OpenChannelHydraulics i .l I . l
TerrvW. Sturm Georgia Ins'tut. oI Te.hnology
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McGraw-Hill Seriesin Water Resourcesand Environmental Engineering CoNsuLTtNcEDrroR George Tchobanoglous, Univenity
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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
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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 <-r (c)
ll/ = pg dA ds (d) FIGURI'I.2 (c) rivcrreach:(d) streamline (a) arbilraryconttolvolunle;(b) strealn(ube; Conkolvolurnes = arca of the control surface (cs) ity vectorl n : ounlard normal unit vectorl and A I 2 sumsup the values of Equation side of riSht hand Tie volume integralon the given by pdV ln the surface mass element each the propertyper unit massb over flux through an clementhe mass (pY d"4 rePresents n) integralin Equation 1.2, vector with the unit vclocily the product of The dot tal aread,4 on the control surface. perpcndicularto vclocity the of (Y compon.nt the n) determines out$ard normal the surface' through the can carry PfoPerty the surfacesince only that comPoncnt for in$'ard and ncgative ard fluxcs for outtr positivc product is Furthermore,the dot products the of sulns up irltc!ral thc sulfacc Thus, volume. fluxes inlo the conrrol to the strrfacc the control flux ovcr tllass and the !'iYc the propertyper unit massb
8
C H A P T E R1 : B a s i cP r i n c i p l e s
net outward flux of thc properly.In sunlrnary,Equation 1.2 slatesthat (he time rate of changcof the systempropeny is the suntof the tinte rate of changeof the prop_ eny insidethe control volume and rhe net outward llux of the property rhrough the control suface. The Rcynolds transpon rheorem can be applied to the properties of mass, momentum. and cncrgy to obtain the control volume fbrm of the corresponding govcrning conservationequations.The control ',olume forms of the equrri.,n. cai be simplificd for the caseof steadv.one-dimcnsionalflow and used in the analysis of many opcn channelfloll problcms. In the case of mass rn, the propen),I = rr and it follows thar dBldt : 0 and b - dBldn = l. so that
d t
I
+ i),,oov j,r(v'n)d"t
(1 . 3 )
whichmeanssimplythatthe timc rateof changeof massinsidelhe controlvolume in the firsttermmustbe balancedby thenetoutwardmassflux throughrhecontrol surface expressed by the secondterm.Now,in thecaseof steadyflow of an incompressible fluid for the one-dimensionrl streamtube shownin Figure1.2b,we have t h ef a n r i l i r rf o r mo f t h ec o n l i n u i r e ) quttion:
o o= 0 = : 0 . , , - : 0 , " /.tn."l
( L,1)
in which IQ - summation of the volume fluxes in or out of the control volume. The mean cross-sectionalvelocity, {. is defined as the volurne flux divided by the cross-sectional area of flo* perpcndicularto the streamlinessuch that the yolume flux can be written as
0:
v,e |,,,,,ae:
(1.5)
in which u, is the point velocity in the streamlinedirection; { is the nrean crosssc\'tionalvelor it) : rnd A is the cro\s-\eclionalrrea of flow. Equation I.3 also can be written in differential form for the general case of unsteadyopen channel flou, of an incompressiblefluid. If the control volume is consideredto have a differential lengrh lr, as shown in Figure 1.2c, then as Ar approacheszero, Equation 1.3 becomes
aA ao at
i).\
n
(1.6)
At anycrosssection,the time rateof change of flow areadueto unsteadiness asthe freesurfacerisesor falls mustbe balanced by a spatialgradientin the \,olumeflux Q in theflow direction.For sreadyflou..dAldris zeroby delinitionand6el0r then alsomustbecomezero,which irnpliesthatthe volumeflux e is constantalongthe channel,in agreement with Equation1.,1. The differentialform of the continuitv
C l i A P l t R l : B r r s iP c rinciplcs
c q u a t i o na s S i \ c n b ) [ ] q u l : r t i o ln6 w i l l b e a p p l i c di n t h c n u n l t ' r r . r trl i n a l r r i so f uDstcadyof^..r chrtrnelflorv in Clrlpler 8 I f $ c t u r l : r ( \ * t o t h c p r o p c r t l ( ) f m o m c n t u n ll,h c f u n d i l n l c i "r l p r ( r p c n vB i n t h e R e , " - n o l t ] i. . r r r . ; ) 1 ]tlh( c o r c r nb c c t t m c sa v c c { o rq u a n l i t \ d c l r ri J b } t h c I r n c a r m o m c n t u mt t = r ; , \ ' ,i n r r h i c f tr t r - m a s sa n d V = v c l c r i t ) \ c ' c l { r .f.h t l o l r l d ! ' n v a t i v c d B / d l i s c r a e t l l l h c \ r ' c t ( ) rs u m o f f o r c c sI l ' a c t i n g o n t h c c o r r { r o \l ( ) l u n l e a c c o r d i n gt o N c r r l o ns s c c o n dl a w .l n t h i sc a s e d. B / d , x = \ ' a n dt h c R c ; n o l d st r a n s v c c l o r e q u a l i o nu' i r i c h c a n h e l o r t { h e o r c n f o r a f i r e d c o n t t o l\ o l u l l ] cb c c o t n c sa w r i t t c na s
: ! ' : ,JI: J,,i vprrv+ f
ou ,un1t'..;
(1.7)
I l u u a t i o n1 . 7 s l r l t c st h r l l h c v r ' c l o t' u n r o f f o t c c sa c l i t ) 8o n l h e c o n t r o lr o l L r n l ci s c q u a lt o t h c t i t ] t cr a t eo f c h l n 8 c o f I r e l r n l o n r e n l t t ri ln] s i d ct h ec o l l ( r o \l o l u m c i t u s t h e n e t n o m c n t u m f l u x o u t o f t h e c o n t r o lr o l u n l e t h r o u g hl h c c o n l f o ls u r f a c cI.n fact, this equation can bc thought of sinrply as Novton's sccondlaw applied to a lhrce fluid. It is crucial lo notethat Equation1.7is a vectorequalionthatrcPresents with the approPrilte comdireclion writlcn in each coordinate separateequalions, ponentsof each vcctorquantlty. For the special case of the streanltubecontrol volume in Figure l.2b' thc steady,one-dimensionalform of the momcnturnequationin the streamdirection,s' i s g i v e nb y
n)oaJ,nu,(v
- > (FpQv,),^ t lBpQv,),,,,
(1.8)
directionl { is the nreanvclocin which u, is the point velcrity in the strcan'ilube coefficient to accotrnlfor a nrrnuniftrrnr flux correclion p thc momentunl and is ityl eqtrationas givcn by liquation 1.8 statcslhat velocity dislributjon.The I'ltorncrrturn the veclor sum of extcmalforcesin ihe strt'rtntubcdirection is cclull to lhe nronlcnflux into t u m f l u x o u t o f t h ec o n t r o lv o l u l r t ci n t h cr , J i r e c t i onnt i n u st h c n l o r r l i r l t l r m the control volume in the s dircction. The rnomentumflux correctioncoelllcicntp in Equation 1.8 is dcfincd by
t ^ lu;dA J.
'
v:A
(1.9)
to correcl for the substitutionof the mean velocity squaredfor the point velcrity squaredard bringing it outside the integralin Equation l.8 ln turbulentflow in prismaticchannels,the valueof p is not significantlygreaterthanlhe valueof unity, wbich is the raluc for a unifonn vclocity distribution.In othcr opcn channelflow situationssuch as imrncdiatclyd.'rrnstrfamof a bridge pier' or in ;r rivcr channel with floodplain flow, lhc valuc (,f fJ ( il'rlol be takenas trtiity hccattseof thc highly n o n u n i f o r t r\ e l o c i t y ( l i \ l r i l , f l l i ) n si r r l l ' " , : s j t u a t i o n s .
l0
C H A P T E R I : B a s i cP r i n c i n l e s
It is imponant to note that thc volunre l1ux. Q, has bccn subsrirutedfor A{ in Equation 1.8and that the renraining{ in the momenturnflux term is dre componenl of mean velocity in the directionin which the forces are summed.The outward volume flux takesa positivesign from (V . n) becauscof the positiveoulward unit vector, ard a negativcsign goes with the inward volunte flux for thc sane reason.The sign of { dependson tl:e chosenpositivedirection for the force sunrmation.lf the forces are being summedin a directionjr that is dilTerentfrom the streanrtube direction. the volunreflux remainsunchangedbut the contponentvelocitv is taken in the r direction rvith the appropriatesign. [n the r direction, Equalion L8 becomcs
)F. - ) (BpQv)*, >(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<1 .\ltt-hanit.s,61hed. Ne\r Yorkt John W i l e y & S o n s .I n c . . 1 9 9 7 . Rouse. Hunler. erJ. Enginurittg |lvlraulics. Io*a Citl: Io$a Institutc of Hydraulic R e s e a r c h1. 9 5 0 . Rouse. Hunter. and Sinron Ince. Hislr;n oJ Iltdraulics. Io$t City: Io$a Inslitute of Hydraulic Research.I 957. White. F. M. Fhtid Mecharit's..lth ed. ^vewYork: \lcCraw,llill. 1999.
EXERCISES l.l.
Classifyeachof lhe followiDgflows assteadlor unsleadtfrontlhe vieu'pointof the obscrver: FIo* (.r) Flow of riveraroundbridgepiers. 1r) Movementof flood surgedownstream.
Obscner ( I ) Sranding oDbridge. (2) ln boar,drifting. ( l) Standing on bank. (l) Moving\\ith surge.
1.2. At thecrestof an ogeespillway.as shownin Figurel.lc. \\ouldyouexpectthepres, sureon the faceof the spillwayto be greaterthan,lessthan.or equalto the hydroslaticvalue?Explainyour answer. 1.3. The river flow at an upstreamgaugingstationis tneasured to be 1500nrr/s,and at anothergaugingstation3 km downstream, rhe discharge is nreasured to be 750 mr/s at the sameinstantof lime. lf the riverchannelis unifonn.with a \l'idthof 300 m, estimate therateof chlngein the watersurfaceelevation in meterspcr hour.Is it rising or falling? 1.4. A pavedparkinglot sectionhasa unifonnslopeovera lengthof 100m (in the flow direction)from the pointof a drainageareadilide to the inletgrate.whichextends acrossthelot widthof 30 m. Rainfallis occurringat a uniforminrensity of l0 cm,4rr If the detentionstorageon the pavedsectionis increasingat the rateof 60 m],/hr,what is the runoff rate into the inlet grate? 1.5, A rectangular channel6 m wide with a depthof flow of 3 nr hasa meanvelociryof L5 m/s.The channelund€rgoes a smooth.gradualcontraction to a \\idth of 4.5 m. (d) Calculate thedepthandvelocityin rheconrracred section. (b) Calculate the netfluid forceon the wallsandfloor of thecontraction in the flow direction. In eachcase,identifyanyassumptions thatyou make. 1,6. A bridgehascylindricalpiersI m in diameterandspacedl5 m apart.Downsrream of the bridge where the flow disturbancefrom the piers is no longerpresent,the flow depthis 2.9 m andthe meanvelocityis 2.5 m./s. (.1) Calculatethe depthof flow upstreamof rhe bridgeassumingrhatthe pier coefficientof dragis 1.2. (D) Detcrmine rheheadlosscausedby thepiers.
CHAPTT,R l : B l s i cP r i n c i p l e s I 9
l . ' 1 . A s ) n r m c l r i cc o n r p o u n dc h . r n n e lr n o v c r b a n kl l o w h a \ a n t a t nc h r n n c l q ' r t n a o o r , t o r n r v r d t ho f - ' 1 0 m , s i d e s l o l c s o f I l . l n d a f l o r r d c | l h o f I r n . l h c f l o o d p l a r n st , n e i t h e rs i d e o f t h e n r r i n c h a n | r ' l r r e - 1 0 0m \ \ i d e a n d f l o w i n g a l a d c p t h o f 0 . - 5n r T h e n r e i l n! e l o c i l \ i n l h c r r , r r r t h a n n e l r s | 5 n r / s .u h i l e ( h e f l o o d p l a i nf l o * h r , , . r m e a n v c l o c i t y o f 0 - l n t / s . . \ : r u r n l n t t h t t t h . ' v e l o c i t ) v a r i a t i o nw r t h i n t h e m a r n c h l n n c l a n d l h e f l o o d p l i i n \ u h \ e c t l o n ! t s t n u c h \ n l a l l e r t h a n l b e c h a n g er n n t c J n v c l o c r t i c sb e l r r c c n s u b s e c ( i o n sl.l n d t h c \ i l l u e o f l h e k t n c l i c c n e r g y c o r r e c t l o n coefficicnt tr.
1 . 8 . The po*,er la\\'\elociry distnburion for fully rough. lurbulent flow in an olxn chan, nelis gi!en by
' '(*',
)
in \\hich l. = point \elGity at a distlnce : from the bed; l. = shear velcrirr bJp)t4; ta = bcd shcar stress:p = fluld densityik, = equi\alenr san
l,
/z\ ,.=*'n\,0/ in which x = the von Karman conslant = 0.40;q = t/30; and the other val-iablesare the sameas defined in Exercise L8. Show thar a and B for rhis disrributionin a very wide channelare given by a - l - 3 e l - 2 € j B : l + e 1 in *hich e : (u.",/V)
l; l.-
: nraximum velocity;and V = mean velocity.
1.10. In a hydraulicjump in a rectangularchannelof width b, $edeprhafterthe jump _r.,is kno$ n to dependon the following variables:
,rr =,fl_yr,q, c] in which)r : depthbefore$e jump; 9 = dischargeper unit width : eh; and g = gravitationalacceleration. Complererhe dimensionalanalysisof the problem. 1.11,
'ihe
backwater Cy caused by bridgepiersin a bridgeopeningis $oughtro dependon the picr diameterand spacing,d and .r, respectively;downstreamdeprh,)/o: downstreanrvelocity,V; fluid density,p; fluid viscosiry,p; and gravitarionalaccele.arion, g. Complete rhedinlensional anaJysis of theproblem.
l0
C r { A P T E R l : B a s i cP r i n c i p l e s
1.12. Thc lonSiludinalvclocity. u, near the fl\ed bed ofan opcn channeldependson the dis' tanceiiom the bed.:: thc kinematicviscosity.!i and the shcarvclocity. ir' : (relp)o5 in which ro is the $all shearstrcss.Develop the dimcnsional anallsis for thc point veloclty. r. 1.13. ln lhc rery slow motion of a lluid around a sphere.the drag force on the sphere,D, dcpendson thc spherediameter.r./:the velocity of lhe approachflow. V: and the fluid viscosity.p- Complele the dinlensionalanltlysis-How man) dinlensionlessgroupsare thereand what are lhe il]lplicationsfor the correspondingrllues of thc group(s)?Why *as the fluid dcnsity not includedin the list of variables? weir. 0, is a function of lhe headon the \\'clt crest. 1.14. The dischargeover a sharp-crested H: the crest length. a; thc hcight ol lhe crest,P; density.p; viscosity.p: surfaceleng. Cirry out the dimensionalanall sis using p. sion. o; and gravitationalacceleration. g, and H as repeatingvariables.If it is known that 0 is directly proponional to crcst Ienglh.L, how would you aller the depcndentlI group?
CHAP'II'R 2
SpecificEnergy
2.1 ENERGY Dlrl-INITIONOF SPECIFIC 'fhe
concept of specificenergyas introducedby Bakhneteff ( 1932) has proven to be rcry uscful in the analysisof open channeltlow. It arisesquite naturally from a considerationof steadyflow through a transitiondefined by a gradual rise in rhe channelbottom elevation,as shown in Figure 2. | . For given approachflow condilions of velocity and depth, the unknown depth,yr, after a channelbottom rise of height Az is of interest.If for the moment we neglect the energy loss, the energy equationcombined with continuity can be written as oz f I
r-.,1 zE^ t
02 !2'
t- ^: z E/1)
( 21 )
in u'hich y : depth; 0 = discharge;A = cross-sectionalareaof flow; and A; = zz it = change in bottom elevationfrom cross-sectionI to 2. Now, it is apparent that the sutn of depth and velocity headmust changeby the amounl Az and that the changemust result in an interchangebetwecndcpth and velocity head such that the energy cquation is satisfied.lf spectfc encry.r'is dcfined as the sum of depth and velocity head, it follows that the possiblesolutions of the problem for the depth depcnd on the variationof specific energy with depth. In fact, there are two real solutionsfor the depth in this problem, and Ihe plot of depth as a function of specific energyclarifies which solution will prevail.Such a plot for constantdischarge Q is called t\e specifc energl,diagron. A more formal definition of specifc energy is the height of the energy grade line abovethe channelbottom. [n uniform flow, for example,the energygradeline, by defrnition, is parallelto the channcl bottom, so that th€ specificenergy is constantin the flow direction.Thc componentof thc gravity force in the tlow direction
).1
22
C H { P T E R 2 : S p e c i f i cE n e r g v
v?tzs - EGL
-l---fz=?
o ------>
--->
Q
FIGL RE 2.1 Transitionwith bottomslep.
is just balancedby the resistingboundaryfriction- ln Figure2.1, the sPecificenergy dcireases in the flow direction. but it would be equally possiblefor the specinc energy to increasein the flow directionby dropping ratherthan raisingthe channel bottom. The total energyalways must remain constantor decrease,but the sPecific energy can increaseas well. In gradually varied flow. a continuouschangein specific energy with flow direction leads to a classificationof gradually varied flow profiles, in Chapter 5, according to the interchangebetweendepth and velocity iead. We show that the rate at which specific energy changesin the flow direction in gradually varicd flow is determined by the excess or deficit of the work done by resistance' -gravity in comparisonto the energyloss due to boundary Blcause the specific energy arises in connection with the determinationof depth changesin one-dimensionalflow, certain restriclionsare inherentin its definition. First, the spccificenergyis defined at cross sectionswhere the flow is gradually varied, so that the depth is identical to the Pressurehead at the channelbottom: that is, the free surface representsthe h)draulic grade line. What happens bet\r'eentwo points at which specific energy is defined is not restrictedby this assumption,however,as evidencedby the situation in Figure2 1. Second.the water surfaceand energygradeline are assumedto be borizontalacrossthe crosssection, so that a single value of velocity head correctedby the kinetic energyflux correction coefficientc sufficesfor the entirecrosssection With thesetwo restrictionsin mind, the definition of specificenergy'E, becomes aV'
12.2)
= nlean crossin which 1 = flow depth:a : kinetic energyflur conection:and V sectionalvelocity. A third resriction on the definition in Equation 2 2 occurs in the caseof a channel with a large slope angle g, as shown in Figure 2.2 In this case it no longer is obvious how the depth should be measured(venically as ) or perPendicularto the channel botlom as d) nor, in fact, whethereither of thesedefinitionsof depth is the correct representationof the pressurehead,p/7. This can be clarified by considering the force'balancebetweenthe gravity and pressureforce perpendicularto the channel bottom in Figure 2.2, wherebyp/7 : r.1cosa- in which 7 is the specificweiSht
Ctt,.t,]rRI
5 1 . - t r f i [i . r r c r g y 2 l
l W = , /\ r d
tlG L]RE2.2 Depthandprcssurc hcadon a sleepslope.
of the fluid. Funhcrmorc, ir should bc nored frorn the geometryin Figure 2.2 rhat d : ,ycos0.The coffect cxpressionfor spccificencrgymust bc written as ^ ,: _dcosd,
aV2 ,r'-
, ^ + -o2Vgl )co\.d
(2.3)
As a practicalmatter,cos20doesnot varyfrom unityby nrorethanI percentif g ( 6", so thattie approximate form shownin Equation2.2 is valid for all exceptthe sleepest channels, suchas a spillwaychute.
3i,1crrlc ENERGyDIAGRAr\{ Now we are ready to consider the actual functional variationof depth y with specific energy,E, in the graphical form called the specifc energy diagram. At ftrst, it will be convenientto consider the case of a rcclangularchannelof wjdth b. The flow rate pcr unit of width 4 can be dcfined for rhe rectangularchannelas e/b, where O = total channeldischarge.Continuity rhen ailowsus to write rhe velociry, V as q/y, and so the specificenergy for a rectangularchannetwith a - I is E =j -+
qzgy'
\2.4)
It is apparentfrom Equation2.4 that thereindcedis a uniquefunctionalvariation between) andE for a constantvalue of 4, and it is sketchedas the specificenergy diagra;nin Figure2.3. Note from F4uation2.4 t}at, as I,becomesvery Iarge,E approaches of theupperlimb of tie 1l so thatthe straighlljne -),= E is an asymptote spccificenergycurvesshownin Figurc 2.3.In addition,it can be shownthar,as _v approacbes ztro. E becorncs iDfinirclylrirgc,inplyingrhar$e E axisis iin asyDrll( )lc ofthc lorvcrlintbofthe specificcr gy curve.Betwccntheset'r'olimits,thespeciirc
C H { P T I R 2 : S p e c i f i cE n e r g y
o
o
E
c
E
2
E
l
SpecificEnergy,E FIGURE 2.3 Specificenergydiagramfor a transition$ith a smooth,upwardbotlomstep.
energymust havea minimum valuefor a givenvalueof flow rateper unit of width r/. In other words, flows with a specific energy lcss than the minimum value for a giYen 4 are physically impossible. The critical depth, -r'.,corresponding to the condition of mininrum specific energy, E.. can bc found by differcntiating the expression for sPecihc energy in Equation 2..1with respectto -vand setting the result to zero: dt dt
q.
- "n _ r
(2.5)
B.\,'
Now for the critical depth. -r'..,we have
"
Iq,-l',, Lel
(2.6J
which indicatesthat critical depth is a function of only the flow ratc per unit width q for a rectangularchannel.Funhermore,with the help of Equation 2 6, the value of minimum specificenergy,E , is given by F
= \ , +
q l8),
!
3
(2.1)
;
and shown in Figure 2.3 as the locus of valuesof critical depth and minimum specific energy for each specific energy curve defined by its own unique value of 4.
