for the fully developedinllow and a typical value of q is I x l0 r. These experimentalvaluesresult in a value of eof approximately-0.1 and a relativeerror in the sequentdepth ratio of lessthan l0 percentat a Froude number of I0. lf the effcct of boundaryshearis relativcly small for hydraulicjLrmpsin smooth channels,it may not necessarilybe negligible in the caseof a channelwith significant boundary roughness.Expcrimentsby Hughesand Flack ( I984) confirm this to be the case for both strip roughnessand gravel beds. Their laboratory results showed that both the lcngth and sequentdepth of a hydrauljcjunrp are reducedb1' large roughnesselements.A bed of j to j in. grarel. for exampJe,rcsultedin a l5 percen'.reduction in the sequentdepth ratio predictedfor a smooth channel at a Froude number of 7. The effect of boundary shearon the hydraulicjump is sinrilar to the effect of fonn roughnessprovided by baffle blocks on the floor of a stilling basjn. Thc obstructioncausesa lower sequentdepth ratio at the same Froude number and

CHrtrt n -l: \'lomentum 13

(a) HydraullcJurnpwrthBall e B ocks

i

o

MomenlumFunction,M jn MomentumFunclion (b) Fleduction T'IGURE3.IO jump dueto theextemalforceof blockson in momentum functionfor a hydraulic Dec.ease volume. the control rlakcs the jump position rnore stable.Thc effcct of the (ibstnrctionon the monreni n F i g u r e3 . 1 0 ,i n w h i c h c ) c a r l yt h e d e c r e a sien t h c v a l u e t u r l b a l a n c ei s i l l u s t r a t e d of the nromentum function from thc supcrcritical to subcrilical slate in the hytiraulic jump must bc exactly equal to the drag forcc of thc obstruction,pr, dir idcd h;' the fluid specific wcight, 7.

1,1 Crapr rp J: I'lomentum

Iirlr-luc BASTNS Hydraulic junrps are used extensivelyas energy dissipation dcrices for spill*ays becauseof the largc pcrccntageof incoming cncrgy of thc supcrcriticalflo\\' that is l o s t ( s e eF i g u r e3 . 4 ) .T h e s t i l l i n gb a s i n .l o c a t e da t t h e d o ! \n \ t r e a me n d o f t h e s p i l l way or the spill\\,aychute,usually is constnlctedoI concrete.It is intendedto hold the jump within thc basin,stabilizeit, and reducethe lcnsth rcquiredfor the jump to occur.Thc rcsulting lou -velocity subcriticalflow rclcascddou nslreamprerents erosion and undernriningof dam and spillway structures. G c n c r a l i z e dd c s i g n so f s t i l l i n gb a s i n sh a v cb c e nd e v c l o p c db y t h e U . S . B u r . a u of Reclamationand otbers.basedon expcrience.fieltl obsen ations,and laboratory n r o d e ls t u d i e s S . p e c i a la p p u n e n a n c easr e p l a c e d* , i t h i n t h e s t i l l i n gb a s i nt o h e l p achieveits purpose.Chute blocks placedat thc entranceto the stilling basin tend to split the incoming jet and block a ponion of it to reducethe basin length and stabilize the jump. The end sill is a gradualrise at the end of the basin to funher shorten the jump and preventscourdownstream,which nray result from the high velocities that developnear the lloor of the basin.The sill can be solid or dentated.Dentation dilfuses the jet at the end of the basin. Baffle blocks are placed acrossthe floor of the basin at specifiedspacingsto funher dissipateenergy by the impact of the high velocity jet. llowever, the blocks can be used for only relatircly lorv velocitiesof incoming flou,: otherl ise, cavitationdamagc may result. With referenceto the typesofjunps that can form as a function of the Froude number of the incoming flow (see FiSure 3.8). the Bureau of Reclantationhas developedseveral standardstiJling basin designs (U.S. Burcau of Reclanration, 1 9 8 ? ) ,t h r e e o f w h i c h a r e s h o w n i n F i g u r e s- 1 . 1 1 . 3 . 1 2a. n d 3 . 1 3 . F o r i n c o m i n g Froudenrrnbers from L7 to 2.5. thejunp is weak and no specialappurtenances are required.This is called the Trpe I basirt. In the Froude number rangc from 2.5 to ,1.5,a transitionjump forms with considerable\\'aveaction. The frpe IV basin is r e c o m n r e n d efdo r t h i s j u m p .a ss h o w ni n F i g u r c3 . I l . 1 1h a sc h u ( eb l o c k sa n d a s o l i d end sill but no brffle blocks.The recornnrended taillr'aterdepth is I 0 pcrcentgrcater than the sequentdepth to help preventsweepoutof the jump. Becauseconsiderable u'ave action can remain downstreamof the basin. this junrp and basin are sometines avoidedaltogethcrby widening the basin to increasethe Froudenumber.For Froude numbersgreaterthan.1.5.either frpc /11or Trpe II basins.as shown in Figu r e s 3 . 1 2 a n d 3 . 1 3 , a r e r e c o m m e n d e dT.h e T y p e I I I b a \ i n s h o w n i n F i - l u r e3 . 1 2 includes baffle blocks. and so it is linrited to applications $here the inconring velocity docs not excecd60 ft/s. For lelocities exce'eding60 ft/s. the Type Il btsin shown in Figure 1.13, u'hich has no baffle blocks and a dcntaled end sill. is suggested.It is slightly longer than the Type III basin. and the tailtvrrer is recommendcd io bc 5 perccntgrcaterthan thc scqucntdcpth to help prevents\\'ccpout. Matching the tailwaterand sequentdepth curvesover a range of opcrrting dischargcsis one of the most importantaspectsof stilling basin dcsign.lf the tailu ater is lower than the sequentdepthofthejunp. thejunp ma) be s$ept out ofthe basin, $hich then no longcr scrves its purposc bccausedangerous erosion is likelr t0 occur do*,nstream of the basin. On the other hrnd. a tailwater elevationthat is higher than the sequentdepthcausesthe junrp to back up againstthc spillway chute

t^^

-

-:t-

_:-: o

(alTtt€

lV Aat,.

D l.€es

3

.

6

6

7

:t ilf t

a t

s I

'\ - , -r,. - ' o1^' z '.

b

3

lr.

)-

s al AIJ 3 l

3

2

2 (b)M,nn,!nra l*are,O€ps's

6

6

-':!t {, s

1

5 -13

3

r-IGL RE 3.II Tlpe lV rtilling basin charactcristicsfor Proudc nurntrrs bct$ecn 2.5 and4.5; r/,, d, : s c q u e n td c p t h s( U . S . B u r c a uo f R e c l a n a l i , r n1 9 8 7 ) .

15

e l

(b) Mi.,mum Ta lwalerDePlhs

L l i .";

6 -l

6

elt (c) ae'ghl ol Ba{ib al@ks and End Srl L

i

t1

e

i r

I

rl3

1

:l

I

f l6

t8

F I G U R E3 . I 2 for Froudcnumbcrsabove'15 whcreincoming\elocTypcIII stillingbasincharactcristics = 1987) yr ity is < 60 ft/s;d,. d, sequentdepths(U S Bureauol Reclatuation 16

" , . 5 ; s 4/

lar TrtE I Bis,^ o mc.s,o^s

20

t l '5 .{ l_-

Al*- '6

i l

3 l 12

12

a

(bt Mr. mln

T3l*aler

o.plhs

5

5

3

3

FIGURI] 3.13 5; /r. d: = \cqucntdeplhs for Froudenunlttrsabove'1 TypcIl stillingbasincharaclcristics ( [ J . SB. u r e a u o f R e c l r : n r t i o1n9 8 7 ) .

't'7

?8

CHAPTIR

-]: \lorttcnlum

"drown out or be subrnerged,so that it no longer clis-ripltesas much cnergy. and The idcll situationis onc in \r'hichthe sequcntdepthsperfectlr ntltch the tail\r'ater ovcr the full rant!- of opcratingdischarges,but this is unlikcl\ to occur.Instead,the basin lloor elcration is set 1()rr]atchsequentdcpth and tail\\aler at tllc naxinrum designdischargeat point A. as shownin Figure 3. I -la. and the basincan bc widened as shown in the figure to help implovc thc ntatch at lower dischargeswhile erring on the subnrergedsidc ratherthan the s$'ccp-outside. Il the scquentdepthcurve is shapedas sho$ n in Figure 3. 1,1b,the tailrvaterand scqucntdcpth would havc lo be nratchedfor a lowcr dischargcthan the maximum. such as point B in the figurc, to ensuresullicient tailwaterfor all discharges. S e t t i n gt h e f l o o r c l $ ' a t i o no f t h e s t i l l i n gb a s i na n d s e l e c t i o no f t h e t y p eo f b a s i n to use dcpendson predictingthe flou and velocity at the toe of the spillway and hcncethe encrgy lossover the spillway.Some generaldcsign guidanceis provided i n t h c D e , s l g no f S n a l l D a n s ( U . S . B u r c a uo f R e c l a m a t i o n1 9 8 7 ) .l I t h e s t i l l i n g basin is locatcd irnmediatclydorvnstreamof thc crcst of an ovedlow spillway or if the spillway churc is no longer than the hydraulic head, no loss at all is recommendcd.Here. the hydraulicheadis dcfined as the differencein elevationbetu'een the reservoiru ater sudaceand the downstreamwater surfaceat the entranceto the stilling basin. lf the spillway chute length is between one and five times the For hydraulic head,an energylossof 10 pcrcentof the hydraulic headis suggested. percent head. a 20 loss five tintes the hydraulic lengths in cxcess of spillway chute of hydraulic head should be considered.For more accurateestimatesof head loss, the equationof graduallyvaried flow can be solved along a spillway chuteof constant slope,as describedin Chapter5, exccpt in the vicinitl of the crestwhere the flow is not gradually varied and the boundarylayer is not fully developed.For this region,the two-dimensionalNavier Stokesequationsin boundary layer form must be solved numerically (Keller and Rastogi 1977).

3.4 SURGES in a discussion of unsteady rightfullybelongs a consideration of surges Although flow surgescan be analyzedby the methodsof this chapter by transformingthem from an unsteady flow problem to a steady one. This transformation, as shown in Figure 3.15, is accomplishedby superimposinga surgevelocity, 4, to the righl so that the surgebecomesstationary.From this viewpoint, which is that of an observer moving at the speedof the surge,the problem is nothing more than the steady-flow formation of a hydraulicjump. Surgesoccur in many open channelflow situations.The abrupt closing of a sluice gate at the downstreamend ofthe channel,shown in Figure 3.15,would create a surgeas shown.Other examplesinclude the shutdo$n of a hydroelectricturbine and the resultingsurgein the headrace,a tidal bore, and the surgecreatedin the downstream river channel by an abrupt dam breali,

.,\

\ 0.2.In Figure4.8, rhe horseshoedata for thi velocityratio V/V,, whereV = partly full velocityand 4 : full flow velocity,are.olnpurid with relationshipsfor circularand horseshoe conduits.The horseshoe data iollow Manning,sequation with constant^rat low relativedepthsbut thenapproachthe pomeroy( 1967)empir_ ical relation for circular sewers(V/_V,: 1.g9j(114; ttal ar largeielative. depihs. (The theoreticalrelationshipfor VtVi with consrantManning's i essentrallyis the samefor circularand horseshoe conduits.)Thus,in Figure-4.g, the reductionin velocity observedin both circular and horseshoeconduitsfor depths greaterthan half full seemsto correspondto an increasein flow resistance due to the shapefactor, but the velocityreductionis not as large aspredictedby Camp(1946). Nonuniformityof the openchannelboundaryin the diiection'oi flow, in either . plan or.profile view, necessarilycausesa changein the velocity distributionand hydraulicresistance to flow. As an example,the development of tireboundarylayer in a supercriticalflow dischargingundera sluicegateresultsin a deceleration and a changein surfaceresistance due to the nonuniformityof the flow crosssectlon.In graduallyvariedflow (e.g.,a gradualnonunifonnityin rhellow direction),the flow resrstance com:nonlyis assumedto be the sameasthatobtainedin a uniform flow at the samedepth.The enor associatedwith this assumprionmay be small in most cases,but it is essentialthatmeasuredvaluesof Manning'sn for leneralengineering

CHAPTIR 4i UnifonnF]ow

ll2

1.0

/1"->z A

Tl;T7a

vlvwli

*

0.8

+ \1-_--ll t 100ft). The n valueis lessthan that for minor streamsof similardescriplion. b€cau\ebanlsofferle\seffe{ti\ere\istance. a. Regularsectionwith no boulders or brush b. Irregularand rouShsection

\linimum

Normal

Maximum

0.0i0

0.035

0.050

0.010 0.015 0.0,r0

0.030 0.03-s 0.040

0.0,10 0.0,15 0.050

0.0-15 0.0-'15 0.or0 0.o15 0.0t0

0.050 0.050 0.060 0.070 0.Ic)0

0.070 0.060 0.080 0.1l0 0.160

0. 0

0.150

0.200

0.0-r0

0.0.10

0.050

0.050

0.060

0.080

0.080

0.100

0.120

0.100

0.120

0.r60

0.0r5 0.035

0.060 0.100

Source:Chow 1959.Usedwith pe|missionofChow eslare.

methodsby Honon,EinsteinandBanks,andLotterfor Chow( 1959)presented valueof Manning'sn for a singlechannel; thatis,for themain obtaininga composite channelonly of a compoundchannelor a canal*ith laterallyvaryingroughness. The Hortonmethodis basedon the assumption thatthe velocitiesin eachwettedperimeter subsection areequalto one anotheras$'ellasequalto themeanvelocity valueof Manning'sn, denoted The resultingcomposite of thewholecrosssection. n", is givenby f

v

l:l

I > P,'l''l = '='

,.. IL .

P

I)

(4.27)

in which P,, n, : wetted perimeterand Manning's n of any sectioni; P = wetted perimeterof the entire crosssection;and N - total numberof sectionsinto which

CHApTER 4 : U n i f o r mF l o w

ll9

(he wetted perimeteris divided. The Einstein and Banks ntethodassumesthat rhe total resistingforce is equal to the sum of the resistingforcesin eachsubsectionand thc hydraulicradiusof each subsectionis equal to the hydraulicradiusof the whole section.The result is given by

f x lr,2 I fz t p t "' t r l l

I

.

L

P

)

(.4.28)

Lotter's fornrulais basedon writing the total dischargeas the sum of the discharges in the subsections: PR5/] A

p {r5l

'\-

','

(4.29)

I

(1972)derivedanotherformulabasedon Finally,Krishnamunhy andChristensen velocitydistribution, the logarithmic which givesn" as P,tli2 ln n, lnn, :

(4.30) P,Yli'

in which-v,= flow depthin the ith section.Morayedand Krishnamurthy (1980) usedcross-sectional datafrom 36 streamsin Maryland,Georgia. Pennsylvania, and Oregonat U.S.GeologicalSurveygaugingstationsto testthe four formulasjust given. An averagevalue of the slope of the energygradeline obtainedfrom the measured depthandvelocitydistributiona[ a crosssectionwasusedto obtaina "measured"compositevalue of Manning's n to comparewith the formulas.The resultsshowedthat the meanerror betweenthe computedn. and the measuredz. was by far smallestfor the Lotter formula.

4.9 UNIFORM FLOW COMPUTATIONS WhethertheManningor Chezyequationis used,thereexistsa uniquevalueof the uniform flow depthfor a givcn channelgeometry,discharge,roughness, andslope. This depth is called the normal depth, and its magnituderelativeto the critical depth determineswhetheror not uniform flow is supercriticalor subcriticalfor a givensetof channelconditions.If the normal depthis greaterthancritical,thenthe uniformflow is subcritical andtheslopeis classihed asmild.For a steepslopethe normaldepthis lessthancritical depth.The actualclassificationof a givenchannel

120

C H A P T E R4 : U n i f o r mF i o w

slope can change with the dischrrgeas the relativemagnirudesof nornraland critical depth change. The conrputationof normal depth using Manning s equation proceedsby rearrangingthe equationas r 5,-r

uQ KNS,

(1.31)

.

in which the right hand side is conrpletely specified b1 design conditions.The designdischargemay be setby flood frequencyconsiderations;the roughnessoftcn dependson the choice of a stablelining; and the slope is a function of the topography. Equation.1.31can be solvcdby trial or by a nonlinearalgebraicequationsolver for a known geometry.In the caseof a trapezoidalchannel.for example,the equation in nondimensionalform becones

[[ b. (\ ' * rby/))1 " [' * ]1r l

+,,')'i'-l"' o

AR]

1

b8'3

nQ t' (l :A8 l

(.1.12)

l

in whichb - channelbottomwidth;rz : sideslope ratio:and_r0= normaldepth. As presented. theequation canbe usedin SI or Englishunirssimplyby substituting the appropriatevalueof K, andunits for C and b consisrentwith K,,i thar is. 4 : l . 4 9 f o r Q i n c f s a n db i n f t w h i l eK " - l . 0 f o r Q i n c u b i cm e r e rps e rs e c o n ad n db in meters.Equation4.32is shownasa graphical solutionfor normaldepthin Figure 4.9 (Chow1959).A similarsolutioncanbe developed for a circularchannel. andit is includedin the figurewith thediameterasthenondimensionalizing lengthscale. When the flow is in the fully roughturbulentregime.Manning'sequationis appropriate for computation of normaldepth,but for the transitional and smooth turbulentregimes,theChezyequationshouldbe used:

trp''. -

t'' Qf (88s)'/r

(.r.33 )

in which/ = the Darcy-Weisbach frictionfactor It has beenplacedon the right handsideof theequation, althoughit depends on theRel noldsnumberandrelative -1.33 u hich in tum arefunctionsof theunknownnormaldepth.Equation roughness, can be solvedfor normaldepthby assuminga valueof ./ and iterating\iith the Moody diagramor Equation4.18 (the Colebrook-White equation)with the pipe diameterreplacedby 4R andthe constant3.7 replacedb1'3.0, so thatthe first term on the right handsidereflectsthe Keulegan constant as /i,/l2rt (Henderson 1966). The iterationrequiredto solveEquation.l.33may havediscouraged its usein the past,so that Manning'sequationoftenhasbeenusedwitiout consideration of the unknownvariabilityof Manning'sn outsidethefully roughflow regime.An alternativeformulationof the Chezyequationfor the smoothturbulentcaseis consideredin the next section.

C H A P T E R4 : U n i f o r m F l o w

t2r

0.1 ARz3lbw3

or

AR2J3/das

FIGURE4.9 (Chow normaldepthin circular, rectangular, andtrapezoidal channels Curvesfor calculating 1959\.(Source:L'sedxith permission of Chorrestate.)

4.to PARTLY FULL FLOW IN SIVIOOTH,CIRCULAR CONDUITS ln thecaseof PVC plasticpipeusedfor gravitysewersanddetentionbasinoutlets, theChezyequationwith theDarcy-Weisbach/rather thanManning'sn is preferred. *ork by NealeandPrice(1964)hasshownthatPVC pipecanbeconExperimental sideredsmooth.Furthermore,their resultsindicatea relatively small effect due to shape.The relation for/ in smoothpipesis givenby I r

= r.o los(R€y'/)

0.8

(4.34)

number;d- pipediameter:and? = kinematic in whichRe : l/d/z is theReynolds viscosity. If we replaced by 4R in the Reynoldsnumber,whereR is the hydraulic from the Chezyequation,then Equation,1.34can be radius,and/ b1 8gAzRS/Q2 variables: recastinto one u ith a moreusefulsetof dimensionless

q+ : qlla,kds)1/?l: Re+ : d(gdS)r r,/yl and r/d. the relative deDth.

122

C H \ P T E R 4 : U n r f o r n rF l o w

Y/o= 0.8

15

liI

7rl

+# )E

10

0.6

llll

--.r--

T]TI

o.4

r.lt

0.3

J-f]+ TtTu 0 1E4

o.7

1E5

0.2

1E6 Re.

F I G U R E4 . T O Discharge capaciry of smooth, circular conduits nowingpanlvfull.

The resultsof plotting(4.34) in termsof thesedimensionless variablesis shownin Figure4. 10.Thisfigurecanbe usedto find the parrlyfull flow depthin a smoothpipe withouttrial anderror.

4.tl GRAVITY SEWER DESIGN The designof stormandsanitaryseweninvolvesthe derermjnation of panlv full flow capacityfor a givendesigndepthor normaldepthfor a givendischarge in cir_ cularconduits.The designis basedon discharges determined eitherby population estrmates and conesponding wastewater ratesper capitaor by hydrologiccalcula_ tionsof peakrunoffratesdueto stormevents. Because pressurized flow is avoided, especially in sanitarysewers, thedesignprobremis to selecta conduitsizethatwill flow panly full for the designdischarge.Even in storm sewers,undesirableflow conditionscan developas full flow is approached. When the relativedepthor filling ratio,,r/d.nearst.0, air accessto the freesurfaceis reducedwith intermittent openingand closingof the section(Hager1999).Sucha condirion,referredto as sluggingh culvert hydraulics.resultsin streamingair pocketsat the crown of the pipeandpulsations thatcoulddamagepipejointsor causeundesirable fluctuations in discharge. The only practicalway of avoidingthesedifficultiesrn sewersrs to designfor panly full flow.

CHApTER 4: Uniformflow

123

A furthercomplication of the circularcrosssectionoccursdueto changesin geometryas the pipe fills. The wettedperimererincreases morerapidlythanthe cross-sectional areanearthe crownof the pipe with the resultthatthe discharge capacitydecreases asthecrownof thepipeis approached. Thiscanbe seenin Figure 4.9, as the curve for normaldepthreachesa maximumin ARrl and then decreases 1.0.ln effect,therearet*o possible as1/dapproaches normaldepthsnear the crownof the pipe,andthe upperone is unlikelyto occurwithoutsluggingor filling thepipe. It is soundpracticeto avoidthesedifficultiesby designing thepipefor a filling ratioof about0.8 or lessat maximumdesignflow. Olderdesigncriteriamay have specified1y'd= 0.5 as thedesignfilling ratio.but thisdoesnot makeefficientuse Theinitialpanof thedesignis to calculate of thepipecapacity. a pipediameterthar u'ill carrythe maximumdcsigndischarge at. say,.r'/d- 0.8.This corresponds to a : 0.305from Manning'sequation, valueof rrQlK,Sr/2d8lr ascanbe verifiedfrom Figure4.9.The initialdiameterthenis calculatedfrom

I ---irro lr 8 d = 1.561 | L/("S'.1

(4.1s)

assuming fully roughturbulent flow,whichcanbe checked asdescribed previously. If Manning'sequationis not applicable,thenthe Chezyequationwith theColebrookWhiteexpression for thefrictionfactorcan be used.The initialdiameterusuallyis roundedup to thenextcommercial pipesize,andtheactualflow depthis computed for thecommercial diameter. The uniformflo$' equation canbe solvedby trial and error,with a computerprogram,or graphicall)'usingFigure4.9 or Figure4.10,as appropriate,to find the normaldepth. The secondpart of the designis to check for the occurrenceof self-cleansing velocitiesto preventthe build-upof depositsin the sewer.It is desirable to havea minimumvelocityof at least0.61m/s(2.0fVs) to scoursandandgrit from thepipe at maximumdischarge, althougha valueof 0.91 m/s(3.0ftls) is prefened(ASCE 1982).Velocities aslow as0.30m/s( 1.0fVs)at low flowsaresufficient only to preventdeposition of thelightersewage solids.accordingto theASCEmanual.Hager (1999)recommends a minimumvelocityof 0.60 to 0.70n/s (2.0to 2.3 fVs).Once the normaldepthhas beendeterminedfor the selectedcommercialpipe diameter, the actualvelocity follows from Qd".,""/A,u hereA is the cross-sectional areacorresponding to the normaldepth:andQa."g"is the designdischarge. An altemative approach velocities to self-cleansing is thenotionof equalselfcleansing,so that nearly the sameaverageboundaryshearstressoccursat both maximumandminimumflows.This may not alwaysbe possiblewithoutincreasing theslopeof thepipe(ASCE1982).Hager( 1999)suggests a criticalshearstress r. of about2.0 Pa (0.O42lbs/ft2) for self-cleansing in separate sewersystems. The corresponding criticalvelocityand its variationwith filling ratioare obtainedby settingtheshearstressr0 : rc so thatthe slopeS = z./yR in Manning'sequation. Solvrngfor thecriticalvelocity,4, the resultin dimensionless form is V-nYp lRl'" " - r ^ u . . , ' ola)

(4.36)

124

C H A P T E R4 : U n i f o r m F l o w

1.0 0.8

0.6 \

o.4

..-

A./AI

o.2 0.0 0.2

0.4

0.6

0.8

NAt and Vc' FIGURE4.II criticalvelocityfor self-cleansing of circularsewers. Dimensionless

in which a.. : critical valueof shearvelocity = (r,/p)t12.'fhedimensionless velocityis a uniquefunctionof the filling ratio,1/d,as shorvnin Figure cleansing 4.1I . However, its valuedoesnotchangesignificantly fromabout0.8 for -r'/d> 0.4. althoughclearly,from Equation4.36. the criticalvelocityitself dependson pipe Also shownin Figure4.ll as a designaid to assistin diameterand roughness. theactualflow velocityis a plot of A/Ar,in whichA = panlyfull flow determining arcaandA, = full pipeflow area= rd2l1. For Vl = 9.3,, : 0.015,andr. : 2.0 Pa,the critical velocity increasesfrom 0.68 rnls (2.2 ftls) to 0.86 rnls (2.8 ftls) as from 0.5 m ( 1.6 ft) to 2.0 m (6.6ft). thediameterincreases EXAMPLE a.l. Find the dischargecapacityof a 24 i,n.(61 cm) diameterPVC storm sewerflowing at 80 percentrelative depth if the slope of the se$er is 0.003. Assumethalit is smooth. Soratiorl, First find the geometricp.opertiesof the sewerat )y'd= 0.8.The angle0 is c o s ' ( t - r l ) : : c o . - ' 1 -t 2 x 0 . 8 ) = 4 . 4 2 8 r6a d \ dt

d:2

Then the area and wetted perimeter can be determinedfrom the formulasgiven in Table2- l: )1

)1

A = ( d - s i n o ) : - 1 4 . 4 2 8 -6 \ i n ( 4 . 4 2 8 6 - )El : = 2 . 6 9f r r ( 0 . 2 5m r ) tt ,l

',

P = 0 : = 4 . 1 2 8 6x : = 4 . 4 3 f t ( l . 3 5 m )

I

I i

C H A P T E R, 1 : U n i f o r m F l o w

125

so thar R = A/P = 2.6911.13- 0.607 fr (0.185 m). The fricrion facror comes from llquation 4.3.1lbr \mooth surfaces\r'ith JR as the length scalein th€ Reynoldsnumber, Re. This requires trial and error \\ith Chezl's equation beginnning *ith an assumed value of/. Forexanrple. assurne/= 0.015. then solve for Q and Re:

|_ \/;AVRS

/;6 .

= ./ \

r _ . :

^ _ -- .' :.6qJ . V0.007 ,1 0.001

U . UI )

: 15.I cfs (0..128 mr/s) Re:

(O - A )JR

( r 5 . 1 / 2 . 6 9x1J)X . 6 0 7

: Ll3 x 106

For this valueof the Reynoldsnumber.Equation 4.3.1gives/: 6.9114 6t ,rial,which is usedin thenextiterarion. In thenextiterarion, mr/s).Re : L30 O: 17.3cfs (0.,190 x l 0 6 , a n d / = 0 . 0 1I l . I n t h e f i n a il l e r a l i o n . O =1 7 . 5c f s ( 0 . , 1 9m 6 ] / s ) ,R e = 1 . 3 2X 106,and/: 0.01I l. $,hichis the sarneas rhepreviousvalue.Checkwirh Figure4.10 b y c o m p u t i n g R e:* 2 x ( 3 2 . 2x 2 x o . o o - l l r v l . 2x l 0 5 = 7 . 3 x l 0 { . F r o m F i g ure.l.l0, readQ* = 10.0andtherefore Q = 17.6cfs (0.499mr/s),which is acceprable graphicalenor. considering the Thefinalanswer is Q : I 7.5cfs (0..196 mr/s).Notethat valueof Manning'sa from Equation4.20 is 0.0090,but this will vary the equivalent with theReynoldsnumber ExAMPLE 4.2. Find theconcrete seuer(n : 0.015)diameterrequiredto carrya maximumdesigndischarge of 10.0cfs (0.28-3 mr/s)on a slopeof 0.003.The minimum expected discharge is 2.5 cfs (0.071mr/s).Checkrhevelocityfor self-cleansing. Solzfibz. First, estimatethe diameterfrom Equation4.35:

I ao l'* d = r . 5 o. l _ ' , l - r . 5 6 . 1 0 . 0\ 1r50 . ( r . 4. 0q . 0 0 ri r/ l ' ^ LK_S''l = 1.96 ft (0.597 m) Roundthediameterup to thenextcommercial pipesizeof 2.0ft (0.61m) andsolvefor the normaldepthof flow. First,computerhe right handsideof Equarion4.3I ; nQ Ir 1.495

0 . 0 1 5x l 0 : 1 . 8 3 8 r/r 1.49x 0.003

Then setup a table as follows with assumedvaluesof )/d from which 0, A, and R can be computedusingTable2- l. Iterateon ,r'// unril ,,lRr/r: | .838. tld

0.6 0.8 0.76 0.'762.

0, rad

A, ftr

P,f.

R, ft

ARut

3.544 1.129 4.235 4.215

1.968 2.69.1 2.562 2.569

3.5.14 ,1..129 1.235 1.215

0.555 0.608 0.605 0.605

1.319 r.933 r.833 L838

This last iterationis consideredacceptable: therefore.yo : 0.762 x 2 = 1.52fI (0.463 m) and,V : 1,0/2.569: 3.89ftis (1.18rn/s).This is consideredmore than

I

126

C H A P T E R- 1 : U n i f o r mF I o w

adequat€for self-cleansing at a maximumdischarge. At the rc:nimumdischarge of 2.5 cfs (0.071mr/s;.calculatethe normaldeprhusingFigure-!.9: ARi t d3.r

nQ L , l 9 SI , r d 8 , r

. 0 1 5x 2 . 5 1..19x 0.003r'rx 23r

: rr 07:

from which -volais approximately 0.33 and-r,o= 6.66 1t.Now. from Figure4. I l, A/Ar = 0 . 2 9 a n d A : O . 2 9 x . , I x 2 1 l ,=1 0 . 9 1f t r . T h e n V : Q l A = 2 . 5 t t ) . 9:12 . 8 f t i s .O b r a i n I}lecriticalrelocityfrom Figure4.1I in which Vl = 0.75and.from F4uarion 4.36,

r.+9x\6.o4r8/L9+xz'u = 1.2ftls (0.67m/s) v,: o'7s{:!::! ) = ().t5x n v g

x \,4t 0.015

in which r. : 0.0.118lbs/ftr(2.0 Pa).The actualvelocityis rrell abovethe critical value.so this is a satisfactorv desien.