C H \ P T i R 2 t S p . c r f i lcr n e . g y 2 5 B c c a r r s cb o t h r , a n d E i n c r e a s ca s z 7i n c r c a s c st,h c s p c c i f i c( ' n c r g ) , c u r \ e sm o v e u p u a r d a n d t o t h r ' r i g h ti n F i g u r c2 . 3 a r q ri n c r c a s r sa n d 1 7 ,> q , T h c p h t s i c a l n t c a n i n go f t h c s p c - c i f icc' n c r g 1d i l g r ; r r l i s n o t n c ' i l r l ys o c l c a ra s i t s r n r r t h c n r a t i cianlr r - r y J r c l a l i Iot ni s. o h r i o u s f r , r r nF i g u r c2 . 3 t h r t . l n a l h e m a t i c a l l y . { h c r c r r c t \ \ o p o s : i b l ev a l u e so f d c p t h f o r a g i r e n r a l u e o f s p c c i f i cc n e r g y T . he phy,sicalmcrrningof thr'sct\\ o dcplhsbt'conresclcar fronr a rcarrangt'rncnt of EquaIi o n f . 6 a s
( 28 ) f r o m * h i c h u c c a n c o n c l u d ct h a t t h e c r i l i c a l d c p r h c o n d i r i o ni s s p e c i f i e db y r h e v a l u eo f t h c F r o u d cn u r n b c rF. . b e c o n t i n gu n i t y . l f $ c ' f u r t h c ro b s c n et h a tt h e c c l e r i t y c. , o f a r c r l s r n l l )a m p l i t u d ed i s t u r b a n c e a l ( h e w a l c r s u r f a c ci s ( g r J l l , a p h y s i c l l i n l e r p r c t a l i o n o f t h e n t e l n i n go f t h e t w o I i r n b so f t h c s p e c i f i ce n e r g yc u r v e si n F i g u r e 2 . 3 i s p o s s i b l eF. i r s r ,u e a s s u r n ci n Figure 2.{ that a small anrplitudcdisturbanccin shallow waterof dcprh} is propaSated at a celerity c relativeto still water. If we superimposea ',elocily c in the opposite direction, this becomesa steady flow problem with conslantenergy,so that (.!' + i/28) : constantand thcrefore
dr,+9dc=o I
(2.e)
Then, with the aid of conlinuity for steady flow, c) = conslant;and we have that c d_r'* _r'dc= 0, which can be combincd with Equation 2.9 to proverhatc = (B))rn with respectto the still water as a refercnceframe. Now, for depth y < r,., the Froude numbcr, y/c, must be greaterthan onc and vcfocity the V > c. ln other words, the flow velocity is greaterthan the cclerity of a small surfacedisturbanccand so swccps any disturbancedo$'nstream. This flow regime is called supercritical,or rapid,Jlow,and charactcrizedby relativelysmall depthsand large vclocities,as can be seenin Figure 2.3. The upperregimeof flow, on the other hand, has.r'> -r; and the Froude nuntber lessthan unity.Therefore,the flo*' velu--ity, V < c, and wave disturbance s can trar el both upstrcamand downstream in this regirne,uhich is callcd subcritical flou.. Subcriticalflow has relativcly large depths and small velocities:for this reason.jt also somelimesis called tranquil Jlo*,. \4'c can concludethat subcrilical flo*' is a flow reginrein which the
t't(;t RF:2.4 Watersufacedj\lurbnn.eof snrallamplitudc\\,ithcelcrit) c.
?6
C H A p T E R2 : S p e c i f iE cnergl,
depth control, or boundarycondition,can exen its influencein the upstreamdirec_ tion, while in supercriticalflow, a control can influence the flow profile only in the downstreamdirection.These observationsbecome important laier when we con, sider the computationof flow profiles. Finally, we can retum to the original problem posed by the transitionin Figure 2.l. lf t}le upstrearrrflow is subcriticat,as indicated b1,point I in Figure 2.3, which depth is the proper solution for point 2, for which Ez: Et - A;? The lower depth can be _r2, reached only by a decreasein specific energy to its minimum value, followed by an increasein specific energy.Becausetiis is physically impossible,lhe correct solution for the unknown depth is the subcriticalone, y.. As the flow passesover the rise in the charncl bonom, the depth will decreaseard the $.ater surfaci elevationwill dio.
2.3 CHOKE A limiting condirionfor the transitionshownin Figure2.1 occursif A; ) ,l;., whereAz"is thedifference betweentheapproachspecificenergyandthentinimum specificenergy.If this differenceis exceeded, it would appearthat the specific energymustbecomelessthanthe minimumvalue,a conditionalreadyshownto be impossible. In response to thisdilemma,the flow responds with a risein the water surfaceand the availablespecificenergyupstreamof the transition. as shownin Figure2.5.ln fact,rhe specificenergyrise in Figure2.5 is just sulficientto force flow throughthe transitionat the criticaldepth.Any furtherincreases rn Az will causea corresponding increase in the upstreamspecificenergy.whilethedepthin the transitionwill remaincriticar.This condition,referredto;s a criore,i ustrates quite dramaticallythe extra degreeof freedomaffordedby the adjustment of the free surfacein openchannelflow. The stepheightrequiredtojust causechokingcanbe developed from theenergy equationappliedfrom the approachsectionto the critical sectionoverthe steo:
Az.: E, - Q:
E, - 1.5.r.
(2.r0)
If E4uation2.10is dividedby the approach depth,..!r, theresultfor thedimension_ lesscriticalstepheightdepends on theapproachFroudenumber.F,. alone:
Fi r + 2 - r . s F, i
( 2 r. l )
Equation 2.1 1 is ploned in Figure 2.6. For an approachFroude number of0.l, for example,rhe critical srepheight for choking is 6g percentof the approach depth but rapidly becomesa smallerfraction of the approachdepth as the approach Froude number increases. EXAMPLE 2.1. For an approachflow in a recrangularchannelwirh deprhof 2.0 m (6.6 ti) andvelocityof 2.2nts (7.2ft/s).determinethe depthof flow overa gradualrisein t}lc channelbottomof .l: = 0.25m (0.82fi). Repeatrhe solutionfor l: = O.SO - (t.O+ttl.
i
o
Y1' ft
yc= (213)E c
fc
o
E
"
F
I
E
|
SpecificEnergy,E FIGURE 2.5 with a smoolh-upwardbottomstep' Chokingin transition
1.0 0.8 0.6 -h
0.4 o.2
o.o
08 0.6 04 o.2 F1 FroudeNumber, APProach
1.0
FIGURE2.6 Critical step height for choling in a trlnsition \ ith a bollom slcp
21
28
c nergy C T A P T E R2 r S p e c i f i E or flow is supercritical to knowwhetherthe approach So/|rtbt. First,it is necessary thecriticaldepthfor mosteasilyby sinrpl)calculating whichis ascenained subcritical. ftr/s): a flow ra(eperunit width of q = I x 2.2 : '1.'lm:/s ('17.'1 r ) . : ( , 1 . . 1 ' l 9 . 8 1 ): ' 1 . 2 5m ( 1 . 1 0f t ) bccause -r',) t,. The from which it is obviousthat the approachflou'is subcritical, flow Froudenumb€ralsocouldbe calculated: approach F ! : 2 . 2 / ( 9 . 8x12 ) ' r : g 5 flo* andthe sectionof maxiNow the energyequation$rittenbclweentheapproach m u ms l e ph c i g h (t 0 . 2 5m ) i s E t - 2 + 2 . 2 1 1 t 9 . 6:2 2 . 2 5- =0 . 2 5+ r . , - . 1 . 4 r i 1 ( t 9 .x6 2] ' l ) whichcanbc solvedby trial. Only rootslargerthanthc c.iticaldepthof I l-5 m ('1.10 ft) are sought.'fheresultis r', : 1.62m (5.32ft). Notethatthe absoluleelevalionof t h ew a t e sr u r f a cder o p sb y t h ea m o u n(t2 - 0 . 2 5- L 6 2 ) : 0 . 1 3 m ( 0 . ' 1 3 f t )l f t h e s l e p specific specificenergyis theapproach to 0.5 m ( 1.61ft). theavailable heightincreases energy(2.25m) lessthe srepheightof 0.5 m. or L75 m (5.74ft)' whichis lessthanthe m i n i m u ms p c c i f iec n e r g yo f ( 1 . 5x 1 . 2 5 ): 1 . 8 8m ( 6 . 1 7f t ) .T h i sm e a n tsh a te c h o k e occursin which the depthover the slepbecomescritical(1.25m) and the upstream asgivenby lhe solutionof depthincreases 8 ( 7 , 8 1f r ) , - 1+ 4 . { l ( t 9 . 6 2 x r l ) : 0 . 5 + 1 . 8 8: 2 , - 1m increase in depthof Thc resultis lr : 2.17m (7.12ft). whichresultsin an upstream 0.1?m (0.56ft). Thecrilicalstepheight.which$illjust causechoking.canbe obtained fronrFigure2.6 or Equation2.I I for a Froudcnumberol 0.5 as 1:,-/-r ' : 0 18.from = ( l i ) . 0 . 3 6 m l . l 8 w h i c hA : .
2.4 DIAGRAM DISCHARGE Transitionsin channelwidth also can be analyzedby the specificcnergy concept. For the rectangularchannel.however,it is no longer true that the flow r31eper unit width q remains constant. Supposethe channel uidth changes from b' in the approachsubcriticalflow to b" in the contractedsection.as shown in Figure 2.7a With negligibleenergyloss.the energyequationsimply statesthat Er : 8., but this requiresthat the flow regime move from one specific energy curve do\r'nward to anotherthat is appropriatefor the new value of 4, as shown in Figure 2'7b by the points I and 2. An altemativeway of viewing the changein flow regime in a width contraction can bc gainedby writing the energyequationand noting that the quantity that remainsJonstantin this instancejs not q but ratherthe specificenergy,E (ncglecting energylossesand assuminga horizontalchannelbottom).Therefore.if a discharge function for a given specific energy,8,, is defined by
(1.12)
( a ) P l a oV r e wo l W r d t hC o n t r a c t r o n
i o
o
SpecificEnergy,€
Discharge per Unilof Width,q
(b) UnchokedContraction
o
o
SpecifjcEnergy,E
Djscha.geper Unito, Width,g (c) Choked Contraclion
FICURE2.7 Speciricencrglanddischarge diagrams for contraclion in r,..idth.
29
30
C H . \ P r ' [ R2 i S p e c i f iE cnergy
then it is obvious thlt thcre is a unique functional rclation bctweenthe discharge per unit width 4 and depthr for the rectangularchannelfor a constantvalueof specific energy.The funclion is shown in Figure 2.7b alongsidcthe spccificenergydiagranr rr ith the approachand contractedsectionsidentilied as points I and 2. respectivel\'. Two specificenergy cur\ics are shown: one correspondingto the upstream width b, and flOw rate per unit width qt: the other for the contractedsection\\ith widrh b. and flow rate per unit width q.. The decreasein depth from point I to point 2 occurs at constantspecificenergy,as shown in the spccific cnergy diagram,and correspondsto ln incrcasein dischargeper unit *idth in the dischargediagram. The dischargefunction given by (2. l2) has a nraximum thilt can bc found by setting dgld,r': 0 and solving for }. to obtain (2/3)E,. This is preciselythe relation for critical depth derived previously,which means that critical depth not only is the dcpth of mininrum specificenergyfor constant4 but also can be interpretedas the deprh of maxirnum dischargefor a given specific cnergy.In Figure 2.7b, the critical depth associatedwith the given specificenergy in the specificencrgy diagram has been transferredacrosshorizontally to the maximum 4 in the discbargediagram. Figure 2.7b also shows that the position of point I on the approachspecific energy curve determinesthe availablespecificenergy and establishesa single discharge diagram for that value of specificenergy because-r'= E u'hen q - 0 in the dischargediagram. Choking can be causedin a contractionby decreasingthe width to a value such rhar rhe availablespecificenergyno longer is sufficient to passthe flow throughthe contraction without an increasein the upstrcamdepth. This is illustratedin Figure 2.7c by the points 1, I', and 2. Point I must move up the specificenergycurve to the point l' upstreamof the contractionwith an increasein specificenergyin Figure 2.7c. This establishcsa new dischargecurve in Figure 2.7c with a new value of maximum dischargeand a new critical depth, shown by point 2. The flow regime passesfrom the new upstreamdepth )r. to y.. in bolh the specificenergyand dischargediagramsbut in differentways, as shown in Figure 2.7c. Also apparentfrom in the Figure 2.7c is that, once the choking criterion is exceeded,funher decreases point l' to continue increasing asymptotidou nstreamwidth b. causethe depth at call), to the straightline ) = t as the approachvelocity headbecomesnearly negtwoligible. In this instance,the critical depth in the contractedsectionapproaches for a rectungular channel. thirds of the approachdcpth Another interpretationof the dischargediagram is shown very clearly by the example in Figure 2.8, in which flow from a resen'oir into a short horizontalchannel or over a broad-crestedweir is controlled by a sluice gate.The reservoirlevel establishesthe hxed value of specific energy, and raising the sluice gate in the channel causesan increasein dischargeas the depth of flow upstreamof the gate decreases.Simultaneously,the depth downstream of the gate increasesto maintain the same discharge.The discharge reaches its maximum value when the upstream depth becomescritical. Beyond this value, the gate no longer has any influence and the dischargecannot be increasedfurther without raising the reservoir level. At the maximum discharge,the depth in the rectangular,horizontal channel becomestwo-thirds of the head in the reservoir if the approachvelocity head is negligible.
C I A r T r , R 2 : S p c c r f i cl i n e r g r
-.1I
-j}q
!'t ctrR!t 2.8 Dirchrrge drrgram for flo* under a sluice gatc on a brold cr!'stcd\,"rir.
If the approachflow to a contractionis supercriticll,spccificencrgy analysis s t i l l a p p l i e si n t h e g e n c r a lc a s ew i t h o u t c h o k i n g ,b u t o b l i q u es t a n d i n gw a y e sc a n c o m p l i c a t et h e a n a l y s i sI.f c h o k i n g o c c u r sd u e t o a c o n t r a c t i o nt w , o limttiig cases are possiblefor a supercriticalapproachflow, as shou'nin Figure2.9.Choking condition A is causedby the occurrenceof a hydraulicjun.rpupstrearnof the contraction follo$ed by passagethrough the critical depth in the contractedopening. Choking condition B, on the other hand, is the resulrof going directly from the supercriticalstatelo the critical depth for the contraction.Betweenconditions,4 and I, choking may or may not occur (point 3 or 2', for exarnple).Thesetwo condjtions are analyzedin more detail in the following chapter.
2.5 CON'IRACTIOn-S AND EXPANSIONSWITH HEAD LOSS The general equation goveming contractionsand expansionswith a subcritical approachflow at cross-sectionI is the encrgyequationwirh headlossesincluded, as given by
, ,* # , = a za- r-:# . r , l * r -*
o:
:
( 2r . ] )
in which A; is positive for an uprvard step. Encrgy lossesare consjdcredand exprcssedas a minor loss coefficicnl, (r, times the diffcrencein velocity heads bctwecn tirc two crosss. ctions.The rbrupt expirlrsionhils Ihc highestencrgy Joss
l2
C H A P T E P l : S p e c r f i cE n c r g v
i
o
'r
SpecificEnergy,E
FIGURE 2.9 Choking modesfor contractionwith supercriticalapproachflow.
becauseof flow separationand viscousdissipationof mean flow energy in the separatedzone. Henderson( 1966)has shown from a combinedenergyand momentum analysisthat the expressionfor the head loss in an abrupt open channel expansion is given by
,,:Il('-l)'.
b2 2Fibi( a1 u1
bl
,l
\2.14)
theapproachandexpandedsections,respecin which the subscriptsI and2 represent that the dePthat crosstively,as shownin Figure 2.10.Equation2.14 assumes 2 andthat the pressure distributionat sectionI equalsthe depthat cross-section zone. acrossthefull widthbr, includingtheseparation 2 is hydrostatic cross-section 2 and 3, andthe The momentumequationthenis writtenbetweencross-sections energyequationfrom I to 3 givesthe headloss.The first term on the right hand for headlossin an abruptpipe expansideof (2.l4) is identicalto theexpression sion, wbile the secondterm is the openchannelflow term with its dependenceon FroudenumberF,. For F, ( 0.5, the secondterm is small comparedto the first, so that it can be neglectedunderthis conditionandy1=,12:1,. Then, for an expression with Equation2.l4 for headlosslike thatgivenin Equation2.13to be consistent with the secondterm neglected,Kr mustbe givenby
CllAPltR
l : S l ) c c r f i lc] n ( ' r g y
ll
2 F r c LR n 2 . 1 0 Plan \ icw of abrupt opcn channelexpensron.
'i,: ( F r < 0 . 5)
(r.r5)
| +!\ b.
in which K. varies from approxirnalely0.8 to 0.05 as b,/b. increasesfrom 0.1 to 0 . 9 .G r a d u a lt a p e r i n go f t h ee \ p a n s i o na t a r a t co f l : 4 ( l a t e r a l : l o n g i t u d i n rael )s u l t s in a head loss coefficientthal is onl\ about 30 percentof the valuegiven by Equat i o n 2 . 1 5 . E n e r g y l o s s e sa r e s m a l l e ri n t h e c a s eo f c o n t r a c t i o n tsh a ne r p r n s i o n s . H e n d e r s o n( 1 9 6 6 ) r e p o n sv a l u e so f e i t h e r 0 . l l o r 0 . 2 3 t i m e s t h e d o w n s t r c a m velocity head, dependingon whether the contractions are rounded or square edged,respectively.For rivers,the HEC-RAS manual (1998) suggestsa value for r(. of 0.3 for gradualexpansionsand a value of 0.1 for gradualcontractions.The default values for WSPRO (Shearmanet al. 1986; Shearman 1990) are 0.5 for expansionsand 0.0 for contractjons. The actual effect of headlossesin the specificenergy analysisof contractions and expansions dcpends on lheir relative magnitude in comparisonwith the approachspecificencrgy.In a contractionfollowcd by an expansion,as in thc open channel venturi nrcter shown in Figure 2.11, or in a bridge contraction,the contractionenergy loss nraybe considerablysmallerthan the expansionloss,as shown in thc specific energy diagram.The overall effect of the total head loss is an upstreamapproachdcpth at point I that is largcr lhan the downstreamtailwater dep$ at point 3, evcn thoughchoking is not occurring.As the tailwateris lowered from point 3 to 3' for the sametotal discharge,choking occurs at some point, as shown in Figure 2.1I at point 2'. Choking also can occur as {he contractedsection width gets smaller for the same total dischargeand the same tailwater.Funher decreasesin contractedwidth causethe depth to remain critical in the contracted seciion,although critical depth itself also is increasingas the width b2 decreases and 4, increases.Tlris causesbackwater,a rise in upstreamdepth.In this case,thc do$ nstreamof the contractcdsectionfollowcd flow regime passesto supr:rcritical by a hydraulic jrrnrpto the fixed tail\\,a(er.
C H A P T E R2 : S p e c i f i cE n e r g y
1
3 , 2
I
.c
31 ,/)
o
o
4 2'
0
SpecificEnergy,E
FIGURE 2.I1 Open channelcontractionfollowedby an expansionwith headloss.
2.6 CRITICALDEPTHIN NONRECTANGTILAR SECTIONS Specificenergyfor nonrectangular mustbe formulatedbeforederivingthecrir sections ical conditionasthepointof minimumspecificenergy.Specificenergyin ary nonrecurgular sectionof areaA anddepth.1;asshownin Figure2.12,canbe expressed as
E
=d . 'r ' * o2sA'
(2.16)
Differentiating withrespect to y andseningdEld) - 0 results in dE dy
ou- dA gA" d)
(2.11)
CHlttfR I
5 1 ' ,r f i . f n . r g y
t5
l'l(;L RE 2.12 proPcnics of gcncrllnonrectengular scclion. Ccom,,-tric F r o m I ' i g u r e2 . I 2 . r | e s e ct h a td . 4 / d r- B , i n \ , ' h i c hB h l i s t h c t o p w i d t h a t t h ew a t e r surfaceand a functionof l The condition for ntininruntspecificenergy'and criticai d c p t ht h e ni s
ao2B-
I
sA:
(2.l8)
in which the subscriptc indicatesthat A and B arc functionsof the critical depth, y.. If *e define the hydraulicdepth D = A./Band substituteV = UA, the Ftoude channel is defined and has the value of unity at tie number for a nonrectangular c r i t i c a Jc o n d i t i o n : F -
v
_
kDlel'r'
(2.t9)
The value ofthe minimum specificencrgycan be obtainedfrom Equations2.16 and 2 . l 8 a n d i s e i v e nb v
6,=!"+
D. 2,
(2.2O)
is inrolvedin the determicxplicitlybut nevedheless in which a docsnot appear E. nationof .)"andthcrefore channelis a matterof The conrputation of cdlicaldepthfor thc nonrcetangular shape.The solvingEquation2.18for thc geometryof a panicularcross-sectional triangular, circular,and neededfor thetrapezoidal, geometric elements appropriate arelistcdin Table2-1. An exactsolutionis availablefor paraboliccrosssections andcircularsections both tie triangularandparaboliccases,but the trapezoidal algebraic equationto obtainthecriticaldepth. requirethe solutionof a nonlinear termsis possiblefor both the trap€A graphicalsolutionin nondimensional section,for cxample, 1966).The trapezoidal z-oidaland circularcases(Hendcrson of theapproprile gcometrcquircsthe solutionof Fquation2.l8 aftersubslitution nc expresslons:
c'8 I
_11,,,.)l' _ li =1",.(b B,
(b + ?my,)
(2.2t)
36
CI|APTER 2: SPecificEnergy
TA BI,E 2-I
: Geometricelenrenlsfor channelsofdifferent shapelv flon depth)
\l'ettedPe.imetcr,P
Area, A
Section
Top Width' A
Rectangular
I lv 1-|
b+2r
I
D
Trapezordal
\l'/,
J 1 b+ , | 1 r )
b+2r(l +n:)ra
b+2mr
mj2
2}.(I + rnl)l/r
lnt\
Adl2
d sin(012)
(A/2)t(l+ rr)14+ (l/.r)ln (; + (1t.r:)rn)l
a'o/r')r'
l J '
-w1' Triangular
Circularr
,L@_1 (0
sind)d:/8
ParaboliC
(u3) Br-
Tl-T/-
L\lr--l t8=2cosrIt-2(yd)) $-4rB
in which b is the bottom width of the trapezoidal section with side slopes of ,r; I (horizontal:vertical). To present the solution of Equation 2.21 graphically, the following dimensionless variables are defined for the trapezoidal channel: om)lt 4*=;L
tgfu)tlz 6stz'
'v ' :
m\', '' b
(2 22)
C r t { P r E R2 : S p c c r l Li cn c r g y l 7
0.1 0.01
0.1
10
z
F I G U R E2 , I 3 Crilical deplh for trapezoidaland circulatchannclsi2.,,, = Q/lgt'fn]: 7t.p = Qrnr'llirrbtcl (lJendcrson.1966).(Soa'cp. OPE^'CHANNEL FIAW b\ H?nder' son, @ 1966 Reprintedbr pernissionof Pnntice'Hall, Inc., Upper Soddle Ri|er NJ )
Equation 2.21 can be nade dirncnsionless with thcse rariables to produce
f r ' ' i l+ v ' 1 1: r t-tnP /t \ ,
!
-),.,\Ll - ' l
(2.23)
This relation, plolled in Figure 2.13,can be used to find critical dep(hdirectly for a lrapezoidalchannel.A similar relationhas been der eloped for the circulaJcase, also plotted in Figure 2.l3 (llcnderson 1966). For the circular sectionthe dimcnsionlessvariablesare redefinedas
2.,,,
o
(2.24)
( g l o ) ' td t t '
Thc Ialue of a has L^-cnshou'nas unity in the defiin *hich d - conduitdiarneter. assumPtionfor a prisrnaticcharnel. nition o[ Z in Figure2.13,which is a reasonable The minimum specificcnergy can be determined and ploned for the trapez-oidaland circular sectionsas wcll (Henderson1966). For the trapezoidalsection with the dimensionlessvariablesas definedin Equation 2.22 and wittr E' = mE"/b, Equation 2.20 in dimensionlessform is given by
I'(l + r') j(l
-r z!
(2.2s)
l\.'ow,bcciritsebolh E' lrd Z are uniquefutictionsof r''. E' can be givcn as a funct i o n o f Z , r s i n l - i g o r e2 .1 4 .A s i r , j l a rr e l a t i o nc a n b e J ( \ c l o P c df o r t h ec i r c u l a rs e c ' t i o n , a l s o s h o * n i n l - i g u r e2 . 1 . 1r,l i t h E ' - L l t l .
38
C H A P r F . R2 : S p e c i f i cE n e r g y
d = diameter;b = botlomwidlh;m:1 IH:V) = sideslope
T'TGURE2.14 lr'linimumspecificenergyfor trapezoidal andcircularchannels 2,,,, = el[g]t.f1l; Z\,"p: Q#nllgtEbsEI (Henderson,1966).(Source:OpEN CHANNELFLOw bt. Henderson, @ 1966.Reprintedby pennissionof Prenrice,Hall.Inc., lJpperSatldleRi,er NJ.)
E x A l r p r - E 2 . 2 . F i n dt h ec r i t i c adl e p t hj n a t r a p e z o i dcahl a n n e$li t h a 2 0 f i ( 6 . 1m ) bonomwidthand2:l sideslopesif I : l00Ocfs (28.3mr/s). Usethebiseclionrech_ niqueandcomparerhesolulionwirh thatfrom Figure2.13. Sol,/tbr. The bisectionproceduredevelopedin AppendixA can be usedto find crir_ ical depthif thefunctionFCr)is properlydcfined.Theequationro be sarisfied is Equa_ t i o n2 . 1 8 s. ol r k e F L v rt o b e
r 0 ) :Q
^ l : i l :
?
r,2.261
The VisualBASIC prognmY0YC that solvesfor criricaldeprhis givenin AppendixB. Data is enteredthrougha sepaIate form module,shownin FigureB.l. The datainput is passedto themainprocedure, Y0YC in the paEmeterlist.The nlainprocedure establishes the initial intervalfor therootsearch(Yl andy2) andthe specifiedrelativeenor crirerion ER. It tlen callstle BISECIION subprocedure. whichin tum callsthefunctionsubproce, dureF for eachiteration.Because thesamesubprocedure is usedto computenormaldepth, for whicha differentfunctionis required,theappropriate iunctionis specifiedby thevalue of the variableNFUNC.Norelhat the criticaldeprhelaluarionrequiresonly rhe channel geometn.par.unerers (b andn) andtie discharge, e, whilerhenormaldeplhcompuhrion (to b€ drscussed in Chapter4) alsorequiresthechannelslope.s. androughness coelicient, n. The final resultis storedin the variableYC. \,,hich is passedbackto $e form module_ The resultfor thisexampleis 3.?40ft (1.14m). r,r.hich canbe checkedwith the sraphical te.inique of Figure2.l3 by calculating {-n: Z , o o= 1 0 0 0x 2 1 1 / ( 3 2 . 2I 1x 2 0 5 r ; = 6 . 2 9 Then,from Figure2.13,ny,lb = 0.17andr. : 3.7 fr ( l.l m).
Cll^rruR
2
S p t c r f i cf : n c t g y
l9
2.7 OVI.-RI]ANKIII,O\\' I n s o r n c\ i t u l t i o n s . t h c f o r c g o i n g! ' l ( ' l l l c l l t i ! rr\ c l a t t t r n s h i pfso r ( h c o c c u r r e n c eo f , . r ' i t i c afl o w n o l o n g c r a p p l y i n t h t - l i i r n l g i \ c n . O n c c r l t n p l c ' o f i n l t l r r ' \ 1i s r l v t r r r r e r b a n kl l o u , a s h r r l l o * f l t t r vo r c r r r i c l cf l o o < i p l a i ncso n t b j l l c d\ \ i t h I n l a i l l c h i l n r c l f l o w t h a t i s o u t o f b . t n k .I n t h i s c r s e . i l n o l r . r t r s ci sr p c l m i s s i b l cl o n c S l t ' c ot . o f I a r g cn o n u n i f o r n t i t i cbsc t * e r . ' nt h e I e l o c i t i c si n l h c o r e r b r n k r n d n i i l i n bccause t d c p t h r t o o b t a t na n c h a n n e l\ V h c n E q u a t i o n2 . 1 6i s d i i f i t c n t i a t i d u i t h r e ' s p e ct o t x p r c s s i o nf o r d r i t i c a ld c p t h .t h e \ a r i a t i o no f a * i t h r l i l o : t b c c o n s i d c r c d : dE
:' orl
nQtB
,
,rrr
-
Qt
do
irrr, ,rr.
( ? . 2 )1
N o w i f d l - l d r i s s c l l o z c r o ,a c o n l p x r u r lcdh a n n c lF r o u d cn u n l b c rc a n b c d t f i n c d l B l a l c x ka n d S t u r n r .l 9 8 l ) :
' ..=(-s*3-##)
( 2 . 28)
The firsr tcnn on the riSht handsidc of Equltion 2.28 leadsto the conventionaldefthe contributionof a inition of the Froude number.while tre secondtenn rePresents nonconstantvalue of thc kinetic entrgy concction coefiicient,rr. The cross-sectlon is divided into a main channeland floodplainsubsectionsfor the computationof a, that lhe encrgy gradeline is horizontalacrossthe which dcpendson the assun'lp(ion crossstction so that the energygradc line slope is the sanlc in each subscction.It funher is assumcdthal the slopeof the energy gradeline' S., can be fomrulatedas S, = Q' lP, in which K is the total channelconvel'anceas dcfined by a uniform flow fonrula suchas Manning's cquation.which is discussedin nore detail in Chapter4. The conveyancedependsonly on the Seometricand roughncsspropertiesof the we have Qrl,\3 = QlRi. or crossscction.Under theseassumptions,
9 , - k, o K
(2.29)
c o l l \ c y a n c e iO : l o l a l d i s f l o w r a t e :t , - s u b s c c t i o n i n r v h i c hp , - s u b s e c t i o n = total con\eyance It,. The definition of a can be expressed: chargeland K
: lv ,/a,ral "^ - | , _ _ (QlA)'A
(2.30)
area.In Equation2.30, areaandA = totalcross-sectional in whicha, - subsection differencein velocity to o is the primary contribution thatthe it hasbeenassumed i n t o E q u a t i o n2 . 3 0t h c nl e a d s E q u a t i o 2 n . 2 9 f S u b s t i t u t i o n betwccn subsections. t o t h cd e f i o i t i o n
>(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<.edure FC shown in Appcndix B in lhc program Ycornp.Olherwise,thc eraluation ofcritical depthprcreedsas in Erantple 2.2 using $e
40
C H A P T E R2 : S p c c i f i cE n c r g l ' biscctionsubprocedure.Note that the unknoun rarilble sought in the bisectionsubpro, cedureis the crilical \\'atersurlaceelevationr!lher lhan the critic,,l depth.The code ntodule in thc appendix requiresa data ille of the cross sectionground points and rhe subsectionbreakpointsand roughnesscoeflicientsas shorvn.The program output for O = 5000 cfs (l.ll mr/s) gives critical depthsof 5.182 ft (L579 m) and 6.7.10fr (2.05,1m). For 0 : 4(X)0cfs ( I l3 mr/s).therers only the main channelvalue of crirical deprhequal to '1.180ft ( L-165nr). \r hile only the overbankcritical dcpth of 7. l9J lr (2.193 m) exisrs for P : 6566 .tt ( 18.1mr/s).
START
Calculatebank-fullFroudeno.,Ft, and upperlimitingdischarge,QU
No
YeS
( > QU?
Compuje lower criticald€rpth,YC 1
Compule upper criticald()pth,YC2
YC2 ( U p p e rb, may or maynotocclrr for this O)
YC1 y( does not {LOWer occur to ' t h i s Q )
/^^-^,n^
t-{^.
t'lt'
F*< Fj?
No 'I
FIGURE 2.2I Flowchartfor finding multiple critical depths.
-
1 io
OL=OU
STOP
C I i A P T I R 2 : S J ^ - c r l iEc n c r g y
1'l
T h c p r o c c ' d u rl e o r i c f o r c ; r l c u l e l r n ng t u l r i p i ec r i l r . a l d c p ( h sr n r h ep r o g r a mY c o n r p r n A p l r n J i \ B r s i l l u \ t r i l c d b ] r h e l l o \ r c h l n i n F i g u r eI l l T l r e r r l u c o f t h c u p p r r l i m , i t r n Sd i \ . h l . g c p. l , / ,r c l l t r r ct r t h c g i r e n p d e t c m t i n c s t h c c r r r l c n c c o f r l o u c r , j l u p p c t c n r r . r l r l c I r h ,r h r c h r \ r h r n c ; r l ! r i l ; r l e dT h c u p i ^ - rc r i t i c a ld c p t hi s d c ' s r g n a r cads y C 2 . u h r l e r h c l o \ \ c r c r r l i c adl . l n h i \ Y C I T h c i r v a l u c sa r e s e rc q u a lt o ( | ) t o l n d i c a l ct h 3 l t h . ' \ d o n o t c r r r t . ' l l c r a l u e o f l " i s l h e c o n ) p o r i n dc h l r n n cFl r o u d cr u r n b c re \ a l u a l e d a l i d c p t ho , i . 0 l t t m e sl h c b l n k f u l l d c p t ht o r i c r c r r n i n icf t h e F r o u d cn u r n b e r r r r r . r c a s r n ! . 3 s r f l t h c c a s co f m u l { r l ) l !c' n l l c a l d e p ( h s , o r n o t l f r t i s n o t l n c r e a s i nagb o \ e b a n k 1 u l ld c l l f L r h ! ' Do n l ! o n c c r j r i c a ld . p t h e x i s r sa n d Q L = Q U . , l f r s i n c r c a s i n gr ,h c nr h c r r t r \ i n l ! n r \ a l u e o I t h c c o I l ] p o u n dc h 3 n n e lF r o u i l en u m b e r F . " o . . t \ n c c d c dt o c a l c u t a l e lhc lo\\ cr ljl]1itingdi\charge.O1-,for the possibleca-scof ntultiplecrrtjcaldcpthsfor the g i\ en 0. Oncc borhO U and Ql, arc known. thcn decisjonsare madeatxJUIlhc c\ rstcnLe of onlr a Io\\r'r crilical dcpth. of only an upper critical dr'prh,or of both.
(Only upper yc occurs)
Compuleuppercritical deplhYC2 (Bolh y;s occur)
!-IGURE 2.21(cotltinue\
48
C H A p T L R 2 : S p e c i f i cE n e r g y
2.8 }VEIRS Thc ocrurrenceof critical deprh is put ro good use in thc design of open channel flow measuringdevices.By creatingan obstructjon.criticaldepthis lorced to oc-curand a uniquerelationshipbetwecndepth and dischargeresults.This is the principlc upon uhich weir designis based.The very exrensivesct of experimentalrcsultsdo.eloped for weirs accountsfor ticir continuing popularity as flow measuringclevlces. Sharp-Crested Rectangular Notch \\'eir Thc sharp-crcstedweir equation can tr derived with respecrto Figure 2.22 by first assuming(l) no headlosses,(2) atnlosphericpressureacrosssectionAB, and (3) no vertical contractionof thc nappe.Under theseidealizedassumptions,the velocity along any streamlineat sectionAB is giren by u : (2ghrtE,where i : venical distance below the energy grade line. This veltrcity distribution can be integrated over the cross-section A,Bto obtain a theorericalvalueof dischargeper unit of width, q,: 2,p 2 s '
,1,:
(
l
{zgn an
(2.37 )
ae
in which Vo - approachvelocity; and H - approachht-adon the crest of thc weir. Carrying out the integration,the resuh is
. #)'- (#)"]",,' ,,:1r4[('
(2.-r8)
V2 n s
I
-------
EGL
B
FIGURE2.22 Idealized flow over a rectangular, sharp-crested weir.
C l J , \ r , l E R2 : S p c c i f iE c nrrgy
g
I f t h c t c r r i n s q u a r cb r a c k c t s w . h i c h c r p r c s s c sl h c c f f c c to f t h c a p p r o a c hv c t c r c i r y h c - a di.s c o r r b i n c du i l h c o n t r a c t i o na n d h c a d l o s sc f f c c l si n l o a d i s c h r r g ec o c f . f r c r c ' n tC , J ,t h c a c l u a lI o r a ld i s c h l r g ci s g i r c n b y
Q:
2 - Y 2 g C ' l , H '' a
(2.39)
i n u h i c h L i s t h c l c n g r ho f r h c n o l c h c r c s rp c r p c n d i c L r l raor t h c f l o w ; a n d H i s r h e h c a d n r c a \ u r c da b o t c l h c c r c s r .T h e d i s c h a r g cc o e l l l c i e n ct a n b e d c t e n l i n c d o n l y b y c x p c r | I c r ) t s .I n t h c L l n i l c ' dS t i t t c si,t i s c u s t o o t a r y1 0s i m p l i f yF i q u a t i o n2 . 3 9a s
(2.10)
Q'clH1l
i n r rh i c h Q j s i n c u b i c f c c r p c r s c c o n da n d l / i s i n f c c l , b u t r h ed j r n e n s i o n l c sf o srm of C, in (2.39) is prcfr-nt'd. W i t h t h c g c o n r c r r i cv a r i a b l c sd c f i n c da s i n F i g u r e2 . 2 3 ,a d i m e n s i o n arln a l ; s i s for thc coefficientC, yiclds
,,=t(:,:-t f'*",*.)
(2.4t)
in *'hich 1- is the crest length perpcndicularto the flow, b is the approachchannel width; H is the heaclabove lhe notch; P is rhe hcighr of thc norchcresr above the channelbonom; Re is the Reynolds number; and lve is rhe Webernumber,One of the ea-rlicstexperimentalrelationsfor C, was given for thc suppressedweir (Ltb : l) by Rchbock (l{enderson 1966),in which he neglecredviscousand surfaceten_ sion effects so that Cd was given as a funcrion of H/p alone:
C a = 0 . 6 1 1+
0.084
P
(2.42)
ln the suppressedweir, there are no laleralcontractioneffectson the werr oappe,so that the coefflcientof dischargedoes not depcnd on l/b. Funiermore, Rehbock's formula rcflects no influence of H/L on tbe discharsecoefllcient. Basedon expcrinrentalresLrlts obtainedat CeorgiaTech. Kindsvaterand Ca-rter (1957) proposed that rhe Reynolds number and Weber number effects can be included in thc hcad discharge relationship by naking small conections to the head, H, and crest lcngth, l. By doing so, they derived from their expcrimenral resuftsan effective coefficient of discharge,Cd., that dependedonly on H/p and Ltr, as shown in Figure 2.23. Their relationshipis given in the form of Equation 2 . 3 9a s
Q=
;V2sc&1,H3,/'
(2.43)
in which
L,=L+k"
(2.44a)
H,- H + k,
(2.44b)
whcreC, and,t. aregivenin Figures2.23bard 2.23c,respectively, andfrHwasfound to bc nearlyconstantwith a valueof 0.001rn (0.003ft). The crest-length correction,
50
c ncrgy C H c P T E R2 : S p e c i f i E
(a) Definition Sketch
0.010
0.80
t ) o =t ' y
0.008
o . 75 0.8 -(
0.70
0.006 E -l
O
0.65
0.6
0.004 0.002
0.4
0.60
o.2-
0.55 0.0 0.5
1.0 1.s 2.0 2.5 H/P
(b) Coeff,cientof Discharge
0.000 -0.002 0.0
0.2
0.4 0.6 L/b
0.8
1
(c) Crest LengthCorrection
FIGURE2.23 weir (Kindsvaterand Caner, 1957). Head-dischargerela::onshipfor a sharp-crested (Source: C. E. Kirdtater and R. W. C. Carter, "Discharge Characteristics of Rectangular Thin-Plate Weirs," J. H'-d. Dir'., @ 1957, ASCE. Reproduced by pemtission of ASCE.)
m (0.014ft). asshownin Figure /<.,is maximumat Lh = 0.8with a valueof 0.0O43 2.23c.Equationsfor Cr" basedon the Kindsvater-Cmerdataare givenasa function of the lateralcontEction ratio,L/b, and the venicalcontractionratio, H/4 in Table 2-3. Kindsvaterard Cafier(1957)foundthattherewasa negligibleinfluenceof H,4, on the dischargecclefficient. weir notches, not Kindsvater andCarter( 1957)constructed tbeirsharp-crested with a knife edgebut with an upstream squareedgehavinga top width of 1.6mm weir shouldbe (1/16in.) and a downstream bevel.The headfor the sharp-crested measuredat a distmce of threeto four timesthe maximumheadmeasuredupstream of the weir platerBos 1988). 'lllle weir,or full-widthweir,with Ub - |.O musthaveprovisions suppressed for aerationof tie undersideof the nappe,becausesome air is entrainedby the pressurern the nappe,whicb affectsthedischargecoefficientdueto subatmospheric pocketunderneaththe nappe.Undesirableoscillationsof the nappealso can result
CllArrLR
::
S p c c i f i cl ] n e r E y
5l
lAtlt ti:-.1 Cormci.nts
of discharge for thc Kinds\ alcr-Carlcr fornrula
Ub
Sdlr"i
C,,,
I
0 6.ll - 0 075 IVP
0.9
0 599 * 0 (ril 11./P
0.8
059? - 0(X5 lL?
0 7
0 595 + 0 030 H/P
06
0 5 9 1 + 0 0 1 8H / P
05
0 5 9 2+ 0 0 l t H P
01
0 5 9 ! + 0 0 0 5 8l l P
0l
0 590 , 0 0010ll,?
02
0 5ll9
0 00llt ltnr
0 t
0 588
0.0021lvP
0
0 58? - 0.0021 Fl,?
D a r n f t o n r K r n d s \ a r e ra n d C a n € r l 9 5 l : B o r l 9 8 E B r a t . r a n d K r n S l 9 1 6
from irregularair supplyralesto the pocket.To ensurefull aeration,Bos ( 1988)suggestslhal the tailwalerremainat least0.05 m (0. l6 ft) below the \\'eircrest. a lim' Kjndsvaterand Cartcr ( 1957)recomnrended For prccisemeasurelnents, weir (0.3 then the ft). If l1lP excecds5' itation of H/P < 2, with P no lessthan 9 cm definitely itself no longer is thc control section,and so large valuesof /J/P should be avoided.
Sharp-Crested Ttiangular Notch 11'eir weir defined in Figure 2 2'1aprovidesa preThe triangularor V-notchsharp-crested cise nleasurementof dischargeover a wide range of dischargesUtilizing the same approach as for the derivationof the head-dischargerelationshipfor a rectangular sharp-cresredweir, it can be shown that the head-dischargcrelationshipfor a Vnotch weir is given by e A z O= C'iVze nn"1ns
(2.4s\
or partially in which I is the notchangle.The weir can eitherbe fully contracted contractcdfor narrowapproachchannels.Equation 2.45 can be presentedin the head,H. : formin whichthehead,H, is replacedby aneffective Kindsvater-Caner and of Reynolds C* beconlesindependent of discharge 11+ t,, andthecoefficjent Webernunibereffects(Bos 1988).Thc valueof Cr., shownin Figure2 24b,varies 0.58and0.59asa functionof 0 only,providedtlratH/P < betweenapproximately O.4 andP/b 4 0.2 to ensurethe fully contractedcase.Thc valueof ,t, variesfrom I to 3 mm (0.0033to 0.01 ft) as a functionof the notchangle,0, as shownin
5l
J : S p . ' c i f i cE n c r g v
CIr rltLx
0
(a) DefinitionSketch
0.61 0.60 0.59 I o 0.58
\
0.57 0.56
0
20
40 60 80 100 120 d, degrees
(b)Coefficient of Discharge
o
20 40 60 80 roo 120 ,, degrees (c) HeadCorrection
FIGURE 2.24 "Discharge Meaueir (Bos' 1988) {Solrce:Bos'M. G l9SS sharp-crested Triangular, Stntclures,"ILRI Puhlication20. 3d Reised Edition.Wgenitgen' TlrcNetlrcr' surements lands,320 p.)
Figure 2.24c. For a partially contractedV-nolch weir, sufficientdata for the dischargecoefficientis availableonly for the caseof € - 90o,and the dischargecoefficient varies wrth H/P and P/b for this case, as given by Bos ( 1988).
Broad-Crested Weir The broad-crestedweir has a finite crest length parallel to the flow. In addition' the crest is long enoughthat parallel floq'and critical depttroccur at some point along the crest, as shown in Figure 2.25 for a rectangular, broad-crested weir. If the energy equation is applied from the approach flow to the critical section on the crest and energylossesare neglected,we have
..
"
Q' 3f(o/t)'l'' zsei 2l s l
(1.46.)
in which H" is the energyhead on the crest as shownin Figure 2 25a1that is. 11": H + Vll2g, in which Vois the approachvelocity.ll the energylossesare absorbed
CllaPItR 2
S p c c i f i cf n c r g y
5l
v(rzs
r , !
l
*
.
!