4.12 COMPOUND CHANNELS A compoundchannelconsistsof a main channel,which carriesbaseflow and frequentlyoccurringrunoff up to bank-fullconditions,anda floodplainon oneor both sidesthat carriesoverbankflow during timesof flooding.The \{anning's equation is written for compoundchannelsin termsof the total conve)ance,,(, definedby andS is theslopeof th: energygradeline, Q/SI/r,in which O is thetotaldischarge which is equal to the bed slopein uniform flow. Becauseof the significantdifferencein geometryand roughnessof the floodplainscomparedto the main channel, the compoundchannelusuallyis dividedinto subsections th3r includethe main channeland the left andright floodplains,althoughthe floodplainsmay haveadditional subsectionsfor varyingroughnessacrossthe floodplain-If it is assumedthat the energy grade line is horizontalacrossthe cross-sectionfor one-dimensional flow, then the slopeof the energygradeline mustbe the samefor eachsubsection of the compoundchannelas well as for the whole crosssection.From continuity, Q : , Qt, so it follows from equalityof the slopesthat K : tt.. in which Q, and,t, representthe dischargeand conveyancein the ith subsecrion.respectively. Therefore, the total conveyancefor a crosssectionis computedas the sum of the conveyancesof the subsections. For Manning'sequation,for example,tie subsection conveyanceis k, = (K1n,)A;R,23,so that conveyancerepres€ntsboth geometric effectsand roughlesseffectson the total conveyanceand total discharge.As discussedby Cunge,Holly, andVerwey( 1980),it is misleadingto !-alculate,for a compoundchannel,a seriesof composite valuesof Manning'sn from Manning'sequation for increasingvaluesof depth and discharge.The result is likely to be a compositen value that variesin an unexpectedmanneras deprhincreases, because this approachlumps both roughnessandgeometriceffectsinto Manning'sn. What is soughtinsteadis a smoothfunction of increasingconveyancewith increasing

Cri\prEa .lt UnilbrmFlow

12.1

dcpth and dischargeobtainedby defining the total conr eyanceas the sum of con_ vcyances in individual subsections.This is referred to as the divided-channel netlod. Some difficulty ariscs in the divided-channel method when the hvdraulic radius and \r,ettedperimeterare defined for the floodplain and main chrnnel subsections.The customarydivision into subsections,as shown in Figure 4.12, uti_ lizes a vertical line between the subsectionsalong rvhich the wetted perimeter often is neglected.This is tantamountto assuming no shear stressbetween the main channel and floodplain flows. In fact, significant momentum exchange occurs between the faster moving main channel flow and the floodplain flow, so that the total dischargeis less than what would be expectedby adding the dischargesof the main channel and floodplains as though they acted independently ( Z h e l e z n y a k o vl 9 7 l ) . M y e r s ( 1 9 7 8 )a n d K n i g h t a n d D e m e t r i o u( 1 9 8 3 )m e a s u r e d the apparent shearforce on the vertical interface bet$een the main channel and floodplain and found it to be significant.Furthermore.the mean velocity for the whole cross section actually decreaseswith increasing depth for overbank flow until it reachesa minimum and then begins increasing again as demonstratedby field measurementson the Sangamon River and Salt Creek in Illinois by Bhowmik and Demissie(1982). The minimum in the mean velocity for the total cross section occurredat an averagefloodplain depth that was 35 percentof the averagemain channeldepth. Severalattemptshave been made at quantifying the momentumtransferat tle main channel-floodplaininterfaceusing conceptsof imaginary interfacesincluded or excluded as wet(edperimeterand dehned at varying locations.with or without the consideration of an apparent shea_rstress acting on the interface. Wright and Carstens( 1970)proposedthat rhe interfacebe included in the wertedperimeterof the main channel and a shear force equal to the mean boundary shear stress in rhe main channelbe appliedto the floodplain inrerface.yen and Overron0 973), on rhe other hand, suggesredthe idea of choosingan interfaceon which shearstressis in fact nearly zero, This led to several methods of choosing an interface, including a diagonal interface from the top of the main channel bank to the channel centerline at the free surface and a horizontal interface from bank to bank of the main cbannel, as shown in Figure 4.12. Wormleatonand Hadjipanos (1985) compared the

Centerline I

FIGURE4.I2 Compound channel withdifferent (H : horizonr.al; subdivisions V = venical;D : diasonal).

l:8

n low CraPrEt 4: UnifonF

in predictingthe sePaandhorizontalinterfaccs of the vertical,diaSonal. accuracy flunleof expcrimental in an measured ,u," tuin channelandfloodplaindischarges main channel width to of floodplain ratio . iOtt t .Zt m (3.97ft) andhavinga fixed or included fully was eiiher intcrface of the half-widthof 3.2.The wettedperimeter results The channel the main of perimeter excludedin the calculationof wctted choiceof interfacemightprovidea satisfac,no*.d th"t, eventhougha particular nearlyall thechoicestendedto overpredict of totalchinneldischarge' tory estimate It the floodplaindischarge' and th. ,"p".rt. mainchanneldischarge underpredict of the kinetic calculation in the were magnified uas furthershownthattheseerrors coefficient. energyflux correction been Severalempirical methodsfor rleterminingdischargedistribulionhave at facility channel flood in the collected data basedon experimental developed, Merand Wormleaton by described as England. Wittingford, Research, Hv,lraulics a totalflow reit tt990t.Tnecnannelis 56 m (184ft) long by l0 m (33 ft) widewith width to floodplain of the ratio the experiments, (19 In cfs). capaciryof Ll ml/s (floodplain depth relative the and to 5 5, I raried from -iin "|ronn"thalf-width developed depth/mainchanneldepth)variedfrom 0 05 to 0 50 Two ofthe methods fromthisdatainc|udeaconectiontotheseparatemainchannelandfloodplaindisandM-enen(1990)applied .t'ri!"t .otput"O Uy Manning'sequation \f,/ormleaton andfloodplaindischarges main channel the to O index the a coiection iacto, called (vertical, diagonal'or horizontal)' interface of choice by a particular calculated wetted from excluded or PerimeterThe @ index was which was eitlei included componentof fluid streamwise to the force shear boundary definedas the ratio of main channeland calculated The force' shear of apparent weight as a measure O index for each of the root square by the multiplied when flooiplain discharges, measureddisto when compared improvement considerable subsection,showe-)o:

s.(so

(5.6)

l6:

C H \ P r ! R 5 : G r a d u a l lVya r i c dF l o w

I n a d d i t i o n .i t i s a p p a r e ntth a t t h e v a l u eo f t h c F r o u d e n u m b e rs q u a r e dr c l a t i v et o unity is determined by the magnitude of thc local depth relative to the critical d c p t hr ' , :

-r'( -r',: )>),;

F: > I Fr So, and F < I so that d_)/dr < 0. As I where,r'. a mild slope, 2 on in the upstream direction, dr/dr approacheszero, so we have an approaches,ln asymptotic approach to normal depth from below. In tie downstream direction, the M2 profile approaches critical depth where F : l, but the mannerin which it does is not immediately obvious. However,if we consider a mild slopefollowed by a so steepslope.S" > S0upstreamof the slope break,where critical depthoccurs,while dou nstream of the slope break, S" ( Sobecause_r > r'o on the steep slope. It can be reasonedthen that S0 = S. at the slope break and boti the numeratorand denominator of (5.,1) approach zero, so that d]y'dr is finite as the water surface passes through critical depth.In region 3 on a mild slope,u here _l ( _r.( _yo, S. > Soand F >1, so that d,I/dr > 0. As _yapproaches,r. in the downstreamdirection, F approaches I, and d_ry'dr approachesinfinity, although a hydraulic jump would occur before that happens.In the upstream direction, both the numerator and denominator of (5.4) approachinfinity as the depti approacheszero, and dy/dr approachessome positive finite limit that is of no practical interest,since there would bc no flow for no depth. It is of interestto note that both Ml and M2 profiles, which are subcritical, approachnormal depth rn the upsteam direction,as controlledby the value of the dou'nstreamdepth.The other profilesin Figure 5.2 can be deducedin the sameway as for the mild slope.In contrastto the Ml and M2 profiles, the two supercritical

z o o, -9

o c .!

a

J

o

(-)

I

o o) a

) o I

o

o

6

6 6 ?_

G a l :

ui t & ? p :

163

16.{

C H A P i F R 5 : G r a d u : l l r V a r i c dF l o w

profiles,S2 and S3. approachnornraldcpth in the r/orlrrslrrrrnrdircction. as detcrmincd by the value of the upstreamdepth. Composite florv profiles for a variety of flow situationscan be sketched as s h o w n i n F i g u r e- 5 . 3 I. n F i g u r e 5 . 3 a .a m i l d s l o p ei s f o l l o w e db y a m i l d e r s l o p e . If the downstrcam slope is rcry long. with uniform flow establishedas the control, then the depth rnust remain at rormal deplh all thc way to the upstrcanl

M2 -

-r-

-s2

CDL

f"lilder (very long) (b)

(a) Reservoir

(c)

(d)

Reservoir

Reservoir

Mitd (very bngl

t

NDL CDL

Free overfall

(e)

Mitd (short)

(f)

,j n? (s) FIGURE5.3 flow profileswith variouscontJols. Composite

(h)

-I

C H A p T E R5 : C r a d u a l lV y a r i e dF l o w

165

slope. This is becauscthc nlild slope profiles cannot approach normal depth in t h e d o w n s t r e a md i r e c t i o nb u t o n l y d i r e r g ef r o m i t ( i . e . ,M I a n d M 2 ) . A s a r c s u l t , the upstream ir{l protrle does not bcgin until thc upstream slope is rcached.FolIorving the same reasoning.the stecp slope followed by a steeperslope in Figure 5.3b must have an S2 or S3 protile on the upstream slope that reachesnormal d e p t h a n d r e m a i n s t h e r e ,i l t h e s l o p e i s v e r y l o n g , u n t i l t h e b r e a k i n s l o p e i s reached. The occurrenceofcritical depthis a very imponant control, sho*n at the break betweena mild and steepslope in Figure 5.3c. Basedon th€ precedingreasoning. the water surface must approachsomc finite slope as it passesthrough critical dcpth. Critical depth also occursat the entrancefrom a reservoirinto a steepslope and at a free overfall, rvherethereis a similar releaseor accelerationof the flow, as s h o w ni n F i g u r e s5 . 3 da n d 5 . 1 e . The entrancefrom a reservoirinto a mild slopc is shown in Figures5.3e and 5.3f. For the long mild channelin Figure 5.3e, the control is normal depth at the entrance,if the channelis very long (hydraulically).but s*itches to the tailwater depth if the channelis short as in Figure 5.3f. Flow profileson a mild or a steepslopewith a sluicegate installedmidway along the channel are shown in Figurcs 5.3g and 5.3h, respecti\ely.In Figure 5.3g, the sluice gate forces an M I profile to occur upstreamand an M3 profile to emerge from undcr the gate downstream.The M3 prolile has an increasing depth until tbe momentum equation is satisfiedfor the sequentdepth occuffing in the downstreamM2 profile. The result is a hydraulicjump (HJ). A similar situationis shownin Figure5.3h, except that the slope is steepand there is an 53 profile upstream of the jump and an S l profi le dor,'nstrean of the jump controlled by the position of the tailwater.

5.4 LAKE DISCHARGEPROBLEM The flow situationsillustratedin Figures5.3d, 5.3e, and 5.3f lead to an irnponant problem if the dischargeis unknown.becauseit is unclear whetherthe given slope in fact is mild or steep.lf the headH at the channelentranceis given,we can write the energy equationfor the steepslope in Figure 5.3d betweenthe upslreamlake \!ater surface and the channel enlrance where the depth is critical (neglecting losses)io cive O1 ''

).a:

(5.9)

For depth equal to the critical deprh.the Froude number must have a value of I so that ctO1B.. l

(5.l0)

-

166

CHAPTER 5: GraduallyVaried Flow

On theotherhand,if theslopcis mild andthechannelis verylongasin Figure5.3e, the entrance depthis normaldepthandtherelevantequations for solvingfor Q andthe entrancedeDthare H::"0

+ 2eAa 4

(5.1r)

o =f e o n i ' s ; '

( 5 .l 2 )

in which 1o is the normal depth.Which of the tu.o conditionsprevailscan be determined by assumingthat the slopeis steepand solving Equations5.9 and 5.10 for the critical depth and critical discharge,,r'. and Q.. These values of _v.and Q. then are substitutedinto Manning's equationto calculatethe critical slope.lf the bed slope So > S., then the slope indeed is steep and rhe discharge is Q.. On the other hand, if S0 < S., thenthe slopeis mild and Equations5. I I and 5.l2 musrbe solvedto determine the actual Q, which will be less than O.. In case the slope is not very long, the normal depth,1,o in Equations5.11 and 5.12, must be replacedby an entrancedepth, * which can be determined only from r.r'atersurface profile computation. ln _v. 1,0, that case,Eguation5. l2 is replacedby the equationof graduallyvariedflow, which must be solyednumericallyas shownin the following section. E X A M PL E 5 . l A very long rectangularchannelconnectstwo reseryoirsandhasa slop€of 0.005.The channelhasa widthof l0 m (32.8ft) anda Manning'sn of 0.030. If the upstreamreservoirsurfaceis 3.50 m ( I 1.5 ft) abovethe channelinlet inven and the downstream reservoiris 2.50 m (8.20 ft) abovethe outlet inven.determinethe dischargein thechannel. Sotuttba. Initiallyassume thattheslop€is sreep.In thiscase,Equarions 5.9and5.10 are panicularlysimplefor a rectangularchannel.They become ) ) ': r , = : H = ; x 1 1 . s . 1 2 . 3 3m ( ? . 6 6f t ) q = Vs.v: : Vs.st x 2.33r:

I l.t mr/s (r20 fr?/s)

in which Il : upstreamheadof the resenoir surfacerelativeto the channelinven and q : dischargeper unit of channelwidrh.The crirical slopecanbe computedfrom nlo ,

j

K:A:R"3

0 . 0 3 'x ( t 0 x l l . l 4 ) ,

( 1 0x

L o .r ( r or u . r r r ' ^( r 0 + 2 2x .23.33) 3 ) l ]"

= 0.01I

Now,sinceSo< S,,theslopemusrbe mild. In rhatcase,Equations 5.1I and5.12must be solvedsimultaneously: o2

'"

J . ) = ! n i - - - - = v ^ +

2eA(,

Q. Q, : to * 19.62x(l0x-ro)' rs6D( *

K' p = 1;q;r5 'r' = rc

(10 x -vo)51r );/' x ( 0 . 0 o 5 ) r:' 1 0 9 . 4 (10+2.y0):1 (10+2xyo)' r

CuaprEr

5: Cradr-ralllVaried F'loq

)61

By trial and eror. assutuea ralue of r,, ( o u

500

q

I Hydraulic jump

550

600

650

700

Distance,m (b) Locationof HydrauticJump FIGURE 5.6 Water surfaceprofiles and momentumfunction for the location of a hydraulicjump in Example5.1. So/zrion. We assumeat first thal the mild slopelengthof50O m (1640ft) qualifiesit to be hydraulicallylong, so the dischargeis controlledby normal deprhon the mild slope and it is l0l mr/s (3565cfs), asdeterminedin Example5.1. Thii meansthat the cntical slopeof 0.011 in Example5.1 still is valid, andtherifore the downstreamslope

IU0

C l r , \ P T E R5 : G r a d u a l l rV r r i c d F I o u

---_NDL

Tailwater

Reservoir

_ $

;HJ - . H J--

r-

cDL

NDL

T.ICURE 5.7 watersudaceprofilesfor increasing Possible tailwater\\'jthnormaldepthcontrolon a mild slope.

of 0.02is steep.The criticaldepthof 2.18m (7.15ft) is the samefor the sreepslope. but its nomraldepthneedsto be calculated from \'lannings equation: . l o l . ,] "

0.010

tol _ -) r r )-

: llo - 2'o_:' Lo oot

from whichr.o= 1.78m (5.8:1 ft). The downstream lile levelis abovebothnormaland criticaldepthon the steepslope,which meansan Sl profile.as sho*n in Figure5.6a. At the upstream end of the slope,criticaldepthrrill occurat the bfeakin slope.One possibilityfor thecomposite watersurfaceprofileis ln M2 on the mild slopelblloued by an S2 on the steepslopeanda hydraulicjump to theSI profile.Otherpossibilities lakele\elrises.At somele\el.rhehvdraulic aresho\rnin Figure5.7 asthedownstream jump andthecriticaldepthwill be drouncdout,andtheSl profileu ill or,'cur alongthe entiresteepslopeandjoin the M I profileon the mild slope.Whichof thesepossibilitiesactuallywill occurcanbe detennined only b1'a watersurfaceprolileconrputation. The computerprograrnWSP in AppendixB. lrhich usesthe directstepmethod. wasappliedto thisproblemu,itha downstream lalielevelof 5.0 m ( 16..1 ft). asthetailwatercondition.First.the M2 profile wascomputedupstreamfrom c.itical depthal the break in slope,then the 52 proflle was computeddownstreamfrom the same point. Finally, the Sl profile u'as computedupstreamfrom the do*nstream lake level. The resultsareshownin Figure5.6a.The locationof the hydraulicjump is determined in Figure5.6bfrom the intersection of the momentumfunctioncurvescomputedat each stepof the watersurfaceprofilecomputalion. The lengthof the.jumpis neglected so the locationis at the uniquepoint whereboth the momentumequationand the equation of graduallyvariedflo* for the 52 and SI profilesaresatisfiedsimultaneously. As a checkon whetherthe mild slopeis hydraulically long,99.9percentof normal depthis reachedat r : 65 m (213ft) downstream of thechannelentrance, so the slopein fact is long enoughthat the controlremainsat the entranceto the ntild slope. The 52 profile reachesnormaldepth$ithin 0. I p€rcentar .! = 595 m ( 1950fr), which is upstream of thechannelexit,so it canbe considered hydraulically longas uell. The hydraulicjump alsooccursat -r = 595 m (1950fi).

CHAPTER 5: Gradualll Varied Flow

l8l

5.8 NATURAL CHANNELS The nrethodof depthdeterminedfrom distanccis used in naruralchannelsby solving the equation of graduallyvaried flow in the form of rhe energyequationwritten from one stationto thc next:

w S+. a r l :

W S+r , , *

- O,

(5.27\

in which the terms are definedin Figure 5.8. This, in effecr. is the integratedform of Equation 5.3, exceptthat minor lossesare addedto the boundarylossesin ft,:

n"=s"t+",1 +-+ tB

(5.28 )

l'8

in which S" = meanslopeof the energygrade line; L = reach length;K. = minor head loss coefficienti and a is evaluatedby Equation 2.1 l- The form of Equation 5.27 is written for cross-section numbersincreasingin the upstreamdirection.The solution is obtainedby iteratingon the differencebetweenthe assumedand calculated water surfaceelevations,using a mcthod such as inten al halving or the secant

I d2V2212g

EGL dy12t29

Bed

Datum FIGURE 5.E Dellnitionsketchfor theslandard srepmelhod(U.S.Army Corpsof Engineers 1998).

-I

I8:

C H A P T E R5 : G r a d u a l l yV a r i e dF l o w

method. The programs HEC-2 and HEC-RAS (U.S. Arnry Corps of Engineers 1998) use the secanrntcthod for solution. When apptic,dto naturalchannels,this orcralf solution procedureis referredto as the slarrdrrrristep nethod and also is used by WSPRO (ShearrnanI990). Rhodes (1995) applied the Newton-Raphson techniqueto the ircrilrionrequiredin the srandardstep nerhod and illustratedthe nethod lbr the panicularcasesof prismaticrecrangularand trapezoidalchannels. The defaultvalueof the nrinor herd Ios\ co€fflcienr,/K,,in a5.2g)is takento be 0.0 for contractionsand 0.-5fbr cxpanrionsby \\ SPRO I Shiumrn 1990).In HEC-2 or HEC RAS. the recommendcdvaluesof K, are 0.I and 0.3 for sradual contrac, tions and cxpansions.respectively.and 0.6 and 0.8 for lbrupt c"ontractjons and ('\prniiOn\. The computationof thc mean slope of the energy grade line can oe accom_ plishcd by severaloptional equarions.[n general,S, = (O/Kjr. in which K is the conveyanccfor any particularcrosssection.To obtain thc mean valueof S, for two cross sections,the following optionsare avaiiable: l. Averageconveyance

f K ' . +K , l ' t

1

(5.29)

l

2. AvcrageEGL slope

s" -

S"' *- 5., ',

(5..30)

J. Gcometricmeanslope

s"

_Q' K'K,

( 53 1 )

-1. Harmonicmeanslope )( ( J't r

J'l

(5.-12)

NlethodI is usedas a defaultby HEC-2and HEC-RAS,while rnerhod3 is the delaultusedby WSPRO.Method2 hasbecnlbundto be mostaccurate for M I pro_ files,whilemethod4 is besrfor M2 profiles(U.S.ArrnyCorpsof Engineers 199g1. The conrputation of watersurfaceprolilesin naluralchannels mustproceedin the upstream directjonfor subcritical profilesand in thedownstream directionfor supercritical profilesbecause thecontrolis locateddownstream for subcritical and upstream for supercritical profiles.Whethera profileon a givenslopeis subcritical or supercritical depends on whetherthedepthis greateror lessrhancriticaldepth, *'hich is detennined by thedischarge andthe boundarycondition.

C H \ P T E R5 : C r a d u a l lVy a r i e dF l o w

t83

In a natLtralchann!-ldivided in ro subrcaches. the norntaldcpth changesfor each subrcachas the slope,roughnes, L, the charnel length, the control is at the downstream end of thechannelwith subcritical flow in the entirechannel. Otherwise, the flow is subcritical upstream ofx" andsupercritical downstream, asshownin Figure5.16.

FIGURE 5.16 Spatrallyvariedflow u ith lateralrnflou.

194

CHAPTER 5: GraduallyVaried Flow

F__

4______l

F I G U R E5 . I 7 Spatiallyvariedflow wirh lateralourflou from a sidedischargeweir

In thecaseof lateraloutflou suchas in thesidedischarge weir shownin Figure 5.17,the directionof the Iateralmomentumflux is unknown.Furthermore. because the weir is a localdisturbance, energylossesalongthe weir arerelatively small.For thesereasons,the energyapproachis usedmoreoften thanthe momentum equation.Thercfore,if we assumethatdtldr = 0, on differentiationof the sDe_ cific energy,E, with respectto .jr,w.ehave

dy o(-.),'(#) gbt)'' - Q'

d-r

(5..13 )

for a rectangularchannelof width D. Equation5.43 can be placedin the form QtV

dl_

gA

dr

t-F'

and it only remainsto specify 4. : sharp-crested weir as

-dQltr

(5.#) from the dischargeequationfor a

.ln

Qt: f:c'Vzs()-P)"

f5 l5l

in which C, = weir dischargecoefficient, : (2/3)Co from Chaprer2. Because*,e assumethe energygradeline to be horizontal,the energyequationgiyes the dischargeat any sectionas

O: by\6s@- ),)

(5.16)

C r r A p r E R 5 : G r a d u a l l rV a r i e dF l o w

195

in whichb = width of thechannelandE = knownconstantspecificenergy.Sub) n d( 5 . . 1 6i )n t o( 5 . - 1 4a)n di n t c g r a t i ntgh,er e s u l a t so b t a i n ebdy D e s t i t u t i n (g5 . 4 5 a (Benefleld, 198'1) is and Parr Judkins. Marchi '{C r

)F

b

E

lP

F ./ -

P V y

t

P

t\rn

.l

t -

V E

\'

P

'

con\lant

().4/)

in which C, = weir dischargecoefllcient; [ : specific energ) of the flow; p = hcight of weir crest above channelbottom; and b : channelu'idth. The subcritical case is shown in Figure 5.17. but it also is possibleto have a supercriticalprofile either alone or tvith a hydraulicjunp (seethe Exercises). Hager( 1987)shou ed that the outllow equationuscd by de \tarchi is exactonly for srrall Froude nunbers. He developeda generalizedoutflow equation for side dischargeweir flow that includesthe efltcts of lateraloutflo*'angle and longitudinal channelridth contraction.Hager (1999) gires gencralsolutionso[the iree surface profile for the enhanr'r'doutflou cqurtion.

REFERENCES Mttnning'sRoughness Coefi' Guidefor Selecting G. J..Jr.,andV. R. Schneider. Arcement, cientsfor Natural Channelsand l-lood P/clas.Repon No. FH\\A-TS-84-204 Federal NationalTechnicallnformation of Transportation, High$ayAdmin.,U.S.Depu'tment VA: 198'1. Service,Springfield. B. A. Htdraulicsof OpenChannelFlotv.NewYork:NtcGrau-Hill l932. Bakhmeteff. and Algorithms.Neu'York: McGrawBeckett,R.. and J. Hurt. NuntericalCalculations HiI. 1967. Benefreld,L. D., J. F. Judkins.Jr, andA. D. Parr.TreatmentPlont H;-draulicsfor Environ' Inc., 1984. Englewood Cliffs,NJ: Prentice-Hall, me tal Engineers. Methodsfor Engineersuith Persona!Computer Chapra.S. C., and R. P Canale.A'r,rnerical New York:Mccraw-Hill,1988. Applications. Chow,V. T. Open ChannelHtdraalics.New York: McGraw-Hill, 1959. of Davidian,J. "Computationof WaterSurfaceProfilesin Open Channels."In Techniques 3. Applications of Survet, Book of the U.5. Geologica! lnrestiqations WateLResources DC: GovernmentPrinting Ofiice, 198J. Hydraulics.Wa-shington. 'Lateral Outflow Over SideWeirs."J. Hydr Engrg, ASCE I 13,no. 'l ( 1987)' Hager,W. H. ,191-504. pp. Hager,W. H. Waste$aterHrdrculi.r. Berlin HeidelberSrSpringerVerlaS,1999. Henderson.F. M. Open ChannelFlox'.New York: Macmillan, 1966 Floodplain Hydrology and Hydraulics,2nd ed. New Hoggan,D. H. Computer-Assisted York:McGraq-Hill. 1997. Accuracies of Gradually-Va.ried of "Simulation N{aflin,C. S.,andD. C. Wiggen.Discussion J. Hyd. Div., ASCE 101, no HYT (1975)' FIow,"by J. P Jolly and V. Yevjevich. pp. l02l -24. Prasad,R. "Numerical Method of ComputingFlow Profiles." J. H,"d. Div-' ASCE 96' no. HYI (1970).pp. 75 86.

196

C H A P T E R 5 : C r a d u i r l l vV a r i c dF l o \ r '

Rhodes,D. F Newton,Raphson Solutionfor Craduall!\aried Flow..'J.IA.r/r:Res.ll. n o .I ( 1 9 9 5 )p,p . 2 l 3 - i 8 . Sheannan.J. O. Ust,r'slltuual for 19Sl'RO-A Conlurer .\lodeIitr ltder Sudacepro\ile Contputations. Rcpon FH\\A,lP-89027. FederalHish\ray Adminrstr!jon. U.S. Depanmenr of Transponalion. 1990. Sturm,I \\'.. D. M Skolds,andM. E. Blalock.'WaterSurfaceprolllesin Conrpound Chan, nels."Proc.of the ASCEHtd. Dit. Sptcioln^ Conl, [{rdraulicsand Hydrologvin rhe SmaliCornputer Age.LalieBuenaVista.Florida.pp. 569-71, 1985. U.S.Amrl Corpsof Engineers. HEC-RASHydraulicReference \,lanual.I.ersion 2.2.Davis. CA: U.S.Army Corpsof Enginc-crs. HydrologicEngineering Cenrer,199g. Yen. B. C. OpenChannelFlow Equarions Rerisired."J. Engrg. il,lech. Dnr, ASCE 99. n o . E l l 5 ( 1 9 7 3 )p. p . 9 7 9 1 0 0 9 .

EXERCISES 5.1. Prore from rheequation of graduallyvariedflow thar52 and 53 profilesas).mprori_ call),approach nont)aldepthin thedownstream direction. 5.2. A reservoirdischarges intoa longtrapezoidal channellhat hasa bottom\r.idthof 20 fi, sideslopesof 3:l, a Manning'sn of 0.025.anda bedslopeof 0.001.The resenoir watersurfaceis l0 ft .rbovethe invenof thechannelentrance. Determine thechannel discharge. 5,3. A reservoirdischargesinto a long, steepchannelfollowed by a long channel\r,itha mild slope.Sketchandlabelthepossibleflow prolilesas lhe rail$aterrises.Explain hou you coulddetermine if thehydraulicjump occurson the steepor mild slope. 5,4. Computethe watersurfaceprofile of Table5- I in the lext usingthe methodof numerical integrarionwith rhe rrapezoidalrule. Use the samestepsizesas in rhe uble and determinethedislance requiredto reachI dcpthof | .74 m. Discus(rheresults. 55. A rectangular channel6.lmwidewithn = 0.014is laidon a slopeof 0.001andrerminatesin a freeoverfall.Upstream300 m from theoverfallrs a slurcegatethatproduces a depthof0.47 m immediarely do\{,nsrream. Fora dischargeof 17.0mr/s.wirh a spread_ sherl computethe watersurfaceprofilesand the locationof the hydraulicjump using the dircct srepmerhod.\'erify with rheprogramWSp,or with a programtharyou u.rite. 5.6. A very wide rectangularchannelcarriesa dischargeof 10.0 mt/Vm on a slopeof 0.0O1with an z valueof 0.026.The channelendsin a free overfall.Compurerhedis_ tancerequiredfor the depthto reach0.9y0usingthe direct stepmethodand compare the result with that from the Bressefunction. 5.7. Derive Equations5.38and5.39 usingthe Bressefunction. 5.E. For a very widechannelon a steepslope,derivea formulafor rhelengrhof an 52 prohle from criticaldepthto l.0l yousingthe Bressefuncrion.Whar is rhislengrhin metenif the slopeis 0.01,rhedischarge perunirof widtr is 2.0 mr/Vm,andMannjns'sn is 0.025?

CHAPTER 5: Cradually Vaned Flow

t9't

channelofbottomwidth l0 ft with sideslopesof 2:l is laid on a slope 5.9. A trapczoidal watersurface of 0.0O05andhasan n valueof 0.0'15.It drainsa lakevtitha constant ends in a free If the channel of the channel entrance. the inv€rt levelof l0 ft above in the channelfor channellengthsof )ff) and 10,00O thedischarge overfall,calculate ft usingtheWSPprogram. 5.10. A 3 ft by 3 ft boxculvertthatis l0Oft longis laidon a slopcof0.00l andhasa Manendof theculvenis a freeoverfallFora discharge ning'sa of0.013.Thedownstream usingthewSP program,andtheheadupstream entrance depth calculate the of 20 cfs. losscoefftcient of 0 5 for a of theculvertusingtheener8yequationwith an entrance entrance.Comparethe resultwith the headcalculatedfrom an assump square-edgcd depthequalto normaldepth. tion of a hydraulicrllylongculvenu ith an entrance of 5.11. UsingIIECRAS.compulethewaterJurfaceprofilein SomeCreekfor a discharge 10,000cfs. Bcgin with a subcriticalprofile and a downstream*'ater surfaceslopeof 0.0087 as a boundarycondition.Then do a rnixed flow analysiswith an upstream reach lengths, boundarycondition of critical depth. The cross-sectionSeometr,v" roughnessvalues,andsubsectionbrealpointsare showniDthe following table.Anawhereanyhydraulicjumpsmay occu. lyzethe resultsindicating The upstreamcrosssectionfor SomeCreekat River Station6000(ft) is given by X (ft) 0 0 36

99 I l0 I l9 |]3 1,13 150 t54 155 160

r68 r88 l9l 200 205 :10 229 ?5d 266 2'7 6 305 31,1 380 380

Elevation(ft) ,165 ,161 458.8 .158 ,r5?.8 458.3 .158.1 ,155.9

n

0.055

0.065

.r55.8 -l )).)

455.3 455..r :15,1 :152

0.0,10

,r50.3 ,150.2 .1505 .r5t.5 152.1 .15.1.5 :l)).-1

455.6 455.3 456.3 .158 ,157.8 ,158 .161 ,r65

0.065

0.055

l

l

C H A P r E R5 : G r a d u a l l\l' a r i e dF l o w

198

Al subseguent staThe lefrandrightbanksareat X: 150ft and210ft, respectively. with a uniformdecrease in elethecrosssectionshouldbeadjusted tionsdownstream. vationfrom lheprevioussectionas follows: Rirer station (ft)

Decreasein elevation(ft)

2.0 6.0 2.8 6.0

{qlo 3000 1500 1000

5.12. Computethe watersurfaceprofile in the Red Fox River for Q : 1000cfs, for which andfor O : 10,00O cfs *ith watersurfaceelevationWS: 5703.80. thedownstream = areshownhere,and the elecross sections lys 5?15.05.The stationsfor the four vations(a andn valuesaregivenin thetbllowingtable(Hoggan1997) Crosss€ction

Station (ft)

0 500 900 t300

I l l I Cmss s€cIionI

x (n) z (ft)

x (fo z (ft) ?0 30

ls l+ 0.100

60 n0 , r1 5 610 650 655 660 670 675 690 69't 700 7lo 7IO 9,10 1020 |215 1235 I5?5 1590 1615 1630 1635

Cmss section2

:0 18 @l 11 16 0.050 @ l{ ll ll : 0.030 l 0 0.1 0.8 r (9 I] r-1.5 0.050 @ 1.1 t,l ll rl 0.10 l{ t6 20 15

@ = Subsertionbreakpotnt.

l0 .10 50 ll0 100 295 415 ,155 505 575 585 596 6t 5 615 6 . 1 9.10 I 180 I 195 1205 t225 1245 1250

25 21 :2 0.10 :0 t0 lrt 17 C"r 16 l3 0.05 9.5 @ 5 .1.2 .1.5 0.01 16 0l 8 @ 18.5 l8 t8 20 0.10 22 21 25

Crosssection3

x&) z (rl .10 90 260 310 -170 '120 ,160 500 530 550 560 580

25 0 .t 0 2{ @ 22 0.05 20 18.7 @ 15 I L2 T.l 1.5 0.03 ll 17.8 19

6m 20 850 22 865 875

2{ 25

Cross seclion 4

-x(fo z (ft)

@

0.05

30 15 r30

26 2.5 21

0.10 @

0.05 3-r0 360 310 .r00 . tl 0 ,160 610 650 675 700

23 1,1 9.5 9.8 0.036 13 22 @) 0.05 22 @ :.{ 25 0.10 26

C H r p r l a 5 : C r a d u a l lV y a . i e dF l o w

199

5.13. The cross-secrion geometry fbr RonringCreckfollows: X (ft)

Elerarion{fl)

,1 l0 20 l0 .10 12 .16 50 5'{ 58 62

10.0 9.5 9.-3 9.J 9.1 7.0 6.1 6.0 6,I 6.1 6.0 1.1 6.3 8.3 8.9 9.0 9.5 9.3 9.6 I0.0

70 '72 76 80 90 t00 It0 Il6

050

.035

.060

.0-r0

The measured$ater su.faceelevationis g.g ft. (a) Manualll calculate rhenormaldischarge for a slopeof 0.0OOg. (b) Manuallycalculare the valueof a andihe specificenergy. (c) ls the flo\ subcrilical or supercritical? r J r V e r i f y ; o u rm . r n u ac la l c u l a i i o n wri r hr h eH E C _ R A p Sr o g r a m . 5.14, Write a compulerprogramin the language of your choicethal computesthe water surfaceprofile in a circulalculvertusingthe methodof inr.grution by the trape_ zoidalrule. 5.15. Writea computerprogramin the language of yourchoicethatcompures a warersur_ faceprofile in a trapezoidalchannelusingthe iounh_orderRunge_liuna metiod. -fest ir wirh rheMl profileof Table5_l. 5.16. For the floq over a horizontalbed with.constanl specificenergyand discharge decreasing in rhedirecrion of flow,derivetheshapes oi.the.uU".irluf andsupercrit_ icalprofilesfor a sidedischarge weir asshownin Figure5.1?. 5.17. Derivetheenersyequarion for sparially variedflow in the form of Equation 5.44,but do not assumethat Soand S,. the bed slopeand slopeof the energygradeline, are equaito zero_Comparetheresultwith Equation5.40anddiscuss. 5.18. A recrangular sidedischarge weir hasa heightof 0.35m. Ir is locatedrn a rectangular channelhaving a c,idthof 0.7 m. If the downsrream deprtri, O.li m to, a aiscf,u.g. of 0.27mr/s.how longshouldtheweir be for a lateraldiicharge oi O.Zf rn,lrf

2U)

C s apren 5: Gradually Varied Flow

5.19. A concrete(n : 0.013) cooling tower collection channelis rectangularwith a length of 45 ft in the flow direction and a widlh of I I fl. The addition of flow from above in the form of a continuousstreamof dropletsis al rhe rale of 0 63 cfs/ft of length Find |}le location of the critical section and compute the r''aler surface profile How deep should $e collectionchannelbe?

CHAPTER 6

HydraulicStructures

6.t INTRODUCTION In this chapter,we considera limited set of hydraulic strucrures(spillways. cul_ verts.and bridges)that provide \\ ater con\evanceto protect some other engineer_ rng structure.Spillways are used on both largeand small dams to passflood flows. thereby preventingovenoppin-sand failure of the dam. Culveni are desisned to carr,v peak flood dischargesunder roads,aysor olher cmbankt,.nt, to ir.r"nt enrbankmentoverflows.Finallv. bridges convey rehicles over u,ateru.avs. but thev nrust accommodatethrough-flous of flood\ aters uithout failure ,1ueio orenop_ ping or foundationfailure by scour. Of prinary imponancefbr the hydraulic structuresconsideredin this chaprer is the magnitudeof backwaterthey cause upstreamof the structure for given a design discharge;that is. the head-dischargerelationshipfor rhe structure.ln general,this relationshipcan assumethe fornr of weir flow. orifice llow. and in the case of culvens, full-pipe flow. Each tl pe of flou has its own characterisricdependence bctweenheadand discharge.For spillways.rhe pressuredistributionon rhe face of the spillway also is imponanr,becauseof rhe possibilityof cavitationand failure of the spillway surface. Both graduallyvariedand rapidlv variedflows arc possiblethroughthesesrructures, but one-dimensionalnrethodsof analtsis usually are sufficient and qell_ developedin this branchof hydraulics.Essenrialto rhe ..hydraulicapproach" is the specificationof empiricai dischargecoefficientsthat have been well established by laboratoryexperimenrsand verified in the fierd. The determinationof contrors in the hydrlulic analysisalso is imponanl. and critical deprh ofren is the control ol. interest.The energyequationand the specificenergydiagramare useful tools in the hydraulic analysesof this chapter.

?0r

C H A P T E R6 : H y d r a u l i cS t r u c t u r e s

6.2 SPILLWAYS The concreteogee spillway is usedto transferlarge flood dischargessafely from a leservoir to the downstreamrivel usually lvith significantelevation changesand relatively high lelocities. The characteristicogee shape shoqn in Figure 6.1 is based on the shape of the undcrsideof the nappe coming off a ventilated,sharpcrestedweir. The purposeof this shapeis to maintain pressureon the face of the spillway near atmosphericand well abovethe cavitationpressure. As an initial depanureon the task of developingthe head-dischargerelationship for ogee spillways, it is useful to use the Rehbock relationship for the disweir given previouslyin Chapter 2 as Equachargecoefficient of a sharp-crested tion 2.12. For a Yery high spillway. the contribution of the term involving H/P becomessmall and the dischargecoefficient,Cr. approachesa ralue of0.6l l; however,this value of C, is defined for a headof H' on a sharp-crestedweir as shown in Figure 6.1. If it is convertedto a value defined in termsof the head,H, which is measuredrelativeto the ogee spillway crest,then Cd = 0.728 becauseH : 0.89H', as shown in Figure 6.1 (Henderson1966).As a result.C - Ql(LHrtz) hasan equivalent value of approximately3.9 in English units for a very high spillway. For lower spillways, the effect of the approachvelocity and the venical contraction of the \r'atersurfaceintroducean additionalgeometricparametergiven by HIP or its inverse, in which P is the height of the spillway crest relative to the approach channel. Funhermore, the design value of the dischargecoefficient is valid for one specific value of head, called the desigrr head. Ho, becausethe pressure distribution changesfrom the ideal atmosphericpressureassociatedwith the ogee shapewheneverthe headchanges.As the headbecomesIargerthanthe design head, the pressureson the face of the spillway becomeless than atmosphericand can approachcar itation conditions.Pressures are largerthan atmosphericfor heads less than the design head.On the other hand, the risk of cavitation at headshigher than designhead is counterbalanced by higherdischargecoefficientsbecauseof the

Concretespillwaycrest conforming to the undersideof nappe of sharp-crestedweir FIGURE 6.7 The ogeespillwayandequivalent weir sharpcrested

CHAPTER 6: Hydraulic Stn_rctures

1.04

i t l Upstreamface slope:3 on 3

1.03 1.O2 \

tt\

'1.01 o

203

'.'