Weir (a) DelinitionSkelchof Broad-Cresled
1.1
1
1.0
o 0.9 T I
t-/
- 0.848 I L -l Shorl Broad; r cresled- FI c.esled
0.8 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1. 0 1 . 2 1. 4 \ Hll
1. 6
(b) Coefficientot Dischargefor H llH + PJ < O.35 F-IGURE2.25 rreirs(Bos,1988) (Source: andshon-crested for broad-crested of discharge Coefficicnt ' ll RI Publicalion20' 3rd Retised Slructures Bos,t'!. G. Ig88 DischorgeMeasurentents Edition, llageningen,The Netherlonds-120p \
in a dischargecoeffrcient, Cr, and we solve for Q in terms of the h€ad, H, the r e s u l ti s
o=c"c,:f]rl'".n.' J L J
(.2.41)
I
in which the approachvelocity coefficicnt C, : (H,/nr? liquation 2.47 can be solvcd for the discharge,assuning that C, - l, and lhen the approachvclocity hcad
c nergy 5 , 1 C H A P ] t : R2 : S p c c i f i E can be calculatedto updatethe value of Cu for a secondcalculation of Q. Altematively, C, can be relatcdto the variableCdA'lA F in which C, - s eir dischargecoefficient; A' = LH = flow areain the control sectionof the weir $ ith a water surface area hcight conespondingto the upstreanthead,H; and A, - flou cross-sectional = + for a suppressed weir L(H P) H is measured in the approachsection where (Bos 1988).The resultingrelationshipbetweenCu and CrA'lA, is
coe'
lcl''
- t)'i'
(r.48)
C, 0.385
At
Equation2.48is plottedin Figure2.26so thatC! canbe estimateddirectly. variable,1,thelengthof the geometric weir,an additional For thebroad-crested into thc dinensionalanalysisfor Cr' andit crestparallelto the flow,is introduced canbe shownthat / H H\
c,: f\p.T )
(2.19)
as givenin Figure2.25b(Bos 1988)for Hl(H + P) < 0.35 In fact,whetherthe weir behavesasexpecteddependson the valueof l1.4 The following broad-crested rangesof behaviorcanbe delineated: 1. 0.08 < H/l < 0.33, broadcrested. 2. 0.33 < H/l < 1.5, shortcrested. 3. H/l > 1.5. shamcrested. 1.20
1. 1 5
d
1.10
1.05
,/ 1.00
I
0.1
o.2 0 . 3
0.4 0.5 cdA'/41
0.6
o.7 0 . 8
FIGURE 2.26 weir (Bos. 1988) (Source.' Approachvelocity correctioncoefficientfor a broad-crested "DischargeMeasurements Structures,"ILRI Publication20' 3rd Revised Bos,M. G. 1988 TheNetherlands,320 p.) Edition. Wageningen,
C l i { p r r , R 2 : S p c c r f il ci n c r g y 5 5 In thc rangcof broird,crcstc'd bchavior,tltc crest is long cnoughin the llo* dircction to obtain plrallcl flow at the critical s('ctionand a theorcticrldischargcctxlficicnt. Ca ,- I 0. but friction Iosscsrcduce thc cxpcnntcnralvlluc of thc dischar.gc c<xfl.i, cient. Cr. t0 0.8{li as long as the $cir rcntainsbroad crcslc,tl.rndH/lH 1- pl ::0.35. In thc shon-crcstcd rlngc of bcharior.thc flow is cun,ilinearalongfrc cntrr!,crcstof thc rr r'rrlnd thc c(^*ffici('nlof dischrrgc.rttrrrll; incrcrres.r, shoun rn Figurc2.25b. T h c a d v a n r a gocf a b r o a d - c r c s l c w d c j r i s . t h a tt h c t l i l r v a t c rc a n b c a b o v et h e c r c s lo f t h c * c i r u i t h o u ta f f c c t i n gt h e h c a d - d i s c h a r g r cel a t i o n s h iapsl o n ga s t h ec o n trol scclionis unaffected. The ljmir of tajl\r,arcrheighr,l/,, abo\e rhccresrof rhe wcir so that the dischargedocs not dccrcascby more than I pcrcentis called the rnodl, lar linit. The modular Iinrir usually is exprcsscdin rcrms of the rario 11./H,u,hcre l/ is the upstrcamherd on rhe qrestof.rhc wcir, and it has a vaiueof H,./H .. 0.66 f o r a r t c t r n g u l a rb r o a d - c r c s t c*dc i r ( B o s 1 9 8 8 ) .
2.9 ENERGYI'QUATIONIN A STRATIFIED FLO1V [-et us supposenow that the flows over obstaclestiat we have been considerine in this chapteroccurar rhe bottom of a deepreservoir of depthD as a resulrof r plringing gravitycunentof higherdensity,as shownin Figure2.27.The ambientdensityin the reservoiris p, and that of the gravity cunent is pr. Such flows occur naturallyas the resultof a densitystratifyingagentsuchas temperatureor saft.If a small obsta_ cle of height Az is on the bottom of ttre reservoir,rien we can write the cncrgy equation for the lower flow layer as before, taling into accounl the additional Dressureof
FICI,RE 2.27 Tu'o layerdcnsitystrittificdtlow overa stepin thechannelbotlom.D >> )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,
.,\
\<,
// ./'
/ S e q u e ndtc p t hl o r b a s i nw i d l h= 2 b
\Sequent cteptntor basin width= D
O,n-
Discharge
(a) CaseA
ir"g ,4.'
o LU
,"'\
s"qr"n,o"p,n
Omar Discharge (b) CaseB FIGURE3.I4 Matchjngiailwaterelevation ro sequenldeplhof a hydraulicjump (U S. Bureauof Reclamatron1987).
19
C H A P T E R3 i \ ' l o n t e n t u m
-l
(a) N.4oving Surge
-----+ V1+ V"
(b) Surge t\"ladeStalionary FIGURE 3.15 jump in (b) The movingsurgein (a) is reduccdto the staiionary
By making the surge stationaD'.the steady-flowform of the continuity and rno*"ntur equationscan be applied to Figure 3. l5b. Thc continuity equationfor a rectansularchannelof unit width is
(V'+Y,)r'':(%+%),v,
( 3 .l 5 )
which can be rewritten in the form
(3.16) In this form, the continuity equation states that ihe net flow rate through the surge given by the raie of volume increaseeffectedby the surgemovement' is -The momentum equation writlen for the stationary surge in Figure 315b becomes
(v,+4)r
-i
: tl tl ' \, /' ,-, i) /r \
( 3 .l 7 )
This is of the same form as the hydraulic jump equation except tiat the Yelocity of yr + flow, V,, has been replacedby ( 4) ln effect, the left hand side of the equation by the moving observer' Because-r'rl1't) I' seen as ,"pr"r.nta the Froudi number
Crl \f Tl,R I
l\'lonrcnturr 8l
t h e F r o u d cn u r r b c r . r ss c r ' nb y a n o b : e r r e r n r o r i n g r r i t h t h c s u r g ci s s u J x - ' r c r i t i r a l c \ c n t h o u g ht h r l l o r r i r rl n r n to f t h t 's r r r g c o u l dh c , . u l ^ - r c r i t i col rrls t r b t ' r i t i c a sl s e c n b ) l \ t . r o n l r y o b s c r r r ' rI l a l s oc l r nb c c o n c l u c j ctdl r r t t h c l l ( l ! \ l t h i n d t h c s u r g ei s : u r . L r i t r c al rl o n rl h c r i c r ip o i n to f t h r r r o r i n g o b s c n r r .. A f u n h c rc o n c l u s i o tnh a lc a n b c d r l r r n [ r o n ] E q u i t l j o n3 . 1 7i s t h a t .r s _ r , / r 'a, p p r o a c h cz' sc r o f o r a n i n f i n i t c s i n r a l s u r f a c ed i s l u r b a n c ct h ' .c c r - l c r i t o y f t h a t c j i s t u r b l n c icn s t i l l $ a t e r ( V , = 0 1 i s g i ! e n by (gr',)r'r as r.lcrircdprc'riouslyin Chaptcr2 from ln cnc'r8yargun)r'nt E q u a t i o n s3 . 1 6a n d3 . l 7 p r o r i d co r l y t r v oe q u l t i o n si o t h c t h r c eu n k n o w n s\;. t . V.. and V,. Thc third equationrcquired for :olulion ofrcn conrr'sfrorn a spt'cifi,'d b o u n d a r yc o n d i t i o n I n t h c c a s eo f a g a t c s l a r n n r i n sgh u tu t t h c ' d o i r n s l r c r ncrn d o f t h c c h a n o e il n F i g u r e- 1 . 1 5f .o r e r a r n p l e ,l h c n c c c s s a Dc o n d i t i o ni s V , : 0 . l : x A \ l P l . E - 1 . 1 . A \ t e i l d yf l o r ,o, c c u r si n a r c e t r n g u l ca hr r n n eul l s t r c a m Lra f sluice gnlc.Tlc \el(x'it)'is 1.0nVs (-'l-l fl/s) rnd thc dcpthof flo* is -'l0 rn (9 8 fr) jusl upstream o f t h cg l l f . l f l h es l u i c eg a t cs u r l , l t ' niIs s l u r t r n r cs d h u t$ , l ) a la r et h ch c i g h t lnd slo-cdof lhe ul)\trtlnl\urge? F-rlualion I I6. andaftersubslllulion So/!r/rrn. Fronrcontinuity, ofr, = -10, v, = 1 9, and 11 = 0, *e obtain:
1.0 ): 30 Equalion -1.17. the morrcntunrcquation,then gives
( l o + Y , ) '=j t t t t i ^ ( ' . . ' - ) 1 . 0\ 3 . 0/ Tlesel*o cquations canbesolved by trialby firstsubstiruting a valueof ,r',in thesec, ond cq!ation that is Sreaterthan,rr and solr ing for V,, * hich thencan be comparedwith the value of 4 from lhe llrst equation.Iteralion is continueduntil the valuesof y, are equal.Alternatively.a furction could be forrred from (3.l6) and (3.17) by substi{uting y, from (1.l6) in(o (1.17)and rearanging so thal the right hand side is zero.The zero of thc function tien could be deternined from a nonlinearalgebraicequationsolver.ln eirhercase.rhe resulris _yr= 3.58 m (11.8 fr) and y, = 5.20 nVs (17 I ft/s).This speed ofthe surSeis $hal eould b€ seenby a stationaryobsener, *,hile an observermoving w i t h t h e s p c c d o f t h c s u r g e* o u l d s e ea F r o u d en u r r r b eor f ( y t + l ' , ) / ( g _ v , ) 0 J l:. t 4 i n front and a liroude nurnberof V,l(S:)nt -.0.88 bchind the surge.
3.5 I}RIDGE PIDRS canbeuseful whenapplied Monrenturn analysis totheobsruction caused bybridge piers in river flow. Thc resultingobstnrctionleadsto backwatereffectsupstrcantin subcriticalflo*'and can cvencausechoUng. Two types of flow are shown in Figure 3.16. Type I is a subt-riticalapproach fl, w rvith a d('crcasein d' pth whcn prssing thlough the constdclionwith the flow r c r n a i n i n gs u b c r i t i c a lI .n l y p e I I f l o u , c h o k i n go c c u r su i t h c r i t i c a ld e p t hc x i s l i n g
E2
C H A P T E RJ : N l o r u c l r t u r n
(a) PlanViewof BridgePiers
Ya > Yca
(b) Profileol Type I Flow
ht'=ft-fq
Y4 > Yc4
(c) Profileof Type ll Flow
FIGURE3.I6 Flowbetween bridgepiers.
in theconstriction. ForTypeI flou themomentumequation canbe writtenbetween \ e c t i o n sI a n d4 . i n F i g u r e3 . 1 6 t. o g i r e M1=Ma+Dfy
( 3 .l 8 )
in which M, and M., are the momentum function valuesat sectionsI and 4, and D is the drag force exenedby the piers.For known conditionsat the downstreamsection 4 and with the drag force given as D : CrpA,Vil2, the changein depth or backwater,T = 0, - )a) can be determined.In the erpressionfor drag, An is the
C - l t \ fr r , Rl : \ l o n r r n t u r l
E]
CDa/s OB 04 o2 0 1 0 05
*
0 01
0 001 0 01
01 Number, Fa Froude Downstream
F r G t i R E3 . 1 7 Solution for back\aler causedb) bridge piers in Type I flow.
frontalareaof thepier and Co is lhe drag cocfflcient with a vrluc bctweenL 5 and 2.0 for a blunt shrpe.Substitutinginto E{luation3. l8 for a reclangularchannel.\\'e have
)i 2
q:
';
8_r'r
2
q' go
Cpar,Vl 7gs
(3.l9)
. q u a t i o n3 . 1 9 c a n b e n o n d i m e n i n w h i c h c = p i e r w i d t h ,a n d s - p i e r s p a c i n gE sionalizcdin tcrms of tbe dou ns{rclm Froude number,F", to produce
- l r l + l ) ( -l + " ,2 ) F;= ^ s r tA I
(3 20)
Da/
i n w h i c h I : , i A { , w h i c h i s r } r er a t i o o f t h e b a c k w a t etro t h e d o w n s l r c a md c p t h . Equation3 l0 is plottedin Figure 3.17. from which thc backwalercausedby piers can be estimated,providedtheir coeflicientof drag is knou'n. E x A M P L E 1 . 2 . A b r i d g ei s s u p p o n ebdy e l l i p t i c apl i e r sh a v i n ga w i d t ho f 1 . 5m of 2.0 If the (4.9ft) anda spacing of 15.0m ('19.2ft). Thepiershavea dragcoefflcient (7.87 (6.23 ft/s), respecf0 and 2.40 m/s velocity are 1.90 m and downstream depth causcdby the piets? tivcly,whatis l}icbackwater Solltioz. First,the valueof Coalr = 2.0(1.5)/i5= 02 The do*nslrealnFroude x 1.9)o:- 0.56.FtonrFigure3.17,lr'flr'ois appLoxinlatell'0.04, nunrberis 2.4/(9.81 from whichthe hackwaterI'i .- 0.01(1.9)- 0.076m (0.25ft). If Equation3 20 is thevalueof /r'flro- 0 O1l, shich conllnntllie graphicalsolutions,,hedr;,rrrcrically,
8.1
Cn,rptt'r J; \lonlenlu ) Whilc this biick\rar!'rvaluc nral sec sl]rall.it could rcprc\!'nta significant incrcascin tht'area llurled up.treant of thc bridgc in vcry llrl. *idc lloodplains..'\lsoclear liorn Figure 3.17 is that thc snaller are the flow blockasc (o/.\)and thc do*n:lrcam Froucle numbcr. thc le\s back$atcr lhlt s'ill dcrelop.
3.6 SUPERCRITICALTRANSITIONS Transilionslbr u'hich the approachllcxv is sttpcrcriticrloffer a dcsien challcnge bccluse of thc existcnceand propagationof standinguave fronts. The rcasonfor the occurrenceof standingwavc fronts in supercriticalllo!\' can be visualizedfrorn the vic\\point of an observerLidingon a small panicle or disturbancetnor itrg ut r c o n s t a n st p c c d .y . i n s t i l l u a t e r ,a s s h o u ' ni n F i g u r e3 . 1 1 3A.t c a c h i n s l a r t o f t i n r e .
(alV
(b)V- c
(c)V> c
I'IGURE 3.I8 P. at speedV in still water(c : *ave ceJerity). Movementof a point disturbance,
C l t A P T t - R- l : M o n r c n t L r m
s5
the disturbance.P. scndsa circular wave front outuard that movesat a spccdcqull l o l h c w a v ec c l c r i t y c, . l f V ( c , a s i n F i g u r e3 . | 8 a , t h e w a v ef r o n t so u t d i s t l n c ct h e nroving disturbance.so that no pileup or addition of rvave fronts occurs.(Jn thc o l h c r h a n d .f o r c a s c so f l ' ) c . a s i n F i g u r e 3 . l 8 c . t h c d i s t u r b a n c eP,' n l o v c sf a s l e r t h a nt h e t l a v e f r o n t s . ' f h cr c s u l ti s a n a c c u m u l a t i o no f w a v c f r o n l s ,t h c o u t t r l ( r u s o f w h i c h f o r r n sa s t r a i g h lti n c a t a n a n g l ! 'p t o t h e p a l h o f t h e d i s t u r b a n clch a t c a n be dc'finedas a standin8\\ave flont. In br't*een thcse(\4o extrcmes,V : c. and the standingwave front is pcrpcndicularto thc path of thc disturbance,P, as s[toun in t, Figure 3. l8b. lf the disturbanccis infinitcsirral, so that c - (3-r)0 whcre ,r = flow depth, lhen obviously V/c is thc Froude numbcr and case (a) in Figure 3 18 is for sub,criticalflow, whilc case(c) rcpresenlssupercriticalflow. Thc supcrcriticalcasc in Figure 3.18c can be analyzedin more dclail to determ i n e t h e a n g l eB . [ n { h e t i n r et r , t h e d i s t a n c em o v e d b y P o i n tP t o p o i n t A i s V t , while, at the sameIime, lhe inilial wa\e frolt grows front P to point I over a radial d i s t a n c cg i v c nb y c l , . T h e n .s i np = l / F , u ' h e r cF i s t h c F r o u d en u n l b c rl.f t h c v i e w point of the obscrveris changed,the 0uid moves at a spccd y in the supcrcritical caseand any boundaryinegularity,such as that causedby a changein wall direction in thc contractionshown in Figure 3. 19. gives rise to a standingwave front at an angle, B,, relative to the original flow direction. This analysis indicatesthat larger Froude numbersrcsult in smaller angles of deflection of the standingwave
Plan
V2sintpt- el
Y1sin P1
Sec A-A FIGURE 3.I9 flow \aith standingobliquewaves. in a supercriticai Straightwall contraction
86
C r { A P T E3R: M o m e n t u m
front,but the possibilityof a finite heightof the standingwayefront is not considered.In thiscircumstance, the analysismustbe modified.
Designof SupercriticalContraction Considera straight-walled contraction,as shownin Figure 3.19, with a wall angleof 0, an approachsupercriticalFroudenumberF,, and contractionratio r (- bllb). Standingwavesof finite heightareformedat the initial changein wall directionhavingan angleof pr. They meetat the centerlineof the contraction and are reflectedback to the wall with an angleof (.82- 0). The goal of good contractiondesign,as outlinedby Ippen and Dawson(1951),is to choosethe valueof 0 for givenvaluesof Froudenumberandcontractionratio thatminimize the transmission of the standingwavesdownstream. This canbe accomplished if the combinedlengthof the first two setsof standingwavesterminatesprecisely at the physicalend of the transition,so that subsequent reflectionsdownstream are cancelledout by the negativedisturbances emanatingfrom the end of the contraction. Tbisdesignproblemcan be solvedby applyingthemomentumandcontinuity equations acrossthe wavefrontsin muchthesameway as for the hydraulicjump. With reference to sectionA-A in Figure3.19,the continuityequationacrossthe wavefront is givenbv V 1 y ,s i n B , : V r _ rs, i n ( B ,- 0 )
(1.21)
and the momentum equation componentsparallel and perpendicularto the wave front are given by, respectivell,.
V,cosB,- V,cos(p,- 0)
= sinr' f l# (; . ')l''
(3.22) (3.23)
Now Equations3.21,3.22,and 3.23 can be solvedfor Br, Vr, and,r',,given the valueof the contraction angleI and the approach FroudenumberFr. With these results,the solutioncanbe repeated acrossthe secondsetof standingwavefronts in Figure3.t9 to obtainp2, V.,.and_r',.FirstEquation3.21is dividedby Equation 3.22to yield t a np r )2 _ ran(pt - 0) )r
(3.24)
whichis substituted into Equation3.23to obtain
tanB, f l[l nnp, s r n p-r * _' / \Jl ' ' r ' L ; * ' t B - o f \ t u n t p-, o l
(3.25)
Ct^PT€R J: l{onrcrrtum 87
F i n a l l y ,E q u a t i o nJ . 2| i s * r i t t c ni n l c n n s o f t h c u p s l r e a ma n d d o w n s ( r c a mF r o u d e numbcrsrclative to the first wavc front, F, and Fr, resp€cti\cly, to givc
Fr= Fl
sinBl
sin(p, 0l
I
(i)"
(3 26)
E q u a t i o n3 . 2 5 i s s o l v e df o r p , f o r g i v e n v a l u e so f d a n d F , . T h c n t h c v a l u e so f - r ' ,a n d F , c a n b e o b t a i n c df r o m ( 3 . 2 4 ) a n d ( 3 . 2 6 ) . r e s p c c t i \ e l y .I f t h i s s o l u t i o n p r o c e d u r ei s r c p e a t c da c r o s st h e s e c o n dw a v e f r o n t , t h e r a l u e s o f B . . % , a n d - l . r follow. T h e s o l u t i o nj u s t o b t a i n e dd < r s n o t n e c c s s a r i l ym i n i r n i z c t r a n s n r i s s i oonf wavcsdownslream.An additionalcondition rcquircd is for thc toral lcngrh of rhe t r a n s i t i o nL,. t o b c e x a c t l ye q u a lt o l h e s u m o f t h e l e n g t h so f l h e t u ' o s c t so f s t a n d ing wavefronts,L, and Ia: ,
b,-bt
2 ran0
bl
2 t a nB l
br 2 tan(p2- 0)
( 3 . 2)7
However,as shownby Sturm (1985),the conditiongiven by (3.27) is entirely equivalent to satisfyingcontinuitythroughthetransition,as givenby I r
b, br
F./v,\r/2 Fr\yr ,/
(3.28)
With Equation3.28,the solutionproceduredeterminesa uniquevalueof r for minimization of wavetransmission, aswell asBr, Vr, andyr, whenvaluesof d and F, are given.The solutioncurvesar€ shownin Figure 3.20. Solutioncurvesof ( 1982).For a given d = /(r, F,) alsoaregivenby Harrison( 1966)andSubramanya F,, eitherr or 0 canbe givenbut not both(llarrisonI 966; Sturm I 985).For example, fromgivcnvaluesof r andF,, Figure3.20detcrminesthe uniquevaluesof 0 andyr, whileEquation3.28can be solvedfor the corresponding Fr. The resultis n r i n i l n i z a ( i o fnw a v et r a n \ n r i s 5 i o r . In thelo*'erhalf of Figure3.20,the chokingconditions,4andB described in 2 areshown.Chokingcondition,4is basedon theoccurrence Chapter ofa hydraulic jump upstream of ihe contracljon followedby passage throughcriticaldcpthin (he contJaction. Cuwe I is obtainedby conservingthe momentumfunctionfor the jump upstream hydraulic of the contraction and specificenergythroughthe contractionitself.Thesecondchokingcriterion,givenby curveI in Figure3.20,is for thecaseof F, becomingequalto I, so (hatthe flow goesdirectly from the approacb supercrilrcal flow to critical depthin the contractionbut witlr energyloss included. CurveI is derivedfrom the solutionprocedurejust rlescribedfor F, approachinga valueof unitysothatenergylossis inhercntlyincluried, as shownby Sturm( 1985). lf 0 liesbet*ecncurvesA and8, chokingmayor may not occur,depending on the existencc of a hydraulicjunrp,but if it is tb therj8htof curveB, chokingdefinitely will crccur.