-:\ -.]

1.00 3on1 0.99

\ 3on2

0.98

0.2

0.6 0.5 0.4 -;.0.3 J-

o.2 0.1 0 o_70

0.90

1.00

C/Coin whichCo = 4.03 FIGURE 6.2 Dischargecoellicient for rhe WES srandardspillway shape(Chow lglg). lsource; (Jsed h h pen istiottol Cltov.estate.\ tower pressureson the face of the spillway. In other uords, the spill*.ay becomes more efficient becauseit passesa higher dischargefor rhe same headwith a larger value of rhe dischargecoefficient.The spillway discharge coefficientrs given in Figure 6.2 for the srandardWES (Waterwaysd^p".im.ni Station)or.erflowspillway in terms of the influenceof the spillway height relative to the design lead, PlH,,, and the effect of headsorher rhan the disign-head as indicatedby H"lHu, in which H, is the design total head and H" is the actual total head on the spillway crest,including the approachvelocity head.The dischargecoefficienr. C. wittr p in cubic feet per secondand both L and H. in feet is definJd bv

(6.1) in which L is the net effectivecrestlength.The inset in Figure 6.2 sho\rsthat a slop_ ing upstreamface. which can be usedto preventa separation eddy thal mlght occur on the venical face of a Iow spillway,causesan increasein the dischargecoefficient

2 0 . 1 C H A p r [ R 6 i H y d r a u l iSc t r u c l u r c s for P/I!,,< 1.0. The lateral contraction causcd by piers and abulmcntslends to reducethe actualcrestlength.L', to its ellectivc value,L:

L : L'

2(.\'4 + l(.,)H"

(6.1)

in which N : numberof piersl K, : pier contractioncoelficicnt: and K, = abutment contractioncoelficient.For square-nosedpicrs. K, = 0.02. while for roundnosedpiers.Kn = 0.0 | , and lor pointcd-nosepicn. K, - 0.0. For squareabutments with headqalis at 90' to thc flow direction. K., = 0.20. uhile lbr rounded abutm e n t s\ \ i t h t h er a d i u so f c u r v a t u r er i n t h e r a n g e .0 . l 5 H t < r 3 0 . 5 / J / . K , , ! 0 1 0 . Well-roundedabutnlentswith r > 0.5H, have a value of K, : 0 0 (U S. Burcau of R c c l a m a t i o n1 9 8 7) . A well-estlblisheddesignprocedurc.u hich has been developcdby the USBR (U.S. Burcau of Reclamation)and the COE (Corps of Engineers).takcsadvantage of the higher spillway efficiency achiered for herds greaterthan the clesignhead. Essentially.the designprocedureinvolres sclcctinga design head that is less than the maximum headto conrputethe spill* ay crcst shape:this is called tudertlesigtt' pressureson the face irrg the spillway crest.Tcstshave shown that subatmospheric of the spillway do not exceedabout one half the design head when H'.,/H./ does not excced 1.33.This is shown in Figure 6.-1,in which the actualpressuredistribution on a high spillway with no piers is given for HlHu varying liom 0.5 to 1.5 where H - H". At HlH,t = 1.0,the pressuresindeedare very closeto atmospheric. -0.2Ht where X The minimunr pressurefor HlH, = 1.33 is 0.'1-jl/, at X : 0.0 at the centerlineof the spillu'ay crest. Instead of arbitrarily setting H"/H./ = l.l3 at the maximum head' Cassidy ( 1970)suggeststhat a betterdesignprtredure is to establisha minimum allo\{able pressurcon the spi)lwaytaceand then deternlincthe designhead The pressureson spilluay faccsare not constantbut 1'luctuatearounda mean raluc. so the COE norv rccomnrendsa more conscrvativedesign procedurcttll-not allo\.'!ingthe average pressurehead to tall below 15 t] to l0 ti. cven though calitation may not be incipientuntil a pressurcheadof 25 ti is reachcd(Reescand Maynord 1987) ln this design approach.the ntinimum allos able pressurc'head becomesthe controlling featureolthe designofthc spillqav crcst. ratherthan a flxed valueof H"lHr. Once the design head is determined. the actual shape of the spillwa-r-crest downstreamof the apex.in what is callcd the doru.stn'an .luodruttt.is given bv: 'Y X' : K,rH','t

(6..3)

in which K.,, = 2.0 and a = 1.85 tbr negligible approachI'elocity; H; : desiSn hearl: and *. f lrc nreasuredfrom ihe crest axis as shown in Figure 6.'1. The upstreontquatlrantof thc spillway crest is construcledfiom a compoundcircular 1o tbrm the standardWES ogee spillway shape.The curve, as shown in Figure6.21. 0.0,1H, radius curve was added in the 1970s resulting in a sliSht increasein the spiff way coetlicientin Figure6.2 for H,/H, > 1.0 and PlH,t> 1.33. Rccseand Maynord ( 1987) proposed.instead,a quaner of an ellipse.u'hich is tangentto the upslreamface. for the shaPcof the upstrcamquadrantas shorvnin Figure 6.5a.The dischargecoetilcientsfor this shapeare given in Figure6 5b for a

0.6

o.4 HlHd = 0.50 -

o.2 H/Hd - 1.0O, -[

l I

-o

M

0

4.2

a ^-a o 4.4

ut-

HtHd=1y/ \

l/1-1

\i'#

{.6 -{.8 '1.0

-0.2

0

o.2 0.4 0.6 Horizontal Dislance DesignHead

(+) 0.8

1. 0

1.2

!.IGL RE 6.3 Crestpressure on WES high-overflow spillu,ay-nopiers(U.S.Army Corysof Engineers. 1 9 7 0H . y d r a u l iD c esign C h a nI l 1 , 1 6 ) . Axis both quadrants R = 0.2QHd

i--'----*x Y.as = 21185y

\\ B = O.04Ha

I

Y

q Crest FIGURE 6.4 StandardWES ogeespillwayshape(U.S. Army Corps of Engineers. 1q70. Hldraulic DesignChnrtI I l- l6).

205

10.0 8.0 6.0 4.0

o-

'rb

2.0

E

Hs o L'a dto

1.0 0.8 0.6 o.4

0.2

I

0.'15 o . 2 1 0.23 0.25 0 . 2 7 0 . 2 9 0 . 1 2 0 . 1 4 0 . 1 6 0 . 1 81 . 9 0 2.'to 2.30 BlHd

AlHd

Xz A2

@_ y)2 82 Coordinaleorigin , A

r-'l

yt as= g"r1fi85y

sketch Definition (a) CoordinateCoetficients FIGURE 6.5 Ellipticalcrestspillwaycoordinatecoefficientsanddischargecoefficients'venicalupstream 1990,HydraulicDesignChan I I l-20) face(U.S.Army CorPsof Engineers,

26

c! I

l

t

1

q

I

l

q qqlq l

l

t |

|

I

10

t L

t

fr ++ it I

il

E+aHT 1 J

l

o 3.2 3.4 3.6 3.8 4.0 4.2 4.4 c = o/LH:t2 DischargeCoetficientVersusP/Hd

3.0

3.2

3.4

3.6 3.8 c = a/LH!2

4.0

4.2

(b) DischargeCoetficients, VerticalUpstreamFace FIGURE 6.5 (continued)

207

90

\

80

Cavitati( )n zone 70

\ 60 T

50

40 No cavitation zone

5tt

I

30

\

20tr -15tt 20 1.1

1.3

1.5

Note;

Hd= DesignTotalHead,ft He = ActualTotalHead, tt (a) No Piers FIGURE 6.6 safetycurves,no piersand with piers(U.S.Army CorPs Ellipticalcrestspillwaycavitation of Enginecrs.1990,HydraulicDesignChan I I l-25).

208

CHcPtrl, h

H ) d r l r u l iSc l r u c l u r e s

209

Cavitatlon

Note: Hd = DesignTolal Head, tt He = ActualTotalHead,tt (b) with Piers FIGURE 6.6 kontinued) vcrtical upstreamface.Reeseand Maynord also developeda setof cavitationsafety curves in which the design head is determinedby the allowable cavitationhead. Thesc are given in Figure 6.6 for elliptical crest spillways with and without piers. Insteadof selectingH 1H d as | .33,a tial designheadcan be chosenfor a minimum l5 ft. Then from Figure 6.6, the value of HlHoand the maxipressurehead of mum headH" can be obtainedto compare with the given value.

210

C H A p T ! . R 6 : H ) ' d r a u l i cS t r u c t u r e s ExAt\tpI-E 6.1. Fbr a nraxinrunrdisch.rrgeof 200.000cfs (5666 mr/s) and a maximum total head()n the spillqay cresl of 6J fl (19.5 nr). deleft)ine the crest length with no piers,the nrininrunrpressureon lhc.resl. and the dischargeat thc dcsign headfor the standardWES ogee spill\\'ay.The herghlof the spillwny cresl,P. is 60 ft ( l8 m). ,So/arioa. For this example,usc thc dcsign proccdureof settingthe ralio oi the maxi munr head to design head lcl the value 1.3-1.so thc dcsign head H,/ = 6.1/1.33= ,18ft (1,1.6m). Also caleulatethe ftio PlH,t - 60/.18= 1.25.Then, from FigLrre6.2 for the standardWES high-overflor"spillway (con]poundcircularcune for upstrcamcrcst).the v a l u e o f C T C , , : 1 . 0 2a n d C : 1 . 0 2x . 1 . 0 - l = 4 . 1 1 .N o w ' l h e r e q u i r e dc r e s ll e n g l hi s

.

L

o-,.

:00.000

cH)'

Lll x (61)'

-

i

- c)ll{,lem)

headis 0.,13H,i, soP"r,d/y FrornFigure6.3 lor II,lHo = 1.33.the nrininrunpressure = -20.6 ft ( 6.3 m). rrhichis an acceptable lalue.ljowever,ifa lcssnegoti\cpresNow theshapeof thespillway the valueol H"/H, canbe adjusled. sureheadis desired, crestis designedfor the dcsignhead.Hr, of ,18fi (1,1.6m). For example.thc shapeof ponionof thecrestwilh X and y in feetis 8i\en by the downstream X r 8 5: 2 . 0 f l ? 8 s=f ( 2 0 x 4 8 0 3 5 ): f 5 3 . ? l y coefflcient asobtained The discharge at thedesignheadwill havea differcntdischarge from Figure6.2.For HJH,j = 1.0,C = ,1.01andthedesigndischarge, Br. is givenby Q1: 4.ol x 95 x 48r1': l2?,000cfs (3.600mr/s) coefficientis takenfrom To designthis spillwayfor an cllipticalcrest.the discharge is determined, or specified, usingFigure6.6. Figure6.5,andthe minimumpressure

6.3 SPII,LWAY AERATION Eventhoughthe shapeof ogeespillwayscanbe dcsignedto minimizethe risk of in the spillwaysurfacesometimes damagedue to cavitation,smallimperfections pressure dropsthat may be and conesponding can leadto localizedacceleration unacceptable. The costof providinga spillwaysurfacethatis smoothenoughor is maybecomeprohibitive. This hasgivenrise strengthened by surfacereinforcement to the useof artihcialaeratjonon very highspillwaysto introduceair at pressures cavitation. pressure nearthe spillwayface,thuspreventing closeto atmospheric in The conceptof artificialaerationhas stimulatedinterestin self-aeration, with the atmosphcre leadsto which the naturalentrainment of air at the interface on theface bulkingof theflow with the commonlyobservedwhite-waterappearance of high spillways.Early work on naturalsurfaceaerationof spillwayswasdoneby StraubandAnderson( 1960)in a 50 ft ( l5 m) longby L5 ft (0.46m) wide flume with slopeangles,0,varyingfrom7.5' to 75o.A sluicegatewaslocatedat theflume entranceandadjustedto achieveuniform flow andaerationconditions.The air concentrationdistributionwas measuredand shown to have two distinct regions:a lower, bubbly mixture layer and an upper layer consistingprimarily of spray.

C H A p T E R6 : H l d r a u l i cS t r u c t u r c s 2 l l Bccausethe depth bccon)esill defincd in aeratedflorv. Strauband Andersonused a rcfcrencedepth.r'0,u'hich was thc unifornr flow dcpth of nonireratedflow. h conespondedto a measurcdChczy C valueof90.5 in English units for their !'rpcrintcnrs. The eflectire dcpth of water,r,,,,which was defined by J; ( I - q)dr., in which C, representsthe poinl air concentrationin volume of air pcr unit total volume, was relatedto the referenccdepthand mean air concentration.C,,,,by the relation

l r = r . 0- r . 3 ( c ,- 0 . 2 5 ) :

(6..+)

The effective depth of water also could be defined in terms of continuityas q/V, in which 4 = flow rate per unit of width and V - mean velocity.The nteanalr concentration\\ as determinedfiom a best fit of the experimentaldata in terms of the slopc of the spillway.S (: sin 0), and the flow ratc pcr unit of u,idth,.11

C . : 0 . 1 4 3 ' " r , r *( ) - / + 0 . 8 7 6 \ q

(6 5)

Equation6.5 appliesfor a rangeof air concentrations from 0.25to 0.?5,andq has unitsof cubicfeetpersecondperfoot.Forexample,for a spillwayslopeof 75' and a flow rareper unit oI widrhof 600 cfs/ftr56 m'/5/mr. thc meanair concentration wouldbe 0..15(or 45 percent), defincdastheratioof volumeof air to totalvolume. Thecorespondingeffectivedepthof waterfrom Equation6.4 *'ouldbe 95 percent of the reference depth.The effectivedepthof watershouldbe usedin themomentum flux term in the momentumfunctionfor the designof a stillingbasinat the baseof the spillway(Henderson 1966).The hydrostatic forcetermin the momenrurnfunctionfor rheaerared flo$ becomes (r^ rrll2rt C)1. Whetherthe air concentration predictedby Equation6.5 can be achieved depends on the thelengthof thespillwayface.In general,thepointof inceptionof wouldnotbe expected surfaceair entrainment to occuruntil theboundary layerhad grownto the pointof intersection with the freesurface.KellerandRastogi(1977) solvedthe boundarylayerequationsnumericallyon a standardWaterwaysExperimentStationspillwaywith a verticalupstream faceto obtainvaluesof thecritical distance,r., for the lengthof the boundarylayer measuredfrom the crest.Wood, Ackers,andLoveless( 1983)developed anempiricalformulafor .t froma multiple regression analysisof KellerandRastogi's results: x :. :

r,

I

a

'"t\'it:I

t l A

loTr'1 I I

sor7r

(6.6)

in which S : spillwayslope- sin0;g - flow rateper unit of width; and &. = roughness heightfor thespillwaysurface. Fromthisequation, we canconclude that the distancerequiredfor inceptionof surfaceair entrainmentdependsprimarily on the slopeof the spillwayandthe flow rateper unit of width. For a concretesurface roughness heightof 0.005ft (0.0015m), andfor a spillwayhaving4 = 600 cfs/ft (56 m3/s/m)and 0 : 75", as in the previousexample,the lengthof spillway requiredfor self-aeration to commencewould be approximately 550 ft (168 m), whichcorresponds to a spillwayheightof 531 ft ( 162m).

Il

C H A P I E R 6 : l l ) ' d r a u l i cS t r u c l u r e s Qa I I

+

F I G U R E6 . 7 Definitionsketchof a spillwayair ramp For sontcspillways,even thoughthey are high enoughfor sclf-aerationsurtlce nraybe insufficicntto preventcavitationon the facc of the spillwly. air entrairtntent the crest.whcre it may not occur at ali. Undcr thesecircuntst'tnecs. ncar espccially have becn usedto inducean air cavity that allo$ s entrainrncntof air rantps aeiation pressure on the undersideof the jet coming off the aerutionrrnrf' atmospheric near is shown in Figure 6.7. in which the air is supplied air ramp typical oi a A sketch through lateralwedgesat thc edgeof the spillatmosphere from the cr!ity to the air u,ay chute or through rccessesor ducts underneaththe ralnp that are fed by chintneys. TurbulencecausesdisruPtionof the lvater surfaceon the undcrsideof thc nappe and air is draggedand entrainedinto the jet, which then is nixcd with the flow downstream.The pressurcin the cavity below the nappewill be slightly less than atmosphericbecauseof head lossesin the air delivery system.so that thc trajectory lnd length of the jet will be different from that of a freejet ' With rcferenccto Figure6.7, a dimensionalanalysisof the problcm leadsto the following expressionfor the length of the jet. 1-'coming ofT the ramp:

t

;

l

l

n

-

l

j /Lt p, . ne we. rlmPgconrctrv

( 6 . 7)

on thespillway flow dcpthandvelocity'respectivel)'. in whichft andV = approach drop in the air Froudenumber: V/(gh)D5lI'p" = pressurc chute;F = approach pressure; Re - Reynoldsnumber: l',/v: andWe cavityrelativeto atnospheric l'l(o/pl)or. The ReynoldsnumberandWebernumbereffectstendto be smallin the prototypespillway,so that for a fixed ramp geometry,the primaryvariablesof pressure differencelt has interestarethe Froudenumberandthe subatmospheric spillwaysthatthe air flow perunit of width from testsof prototyPe beensuggested (de S Pinto1988) of spillwiy q.: kvL, wherek is a constantof proportionality It followsthenthat q. L (6.8) C^: k; q

ratio on the left-handside of F4uation 6.8 is equivalentto the air conThe drscharge centration,C., as shown,which should be 5 l0 percentto preventcavitationdamage,basedon past exPerience(de S. Pinto 1988).Thus, providedthe constantk is

6 : l l y d r a u l i cS l r u c t u r e s 2 1 3 CHApTER known fronr prototype experience,the required value of /fh can be deterntjned fronr Equation 6.8 for the de:ired air conccntration.Then, front the relalionship given by Equation6.7 fronr phl sical nrodcl studiesor numericalanalysisof the jet Jp, can be detertrajcctoryfor a given ramp geometry,the requircdundcrpressure known valuc of the Froude number. value of Z/fi and the for the specificd mined Finally, the air delivery systemcan be designedto provide thc air flow rate $ ith the specificd pressuredrop. The value of k in Equation 6.8 has bcen detemined to be 0.033 from the Foz do Areia prototypc spillway tests (de S. Pinto 1988),but it can vary for difterent flow conditionsand diffcrent ranp geomctrics.What is requiredis a ntodel study w i t h a r e l a t i v e l yl a r g es c a l e( l : 1 0 t o l : l 5 ) t o c l i n i n a t eR e y n o l d sn u n b e r a n d W c b c r number effectsand so detemrine soecificdcsign valuesof t.

6.4 STEPPEDSPILLWAYS Steppedspillwayshave been used extensivelyaroundthe world sinceantiquity, but they became very popular in the past few decadeswith the advcnt of rollercornpactedconcrete(RCC) and gabion constructionof dams (Chanson 1994a). They provide good surfaceaerationbut also increasethe energydissipationin the flow down the spillu,ayin comparisonto a smooth spillway.This latter featureof steppedspillwaysmay rcduce the cost of the downstreamstilling basin. Steppedspillways can operate either in a nappe llow regime or a skimning flow reginre.In nappe flow, u,hich tcnds to occur at lower dischargeson flatter spillways,the flow consistsof a sericsofjets that strike the floor of the succeeding steps.Eachjet usually is follorled by a partial hydraulicjump. In skimming flow. the jets move smoothly without breakupacrossthe steps,which act as a seriesof roughnesselenrents. A recirculatingvonex forms on each stepin which energydissipates.The skimnringflow regime is shown in Figure 6.8. Rajaratnam( 1990) suggestedthat thc onset of skimming flow occurs for values of _r'./iexceeding 0.8,

FIGURE 6.8 Dellnirionskcrchof r \teppcd.lillq rl

2l.t

c truclurcs C H A P T E R6 . H y d r r L r l i S

'1.0 chrislodoulou(1993) .,,.,

I I

u.5

0.0

0.2

0.1

0.3

0.4

0.5

FIGURE 6.9 spillwayin skimmingflow withN steps(Rice ModelstudyresulL\for headlosson a stcpped "Model Stuh ofa Roller Con' and Kadavy 1996t.(Source.C. E. Riceattd K. C. Kadat':-. pacted ConcreteSteppcdSpillnar" J. Hvdr Eagrg-,A 1996. ASCE.Reproducedb'- pernissionoJASCE.)

where _v.is the critical dcpth for the flow on the spillway and lr is the hcight of an individual step. 'fhc amount of energydissipationthat occurs on a steppedspillway lor skimming flow is one of the prirnarydesignvariablcs.Christodoulou(1993) suggested that the energy head dissipated.,\H, in ratio to the totll head. H,,. upstreamof the = critical dcpth; N : number dam relativeto the toe is relatcdto r'./N/r,in which _r',. : the heightof eachstep,as shown in Figure 6.9. Rice and Kadavy of steps;and ft ( 1996)haveconfirmed the validity of Figurc 6.9 for a physical model of the Salado Creek spillway in Texas.Basedon lheir datapoints,the Christodouloucurve in Figure 6.9 is valid for Vl valuesin the rangeof 0.7 to 2.5. r',/lr < .1.5,and r. /Mr < 0.5. Chanson (1994b) analyzedexperimcntaldata for stepped spillways from a large numberof investigatorsand conrparedthe resultsfor relativeenergyloss with an analyticalformulation for uniform flow conditionsgiven by

At/ H^

.nt d + 0 5Ct: r CrL"t -

l

l.) +

H.n,.

(6.9)

in rvhich C, = /8 sin 0):/ - friction factor; 0 : tan I lh/l): t/,/,,,,,= dam crest height abovethe loe: and _r,.: critical flow depth. He found rcasonableagreement with the cxperimentalresults,consideringthe degreeof scatter.using/ = 1.0(nonaeratedflow) and 0 - 52" over a \rerywidc rangeof H,,,,,,[',from approximately2 to 90. Usually, H t,,,. = Nh, so Equation6.9 correspondswith the variablesof Fig-

C H \ P T E R6 : l l y d r l u l i cS t r u c t u r e s 2 1 5 6 9 rnustbe used with ure 6.9 exccpt that it covcrsa wider rangc in -1./Nir'Eqr'ration of aeration' c,rt. bc."urc ol the uncertaintyin tlle friction factor due to the cffects as well as Stcppedspillwaysoffcr the advantaSeof elhanced air cnlrainn'lent entrainnent "ncrgy iirrip,rtion. ih.n.on (199'1b)shows that the inccPtionof ilir spillway .,..ui, in a shortcr distancc on a stepped sPillway than on a snooth the equilibrium becauseof the nttlrerapid rate of bcundary-la)er gro\\'th Howevcr' printarily is a air concenlrationis similar on steppedand smooth spillrlays and refer to the function of stope.For more detailson the clcsignof steppedspillways ( comorehcnsivctrealtnentof the subjectby Chanson 199'1b)'

6.5 CUI)/ERTS among the []ost Culverts seem to be simple hydraulic structuresbut in fact are occur in them complicatedbecauscof the \l ide variety of flow conditionsthatcan time A culof function Flow can be graduallyvaried or rapidly varied and also a as in conditions ven can flow full, in which case it operatcsunder pressure-flow can flow channel pipe f1ow.or it can flow partly full, as an open channel The open gradof a Le ,upe.criti.ol or subcritical,and its analysismay include computation thc outlct is ually varied tlow profilc or a hyilraulicjump Culvertsflow full when headwater very high submerge,tdue to high tailwaterbut also may flow full for a of submergence the In both tull and partly full flow' u ith thJ outlet unsubnrerged. of flow that type the inlet or outlct is an important criterion in determining the of a culvl:rt flow is occurs.Perhapsthe most inportant distinguishingcharacteristic control' the hcadinlet u,hetherit is under inlet or outlet control ln the casc of the inlet including dischargerelation is deternincd entirely by the inlct geometry' for inlet conareo,e.[. rounding.and shape Tailwaterconditions are immaterial not is affected relation trol. In;tlet control,on the other hand,the head-discharge as and area shape' only by the inlet but also by the barrel roughness'lcngth, slope' are sumct)ntrol outlet *.il oi,n. tailwrter elevation.Theseinfluenceson inlet and culvertswith a marized in Table6- I . lnlet control generallyoccurs for short' steep with high culverts free outlet, while outlet control prevailsfor long' rough-barreled tailwater conditions. deterCulvert design usually is basedon the selectionof a designdischarge may be example' for mined from freq-uencyan;lysis. lnterstatehighway culverts' the to limit is sized designedto carry th; 100 year peak discharge The culvert preventoverheaJwaterresultingfrom the designdischargeto a specifiedvaluc to its is detertnined' size topping the highwiy embankment.Once the design culvert disincluding p.*o.ilun.. riay be analyzed over a wide range of discharges' by a plot iharges that overtopthe embankment This analysiscan be summarized Thisstepis curve relation.called lhe performance of tltleconrpletehead-discharge inlet or outimponant io accuratelydeterminewhether the culven operatesunder peak selected on a is based let control for the designdischargeThe design process both inlet which in dischargein steadyflow, and a conservativeapproachis taken

:t6

C H A P T L R6 : H ) ' d r a u l i cS l r u c t u r e s TABI,E 6.I Factors influencing

culvcrt pcrformance

Factor

Inlet Control

Oullet CoDtrol

H e a d ! \ a t e re l e \ a t i o n Inlel area I n l c t e d g e c o n f ig u r a l i o n Inlet shape Barrelroughness BalTel area BalTel \hape Barrel lengrh Barrcl slopc T a i l w a l e re l e v a l i o n s.rh.:

Daraftom FederalHrghqavAdministfutrcn( 1985).

c

.9 -9

LU

3

! I

Qc

Qee

Discharge,O FIGURE 6.IO Culven performancecurvesfor the determinationof inlet or oullet control (FederaiHighwayAdministralion 1985).

and outlet control head dischargerelationshipsare checkedto determinethe limiting control. The higher head resulting either from inlet or outlet control is comparedwirh the allowableheadwaterelevation.If, at the designheadwateras shown in Figure 6.10, for example,the inlet-controldischarge,Qr., is lessthan the outletcontrol discharge,Ooc, then the inlet capacityis less than the barrel capacity,and the inlet controls the head-dischargerelation at the design condition. This is the

C H A p T E R6 : I l l d r a u l i cS t r u c t u r e s 2 1 1 sanreas choosing the highcr head for a givcn discharge.as can bc sccn in Figure 6 . 1 0 .A s t h e h e a di n c r e a s eisn F i g u r e6 . 1 0 .t h e c u l v c r tr e m a i n si n i n l e t c o n t r o lu n t i l the intersectionbetrveenthe inlet-controland outL't-contlolcurvcs, bcyond which it is assumedto be in outlct control. The hcad-dischargerelationshipof a culven follows rvell-known hydraulic bchavior.The culven nay ng1ns 3 seir, an orifice, or a pipe in prcssurellow. For as a ueir at the inlct and the discharge an unsubrnergedinlct, the culvert-opcrates is proponional to the hcad to the : po\\cr. tf the inlet is submergedand thc culven is in inlet control, orillce flow occursand the dischargeis proportionll to thc head to the j power This neans that the hcad incrcasesmore rapidly rvith un incrcrse in dischargethan for rreir flow. ln pressureflow, the hcld-dischargerelation is dclermined by the etTectivchead,which is the differencein total head betwcenthc headu'aterand tailwater. The U.S. Gcological Survcl' (Bodhaine 1976) classifiesculvert llow into six types, dcpendingprimarily on the headwaterand tailwatcr levels and whcthcr the slope is mild or steep.Thesetypes offlow also have been givcn by French ( 1985), but Chow ( 1959)used a differentnumberingsystcmfor the same six types of flow. Additional types of culvert flow can be idcntified;however,a simpler classification relationship.In this classifidcpcndsonly on the type of hydraulic head-discharge cation, the most inrponant criteria are whetherthe culvert is in inlet or outlet control and whether the inlet is submergedor unsubmerged.Submergenceof the inlet occurswhen the ratio of inlet headto heightof the culvert,HW/d, is in the rangeof 1.2 to 1.5, with the latter value usually taken as thc submergencecriterion. Inlet head,HW, is dcfined as the height of the hcadwaterabove the inven of the culvcrt inlet, as shown in Figure 6. I 1.

Inlet Control Sevcraltypesof inlet control are illustratedin Figure 6. I L In Figure 6. I la, both the inlet and outlet are unsubmergedon a steepslope.Flow passesthrough the critical depth at the inlet and the do\\'nstreamflow is supercritical (52 curve) as it approachesnormal depth.This is U.S. GeologicalSurvey(USGS) Type I flow. The outlet is submergedin Figure 6.1lb. which forces a hydraulic jump in the barrel As long as the tailwater is not high enoughto move the jump upstreamto the inlct. relationshipdoes not the culvert remainsin inlet control: that is. the head-discharge change.In Figure 6.I 1c,the inlet is subnergedand the outlet is unsubmcrged.Critical depth occurs just downstreamof the inlet, but the culven is in orificc flow (USGS Type 5). Both the inlet and outlet are subnergedin Figure 6. I I d. and a vcnt must be provided to preventan unstableflow situation,which oscillatcsbetween full florv and partly full flow. With the vent in placeand the hydraulicjump remaining downstreamof the culvert entrance,this remainsinlet control with orifice f'low at the entraDce. The head-dischnrgerelationshipsfor inlet control are basedon either weir flow for an unsubmcrgedinlet or orifice flo$ for a submcrgedinlet. ln other words, only two types of flow occur in inlet control in terms of the type of head-dischargerelationship

218

C H A P T E R 6 : H y d r a u l i cS t m c t u r e s

(a) OutletUnsubmerged

(b) OutletSubmerged, InletUnsubmerged

(c) InlelSubmerged Mediandrain

(d)OutlelSubmerged FIGURE6.I I Adrninistration 1985). Highway Typesof inletcontrollFederal andorificeflow, which we referto hereasType thatgovems:( 1) inletsubmerged on a steepslopewith $ eir flow, whichis calledType lC- 1I andinlet unsubmerged relationfor weir flow (lC-2) is derivedfrom theenergy IC-2.The head-discharge the to the criticaldepth section,neglecting equationwritten from the headwater approachvelocityhead:

o: H W - . t . + (+l & ) ; f t

( 6 .l 0 a )

in which HW : head above the inven of the culvert inlet: r; : critical depthl A" = flow area correspondingto critical deptht and K" = enlrance loss coefftcient.An additional equation is neededto eliminate the critical depth- ard it comes from the condi-

CHApTI,R 6 : l l y d r a u l i cS t r u c t u r e s 2 l g tion of scttingthe Froudenumberequal to unity. Equation6. l0a can be rearranged to solvefor the discharge,Q:

q:

coe,^r/-Zg@ - ,)

(6.r0b)

or it can be placed in the fonn of a u eir equation.Note that the coellicient of dis_ chargc.C, : l/(l + 4)r/2. The USGS (Bodhaine 1976)dcvelopedvalues for the coefficientC, as a function of the head to diameter ratio,Ml/i, for circular cul_ c u l v c d sw i r h a s q u a r ee d g e i n a v e r t i c a lh e a d w a l l C , r :0.93 for l9l1f.Fo.-pip" HWld < 0.1, and it decreasesro 0.80 at IIW\d - 1.5,where the entrance becomes submerged.The coefficient C, can be corrected for bel.elsand rounding of the entranceedge. For a standard45" berel with rhe ratio of bevel heighr to-culven diantcterwy'd- 0.0.12,the correctionto the coefficientCd is approxintately l. | . For machinetongue-and-groovc reinforcedconcrctepipe from tg to 16 in. in cliameter, the value of C, - 0.95 rvith no sysren)aricvariationfound bctweenC, and HW/d. For box culvens set flush in a vertical headwall,the value of C, - 0.95 for USGS Type I flow (lC-2). Once the inlet is subnrerged(Type IC_I), the governinghydraulic equarion is the oriflce-flow equationgiven as

O : c,A.\/2s(Hw)

( 6 . 1)l in whichC, : coefficient of dischargc; A, = cross-sectional areaof inlet:andHW = headon the inlet invertof the culren. Somevalues of C, for orifice flow are givenin Table6-2 for variousdegreesof roundingwith radiusr andfor bevelsof heightw asa functionof HW/d.Thepurposeof bevclsor roundingis ro reduce the flow contraction at the inretof rhecur\ert to obtaina higherdiscf,arge coetticient. The FHWA (FederalHighwayAdministration) developerl head_discharge relation_ shipsfor inlet controlusingbcvelsof -15.or 33.7"uith rry'bor y+,ld= 0.042 and TAI}LE 6.2

Orifice dischargecoefficientsfor culverls [e = CaA.(25 lllr\tnl rlb, r/d;

1.5 t.6 |.'7 t.8 1.9 2.0 25 3.0 ,{.() 5.0 Sura/

0.1.1 0..16 0..17 0.,18 0..19 0.50 0.51 0.5.1 0.55 (r.)/ 0.58 0.59

0..16 0..19 0.51 0.52

0.5.r 0.55 0.56 0.59 0.61 0.62 0.63 0.64

D d r a i r o m B o d h a r n e1 l q l 6 r

0.{9 0.52 0.5J 0.55 0.57

0.s8 0.59 0.61 0.6-1 0.65 0.66 0.61

x)lb,eld

0.50 0.53 0.55 0.57 0.58 0.59 0.60 0.6,1 0.66 0.67 0.68 0.69

0.50 0.53 0.55 0.57 0.58 0.60 0.61 0.6{ 0.67 0.69 0.70 0.71

0.t0

0.t4

0.5r 0.5,{

0.51 0.5.{ 0.56 0.57 0.58 0.60 0.62 0.66 0.70 0.?I 0.12 0.73

0.56 0.57 0.58 0.60 0.6t 0.65 0.69 0.70 0.71 0.'t7

2 2 O C H A p T E R6 : I l l d r a u l i cS l r u c r u r e s 0.083, respcctivcly,where n is thc height of the bo,el; b is the hcight of a tor cul vert; and r/ is the diarrleterof a circul.rrcujrert. The 15. bevel is recontmend.,dfbr easeof construction(Fcderal lligh*ay Adnrinislrationl9li5). Fronr .l.able6_1.we see that thcse two sttndard berels increasethc dischargecoellicient by approxi mately l0 to 20 pcrcentin comparisonwith a square,cdgeinlet (r - 0; x. : 0). For a grooved-endconcretepipe culvcrt, bevcls are unnecessary, becausethc sroovc gives about the same inrpro\entcnt in the dischargccoefficient. Bctween (he unsubmergedand submcrgedportions of the inlct control hcad_ dischargeequations,a smooth transitioncurve connectsthe two. Bascd on cxten_ sive experimentalresults obtained by the National Bureau of Standards.(.ust-fil power relationshipshavc been obtainedfbr both the unsubmergedand submerged portionsof thc inlet control head-discharge rclltionship. For thc inlel unsubmr-rged. two fbnns of the equalionare rccommended:

o)u -o5s -t fr =v i' E .+ ^ l' L Adosl Hw

/

I o1,l

"la;u:1

t6.l2a)

{6.12b)

in which HW - heaciabovejnvertof culvertinlet in fcet;E : minimumspecific energyin feettd = heightof culvcn inlerin feet;Q = designdischargein cubic fcct per second;A : full cross-sectional areaof barrelin squarefcet;S : culven banelslopcin feetper foor; andK, M - constanrs for differenttyrrcsof inlersfrom T a b l e6 - 3 . E q u a t i o n6 . 1 2 ai s F o r m I r n d p r e f e r r e dE: q u a t i o n 6 . 1 2 bi s F o r m 2 , whichis usedmoreeasily.For the inletsubmcrged, the best-fitpowerrelarionship is of the form H \

/

: ' L| oO ll , :l + Y - o 5 s

(6.l3)

in which c and f are constantsobtainedfrom Table6-3 for e, A, and d in English u n i t sa s f o r E q u a t i o n 6 s . 1 2 .F 4 u a t i o n s6 . 1 2a p p l y u p t o v a l u e so f e / ( A t l o t l = 3 . 5 , while Equation6. | 3 is I alid fot QllAdo 5) > 4.0. lnlet control nomographsbascdon Equations6.l2 and 6.13 hare bcen developcdfor manualculvert designand can be found in HDS 5 (FcderalHighu,ay AdminisrrarionI985). A ful inlet conrrol cuNe can be developedgraphicaili by connectingEquarions6.12 and 6.13 with sntooth curvesin the transitionregion.For conlputcrapplications.polynomialregressionhas beenappliedto obtain best-firrelationshipsfor the inlet control curve of the form

HW d

= A + BX + CX2 + DXr + EXl + FX5 - C.S

(6.14)

in which C" = slopc correcrioncoefficient;S : culvert slope; and X : e/tBd1t2), where 0 = dischargein one barrel: I = culven span of one banel; and d : cul_ vert hcight. Thc polynonrialand slope correctioncocfllcientsare availablein Fed_ e r a l H i g h w a yA d r l i n i s r r a r i o n( | 9 8 2 a n d 1 9 7 9 ) .