88
C H A P T E R3 : M o m e n t u m
'10
20
Angle,6 Contraction (a) Ratioof DepthsThroughContraction
0.8
E o.o
fL
c
o.2
0
1
0 2 0 Angle,d Contraction
3
0
(b)Contraction Ratio,r TIGURE3,20 Supercritical contraction with theminimization of standingwaves: A, B = chokingcriteria (Sturm1985).(Source:T. W Sturm,"SimplifiedDesignoJ Contractions in Supercritical Flou," l. Hydr Engrg.,@ 1985,ASCE. Reproduced bypermission of ASCE.) with chokingconditionB canbe derivedby writing The energylossassociated the energyequationbetweensectionsI and3 in Figure3-19,includingthe unknown energy loss AE. Then,the specificenergyat section3 is setequalto its minimum valuefor whichF.,= l, andtheeguationis solvedfor A,Ein dimensionless form to produce(Sturm1985)
C H A P T E R3 : M o m e n t u m
e: (';9 A
E
I
'-T - t \ , . )
F i 3 / F ,\ " 1
l
(3 29)
inwhichr-:criticalcontractionratiogivenbycurveBforwhichFl:lValues of Froudenumof fflf' "tongcurveI exceed0 1. or l0 percent'only for values 4' ber F' in excessof approximately channel hasan approach contraction reclangular ExAItPLE 3.3. A straiSht-walled has flow ('19 approach f0 The m width of l 5 (9.8 contracted and a ft) m width of 3.0 of values the (9 wha{ are 8 fvs)' u O.ptttof O.lOnt (0.j3 f0 and a vel*-ity of 3 0 n/s be to and length angle contraction the ubar should depthandvelocityand downstream occur? choking Will $a\es? standing of minimizetransmission = 3'0(981 x 0l)05 : 30' Froudenumberis V,/1g,11)05 Solution. Tl,e approach = x* t i t e r = b , / b , = ^ i . s l 3 : 0 . s ] - t t . n .t ' r o mF i g u r e3 2 0 ' d I I " a n d I i / - r ' r : 2 5 a p p r o--28' Fr 3 Equation = (082 ft)' From = m 025 25 x 0l it"i.ry' ,o'rr'i, h : x 5
: 152 x (981 0 2s)0r iii'iiijO,rt,l' i': (l/0.5)x 3.0/251 1.52andsov1 fron (3 27)with0 = I l' is (br : 2 3&;;i7.sl ftls).Thelengthof dret'ansition
in theregion b.)/(2tan0) = (3 - l 5y(2tanl l") = 3 86m (127 ft) Thesolution-lies < is requiredfor 5' that 0 Note possible is iJ,*."n "u*.t e andI sothatcholing = would necessiJo 0 example' tn this any circumstance notto occurunder choking Froude approach for an 067 .' to approximately ratio' cont.action the i"i. iitiui"g (28 1 ft) m 57 to 8 lengrhwouldincrease numberof I andthecontraction
Designof Supercritical Expansion flo$ for supercritical it maybe desirableto designan expansion In sonreinstances, gates' spillways' sluice from flow issues at pointswherehighvelocity,supercrirical if theexpanby Chori (1959),the flow will separate or'rt."p .hut.r. Ai described to expand forced is the flow if sion isioo abrupt;andthe transitionmay be too long of the the walls from tmanate t- g*Ouuffy.In addition,local standingwavesmay downstream. propagation funher traniition and cornbineat the centerlinewith andanaRous",Bhootha,andHsu(1951)studiedthisproblemboth€xperimentally most The 3 21 Figure in lytically.They suggesta two-partwall curvature,asshown given by is expansion efn.ient strapeior the divergentponion of the
. = 1 [ ] /l - ' 1 " * r l
br
2 1 . 1\ b r F r/
I
(3.30 )
'r : of theexpansron; in whichi = lateralpositionof the \\ all fromthecenterline beginning the : from measured uoorou.h.hunn"lwidth;.r longitudinalcoordinate flow This curvecon= approach the of number Froude Fr and .*pun.ion, oi rh. portion of tinuesdivergingin the downstreamrlirection,which requiresthe second lt example' 21' for 3 the ransitio-n,u-ullg"ot"try downstreamof point P in Figure negaand positive the "onrirt, of u ,"u"..1".u-utur" obtainedfrom an analysisof from the wall to Promotecancellationof their effectsand elimitive disturbances nationof the propagationof excessire standingwaves'
90
C H A P T E R3 : M o m e n t u m
2.5
o
1 2
3
4
5
6
7
8
9 1 0
1 1 1 2
xllb1F1) FIGURE 3.2I Generalizedboundarycurves for expansion(best fit of Rouse et al. (1951) curves by Mazumdarand Hager(1993)\.(Source:H, Rouse,B. U Bhootha,and E. y. Hsu, ,,Designof ChannelExpansions,"A 1951,ASCE.Reproduced by permissionof ASCE.)
The length of the first portion of the transition,1,, and the total length of the transilion.1.. are given by Mazumder and Hager I l9i3 r ro be
L"
44:
o't"
(3.3I )
L.
(3.32)
a6=t+325(r.-l)
in which r" = expansionratio : brlbr,andthe lengthsL, andL,are asdefinedin Figure3.21.Thegeometryof the reverse curvature downsrream of pointp is given approximately by a bestfit of the odginalboundarycun,esof Rouse,Bhootha,and Hsu (1951)obtainedby MazumderandHager(1993):
b , / 2- ; "
= s i n9 0 '
'l
in whichr, is determined from Equation3.30for x = l" givenby Equation3.3l. MazumderandHager(1993)experimentally studiedexparsions designedaccording to the generalizedRouseet al. boundarycuryesand concludedthat the maximum Froudenumbercanbe as muchas2.5 rimesthedesignFroudenumberwithout significantly changingthe waveheightsor the pattemof standingwaves.As a
C H A p T E R3 : M o m e n t u m 9 l practical maner, this means that the expansioncan be shoner, becauselt can be designedfor a smaller Froude number
REFERENCES Arbhabhirama. A., andA. Abella."HydraulicJumpWithinGraduallyExpanding Channel.,, J . H _ y dD. i l : , A S C E 9 7 n , o .H Y I ( 1 9 7 1 )p, p . 3 t - - . 1 t . B r a d l e y , J . N . . a n d A . J . P e r e r k a . ' T h e H y d r a u J i c D e s i ignngoBf S a st ii n sH: y d r a u l iJcu m p s on a HorizontalApron(BasinI)." J. H-rd Diu, ASCE83. no. Hy5 (1957),pp. l_24. Chow,VenTe. Open-Ch.tnnel H\drarlics. New york: McGraw_Hill. 1959. Harleman. D. R. F Discussion of "Turbulence Characteristics of the HydraulicJump,,,by H. Rouse.T. T. Siao,andS. Nagaratnam. Tra sacti.,nsof rhe ASCE.vot. 124.1959. pp.959-62. Harrison,A. J. !1. "Designof Channelsfor SupercriricalFlor,..,proc. Inst.of Civil Engrs. 35 (1966).pp. 475-90. Hughes,W. J., and l. E. Flack. "Hydraulic Jump propeniesover a Rouqh Bed.,,,1.Hydr f n 3 r g . .A S C E l l 0 . n o . I 2 { t q 8 4 ) p . p.t755-7t. Ippen,A. T., and J. H. Dawson."Designof ChannelContractions..'ln High Velociry* Flou.in OpenChannels:A Svnposium,Transactions of rhe ASCE.I,ol. I 16. 1951.DD.326,46. "Design Keller,R. J., and A. K. Rasrogi. Chan for predicringCriticalpoinron dpillwavs... "/.l/1d Dnr, ASCE t03, no. Hy12 (tgii), pp. t4t1-29. "Radial Khalifa,A. M.. andJ.A. McCorquodale. HydraulicJump...,/.lll. Dil, ASCE 105, no. HY9 (1979).pp. 1065-78. Kolose us,H. J..andD. Ahmad."CircularHydraulicJump.,, J. H-r.dDiu..ASCE95,no.Hy I (1969),pp. 109,22. Lawson,J. D.. and B. C. Phillips."CircularHydraulicJump|.J. Hydr Elgrg.,ASCE 109, n o . 4 ( 1 9 8 3 )p. p .5 0 5 - 1 8 . Leutheusser. H. J..andY Kanha."Effectsof InflowConditionon l{ydraulicJump.,,./. Hr.r/. Dnr, ASCE 98, no.HY8 (1972),pp. 1367 85. "Hydraulic Mass€y,B. S. Jump in Trapezoidat Channels, An ImprovedMethod.,,Water Power(lune 196l). pp.232-3'7. Mazumder. S. K.. andW. H. Hager.'.Supercrirical Expansion Flow in RouseModifiedand Reverse Transirions." l. Htdr Engrg.,ASCEI 19,no. 2 ( 1993), pD.201_19. Rajaratnam, N. 'TheHydraulicJumpasa WallJer."J. Hrd. Dir. nSfE 9l, no.Hy5 ( 1965). pp.10731. Rouse,H., B. V Bhoorha. andE. Y Hsu...Design of ChannelExpansions.', Transactions of t h eA S C Er. ' o l .I 1 6 . 1 9 5 t ,p p . 1 3 6 9 - 8 5 . ..Turbulence Rouse,H., T. T. Siao.and S. Nagaratnam. Characleristics of the Hydraulic Jump.""r.Hrd. Dir,.,ASCE8.1,no. Hy | ( 1958),pp. 1528_ I ro 1528_j0. Silvesler, R. "Hldraulic Jumpin All Shapes of HorizontalChannels." "/.Hrd Dir,..ASCE 9 0 ,n o .H Y I ( 1 9 6 4 )p, p . 2 3 - 5 5 . Sturm,T. W. "SimplifiedDesignof Conrracrions in Supercritical Flow..,J. Htdr Engrg., A S C E I I l . n o .5 ( 1 9 8 5 )p, p .8 7 1 - 7 5 . Subramanya. K. Flow iD OpenCharrrels, vol.2. New Delhi,lndia:TaraMccraw_Hill,I9g2. Thiruvengadam. A. "HydraulicJumpin CircularChannels.,'Warcrpover (December1961,)pp. 496-9't. U.S.Bureauof Recfamation. Department ofthe InteriorDeskr of SnallDams,}rded.,a WaterResources TechnicalPublication. Washington, DC: U.S. Govemmentprintins Office.1987.
92
C H A P T E R3 : M o m e n l u m
EXERCISES channelwith a basewidthof 20 fl 1 . 1 .A hydraulicjump is 1
atd sideslopesof2:1. The upstreamdepthis 1.25ft and O = 1000cfs. Find the dou nstreamdepthandthe headlossin thejunlp. Solveby Figure3.2 andverifyb1 manualcalculations. Comparetheresultsfo. thesequent depthratioandrelativehead loss with thosein a rectangular channelof the samebottomwidth and approach Froudenumber
jump in a 3 ft diametersrormsewerwirh 3.2. Detenninethesequent depthfor a hydraulic a flow depthof 0.6 ft at a discharge of 5 cfs. Solre by Figure3.3 andverifyby manual calculations.
3.3. Derivetherelationship betweenthescquent depthratioandapproach Froudenumber for a triangular channelandverify with Figure3-5.Repeatthederivation for a parabolicchannel.
3.4. A flume with a triangularcrosssectioncontainsuater flowingat a depthof 0.l5 m ard at a discharge of 0.30mr/s.The sideslopesof the flumeare2:1.Determine rhe sequent depthfor a hydraulicjump. -1.5-
A parabolicchannelhasa bank-fulldepthof 2.0 m anda bank-fullwidthof 10.0m. j ump in thechannelis I .5 m for a floq If thedownstream sequent depthof a hydraulic rateof 8.5 mr/s.whatis the upstream sequent depth?
3.6. A hydraulicjump occurson a slopingrectangular channelthathasan angleof inclination,6. The sequent perpendicular depthsaredr andd, measured to thechannel bot, tom. Assumethatthejump hasa length.L' and a linearprofile.Derivethe solution for the sequent depthratio.andshowthatit is identicalto thesolutionfor a horizontal slopeifthe upstream Froudenumber,F,, is replaced by thedimensionless number. G , , g i v e nb y Gr=
Fr L, sin9 dt- d,
3.7, The discharge of waterovera spillway40 ft wide is 10,000cfs inroa srillingbasinof the samewidth.The lakelevelbehindthespillwa),hasan elevation of 200fr, andtbe riverwatersurface elevation downstream of thestillingbasinis 100ft. Assuminga l0 percentenergylossin the flow down the spillway.find the inven ele!ationofrhe floor of the stillingbasinso thatthe hydraulicjump formsin the basin.Selectthe appro(USBR)stillingbasinand sketchit showingall priateti.S. Bureauof Reclamation dimensions.
3.8. A spillwaychuteandthehydraulicjump stillingbasinat theendof thechurearerecta,rgularin shapewith a widthof 80 ft. The floor of the slillingbasinis ar an ele\'ation 787.6ft abovethe datum.The incomingflo$ hasa depthof 2.60 ft at a design discharge of 9500cfs.Withinthebasinare l5 bameblocks,each2.5fr highand2.75 ft wide.
C H A P T E R3 : M o m e n t u m
9l
(a) Assumingan effectivecoefficientof dlagof 0.5 for lhe baffleblocks,basedon velocityandcombinedfrontalareaol-lheblocks.calculate the theupstreanr depthwithoutbaffleblocks. depthandcompare\r irh the sequent sequenl (b) what is the energylossin the basinwith andwilhouttheblocks? (c) If thetailwaterelevation for Q = 9500cfs is 797.6.will thestillingbasinperform asdesignedlExplainlour answer. flume in thc laboratory, thedcpthjust downstream rectangular 3.9. In a shon,horizontal. endof theflune is I -0 cm andthedepthjust upstream of a sluicegateat theupstream of thesluicegateis 60 cm. The \r idth of th€flumeis 38 cm. If the tailgateheightis l5 cn, overwhich thereis a free orerfall.will a hydraulicjump occuror u,ill it be submerged? channel,andit is controlledby a sluice 3.10. A steadyflow is occurringin a rectangular velocilyis 3.0m/sec.If the gate gate.The upstream depthis 1.0m. andtheupstream deterrnine thedepthandspeedof lhe resultingsurge. is slamrned shutabruptly. channelare 8 anddownstream of a sluicegatein a rectangular 3.11. The depthsupstream for a steadyflow ft and 2 ft, respectively, (a) what is the valueof the flow .ate per unit of width q'l (b) lf 4 in part (a) is reducedby 50 percentby an abruptpanial closureof the gate. what will be the heightand speedof the surgeupstreamof the gate? 3.12. Write boththe momentumandenergyequationsfor thesubcriticalcaseof flow through bridgepiersof diametera and spacingr. lf theheadlossin theenergyequationis w.iG tenas KLVll2g. in which K. is the headlosscoefficientand yr is the approachvelocity, sho$,thatKr = Clh for the specialcasethat(yr - !r) is very small(Figure3.16). 3.13, For a riverflow betweenbridgepiers3 m in diameterwith a spacingof 20 m, determine the backwaterusing the momentummethodif the downstreamdepth is 4.0 m and the downstreamvelocity is 1.9 nts. Assumea coefficientof drag of 2.0 for the bridgepiers. 3.14. A st.aighFwalledcontractionconnectstwo rectangularchannels12 ft and 6 ft wide. The dischargethroughthe contractionis 200 cfs, and the depthof the approachflow is 0.7 ft. Calculatethe downstreamdepth,Froudenumber,andthe lengthof the contractionthat will minimizestandingwaves.Will chokingbe a problem? the \aave-front anglesBr and82.Whatvariables do they 3,15. For Exercise3.14,calculate variables dependon? Producea generalizedplot for B in termsof the dimensionless on whichit depends. 3.16. A supercriticaltransitionexpandsfrom a width of 1.0 m to 3.0 m, and the approach flow depth and velocity are 0.64 m and 10 m/s for maximum design conditions, respectively.Calculatethe downsream depth, and design and plot the fansition geometry.What wouldbe the lenglh of the expansionif it weredesignedfor a Froude numberlhat is 40 percentof the designvalue? 3.17, Write a computerprogramthat finds the sequentdepthfor a hydraulicjump i[ a trapezoidalchannel.First,computecritical depthanddetermineif the given depth is
9.1
CHAPTER,I:
i\lonlentunt
subcriticalor sup!'rcriticalto lifiit thc root scarch.Then. use the biscclion mcrhod lo flnd the sequenldepth. -1.18.The foll.|\ring datr for a hr draulic ju ntp have beenmcasured in a laboruon fl unte bl two different lab tcams. Tle llume has a width of 38 cnl. Tle upsrrcantsluice gale was set to producea supercriticalflow for a givcn nteasurcddischarge.and the tailgate was adjusleduntil the hydraulicjunrp was positioneriat the desired lcration in the flumc. Thr'dcpths nreasuredby a point gauge upstreamand dorvnstreamof rhe Jump,]"r and r'.. respecli\elv, arc givcn in the following table as is the dischargenreasuredby a calibrlted bend mctcr with rn uncerlaintyof a0.0(X)l mr/s. Thc estinrated uncertaintyin the upstrcamdepthsis :10.02cnr. while the downstreamdeprhshave a Iargereslimaleduncenainty of :t0.30 cm due to surfacewives. Photographsof the flume and hydniulicjumps at selectedFroudenu bers are shown in Figure -j.22.
(a) Froudenumber: 1.9
{b) Froudenumber: 4.4
(c) Froudenumber: 7.2 FIGURE 3.22 Hydraulicjump with differentupstream Froudenumbers(phorographs by G. Srurm).
C H A P T E R3 : M o m e n t u m Team A )r, cnl
L,l1 l.61 2.22 I t0 l.30 1.38 l.59 2.10
.Yrtcm
r5.8 12.3 I0.2 r3.3 I1.0 l l.9 9.9
95
Team B Q. nrr/s
0 0 r65 0.0165 0.0t65 0.0r65 0.0131 0.0l]I 0.01-tI 0.0t31
)ti cnr
1.76 2.20 2.61 t.2l 1.70 t.87
J2, cm l5.1 14.5 12.8 I1.0 13.0 I1.7 10.9 10.0
q, mr/s
0.0166 0.0166 0.0166 0.0r66 0.0125 0.0125 0.0125 0.0125
(a) Calculateandplot thesequent depthratiosasa functionof theFroudenumbers of the experimental dataandcomparethemwith the theoretical relationship for a hldraulicjump in a rectangular channel. (b) Calculateandplot thedimensionless energyloss1E/E, as functionof Froude numberfor the experimentaldataandcompareit \\ ith the theoretical relationship. (c) Estimatethe experimentaluncenaintyin _y,/_yl and the Froudenumberand plot error barson your graphs.Doesthe estinrateduncenaintyaccountfor the differencesbetweenmeasuredand theoreticalvalues?Are the resultsfor TeamA and Team B consistent? (d) Basedon thephotosin Figure3.22,describethe appearance of thejump asa function of Froudenumberand indicatethe relativeenergyloss IE/EI for each photo.
CHAPTER 4
Uniform Flow
4.1 INTRODUCTION the oftenis usedas a designconditionto determine Uniformflow in openchannels of The designdischargeis setby considerations of artificialchannels. dimensions analysis, and the channelslopeandcross-sectional risk andfrequency acceptable andavailability of land.Specby topography, soil conditions. shapearedetermined coefficient resultsthen in a uniquevalueof the depthof ificationof the resistance uniform flow, known as the normal depth. The determinationof normal depth thepositionof thefreesurfaceandthe requiredchanneldepthnecessary establishes The resistance coefficientis a to completethe designof the channeldimensions. theattentionof vital link in thisdesignprocess, andits estimationhascommanded of its variationwith hldraulic engineers sincethe l9th century.An understanding the surfaceroughnessof the channeldevelopedslowly, and only in recenttimes ha\.eotherfactorsthatinfluenceits valuebeenstudied. of conduitsflowingfull is oneof themostextensively The hydraulicresistance studiedareasin hydraulicengineering,but many difficultiesremainfor the caseof flow resistancein openchannels.In the caseof full pipe flow, Nikuradse'sexperiwork by Colebrook roughenedpipes.and the subsequent ments on sand-grain ( 1939)andMoody ( 1944),led to the development of the frictionfactor-Reynolds number plot. now known as the Mood,"diagranr,in which relativeroughnessis a successby parameterThe Moody diagramhas beenappliedwith considerable practicing engineersto the problem of determining pipe flow resistance.FIow in openchannels, on theotherhand,hasbeenmorediffrcultto quantify. resistance in openchannelflow, and the is encountered A much uider rangeof roughness exra degreeof freedomofferedby the free surfacein openchannelflow givesrise shape,and surfacewaves. to tie complexeffectsof nonuniformity,cross-sectional The importanceof the resistancecoefhcientgoesbeyond its use in channel design for uniform flow. The computationof flood stagesin graduallyvariedflow 9'7
98
Flow 4: Uniforrn CHAPTUR
and of the movementof translatoryu ar es in unsteadyflow dependon an accurate estimateof the resistancecoefficient. \luch of our presentunderstandingof the resistancecoefficientis due to a combination of theory and experimentapplied to uniform flow. but much renlainsto be leamed about flol resistancein gradually varied and unsteadyflow. Determiningflow resistancein morable-bedchannelsis especiallychallenging becauseof bed forms, such as dunes and ripples, that createform resistancethat varies with the flow conditions.Funher discussionof this case can be found in Chapter 10.
4.2 ANALYSIS DIMENSIONAL values work in establishing of the significantroleplayedby experimental Because problem' of the analysis with a dimensional to begin it is useful of flow resistance, of the nrean For a channelof any generalshape,we write the functionaldependence (Rouse 1965) 70 as boundaryshearstress (4.1) ro - f r(p, tL,e, V, R, k, C, N, U) accelerationl in wbich p - fluid densitylp = fluid viscositylg = gravitational y = meancross-sectional flow velocity;R : hydraulicradius,which is a characteristic length scaleof the flow, defined as flow areadividedby wettedboundary perimeter;andk - measureof roughnesselementheight.The lastthreeparameters The parameterC reflectsthe bn theright ofEquation4.1 alreadyare dimensionless. degree of nonunifonnityof flou'; the N indicates shape; effectof cross-sectional relaof the ftrnctional analysis Dimensional effects. unsteadiness andU represents 4.1 yields tion givenas Equation
#:r(*
: r y R " : * , F , cN , u )
(4.2)
in which Re = Reynoldsnumber;Rr = relativeroughness;F = Froudenumber: and C, N, andU alreadyhavebeendehned The lengthscaleusedin the Reynolds is four timesthe hydraulicradius,andthis will be numberandrelativeroughness justified subsequently. From the control-volumeform of the momentumequation equation,which ipplied to a steady,uniform pipe flow and the Darcy-Weisbach length scale,the as the pipe diameter of in terms facto( friction definesthe /, 4.2 becomes Equation left of parameter on the dimensionless dependent ^t./ 2
e
(4.3)
meanvelocityEquation4.3 can be takenas the in which V is the cross-sectional friction factor/, and we want the definition of/ the Darcy-Weisbach dehnitionof flow The functionalrelationof Equation4'2 channel same for open the to remain givesvaluesof the friction factor'/. in which diagram, Moody for the is the basis
C U A P T E R. l : U n i f o r m F l o w
99
with the influpipeflow as a functionof Reynoldsnumberandrclativeroughness parameters by the remainingdimensionless in Equation4.2 encesrepresented neglected. In openchannelflow, the Reynoldsnumberoften is large,so that the parameter flow is in the fully roughturbulentregimeandtheprimaryindependent is therelativeroughness.
!1.3
MOMENTUM ANALYSIS uniformflow. asshownin Figure Considera controllolumc of lengthAl in steady, 4.1. By definition,the hydrostatic forces,F,, and F,,, are equalandopposite.In addition,the meanrelocity is invariantin the flow direction,so thatthechangein reduces to a balance momentum flux is zero.Thus,themomentum equation between thegravityforcecomponent in theflow directionandtheresistingshearforce: (4.4) TALL sin0 = roPLL in which 7 = specificweightof the fluid;A = cross-sectional areaof flow; ro = meanboundaryshearstressiandP : wettedperimeterof the boundaryon which theshearstress acts.If Equation4.4 is dividedthroughby PAl, thehydraulicradius R : AlP appearsas an intrinsic variablefrom the momentumanalysis.Physically, il representsthe ratio of flow volumeto boundarysurfacearea,or shearstressto unit weight,in the flow direction.Equation4.4 canbe writtenas ro : 7R sin0 : 7RS
(,1. s)
if we replacesin0 with S : tand for smallvaluesof 0. Funhermore, if we solve Equation4.5 for the bed slope,whichequalsthe energygradeline slope,lrrll,,in steadyuniform flo*, and expressthe shearstressin termsof the pipe flow definition of frictionfactor/from Equation4.3,we have ht
='o L t R
_fpv'18: t' v' yR
4R 29
Fp2=
W =.lAaL FIGURE4.1 Forcebalance in uniformflow.
(4.6)
100
C H . \ p r E R4 : U n i f o r mF I o w
from which it is evident that the appropriareiength scale in the Darcy-Weisbach equationfor open channelflow is 4R to replace the pipe diameter.It would seem reasonableto use4R as the length scalein the Reynoldsnumberand relativeroughnessas well. The unexpectedbenefit of this approachis that friction facrorsin rur_ bulent open channelflow are similar (but not exaclly the same)to those obtained from the Moody diagram developedfrom pipe flow results.In other words, the hydraulicradiusembodiesmuch of the effect of channelshapeon friction factorbut not all of it. The effectsof nonunifornrshearstressdistributionand secondarycurrents also are relatedto shapeand must be accountedfor separatelv. Before applying uniform flow formulas to the design of open channels,the backgroundof their developmentis considered.First, Chezy's and Manning's for_ mulas for steady,uniform flow in open channelsare presented.Then, rne equanons for the friction factor as a function of Rel,nolds number and relativeroughnessin pipe flows are reviewed and extended to open channel flow. Finally, the effects of the Froudenumber,nonuniformity,and cross-sectional shapeon open channelflow resistanceare explored.
4.4 BACKGROUND OF THE CHEZY AND MANNING FORMULAS While Equation4.5 givesa formulafor the calculation of meanshearstressin uni_ form flow, the problemof determiningthe depthof uniform flow for a given dis_ chargerequiresan additionaluniform flow fornrula. Historically,such formulas havebeenpresented for velocityof flow as a functionof hydraulicradiusandslope. If Equation4.6 is solvedfor velocity,we have r/2
" = [ 8 P ] vRs= cvRs L;l
(4.7)
in. which C is called the Chezy C in honor of Antoine Chezy,who first proposed this formula.Chezy,a Frenchengineer,was chargedwith the task of de;ig;ing a watersupplycanalfrom theYvette River to the city of parisin 176g.His final rec_ ommendationsin l7?5 containedthe Chezy formula wrinen in termsof ratios of velocitiesof two rivers;and in a later memorandumin 1776,he savethe formula for velocity as we now know it. He presenteda conslantualuefo-rC, but he real_ ized that it variedfrom one river to another Unfortunately,Chezy'swork was not publishedand so did not becomewidely known unril after 1897,when it was pub_ lishedby Herschel in theUnitedStates(Biswas1970).Du Buar,a contemoorarv of Chezy,arrivedat the sameuniform flow formula somefour yearslaterthanChezy. althoughhe concludedthat the effectsof boundaryroughnesscould be neglected. His work waspublishedin an eady book on hydraulics.Many otheruniform flow formulasof the "universaltype" with no variationof the coefficientswith roush_ nesswereproposedin the early l9th centurysuchasthoseof Eytelweinand prJny (Dooge1992). Within this contextof extensivework on uniform flow in the first half of the lgth century RobertManningbeganhis careeras a drainageengineerin 1g46.He
CHAPTER 4: UniformFlo\*
l0l
wasself-taught andgreatlyadmiredtheFrenchwritingson hydraulics. The uniform flow formulabearingManning'snamewasnot proposed until theendof his career, whenin 1889at the ageof 73 he presented it in a paperwhile he was still chief engineer of theBoardoi Worksof lreland.His formulawasbasedprimarityon the work of Darcyand Bazinon outdoorexperimental pioneering canalsfrom 1855to 1860.This work u,aspublishedby Bazinin | 865 afterrhe deathof Darc),and it thattheChezyC depended showedconclusiyelv on thenatureof thesurfacerough(Dooge1992). nessof thecanalboundaries In his 1889paper,Manningselected seyenwell-knownuniformflou.formulasfor velocityin an openchannelexpressed asa functionof hydraulicradiusand slope.He calculated the velocitiesovera rangeof hydraulicradiifrom 0.25 to 30 m from eachformulafor a givenslopeandanalyzed the meanof theresults.From thesepreliminaryresults,he concludedthal the velocitywas proportionalto the hydraulicradiusto the; powerandto theslopeto thej power.but he realizedthere might be a moregenerallyapplicablevalueof the exponenton hydraulicradius. theresultsof som€selected He thentookthecrucialstepof analyzing experiments of Bazinon scmicircularcanalslinedwith cementand with a sand-cement mixture.Manningconcludedthat the exponentin both caseswas very close to the fraction]. The resultinguniforn flow formulawasgivenasformulaV in rhe 1889 paper:
y : crR:/lsr/l
(4.8)
in which the subscripton C has beenaddedto distinguishthe coefficientfrom the Chezy C. Manning proceededto comparethe resrlts of this formula u,ith 170 observations. of which I04 \r'erethoseof Bazin.He concludedthatthis formula performedbetterthan tlroseof Bazin and Kutter,the latterof which was very popularat thetime. Manningwas dissatisfiedwith formulaV howeverbecauseof its lack of dimensional homogeneity and the necessity of takinga cuberoot in the evaluation of the velocity.He thereforeproposeda secondformula,formulaI, which overcame theseobjections,althoughit usedthe barometricpressureheadto achievean artificial nondimensionality. It performednearlyaswell asformulaV andseemedto be Manning'sformulaof choice.Ironically,this formulahasbeendiscardedand forhis paperwith the following mulaV bearsManning'sname.Manningconcluded "Althoughtheauthormakesno pretension statement: to mathematical skill. he may, in conclusion, be allowedto expressthehopethattheresultsof his labors.suchas a science, in thestudyand practice theyare,mayadvance, evenin a smalldegree, of whichhe hasspenta long professional life." The dissemination of the uniformflow formulathat now bearsManning's namewasgreatlyenhanced publication bv Flamant's of it in his l89l textbookand his referenceto formulaV as Manning'sformula.A carefulreviewof the historical recordby Williams(1970),however,showsthatsomel0 investigators proposeda formula of this t),pe.The first suggestionof the exponentof ] on the h1'draulic radiusactuallywasmadeby the Frenchengine€r Gaucklerin 1867.Gauckler'sformulaalsowasbasedon DarcyandBazin'sexperiments but it neverreceiredwide acceptance, partlybecause of the widespread useof a formulaproposedby GanguilletandKutterin 1869for Chezy'sC. Thisformulafor C wasverycomplexand
t02
CHAPTER -1: Uniform Flow
hada dependence on theslopeanda singleroughness coeflicient, n, calledKutter's rr.This wasthe resultol attempting to reconcileDarcyand Bazin'sdataon small canalsof nroderate slopewith the observations of Humphrel andAbbotton the MississippiRiver for very smallslopes.Manning,in fact, eliminatedHumphrey andAbbott'sdatafrom his 170observations because of thedifficultyhe perceived in measuring suchsnrallslopesandshowedno smalldisdainfor rhecomplexityof Kutter'sformulain his 1889paper The final ironic twist in thedevelopment of whatis nol knownasManning's by Flamantthat Cr in Manning'sfonnulaV couldbe formula wasthe suggestion as the reciprocalof Kutter'sn in metricunits.Sereralsubsequent expressed texts King ( l9 | 8) advocated repeated thisassenion,andtheAmericanhydraulician this "Manning'sr," What we now knou'as Manning'sforstepwhile referringto n as reallyshouldbecalledthe Gauckler-Manrtirtg mufa,whichWilliams( 1970)suggests fornulq. ts written todav as
y: &4:,r5r,: n
(1.9)
in which V is velocity;R is hydraulicradius;andS is bedslope.The valueof K, I .0 with R in m and y in n/s, andK- = 1.49for R in ft andV in ft/s.Thelattervalue in whichManning'sn maintains the samevaluein either comesfrom a conversion units. so that the dimensional units of Koln, originally mr/t/s,have(o SI or English : (3.28 to ftr/3/s by the factor ftlm)r/r 1.49. That Equation 4.9 has be converted enduredfor morethana centuryasa unifomrflow formulau ould seemto indicate thatManning'slaborswerenot in vain,althoughthe formulathatbearshis name probablywouldbe surprising to him.
4.5 LOGARITHMIC
FORMULA FROM MODERN FLUID MECHANICS
designpipesfor the conditionof full pipe Thefacilitywith * hichmodemengineers on the velocitydistribuflow is duein largepan to Prandtl'spioneeringresearch tion in turbulentboundarylayers.Prandtl'sturbulentmixing lengthconceptand resultin the logarithmicvelocvon Karman'ssimilarityhypothesis for turbulence itv distdbution tn -
(4.l0)
in which u. is the shearvelocity- (rJ t'2:'x: von Karman'sconstant= 0.40t andrr and: are the pointvelocityanddistancefrom :0 : constantof integration: analysis thewall, respectively. In thecaseof a smoothwall surface.a dimensional for 3oasa functiononly of a. andkinematicviscosityl sho\\'sthatu.:o/zis a conEquation4.10canbe rewrittenas stant:therefore.
ln5+e,
(4.1r)
C H.{ PTER 4: Uniform Flow
l0.l
in which A, is a constant detennined by Nikuradses experiments on smoothpipes to ha\e a valueof 5.5.Equation,1.I l, knownas the lav'of tlte lra11,strictlyspeaking appliesonly to a near-wallregionwhere:/h < 0.2, in whicb/r is ihe boundarylayerthickness. This regionis calledthe logarithmicoverlaplater, in whichboth viscousandturbulentshearstresses are imponant.lt unexpectedly canbe applied over nearlythe full thickness of the flow.VeD nearthe wall, in the viscoussublayel only viscousshearapplies,andthe lat of the wall simplifiesto v
( 4 .I 2 )
velocitydistribution givenb\ Equation4. l2 appliesonly for The viscous-sublayer u,:/v 1 5 but often is appliedup to its intersection with the logarithmiclaw, at u.zlv : | 1.6(Roberson andCrowe1997).Sereralinvestigators haveimprovedthe fit of the velocitydistribution in thetransitionfrom rhelaminarsublayer to thelogarithmicoverlapregion,andtheserelationsare summarized by White(19721). The completelaw of thewall for the velocitydistributionis illustrated in Figure4.2.
20 :l+
Llnear U'= Z* viscous I j sublayer, --4 Eq.4.12
I
\ 10
5
0
1c ?=zu'
FIGL RE 4.2 Velocity profilesin turbulentwall flow (White 1999).\Source: F White,Fluid Mechanics, 1e, @ 1999,McGravt-Hill.Reproducedwith pemission ofThe McGrav-Hill Companies.)
104
C H A P T E R4 : U n i f o r m F l o w
nearthewall,earlyobserIn the outerregion,far from the viscousinfluences showedthata velocitydefectlaw vationsof boundarylayersin pipesandchannels (Daily andHarleman1966): is applicable - -
l K
Ur
z
ln:+A,
(4.13)
h
in whicha.", is themaximumpointvelocityat theouteredgeof theboundarylayer Eguation4.13 is applicable for smoothor and i is the boundarylayerthickness. into the logarithmicoverlaplayer roughwallsandcanbe extended of theoryandexperiment thatled to thelogarithmicvelocity The combination distributionfor turbulentflow replacesthe exactintegrationof the momentum whichis possibleonly in thecaseof laminarflow.The resultfor laminar equation, law for the frictionfactor,/ = 64lRe.Our objectiveis to obtain flow is Poiseuille's logaa relationfor the frictionfactorin turbulentflow basedon the semiempirical givenby Equation4.1l. The logarithmicvelocitydisrithmicvelocitydistribution it over the flow into a resistance equationby integrating tributionis transformed thicknessto obtain an expressionfor the dimensionlessmeanvelocity, y/r.. The relationfor meanvelocitycanbe expressedin termsof the friction factor,f,by rearrangingEquation4.3 as
v
E
(4.r4)
pipe,asfirst givenby Prandtl(Roberson andCrowe Theresultfor a smooth-walled 1 9 9 7 )i.s (4.15) For a rough-walledpipe or channel,the viscoussublayeris disruptedby of thesublayer itself.In this if theyarelargerthanthethickness roughness elements elements, case,theviscosityno longeris imponant.but theheightof l}Ieroughness l, becomesvery influentialin determiningthe velocityprofile.A dimensional analysisindicatesthat the velocitydistributionshoulddependon:/t and the dependencemust be logarithmicto satisfy the overlapof the outer, or velocityfor a defect,law into the inner,or wall, region.The resultingvelocitydistribution roushwall is
ln: + A,
(4.t6)
pipesin roughened the valueof .4, to be 8.5 for sand-grain Nikuradsedetermined .1. > which l,Av 70.If Equation l6 is integrated over fully roughturbulentflow,for velocity, we can derive the friction factor pipe section to obtain the mean a cross relationlor fully rough turbulentpipe flow: - l . - l l n d. * 1 . 1 4 ( 4 . 1)7 ^ ti vf roughness heightfrom Nikuradse's in whichd = pipediameterand/<,- sand-grain exDenments.
C H A P T E R4 : U n i f o r m F l o w
105
ln betweenthc turbulentpiperelationsfor frictionfactorfor smoothandfully roughconditionsgivenby Equations,l.l5and4.17,respectively, is a transitionalrough regime,definedapproximately by 4 1u.k,/v 170. The behariorof the friction-factor relationin this transitionregimedepends on the typeof roughness. pipesand comIt is different,for example,for Nikuradse's sand-grain roughened mcrcialpipes.Colebrook( 1939)ht a transitionrelationfor commercialpipesthat is asymptotic to boththe smoothandlully roughfriction-factor relation::
| r=-z Vf
^ .I o El kl -,*l d+ 2 . s t l L s.t
,l R.VFI
( 4 .l 8 )
is expressed in which the commercialpipe roughness as an equivalentsand-grain roughness by determiningthe sand-grain roughness heightthat would give the pipein fully roughturbulentflow. Equasamefrictionfactorasfor thecommercial tion4. 18is thebasisfor the Moodydiagramshownin Figure4.3 (seeRouse1980). Keulegan(1938)appliedrhe logarithmicvelocitydistributionto flou'in open channels.He proceeded to integratethe Nikuradsevelocitydistributionfor fully roughturbulentflow (Equation4. l6) overa trapezoidal openchannelcrosssection to obtain,for the friction facto( I
Ur
:.03 foe - | + 2.2t - z.oj tosS k, k,
(4.19a)
in whichthevalueof 6 : 12.26on therighthandsideof (4.l9a).In realin, f varies slightlywith thechannelshape,buta valueoff - l2 is recommended b1 anASCE TaskForce(1963)andthe slopeof 2.03oftenis roundedto 2.0.Keulesanderived theexpression for { for rectangular channels to be
i : " - o [ ' ( '. r ; ) - i . ' o )
(4.19b)
in which b : channelwidth and,t : flow depth.For the aspectratio b./,Ivarying from 5 to 100,for example.{ takeson valuesfrom 12.6to I l.l, respecd\,ely. The relationshipbetweenManning'sn and Darcy-Weisbach's / now can be obtained fromtheirdefinitionsto determine theapplicability of Manning'sequation:
,= *fir/"'n'iu
(4.20)
in whichK, = L0 for SI unitsand1.49for Englishunits.If we substitute Equation 4.19ainto Equation4.20with f = 12andtheslopeof2.03roundedto 2.0.we have
K" /R1"u n
&1,"
(&)" \r,/
(4.2t)
z.or"e(rz"a)
which hasbeenplottedin Figure4.4 for both Englishandmetricunits.Or er a fairly wide rangeof valuesof R/ii,. the valueof a/tli6 is constantandthereforeoot a function of flow depth,an essentialassumption of Manning'sequationin which the depth
a^4elou r. sseuqonoS rf) \t
o
o
O
o
o
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N
o
o
rO
o
(o O
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o
rl O
o
o
o
CV O
o
o
@@t eOOO OOOO
o
o
o
N O O
o
O O
oO O
9 9 t-- t
,:-
T-:
L
!
a 8 a
t
-
l n e 8L
8
t E Bt
: :
2r
s r
?
r1
-
t
> -
=
i! * tE_ l9 t6 8
E d
l !
s -
E } 5a .: !.
te-
> 3
;6,
r-----=
'-
H
: q
!
O
,
b
o
:
)
q
@
o
F
q@ qt
f
q
)
o$ o
C
o c
!62 a\ l\z^ ; i |l '
l_:,/
106
I
D
c i
q
. ' rN oL ' : q c j q
-) iG) t( >i
.tolceluoUcuf
v : a r
CHAPTER 4: Uniform Flow
lo'7
0.1
-( Ks in
i l
meters
L]l t (
0.01
10
100
sln
e)
1000
R/ks FIGURE 4,4 Manning'sn variationwith relativesmoouness. dependenceof the velociry for a given roughnessheight is assumed to be contained enurely In lhe Rrr rerm.The minimum valueof cn : n/t,r/6in Figure4.4 is 0.039 for merrc unrtsand c, - 0.032 for English units at R/1, : 33.7, although thesevalues varl slightly with rhe constantsassumedin the Keulegan equation(yei 1992a).More -+5 generaily, the value of r(r/6 can be shown to be within percent of a constant
a r3nq9o.fR//<,givenby 4 < R/t, < 50Oasshownby yen ( 1992a).Hager y.alu^1gv9r (1999)givesthe limits on theconsrancy of n with depthandsoon rie rangeofapplicability of theMannjngequarionto be 3.6 < Rlk,< 3t0. Thelimitationofiully rough turbulent.flow for the lr4anningequationalso is implicit in the comparison with Keulegan'sequation.This limitationrequiresa.t/z > ?0, which can be translated into the limit .-
f
u
I v L ",
6
-'l
VpRS I aVs I I
| >- ') -1 Y r n I
I
(4.22)
usingthe minimumvalueof ngt/1|(K,,k,',u) - 0.122from Equation4.21to substitute for,t. Forexample,a 2 ft (0.61m) diameterstormseweiwithz = 0.015flow_ at a slopeof 0.001wouldexceedrhe limit givenby the inequalityin lie^l^ust!il (,1.22)andbe in the fully roughturbulentregime. The literaturecontainssomedisagreement aboutthe valueofcn = n/lll6 asdiscussedby French(1985),panlybecause its valuedepends on whethertheunitsoft, are melric (m) or English(ft). The Stricklervalueof c, is givenby Henderson( 1966j to be 0.034in Englishunirs(0.041in SI units)basedJn rneasurements madeby Strickler(1923)in gravel-bcdsrreamswirh t : dso,the diamererfor which 50 peicentof thesediment particles aresmallerby weighi.tne minimumvalueof c- from
108
C H A P T E R4 : U n i f o r m F l o w
10
1 R6/dsa - - Bathurst (ks= 2.4 d8a) - ' - ' ' L i m e r i n o( ks s = 3 2d e r ) _ Bestfit (ks= 1.4dsa)
(1990) r Dickman .Thein(1993)
FIGURE4.5 flow with largero!ghness elements. Frictionfactorin fully roughturbulentopenchannel equationis 0.032in Englishunits(0.039,SI) as notedearlier,which Keulegan's thattheeffectivesizein thegravelwell with theStricklervalueconsidering agrees bed streamis largerthanr/rodue to bed armoring,as arguedby Henderson( I966). includingHager(1999),givethe Stricklervalueof severalothersources, However, againu'hen c" : 0.039in Englishunits(0.048in SI units).Thispointis considered coefficientfor rock riprap laterin this chapter. discussingthe resistance of Darcy-Weisbach's Somelaboratoryand field measurements / are compared with Keulegan's relation(Equation4.l9a) in Figure4.5.The datapointsby Thein (1993)andDickman(1990)weremeasured in a 1.07m (3.5ft) wide tilting flume combinations of with a coarsegravelbedin whichuniformflow wassetfor several depthandslope.The Bathurst( 1985)datain Figure4.5 wereobtainedfrom highthe riversin Britain.Limerinos(1970)measured gradientgravelandboulder-bed resistancecoefficientin I I gravel and cobble-bedstreamsin California.The data in Figure 4.5 are presentedin terms of the bed friction factor,r, and the bed hydraulicradius,Rr, to indicatethat the flume datahavebeencorrectedfor the in Figuretl,5 (whereRrldro By comparingtheintercepts effectof smoothsidewalls. : 1.0)with the Keulegan sand-grain roughness, constant, a valueof theequivalent as a multipleof dro,which is the sedimentgrain size for k., can be determined from which84 percentof the sedimentis smallerby weight,It canbe determined Figure4.5 thatl./4r hasa valueof 1.4for the lab data,2.4 for the Bathurstdata, thatk,ldu = 3.5,basedon and3.2 for the Lirnerinosdata.Hey ( 1979)concluded that wake interferdatafrom severalsourceson gravel-bedstreams,andsuggested
CHAPTIR .l: UniformFlow
109
10
s J > t
a
ool
0.1
0.01 0.1
10
100
ak" FIGURE4.6 Comparison of field datafrom Blodgett( 1986)on flow resisrance in boulder,cobble,and gravel-bed streams withKeulegan's equation forManning's z. encelossesdownstream of thelargerroughness elements mayaccountfor thelarge valueof 1.. Resultsfor the friction factor in gravel-bedstreamsalso can be presentedin tcrmsof Manning'sr, accordingto Equation4.21. Data assembled by Blodgen (1986)for boulder-.cobble-,andgravel-bed streamsin the westernUnitedStates are plottedin Figure4.6 in termsof /<,,which can be determinedto be 6.3 drofrom fittingthe Keuleganequation(Equation4.21)for Manning'sn. Only the valuesof d50were reponedby Blodgett,becausethe interestwas in obtaininga relationship for rock-ripraplined channelsbasedon naturalchanneldata.The datain Figure4.6 are givenin termsof the averageor hydraulicdepth,D, insteadof hydraulicradius because Blodgettfoundthemto be virtuallyidentical. The Blodgettdaraincludethe databy Limerinos(1970), who reporteda standarddeviationin the percentdifferencebetweenmeasuredandfitted valuesof nto6e ! 22 percentwhend5owasused as the characteristicgrain sizeand t l9 percentwhen dro was used.