=.?

g".9 r.E

6 Z

5-

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a

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c

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J

a

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z ;

a

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l

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&&.

t

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a

V

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a z !

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V =

a

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:

.i. 221

-t r .: .-r r: 2 .t c !i 9 r 1 I I r - - 3 - -

q. q I 9 -

!

r---,: r: c r ir-c r:-r - - a i - 3 - . -

rt ar

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tl

: - '2 a=?oti qdfl*: .izi.!a

i-

- E4ei4* ; r o r f . =r : : . !

z9 9== ;* i+ ! =; * i - i

!x;Ee; a = + v =_i" 76 ::qyi , .

9

e : - = Y q E = . 9 ! E - i d E i ; : : Z - . - . i i

- i o > r 5 " > r 6 - - 9 3" .9 6 i

o u t

e33E Eii4 i4n-a €E36 brgS

I= ji -= ad $. ::u Sn :u :u at: i -= . *= ,E 1a . .- ,-. .. E= ,-11=3- +. 1!: . :+: +- :- :uIEGgEdg

Ei l uzzE zzE EgX; ' L ' i"!:E i :i €a ;EEE9 9 ; ; i A ; i E Z ) ; ; Eei i i Ff r =3i ' ; s ; E 3 , r3is .a

> -

o

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U

U

€

€

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g 9

:

F

o

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a

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9 t

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A

a

C HAprl-tR6: t.lydraulic Structures 223 Walersurface(W.S.)

w.s.

FIGURE 6.I2 Typesof outletconlrol(FederalHighwayAdministration 1985).

Outlet Control Typesof outletcontrolare shorvnin Figure6.12.Flow condition(a) is the classic full-pipeflou in whichpressure flow occursthroughout the banel.In flow condition (b), the outletis submergcdbut the inlet is unsubmerged for low valuesof headwater because of the flow contraction at the inlet.The outletis unsubmereed in flow condition(c),but theculvenstill flowsfull dueto a hish headwater. ln tlow

221

E R 6 : H l d r a u l i cS t r r t c t u r e s CHAPT

partly full ncar coldition (d), the ou(let not onl)' is unsubtncrged'thc barrel flo\\ s thcoutletandpassestlrroughcriticaldepththcre.I-.inally,inflorrcondition(e).both flow that is subthe inlet and outlet are unsubtnergcdand we havc open channcl while critical on a mild slope Flow conditions(a) and (b) are USGS llow Typc:1' (e) for conditions(c) and (d) can be consideredUSGS flow Type 6 Flo$ condition uhether the on depcnding or 3' 2 Type flow USGS is cither flow op"n.h"nn.i than crilical downstrcamcontrol is critical depth(M2 profile) or a tailwatcr Ereatcr (a)' (b) (c)' conditions llow next' shown As respectivcly profile), (M or M2 I depth condition (d)' ,nl (i) ,tt can be rreatcd as full flow with sonte adjustnent for with submerged Hence,we refer to theseflow t1'peshere as OC-l for outlet control so it is clasinlet' unsubnrerged has an hand' (e), other the on condition inlet. Flow sified OC-2. Flow conditions (a)' (b)' and (c) (Type OC- 1) all are govcmed by thc cnerg)' \ \ r i l t c n f r o n t t h c h e r d u u l c rl o t h c t a i l \ \ J l c r : c(ruf,lion

Hw--7w-sor+ (r . r,. th) #

( 6 r. 5 )

= slopelL = in which IW: tail\\'aterdepthrelativeto the outlet inverti S0 culven = factor: friction culven length; K" - entranie loss coefficient;/ Darcy-Weisbach culven = and culvert cross-scctionalareai 0 R = lull-fl-ow hydraulic radius;A form in the writtcn and reananged be can equation This discharge.

u-^

2g(Hw-Iw+s/-)

( 6 .l 6 )

on therighthandsideof Equation6 16is in thenumerator The termin parentheses of theheadin elevations it is thedifference because H"u, calfedtheeiectit'e heur1, (H 6 12)can Figure in H.r based on nomographs waterandt;lwater. Outletcontrol emphabe It should 1985)Administration (Federal Highway be foundin HDS-5 6 16 6 15 and in Equations appears slope culvert why the sizedthattheonly reason of the culven the invert to relative HW, head. the of the d.finition of is because througha inlet.As long as the effectiveheadis the same,the full-flow discharge slope' the barrel of regardless the same will be culvertof specifiedlength is written in termsof ManTtre treaOloss tenn in Equation6.l6 sometimes in whichp/4R is replaced equation, ning'sequationinsteadof theDarcy-Weisbach as follows: L '4R

2gn2L t,: R4 r

(6.17)

in which n : ManninS'srt valuefor full flow, andK^ I 0 for SI unitsand l'49 for Englishunits,as in Chapter4. Typicalvaluesof Manning'sn for culvertsare shownin Table6-.4.

C l l ^ p T r ' R 6 : llydftrulic Stnlcturcs

225

1tBt_E 6-.1 Rccomnrended

I\Ianning's n values for selected conduit-s

Tl pe ofconduit

\\'all and joinl description

\lanning's a

Con.r.re pipe

CclodJoinls,\mrxnh $i{lls CoodjLrinrs. rou!h walls Poor.joints. roush!\ all\

0 . 0 1 l. 0 . 0 t 1 0 . 0 1 .01. 0 r 6 0 . 0 r 60 . 0 1 7

Con.rr'lebox

GoodJoinrs.\m(xnhijnishcdralls Poo.joinrs.rough.unlini\hed\r alls

0.012 {)) 0l5 0 . 0 r 10 . 0 1 8

('()rrusrredmelalpipesandboxes. lnnular corRrgrtions

2 i by 1 in. corrueations 6 b) I In.corru-qalions 5 b) I in. corrugarions I by I rn.conxlalions plrte 6 b) I in. slrlrc{ural plate 9 b1 l I in. structural

0.02711.021 0.0250.022 0.0260.025 0.02rJ 0.02? 0.015--().013 0.0170.033

( i r r r u - ! 3 t e d m e t a l p i p € s .h e l i c a l

2 i by j in. corrugarions. 2,1in. plite ridth

0.0t2 11.02.1

at ll in. spacing. ] by.,lin. recesses g{x loinls

0 . 0 1 20 . 0r 3

c o r r u ! a l i o n s . f u l 1c i r c u l a r f l o r Spiral rib nlelal pip€

i - a r . . e D . r a i r o m F e d e r aHl i C h w a A y d m i n i s r r a r i o( 1 n9 8 5 ) .

Valuesof the entranceloss coefficientfor outlet control are given in Table6-5. The value of K" for a squareedge in a hcadwallis 0.5, while for bevelededgesand the groove end of concretepipe culverts,K" = 0.2. On box culvertswith a square edge. a small reductionin K" to a value of 0.4 is obtainedfor wingwalls at an angle of 30'-75' from the centerlineof the banel; otherwise,wingwalls have either no effect for concretepipcs or a detrimentaleffect if constructedparallelto the sides of a box culven. The flow condition(d) in Figure 6. l2 actually requirescomputationof the subcritical flow profile from the outlet to the point where it intcrsectsthe crown of the culvert. Numerousbackwatercalculationsby the FHWA, however,led to a simpler procedure for manual calculations.A full-flow hydraulic grade line is assumedto cnd at the outlet at a point halfway betweenthe crirical depth and the crown of the culvert. Cy.+ d)/2, and is extendedto the inlet as though full flow prevailedthrough the entire length of the culven. Then the full-flow equation,Equation6.15, can be used to calculatethe head-discharge relationwith flV replaccdby (),. + d)/2. If the tailwater is higher than (,". + Al2, then the actual tailwater depth is taken as rhe value of ZW. ln computerprogramssuch as HY8 (FederalHighway Administration 1996). the water surfaceprofile for condition (d) is computed until it reachesthe crou,n of the pipe, after which full-flow calculationsare made.Thus, it is given a special lype 7 in additionto USGS Types I through 6, which are used in the program. Since it is a mixture of OC-l and OC-2, as defined here, it shouldbe given its ou n designationof OC-3 in HY8.

226

CBAPTLR 6: Ilydraulic Structures

'I'ABLE

6.5

llntrancelosscocfficienf,s: Outlct control,full or partly full entranceheadloss,where

/v,\

H ' , = K' \I 2 s il T]-peof \lructrrreand dc\ign of eotrance Pipe,concrete Projectingfrom fill, socketend (grooveend) Projecting from fill, squarecul end Head$allor headwallandwing$alls Socketendof pipe(grooveend) Square edge (radius= ll d) Rounded Mitered lo confonn to fill slope End seclionconforminglo fill slope Bevelededges,33.?'or 45' bevels Side-or slope-tapered inlet Pipe,or pipe arch.corrugatedmetal Projecting from 6ll (no headwall) Ileadwallor headwalllrndwingwalls, squareedge Miteredto conform to fill slope.plved or unpavedslope End sectionconforminB10fill slope Belelededges,33.7' or.l5" bevels Side or slope'tapered inlet Box. reinforcedconcrete (no wing' alls) lleadwallparallelto embankment Squareedgedon threeedges Roundedon tbreeedgesto radiusof .r barreldimcnsion,or b€veled edgeson threesides Wingwallsal30'-75' to barrel Squareedgedat crown Crown edgeroundedIo radiusof * barretdimension,or be!eled top edge Wingwallat l0'-25'to banel Squareedgedat crown Wingwallsparallel(extension of sides) Squareedgedat crown Side or slope-tapered inlet

Co€mcient i'.

0.2 0.5 0.2 0.5 0.2 0.'l 0.5 0.2 0.2 0.9 0.5 0.1 0.5 o.2 o.2 0.5 0.2 0.4 0.2 0.5 0.'7 0.2

Sdlr"r Data irom FederalIiiShway Administrarion( 1985).

Outletcontrolcondition(e) (TypeOC-2)in Figure6.l2 requiresthecompuration of a graduallyvariedflow profile from the outlet proceedingupstreamto the culvertixlet.This will be eitheran M2 or an Ml profile.Ar rheinlet,the velociry headandentrancelossesfrom Table6-5 are addedto tie inlet flow depthto obtain the upstream headwater, HW. Theflow profileis computed in HY8 usingthedirect stepmcthod.

C I r A Pr E R 6 : I l y d r a u l i cS t r u c t u r c s 2 2 1

Cn= krC,

3 . 10 d. 3.00 Gravel

2.90 0.'16 0.20 0.24 0.28 0.32

1.00

HWt/Ll

(a) DischargeCoeflicientfor HW./Lr> 0.15

3 . 10 3.00 Paved,,,/ 2 . 9 0/ d-2.80 2.70 2.60 2.50 0

\

090

c ravel

0.80

I

Paved

\\

0.70 0.60

/o'i*'

2.o 3.0 4.0 HWr,t\ tot HW/L,< O.15 (b)Discharge Coefficienl 1.0

0.50 0.6

0.7

0.8 0.9 ht/Hwr

'1.0

(c) SubmergenceFactor

6.13 I.'IGURD (Federal l9E5). HighwayAdministration for roadwayovertopping coefficients Discharge

Road Overtopping When the roadway overtops,the roadway embankmentbehaveslike a broadweir is writfor a broad-crested weir,asshownin Figure6.13.Theequation crested case as ten for this (6.18) Q = C,L(Hw,)t' in which Q : overtoppingdischargein cubicfeetper second;C. = weir discharge coefflcient;L = length of roadwaycrestin feet; andHW, - headon the roadway crestin feet. Figure6. l3a givesthe dischargecoefficientfor deepovertopping,and Figure 6.13b showsits value for shallowovenopping The correctionfactor tr in of the weir by the tailwaterAn iterativeprocedure Figure6.13cis for submergence

128

C r i ^ p r r R 6 : l l y t l r ' l l u lS i ct r L t c t u r c s

h a s t o b c c n r p l o y c d( o d c t e r n ) i n et h c d i v i s i o n o l t l o \ \ b c t \ \ ' e e nt h e c u l r e n l n d c r n b a n k r D eo n vt c r f l o w .D i l l c r c n th c a d u a l c rc l c v a t i o n sa r e l s s u m c du n l i l t h c s u m o l ' flow thc culvcn and crubanknrentovcrllow cquals thc spccilicd ciischargc.

Improved Inlets W h c n a c u l | c r t i s i n o u t l e tc o n t r o l .o n l y m i n i n r a li r n p r o r c n l c n t cs a n b c r n l d c 1 o incrcasethe dischargcfor a given headwalcrelcvation. Bevcling of lhc cnlrunce reduccsthc cntrancchcad loss.but the barel friction loss is likelv lo be the doninant head loss.Thc barrel friction Iosscan be rcducedb1 using culrcrts ftrbricated from materialshaving lowcr valucsof N{anning'srt. but this becomesan ecL)lonric i s s u e .O n t h e o t h e rh a n d .a c u l \ ' c r tt h l t i s i n i n l e t c o n t r o l i s a n r e n a b lteo c o n s r d e r ablc inrprorcmentin perltrrnranceby dcsign changesto thc inlct itsclf. The purposc of inrprovedinlets is flrst to reduce the Ilow cootraction.which increasesthe effectivc flow arca as well as decreasesthe head loss that occurs in severecontractions.In addition,improved inlcts can include a/n//. or depression. that increasesthe head on the throat of the barrcl. uhere thc control section is l o c a t e df,o r t h e . a m e h c r d u l t e r e l e v r l i ' r n . At the first level of inlet improvement,the inlet edges can be beveled.The dcgreeof inrprovementcan be seen in Figure 6.1,1,which is a set of inlet control

3.0 Mtitercd\// i

Thin edgeproiectinsf I

2.0

i -

'1.0

a

z

7

'1.0 2.0

z7

4

28.1{6 28.595 22.121

20000.000 55t8.52.1 240.000 .100 l.611 8781t3.{0 2-10.{X}0 7-t0.000 .30,1 .0005 I.708 .001 l bul also the physical sta_ bility linrit i'rposed br roll w.ves for numericalsrabilityio be achie,'ed.This is the reasonfbr the statcmentin sorneestablishednumerical codes using the irlplicit m!'thod that lhey do not apply to supercriticaiflorv (e.g., BRANCH). I f d i f f i c u l r i c so c c u r i n t h c a p p p l i c a t i oonf t h e p r e i s s m a n sn c h e n t ee. v c nt h o u q h the stabiliry limirs on t and the Vedernikov number are satisfied. then oth-er sourcesof thc difficulrjes rrusr be sougbt. The stability analyses.for exanrple, assumea uniform grid spacing in the flow direction,whereasthe spacing is likely to be nonuniform in applications to rivers. The irregularity of the cross_sectlon gcornetry.the occurrenceof rapidly varied flow. and the applicationof the bound_ l r l c o n d i t i o n sa l l c o u l d c o n t r i b u l er o p r o b l c n r su i t h t h e i n p l i c i r n t e t h o d ;n c v e r _ thcless.it has been widely used successfulJy in severalcslablishedcocles(UNET ( U . S .A r n t v C o r p so f E n g j n e e r s1 9 9 5 ) B . R A N C H ( S c h a f f r a n e kB. a t t z e r a . n dG o l d _ b e r g l 9 8 l ) . F L D W A V ( F r e a da n d L e w i s 1 9 9 5 ) ) .

8.6 CONIPARISON OF NUI\{ERICALMETHODS From the foregoing presentationof the numerical method of charactensticswith specifiedtime intervals(MOC-STI). severalexplicit finite differencemethods,and the implicit finite difference method (preissnann),it is apparenttbat an obvious advantageof rhe inrplicit nlethod is its uncondirionalstabilitywith no limits on the time step.In addition.the compactness of the preissmannimplicit schemern panrc_ ular allows it to be applied with spatialsrepsof variable length. As a result, the Preissrnann schentehas beconrevery popularfor applicationsin large rivcrs suchas routing of flood hydrographsor dam-breakoutflows. Reachlcngths in such appli_ cationsare variablebecauseof changesin channelgeomerryanJ roughnessin ihe flow direction.In addition.the absenceof a time steplimitation is advantageous for flood hydrographsthar have long time basesto avoid a largecomputationaltirne. Amein and Fang ( 1970) applied the box (preissmann)implicit schemeto rhe routing of a flood on the Neuse River from Goldsboroto Kinston, North Carolina, which is a river reach having a length of 72 km. The upstreamboundary condition was specifiedto be the measuredstagehydrograph,while the downstreamcondition was the measuredrating curve. lnitial conditionswere determinedfrom back_ water calculations.staning with the measureddownstreamdepth. For comparison of the methodof characteristics(MOC), explicit, and implicit methods,a compos_ ite channelcrosssection was assumed,with geometricpropertiesdeterminedas an averageover the entirereach.The computedresultsfor all threemethodswere comparedwith thc measuredstagehydrographat Kinston for two different floods over a tinre period of about l5 to 20 days.The resultsshowedsimilar accuracyin com_ parison with the obsen ed hydrographs,but the implicit method was much more effrcient.The explicit method requireda time stepof 0.025 hr for a subreachlength of 2.4 or 4.8 km ( I .5 or 3.0 mi) to maintainstability.Time srepsof as large as 20 hr

310

C H A p T E R8 : N u n t e r i c S a lo l u t i oonf l h e U n s t e . d F y l o \ rE q u a t j o n s

were possiblefor the implicit nrethodwith a subreachlengrh of :1.8knr, althougha sonrewhatshoner tinre siep might be desirableif ntore rapid changesare taking place in stageor discharge.For the samesubreachlength of -1.8km and a time step of 5 hr in the implicit method,the computcrtime \.\'asmore than four tinresgrcater for the explicit ntethodrhan for the implicit method. Price ( 197.1)conrparedthe N{OC, cxplicit. and implicir mcthodsfor a nronoclinal wave, ll hich is a translatorywave similar to the front of a flood wave in very long channels.It approachesa constantdepth very far upstreamand a smalierconstantdepth downstreamwith a wave profile in betweenthat does not changeshape as it travelsdorvnstreamat a constantwave speed,c,,.The monoclinalu'aveis a stable, progressivewave fornt that resultsafter long tintes !\hen an initial constant depth is increascdabruptly to a largerconstantvalueat the upstreamend of a riyer reach.If the *ave profile is gradually varied in a witle pri,marrc channel,fiere is an analytical solution for the profile (Henderson1966;. The monoclinal wave is useful for numcrical comparisonsbecauseit retainsthc nonlinear inertial ternrsin the full dynamic equationswhile having an analyticalsolution. lt has a nra.rinrum speedof (V * c) and a mininrum speedequal to that of the ..kinematic"wave,discussedin the next chapter,for which the inenial terns and the d_trldr term are small in comparisonto the bed slopein the momentumequation.Of interestin this chapter, however,is the comparisonmadeby Price betweenspecific numericalsolution techniquesand the analytical solution for the monoclinal wave. He selectedan upstreamdepth of 8.0 m (26.2 fr), a downstreamdeprhof 3.0 m (9.8 fr), anclchannel slopesof0.00l and 0.00025over a roralreachlength of 100 km (62 mi) having a Chezy C of 30 mr/2/s.Thesedataresultedin monoclinalwave speedsof 3.31 m,/s (10.9 ftls) and 1.65 m/s (5.,11frls) for a very uide channel*iLh the slopesof 0.001 and 0.00025.re\pectively. Price cornpared two explicit techniques (Lax-Wendroff and the leapfrog schcme),the method of characreristics, and the implicit schemewith the analytical solution of the monoclinal wave. Price found that the expljcit and methodof characteristicstechniqueshad the leasterror when At/At u,as approximatelyequal to the maximum Courant celerity, y + c; that is, a Courant number equal to l. The implicit method exhibitedthe smallesterror for Ar/At approximatelyequal to the monoclinalwave celerity.This resultedin a largerpossiblerime stepfor the implicit method than for any of the other methods and so greater computational efhciency. Furthermore,Price determinedthat the error in the implicit method is much less sensitiveto changesin Al for a fixed value of Ar.

8.7 SHOCKS In the hydraulics of unsteady open channel flow, shocks are the same as moving surgesat which there is a discontinuity in depth and velocity. ln the method of char_ acteristics, the shock conesponds to an intersection of converging positive characteristics at which the methods of gradually varied flow no longer are applicable becauseof strong vertical accelerationsand a pressuredistribution that no longer is hydrostatic at $e shock itself. Across the shock, both mass and the momentum func-

C H { p T E R8 : N u r n e r i c S a lo l u t i o n o f t h U e n s t c a dFyl o wE q u a t i o n s 3 2 1 tron nust be consened,as discussedin Chaptcr3. On eitherside of the shock,grad ually variedunstcadyflow usuallyexist-sand can be treittedusing any of the nunrer_ ical ntc(hodsin this chapter The difficulry then is in compurir; rhe disconrinuity causcdby the shockitself.This importanrproblem ariscsin'dam_irear wave fronts, ralld opcrationof gatesin canal systems.and transientsin the headraceor tarlrace of a hydrocicctricplant that occur upon rapid stanupor shutdoq,n of the turbines. Thereare two methodsof solving the problcm of shockconlputation: shock fit_ ting and shockcapruring,also known as .,con]pulingthrough.,, In the first method, the positionof the shockfront at time I - Jl is computeduiing the methodof characteristicscombinedwith the shock compatibility equations,ivhich srrnptyare the continuityand monientumequationswritten acrossthe shock or surgeas gtven previously by Equations3.12 and 3.13. Six unknowns are found at r i Ar:1fre Oeptir and velocity.at.theback of the surge,r, and V,; cleprhand velocity at the front of the surge.r'. and %; the speedofthe surge.l/.; and the positionofthe surger,r.,. How_ ever.only thrce equationsare given by rhe two shock conrparibitity equahonsand thc ordinarydifferentialequationfor the parh of lhe shock,V = d,"/dt.-Fo., .rrg. advancing in the positive r direction, two forward characteristics and one backward characteristic can be sketchedfrom the unknown positionand time at point p in the x-l plane backwardto time ievel ,tAr, as shown in Figure g.g. Each of rhesecharactensticshas two equationsassociated* ith it, as OescriUea in Chapter7, and three more unknown valuesare introducedas the r positionsof the intersections of these characteristics with the krown time line. In all, a total of nine equattons can be solvedfor nine unknownvaluesto obtain not only the new position of the shockbut also the depth and velocity on both sides of the shock. These latter variables then can be used.as intemal boundaryconditionsro solve the SainFVenant equationsfor the gradually varied flow regions both upstream and downskeam of the ihocx.

( k+ 1 ) d t

FIGURE 8.8 Shockfitting usingcharacteristics (Lai l9g6). (Source:Figurefrom ,,Numcica!Modelingof Unstead,v OpenChannelFlow,,by Chinrutai in ADVANCESIN HyDROSCIENCE,Volune 11, copyright@ 1986by Acatlemicpress,reproducedb,-permission ofthe pubtisher)

322 CHApTER 8: Numerical Solurion ofrheUnsteady FIorrEquations In the secondmcthodof compuringshocks(shockeapturing), the numerical solutionprocedure for rhe Saint_Vcnanr equutions is simplycorniuredthroughthe surgewith no specialtreatntentof rhe discontinuity. IfArriinana D.Fa.io 0909) appliedthecquivalent of the Lax diffusiveschemeon a ,taggered grid to the prob_ Icmofhydroelectric loadrejectionin thc headrace dueto slirit.rown"of turbinesand showedgoodagreement with measured watersurfaceprolilesof an uDdularsurge. Manin andZovne(197l) usedthc nterhodto showreasonable agreement bet\\,een computed for the propagation of shocks due to an iisrantaneous _solutions dam breakin a horizontal frictionless channelwith theanalyticalsolutionof Stoker,dis_ cussedpreviously in Chapter7. TerzidisandStrelkoff( 1970)demonstrated the use of the Lax diffusivescherneand Lax_Wendroff schemein computingthroughthe propagation of a shockwavein nonuniformflow.NumericalOissipatLn cauiedby thenunterical methoditselftendsto smoorhrbeabruprdiscontiuuiiy in theLax dii_ fusivescheme,whjle anificialdissiprtionmay be requiredfor rh! Lax_Wendroff schemeto smoothoscillationsbehindthe shock,althoughTerzidis and Strelkoff achievcd simply using a time step equal to eight_tenths the value ^similar.results requiredfor srability. On theorherhand,useof nondiisipative ;ethods suchas rhe leapfrogschemercquiresan anificialviscosityto oampenthe osciltations. In the Preissmann method,takingthe weightingfacto;g > 0.j introduces Oissipation rhat may avoidoscillations on the backof the shockresultingfrom hydroelectric load re.lection in a turbineheadrace; however, a valueof0 : I ifully im;licit) maycause excessive damping.Wylie andStreeter(197g)showedthata valueof g _ (i.O p.o_ ducedgoodagreement betweenrheimpricitmethodandthemethodof characterisf:i ,h. hydroelecrric loadrejectionproblem.For very abruptshockssuch as lj.r tnosethatoccurdownstream of a very large,rapiddam breakandfor transcntical flow, the Preissmann methodno longermiy be useful,andexplicitschemes have beendevclopedfor this case,as dcscribedpreviously(Fennema and Chaudhry 1987;Jha,Akiyama,andUra 1995:Meselhe, Sotiropouios, andHolly 1997).U,hile not asimponanlin lhe graduallyvariedflou regions.ir is imperative rhar !e ::m1y rne govemlngequatrons be wrinenin conservation form for computingthroughthe shockto conserve themomentumfunctionandmassflux. ExAMpLE 8.2. A hydroelecrric rurbine decreases its loadlinearlyfroml0OOcfs 3.r,/:l^ g zerodischarge in t0 sec. The t "uOru". .t -n.i i, irJpezoiaatwirh a 128 length of 5000fr (1520m),a bonomwidrhof 20fr (6.1m),sideslopelof 1.5:1, Man_ a bedrtopeof 0.0002. compure rhedeprh hyjrographr ar.r/L _ l,:8: I ^ 9 ?ll. :ld u.v, rr.z.u.4.u.o,u.6.and LU ustngthe methodof characteristjcs with specifiedtime Intervalsandthe Lax diffusivescneme.

Solalron. The channellength is dividedinto 50 spatiatinrervals and rhe trme step is selectedso that the Courantnumberis S I for all grid nodesat the currenltime level, as in Example8.l. The upstreamboundaryis a re-seruoir, as in Exanrpleg.l, and the downstreamboundaryconditionis a dischargehydrographu ith a Iinear decreasein turbine dischargefrom the steady_srate ualu" oi IOOO "is ()g.3 mr/s) lo zero in l0 sec.Ar time t : 0, the waterin the headmceis in steadyuniform flow with a normaldepth of 1.66ft (2.33m) anda criticatdepthof 3.85ft (i.tZ mt. resultsareshownin Figure g.9afor the Lax diffusivemethod, andFigureg.9b - .The for the methodof characteristics. An abruprincreasein depthat theiurbrneis tottou.ea

10

8

6 0 is equivalentto developingan inequalitl such that the numeratorof Co in (9.1 I ) remainsnonnegative,so that the following limit on A/ must be satisfied:

Lt > 20x

( 91 . s)

If 0 is the wave travel time dcfined by Ar/V", where V" is a representativevalue ofthe wave travel speed,then (9.15) can be viewed as a limit on ,l.r for a value of Al determinedby the required discretizationof the time of rise of the inflow hydrograph:

v^t 2X

( 9 .l 6 )

Cll,\prER 9: SimplifiedMethodsofFlow Routing

341

Weinrnannand Laurenson (1979) suggesta less scvereliIrrit, in which A/ on the r i g h t h a n d s i d e ot h f e i n c q u a l i t y i n ( 9 . 1 6 ) i s r c p l a c c d b-yrfi s e t i m e o f t h ei n f l o w hydrograph.Cunge ( i 969) showsfrom r rrrhrjiry anal)\i\ rhrr the condition for stability is X < 0.5 and funher suggeststhat X > 0 for thc physical interpretationof wedgeand prism sroragero make sense.However,ponce and Theurer (19g2) argue that negativevalues of X arc possible.This is discussedlater in more detail for an exlensionof the Muskingunr mcthod called thc M u.skingun-Cungeor Muskingum dilfusion schene. While the Muskingunr method appearscomplcte, it dcpends strongly on the parameters0 and X. In peneral,thesearc takento be constantfor a given nver reach, and the original method of estinating them rcquiresmeasuredvaluesof inflow and outflow for the river reach under consideration.Becausethey essentiallyare calibration constantswhen deterntinedin this way, thcre is no assurancethat thev will havc the santevalues for a flood differentfrom the calibrationflood. If it is assurnedrhat the complele inflow and outflow hydrographshave been measuredfor a given river reach,then the cumulativestoragecan be computedfrom a rearrangement of Equation 9.7 as

s , - s ,+ ! r ' r t . , t , o . o , )

(9.17)

Rcpeatedapplicationof (9. l7) for successivevaluesof time allows the determina_ tion of the cumulatire storage,S, at any time r. The initial value of storageis usu_ ally taken as zero. Then, accordingto Equation 9.6, we seek a linear relationship betweenrelativestorage,S, and rhe weightedflow value ))O}, whicir {X1 + (l also can be computed from the inflow and outflow hydrographsas a function of time. However, river relationshipsgencrally display .orni d"g... of hysteresis becauseof greaterstoragcon the rising side of the hydrograph lBras 1990).Thus, as a practicalmatter. the valuc of X that producesthe best single_\,alued relation_ ship, or narrowestloop, is detcrmincdby trial and enor, and thi slope of the best_ fit straight line gives the value of 0 as requiretlby Equation9.6 an; illustrated in Figure 9.4 for Example 9.2. As an alternativeto the graphicalmethodfor estimating0 and X shown in Fig_ ure 9.4, theseparameterscan be determinedby a least_squares parameterestima_ tion technique.Singh and McCann (1980) show that the least_squares method of minimizing the differencebetweenobservedand estimatedstorageis equivalent to maximizing the correlarion coefficientbetween .t and the weighted flow in the graphical method. The Jeasrsquares techniqueseeksto minimiie rhe error func_ tion. E. siven bv

l A t ) + B O ) + S -, S , ] ,

( er.8 )

in whichA : 0X: B : 0(l !; S, : inirialsrorage; andSr = observedretarive storf,ge at thejrh time srep.Gill ( 1977)proposed sucha tech;ique,andthe values ofA andI aregivenby Aldama(1990).The summarion takespiacewirhj running

3.11 C H A p r E R 9 : S i m p l i f i e dM e t h o d so f F l o w R o u l i n g

A

x 2 0 I

+

-€

x

5.0E+05

X=0.1

-'c

X = O.25

' +-

X=O.4

1.5E+06

1.0E+06

2.0E+06

Slorage,S, m3 FIGURE 9.4 0 andX, Example9 2 Muskingunr Graphicalmerhodfor determining

from I to N observedvaluesof inflorv,outflow, and relativestoragefor a given river reach.The error.E, is minimized by differentiatingwith respectto A and B and setgive the ting the results to z-ero.The rcsulting equalionsare solvcd for A and B to

followingexpressions:

A=

(9.19) rroto >r- : / i ) o ; ) ) l :t() + ( N : O ; - ( : O , ) t>) 4 S i+ ( : I r >O t- N > t t o t ) > o ' s , )

( 92 0 ) t r >t t o t- : / ; : o , ): s , a t(: + (>rj2oj N>rroj):rt + (N:/,r (:/,)'):qs,l (9.2 |) + 2> tr> ot: ItO) c - N f> r: >o; , 12r,o,)1.t - ( : 4 ) : > o ; - : / j :( : o r ) l

8:

Once A ald B are computed'0 and X can be determinedfiom

0=A+8.

X=--

A

A + I J

\9.22\

is an exiensionof the Muskingummethod' technique The Muskingum-Cunge propandgeomctrie in which the valuesof 0 andX canbe relatedto the discharge

C H A f T E R9 : S i D r p l i f i eMde r h o dosf F l o wR o u r i n g 3 4 3 ertiesof rhc channcl.To achicvea betrerundcrsrandingof how this is accomplished, -are the kinernaticwave and dillusion routing tcchniques considerednext. EXANtpLE 9.2. Utilizethcobsen.ed inflowsandoulflo*,sfor a riverreachglvenin Table9-3 (Hjclmfeltand Cassidy1975)ro ob(ainraluesof d andX using both rhe graphical methodandrheleast-squares nrcthod.Thendcrcrmine theroutingcoefficienrs androutetheinflow hydrograph throughthe riler reach. Solrttion. 'lhe slorageis calculatedfrom lhe avcrageinllow andoufflow raresover a singlctime stepusingEquation9.17and accunulatcd beginning wirh zeroinitialstor, age as shownin Table9-3. Then variousvaluesof X are substituted to obtainthe weightedinflowandourflowquanrily.X1 + 0 - X)O.ar rhecndofeachrime srep.The srorage.is relaledto rhis quanrityby F4ualion9.6. so the plor shownin Figure 9.4 allowsthe determination of the inverseslope,rvhichis equalto the Muskingumtime constant, d. Figure9.,1showstheresultsfor valuesof X _ 0.10,0.25.and0.40.By trial anderor, the narrowest loopoccursfor X approximately cqualto 0.25wlth Inferseclion of the risingand faltinglinrbsaboutmidwayalongthe storage axis.The best_fit Iineof rhedarafor X : 0.25givesa valueof 0 : 0.92daysfrom igure 9.4,anclthese arerheresulrsfor rhegraphicalmethod.Alrernatively, Equations 9.l6 rhrough9.22can besolvedfor thedaragivenin Table9-3 ro producerhev;luesX = 0.243and : 0 O.g97 days.Thesela(er valuesarechosenas the Muskingumparameters, with .1, : 0.5 days to calculatethe Muskingumroutingcoefficientsfrom Equations 9.ll through9.13, with theresult C6 = 0.034,

C' : 0.504,

C.t: 0.462

For thiscase,Al > 2dX so thal Co > 0. Finally.lhe solutionof the rouung equauon, Equation9.10,canproceedasshownin Table9-.1.Theoutflowinjtially ls assumed to TABLE 9..] Computatiortof storageand Muskingum parametersof Example9.2

xr+(l-x)'o 0.0 0.5 1.0 t.5 2.0 3.0 4.0 4.5 5.0 f.)

6.0 6.5 7.0

2.2 28.4 ll.8 29.'7 25.3 20.4 16.3 t2.6 9.3 5.0 ,1.I 3.6

2.0 '7.0 Il.? t6.5 21.0 29.1 28.4 23.8 t9.4 l5.l .2 8.2 o.4 5.2

8..1 2l.5 30.1 30.8 2'7.5 22.9 lE.,{ r.1.5 I1.0 8.0 5.9 4.6 3.9 3.0

.1.5 9..1 14.I 20.3 26 6 28.E 26.1 21.6 t'7.1 t3.3 9.1 7.3 5.8 4.9

0.00E+00 L66E+05 +05 6.89E 1.388+06 I.838+06 1.87E+06 I.62E+06 1.29E+06 9.76E+05 7.008- 05 .1.73 E+ 05 3.0?E+ 05 t.888+05 |.0.18+05 2.l6E + 0,1

2.0 7.8 13.4 t 8.0 24.6 28.7 27.6 2 3|. 18.7 11.7 10.8 7.9 6.2 5.0 4.4

2.1 8.9 15.9 20.3 25.1 28.2 76.1 2t.9 t'7.7 t3.8 l0.l 7.4 5.8 4.8 4.1

2.1 10.0 18.4 22.6 26.3 27.6 25.2 20.8 t6.7 12.9 9.4 6.9 ).J 4.6

3.1

Cuaptrn

314

9 : S i m p l i f i e d\ l e t h o d s o f F l o w R o u t i n g

TABLE 9-4

l\tuskingumrouting (C0= 0.03J.Cr = 0'504'C: = 0'462)of Exanrple9-2 Ctxor

Crx lr

Cox lz

1, nr_r/s

t, days

2.2

0.0 0.5 1.0 1.5 2.0

28..1 31.8 29.1 25.3 20.4 r6.3 12.6 9.1 6.7 5.0 4.1 1.6 2.4

3.0 3.5 4.0

,r.5 5.0 5.5 6.0 6.5 ?.0

L0l l.ll

Ll I 7 . ll t.1.12 16.03 l,1.9rl t2.'/6 10.29 6.22 6.35 .1.69 3.38 1.52 2.01 1.82

0.50 0.97 r .09 r02 0.86 0.70 056 u.,13 0.32 0:3 0.1? 0.l,l 0.12 0.08

.r lE 9.ll

r:.09 lt89 12.16

r0.62 E.89 7 .1 8 5.59 3 . 17 2..18

0r.mr/s 2.2 :.62 9.19 19.19 16.lE 17.93 :6.1,1 ll.00 19.2'7 I5.56 1 2 1. 0 9.14 6.88 5.17 1 . 37

35 outflow

30 25 lQ 20 E

o ts 10 5

X = O.243 1'= 0.897days

,\-Observed

\i-

I I

I

Calctrlatedoutllow

\ o

/

\

/

l

(

\ ooJ"*"o,n,lo*

l"t ,.J

\

o\

D-.-]-!

0 l, days FIGURE 9.5 9 2. for Muskingumrouling,Exarnple lnflow andoutflowhydrographs from one time stePto the next. The be equalto the inflow, and the routing progresses andobsenedoutflowscanbe resultsare shownin Figure9.5 in whichthecalculated 4 percentof the observcd peak is within approximately compared.The routedoutflow generally fit of the dataon the falling is a betler peak. There to same time valueat the sideof the hydrographthan on the rising side.

C H A t , t € R 9 : S i m p l i f i e dl \ , l c r h o dosf F l o * R o u r i n g

1.15

9.3 KINEN{ATICWAVE ROUTING As shoun in Equation9.3. the monrenlurncquationis simplilied in kinentatrcwave routing by neglectingboth thc inertia terms and the pressuregraclienrterm, so tbat it becomes

S,, .- S,'

(9.23 )

Equation9.23 is incorporatedinto the continuity cquationgivcn by (9.2) to obtain the kinematicwave equation.One interprelationof (9.23) is rhat unifornr f.lorvcan be assumedin a quasistcadyfashionfrom one time stepto thc oext orer each reach length in a finile difference numerical solurion of thc continuity equarion.Thus, Equation 9.23 is cquivalent to exprcssingthe dischargee in ternts of a uniform flou fornrulasuch as Manning's equation,which cln bc rearrangedas

(9.21) in which bo = constantfor a wide channelof constantslope and rougnncss:A = cross-sectional flow area; and exponenta - 5/1. Under theseconditions. the dis_ charge0 is a funclion of cross-sectional areaA alone so that

do i-abaA"'=av

(e.2s)

in which V : Q/A : mean flow velocity.The significanceof (9.25) cln be seenby writing the continuity equationgiven by (9.2) as

aQ d.r

aQ a.\

dA at

/,r,A\ tQ 0 \Oe/ u,-

(9.26)

uhich c;rnbe rearrangcdIn the form

ao

/,to\ ho

dr

\d,4 / it

(9.27)

We assumea uniquerelationship bctweenstage(or area)and discharge,so that dQld4 is an ordinaryderir,ative. The physicalmeaningof deld,.1can be shownbv settinSrhetotaldifferentialof Q to zeroresultingin

d- oa: ! 9t

do

dt+.-dr=0

(e.28)

(9.27)and (9.28),ir is obviousrharde/dA = dx/dr,u,hichcan be On c,rmparing inteipreted asan absolutekinematicwavecelerity,c,, in the kinematrcwaveequaticn. whichnow canbe rvrittenas dQ dQ aQ -=-+c,-=(, dI

A/

Ax

(9.29)

3 , 1 6 C H A p T E R9 : S i m p l i l i eld\ ' l e l h o dosf F l o NR o u l i n g From the fbregoing. rlc can conclude that Equation 9.29 has a single lamily olc h a r a c t e r i s t i casl o n g $ h i c h t h e d i s c h a r g cQ : c o n s t a nitn t h e . r l p l a n e* ' i t h a p o s itive slope given b) the kinenraticuave cclerity,r'^.The characteristicsarc straight lines becauscof the a\sumptionof a uniquc dcpth-dischargerelationshipso that a given characteristichas a conslantvalueof dischargeand depth and. therefore.constant wa!e celerity. In tcrms of our discussionof characteristicsin Chapter7, an observermoving at the spcedc/,along a particularcharacteristicpath would seeno change in the dischargeQ associatcdwith that characteristic.In othcr words, the partial differential equation (9.29) can be expressedin characteristicforrn by the following pair of equations:

0 = constant dr

(e.30) ( 9 . 3 )1

d,l

in which the constant and thc kincmatic wave cclcrity in gcneral are different for eachcharacteristic.Furthermore,frorn (9.25),(9.27),and (9.29). the absolutekinematic wave celeritl, cu,can be expressedas

-'

dQ d4

I d o 8 d y

(e.32)

in which B = water surfacetop width. Equation9.32 statesthat the kinematicwave celerity can be determinedfrom the inverseof the flow top width times the slopeof relationship,which was shown to be equi\alent to a constanl the stage-discharge times the mean velocity.The constantd - ;, using Manning s equation.and, = l, from Chezy'sequation.Implicit in this estimateof the wavecelerity is the existence of a single-valued,stage-discharge relationship. In general.r:j,u ould be expectedto vary with depth and thereforewith Q: however, a simplification often is possible,in which c* is assumedto be constantand equal to a referencevalue correspondingeither to the peak O or an averageQ for the inflow hydrograph.Under theseconditions,the characteristicsbecomestraight, parallel lines, as shown in Figure 9.6. Along thesecharacteristics.a specificvalue of 0 (or depth) is translatedat the constantspeedof cr. Therefore,the kinematic wave equationfor constantwave celerity is line:[ with an analytical solution representedby a pure translationof the inflow hydrograph. When allowed to be rariable,it is clear that c* will increasewith increasingQ and also with increasingdepth,,I. Therefore,higher dischargeswill move downstreamat a higher speed,resultingin a steepeningof the wave front, or leadingedge of the kinematicu ar e. (The rising limb of the hydrographalso s ill steepen.)However, attenuationor subsidenceof the kinenatic wave still will not occur, because of the omission of the prcssuregradientand inenia terms in the momentumequa"dynamic" wave. tion, which are imponant in a Whilc it is well knou'n that river floods usually subsideor attenuate,the question remainsas to the condit;onsfor which the kinematic wave method is applicable, sinceit does not allou for subsidence.If the momentumequationis solvedfor Sr, the following results:

C H A r , r ' E9 R: S i m p l i f i e ldu e t h o dosf F l o wR o u r i n g 347

___-_l I I

FIGURE 9.6 Puretranslation of thc linearkinefialicwave.

5 r - S o-

t

a-r

t'dv

l a v

.8 d.r

8 a l

(e.33)

What is requiredthen is to detcrminethe relative nagnitude of the bed slope,So,in comparisonto rhe remainingthreedynamic terms on the right hand side of (9.33). From an orderof magnitudeanalysis,Henderson( 1966)concludedthat S^ is much larger than the remainingterms fcrrfloods in sreeprivers; while for very iat riuers u ith low Froudenumbers.Soand rlry'd-r are of the sameorder and the inertia tenns are negligible.Equation9.33 can be nondirncnsionalizedin ternrsof a sreaoy-srare unifornt flow I'elocity.Vo;a concspondingnormal depth.I.n:and a referencechan_ n e l r e a c hl e n g t h .L o . N o n d i m c n s i o n a l i z i nagn d d i v i d i n gt h r o u g hb y S or e s u l t si n a dinrensionless equationgivcn bl

s/ So

[ .ti, lar" I 1 1 - 6 516n , r "

I v:" )/ iiv" ay'\ I t v" - +

I s 1 - , , SI ,\ ,

,1r"

.'r )

(e34)

in which)r''= ,r/,r'o; ,l = _r/Lct:V. = V/Vu:andt, _ tV,,/Lo.The dimensionless numbers multiplyingthe inertiatcrmsand pressurcforce terms,rcspectively, can be transformed as

v; _ Fi).n= 1 8Lo5o

LoS,,

I ,\0 1-uSo ,tF;

k

(9.35)

(e.36)

in which Fo = Vol(g.ro)l':: a reft'renceFroude number and k : a klnematic flow number defined by Woolhiser and Liggetr (1967). For large values of &, rhe dynamic terms are small relativero the bed slope. If / 10, the kinemarrcwave approximationis consideredsatisfactory,especially for overlandflow for which /t

118

d e t h o do r f Flo* Rouring C r { A p r E R9 : S i m p l i f i eM

can have valucsin excessof 1000 (\\r-rclhisc'rand I_igrert 1967).Also, Millcr and C u n g c( 1 9 7 5 )s u g g e s r ctdh a tr h e F r o u d cn u r n b c rs h o u l db c l e s st h a nI f o r t h e k i n e nratic $avc cquationto apply, not onl\ becauseol thc forntltion o[ roll waves for largervaluesbut also becausethis is the Iimit at \\ hich thc kjnenlltic wavc celcritv becomcsequal to the dynamic wave celerity,as sho$n by Equltion 9..18luter.Also uscful to note is that, from thc rario of thc coefllcienlsin 19.j5) and (9.j6). the iner, tia termsapproachthc sameordcr as the pressurclbrcc tcrur as thc Froudc nunlber approachesunity. P o n c e ,L i . a n d S i n o n s ( 1 9 7 8 )a p p l i c da s i n u s o i d apl e u r b t r i o n1 0 a u n i F o r m flow and cxanrincdthe arrenuationfacror to derernlinerhe applicabilityollhe kinematic wave nodcl. Ponce( 1989)suggcsledfrom the resultsthat the kinetnaticware m o d e li s a n n l i c a b l e if

1,11,-s! > 8s

( 9. 3 7)

in which Vo and,r,urepresentaveragerclocity and flou depth. respectivcly;So : bed slope;and Z : time of rise of the inflow hldrograph.This criterion indicates that both steepslopesand long hydrographrise tinrestend to favor the use of kinemalic wave routing in which inertia and pressure-qradienttcrms can bc neglecled in comparisonto the resulting quasisreadybalancebctween gravity and friction terms in the momentuntequalion. In spiteof theselimitationson the useof the kinematicrvaveapproximation,ir can be shown that both kinen'taticwave and dynanticuave behavioroccur ln a nver flood wave (Ferrick and Goodman 1998).Hendcrson( | 966) arguedthat rhc bulk of a nalural flood wave of small height nrotes at the kinematic$ave speed,c*. while the leading dge of the wave cxperiencesdl,nanricet'lecrsand rapid subsidence. Becauseth! r.inentaticwave moves in the downstrelm dircctiononly, its specd.r,,. can be comlr.rredwith the downslreamspecdof the dynlrnic uavc, *,hich has been given previouslyas V * c, by taking rhe ratio of thc t$o wavc speeds: C t _

V + c

aF

F + t

( 9 . 38)

in which F = Froude number - Vlc and c: (gr.)r/r.It can be shown from (9.3g) for q = 3/2(Chezy) that c( < (y + t.) so long as F < 2 and attenuationof rhe "dynamic forerunner" will result (Hendcrson 1966). lt $'ould seen then that the kinematicwave movesdownstreammore slowly thanthe dynamic wave fbrerunner unlessF > 2. at which time it will steeDento form a surce. T h e q u e s t i o na r i s e sa s t o $ h c t h e rt h c \ l e e p e ni n s o f t h ; L i n c n t l t i cw a v ec a n l e a d to some stabletbrm befbreactuallybecorring an abruptsurgeas the dynamic tcrms rn the momentum equationbecome imponant. This leadsto thc conceptof a uni_ formly progressivewave called the nonoL.linal rislrrg unr.e shown in Figure 9.7. The monoclinalwave can be conceptualizedas the result o[ an abrupt increasein dischargeat the upstreamend of a ven, long prisntaticchannel.Vcry far upstream. the flow is uniform with a depth of ,r.,and velocitv. Vr. uhile very far oownstream the flow is uniform with dcpth r,, such lhll \, < ! < r,. whcrc \.. = critical depth

C H . \ p r t , R 9 : S i n p l i l i c d l r { c ' t h t t so f F l o w R o u t i n g

l:19

vr ----'--+

(a) lMonoclinal WaveDefinitionSketch

(b) [.4onoclinal WaveCelerity FIGURE 9.7 Monoclinalwavedefinitionandwavecelerityfrom discharse-aJea cune.

for the moving wave profile. A stationaryobserverwould see the depth gradually increasefrom the initial uniform flow value,)i, to the final value,,yr,as the wave profile movesdownstreamwith time. The slopeof the profile is relativelygentle so that it cannotbe considereda surge,but bccauseit doesnot changelorm as it moves downstream,it can be treated using the sarnemethodology as for surges.For a monoclinalwave moving downstreamat a constantabsolutespeed.c., the problem can be made stationaryby superimposinga speedof c. in the upstreamdirection. Tbe continuity equation then is appliedat points i and/along the wave profile, as shown in Figure 9.7a, to yield

( c ^- v ) A 1 = G ^

v)A,: Q,

(e.3e)

in which p, is referred to as the ousrrar dischargerhat is seen by a moving observerwith the speedc,. If Equation9.39 is solvedfor the monoclinalwave spced,theresultis the so-called Kleitz-Seddon principle(Chow 1959),givenby

o,- o. At- A'

(9.40)

350

C r l A P r t ' R9 : S i n r p l i t i c\d4 e t h o dosf F I o \ \R o u l i n 3

r,r'hichcan bc illustrated,as in Figure9.7b. hy rhe slopeol thc \lrright linc bcl\\cen pointsP, and Pr. The curve in Figure9.7b is shownconcarc ul. bccausethc vcl(xity generallyincreascswith stageand llow arealbr flo*'in the nlain channelalone.\\'c can deducefrom thc ligure that c,,,is greaterthan the l'lo$' \ clocit)' at either Point i or/and that the max imunt valueof c,,,occursas the depth.I,. approachesr;. For the specialcaseof a \ei-y widc channel.Iiquation9.'10bccones V1y1- V,.v,

( 9 . . 1)1

With the aid of the Chezy equationfor a vcry u idc channel.the ratio of the monoclinal wave cclerity. c,,,.to the kinertatic wave ce)erityof the initial unilomr flow, cri, can be detcrmined from Equation9..11as

(rr)"- ,

c,,

2 \ r',,/

c*i

3

(9.42)

,)i ,)i

in which c1,has been determinedfrom the Chezy equation and Equation 9.32 as (3/2)yii that is, a - 3/2.It is clear from (9.'12)that the monoclinal wave celerity is greaterthan the initial kinematic wave celcrily as well as the initial unifonn flow and yr. As )i velocity, y,, and it dependsonly on the specifiedratio of depths .r7/,rr approaches -vi, we can see from (9.42) that the monoclinal wave celerily as a lorverlimit, the kinematicwavecelerity of the initial rrniformflow. approaches, Determinationof thc shapeof the monoclinal wave Profile requiresthe use of the momentum equationapp)iedfrom thc viewpoint of a moving observerwith the constantabsolutespeed,cn, who seesa steady,graduallyvaried flow profile. Under thesecircumstances,the equationof graduallyvaried flow for a very wide channel with Chezy friction becomcs (l-

dt_ dr

(9.43) q:

l - -

gI'

in which 1 = depth;x - distancealong the channel;17- flow rate per unit of $'idth - y-y: C = Chezy resistancecoefficient;and 4. = ovcrrun dischargeper unit of width : (c. - D-r. Note that the evaluationof the friction slope dependson the absolutevelocity and discharge,while the Froude number squaredterm 4:/(8)r). which comes from convectiveinertia, is basedon the relatiYeYelocityand overrun dischargeas seenby the moving observerTherefore,the critical depththat defines rhe limit of stability of the monoclinalwave is given by 1. : lqlle)t13,conesponding to the overrun Froude nunber having a value of unity. As ,r'rapproachesl'.' the denominatorof (9.,13)approacheszero and the slopeof the prohle becornesinfinite with the formation of a surse.

C H A p I E R9 : S i m p l i f r c ldv l e l h o dosf F l o wR o u r i r ) g 3 5 1 T h e s u r g et h a l f b n n s a t t h e s t a b i l i t yl i o l i l o f t h e n r o n o c l i n aw l a r e d e f i n c sa mlinrum celerit] that is rcachcd.If thc continuitl, definition of ovcrrun discharge fr..nr Equation9.-19is sinrplifiedfor a rery uide channcl and set equal ro the limitinc value for thc orenun Froude nunrberequal to I uith \,, = 1.,,the result is

- r4i s,: k. r1)r,

(e..14)

Equation 9.4'1can be solvedto obtain the marintum lalue of c,,,= y, + c,, whcre 6, : 1gr';)l/:.In other words. the monoclinal u ave has a matinrum celerity corresponding to the dynamic wavc celerity of the initial unifomr flow, while it has a minimum celerity equal to the kinematic war e celeriry of the initial uniform flow, as shown by Equation9.42.This can be placed in dimcnsionlessform (Fcrrick and Ger,,rdman 1998)and cxorcsscdas

1 g ! 5 1 ( ' _ l ) cr,

- 1\

F,/

(9..1s)

in ri hich F, : Froudenumberof initial uniform flow, and the kinematicwavecelerity hasbeentakenas (3/2)y, from the Chezy equation.The sameconclusionfor the upper limit was reachedpreviously when comparing the kinematic wave celerity and dynamic wave celerity in Equation9.38. Hence, lor a given Froude number of the initial uniform flow, there is a maximurn celerity for the monoclinal waye that decreasesas the Froudenurnbcrincreasesto a value of 2, at which time the upper and lower limits both collapseto the kinematic wave celerity.Settingthe right hand side of Equation9.,12equal to the upper limit giycn by (9..15),rhe stabilirylimir on the initial Froude numbercan be defined in ternts of the ratio of the final and initiol nomal depths(Ferrickand Goodman 1998):

F,

0l'' t)

(9.46)

in whichr,,= )f/,)i.Fora giveninitialFroudenumber,Equation9.46givesthemaximum valueof thedepthratio,beyondwhich the monoclinalwavebeconesunstable andremainsat its maximumdynamiccelerity. Thereis an analyticalsolutionto Equation9.,13for the profileof the stable monoclinalwave(LighthillandWhitham1955;Chow1959;Henderson 1966).The detailsof the solutionaregivenby Agsomand Dooge( l99l ). The resultis S o r -r , : + ar ,ln(r' )t

r',) * a. ln(,r',- ,i) + a,ln(r,

Y0)+ Cr Q.47)

-ti

in which

rl J,()',

r:

)i)(,t,

- Yo)

); r: - r ( 1- - r , ) ( r 7 - r o )

(9.48a)

(9.48b)

-t-52

C H A p T E R 9 : S i m p l i t i e dl u e l h o d so f F l o l r R o u t i n g

Yr

"vi

.1,(li, l,)(ro v

ri)

,)Ji

(9..18c ) ( 9 . 48 d)

(Vi * rtu)'

and C/ - constantofintegration.To obtain the wavc protile at any time Ir, the solution for the profile given by Equation 9.47 is translateda distanceof c,,/r with c. given by Equation9..12.Becausethe profile is infinitely long, some depth sligbtly lessthan the final normal dcpth can be specifiedat,r = 0 with thc constantof integration. q, determinedaccordingly.Alternatively,C, can be determincdsuch that the mid-depthof the *ave occurs at r : 0 when I = 0 and subsequcntlytravelsat the speedc,,,,like all other points on thc wave proflle. l are o f a s t a b l em o n o c l i n aw A g s o m a n d D o o g e( 1 9 9 1 )c o n f i n r e d t h c c x i s t c n c e profile using numericalexperiments.The theoreticalsolutionwas usedto obtain an upstreamhydrographthat then was routed downstreamusing the method of characteristicsfor the full dynamic equations.The resultingrouted hydrographpropagated downstream at the specd givcn by the Kleitz-Seddon principle without to the theoreticalmonoclinal changein shape.Funhermore.there was convergence shapefor a uniformly rising inflow hydrograph.Therefore,thc monoclinalwave is a specialcaseof a dynamic wave of equilibrium shapeat large valuesof time in which kinernaticsteepeningis balancedby dynamic smoothingeffects.It has been usedfor testingnumericalmethodsand evaluatingthe effect of variousterms in the momentumequationwhen comparedto sirnplifiedrouting methods.

9.4 DIFFUSION ROUTING of the flood wavebut only Because kinematicroutingcannotpredictsubsidence graditranslation, it is appropriate to considerthe etlectof includingthc pressure theinertiaterms.With this enttermsof thedtnamicequationwhile still neglecting the momentum equationbecomes simplification,

a\' Jo

(9.19)

dr

Writing.t: QrlKr, in whichK - channelconveyance, substituting into(9.,19) and with respectto time,the simplifiedmomentum equationbecomes differentiating

2Q aQ _ zQ' dK _ Kr at Kr at

d'], atax

(9.50)

The right hand side of (9.50) can be eliminatedby differentiatingthe continuity equation(Equation9.2) with respectto distancer to obtain

C H A I ' 1r , R 9 : S i n r p l i l l e d} l c t h o d so f F l o * . R o u r i n g

dlo

dry

at-

rlr.lt

353

( 9 . s1 )

i n w h i c h w e h a r . ea s s u n t c da r e c t a n g u l acrh a n n e ol f c o n s l a n w t idthB. Substiluting ( 9 . 5 0 )i n t o ( 9 . 5I ) . w e h a re

2 Q AQ_ ?Q. a K 6t

Kl

1 AtQ B ,J.rl

( 9 . 52)

Becausethc convcyancer( is a single-valuedfunction of dcpth 1 and thcrcfore of cross-sectional arcaA. its derivativewith respectto time in the secondtcrm on the Ieft of (9.52) can be transfonrcd. with the aid of the conlinuity equltion, to

AK

dK AA

dK rQ

at

Mat

tL4 d-r

( 9 . 53)

lf we furtherassunethat dKld,l canbe evaiuated from the uniformflow forntula in whichK = 9/Sl]5and rhensubstitute theresulrfrom (9.53)backinto (9.52),we have

ao 0r

do do _

Q_A'Q d y ' . O x 2B5o 3.tr

(9.54)

If dQldA is interpretedas the kinematic wave celerity, c*, as previously, the left hand side of (9.54) is the sameas the kinematicrcuting equation,but the right hand side now has the appearanceof a 'diffusionterm," with an apparentdiffusion coefficient given by D : Q/(2BS). From the behaviorof the diftu sion/djspersionrerm in river mixing probJems.the diffusion analogymakesit clear thar attenuationin O will be producedby this simplified rouringequarionin addition to advectionat rhe kinematicwave speed. For constantwave celeritl'and diffusion coefficient,Equation9.5'1is the govcming equationfor linear diffusion routing for which thereare exact solutions.The sameequationresultsif depth ratherthan dischargeis the dependentvariable.For example, Henderson (1966) gi\,es the Hayami (1951) solution for rouring an upstreamdepth hydrographrhat consistsof a seriesof unit stepchangesin depth. It is instructiveto derive the lineardiffusion equationfor depth from a slightly different viewpoint than rhar used to obtain (9.5.1)to gain further insighr into the limitationsof diffusion routing. If the depth,1, and velocity,y, are written in terms of their initial uniform flow values,r,, and V,, plus small perturbarionsfrom these valuesas _r'- ,), + )' and V : V, + y', then using an order-of-magnitudeanalysis the continuity equationfor a wide channelbecomes

a\'' AV' " + v, dr,' - 'l, -.- = 0 : al d,r dt Likewise,themomentumequationabsentthe ineniatermsis S, ]dr'' l ] ! + s " I/. - "\So 6x

')=o

(9.s6)

351

C HAPrER 9: SintplifiedMelhodsof Flor R()uling

The rario .5rlS,,can be evaluatcdusing the Chczy equation fbr a wide channel. Ncglcctingthe appropriatetcrms in this ratio from order-of-magnitudeconsiderations, thc momcntum cquatlonreducesto

2 *-'

*(

Y:

v,

=,

( 9 . 57)

;)

\.'" The nonlentumequationis differentiated ith respectto -r.andtheternrdy'ldr in t h er c s u l t i n g e q u a t i o ins c l i m i n a t ebdy s u b s t i r u t i of rno m r h e c o n t i n u i tey q u t t i o n . With somealgebraandrearrangement of terms.thc resultis givenby

a6 At

a6 " Ax

a26 3r'

(9.5 8)

'i in which@: .r cr = (3/2){ frornChezy;andD : ( Vrr,)/{150) for a widechannel. It is apparent that,onceagain,we havederivedthelinea|diflusionroutingequation, but it is strictlyvalidfor smalldeviations in deprhfrom rhe inirialdeprh.Neverrheless,of particularintercstis thesolutionto (9.58)for the upstream boundary condition of an abrupt increasein stagefrom the initial value d. to d;. The variable{ couldbe redefinedeasilyin dimensionless terrnsasdd - (v - r.,)/tri- -r). The solu(1959)to be tionto thelineardiffusionequation thenis givenby Carsla*andYeager

t[ /., c,/\ /c,-r\ /..,.,,\l d a - l l e r f ct ; ; . , - / . * p ( ^ f e r f c [ , , ^ . , - J l \\+utl / \t4url

r9.s9r

/)

in which erfc : the complementaryenor function. The solution indicatesthat the wave will move downstreamat the speedcr while spreadingor "diffusing" at a rate controlledby the apparentdiffusion coefficient,D. By dehnitjon of D, more diffusion will occur for smallerslopesand larger valuesof depth. Somecontparisonshavebecn madeby Ferrick and Goodman (1998) to entphasizethe effect of the diffusion term with respectto the inenia terms.They compared the solutionof the linearizeddynamic form of the momentum and continuityequations with the diffusion routing solution for a flood wave. The boundarycondition consistedof an abrupt increasein depth and dischargeat tie upstreamboundary, startingfrom an initial condition of steady,unifonn flow. They found that the initial shock traveleddownstreamwith the dynamic forerunnerat the speedof (y; + c,) and remaineddistinct from the diffusion r,ave profile until after the shockattenuated.Then, the profile celeritiesconvergedand approacbedthe kinemaric \\ave celerity. We must point out an inconsistencyin the derivationof the diffusion routing equation(9.54) that occursbecauseof the assumptionin irs derivationthat d(/d,4 can be evaluatedfrom the uniform flow fornula: that is, b) specificallyinvokingthe bed slopein place of the friction slope.By definition,the diffusion routingequation includes the pressure gradient term as representedby dy,/dr: yet to obtain the final form of the diffusion equation,dr/dr in effect is takento b€ zero in the evaluationof the parameters.so that S0 : S, as in kinematic routing. The secondderivation of the

C H A p r u R 9 : S i n r p l i f i e dN l e t h o d so f F l o w R o u t i n g

355

: .|',for the evaluation linear cjillirsionroutingequation(9.58)further implies that r' of corslunt valuesof c, and D. Thercfore,in the slrictcstsense,Equation9.54 is applicahlconly fbr thc caseof quasiuniformflo* s rvith relativelysmall valuesof the the pressuregradient.whilc thc Iinearforn given bl Equation9.58 furthersuSSests limiution of small deviationsfrom the initial unifonn llow depth. case,in which the paranetersc^ and D are allowed In the variable-parameter to vary with 0 but are obtainedfrom a uniform flow fonnula. Cappeleare( 1997) sho\\'sthat mass(or volumc)is not conservcdin thc routing: that is, the outflow volume under the oulflow hydrographtends to be smaller than thc inflow volume under the inflow hydrograph.lJe proposesa nore accuriilenonlineardiffusion routing nrcthodthat propcrlyaccountsfor the pressuregradienteffecton the evaluation of the variableparamelcrs,but it requiresa more sophisticatednumericalsolution technique.The advantageof the linear approachin u'hich the parametersare constant is that volume is conserved.and the rirer can be divided into a seriesof varying from reachto reachas the physicalcharacteristics reacheswith paranreters of the channelchangeas descrjbedin the next section. ExAMPLE 9.t. A verywideriverchannelhasa slopeof 0.0005,andinitiallyit is flowingat a uniformdepthof 1.0m (3.3ft). The ChezyC = 2.1in SI units(/t : 0.042). to L2 m at the upstream end,wherer = 0, If the depthof flow is abruptlyincreased computethe monoclinalwaveprofile and compareit with the diffusionwavesolution at variousvaluesof time. SollJr.or. The initial flow velocityfollows from a solutionof the Chezyequation y , : c r l t s , l ' = 2 4 x ( 1 . 0 ):r x 0 . 0 0 o 5 r 1 =0 . 5 3 7 ftls) m / s( 1 . 7 6 waveccle.ttycL: Ql2)v,: so thatthe kinematic coefficient thenis

0.805rn/s(1.6,1 ftls).The diffusion

: ?:lt^:^:9 = s37mr/s(5780 r,:,rs) 2 x 0.00rc5 xavecelerityis computed from Equation9-.ll to give The monoclinal 2 ( 1 2 ) 1-2 1 . (. ) . 8 0 5 0 . 8 1 1m s ( 2 . 7 7f t r ) c-^ ] t.2 I The monoclinalwavecelerityis only slightlygreaterthanthekinematicwavecelerity becausethe increasein depthis only 20 percenlof lhe initial deplh.Sucha small for the solutionof the lineardiffusionequationto apply.Equaincrease is necessary tion 9.59is solvedfor a seriesof r valuesat a specifiedtime lo obtainthe diffusion wavesolution.The valuesof time aretakento be 50, 100,200.and400 x l0r secas and sho\rnin Figure9.8.The valuesof d, aredefinedby lr - r',)/(!r ,r;),where,1, The solutionfor themonoclinal _v,arethe initialandfinaldepthsof flow,respectively. w i t ha r = - 6 - 8 1 4 , a 2 = 9 . 2 ' 7 l , a n d u a v e p r o f i l e c o m ferso mE q u a t i o n9s. 4 7a n d9 . , 1 8 constantis chosensuchthatthe mid-depthpoint(dd = .rr = 0.0160.The integration 0.5) travelsat the speedc- from,r : 0 at I = 0. As shownin Figure9.8, the shapeof

C H A P TE R 9 : S i n r p l i f i e dM e l h o d \ o f F l o w R o u t i n g

356

-

€

Diffusionwave l\,4onochnal wave

t in 1000sof sec

-50

0

50

100 xSolyi

1s0

200

250

FIGURE9.E Comparisonof diffusion wave and monoclinal wave prohles at various!imes.

the monoclinalwave profile does not change at successjvetimes. The diffusion wave profile shows increasedspreading due to diffusion as rime progressesuntil it approaches the shapeofthc monoclinalwave.It lags the monoclinalwaveslightly,however.because of the smaller kinematic wave speed. Both the time and correspondingdistance required for the diffusion uave to approach the shape of the low-amplirude ntonoclinal *ave are long, so that a very long river \r,ould b€ necessaryto achieve convergence.The applicability of the diffusion wave solution for this problem dependson rhe ljme being long enough for the initial shock to have dissipated.The rate of diffusion dependson rhe channelslope,initial depth, and channel roughnesswith greaterdiffusion and a[enuation occurring for smoother, deeper flows in channels of flatter slope.

9.5 MUSKINGUM.CUNGE METHOD The Muskingum-Cunge methodis a generalization of theMuskingummethodthat takesadvantage of thediffusioncontributions of thc momentum equation by allowing for truewaveattenuation througha matchingof numerical andphysicaldiffusion.First,a numerical discretization of the kinematicwaveequationis developed to setthestagefor quantifying numericaldiffusionwithinthecontextof theMuskingumrnethod. With reference to thecomputational moleculeshownin Figure9.9, the kirematicwaveequationasgivenby Equation9.29is discretized with weighting factorsX and Y to give

CriAprLR 9: Sintplilied l\'lethodsof Flo*' Routing

357

FIGURE 9.9 Cornputationalmolecule for numerical solution of kincmatic wave rouling problem.

x ( Q l --' o : ) + ( t - x ) ( o i : i Q : , , ) + ..

Y ( Q : - ,- Q : ) + ( t - Y ) ( Q : : t ' Q : - ' )

(e,60)

Treatingthe kinematicwave celerityas a constant,some specialcasesof Equation 9.60 can be considered.For example,a centcred,second-orderfinite difference schemeresultsif x = Y = 0.5. If thesesubstitutionsarc made and Equation 9.60 is rearrangedin the form of the Muskingum routing equation,as given by Equation 9. 1,1.then the routins cocfficientsfor this casebecome

C" ' -I : Co=, r i (,

C r-

I r-

C,

I t c,

(9.61)

in which C, is the Courant number defined by c;tr/Ar. If the Courant number is exactly l, then the coeficients becomeCo = 0, Cr = l, and q = 0: that is. pill = Qf, so that the centeredfinite differenceschemeproducespure translationonly for the Courantnumber equal to I, and thus it reprcsentsan exact solution of the kinematic wave equationfor this specialcase. E x A I l p L E 9 . 4 . R o u t et h et r i a n gl ua ri n f l o wh y d r o g r a ps h o w ni n F i g u r eg .l 0 u s i n g givenby Equation *ave equation 9.60\rith X : I: to thekinematic theapproximation 0.5 andC- = 1.,1. 9.61become Solution. The routingcrxificientsfrom Equations = = Ct: -O.16'7 Co 0.167; Cr l.0i arecarriedout as The valueof trl is chosento be 0.5 hr andthe routingcomputations to be stable.but the shapeof the hydrograph shownin Tablc9-5.The schemeappears

800

X= Y=0.5 600 Inflow-

E 400

o 200

t,

il 4

A \

Outf o*, nrln",';".t ] (

,,Oulllow,

analytical

I

\

\

l,'

\

\

4 T i m e ,h r

FIGURE 9.10 of kinemalicwaveequationfor X = f : solutions andanalytical of nunrerical Comparison = 1.4. 0.5 andC,

TA BLE 9.5

= 1.'l; Numericalsolutionoflinear kinematicwaveequation(X : 0.5i I/ = 0.5;C, = : : 0.167)o{ Example9.4 1.0;C, Co 0.161iC, Numerical,AnalJ{ical Cot It

,, hr

0.0 0.5 1.0 1.5 2.O /.) 3.0 3.5 ,{.0 .1.5 5.0 ).J

6.0

358

0 100 200 300 ,r00 500 600 500 .100 300 200 100 0 0

16.67 33.31 50.00 66.67 83.33 100.00 83.33 66.67 s0.00 13.33 t6.6'7 0.00 0.00

Crxlt

100 200 300 400 500 600 5m ,r00 t00 200 100 0

CtxOr

0.00 1.78 2t.'76 18.04 -51.71 - 7l.'13

-En.r0 99.2t 'n .91 62.01 -28.-57 - I 1.9{)

0,, m1s 0 11 r3l 128 129 .ll9 5:9 595 161 372 2tI t71 1l -t2

o1,mr/s 0 0 100 100 t00 {00 500 600 5C)0 .100 .tm 200 100 0

C H ^ p r E R 9 : S i r r p l i f i e d\ I e r h o d so f F l o w R o u r i n g

359

changes as shown in Figure 9.10. xilh the nunt.rical solution leading the analytical solution on both the rising and ftlling sidesof the outflou hydrograph.The numerical peak outflo!\'is only slighth sntallcrthan the analllical value.The distonion is caused b] the numericalsolulion ilsclf.