4.6 DISCUSSION OF FACTORS AFFECTING/AND
n
The dependence of/on the Reynoldsnumberand relativeroughnesshasbeendiscussedwith respectto the Moody diagramfor pipe flow. The Reynoldsnumber dependenceis not as importantin open channelflow, especiallyin large natural channels,for which the Reynoldsnumberis quite large.If a smooth-walledconduit
ll0
Flow C H A P T E4R: U n i f o r m
is flo*ing partlyfull in thesmoolhturbulentregime.however.Manning'sequltion is not directlyapplicable, because Manning'sn can be expecledto vary with the (Henderson Reynoldsnumber 1966).In thiscase,the useof theDarcy-Weisbach's ( / is preferred,althoughYen 1992a)showsthatfor Re)noldsnumberslessthanthe fully roughturbulentvalues,thereis a narrowerrangeof R/k,,withinwhichManconstant. ning'sn still maybe reasonably in openchannelflow is not aswell relative roughness The dependence of/on it is difficult to assignan equivalent sand-grain in pipe flow because known as roughness height typically found in open values of absolute for the large roughness (1965) importance of roughness concentration, discusses the Rouse channels. height,,t,. He on the equivalentsand-grainroughness shape,and arrangement resultsthatindicatethemaximumvalueof relativeroughness reponsexperimental ( 1980) concentration of 20-25percent.KumarandRoberson occursat a roughness relationfor in obtainingan analytical advances andKumar( 1992)madesignificant the variatjon of relative roughnesswith concentrationand shapefor randomly hasled to a completelygeneralalgoelements. This research arrangedroughness (Kumar and Roberson1980)that roughconduitresistance rithm for determining seemsto work well for artificialroughnesselements,but morelimitedcomparisons with natural channelroughnesshavebeenmade.The analyticalmethodutilizes a heiSht element, theaverage for an individualroughness dragcoefficientdetermined of roughnesselements,and an arealprojectionfactor to describethe projectionof to the flow as a functionof diselements on a planeperpendicular the roughness tance from the boundary.With this information,the equivalentsand-grainroughelenesscan be estimatedasa functionof the concentrationandshapeof roughness elements, ments.The methodcannotbe applieddirectly to in-line roughness however.becauseit doesnot accountfor wakeinterferenceeffects. shapeoccursas a result of flow resistanceon cross-sectional The dependence distriof changesboth in the channelhydraulicradius,R, and the cross-sectional bution of velocityandshear.Therefore,applicationsof the Moody diagramin open channelflow in which pipe diameteris replacedby 4R may not completelyreflect shapeon flow resistance.Kazemipourand Apelt the effects of cross-sectional (1979) suggestedthat two dimensionlessparametersare requiredto characterize coefficient/: shapeeffectson the resistance K. P B\ / / - r [ n ei ., a . a )
(4.23)
"functionof": Re : Reynoldsnumber;k, : sand-grainroughin which F denotes = nessheight;R hydraulicradius;B : channeltoP width:D : hydraulicdepth; andP : wettedperimeterThe first shapefactor,P/8, is a measureof the influence of the sheardistributiononi andthe secondshapefactor,8/D, is a channelaspect ratio. Using the dataof Shih and Grigg (1967)and Tracyand lrster (1961)for smoothrectangularchannels,KazemipourandApelt (1979)showedthat the oPen channelfriction factor,t, canbe obtaineddirectly from the pipe friction factor,t, determinedfrom the Moody diagram:
f. = ofu
(4.24)
C H { r r E R . 1 : U n i f o r r nF l o w
l
in which
' (;)
Qt Qz
_-
/B
(4.2s)
I \
+ n " O'
1 - + 0 . 3I The functionty'2(B/D) is proposed as a besrfit of theexperinrental relationship presentedby Kazemipour andApelr(1979)basedon theeiperimental datafor smooth rectanguiar channels thatthey usedover a rangeof g/D from approximately I to :10.Thcy alsoappliedtheir experimental relationship succersfuily to limiteddata for fully roughturbulentflow in rectangurar channeriandan addiiionar datasetof their own for sntoothrectangular channels(Kazernipour andApelt l9g2). Using Equation4.25,the valueof o for a rectangular channelvariesfrom approximatel! 1.04to Ll0 astheaspecrratio(b/y)increases from I ro 40. Experimental research by Sturm and King (l9gg) on the flow resistance . of horseshoe-shaped conduitsflowing partly full has shownthat the Kazemrpourand Apelt(1982)relations for shapeeffectsin rectangurar channels cannotbe extended to horseshoeconduits.The Neale and price (1964) data for partly full flow in smoothcircularconduitsalsoshowconsiderablescatterfrom Equation4.25. For the horseshoe conduitflowing partly full, the shapeeffectdependson the ratio of depth to diamet€r,-r.y'4with greaterinfluenceat largervaluesof _v/d.A summaryof the resultsis shownin Figures4.7 and4.g. In Figure4.7,thevilue of o for th! horseshoe.cond-uitessentiallyis unity for relativedipths lessthan0.4, but it increases to a value of approximatelyL25 for larger relativedepths.For the smooth, circular conduitdataof Nealeand price(1964),an average valueof q : 1.05occursfor 1ld > 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<--}t I 0823d
: x
0.6
Horseshoe shape
K
\
f 0.4
x
*" x_
a45
o.2
0.0
a
o.2 o.4 0 . 6
0.8
1.0 1.2
1.4
1.6
1.8
2.O
o = fc/fu . Horseshoe (Sturm& King1988) (Neale & Price1964) x Circular FIGURE4.7 (transition) andcircular(smooth) for partlyfull flow in horseshoe Frictionfactorcorrection " Efectson Flow : T.W.StunnandD. King, Shope (StunnandKing I988). (Sorrce conduits by perConduits,"J Hydr Engrg,A 1988'ASCE.Reproduced Resistance in Horseshoe missionof ASCE.)
applicationsbe obtainedonly for uniformflow to eliminatethe effectsof nonuniforrnity. Uniform flow in laboratoryflumesis difficult to obtain unlessthey are very the friction factor for a smoothchanlong.Tracy andl,ester(1961),who measured nel in the 80 ft (24 m) long tilting flume at GeorgiaTech,deviseda procedurefor determininguniform flow depth.Their techniquewas to establishtwo conrol gate positions,one of which provided a water surface profile that asymptotically uniform (normal)depthfrom aboveandthe otheronefrom below.Only approached in this way weretheyableto obtainan accuratevalueof uniform flow depth,evenin a relativelylong flume. asa resultof cross-sectional Nonuniforrnitiesdue to changesin form resistance changesmay be consideredto be Froudenumberdependent.Thus, flow around bridgepiersor flow with a boundarythat haslarge,widely spacedroughnesselementsexperienceswave formation,and Froudenumbereffectson the resistance coefficientare introduced.Supercriticalflow aroundbendsor in contractionsis anotbercasein which the resistanceis Froudenumberdependent.
C H A P T E R- l : U n i f o r m F l o w
0.8
l ll
a
I 0.6
o.4
2<
7
/
.2_
o.2 / t-a
o.2
-
o.4
0.6
0.8
't.2
vNr Constantr' - - - - - - - - - - C a m p( ' 1 9 4 6 )
. Horseshoe data P o m e r o(y1 9 6 7 )
FIGURE4.8 velocityrelationships Relative for circularconduits compared to datafor horseshoe conduits (SturmandKing 1988).(Source: T. W.StunnandD. King, "ShapeEfectson Flow Resisrancein Horseshoe Conduits," J. Hydr Engrg,,A 1988,ASCE.Reproduced bypermission of ASCE.)
Unsteadiness of openchannelflow also bringswith it changesin velocitydistribution andresistanceto flow. The occurrenceof surfaceinstabilitiesin supercritical flow, commonly calledroll wayes,is an exampleof the unsteadiness effect. Rouse(1965) suggestedthat the increasein resistancedue to roll wavescould be relatedto the ratio of the flow Froudenumberto the critical valueof Froudenumber abovewhich instabilityoccurs(: 1.5to 2. for wide channels). Berlamontand ( l98l) confirmedthis formulationand funher assertedthat these Vanderstappen resistanceeffects are more likely io occur in wide channels.They indicatedthat Froude number effects in supercriticalflow may have been overlookedby some investigatorsbecausetheseeffectsare small and independentof Reynoldsnumber when it rs large. ln summary,the effectsof unsteadiness, Froudenumber,nonuniformity,crosssectionalshape,and roughnesselementconcentrationand arrangement, as well as the usual Reynoldsnumberand relativerougbnesseffects,all can be expectedto
lll
C H A P T E R4 : U n i f o r m F l o w
affect open channelflow resistance. Continued use of the Manning's r meanssimplv that we lump all ofour ignoranceabout flo* resistanceinto a singlecoefficient. For example, it is difficult to establish the physical significanceof observed changesin Manning's n with river stagebecauseof the many factorsthat affect it. Engineeringexperiencewill continueto dictate the choice of Manning's r values, but tbey shouldbe verified by field measurernents as much as possible.In the case of turbulent,partly full flow in smoothconduits. the Darcy-Weisbach / may be the preferredresistance coefllcientihowever,the constancyof Manning'sn over a wide range of flow conditionsfor a given boundary roughness,panicularly in natural cbannels,make it a valuable tool for assessingthe effects of open-channelflow resisiance.
4.7 SELECTION OF MANNING'S,, IN NATURAL CHANNELS As mentionedpreviously,thereis no substitutefor experiencein the selectionof Table4-l from Chow (1959)givesan ideaofthe Manning'sn for naturalchannels. variability to be expectedin Manning'sn. The picturesof channelswith measured ( 1984),Bames( 1967), valuesof Manning'sn asgivenby ArcementandSchneider and Chow (1959)areveryusefulfor developingpreliminaryvaluesof Manning's n. Someof thesephotographsare given at the end of this chapterIn addition,for those channelsoutsidethe engineer'sprevious experience,the more regimented procedure suggested by Cowan(1956)is helpful: n = l n b + n t + n ? + n 1+ n a ) m
(4.26)
in uhich n, = the basevaluefor a straight,uniformchannel;n, = corectionfor surfaceinegularitiesia2 : correctionfor variationsin the shapeand size of the cross section;n3 : correctionfor obstructionsin4 : corTectionfor vegetationand flo\* conditions;andm : conectionfactorfor channelmeandering. Valuesfor each ( of thesecorrectionsaresuggested by Arcementand Schneider 1984)for both natural channels andfloodplains.
4.8 CTIANNELS WITH COMPOSITE ROUGHNESS Under somecircumstances, a naturalor anificial channelmay havevaryingroughnessacrossits wettedperimeter;for example,u'ith differentlining materialson the prebed andbanksor vegetated bankswith an unvegetatedbed.The methodologies sentedin this sectionare not meantfor compoundchannelsin which the geometry and the roughnessare signihcantlydifferent on the floodplainscomparedto the main channel.For compoundchannels,it is more appropriateto divide the channel with differentvaluesof the roughness into mainchannelandfloodplainsubsections coefficientto obtainthe total conveyance. as will be discussedsoon.
CHAPTLR 4: Unifonn Flow
ll5
TABI-E {.I
Valuesof the \Ianning's RoughnessCoeflicienta tpe
of Channel and Description
A. ClosedConduit\Flot!ingPanlyFull A I. \telal a. Brass.smoolh b. Steel L Ltl*kbaraDd",elded 2 . R i \ e l e da n ds p i r a l c. Castiron L Coared 2. Uncorled d. WrouShtiron L Black 2. Cal\anized e. Corrugstedmetal L SuM.rin 2. Srormdrain A-2. Nonmetal a. Lucire b. ctass c. Cemenl L Neal.surface 2. Morlar d. Concrete l. Culven.slraightand freeofdebris 2. Cul\ertwith bends,connections. and somedebris 3. Finished 4. Se$erwith manholes, inlel.etc., suarshr 5. Unfinished, sreelform 6. Unfinished, smoorh*ood form 7. Unfinished, roughwood form e. Wood L Sta\e 2. laminared.rreated f. Clay L Commondrainage tile 2. \arrifiedse\rer 3. \alrified sewerwith manholes, rnrel erc. 4. Vitrifiedsubdrain wirh openjoinr g. Brick*ork L Clazed 2. Linedwirhcementmonar h. Sanita4 s€werscoatedwiti se*age slimes.with b€ndsandconnections r. Pavedinren, sewer,smoothboltom j. Rubblemasonry,cemenred
llinimum
Nonnal
Maximum
0.009
0.010
0.011
0.010 0.011
0.012 0.016
0.Ot-1 0.017
0.010 0.011
0.013 0.01,1
0.0t.1 0.016
0.012 0.013
0.0t4 0.0t6
0.01-s 0.017
0.0t7 0.02t
0.019 0.024
0.011 0.030
0.008 0.009
0.009 0.0t0
0.010 0.013
0.010 0.01|
0.011 0.01i
0.01j 0.015
0.010
0.01|
0.0t3
0.01I 0.011
0.011 0.012
0.0t.1 0.0t.1
0.013 0.012 0.012 0.015
0.015 0.013 0.014 0.017
0.0t 7 0.01J 0.016 0.010
0.010 0.015
0.012 0.017
0.01.1 0.0]0
0.011 0.011
0.013 0.014
0.017 0.0t7
0.013 0.014
0.015 0.016
0.01? 0.018
0.0t I 0.012
0.013 0.015
0.015 0.0t7
0.012 0.016 0.018
0.013 0.019 0.025
0.016 0_0:0 0.030
ll6
C H A P T E R4 : U n i f o n r F l o w
TABI,E J-l {Coniinucd)
Typeof Chann€land Descriplion B. Linedor Builrup Channels B- 1. Meral a. Snroothslcel.urface L Unpainted :. Painled b. Conugded B 2. Nonmctal a. Cement L Nert. surface 2. Monar b. Wood L Planed.untreared 2. Planed. creosoted 3. Unplaned ,1. PIankl\ith banens 5. Lined \\'irh roofing paper c. Concrete l_ Trowel finish 2. Floal finish 3. Finished. u ith gravelon botlom ,1. Unfinished 5. Cunite.goodseclion 6. Cunite,*ayv section 7. On good excavatedrock 8. On irregularexcavatedrocK d. Concretebo[om f]oal finishedwith sidesof L Dressedstonein monar 2. Randomstonein monar l. Cementrubble masonry.ptastered .4- Cementrubble masonry 5. Dry rubbleor riprap e. Gravelbo(om u ith sidesof L Formedconcrete 2, Randomstonein monar 3. Dry rubbleor riprap t Brick l. Glazed 2, In cementmortar g. Masonry l. Cemented rubble 2. Dry rubble h. Dressedashlar L Asphah l Smooth 2. Rough j. Vegeral lininS C. Excavatedor Dredged a. 8anh, straighland uniform L Clean,recenrlycompleled
Ilinimum
Nornral
i\[axinrum
0.0 0.0r2 0.011
0.0t-1 0.01,5
0.01.1 0.017 0.010
0.010 0.011
0.0t1 0.013
0.0r3 0.015
0.010 0.01 I 0.01 I 0.0r2 0.010
0.011 0.01l 0.0t3 0.0r5 0.01{
0.0t.1 0.015 0.0t5 0.018 0.017
0.0 0.013 0.0t5 0.0t.1 0.0r6 0.0t8 0.017 0.022
0.0r3 0.015 0.017 0.017 0.019 0.022 0.020 0.027
0.015 0.0r6 0.020 0.020 0.023 0.025
0.0r5 0.017 0.0r6 0.020 0.020
0.0r7 0.020 0.020 0.025 0.030
0.020 0.02,1 0.024 0.030 0.035
0.0i7 0.020 0.023
0.020 0.021 0.03.1
0.025 0.026 0.036
0.01l 0.012
0.0r3 0.015
0.0t5 0.0t8
0.017 0.023 0.0t3
0.025 0.032 0.015
0.030 0.035 0.017
0.013 0.0t6 0.030
o llu
0.500
0.0t6
0.0t8
0.020
0.0r1
0.01l
C H { P T E R 4 : U n i f o r mF l o w
l)pe of Channeland Descrip{ion 2. Clean,afrerweathering 3. Cravel,unifomrseclion, clean 4. With shon grass.few weeds b. Eanh.\rindingandsluggish L No leSellltion 2, Crass.sonreweeds r q u a t rp. l d n t '. n l . D e n . eu e e d r , , a decpchannels L Ea(h borromJndrubble''de\ 5. Stonybotlonandweedybanks 6. Cobbleborlomandcleansides c- Draglineexcavated or dredged L No vegetation 2. Lighrb.ushon banks d. Rockcuts L Smoolhanduniform 2. Jaggedand irregular e. Channelsnot maintained.weedsand brushuncul L Dense,reeds. highasflo\,r'depd 2. Cleanbotlom.brushon sides 3. Same.highestsrageof flow ;1. Densebtush,high srage D , Nalural Streams Fl - Minor slreams{lop \lidrh at flood stage< 100ft) a. Streams on plain L Clean.s(raighr,full srage.no rifis or deeppools Snmeas above.bul nloreslones
i\tinimum
I t'7
Normal
!laximum
0018 0.0t2 0 0:l
0.022 0.025 0.027
0.025 0.030 0.031
0.0:l 0.c!15
0.025 0.030
0.030 0.033
0.030 0.018 0.0t5 0.010
0.035 0.030 0.035 0.0.10
0.0.10 0s35 0.0.10 0.050
0 0t5 0 0t5
0.028 0.050
0.033 0.060
0 015 o.015
0.035 0.0.10
0.040 0.050
0.050 0.0.10 0.015 0.080
0.080 0.050 0.0?0 0.100
0.120 0.080 0 . 1t 0 0.l,t0
0.0t5
0.030
0.033
0 0t0
0.035
0.040
0.033
0.040
0.045
0.035
0.045
0.050
0.0.10 0.0.r5 0.050
0.0,18 0.050 0.070
0.055 0.060 0.080
0.0?5
0.100
0.150
0.0-r0 0.0.10
0.040 0.050
0.050 0.070
0.025
0.030
0.015
-t. Clean.\\indinB,somepoolsand ,t. Sameasabove,bul someweeds andslones
5 . S.rmers above,lower stages.more incffective slop€sandsections
6. Sameas4, bul morestones 'L
Sluggishreaches,weedy.deeppools
8 . Very weedyreaches,deeppools,or floodwayswilh heavystandof limber and underbrush b, Mountainstreanrs,no vegetationrn channel, banksusuallysteep. rees and brushalongbankssubmerged at highstages L Bortom:gravels,cobbles.and few boulders 2. Boltom:cobbleswilh largeboulders D-1. Floodplains a. Pasture.no brush l. ShortSrass
118
C H A P T E R4 : U n i f o r mF l o w
TABLE ,l- I (Continu€d)
'rype
of Channel and Description
2. High Brass b. Cultivated areas L No crop 2. Maturerow cfops 3. Marurefieldcrops c. Brush L Scatleredbrush.heavyweeds 2. LiShtbrushandlrees,in \linter I Lighl brushandtrees,in summer :1. Mediumto dcnsebrush,in \1in(er 5. Mediumto densebrush,in summer d. Trees L Densewillows.summer, straight 2. Clearedlandwith trceslumps, no sprouts 3. Sameasabove,but wilh heavy growthof sprouts 4. Heavystandof limtr€r,a few do\\n trees,liltle undergrowth,flood stage below branches 5. Sameas above,but with flood stage reachingbranches D-3. Major streams(lop width at flood srage > 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-
CHAPTER.1: UniformFlow
129
onal interfaceillustratedin Figure 4.l2 to compute the total conveyancegrve the leasterror in the dischargedistributionin comparisonwith the numericalmodel Othcr methodsfor conrpoundchannel dischargedistribution have appearedin the literature. Bousmar and Zech ( t999) proposed a lateral momentum exchangemodel momentumeguationappliedto the main channelwith basedon the one-dimensional lateral inflow and outflow. They derived an additional head loss term conesponding to the exchangedischargesat the interface.but it has to be obtaincd fronr the simultancous solution of three nonlinea.ralSebraicequationsfor the main channeland Ieft and right floodplains with spccification of two empirical coeFficients.Myers and Lyness 1t ell l suggesteOtwo entpirical power relations: ( | ) the ratio of total discharge/bankfull dischargeas a function of the ratio of total depth,rbank-fulldepth, and (2) the ratio of main channeldischarge/floodplaindischargeas a function of the ratio of floodplain deptlvtotal depth. Stunn and Sadiq ( 1996; measuredan increasein the rnain channel vaiue of Manning's n of approximately 20 percent for overbank flow in comparison to the bank-full value for tuo different laboratory compound-channelgeometries' While it should be apparentthat much researcheffon has been expendedon the problem of dischargeand its distribution in compound channels,a final solution iemains elusive. The methods based on laboratory data are limited to a speciltc range of compoundchannelgeometries.The 3D numericalapproachof Pezzinga ( 199,1),with a more advancedturbulencemodel and more extensiveverificationby experimentaldata, holds some promise for solving the problem ln the interim, eit-her the divided channel method, using a venical interface with the wetted oerimeterincludedfor the rnain channelbut not the floodplain (Samuels1989),or ih. diuid.d channel method with the diaSonal interface that is excluded from wetted perimeter seemsto give the best results.