Exanrple9..1bringsup the nrorcgcneralqucsrionsof nunrerical s(abilityand consistency for variablevaluesofX andfl thatis. doesthesolutionof thefinitediffcrenceequationantplifyandgrowwithoutboundor not anddocsthesolutionconvergeto thatol the originalpartialdiffercnlialequation,respectively? To answer thesequestions,the renainder,R, or differencebctweenthe finite difference approximation of thcdifferential equation andthedifferentialequarion itselfcanbe determinedfrom a Taylorseriescxpansion of the functionO (t.!r, ljr) aboutthe point(llt, lAr). As shownby Cunge( 1969)andPoncc,Chen,andSimons( 1979), the remainder R canbe expressed as

-,). *(.1 ,,)l# ^-.,,,i(j * . ^ . r , ' { i r, c . ) [ ; ( x - ,c . Y )

9 ' ! - o r r , , , r{ e6 2 ) I ,, .",1)

The coefficient multiplying the secondderivativeof Q behaveslike an anificial or numerical diffusion due only to the numericalapproximationitself,becauseit does not appearin the original kinematicwave equation.It is ciear that.for X = I/ : 0.5, the nunrericaldiffusion coefflcientgoes to zero (convergence)and the appro.r*imrtion error is of second-ordcrO(,!r2) unlessthe value of the CourantnumberC. - L ln this case,the coefficientmultiptyingthe third derivativeterm,which causesnumerical dispersion or changesin shape,also goes to zero. For the Couranr numDer not equal to I, as in Example 9.4 and Figure 9.10, numerical dispersionresults,even though numericaldiffusionis not present.Furthermore,for f: 0.5 andX > 0.5, we seethat the numedcaldiffusioncoefficientbecomesnegative.which causesnumerical amplificationofthe solurionand insrability,as proven by Cunge(1969). E x A II p L E 9. s. Roulethe rriangular inflow hl drographof Example9.4 usingthe finite difference approximation of thc kinematicua\'e equationwith X : 0.I and y = 0.5. SettheCourantnumberC- : L0. So/ufion. The routingcoefficienrs arerecalculared from Equation9.60for X = 0.I, f = 0.5.andCourantnumberof 1.0to vield Co : 0 286;

C,:0.428;

c,:0.286

On examinationof Equation9.62 for the remainderof the numericalapproximarion,it is apparentthat the numericaldiffusion coefllcienthas a finile value but the dispersion

CHApTER 9: Sinrplificd luethods of Flow Routing

360

TAALE 9.6

Numericalsolutionof kinematicrrale equation(X : 0.li y = 0,5;C, = 1.0;Co : 0.2857i Ct : 0,42E6:C1: 0.2857)of Example9.5 \umerical, Analltical

0.0 u.) 1.0 1.5 2.O 2.5 3.0 3.5 4.0 5.0 5.5 6.0 6.5

0 100 200 300 .100 500 600 500 4U) 300 200 100 0 0

Ctx ot

Crx I1

Cox Iz

I, hr

28.57 51.l4 85.71 1 1 42 9 1,12.86 l7l .,13 t,12.86 l r{.29 85.71 5 7 .t , 1 28.57 0.00 0.00

0.(x)

0.00 E.l6 l0 90 5 18 l 85.90 4.1,1 I'll.8? 155. t3E.t9 112.95

.12.86 85.71 l2rJ.57 l? L,{3 211.29 2 5 7 I. . 1 )14.29

r7t..r3 128.57 85.71 42.86 0.00

57.01 2n.5.1

O, nrr/s

O} mr/s

0 29 108 202 l0l 100 500 5{l 18.1 195 199 200 100 29

0 0 100 200 100 .100 500 600 500 .100 100 200 t00 0

800 X = 0 . 1 ,Y - 0 . 5 C,- t'0 600 lnllow \

i

E 400

I

-.-

Outflow,analytical

\

,/ //

200

nertcal

\

il

t'ni"*'i" \

\

4 T i m e .h r

T'IGURE9.II of kinematicwareequalionfor X = O l. of numericalandanalyticalsolutions Comparison : = 1.0. I 0.5,andC, tenn goeslo zero.The routingcomputationsareshou'nin Table9-6 and plotledin Fi8 ofthe peakoutflowcaused ure9.1L The strikingresultin Figure9.1I is theatlenuation and not approximahon property of lhe numencal is a diffusion that by purenumerical of theanalyticalsolutionof the kinematicrvaveequatton.

C H \ P T I R 9 : S i n r p l i f i c]d4 c t h 0 r l"'1 l l { ) wR o u l i n g 3 6 1 To generalizethe conlputilion of thc routing crxfficit lrr l,rr lhc specific case o f f - 0 . 5 b u t X v a r i a b l ei.u b s t i t u t i o n sa r c m l c i i s o t h r t l r ( t r r i r t l o9r r6 0 b e c o n r e s

x ( O i - ' - O ) + ( ,rx X O i - ' , 0, 1 . , ) * - : , ( 0 1 . , -, tr,,i j i

O l - ' :) 0 (9.63)

Collecting terms and placing Equarion9.63 in the fornr o! llrr' fr'luskingumrouting e q u a t i o na s S i v e nb y E q u i l t i o n9 . l - 1 ,t h c r o u t i n gc o c f f i c i c l t \. r r

c,,

0.5c,- x I -X+0.5C,

0.5q + x l-x+0.5c, l-x-0.5c,, I -X+0.5C,

(9.64)

(96s) (9 . 6 6 )

Now if both the numeratorsand dcnominatorsof Equations9.6-1through 9.66 are multipliedby the Muskingun constant,0, and contparedu ith the Muskingum routing coefficientdefinitionsgiven by Equations9.1I through9. 13, it is obvious that they are identical if 0 representsthe kinematic wave trarel time given by lr/co, the Muskingum weighting wherec* is the kinematicwave speed,and if X represents factor It follows that the Muskingum methodin fact is a numcricalsolution of the linear kinematic wave equation that shows attenuationof the flood wave through numericaldiffusion, as illustratedby Example9.5. For the specialcaseof X = 0.5 and C, - l, lhe Muskingum method providesthe exact solutionof the linear kinematic wave with pure translation.as detenninedfrom Equations9.60 and 9.61. Under thesecircurnstances.it would seemthat the N'luskingummethod is not very useful unlessthe numerical diflusion that it producesis relatedin some qay to the apparentphysical diffusion and wave attcnuationthat actually occur in a river Cunge (1969) set the numericaldiffusion coeflicientfrom the approximation enor expressedby Equation9.62 for f : 0.5 and variableX equal to the apparent physicaldiffusion coefficientas derivedin Equation9.54.The resultingexpression can be solved for the Muskingum weighting factorX to produce

x:0.5(,-u*og)

(9.67)

with 0 = Ar/c* as before.When X in the Muskingummethodis calculatedfrom method,in which Equation 9.67,thenrethodis referredto astheMastirrgrrn-Cunge theroutingparameters dependin a knownwayon theflow characteristics andchanEither nel properties, so thatnumericalandphysicaldiffusioncffectsarematched. yariableparameters can be usedas is discussed in more or constantparameters detaillarer. (1980)refinedtheCungeapproach Koussis by maintaining thelimedcrivatives as whilediscretizing onlythe.rderivative in thekinenratic con(inuous butstillweighted,

362

C H A p T E R9 : S i m p l i f i eM d c t h o d os f F I o wR o u t i n g

waveequation.He thcn assumeda linciu varia(ionin the inflos hydrograpnovcr me time intervalAt ard obtaincdaltemateexpressions for thc N,lu\kinguntcoefficients:

1 - p C,,

t - p C,

B

c::p-"-r(-*)

(9.68)

(9.69)

(9.70)

in whichC, - Courantnunrber= \tl| and0 = traveltime in reach= .l"r/c,.Fol_ lowingthe sameproccdure of marchingphysicalandnumericaldiffusion,Ktussis developed an expression for theMuskingurnweightingfacrorX gilen by X : l -

C,

,l n /l -l + A + q \ I \ I + l - c",/

(9.7 t)

in which

BSec,^\r

(e.72)

Althoughthisexpression for MuskingumX seemsnrorerefinedthan(9.67).which givesX : 0.5 ( | I), Koussisfoundlittle to recomnrend one formulationover the other.Later.Perumal(1989)showedthatthe conventional andrefinedMusk_ i n g u ms c h e m ew s e r ei d e n r i c af o l r I C " / ( 1 X ) ] s 0 . 1 8 . 1u.h i l e r h ec o n v e n r i o n a l Muskingum-Cunge scheme\r'asslightlyberrcrthanthe refinedschemewhenboth werecompared u ith ananalytical solutiongivenby Nash( 1959)outsidethisrange. Theissueof constant vs.variableparameters in rheMuskingum_Cunge rneth-od alreadyhasbeenalludedto, andit reallyis a questionof wherherc, andi arecal_ culatedasa functionof thevaryingdischarge e to producevariableroutingcoeffi_ cientsor theyaretakenasconstant asa functionof a reference discharse. It should be clearfrom theoutset,however, thatallowingthecoefficienrs to varyi*irh e does not removethe approximation of evaluating thembasedon a uniformflow formula asa functionof the bedslopewith theactualdepthassumed ro be normal.Koussis ( 1978)proposedtheuseof a constant valueof X but a variabled = ,\r/c^with discharge.PonceandYevjevich( 1978)tcstedscveralnrerhods of dcterntining variable parameters andsuggested thatthebestperformance camefrom evaluating c. and,\ for eachappiication of theconrputational moleculefrom eitherrhree-point or four_ porntaverages of c^ andQ (to be usedin thecomputation of ,\). In thethree_point method,0 andc^ aretakenastheaverage of the valuesat grid points(1,k), (i + l, k ) a n d ( i , t + l ) i n F i g u r e 9 . 9I.n l h ef o u r - p o i nmr e t h o da. l l i o u r g r i d p o i n rasr eu s e d in theaveraging process, whichnecessarily requiresiterationbecauiethevaluesat ( t + I , k + 1 ) i n i t i a l l ya r eu n k n o w nT.h et h r e e - p o ianvt e r a gre. a l u e s o f c * a n d eare

C H A p T E R9 : S i m p l i f i c \dl e t h o d so f F l o q R , outing 363 used as staning valucsin the four-point iterationmethcxl.Both methods,however, display somc loss ol'lolume in thc routed outflow h\drograph. whereasthe constantparametcrnrclhodconservesvolunre.Ponceand Chaganti( 199,1)repon sJight improvementsin volumeconservationif ci is computed from a three-pointor fourpoint alerase value of p rathcr than being itself averaged.On the other hand, the variable paranrctermethodsreproducethe expected nonlinear steepeningof the flood wave. In a contparisonbetweenan analyticalsolution and the constantparameter method of routing tbr a sinusoidal inflow hydrograph. Ponce, Lohani, and Scheyhing( 1996)showthat thc peakoutflow and the peak traveltinrevary between I and 2 percentfrom the analyticalvalues. Tang. Knight, and Sanucls ( | 999) investigaredthe r olume lossin the variableparameteri\,luskingum-Cunge nethod in more dctail and confirmedthat the use of an average0 ratherlhtn an average(.( slightly irnprovedthe volume loss (by about 0.I perccnt).In general.greatervolume loss was reponed for the rhree-pointmethods than the four-pointnlcthods.and routing on very nild slopes(S = 0.0001)produced the greatestvolume loss,with valuesup to 8 percent.An attemptwas made to incorporate the correction suggestedby Cappaleare(1997) for including the effects of the pressuregradientternts in the estimation of routing parametersbut only in an approximateway. Some improvementin yolume loss was shown but it dependedon an empiricaladjustmentfactor in the pressure-gradient correctionformula for Q. If we return to the questions of stability and accuracy with respect to the Muskingum-Cunge method,it must bc true that X < 0.5 for stabilityas shown by Cunge 11969), but the criterionfor the routing coefficient C0 to be greaterthan or equal to zero to avoid negativr'outflows (or a dip in the ourflow hydrograph)can be expressedin a different way. Ponce (1989) rcfcrred to A, defined by Equation 9.72, as a cell Reynoldsnumber.In terms of A, the accuracycriterion of Co > 0, *hich is equivalentto the critcrion given by (9.15), becomes(C,, + A) > I from Equations9.64 and 9.67. Bascdon routing a large number of inflow hydrographs with a realisticshapegiven by the gamma function, Ponceand Theurer(1982) suggesteda strongerconditionof Co > 0.33 to ensureaccuracyas well ai consistency (in the senseof removingthe sensitivityof the outllow hydrographto grid size).As a practical matter.this criterion becomes(C, + A) > 2, which definesthe maximum length of the routingreach,&, for given valuesof the time stepand the waye celerity as well as the flow rate,channeltop width, and slope,as follows:

r " = I l t . , r ,+ o - ' \ 2 \

BS,,c 1/

(9.71)

Note from (9.64) that the simple criterion of naintaining Co greaterthan some positive constantbasedon the cmpirical studiesof Ponce and Theurer( 1982)does not precluJe the value ofX from becoming negative. Dooge (1973) as well as Strupczerlski and Kundzewicz(1980) justified marhemaricallyrhe possibiliry of X < 0. ln the conventionalMuskingum method,an additional criterion of C, > 0 often is specified to ensurc the avoidanceof negative outflows, which generally is satisfied for C, < I and X < 0.5. bur Hjclmleldt ( 1985) demonstratedrhar rhis

364

CxAp'rER9: SimplifiedMethodsof Flow Routing

crilcrion can be rclaxed for most realistic in11owsequencesin agrecnrentwith Ponceand Theurer( | 982). Thc additional critcrion does guarantec,hou n'cr, positive outflows for any possiblepositire inflow sequence. Although the Muskingum-Cungemethodgives the exactanalyticalsolution of the linear kincmatic wave routing prob)em for C, = I and X = 0.5, the more usual case is for X < 0.-5.From Equation 9.62, we see that. for f : 0.5, X < 0.5, and q : 1.0.the dispersionterm is zero and the numericaldiffusion cocfficientexists, making the numericalmethodfirst order: that is, the approximationenor is O(l.r). However.under thc same conditions but tbr the Courant nunber not equal to I, numericaldispersionoccursas well as numcricaldiffusion. For this reason,Ponce (1989) recommendsthat the Courant nunbcr be kept as cJoseto unity as possible. not for stability reasonsbut to limit nunrericaldispersion.In fact, stabilitv conditions requireno specific limits on the Courant number,as are dictatedin explicit finite diflcrenceapproximationsof the hl pcrbolic dynamic cquations. The applicabilityof the Muskingum-Cungemethod is linited to flood waves of the diffusion type with no significant dynamic ellects due to backwater,such as loopedstage-discharSe ratingcurves.Ponce( 1989)suggeststhe following criterion for applicability:

" i;1" > t5

(9.11)

in which I, is the rise tinle of the hydrograph;Sois the botlom slope;1o is the average flow depth;and g is gravitationalacceleration.Overall, the Muskingum-Cunge method is a significantimprovementover the Muskingum method becausehydrograph data are not rcquired for calibration, so that it can be uscd on ungauged streamswith known geometryand slope. The variable-paranreter method may be useful $'hereslopesare moderateto lar8e. so that volume loss is acceptable,but largeimprovenlentsin accuracyshould not be expectedover the constant-parameter approach.Correctionsfor the pressure-gradicnt term havc the potcntial to improve diffusion routing so long as the numerical methods remain sirrpler than full dynamic routing;otherwise.dynamic routing shouldbe used in the first place. EXAttPLa 9.6. An inflow hydrographfbr a river reachhasa peakdischargeof 45m cis { I28 mJ/s)at a lime to n€akof 2 hr with a basetime of 6 hr Assumethatthe inflowhydrograph is triangular in shapewith a baseflow of 500cfs ( l.l mr/s).The river reachhasa lengthof 18.0fi)ft (5-190m) and a slopeof 0.0005ftlft. The channelcross with a bottom$ idlh of l(X)lt (30.5m) andsideslopesof 2:1.The sectionis trapezoidal \'lanning'sr for lhe chdnnelis 0.025.Find lhe outflow peakdischarge and time of o(currence for therivcrreachusin-q the Muskingum-Cunge methodandcomparethem 1()rhedynamicroutingnrelhodusingthc methodof characlcrislics. Soll,ltlor. Fir\t the kinenrdtic ware speedis calculated basedon Manning'sequation anda reference discharge of 2-500cfs (70.8mr/s).whichis midwaybetweenthe base ilo!\'andthe perk discharge. For the givenconditions. the .esultingnormaldepthis 5.71fl ( 1.7,1 m) andlhe \clocity.\/.f is -1.93fvs ( 1.20rn/s).lf we considerthechannel to b€ \ery wideasa firstapproximalion. thcnc, : (5/3)y0= 6.55ft/s(2.00m/s).The

\r

C

lcrhodl of Flo* Rouring , R a : S r r r r p l r f r c\ d

165

valuc of ll is tenlali\elv cho\en to be {).5hr basedon discretizalionof the time to peitk, rlhich is equal to 2 hr. Then \r can bc cslimrled fronr the incquality of (9.73):

r . - l / . , - l r* ! _ ) 2 \ "

IJS,,:'' /

: I i o . r , , o . 5 xo 1 6 0+0 I \

:500 n2.8x0.0005x6.55

: 9003 fr (l?'1'1m) Thcrefore, two routingrcaches.eacht\ith a lcng(hof 9000ft (27,13m), can be used. = 1.3I , *'hich is slightly This givesa Courantnumt'erof cr3r/Ir = 6.55r 1800/9000 grcaterthrn unity.andthe \ alre of X froln (9.67)is

x=os(t

^,,*-)

=os(r-

2500 122.8x 0.0005x 6.55x 9000)

:o'rt

furtherslightadjustment-s of Al and If thevalueof eilherCnor X seemsunsatisfactory, andguarantees Co> sincethec.ilcriongi\en by (9.73)is conservative Ir arepossible, 0.33ratherthanCn > 0. \\ hich is all thatreallyis requiredto avoidnegati\eoutflows areconrputed from (9.61)through The valuesof the routingcoefficients in mostcases. (9.66)to yield Co:0.3311

C' = 0.5'10;

C. = 0.121

is solvedstepby stepin Table9-7 whichsumto unityasrequired. Theroutingequation for the first subreachof 9tXlOft (2?,13m). Then the outflow becomesthe inflow for the for * hichonly the finalresultsareshownin thetableq'ithoutthe internextsubreach, resultsate comparedwith dynamic nredialecomputalions. The Muskingum-Cunge (MOC) in Figure9.12.It js af'parent routingresulrs from the methodof characteristics

TA BLE 9.7

routing {C0 : 0.333;Cr = 0.540;C, : 0.127)of Example9.6 Muskingum-Cunge Time, hr

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.-5 4.0 5.0 5.5 6.0 '7.O '7 .5

Inllow, cfs

-500 1500 2-s00 1500 ,1500 ,1000 3500 3000 2500 2000 1500 lmo 500 500 500 5CX)

Co x 12

500 t33 r 166 I {99 1332 I t66 999 8-r3 666 500 313 167 167 161 t67

Cr x 1l

270 8r0 1350 1890 2{10 2 r60 lE90 1620 1350 r080 8!0 540 110 270 270

Ct x Or 64 106 222 34E 174 538 491 429 366 303 239 t76 tt2 '70 64

r = 9000ft; 01, cfs 500 833 t'74E 3736 1236 386,1 3380 2882 2382 1882 I t82 882 549 506 501

.t = 18000ft; 02, cfs 500 6 tI l 0 t99'l 2976 3806 4058 3 727 3258 2'7 63 2261 t'164 1264 819 569 512

366

t V . S r r r r p l i l i c\ lde r h o d .o l F l , , u R , , r r r r r r r

CtrAp

5000 Co- 0.333;Cr = 0 540: Ct = 0.127

Muskingum-Cunge outflow

o 2000

1000

0 Time,hr FIGURE 9.I2 Inflowandoutflowhydrographs for Muskingum-Cunge andMOC routing.

thatthe assumption wavecelerityin theMuskingum-Cunge of a constant methodfails to capturethe nonlinear steepening andflattening on therisingandfallingsides,respectively,ofthe outflowhydrograph. Howcver, thepcakoutflo* ratesagreeverywell.The occunence of the outflowpeakis at r : 3.0 hr, but this is limitedby the time stcpof theapproximate routingmethod. Themethodof characteristics givesa peaktimeof 2.8 hr. Notethatthecriterionof Equation 9.74for diffusionroutingis not met,but reasonableagreement still is obtained betweenthetwo methodsin thisexanrple.

REFERENCES Agsom,S., and JamesC. l. Dooge."NumericalExperiments on the MonoclinalRising t ave;'J. Hldrolog:- 124( 1991),pp. 293-306. Aldama, A. A. "l-east-Squares ParameterEslimationfor Muskingum Flood Routing."J. Hydr Engrg.,ASCE I16, no.4 ( 1990),pp. 580-86. Bras,R. L. Hydrolog;-.Reading,MA: Addison,Wesley,1990. Cappefaere,B. "Accurate Diffusive Wave Routing." "/. Hydr Engrg., ASCE 123, no. 3 ( 1997),pp. 17,1-81. Carslaw.H. S., and J. C. Jaeger.Conductionof Hedt in Solids,2nd ed. New York: Oxford Universityhess, 1959. Chow,VenTe. Open ChannelHydraalicr.New York: McGfaw-HiU, 1959. Chow,V T., D. R. Maidment,and L. W Mays. Applied H.vdrology.New York: McGrawHill. 1988.

CHApTER 9 r S i m p l j i r eM d c r h o dos f F l o wR o u l i n g 3 6 . 1 Cunge.J. A. 'On rhc'Subjccrof a FloodPropagation Conrpurarion l\relhod... "/. Hy,d.Res.l, no. 2 ( 1969),pp. 205..30. Dcrr-ee,J. C. L Littear Tlpory oJ lltdralogic.t_\.rlenj. l_1.S. Dept.Agnculture,ARS, Tech. printingOffice,1973. Bull. No. 1.168. \lhshingron, DC: Government Fe.rick.M. G.. andN. J. Goodrlan.',Analysis of LinearandN,lonoclinal RiverWaveSolu_ lions."J. l/_rdr611916., ASCE 12,1, no.7 (1998).pp.728-11. Gill. M. "Routingof Floodsin RiverChannels." Norcic Hrdr g ( l9?7),pp. 163-70. Halami, S. "On rhe Propagation of Flood Waves.'Bulletin No. /, Diiasrerprevention Research Institute. KyotaUniversity, Japan,1951. Henderson, F.M. OpenCfuuntel['lov. NewYork:]\l;_millan,1966. Hjelmtcft,A. T.. Jr, andJ. J. Cassidy. Hldnlog,-for Engircersand planners. Ames,Iorl,a: press.1975. The lowaSraleUniversiry Hjelmfelt. A. T.. Jr. "NegariveOutflowsfrom Muskin:-umRouring.',",f.Hvh Engrg., ASCE I I l . n o .6 ( 1 9 8 5 )p, p . l 0 l 0 - 1 , 1 . Koussi\,A. D. "Comparison of MuskingurnMethod DifferenceSchemes.'. J. Hyd. Div., ASCE 106,no. HY5 (1980),pp.925-29. Koussis.A. D. "Theoretical Estimates of FloodRouungparanielers...J. Hl.d.D/.u.ASCE l O 4 .n o .H Y I ( 1 9 7 8 )p. p . 1 0 9 - t 5 . Lighthill, M. J., and G. B. Whirham.,,OnKinemaricWaves:FloodMovements in Lone Rivers."Proc.Rof.Soc.(A). 229,no. tt'78( 1955r. po. 281-316. Milf er,W A., andJ. A. Cunge.'.Simplified Equrrionsof iinsreadyFlou...In tlnsteady Flow in OpenChamels,ed.K. MahmoodandV yerjer ich,vol. I, pp. g9 l g2.Fon Coltins. CO: WaterResources Publications, 1975. Nash._I.E. 'A Note on rhe MuskingumFlood Roulin_sMethod."J. 6eoph,r,s. Res.64, no. g ( I 9 5 9 ) ,p p . 1 0 5 35 6 . "Unificarion Perumal,.Mof MuskingumDifference S,-hemes." 1 H_r.rft Engrg.,ASCE I 15, n o . 4 ( 1 9 8 9 )p, p .5 3 6 - 4 3 . Ponce.V M. Engitee g H,-drolo8,r Englewood Clifii. NJ: prenrice,Hall, 1969. Ponce.V- M., and P V Chaganti...Variable-parameter MuskingumMerhodRevisited.,,"/. Hydrologl-162(1994),pp.433 39. Ponce,V M., Y H. Chen.andD. B. Simons.,.Uncondjtional Stabitiryin Convecrion Com_ putations." "/.H1d.Dir,.,ASCE t05, no. Hyg (19t9). pD.1079_g6. Ponce.V M., R-M. Li, andD. B. Simons...Applicabrlirv oi Kinemaric andDitlusionMod_ els.""/.l/_r'dDrr,.,ASCE 104,no. Hy3 ( t978),pp. 353_60. .Anilytical Ponce.V M.. A. K. Lohani,and C. Scheyhing. Verificalion of Muskingum_ CungeRouring." 1 H!-drology, 174(1996),pp. 235_41. Ponce.V M., and F. D. Theurer..Accuracy Criieria jn DrffusronRourrng.,, J. Hyd. Div., ASCE 108,no. HY6 (1982),pp.147-57. ..Muskingum-Cunge Ponce.V- M., andV Yevjevjch. \terhodwirhVariableparameters.,,J. llrd Diq, ASCE 104,no. Hyl2 ( 1978),pp. I66t_{7. Singh.V P, andR. McCann.'.SomeNoteson MuskingumMerhodof FloodRouting.,,"/. Hldrulogylg ( lq80). pp. .l,t1-61. ..Muskingum Strupcze*ski, W., andZ. Kundezewicz. VerhodRevisired.,. I Htdrotogy4g (1980).pp.32742. Tang.X-N., D. W. Knight,andp G. Samuels.,.VolumeConservalionin Variableparameter \ruskingum-Cun8e Method|'J.Hldt Errgrg., ASCE 125,no.6 (1999),pp.610_20. Weinm:.nn.P 8., andE. M. Laurenson..ApproximateFlood RoutingMerhods: A Review.,, J H _ v dD i u , A S C E1 0 5 n , o .H Y t 2 ( 1 9 7 9 )p, p . l 5 : t 3 6 . ..Unsready, Woolhiser,D. A., andJ. A. Liggert. ijne-DimensionalFlo$,o!,era plane_The fusingHydrograpb." llater Resources Res.3, no. -l (1967),pp.753_71.

368

CttAPTIR 9: SirnplifreM d e t h o d so f F l o $ R o u l i n g

EXERCISES 9,1. A stormwaterdetenlionbasin has bonom dimensionsof 700 ft b], 700 ft q'ith inlerior 4: I side slopes.The outflow structuresconsisl of a l2 in. ditmcler p;pe laid through the fill on a steepslope \!ith iln jnlet inren ele\alion of 100 fl at the boftom of the basinand a broad-creslcdweir with a crestelevationof 107 fr and a crest length of l0 ft. The top of the berm is at elevarion 109 fi. The design inflou hydrographcan be approximatedas a triangularshape\\ itr a peak dischargeof 125 cfs at a time of 8 hr and with a basetime of 2.1hr. If rhe delentionbasin inilidlly is empt]-.roltte rhe inflow hydrographthrough the basin and determine the peak outflo\\ rate and stage.How could thcscbc funhcr reducedl Explain in detail. 9.2. Prove that the temporarystoragein reservoirrouting is givcn b\ the areabctween the inflow and outflow hydrographs.For triangular inflo$ and oulflo$, hydrographs. dcrive a relationshipfor the dctentionstorageas a functionof inflow and outflow pcak dischargesand the basetime of the inflow hydrograph. 9.3. The inflow hydrograph for a 25,000 fl reach of the Tallapoosa Riler follo$,s. Route the hydrographusing Muskingum routing with At = 2 hr. -!l : 25.000ft, I = 3.6 hr. and X = 0.2. Plot the inflow and outflow hydrographsand determinethe percenr reduction in the inflol peak as well as the travel time of the peali. ,, hr

0 2 6 8

t0 12 1,1 l6

I8

Q. cfs

100 500 1500 2500 5000 l 1000 22fi)0 28000 28500 26000

t. hr

20 22 24 26 28 30 32 3.r 36 38 40

C, cfs

22000 1'7 5AJ 14000 10000 7000 .1500 2500 t 500 1000 500 100

9.4. Routethe inflow hydrographof Example9.6 throughthe sarneriver, but for a total reach length of 27.000 ft, using the methodof characteristics computerprogram CHAR (on the website).Then use this outflow hydrograph*iti the inflow hydrographto derive0, X, and the Muskingumroutingcoelficientsusingthe least-squares fitting method.Finally do the Muskingum routing and comparethe resultswith the ou(flo\ hydrogrrphfrom dynamicroutrng. 9.5. Derive the kinematicwavecelerity for a trap€zoidaichannel.Plot the ratio of kinerratic wavecelerityto averagechannelflow velocity,cnlV,asa functionof tte aspect :atio b/) for severalvaluesof side slope ratio |n includingrn = 0 for a rectangular channclanddiscuss tberesults. 9.6. For uniform flow at a depthof 5 cm on a concreteparkinglot havinga slopeof 0.01 anda flow lengthof 200 m, calculatethe kinematicwavenumber,t Would kinematic

C H ^ p r I R 9 : S i r r p l i t i c dM e l h o d so f F l o w R o u l i n g

169

= routingapply'lRepealfor a \ery \r'ideriverchannclwith a slopeof0 00l n 0 035' results' Discuss the ur' of 200 lcngth flow the sanle lr depthof 5 m, and lhe hldrographriselime is l5 rnin 9.6,suppose 9.7. Forthc sameparkinglol of Excrcise on Equation9 171 bascd apply nethod rra\c routing Wouldthekinemalic of 300m usingtheanal)ricalkineol er a distance 9.8. Routeanoverlandno\l hydrograph The ovcrlandflow occursorer a 100 rvave celerity \ariable with maticwavesolution = n wideplanestripof conslantslope0 006 andl] 0 0l5 The inflow hydrographis triangulir*'ith a peakof 1.25mr/sat a time to peakof l5 min andwith a basetime of 4imin. The inilialbaseflow is 0.25mr/s Plottheinflowiindoutflowh)drograPhs anddiscussthe shapeof theoutflowhldrograph -ti' the monoclinalwave 9,9. DeriveEquation9.'12and thenshowthat.as) aPproaches \ \ ' 1 \ ' ' e e l c r i t v ' l i n c r r l i ' l i . l h c i n i l r r l .rlcrill approuche' *ave profilein a very wide river *ith an initial 9,10. Conlputeand plot the rnonoclinal : 1 0 n l . T h e s l o p e o f t h e r i v e r i s 0 0a0nl d $ e M a n d e p l h o l a i n a l n r a n d d e p t ho f 1 . 0 n i n g ' s ni s 0 . 0 ' 1 0A.s s u m et h a lt h ed e p l ha t- r = 0 a n d t : 0 i s 3 9 9 m A l s o c a l c u l a t e this to theinitialkinematic$ aYecelerity' wavecelerityandconipare thenronoclinal 9.11. DeriveEquation9.58.the lineardiffusionroulingequatron' 9 60 for thekinematic\'r3\eequation of Equation 9.12. Usingthc numericalapproximation with X : I and y : 0. derile theroutingcoeflicientsandroutethehydrographofExample 9.4 for C" : 1.5.Wirh referenceto Equation9 62' what haPpensto the outflow irydrographif C" : 1.0?Explainwhy the methodbecomesunstablefor C' { l 0 9 60 for thekinemaricua\e equation of Equation 9.t3. Usingthc numericalapproximation : androutethehydrographof Exam: coe fficients rouling the 1.derive wirhX 0 and/ ple 9.4 for C, = 0 67 With referenceto Equation9'62. what happensro the outflow irydrographiic, : l 0r Explainwhy the rnethodbeco es unstablefor C. > l 0' 9.14. Using the numcricalapproximationof Equation9 60 for the kjnemalic\*a\e equation androutethehyd'ographof Examwith X = 0 and y : 0. derivetre routingcoefficients what hapPensto the outflow : 962 Equation l.0 wi$ referenceto ple 9.4 for q for any value of C"? unstable bccome method the + l.0l Does irydrographii C" your answer. Explain methodin termsof l andq' 9.15. Expressthe roulingcoefficientsof IheMuskingum-Cunge of Example9 6 for So= merhodto routethehydrograph 9,16. UsetheMuskingunl-Cunge program CHAR the computer from $ith results the 0.001andcompare 9,1?. Rcpeatthe Muskingum-Cungeroutingof Example9 6 for a very w-idechannelusing the threepoint variableparamelermethodwith cr and i calculatedfrom three-point averagesof O at each time step Comparethe resultswith the constantparameter method. routingusingthe four-Pointvari9.18, Write a computerprogramfor Muskingum-Cunge me$od andapplyil to Example9 6' ableparameler

C I I, { P T E R

IO .j.JJ-*Eeqrfu

Flow in Alluvial Channels

r0.1 INTRODUCTION Rivers,whicharenaturar openchannels, oftenhavemovabre alruvralsediment boundaries at rhebedandbanksthataddanother deg.ee of cornit.^,tyto tt, "rti-

mationof flow resistance. Becausethe sedimentb-editself is'sublectto move_ ment,the flow createsperturbations of that boundary,o.hich amountto sanO wavesthatpropagateeitherdOwnstream or upstream dependingon the flow con_ ditionsand sedimentproperties. The amplitudeof,l. p."urU-utions affectsthe flow resistance and hencethe stageat a givendischarge, ",nif.li',n. sametime the flow condirionscontrolrhe am.plitud-e anO*av.te'r,gttoi-tli" p..trruutiou.. For this reasonalluvialrivershavebeendescribed ^,Aiin,rrtirrr" ^nd sculptor ( V a n o n1i 9 7 7 ) . Aside from rhe problemof additionalflow resistance, the sedimentregimeof open channelflow in a river is responsiblefor bed unj b_k instability, scou. aroundstructures suchas bridgepieri andabutment., O"po.irionunOUu.l"tof nrl habitat,los_s of clarity in the iuaiercolumnand i"friUiti6.,i pf,.i"rynthesis, and transportof adsorbedcontaminants. Suchproblemsasso.iut.dJuith-r"u,-enrranspon are lntertwinedwith the purehydraulic considerations of open channelflow andsodeservesomeattentionin thistext,especially with respectto flow_sediment interactions. This chapterdescribessedimentpropertiesand discusses methodsfor predic! ing bedandbank stabilityby identifyingihe threshold of ,.Oi."ni'rnouern"nt. S.O_ lment ln motton and the bed forms createdare discussed next along with the coupledproblemof flow resistance andstage_discharge pred;.tion.a Uf,"f ou.*i.* of bedloadand suspendedload fanspon equationsand the carculationof total sedi_ ment load are presentedfollowedby a consideration of raou. p.ott"rn, assocrated with bridgesconstructedacrossrivers.

37r

.l7l

C H l p r L R l 0 : I r l o wi n ; \ l l u ri a l C h a n n c l s

10.2 SI'DI}IENT PROPERTIIiS Sornepropcrticsof inclividualsedintentgrains ale importanl lirr cohesionlcsssediIncnts(slDds and sravcls).suchas grain size.shape.and specilicgravity.as well as Iall relocity. which is a function of all the previouslr mentionedproperties.The hehavior of scdinrcntgrains or particlesin bulk mar be ol intercst.too. The bulk specific s cight of scdinrentsclepositedin a lake bed. for exanrple.or rhc grain size distribution of sandsand gravclsin a well-gradedstrcambedaff'ectthc behaviorof the bed as a whole. In addition,fbr clay or cohesivcscdinrcnrs,identifyingthe interactionsof platclctlikeparticleswith variablcsurfacechargeis cssentialto an understandingof thc stabilityof the bed with respectto erosion or resuspension (Dcnnert et al. 1998), but this chaptcrfocuseson rroncohesivcscdintenls.

Particle Size The grain size of a scdimentparticle is one of its most inponant propenies.The American GeophysicalUnion (AGU) scaleclassifiessize rangesas shownin Table l0-l *ith each size class reprcsentinga geomerricseries in which the maximum and minimum sizesin the range differ by a factor of 2. Thc size of sand panicles usually is measuredas the sievediameter,which is the length of the sideof a square sieveopening throughwhich the given particlewill just pass.The size of silts and clays, on the other hand,often dependson sedimentationmethodsand the relationship between fall velocity and sedimentationdiameter. which is defined as the diameter of a sphereof the same specific weight haling the same terminal fall velocin'as the given particlc in the same sedimentationfluid. Thc rclationship betweensedimentationdiameterand fall velocitv is discussedin the sectionon fall velocit\'.

Particle Shape Sand grains in panicularhave a shapethat variesfrom angularto roundeddepending on the fluvial environmentin which they are found. Ri\er sandstendto be wom somewhatby fluvial action and deviateconsiderabl) from a sphericalshape.One way of defining shapeis the so-calledshapefacror (l\{cKnown and Malaika 1950) siven bv

S,F.

a

^/voc

(r0.r)

in which S.F. = shapefactor,and the variablesa, b, and c are the lengthsof three mutualll perpendicularaxes such that a is the shortest axis. In other words, the shapefactor is definedas the length of the shonestariis divided by the geometric mean length of the other two axes.A sphereobviously would have a shapefactor of 1.0 \ 'ith no preferentialdirection of axes.For an ellipsoid with axis lengths in

CltAprER 'I'ABLE

l0:

Flow in Alluvial Channcls

3't3

I O.I

Sedimentgrade scale(AGU) Classname

Size range, mm

Very larSeboulders LarSeboulders lVediumboulders Smallboulders Largccobbles Smallcobbles Very coersegravel Coarsegravel Mediumgravel Fine gravel Veryfinegrarel

4.09G2.0i8 2.0.181,02.{ 1 , 0 2 .5{r l 5r 2 - 2 5 6 256-r:8 12E5,1 61-32 32 l6 lG-8 8-1 .1 2 2.0-t.0 LH.5 0.50{.25 0.250-{.125 0.125 0.062 0.062{.031 0.03r-0.0r6 0.0I6-0.008 0.008{.(x}1 0.004{.001 0.00t-0.001 0.(J0 r0 {.0005 0.0005,!.0002,1

Coarsesand Mediumsand Finesand Very fine sand Coarsesilt Mediumsill Finesill Very fl ne silr Coarseclay \'lediumclay Fineclay Verv fine clay

the ratioof l: l:3, the shapetactorwould be 0.577.A shapcfacrorof 0.7 hasbeen foundto be aboutaveragefor naturalsands(U.S.Interagcncy Committee1957). The shapefactorcanbe determined usinga microscope. Particle SpecificGravity Because the predominant mineralin sandand graveloftenis quanz,the specific gravity(SG) usuallyis takento be 2.65. However,for lesswom sediments thar retainthe mineralogyof the parentrock, severalmineralssuchas feldsoar.mica. barite,andmagnetite, for example, still maybe presentin appreciable quintiries.so thatspecificgravitymay needto be measured at eachinvestigation site.Clay sedimentsgenerally arehydrousaluminumsilicateswith a characteristic sheetstructure havinga specificgravityfrom 2.2 to 2.6 (Sowers1979). Oncethe specificgravityis known. the specificweight,7,, of the sediment soli,Jis simplythe specificgravitytimes the specificweightof water.Sandand gravelhavea specificweighrof approximately i 65 lbs/ftror 26.0kN/m3.The mass density,p., is the specificgravitytimesrhe massdcnsityof water,so quanzsedi_ menlshavea massdensityof 5.14slugs/ftror 2650kg/mr.

-

311

CHApTER l0: Flow inAlluvial Channels

Bccausethe sedinrentgrains of inlerestin sedinrenttrtnspod usually are subincrged, another propeny of interest related to specific gravity is the submerged speciticwcight, which is given by yl = (f. - y) = (SG I )7, in which 7, - specific weight of the sedimentsolid and 7 - specificwcight of water The subrnerged specificwcight of sand grains,for example,is 103 lbs/ftr or 16.2kN/mr.

Bulk Specific Weight As sedimentsare depositcdin relati\ely quiescentenvironments,they occupy a \ olume that includes the pore space filled with water subject to consolidationo\er time. Estimatesof sedimentcarried into a reservoirby weight can be translatedinro volume occupiedonly by use of the bulk specificweight. Such predictionsof the volume of scdimentdepositedas a function of tinreare essentialto estimatesof the useful life of a reservoir,or the time betweendredgingeventsto maintain navigable waterways.The bulk specific\aeight of a sedimentdepositis definedas the dry weight of sedimentdivided by the total volume occupiedby both sedimentand pore space.Lane and Koelzer ( 1953) proposeda relationshiptbr the specific weight of depositsin reservoirsgiven by yo:

yo, I

Blogr

( r0.2)

in which 7, : bulk specificweightin lbs/ftrof a depositwith an ageof r years; 7r, : initialbulk specificweightof thedeposirin lbs/ft'at theendof thefirstyear; (lbs/ftr).For a sedimentthatalwaysis subnrergcd andB = constant or nearlysubmerged, and B have values of and for sand. 93 0 65 and 5.7 for silt, and30 and 70, l6 for clay,respectively.

Fall Velocity The fall velocityof sediment is definedastheterminalspeedof a sediment grainin waterat a specifiedtemperature in an infiniteexpanse of quiescent water It plays a veryimportantrolein distinguishing betweensuspended sedimentload,in which grainsarecarricdin the watercolumn,andbedload, thesediment whichconsistsof individualgrainstransported nearthe bedwith intermittent or continuous contact with the bed itself.Fall velocityis closelyrelatedto the fluid mechanics problem of estimatingdrag arounda submerged spheredue to a fluid flow of specified velocity.The only differences lie in the viewpointof the observer(the sphereis movingandthefluid is at rest)andin whichof therelevant quantities areunknown. In thecaseof flow arounda fixed sphere,theunknownis thedragforce;while for a spheredroppingat terminalspeedin a fluid at rest,theunknownis thefall velocity. In the lattercase,the dragforce mustbe equalandoppositeto the submerged weightof the sphereat terminalvelocityto give pA rtrl

c,;

= ( r ' - r ) rd' o

(10.3)

C H A p T I R l 0 : F l o \ , i' n A l l u \ i a lC h a n n c l s 3 7 5 in which C, - drag coefficientof thc sphere;y and p : specific weight and density of the fluid. respectively:7, : specificweighr of the solid; A,. : frontal areaof the-sphereprojectedonto a plane perpendicularto the palh of the falling sphere(= nd2/1); d: diamererof the sphere;and x7 = fall velocity of the sphere.Solving for the fall velocirv-we have

(10.4) Unfortunately,Equation 10.4 cannot be solved explicitly for the fall velocity becausethe coefficient of drag, Cr, is a function of the Reynolds number (Re : t!;r1lz),where z is the kinenraticviscosityof the fluid. The Reynolds numberobvi_ ously involves the unknown fall velocity.The Co vs. Re diagram for a sphcreis s h o w ni n F i g u r e 1 0 . l . The dilemma of solving Equation 10..1can be overcomein severalways. One approachis to assumea valueof Co, solve for the fall velocity from Equation 10.4 and computethe Reynoldsnumberto usein Figure 10.I to obtain the next valueof C, in an itcrative prtxess.To developa numericalsolution procedureinvolving a nonlinearalgebraicequationsolver,best-fitrelationshipsare availablefor Cr,,such as the one given in Figure l0.l as suggestedby.White (1974):

21

-

+ _

6

Re r+\&;

r

"

n,t

(10.5 )

1E4

1E3

.9 1E2 .9 o

(i

o 1E0

1E-3

1E-2

1E-l

1E0 1E] 1E2 Reynolds Number, Fe

rtJ

I'IGURE IO.I Coefficientof drag for spheres(besr-firequationfrom Whire 1974).

1E4

1E5

316

l hannels C H A P r I R l 0 : F l o wi n A l l u v i a C

u , h i c hi s v a l i d u p t o a R c y n o l d sn u t n b e ro [ a p p r o x i m r t c l y2 / I 0 5 w h e n t h e d r a g crisis occurs as the lanlinarboundary layer changesto a turbulcnt boundary layer and the separationpoint moves further downstreamon the surfaceof the sphere. Iterationor a nuntericalsolution of(10.4) is unnecessarl'.ho*'ever,for the Stokes range (Re < l), for which there is an exact solution by Stokesfor the drag force and cocfficient of drag under the assumptionof negligible inertia terms in the Navicr-Stokesequations;that is, creepingmotion. In this specialcase,Co = 24/Re or rhe drag force D = 3zpr1d. Substitutingthe Stokes solution for drag force on the left hand sideof ( 10.3)and solving for the fall velocity givesStokes'law for the fall vclocity: rr'l

t)sd'

| O,lt l8

v

(10.6)

in which 7, - specificweightof the sphere;7 = specificweightof the fluid;/ : diameterof thc sphere: andv = kinematicviscosityof the fluid.Stokes'la* is limitedto Re < 1, whichcanbe usedto substitute into (10.6)for the fall velocity,l1,,, to obtainthe maximumspheresizefor whichStokes'lawapplies.The resultfor-a quartzspherefallingin waterat 20"C is d.,, - 0.1 mm. whichis a veryfinesand. For sphericalpaniclesoutsidethe Stokesrange,an alternative to the iterative solutioninvolvingFigure10.1,or thenumericalsolutionusingEquation10.5,is to analysisof theproblem.The difficultywith Figure10.I rearrange thedimensional whereas is thatit wasdeveloped for predicting thedragforceon a sphere, theproblem of interesthereis thc determination of fall velocitl of the sphere,andthe fall velocityappearsin the definitionof both CD and Re. However,accordingto the group can be replacedby some rulesof dimensional analysis, any dimensionless in Cbapterl. In this case.a good combinationofthe othergroupsas discussed choicewould be CrRe: because the fall velocityis eliminatedin this group.The evaluation of a reiateddimensionless srouDcanbe obtainedfrom

QJt - t)sd' : c'n; I v'

(10.7)

in which the constantof 4/3 on the right hand side has been moved to the left band side. Now define a more convenientdimensionlessnumber,d", given by

, - f0,lt t

- 1)sdr'Jt: r v-

l l

(10.8)

TakingEquation10.5for thedragcoefficientandplottingRe vs.dr resultsin Figure 10.2,in which the abscissa is calculated from (10.8).The Reynoldsnumber thencan be readdirectlyfrom the figureto determinethe fall velocityoutsidethe Stokesrange. just developed It remainsto applythe methods for spheres to sediment paniclesthat are not spherically shaped. One methodfor accomplishing this taskis to definethe sedimentation diameteras describedin the sectionon sedimentsize. whichrelatesthefall velocityto thediameterof a fictitioussphere havingthesame fall velocity as the givenparticle.Unfortunately, sedimentation diametervaries

C H A p T E R l 0 : F l o w i n A l l u v i a lC h a n n e l s

1E5

1E4

rl)

E

1E2

1E1

1E0

'1E1

1E2

1E3

1E4

FIGURE10.2 particle diameter 1.. asa function of dimensionless Fallvelocity of a sphere for a fluid temperature of 24"C, with Reynoldsnumber,so it hasbeenstandardized and called the sttndardJall diameter lf the fall velocity of a sedinent has been from Figure10.I andEquafall diametercanbe determined measured, its standard by tion 10.4.However,for sandgrains,the sievediametcrd. usuallyis measured takingthe geometricmeanof the sie\e sizesjust passingand retainingthe given fiom the sieve sandgrainin a nestof sieves.What is neededthenis a conversion diameterof the actuals€dimentto the fall diameter,whichdependson the shape factor,as shownin Figure10.3.Oncet}refall diamcteris known.anyof the methfor spheres canbe usedto obtainthe fall velocity.Fonunately, odsjust discussed the fall diameterdoesnot vary significantlyfrom the standardfall diameterover a remperature rangeof 20' to 30"C. diameterto find the fall velocity,the to usingsedimentation As an altemative directlyandgivenin a C, coefficient of dragof sandpaniclescan be determined vs. Re diagramlike thatof Figure10.1.Engelundand Hansen(1967)havesuggestedthe followingbestfit to thedatafor sandandgravel(Re < 101): .D-Re

'.

(10.9)

Equation10.9canbe usedin combinationwith Equation10.4for the fall velocity to obtainan exactsolutionfor the fall velocity,whichis givenby (Julien1995): u. tl.

** - T'

s [ v 4 . o o r r o , i rl l

( 1 0 l.0 )

378

CliAprER l0: Flow in Allurial Channels

10

s.F.= 0.3 05 0.7 0.9

E E 6 E i: 9 1 0 .9 g)

.9

E o

( ' w l

0.1

1.0 Standard FallDiameter, mm

10

FIGURI!TO.3 Relationship betweenfall diameterandsievediameter for differentshapefactorsof natu(U.S.lnteragency rallywomsandparticles Comrnittee l95i). Ex A Ntp L E I 0 . I . Find the fall velocity of a mediumsandwith a sievediameterof 0.50mm (0.0O161 ft) fallingin *,aterat 20"Cby two merhods: (l) usingFigures10.2 and 10.3and(2) from Equation10.10. So/,l/ion. From Figure 10.3.for a sievediameterof 0.50 mm (0.0019 ft) and a shape factorof 0.7,rhesrandard fall diamereris 0.,17rnm (0.00154 ft). Then,we calculare ./, for the sphere*irh fall diameter,dr, as t.os x 9.81 x o.ooo4?3 1'13

d. ' - IL - - t r

^roY

I

-lle

FromFigure10.2,Re - 33 so that bi = 33 x (l x 10-1/0.00047= 7.0 x l0-2 rn/s (0.23ftls). In the secondmethod,which can be usedonly for sandgrains,d. is recalculated fo. the sievediameter, d,, of 0.5 mm to givea valueof 12.6.Then,we substiture into ( 10.l0)to obtain \+id,

.lil = s ;1t\4 + oollttltF - rl = rs v

: 7.0 x l0 : rn/s(0.23frs). from whichhl : 35 x (1 X l0 6)/0.0005 Grain Size Distribution While some naturalsortingoccurs in rivers with the formationof a thin armor lar er of coarserpanicles in the bed under conditionsof degradation,generallya .*.ide

CHApTERl0: Flow in AlluvialChanncls

319

rangeof sizes can be found in transportand in the riverbed. Some measureof the degrceof sorting of thc grain sizesis requiredusing slatisticalfrcquencydistributions. The lognormal probability density function commonly is applied to river sands,with an estimateof its parameters(meanand standarddeviation)being used to characterizethe particle size distributionas obtained from sieve analysis.The lognormal probability density function simply is the normal probability density function applied to the logs of the sievediameters,so it is given by

I

f(o v G -

(10.1l)

in which ( - (log 4 - p)/o, tl, rs sievediameter;p is the mean of the logs of the sievediametersiand o is the standarddeviationof the logs of the sievediameters. The geometricstandarddeviation,on. is used more often to describegrain sizedistributions,and it is defined by logtr, = o. The cumulativedistributionfunction,F((), is usedto relatethe theoreticalprobability distributionof ( 10.I I ) to the resultsof a grain-sizeanalysis.lt representsthe cumulativeprobability that a grain size is lessthan or equal to a given sievediameter, and it is measuredas the cumulativeweight passinga given sievesize as a fraction of the total weight of the sedimentsample.Mathematically,it is obtainedfrom the areaunderneaththe probabilitydensityfunction as

F@= r[' ",',a, y2tr J

(1 0 . 1 2 )

_

in whicht is a durnmyvariableof integration, and 100 x F(O - percentfinerof thetheoretical lognormaldistribution. Shownin Figure10.-1 aretheindividualdata 99 99 99 95 998

loq o.

99 9a 95 90

84.10/.

! e o

50%

c 5 0 R 4 0 o , '"^

o-

15.9%

10 5

02 o1 005 00r

0.001

0.01

0.1 SieveOpening, mm

FIGURE IO.4 Sizedistribution of a sandsampleon log,normalscale.

10

180

CHApTER l0: Flow inAliuvialChannels

pointsof a siere analysisplottedon a lognornralgrid. The abscissavaluesrepresent sievesizesplotled on a log scale,while the ordinatesare perccntfiner valuesplotted on a nomral probabilityscalesuch that a theoreticallognormal cumulativedistribution function plots as a straightline. The actualdata show somecurvatureand deviationfrom the lognornal distribution,especiallyat the tails of the distribution. The data are fitted by drawing a straight line betweenthe 84.1 percentfiner size (dr.r,) and the 15.9percentfiner size (d15e),uhich representsthe distancebetween plus or minus onc standarddeYiationfrom the mean.Expressedin terrnsof .'!,lhe distanceis plus or minus one times log os, as illustratedin Figure 10.4.The intersectionof the straightline with the 50 percentfiner ordinate is definedas the geometric mean sievediameter,dr, as shown in Figure 10.4. shile the intersectionof the curvc connectinSthe datapointsand the 50 percentfiner ordinateis the median size.drn.These may or may not be the same,dependingon the actualsizedistribution data. Both the geometric standarddeviation and the geometric mcan size can be e x p r e s s eidn t e r m so f d r . , a n d r ! ; s u .B y d e f i n i t i o n l, o g o . - ( l o g d s r r l o g d ) = (logr1" logd,re), which can be expressedas

"t-

dso,

4

-

dr

a,-

( r 0r.3 )

ir is immediately apparent thatdn - (d8.1 Then,by cross-multiplyjng, rdr5e)r/r.Fur= (dror/d,.e\l/2. value by back substitution, the of o, thermore,

10.3 INITIATION OI- NIOTION Deternining thestability of thebedandbanksof a naturalalluvialchannel or of the rock riprap lining of a constructedchannelas in Chapter.l dependson the definition of the thresholdof sedimentmovement.In a qualitative sense,sediment grains in a noncohesivesedimentbed begin rolling and sliding at isolated,random locationson rhe bed as the thresholdcondition is just exceeded.The thresholdcondition can be describedin terms of a critical shear str€ssor a critical velocity at which the forces or momentsresistingmotion of an indir idual grain are overcome. The lorces resistingmotion in a noncohesivesedimentare due to the submerged weight of the grain, while in a cohesivesediment,physicochemicalinterparticle lbrces offer the primary resistanceto sedimentmotion. This sectionfocuseson the caseof noncohesivescdiments. If the thresholdof motion is definedin termsof a critical shearstress,r, , it can bc given as a function of the following variables:

r. - f 17,

"y.d. p, t!)

( 1 0 l.4 )

: sedimentgrain in which 7, 7 - subnrergedspecificweight of the sedimentld : sizel and p and g. fluid densityand dynanricviscosity.respectively.Dimensional a n a l y s i so f ( 1 0 . 1 4 )l e a d si r n m e d i a t e ltyo t h e r e s u l t

C H A P T F . Rl 0 : F l o w i n A l l u v i a lC h a n n e l s

r,

(r,

" ( -u , , d \

v)d

't -. t\

381

( 1 0 l.5 )

|

u t

in which n.. : 1r,/p)tt2 = critical value of the shearvelocity; atldv - LLlp : kinematic viscosity.This is the resulr thar Shields( 1936)obtainedmore indirecrll'.The dimensionlesscritical shearstresson the left of ( 10.l5) is refened to as the Srields poranteter,r"., and the dinrensionlessparameteron the right of (10.15) has the form of a Reynolds numbeq which is called the critical boundary ot particle Reyttolds twnbe r, Re.,. Shieldswas an American who, in Berlin in the 1930s,conductedflume experimentson initiationof motion and bedloadtransportof sedimentas affectedby the spccificgravity ofthe scdiment.He utilized sedimentsofbarite, amber,lignire, and graniteto obtain a \\'jdc range in the subnergedspecific weight of sedimenrfrom 590-32,000 N/mr (4-200 lbs/frr). The sedimentgrains were subangularro very angular,with mediansizesranging from 0.36 to 3.4,1mm (0.0012ro 0.01ll fi). He combined his results with those of previous investigationsat the same research institutethat were conductcdon river sandsby Casey ( 1935) and Kramer ( 1935), as well as addingresultsof Gilbert ( l9l4) and the U.S. WaterwaysExperimenrStation (WES) for river sands.He presentcdthe resultsaccordingto the dimensionless groups given in Equation 10.15 as a shadedzone for the beginningof sediment nrotion in what has conte to be called the S/rields diagrum, althorsghit has undergone a numberof revisions.Rouse ( 1939)first presentedit in the English literature and replaccdthe shadedzone with a curve.The Shieldsdiagramis given in Figure 10.5with additionaldata and modificationsproposedby Yalin and Karahan( 1979).

\ 2," 0.1

B\

tr Shields + USWES S.J While-oil Q Gilberl O Yalin V Neill i Kramer a Karahan x Casey

o

'*ffi

v v l

E

Smooth

0.01 0.01

0.1

I

rransit i o n l o ,rllyrough 10 Re'"

100

1000

10000

FIGUREIO.5 The Shieldsdiagram as updatedb)'Yalin and Karahan(1979). (Sorrr..,.M. S. Yatind d E. Karahan. "lnception of Sedintent Transport," J. Htd. Dit.. @ 1979, ASCE. ReproJuted b,pernission of ASCE.)

C H A p I L R I ( ) : l ' - l o t vi n A l l u v i a lC h a n n e l s

As given in Figurc 10..5. the paranetershave an instructivephvsicalintcrpretarion. The Shieldsparantclcr.r... can be interpretedas the ratio of the shearsfess to the submergedweight ()l a grain per unit of surfaceareaat critical conditions,while the boundaryReynoldsnurnber,Re.., represents the ratioof thc graindiameterto the viscous sublayerthickncss(ignoringthe constantin the exprcssionfor the viscoussublayer thickness6 = 11.6r,/a,).Accordinglv.regionsofsmoorh. rransitional, and fully rough turbulentflow over a grain could be expectedas shoun in Figure 10.5as the grain size beconrcslargerrelariveto the viscoussublayerthicknessand the individual grainsprotrudefrom it, creatingboundary,gencrated turbulence. The data for Re." < I in Figure 10.5 were obtainedprimarily for fine,grained silica solids tbat were cohesionless. For this range,in which the boundarylayer is smooth-turbulentor laminar,MantL (1977) proposeda relariongiven by

r , . : 0 . 1( R e . . ) o r

(l0.l6a)

Yalin andKarahan(1979)showedthata separare laminarflou curve,whichis not shownin thefigure,existswhentheboundaryReynoldsnumberexceeds unityand suggested thatthe laminarandsmoothturbulentdatacoincidefor Reynoldsnumbers lessthanunity because the grainsare submerged in the viscoussublayerin both cases.For Re.. ) I, YalinandKarahan( 1979)addeda considerable amount of additionaldatato theoriginalShieldsdata,u,hichincludesdatapointslabeledas Shields,Gilben, Kramer,Casey,and US'rVESin Figure I0.5 as sumnarizedby Buffington(1999).Basedon the additionaldara,panicularlyin rhe fully rough region,theconstant valueof r.. in thefully roughturbulenr regionis 0.0,15, andthe transitioncurveproposed by YalinandKarahan(1979)canb€ fittedby r-. :

)e,(tog Re..)i

( 10.r6b)

i n w h i c h A o = 0 . 1 0 0 ,A r = 0 . 1 3 6 1 ,A : : 0 . 0 5 9 7 7 A , r = 0 . 0 1 9 8 4 ,a n d A o : ( 0.01134 for I < Re." 70 wirh r.. = 0.0.15for Re,. ) 70. However,the acrual limits of the transitionregionare given by Yalin and Karahan(1979) as 1.5 < Re.. ( 40. BecauseRe-. is defined usually in terms of d50,and raking t, : 2dro,these fimits correspondto 3 1 u,k,/v < 80, which is sirnilar to the rangeof 5 to 70 for u,k,/v glen for the transitionregion in pipe flow by Schlichring( 1968). The manner in which Shieldsobtained the critical shearstressfrom both his experimentsand those of othersis a matter of some contro\ersy(Kennedy 1995; Buffington 1999).Shields'original rabulareddara were losr during World War II. and the descriptionsof methodologyin his doctoralthesisare r agueand sometimes contradictory. BecauseShields conlinued his career in machine design in the United Statesafter finishing his doctoraldissenationratherrhan in sedimenttransport, he was unawareof the intpactof his work until near his deathand so shedno light on th,Jcontroversy.The critical shearsrresscan be obtainedeither from visual observationof the thresholdof motion or from extrapolationof measuredsediment transpo;1rates10zero. Kramer's work is basedon the visual classificationof sediment rnotionas ( | ) weak movement.defined as the motion of a few or severalsand particlesat isolatedpoints in rhe flunte bed: (2) medium movement,describedas motion of many sand grains too numerousto be countedbut without appreciable

C r { A p i r ' Rl 0 : F l o q i n A l l u v i a C l hannels 383 sedimentdischarge:and (3) generalmovement,characrerizcdas ntotion of grains o f a l l s i z e s i n a l l p a r t so f t h c b e d a t a l l t i m e s . K e n n e d y ( 1 9 9 5 ) s u g g c s t c dt h a t Shields nra;' have uscdthe visual observationmethodde\ elopedby Kraner in previous experinrcntsin lhe same l'lume.basedon what appearsto have beenaveraging by Shiclds of Kranrer'suidcly varying data for critical shearstress.Basedon analysisof the data ol other investigatorsused by Shields. Bulfington ( 1999)con, cluded that Shieldsprobablydid use the dcfinition of '*eak nlovement"as the criterion for thrcshold conditionsfor the data of Casey. Kramer. and WES, while it "general appearsthat he uscd movement" for Gilbert's data. On the other hand, Buffington surmisesthat Shieldsmay have usedthe ntethodofextrlpolarion of sedinrent dischargeto zero for his own databecauscof the statementin his dissenation that this was the appropriatemethod for uniform sedimenrsand referencesto his data else\r'herein the thesisas being rcpresentative of uniform sediments.Regardless of the method used for obtainingcritical shearsrress.additionaluncenainties e x i s t i n S h i e l d s ' o r i g i n adl i a g r a ma s a r e s u l to f t h e u s e o f b o t h m e a na n d m e d i a n grain sizes; the existenceof bed forms in some of the data, which causeoverestimalion of critical shearstressl the lack of true uniformitl of the sedimentsizes;and the variability of sedinrentangularityof the sedimenrsused (Buffingron 1999).We can conclude that, although the Shields diagram is a ralid representationof the physics of initiation of sedimentmotion, its usersshould recognizeit as a band of d a t aa b o u ta g e n e r a rl e l a t i o n s h i fpo r i n c i p i e n rm o r i o n . As presentedin Figure 10.5,the Shields diagram is not very convenientfor directly estirnatingthe critical shearstress,becauseit appearsin the definition of both the Shields parameterand the boundaryReynolds number.To use the Shields diagram to estimalecritical shearstress,a third dimensionlessparameterthat eliminates the critical shearstresscan be introduced.Such a parameteris given, for example, by J0.I Re..2/z..Jr/r, so that an auxiliary set of curves can be constructed on the Shields diagram, the intersectionof which with lhe Shields curve allows direct determinationof the critical shearstress(seeVanoni 1977).On closerexamination, however.the auxiliary parametercan be recast as the dirnensionless grain diameter d, : 1Re,l/r..lri3that was encounteredin the developmentof a relationship for fall velocity of sandgrains.Accordingly, the Shields diagram is replotted in Figure 10.6 as 2.. vs. d., as suggestedby Julien ( 1995), so that the crirical shear stresscan be determineddirectly,sinced" is a function of only the grain diameter and specific weight, and the fluid specificweight and viscosity.The curve in Figure 10.6 has been converteddirectly from the updated relationshipproposedby Yalin and Karahan(1979) in Figure 10.5. Of particularinterestin Figure 10.5or 10.6 for coarsesedimentsis the critical value of the Shieldsparaneterin the region of fully rough turbulentflow, where it approachesa constantvalue.In this region,a constantvalue of the Shieldsparameter implies that the critical shear stressis linearly proportional to the grain diameter Rouse( 1939)initially indicateda constantvalueof r." : 0.060 in his drawingof the Shieldscurve nearthe upperrangeof Shieldsdata.although someextrapolationwas involved. Laursen (1963), in his developmentof a prediction equationfor bridge contractionscour,useda value of r.. = 0.039, while the value in Figure 10.5from Yalin and Karahan is approximately 0.0,15.Julien suggestedthat the constant value of r." : 9.96 ,un t. where { : angle of intemal friction to account for the size and

384

C A P I l , R l 0 : F l o \ | 'i n A l l u v i a lC h a n n e l s

Smooth \

Transition \

;Io

S

t'c 0.1 :

0.01 0.1

r

l1

rl

ruttYrougn

\-o

\o \

\\1

3,u

l

\ \)

x=

'ii,, I 1 l,:t , ,' 10

100

1000

FIGUREI0.6 of criticalshearstress form of the Shieldsdiagramfor directdetermination An altemate (afterJulien1995).(Soirrce;P Y.Julien,ErosionandSedinetation,A 1995'Canbridge Universit\Press) of Cambrid7e withthepermission (J irersityPress. Reprinted angularityof the grains. ln this formulation,r". variesfrom 0039 for very fine grivet tob.050 for very coarsegravelto 0.054for bouldersin theconstantt'. region in which d- is grealerthan about40. The variabilityof the constantvalueof r.. for largevaluesof the boundary that a rangeof numberand the scatterof datain Figure10.5emphasize ReynoltJs "criticalconditions"shouldform the ShieldsdiagramAccordingly, two additional enor in standard a times the I by are defined which 10.6, in Figure curvesappea.r given there' data the 10.5 and Figure in the curve log unitJbetween Regardlessof the value chosenfor the Shieldsparameter,a corresponding (1938)equationfor from Keulegan's valueo] criticalvelocitycan be calculated velocity, 4.., is relatedto 7'. value of shear critical If the flow. turbulent fully rough equationbecomes with waterasthe fluid, Keulegan's

It2.2Rl v , - 5 . 1 5V r . , ( S G - l ) g d 5 sl o g " l

(10.17)

R = hydraulicradius;andft, = in which SG = specificgravityof the sediment; in Chapter4, from which varies,as discussed equivalentsand-grainroughness, velocity, which is a meao critical that the to note is of interest 3.5du.It l.'4dt4to the flow and therefore radius hydraulic with the varies velocity, crosi-sectionai veloccritical reports of Hence, parameter. Shields value of the the same for deoth depth with a specific correspond sbould grain size varying of sediments ity for instead of is used equation If Manning's applicable. they are which ,ung" ou", Keulegan'sequation with Manning's n expressedin terms of a Strickler-type

C H A P t tR l 0

F I n \ \r n A l l u \ i J lC h r n n c l \

185

lhen the critical $'alervelocity for a vcry wide channt'lcan expression1a : c,,r/]16), be expresscdas

,, -

f,rrtt"

6 - t) r,,tl,r.r'i

( 1 0r.8 )

in rvhich K" = 1.,19in English units and 1.0 in SI unitsl c, - constantin Stricklertype rclationshipfor Manning's rr Qr = c,,dllo),which is equal to 0 039 in English units and 0.0.175in SI unitsl SG - specificgravity of the sedinrent;7.. = cnlical valueof the Shieldsparameter:d 25, then a plane bed resultsand ti value ti may contintreto increase of previously, the mcthod, but as discusscd to Julien according rivers. very large beyond T: 25 in

Karim-KennedyMethod waspresented problemin alluvialchannels to thestage-discharge A thirdapproach to a datau'as applied analysis by KarimandKennedy( 1990).Nonlinearregression the most sigof 339riverflowsand608 flume flowsto determine bise consisting sediment as well as variablesaffectingdepth-discharge nificantdimcnsionless Thc databaseincludedthe laboratorydatareportedby Guy, transportrelationships. (1966)as well as field datafor the MissouriRiver;MidSinions.andRichardson dle Loup, Niobrara,and ElkhornRivers in Nebraska;Rio GrandelMississippi Depthsvariedfrom 0.03to l6 rn (0 1 to 52 ft): River:anricanaldatafrom Pakistan. coveredtherangefrom0 3 to 2.9 rrVs( I .0 to 9.5 ftls);andsedimcntsizes velocities The f'lowresistfrom 0.08to 28.6mm (2.6 x l0-a to 9.4 x l0-2 ft) wereincluded. which/ is the in factors friction flfo ancewas formulatedin termsof the ratio of friction reference is a bed, and/o friction factor for flow over a moving sediment type of given by a Nikuradse-Keulegan factor for flow over a fixed sedimentbed as relationship I

l5

t 2 r , "] l

(10.40)

?sloc2id:]

( 1966)analysisof flow basedon Engelund's in whicht, - 2.5 dro.Itwasassumed. of ripple or dune with the ratio over lowei regime beds,that//0 varies linearly heightto flow depth: f A L:1.2O + 8.92)o fo

( 1 0 .r4)

with the coefficients detcrmined from the river and flume data lt remains to oblain a relationship for A/yo, which was developed in the original Karim-Kennedy

C H A p T E Rl 0 : F l o * i n A l l u v i aC l hannels 401 Inethodfrom data by Allen ( 1978) in terns of the Shiclds parameter.The besr,fit relationshipwas given by

I

: 0.08+ ' , . ( : )

, 8i i ( ; ) ' .

'on(?)'- 8833(;)' ( 10.12)

for z. < I.5 and,1/r'o= 0 for ;. > | .5. Karimand Kennedythenappliedregressionanalysis to theirdatasetro obtaina relationship for dimensionless velocityas a functionof relativeroughness. slope,and//0, whichis givenby

."Gc - Drd,

= 6.683(,/lq)""s"',(

"*'

/ )

( 10.43)

in which SG - the spccificrrarity of the sedintcnti d50= the mediansediment = size;S: bed slope;_Ih deprh:antlflfo is obtainedfrom Equations10.41and 10.,12. For a givendepth,the \elocity can be calculated directlyfrom Equation 10.,13. The bedformsare identifiedas beingin lowerregimefor r" ( 1.2,transition for 1.2< r. < 1.5,andupperrcgimefor r- > 1.5. It is interesting to contpareEquation10.43with Manning'sequationfor a wide channelby rearranging it for SG = 2.65andg = 9.81m/s2to yield an expression for Manning'sa givenby

oo:za3;"(rl)""

( 10.44)

in whichverysmallexponents on S andro hayebeenneglected. EquationI0.44 is in SI unitsandsimilarto the Strickler.equation with an exponenton droof 0.126, whichis closeto theStricklerr alueof;, butwith theveryimportantadditionof the of the bedfonns.This equationemphasizes //o term,whichreflectsthe resisrance the signif-rcant role pJayedby bed formsin alluvialchannelresistance and underscoresthe mistakesthat can be madeby applyingestimates of Manning'sn for hxed-bedchannels from Chapter4 to alluvialchannclswith movablebeds. Karim andKennedyalsode\elopeda simplifiedprocedure for computingthe transitionponion of discontinuous depth-velocity curves.The upperpan of the lowerregimeis assumed to occurat aboutr. = 1.3,whilethelowerpanof theupper = regimeis definedat r0.9.The corresponding depthsfor thesetransirionpoints thenarecalculated from thedefinitionof r.. The lowerregimerelationship is constructedasa straightlineon log-logscalesfrom thecomputed depth-velocity point fbr theminimumdepthto the lorler-regime transition depth-velocity pointwith//0 = 4.5,themaximumvalue.The upperregimerelationship is developed in the same way from the maximumdepthro rheupper-regime transitiondepthwithl7i = L2. Horizontallines aredrawnfrom the lowerto the upperregimerelationshipsat both thetransition poinrsat theupperlimit of thelowerregimeandthelowerlimit of the upperregimeto representthe falling and risingportions,respectively, of the depthvelocityratingcurvesfor risingand fallinghydrographs. Subsequent research by Karim (1995)revisedtherelationship for A/_r'o in terms of the ratioa./x7,theratioof shearvelocityto sedimentfall velocity,which is an

402

l hannels C H A P T E Rl 0 : F l o wi n A l l u v i aC

indicttor of the relalivc contributionof bedloadand suspendedsedimentload to the roral sedimentload. One slatedadvantageof this changc is to include the temperature effect on the bed forrn height,since fall velocity dependson the fluid tcmperature.The resulting relationshipfor J/_r'ois given by

I = -00+*o:sr(l.) 0 . 0 0 u 6tr,,f u . ) '0 0 J r etri( a . -, )o o o r r r { ' , . )

.\'o

1

\wJ /

,/

\

/

\wt /

( 10..r5 ) for 0.15 < uJu, < 3.64, and A/,r'o: 0 for u,/w, < 0.l5 or r,/r, > 3.64. Equation 10.45 is based on only the laboratoryflume data reported by Guy, Simons, and Richardson(1966) and some Missouri River data. Equation 10..15in combination \r'ith Equations 10..10through 10.42 is applied to the full data set of rhe KarimKennedymethod as well as to l3 flows in the GangesRiver, Rio Grande,and Mississippi River to predict depth-velocityrating curves. Mean normalizederrors in both depth and velocity for all data setsare approximatelyl0 percent. More recently,Karim (1999) developedanotherr€lationshipfor A/.r'othat provides a better fit than previousmethodsfor a data set consistingof field data from the Missouri River, Jamuna River, ParanaRivel Zaire River, Bergshe Mass River, and the Rhine River as well as Pakistancanal data.The relationshioof Julien and Klaassen(1995) given as Equation 10.30also perfornredwell for rliis data set. f , x A M p L E 1 0 . J . T h e M i d d l eL o u pR i v e r i nN e b r a s khaa sa s l o p eo f 0 . 0 0 1a n d a mediangrainsized5o= 0.26mm (0.000852 ft). The valuesofdu, : 0.32mm (0.00105 ft) anddeo: 0.48 mm (0.00157ft). For a discharge per unir uidrh of 3.0 ftrls (0.28 m:/s),find the depthandvelocityof flow usingthe Engelundmerhod,vanRijn method, andKa.im-Kennedy method. Solaft'oz. Assumethat the channelis very wide so that R = yo in all the methods. |. EngelundMethod.Assumea valueof y6 = 0.3 ft (0.09 m). Then calculare11 as

0.3 x 0.001 7)65 (v, v)ds,' 1.65x 0.000852

, The velocity is givenby

Y = VeIi,S - s.75loc 2dd5 L6 l ] I or = v 3 2 . 2 x 0 . 3x 0 . 0 0x1 + 5 . 7 s , . r , : l . 8 lf t l s , o00,*l L6 or 0.55 rn/s.From Figure10.15,find r. or useEquation10.34aassuminglower regimebed forms, from which

,. = fLs?t - aoq: Vzs .x1o:r- ooo;: o.or Now calculate)o from the definitionof r. to give r(SG ,lo

l)dro

0.61 x 1.65x 0.000852 : 0.86ft (0.26m)

0.001

C H ^ p r E R l 0 : F l o \ l i n A l l u v i aC l hannels ,l0i Finallr.calculare z7: {r'o = 1.31X 0.86 = I 56 lir/s 10.145nr:/s.1. Because this is smallcrthanthegivenvalueof 3.0 ftr/s(0_lSmr/s), rcpeatfor a largervalueof yi. Forvj, : 6.5 fr (0.l-5m), rl = 0.36and V = 1.50ftrs(0.76m/s).Thenr. = 0.86and i{r = l.l I fl (0.37m) so rharq = 3.p2frr/s(0.2g1mr/s.1. This is cioseenough,bur checklbr lowerregimebedforms.Calcutale7o: TliJ = 62..1x l.2l X 0.001: 0.076lbs/frr(1.6Pa)rod srreampower= rat, = 0.0'16X 2.5 = 0.19lbs/(ft_s) (2.8 N/(nr-s)).Then,for a fall diameterof 025 mm (seeFigure 10.3),rhe Simons, Richardso dn i a g r a r(nF i g u r eI 0 . 1 2 ) i n d i c a r e s d u n e s . s o t h i s i s a s n l i s f a c r o r y s o l u r j o n : r , - L i I f r r 0 . J ?m r r n d V - 2 . 5 0t r - l 1 s 0 . 7 6r n \ ) 2. VanRijnivlerhod. Assumea depthof 1.0fr (0.10m) andfrom conrinuity, V = q/r-o: 3.0/1.0= 3.0f/s (0.91m/s).Thencalculaleu: from l2ri, 5 . 7 5l o s-

3.0 rr^ rn 5 7 sl o s - o o o ; -

- 0 .I 5I f t . ( 0 . 6 , 1.6 6. ,

By definition. rl = !lrl[(Sc - l)sdr,J:0.t53:/0.65 x 32.2x 0.000852) = 0.52. Obtain;,, by firstcalculating d. as

,

f { s C - r r s d l n, l I r . o s . . r 2I.,2 - 6s l2 2), which correspondsto coarsersediments. Finally, the estimation of the shearvelocity,a., from a uniform flow formula as (g-r'f)0 5 introduces errors because river flows seldom are uniform. The slope often is estimatedas the water surfaceslope,but it is very difficult to measure accurately.The estimationof a. affectsthe value of the von Karman constant if it is determinedfrom measured velocity prohles as well as the value of Ro directly.Thesedifficultiessuggestthat measuredvaluesof \ may be more reliable than predictedones. The suspendedsedimenttransponrate is conputed from an integrationof the productof the point velocity and concentrationfrom the referencebed level at : c to the free surfacewhere: : r":

s,:J .co:

( 10.66)

in which C. is thesuspended sedimentconcentration usuallygivenin mg/L or g/L so thatg, in thiscaseoftenis expressed as kg/s/min the SI systemor con\ertedto

, 1 1 , 1 C H A p T E Rl 0 : F l o wr n A l l u v i a C l hannels the English system as lbs/s/ft. The concentrationalso can be exprcssed as ppnr (pans per million) b_vucight, C..nn.,which is related to rhe concenirationin nrg,/L.

C -o". uY

c,.or.

106(sG)c"

r + q.(sc l)

c'..r, t I + C,,(SG-

ij

(lo 67)

in which SG = the specificgravity of sediment;and C, - the concentratron yol_ by ume defined by the r'orume of sedinrentdivided by t-herotar vorume of sedinrent and water. From Equation r0.67, it can be demonstratedthat sediment concentratron_expressed in ppm is equivalentto the units of mg/L (within 5 percent)as long as C, < 0.032 or C,.oo. ( 80,500ppm. Einstein (1950) substitutedthe Rouse solution for suspendcdseclirnent con_ centration(Equation 10.65) and the semi-logarithnricvetocity disrribution(Equa_ tion 4.16) into Equation 10.66to obtain the suspendedsedimint rransponrate. He assumeda valueof x: 0.4 and usedai in the calculationof Ro. The reference concentration,q, was calculatedfor a bed layer with a thicknessof two grain diame_ ters having a bed-load rransponrate determineclfrom the Einstein bed-load function. The grain vetocirl,in the bed layer was taken as the velocity at the edge of the viscoussublayer( l 1.6ll: ) so that

(10.68) The integrationof Equation10.66wasdonenumericallyandpresented graphin ical form. Furthernlore,Einsteinsuggested that the grain size distributionbe dividedinto sizefractionseachwith a representative giain size,4i, and that the suspended sedimentdischargebe computedfor eachsizefraction.The total sus_ pendedsedimentdischargethenis I p,g,,,in whichp, is the fraction by weightof the bed sedimentwith meansized,, The bed_load dischargefor eachsize fraction.alsois weighredby p, and addedto the suspended ,ldirn.nt dischargero obtainthe totalbed-material discharse. Theprincipalcriticismof theEinJeinmerhodology is theuseof rl andr : 0.4 in the Rous€exponentRo.The grain shearvelocitycl-earlyis the appropnate choice for depth-discharge predictorsand bed-loadtransponformulaswhen bed forms are present.However,the full contributionof turbulenceto suspended load,asreflected by the valueof a*, should be usedin the definirionof \. The decreaseof the von Karmanconstantfrom 0.4 to valuesas small as 0.2 for lieavy sedimentconcentrationsalsois norreflecred in theEinsteinmethodology. Vanoni1t946y suggested that the decreasein x resultsfrom dampingof the turbulenceby sediment, especialty nearthe bed.Regardless of thecriticismof the Einsteinmethodology,it is an impoi_ tanthistoricalcontributionbecause of its comprehensive approachindthe introduc_ tion of theconceptof probabilityappliedto sidimentdischarqe estimarron. Anothercomprehensive approach ro theestimation of suipended sediment dis_ . charge.and the conespondingtotal sedimentdischargehas been proposed by van Rijn (1984b).He employedthe paraboricdistributionof the seiiment diffusion coefficientin the lower half of the flow (Equation10.64)and a constantdrstribu-

C ir \ p H R I tl. Flo* in Allurrrl Chrnnels

.115

tion in the upper half of the l1orv(cqual to the maxinrurrrof the parabolicdistribution). purponedly to obtain betteragreementbel\\een nteasuredand predicteddistributions of suspendedsediment.The resulting prcdicted concenrrationdistribution is a combinalionof Equations10.60and 10.65 lbr the upperand lower halves oi the flow depth. respcctively.Van Rijn separatedthe ef-fectsof B and r on the Rouse exponentRn. Basedon the resultsof Colcman ( 1970) for the sedimentdiftu!ion coefllcientin the upperhalf of the flow. \ an Rijn suggesteda relationshipfor 6 sivcnbv

|"'l: F = t +-) l l u . )

(10.69)

tbr 0.1 < l7lrr, < l. The effect of turbulencedamping on reductionin mixing near the bed and thc changein the velocity profile $ as treatedby van Rijn by increasinr the valueof Ro insteadof decreasing,(, so thal R; = & + tr\, in which Ro is defincd with a value of r : 0.4. while A\ representsa mixing correctionfactor. L ltimrtely, the value of ,\Ro was obtainedas a result of fitting velocity and concentration profiles front the laboratorydata of Einstein and Chien (1955), Banon and Lin (1955), and Vanoniand Brooks (1957) for heavy sedimenlladenflows and simplifying the resultsto obtain 08[c-.l01 I 1 |

t Col

(10.70)

in uhich C, - the referenceconcentration(volumetric) and Co = the maximum \olumetric concentrationtakento be 0.65. Equarion 10.70is valid for 0.01 < wrlr, < L The referenceconcentration,C,, is modified somewhatfrom the value usedto deYelopvan Rijn's bed-loadtransportformula. The referencelevel a for determinin-e C,, is assunredto be half the bed-form height. J,. or the equivalentsand-grain roughnessheight,t., if the former is unavailable.Basedon only 20 flume and river data points, the expressionfor C, was determinedto be

4! y{ c" : ao.ol5 d

.

(10.71)

Finally,insteadof usingthe Einsteinapproachof weightingthe sizefractionsto determinethe suspended sedimentdischargefor a sedimentmixture,van Rijn developed an expression for an effectivegrainsizeof the suspended sediment, d, sivenby

h=,

- o o r r (-ol.) ( r 2 5 )

(r0.72)

This resultwasobtainedby makingseveralcomputations usingthe size-fractions methodandthendetermining theeffectivegrainsizethatwouldgivethesamevalue of suspended discharge. sediment Usingtheeffectivegrainsize(Equation10.72)to obtainthefall velocity,thetwo-pansolutionfor suspended sediment concenrrrtion $ith conectionof Ro (Equation10.70),the valueof p givenby (10.69),and the

416

C H A p T E R l 0 : F I o w i n A l l u v i a lC h a n n e l s

Nikuradse fully roughturbulentvelocitvprofile,Equation10.66for suspendcd scdimentdischarge is integrated with a reference conccnrrarion givcnby ( 10.7I ). The numericalintcgrationis simplifiedto obrainan approxinaterelationship lbr rhe suspcndcd scdimentdischarge givcnbv q,:

ItVloC.

( 10.73a)

in which V - mean velocity; _vo- depth: C, = referenceconcentrationiand the integrationfactor /, is crlculated from

I o l ^ -r I o ] , , t;t [;] I a lR; l' ;l lr2 R;l

( r0.73b)

The van Rijn bcd-loadformulais usedto calculatebed-loaddischarge, which is addedto thesuspended-sediment discharge to obtainthetotalbed-material discharge. While the foregoingmethodologycontainsseveralsimplifiedexpressions basedon limiteddatato describethe very complicated interaction betweenturbulenceandsediment panicles,vanRijn obtainedreasonable agreement betweenpredictedandmeasured sediment discharges for severallaboratory andfield datasets. The agreement wasshownto be comparable to the resultsfrom severalothertotal sediment discharge formulas. Usinga discrepancy ratiodefinedastheratioof computedsediment discharge to measured sedimentdischarge, 76 percentof thecomputedvalueswerein the rangeof discrepancy ratiosfrom 0.5 to 2.0.For comparison,the Engelund-Hansen andYangtotal sedimentdischarge formulas,described in the followingsection,had performance scoresof 68 percentand 58 percent, respectively, of the computedvaluesfalling in the rangeof 0.5 to 2.0 timesthe nreasured valuesfor thesamedataset.

Total SedimentDischarg€ just described In contrast to themethodologies for separate calculations of bed-load dischargeand suspended-sediment discharge, total sedimentdischargeformulas correlatetotal sedimenttransportratesdirectlywith hydraulicvariableswithout distinguishing between bedloadandsuspended load.Thisavoidsthedifiicultproblem of definingthedifference betweenthe two typesof loadandof determining the bed-load concentration at somereference level.If suchformulasperformat leastas well asthebed-load./suspended-load formulations, thenthereis muchto commend theiruse,not theleastof whichis a greaterdegreeof simplicity.However,for total loadformulasto be successful, they mustrely on as largea database of field and laboratoil measurements as possibleand be formulatedin termsof physically meaningful dimensionless parameters. The Engclund-Hansen formula(1967)for rotal sedimentdischarge, 4,, was derivedfrom energyconsiderations and the similarityprinciplesdiscussed previ-

C H A p T E Rl 0 : F l o w i n A l l u r i a lC h a n n e l s . l l 7

prcdiction. lt is ously in connectionwith the Engelundmethodfor depth-discharge g r v e no y

r r d ,= 0 . l r l :

I 10.?,1)

in u,hich c, - 2rt/pV)', d, = S,/ttSG - l)8/i0lr/rt and r, : Shields' Parameter, definedwith thc total bed shearstress: r/[(7, 7)dro].The cocfflcientand exponent in Equation 10.74 were obtained from correlalionof sedirnenttranspon data fronr the laboratoryexpcrimentsreponedby Guy, Simons.and Richardson( 1966). antl reasonablygood correlationrvasfound for dune bed forrns as well as transition and upper regimebed forms (Engelund 1967). Yang ( 1972, 1973)developedthe conceptofunit streampoweras an important independentvariable that determinestotal sedinrentdischarge.The unit stream power is defined as the po\rer arailable pcr unit weight of fluid to transport sediment and is equal to the product of velocity and energy slope. VS.A dimensional analysisthat includes unit stream power, VS; fall velocity, lr7; shear velocity, u.: median grain size,dro; and viscosity.v, suggeststhat the independentdimensionless variablesaffecting total sedimentdischargeor concentrationC, are VS/wt, vrrtlrdv, and uJwr. Yang (1971) modified the dimensionlessunit stream power, VSlw,,by subtractingits critical value at the initiation of motion, V.!/wr, in which 4 is the critical vclocity. A multiple regressionanalysisof .163setsof laboratory data for sandtransportin terms of thesedimensionlessvariablesgave the following relationshipfor total sedimentdischarge: rr,rdrn

ij locC, = 5.,135- 0.286log -

v

u.

- 0.457loe -

rNl

(10.75)

t) . ( v s v . s \ + ( t.tsg- o.+os toeab - o.:t+toe - r \ / -\ uj tr't / U v

by weightin ppm = lOEx y,q,/yq. The in which C, - total sandconcentration dimensionless criticalvelocityis definedby

v,

z.)

wl

log(u"d5e/z)- 0.06

+ 0.66

f o r 1 . 2(

u"d^ -i < 70 (10.76) v

and V"/w, = 2.05 for u.drolv > 70. Yang's(1973)laboratorydataset on which ( t966), Equation10.75wasbasedincludesIhedataof Guy,Simons,andRichardson Williams(1967).VanoniandBrooks(1957),andKennedy(1961)aswell as others for whichflow depthswereon theorderof0.03 to 0.30m (0.1to 1.0ft). Tbe coefequation was0.94.Equation10.75was ficientofdetermination f for tle regression verifiedwith Gilbert's( l9l4) laboratorydataandfield datafrom the NiobraraRiver (ColbyandHembree1955),l!'tiddleLoup River(HubbellandMatejka1959),and with the Middle the MississippiRiver (Jordan1965),althoughthe comparisons Loup andMississippirivers s'ere not quiteasgoodasfor the laboratorydata sets. predictors The Karim-Kennedy(1990) methodologyfor depth-discharge describedpreviouslyalso includesa total sedimentdischargeformula obtained of 339 river flows and608 flume flows. from nonlinearregressionusing a database

418

C H A P T E R l 0 : F l o w i n A l l u v i a lC h a n n e l s

Severalphysically reasonabledimensionlcssratios are used \\'ith a calibrationdata set (615 laboratoryand field flows), and nonlinearregressionanalysisis carriedout for the dimensionlesssedimentdischargeand \elocity. The resulting valuesof sediment dischargeand velocity then are comparedu'ith measuredvaluesfor a control data set and lhe least significant indcpendentdimensionlessvariables removed from the analysis.This processis repeatedseveraltimes until the final relationship is obtainedas l o g{ , = l o g

q,

\.{sc- lta

=

+ Z.g:2l:sl 2.2'19 -L {sG

'l

+ r.ooolos - It

I

- Dsd'o

u.-,.. I

l l o -eLl V ( S C - l ) . s d '| o l \,(sc - l kd, I

+ or99bs(*) r.cj

(10.77)

l

I

V(SG - I )sdrol

= discharge in which4, : total volumetricsediment Perunit width; V flow veloc= = grain sediment = specificgravity;dro median ity;y,1 flow depth;SG sediment = normalized velocity. The mean criticalshear shearvelocity;andr.. sizelu. error of Equation 10.77,definedas the meanof the ratios formed by the absolute over sedimentdischarges predicted andmeasured between valuesof thedifferences for the control data 43 percent values,is foundto be approximately the measured includes flow data set setand40 percentfor thecombineddataset.The combined depthsfrom 0.03 to 5.9 m (0.1to 19 ft), velocitiesfrom 0.3 to 2.7 m/s (1 0 to 8.9 ? ftli), drovaluesfrom 0.08to 28.6mm (2.6 x l0-4 ft to 9.4 x l0 ft)' andtotal from 9 to'19,300ppm by weight concentrations scdimentdischarge for the samedatasetsas a simplerpowerrelationship Karim (1998)proposed givenby with the rcsult analysis, employedin the Karim-Kennedy q,

\,{Sc=t4

: 0 00119 [ \,{sc

I '''

- Dsd,

l;]

" (ro?8)

The meannormalizedenor for Equation10.78is 45 percentfor the controldataset, which is not significantlydifferentfrom the performanceof Equation 10.77.The formulafor meannormalizederrorsfor theYangformulaandthe Engelund-Hansen 49 percent, respectively the samecontrol data setare63 percentand andfield datahavingnonuniform Karim appliedEquation10.78to laboratory is fractions. The sedimentdischarge by dividingthe sedimentinto size sediments partial bed armora by in eachsizefractionby Equation10.78multiplied computed ing factorand a hiding factor.The partialarmoringfactoris intendedto accountfor portionsof the bed that arearmoredandunalailablefor transpon'while the hiding factorukes into accountthe shelteringeffectof largergrainson smallergrains.The in eachsizefractionthenaresummed'andthe totalsediment discharges sediment from Equation1078 to thosecomPuted valuesfoundto be comparable discharge usingonly the mediangrainsize,d50.

C H A p r I R l 0 : F I o s i n A l l u r i a lC h r n n e l s 4 t 9

Scveral othcrtotal sedintenldischargclormulas can be found in the liter ture, i n c ) u d i n gt h o s eo f B a g n o l d( 1 9 6 6 ) .L r u r s c n( 1 9 5 8 b )A . c k e r sa n d$ ' h i r e( l 9 ? 3) , a n d B r o w n l i e ( 1 9 8 1 ) .A m o r e c o m p l e l er c v i e w a n d r a n k i n s o f r a r i o u sf o r m u l a sf o r computalionof total sedimentdischargecan be found in Alonso ( 1980),ASCE Task Committee ( 1982),Yang ( 1996),and Bechtelerand Vener ( 1989). In the lasr refer"recommcnded ence,the Karim-Kennedyformula was besl for common use" while "within the forrnulas of Yang and Bagnold, thc range of validiry." were found to "yield the most reliablercsults." Scdiment transportformulas should be chosen thar have a databasewithin which the flow and sedinrentconditionsof interestfit, and se\eral formulasshould be used and comparedwheneverpossible.For example.the Ensclund-Hansenformula is most appropriatefor sand transportin the lou er regime. while the MeyerPeter and Miiller formula should be choscn when there is coarsebed material in bed-load transport.On thc other hand, the Einstein-Broq'nformula is not a good choice when appreciablebed nlaterialis carried in suspension.Where they exist, gauging stations ale useful for developingsedimentrating cunes between measuredsedimentdischargeand eitherwater dischargeor relocit\'. However,the wash load has to be subtractedfrom the measuredsuspendedsedimenrdischarge,and the bed load and unmeasuredsuspendedsedimentdischargeusuallv have to be calculated and added to the measuredsuspendedsedimentdischargero obtain the total bed-materialdischarge(seeColby and Hembree 1955). E x A \I p L E l 0.1. TheNiobraraRiverhasa measured flotvdeprhof I .60fr (0.49m) andmeasured vclocityof 3.5?flls ( L09 nL/s)to giveq = 5.11ft:/s(0.53m2/s)with an energrslopeof0.0017.The mediansediment sized50= 0.27mm (0.000885 ft), deo: 0.,18mm (0.00157ft). and('{ : 1.58.The temperature is 68' F The meantotalsedimentconcentration for theseconditions wasmeasured to be 1890ppmby weight.Calculate the total sedimentdischargcusingthe van Rijn metho.1. Yangme$od, and Karinr-Kennedy method. Solanba. First,calculatesomequantitiescommonlo all threemefiods.For the given y : 1.08x l0 5 ftr/s(1.0 x l0 6 m?/s)andd, is obiainedfrom remperarure, - I d. = dso[(SG ) s / , ' ] ' ' ' : 0 . 0 0 0 8 8x5 1 1 . 6 5x 3 2 . ? r ( 1 . 0 x8 l 0 - 5 ) r l r l r= 6 . 8 1 The fall velocitythenis 8u' ' . ,- ; [ ( r ut0

. o o r j q / ] r o 't-l

8x1.08x10-5 + 0 . 0 1 3 9x 6 . 8 1t ) o t - l ' = 0 . 1 2 9f t l s 0.000885 [ ( l or 0.f,t393m,/s,and the critical valueof Shields'paramereris ;.. : 0.0,15from Figure = 10.0.15 10.6.The conesponding valueofr.. = ["..(SG - I )gd v,

if

Kr-21

I

(l0.l0la)

I

(l0.l0lb)

Thcse cxpressionsfor K, have the effect of collapsingthe scour data for both uniform and nonunifornlsedimentsin both clear-waterand Iive-bedscour' For uniform sediments,V = V, so that the det€rrniningsedimentmobility factor is V'IV. < I for clear-waterscour.For nonuniform sediments,it must be true that l/d > 4; otherwise. 4 is set equal to y.. The depthcffect, which is due to interactionof the surface rollei and the downflow on the upstreamfaceof the pier (Raudkivi and Ettema 1983),is accountedfor b!'

x, - our(;)""

t t ; < 2.6

K, = 1.0

t f ;

> 2.6

(10.l02a) ( 10.l02b)

The sedimentsize effect depcndson the value of b/d.., as given by

[ . r e ;I

'"tL o_ Kl - 0.s1

I

if

,

if

K,i-- IQ

)s aTso

(10.l03a) (10.103b)

size.d''",/l 8 The by thearrnorsediment drois replaced For nonunifonnsedirnents, provides an upper fornlulation and this maximumpossiblevalueof d/bis 2.4, method and Sutherland the Melville to the scourdata.The datarangefor envelope VrlV, 7 to 12, and values from 0 sizesfrom 0.24to 5.24mm,1,/b includessediment expression in the dePth changes (Melville Slight 1997). valuesbetween0.4 and5.2 (" were madeby Mehille (1997)to includewide piers(r''/b < 0.2) as well as intermediatewidth and narrow piers. analysisof live-bedscourat bridge Froehlich(1988)completeda regression givenby relationship a best-fit piersat some23 field sites.He presented r

f,.

1016

[

A l0oE

Ib - o . : z x l.br), Il ' F 9 L' ld:s oI J

(10.10,1)

67 = b cosq + in which r(, : pier shapefactor;(, : skewnessfactor = (b'lb)o ' b' = L^sinl,b = pier width; L" - piet length;-vr = depthof approachflow; F' = facgrain size The skewness median anddro F'ioudcnumbeiof approach'flow; is the sameas Equation1094 usedin the CSU formula The power tor essentially on bldroisvery small,indicatinga relativelyminor influence.Tbe coefficientof an envelope of Equation10.104is 075. Froehlichrecommended iletermination

C H A p ] ' E R1 0 : F l o u i n A l l u v i a C l hannels 43'1 curve oblaincd by addinga factor of safetyof L0 to the right hand side of Equation

t0.104. Comparisonsbetweenseveralpier scour forlnulasand Iaboratoryand field data have been made by Jones( 1983),Johnson( 1995), and Landersand Mueller ( 1996). Jonesconcludedthat the CSU formula envelopedall of the laboratoryand field data tested,but it gives smallerestimatesof scour deprhthan the Laursenand Toch,Jain and Fischer, and Melville and Sutherlandforntulas ar low values of the Froude number.Johnsonfound lhat all four of thesescourfonnulas havc high valuesof bias (ratio of predictcdto measuredscour depth) for yr/b < 1.5,with high valuesof the coefficientof variation(COV) as wcll. For y,/b > L5, rhc CSU formula performed well with a low value of COV and a bias from 1.5 ro | .8, providing a re'asonable factor of safety. In general,the Melville and Sutherland formula overpredicted more than any of the fornrulastesteduith bias values varying from 2.2 to 2.9 for .r,/b > 1.5. for cxample. Landersand Mueller (1996) evaluatedpier scour formulas on the basisof a much more extensivedata set of 139 field pier-scourmeasurenrentsfrom 90 piers at,14bridgesobtainedduring high-flow conditions.Data were separatedinto live-bed scour and clear-waterscour measurements. Although the data sho$ ed considerablescatter,it was concludedthar rhe influenceof flow deoth on scour depth did not become insignificant at large values of the ratio of flbw depth to pier width, as indicatcdby the Melvi)le and Sutherlandformula. In addition, no influence of the Froude number and only a very weak influenceof sedirnent size were found. Both the HEC- l8 and Froehlich scour fonnulas performed well as conservativedesignequationsbut overpredictedthe scourby largeamounts for many cases. Abutment scour Melville (1992, 1997) proposedan abutmentscour formula rhat is similar in form to the Melville and Sutherlandpier scour formula, arguing that short abuG nents behavelike piers.The abutmentscour fornrula is given by

d, - K,rK,KuK,KrKr;

(r 0 . 1 0 5 )

in which K represents expressions accountingfor variousinfluenceson scour depth:K,. = depth-size effect;K, = flow intensityeffect;K, = sedimentsize effect;K. : abutmentshapefactor;K, = skewness or alignmentfactor;andKo = channelgeometryfactor.The depth-size factoris definedby the followingexpresslons: ( 1 0 .l 0 6 a ) (10.106b) K,r - I 0.yr;

.. 2 5 r1>

(l0.l06c)

in uhich -r', : approachflow depth and L, = embankmentor abutmentlength. Theseexpressionsindicatethat scour depth is independentof depth for short abutmcnts (L,,h,r< I ) and independentof abutmentlcngth for long abutments(L,,/-)r>

-tlS

l hannels C s a p r e n l 0 : F l o wi n A l l u v i a C

25). The flow intensity factor essentiallyis the same as for piers, except that the maximum valrc of tl,lb = 2.4 for Piershas been removedto give

..

v r- ( v , v . )

X,=--

Kt= |

-

vt

lor

(10.107a)

V

V

for

(4-t,) _. 0 :fb€! Er.lt grrb 50 tor t.1to r 3 . { Y 1+ 1 2 } / 2 rr3 . r(Y3, llrq!|C, 0, g, b, !. D) !Z rf If rf

.lTlrlY3 sub !I . 0 ftcD lxlt !t < 0 Tb€n 12 ' Y! l1!. t1 ' Y3 lb3((Y2 - Yl) / Y3) < ER:lb.! Etlt

gub

[.:.t I laal sub lurctiotr Di! I l.

!(t, rl'oNc' Q, s' b, D, n) lr siDgl€ ! As glogl. P l! alngl., R t! shgl., sirglr, t . I i ( b + D r a ) P r b + 2 r f . 9 C r ( 1 + ! ^ 2 ) R . l / P t r b + 2 ' ! t l IlxrU[C'1lt.|l O.s 1.5/r' r : O - 9 q ! ( 3 2 . 2' 1a ^ !13a r ' Q -

( 1 . { 8/ 64 1 r 1 ' R ^

( 2/ 3 } ' s ^

( 1/ 2 )

lnd rf ttld ?urctlon

8.2 YCOMP PROGRAM FOR FINDING MULTIPLE CRITICAL DEPTHS INA COMPOUNDCHANNDL Ycompincludethedischarge Q andtheslopeS0,thelatInputdatafor theprogram with tei of whichis usedonly for thecomputationof normaldepthfor comparison

AppENDIx B; Examplesof ComputerProgramsin Visual BASIC

DATA II{PUT

INPUTDATAFILE None AALCULATE

RESULTS

FIGURE B.2 channelusing multiPlecriticaldepthsin a comPound VisualBASICform for determining theprogramYcomp.

critical depth(s).The input dataare placedin text boxesin the Visual BASIC form shownin FigureB.2- An inputdatafile that givesthe compoundchannelgeometry and roughnessis requiredandis specifiedby clicking on the INPUT DATA FILE button.The form codeis givenin Subcmdlnput.A sampledatafile is shownat the pairs' which definethe end of the modulecode.It consistsof the station-elevation with correspondingMancross-sectiongeometry,and the subsectionlocations to ning's z values.The subsectionlocationsmust correspond ground points,and verticalbanftsare not allowed.The modulecodefinds the locationsof the right and left banks,assumingthat the mainchannelconsistsof only one subsection.

AppENDtxB: Examples ofComputer hogramsin VisualBASIC 4jl The resultsarc computedby the modulecode, which includesa bisectionsubprocedurcto solvefor the critical depthsbasedon the compoundchannelfunction given by the functionsubprocedure FC. The programdeterminesif thereare values of critical depthin the main channel(lowerYc) and floodplain(upperyc). It also determinesthe dischargerange,if any,over which multiple critical depthsin the main channelandfloodplainexist.For discharges greaterthantheuppere, only the uppercritical depthoccurs,while only the lower critical depthexistsfor discharges lessthan the lower Q. In betweenthesetwo discharges,both lower and uppercritical depthsoccur.If the compoundchannelhas only one critical depth,then the lower Q equalsthe upperQ.

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As glaglc

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tfs|}ql lad

rf

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BAslc APPENDIx B: Exarnplesof ComputerProgramsin visual Db Dt! Dt! Dt! D!! Dl!

Pr t! strgl' R'r lt glnglc' g1ng1'' a! 8lng1' 63 82 tr DP lt 8bt1' Dx tr SlDgl'' It abtrl'' It| rt thgl' DPD! ls glngl" 8lBg1" Ar at!91' Dl!!tr t! ghglc' Trntf Tlinxl lr glDgl" :ntcg'r L A!

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lot

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D?Dt.DpDr+DP/Dl (Y1 +Y2) >0fb.l ( f 1 + t ? ) a : r + o . 5 . D A r - 12) ' Dp. gc!(Dx r DI + (11

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AppENDlx B : Examplesof ComputerProgramsin Visual BASIC

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8.3 PROFILECOMPUTATION WSPPROGRAMFORWATERSURFACE The programWSP computesnormal andcritical depth,classifiesthe watersurface profile, and computesthe water surfaceprofile in a trapezoidalchannelusing the iirect stepmethod.The VisualBASIC form is shownin Figure8.3. First,the channel discharge,Q; slope,S; bottom width, b; side sloperatio, m; and Manning's n areenteredin the text boxeson the form. Then,the valuesofcritical depth,YC' and normal depth,Y0, are calculatedas in the ProgramY0YC. For the water surface profile computation,the channellength,XL, and the control depth,YCONT' are intered next.If critical depthis the control,zerocanbe enteredin thecontroldepth box, and the program will assigl a depth slightly greateror less than critical

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in VisualBASIC Programs of Computer B: Examples 4'18 ApPENDIX dependingon the profrle classificationin order to avoid startingon the wrong profil;. The aALCULAIE PROFILEbuttonis clicked,and the profile type is classified and shown.The profile is calculatedin Sub PROFILE using the direct step merhodby dividing the depthintervalappropriateto the particularProfiletype into The specificenergyandslopeofthe energygradeline arecalculated 30 increments. in Sub SPENERGY.Inter?olationis appliedto obtain the depth at the end of the specifiedlengthof channel.The channelstationsincreasein the downstreamdirecof whetherthecomputationproceedsin the upstreamdirection(subti,onregardless criticaDor downstreamdirection(supercritical).If normaldepthor critical depthis approachedwithin 0.1 percentbeforethe specifiedchannellength is reacbed'the cbmputationis stopped.The resultsfor depth,YP; velocity,VP; and momentum function,MB are printed at eachstep,XP, in the RESUUIS picture box The output canbe directedto a printerwith the PRINT button.The programaccommodates horizontal,mild, and steePslopes WSP Form Code ODtloa EC,ltctt Dh O rs shgl.. Di! Y0 It glDglc, DL! stiPa Dl.[

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APPENDIx

Examples of Computer Programs in Visual BASIC

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AppENDrx B: Examplesof Computerprogramsin Visual BASIC

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APPENDIXB: Examplesof ComPuterProgramsin \4sual BASIC l P ! N a + 1 -

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