4,13 RIPRAP.LINED CHANNELS As an applicationof uniform flow principles,the designprocedurefor riprap-lined Paintal.andDavenport in NCHRPRepon 108(Anderson, as developed channels of tractiveforce of the method lt is an extension 1970)is givenin this section. (Chow 1959)' design for stable channel Reclamation by the Bureauof developed ( are discussed' 1988) and Conon procedure by Chen Furthei modificationsof the forms a flexrock riprap such as concrete, lining In contrastto a fixed channel failwithout erosion to minor adjusting of advantage ible channellining thathasthe choose philosophy is to The design stability. provide channel ure andcontinuingto the channeldimensionsand riprap size such that the maximum boundaryshear stressdoesnot exceedthe critical shearstressfor erosion.As a part of the design of the riprap is estimated. procedure,the flow resistance Experimentaldata on the resistanceof rock riprap are summarizedin Report 108,andManning'sn is takenas n : O.Oadt5[6
(4.3'1)
130
Crrprtn
4 : U n i f o r mF l o u
in which drn : median panicle siTe in leet This equationis of the same form as = 0 0'1 in English units' Strickler'sequationfor sand t\ ith the constantc, on The critical shearstressrelation, also based rock riprap data,is of the same form as the Shieltlsrelation,u hich is describedin detail in Chapter l0:
(-1.l8) in which r* = criticalshearsressrequiredfor initiationof motionin lbs/ftr and thatthe partid:o = medianpaniclesizein feet.Equation4.38implicitlyassumes (i ciJ Reynoklsnumberis large enoughthat viscouseffectsare unimponant e ' seeChapterl0). Shieldsr." : constant; on boththebedandsidesof trapezoidal areanalyzed distributions stress Shear in NCHRPRepon 108: are adopted relations following the and channels (4.39) (rr),- = l.57RS (ri )-* = I 27Rs
(4.40)
- maximumsidewall in which (rj).", : maximumbed shearstress'and(rD.* (see Chapterl0) is that channels stable of shownfrom thetheory shearst.ess.-Alib given by is motion thetractiveforceratio,K., at impending
rA I rin]9l' : K , : : : : , l l _ . , , 1 rk
L
stn-@I
(4.41 )
: in which0 = sideslopeangle:d - angleof reposeof riprap;16. criticalshear stresson the sidewall;and70. : criticalshearstressfor initiationof motionon the bed.Thetractiveforceratio.K,,islessrhanlinvaluebecauseasmallercritical shearstressis requiredto initiatemotionon the sideslopedueto the gravityforce side slopes'chosen componentdown the slope.Anglesof reposeand suggested stressare approxisheal to critical stress shear maximum of ratios such that the 4' 13' in Figure summarized are banks, bed and the matelyequalon follows: as be summarized procedure can design The riprap I . Choosea riprap diameterand obtain{ and0 from Figure4 13' from Equations4 38 and 4 41' 2. Calculatethe iritical bed and wall shearstresses 4.37. Equation from 3. DetermineManning'sn 4. For a given channelbottom width, discharge,and slope,find the normal depth from Manning'sequatlon. '1 5. Calculaternuiimu* bed and shearstressesfrom Equations 39 and 4 40 and values comDarethem with critical 6. Repiat with anotherriprap diameterand/orbottomwidth until the maximum arejust smallerthanthecriticalvalues shearstresses is simplifiedby ChenandCotton(1988)in FHWA publication This procedure HEC-15ior the specialcaseof channelsideslopesthat are 3:l or flatter In this case,the riprap on the side slopesremainsstable,and failure occursfirst on the channelbed.In addition,Manning'sn is computedfrom the relationshipdeveloped by Blodgett(1986)for the data shownpreviouslyin Figure4 6 Blodgen(1986) obtaineda bestfit of the data givenby
CH\PrER ,1: UnilbnnFlow
42
c. rshed ,o:l-
13l
-
9 a n ^, 6$ ,-
YZ
s38 at
.ri i\(
o
{r 6, c
32 100
10
600
MeanStoneSize,mm (a) Angleol Reposeof Riprap
o2A c,) o -9 o) o 18 9
n d'r|"
!2 61a,01'u
V8g 0.'794+ 1.85 log(R,/d5s)
(4.42)
l3:
C H A P T E R, l : U n i f o r mF l o w
equationu,itha constant of 1.85mulwhich is slightlydifferentthanthe Keulegan the termratherthan2.0. [n addition,Blodgettsubstituted tiplling the logarithnric theywerenearlyequal.Thisequahldraulicdepthfor thehydrrulicradiusbecause in Englishunits tion givesvaluesof c, = r/d{f of approximatell0.0-16to 0.0-1.1 (0.056to 0.054in SI) for 30 < R/dro< 185.ir hich is the upperlimit of applicabilitv.Therefore, Equation 4.42givesslightlyhighervaluesof Manning'sn thanthe Andcrsonet al. equation(,1.37), for which c, = 0.0,1in Englishunits.Recallthat the Stricklervaluefor cnis 0.039in Englishunits usinSthe valuegivenby Hager ( 1999).Furthermore, Maynord( l99l ) dctermined c, : 0.038in Englishunitsfrom (5 < flume experinrents usingrock riprapin the intermediate scaleof roughness R/dr' < l5) and suggested thatthis valuealsocould applyin the lowerrangeof (15 < R/d50<.15). Therefore.Equation,l..l2shouldgive small-scale roughness conservative estimates of Manning'sn for riprapdesign. givenin HEC-l5 can be summarized asfollows: The simplifiedprocedure l. 2. 3. 4.
Choosea riprapdiameter. Calculatethecriticalbottomshearstressfrom Equation.1.38. EstimateManning'sn from Equation4.42 with an assumed depth. Calculatethe normaldepth)0 from Manning'sequationand iterateon Man4.42. ning'sn from Equation andcompareit with 5. Calculatethe maximumshearstresson the bottom as 7_r'oS thecriticalstress.
4.11 GR{SS.LINED CHANNELS Channelsalsocanbe designed for stabilitywith vegetative linings.This hasbeen donesuccessfully by theSoil Conservation Sen ice for manyyears.Vegetative linings areclassifiedaccordingto their degreeof r egetalretardanceasClassA, B, C, D. or E. Permissibleshearstresses are assignedto eachretardanceclass,given in Table4-2 (ChenandCotton1988t A descriptionof eachretardance classis given in Table4-3.The flow resrstance as expressed by Manning'sn valueis presented in HEC-15asa functionofchannel
TABLE4.2 Permissible shear stressesand constant o0 for vegetative linings Retardence Class
B C D E
P€rmissible rr, psf
3.70 2.l0 L00 0.60 0.35
Permissible tr' Pa
1'7'7 100 ,{8 29 l7
oo in Resistsdce Equation
24.7 30.? 36.,1 10.0 12.7
CHAPTER 4: Uniform Flow
133
TABLE l.J
Classificationof vegetalcover as to degreeof retardance(SCS-TP-61) Vegetal Retardance Class
Cover
Condilion
\leeping lovegrass
Excellen! sland,tall (avera8e 30 in-)(76 cm)
Yellowbluestem I\chaemum
Excellentstand,tall (averaSe 36 in.) (91 cm)
Kudzu
Very densegrowth.uncul
Bemrudagrass
l2 in.) (30 cm) Cool sland.tall (average
Nativegrassmix(ure (lirlleblueslem, blueslem. bluegamma,andotlEr longandsho( Midwesl Srasses) \\reepinglovegrass
Good stand.unmowed Goodstand,tall (average 24 in.) (61 cm)
sericea Lespedeza
Cood stand,not woody.tall (averagel9 in.) (48 cm)
Alfalfa
Cood stand,uncul(average I I in.)(28 cm)
\\'eepinglovegrass
Cood stand,unmow€d(averageI3 in.) (33 cm)
Kudzu
Densegrowth.uncut
Bluegamma
Coodstand,uncut(average l3 in.) (28 cm)
Crabgrass
Fairstand.uncut( | 0 to .18in.)(25 to 120cm)
BermudaSaass
Good sland.mowed(average6 in.) ( l5 crn)
Conmon lesp€deza
Goodstand.uncut(averaSe I I in.) (28 cm)
Grass-legunre mirturesummer(orchnrdgmss, redtop,Ilalian ryegrass, andcommonlesp€deza)
Co(d stand,uncut(6 to 8 in.)(15 to 20 cm)
Centip€degrass
Verydensecover(average 6 in.)( l5 cm)
Kentuckybluegrass
(6 to l2 in.)( I5 to 30 cm) Goodsund.headed
Bermudagrass
Goodsland.cut to 2.5 in. height(6 cm)
Commonlespedeza
4.5 in.) (l I cm) Excellent stand,un ut (average
Buffalograss
Coodstand.uncul(3 to 6 in.) (8 to l5 cm)
Grass-legume mixtur€ fall. spring(orchardgrass. redtop,Italian ryegrass. andcommonlespedeza)
Goodstand.uncut(4 to 5 in.) (10 lo l3 cm)
[,espedeza sericea
Aftercuxingto 2 in. height(5 cm) Very goodslandbeforecutting
Bermudagrass
Goodsrand. cut ro L5 in. hei8ht(4 cm)
Bermudagrass
Bumedstubble
Note:Coversclassifiedhale beente(ed in exp€n menralcbannels. Coversweregreenandgenerallyunrform
l3,l
CH A PT8R 4: Uniform Ro\r
slopcandhydraulicradius,R. basedon theworkof Kouwen,Unny,andHill ( 1969). Thcsecurvesareshownin Figure{.1-1.andtheyarebasedon theequationgivenby R
r/6
(4.43)
a o + 1 6 . 1 l o g ( lRs o a . 1
ClassA
oooll
o) .E
\
ol
=
-\.u-
0.01 0.1
1 R, meters
10
ClassB
l_s t6-&t@F H l----r--i
\
\
\1 \ =
c
_l
0.01 0.1
T
: -:-
1 P, meters
FIGURE 4.I4 channels(ChenandCotton1988). Manning'sn for vegetated
'10
C H A p T E R . 1 : U n i f o r mF l o w
135
in which the h)'draulicradiusR is in meters;S = channelslope in merersper merer; and valuesof ao are given in Table4-2. The design procedurecan be sunrmarizedas follows: L Choosea vegetalretardanceclassA, B, C. D, or E and determinethe permissi_ ble shearstress,r,,, from Table4-2.
o) c
-
c
1 R, meters ClassD
ol
'-
o
136
C H A P T E R4 : U n i f o r m F l o w
ClassE
o)
'e c
B, melers FIGURS 4.14 (ContinueA
2. Estimatea flow depthfor givenbottomwidth b andsideslopes(m:l; m > 3) and calculatethehydraulicradius,R. 3. ObtainManning'sn valuefrom Figure4.14or Equation1.43for the appropriclassandthegivenchannelslope. ate vegetalretardance 4. Calculatethe normaldepthfrom Manning'sequationfor the designdiscbarge and comparewith the assumeddepth.Iterateon the depthuntil the correctManning's n anddepth,r'ohavebeendetermined. 5. Calculatethe maximumbottom shearstressas rmax= 7-ro.landcompareit with the permissibleshearstress,tp. Adjust the channelbottomwidth, slope,or vegetal retardanceclassuntil rme = r p. linings such asjute, This designprocedurecan be usedto design temPoral-v fiberglassroving, strawwith net, and syntheticmatsthat are usefulfor stabilizing channelsimmediatelyafterconstructionbeforea standof grassdevelops.The pervaluesfor temporaryIiningsare given in and roughness missibleshearstresses HEC-15andby Conon(1999). ditchhasa bottomtaidthof 1.5m andside roadside ExAMPLE 4.3. A tmpezoidal channellining is a grassslopesof 3:1.The channelslopeis 0.012,andthe proposed legumemixturethathasa heightof l5 to 20 cm. what is themaximumallowabledischargefor this lining? Solation. This exampleillustratesan altematedesignprocedurefrom the one Just given that is useful for selectingthe initial lining. From Table4-3' the vegetalretar-
C H A P T E R4 t U n i f o r m F l o w
137
danceclassis C. andthepermissible shearstress r, = .18Pafrom Table,l-2.Thenthe maximumallowabledepthcomesfrom setlingr.", : y),J = ro and solving for _r.o: T^ YJ
= 0 . 4 0 8m ( 1 . 3 1f t )
9 8 1 0x 0 . 0 1 2
The geometricpropcrties of areaand \retledperimeterof the crosssectionfor this depthare A : , r ' o ( b+ n ) o ) : 0 . , 1 0 8x ( 1 . 5+ 3 x 0 . , 1 0 8: ) l . t l n r r ( l 2 . 0 f C ) p = b + 2 r o t / 1 + , n : 1 . 5+ 2 x 0 . 4 0 8x \ 4 - : 4 . 0 8 m ( 1 j . 4 f r ) T h e nt h eh y d r a u l irca d i u R s = A/P: l.ll/-1.08=0.272m(0.892fr).FromFigure.t.l4 or Equation4..13.rhe valueof Manning'sn is 0.074.The allowabledischargefrom Manning'sequationis Q =
^n"
eR:
', ' -
1.0 / ' l.ll / 10.212; *;
{ 0 . 0 1 2 r) '
: 0.690mr/s (24.4cfs) The allowabledischargeis comparedwirh rhe designdischargero decideif this lining is suitable.The final designdepthis determinedfrom the proceduregiven previously.
4.15 SLOPE CLASSIFICATION Asidefrom its primaryusein channeldesign,thenormaldepthusedin conjuncrion with the criticaldepthof flow is a usefulconceptin classifying slopesas mild or steepandultimatelyin classifyinggraduallyvariedflow profiles.A mild slopeis dehnedas a slopeon whjch the uniformflow depthis subcritical; thatis, normal depth,.ve, is greaterthancriticaldepth,v..For a steepslope,theuniformflow depth is supercritical thesetwo cases, it is obviousthat $o < -r").At theboundarybetween ),0 = )., so that it is useful to definea critical slopeas that valueof bed slopefor which uniformflow wouldoccurat criticaldepth.UsingManning'sequation,the critical slope,5., becomes -
ntQ' K:A:R'/1
(4.44)
in which A. and R" represent the areaand hydraulicradiusevaluated at critical depth.A mild slopecan be dehnedas having a bed slope,So,lessthan the critical slope,S", while for a steepslope,56 > S". The critical slopeis understoodto be a quantityto be usedonly asa criterionfor classification calculated of a slopeasmild or steep. The critical slopeis a functionof the discharge,so that a particularbed slope may be mild at somedischarges andsteepat others.This pointis illustrated easily with a verywide,rectangular channel.For thisshape,the hydraulicradiusmay be
-
l18
C H A P T E R4 : U n i f o r m F l o *
'1.0
n = 0.015
- 0.8 E d 0.6 qt
\ Steep
Mild
E
5 o.a i5 c
) o.2
0.002
0.006 0.004 CriticalSlope,Sc
0.008
0.010
FIGURE4.I5 channel' Criticalslopefor a wide,rectangular
considapproximatedas the depthof flow, andthe Manning'sequationsimplifies becomes slope erablyso that the critical
s.:ld:4]n-,*,
(.4.45)
with increasingdischargeFor example' a from which the critical slopedecreases of *id" ,".tungulu.channelwith a Manning'sn valueof 0 015 hasa bed slope from changes the slope bed slope, value of At this 4.15. as tnJtun in Figure O.OO4 the criti.ild to ,,""p at a disJhargeof 0.216 mr/s/m(2.32 cfslft), which is called the wide reccal dischargi,4.. The minimum possiblevalueof thecritical slopefor limit slope' is called,the this and zero, approaches tungut. "Ttre "ft"unn.tasymptotically wide is very as be classified cannot that channel a rectangular for limit slope for areaandhydraulicradiusfor finite (nao andSridharan1970t.If the expressions is elimiui..,ungutar channelare substitutedinto Equation4 44 and fte discharge becomes slope the critical discharge, and depth critical between natedbithe relation
sn2 lu + 2y,1or'
F"Y'l un )
(4.46)
to zero' Ifthis cxpressionfor critical slopeis differentiatedwith respectto -!candset : value a of has l/6 and tt".inl*urn criticalslope,or limit slope,occursat y./b 2.67g nz ^ (4.47\ Kr, bt,'
C H A p T E R4 : U n i f o r m F l o w
139
1.0 0.8
0.6
/
Mitd
a
sr""p
* 0.4
o.2
I 1/6
o.4
0.8
1.2
2.0
FIGURE 4.t6 Critical slopefor a rectangularchannelin termsof the Iimit slop€.
Theexpression for thecriticalslopein Equation 4.46canbe nondimensionalized in termsof the limit sloDeto Droduce
s. -
sr
(' .'))"
0.375
"
(4.48)
(;) Thisequationis plottedin Figure4.16,from whichwe seerhat,for a bedslooeless thanthe limit slope,the sloperemainsmild for all possibledischarges. The limit slopecan be usedto nondimensionalize theexpression for bedslope 5o from Manning'sequationwrittenin termsof theFroudenumberof the uniform flow F^:
('*'f;)"' (f)'"
so_ 0.375F8 sL
(4.49)
in which Sois the bed slope;S. is the limit slope;and1'ois the normaldepth.For the caseof Fo : l, this equationreducesto Equation4.48 with So = S. anO = _vo ,i.. Equation4.49 is plottedin Figure4.17 for differentconstantvaluesof the
l.l0
C H A P T E R4 : U n i f o n n F l o w
10
1
0.01
0.001 0.01
10
so/sL FIGURE 4.17 Normaf depth vs. slopefor a constantFroudenumberlRao and Sridharan1910J (Source: 'Linit Slopeh LlniforntFIox Conpnatiotts" J' Hycl Div ' N. S. L. Raoantl K. Srilharan, bt pennissionofASCE.) A 1970. ASCE.Reproduced
Froude number. It can be shown that the maximum I'alue of the Froude number occurs at,r',,/D: l/6 and hls a value of
t;
F."-
/Jo
(.1.50)
VS,
channelsuchthata givenyalue a rectangular of designing This raisesthepossibility the channelmay experiany discharge for not exceeded of the Froudenumberis from becomingtoo number Froude prevent the maximum ence.It is desirableto with criticalflow' associated instability' of the free-surface closeto unity because E s A l t P L E , l . 4 . A c o n c r e t e - l i nr e dc t a n g u lcshr a n n ehl a sa b o t t o mw i d t ho f 3 ' 0 m (9.8 ft) and a Mannings n of 0.015.The bed slope is 0.007.and the dischargeis expjctedto vary from zeroto 60.0mr/s(2120cfs). Determineil the slopeis steepor be mild for all At whatslopeuould thechannel mild overthefull rangeof discharges. discharges? So/ation. First, find the limit slopefrom (4.'17): 'c': -
: . 6 7 Rn : ' K- b
2 . o 7^ 9 . 8 1. U l 5 : lo'
t,_tlll4uv
C H A f T E R. l : U n i t b m t F I o w
l4l
Thus. the acrualslope of 0.&)7 is greater rhan the limit slope, and discharsesin both t h e r t t i l da n d s t e r p . l , ' p cu l r r r . i l i c : r r i oanr e p o . , r b l e .S c t j - 0 . 0 0 ?r n F , l u u r r J ln. { g a n d solte tbrJ,. This is a trial,and-errorrolurron \^irh r$o r,rnt.. one tbr \ /, > i/6 and anotherlbrr',/b
./. \{a
Vrrr -r jq' - 18.? rn:. {tot fr: .)
in.$hiehonl.vrheupperrrlue of r'. hasbcenillustrated. The otherralueof4, is 0.020 mr/\ (0.12frr/5,.The r\ro raluesof criticaldischarge. O..(= g.b).are56.1,iirl|!ItSSO cfs)and0.060mr/s{2.1cfs).Bet$eenthesetwo discharges. theslopewill be sreep, ano for O > 56.1mr/sor Q < 0.060mr/s.the slopewill be mild.This canbe secnin Fieure-l.l T for theintersection : 0.007/.00.109 of a venicalIine,alongu hichS,y'S. : L il andthecurvefor Froudenurnbcr: L0. If theslopecouldbeconsrructed to be lessthan the Iinlit slopeoi 0.00,109, rhenit wouldbe mild for all discharses.
.t.16 BEST HYDRAULICSECTION From economicconsiderations of minimizingthe flow cross-sectional areafor a given designdischarge, a theoreticallyoptimum crosssectioncan be derived, althoughmany other factors,includingchannelstabilityand maximumFroude number,may be thc overridingdesigncriteria.Minirnizationof flow areaimplies maximization of velocityfor a givendischargeand,therefore, a maximization of hl,draulicradius,R, for a givenchannelslopeandroughness basedon anyuniform flow formula.The problemcanbe recastrhenas minimizingthewettedperimeter, P, for a fixedcross-sectional area,A, sinceR : A/p. Underthiscriterion.it is clear thata semicircle wouldprovidethebesthydraulicsectionof all. For therectansu_ lar section,the wettedperimeter. P, is givenby p : b + 2) andsubsriruting b-= A/,1,we candifferentiate P with respectto ]. while holdingA constantandsetthe resultto zero:
d P- - d. l A ^ l e . i + 2 \ ' l - - - - 2 = 0 o)
d)'L)
J
r-
(4.5r)
Then,we seethatthe bestrectangular sectionhas,4 : 2).:andb = 2.y,so thata semicirclecanbe inscribed insideit. From the samereasoning (seetheExercises), the besttrapezoidalsectionis onefor which R - _iy'2 andm = l/30J.sothatthe side slopeangle0 : en t(l/m) = 60. andrheshapeis tharof a half-hexagon insideof which a senlicirclecan be inscribed. The besthydraulicsectionmight be desirableonly for a concrete_lined pris_ matic channel.lf it is rectangular,an aspectratio of &/y : 2 for the best section qould meanthat a subcriticalFroudenumberwould be lessthan rts maxlmum value at b/,y= 6 for a givenslope.In addition,secondarycurrentswould be much more likely in the bestsectionbecauseof its small aspectratio.However,once
112
. 1 : U n i f o r nF l o s CHAPTER
channelstability beconlesan issue,thc asp!'ctratio is likely to grcatly increaseto keep tbe shearstressbelow ils critical value.
4.17 NIANNING'SFORI\'TULA DIMENSIONALLYHONIOGENEOUS practice, thereis I nrsging in engineering r is firmlyentrenched WhileN'tanning's clesireto transform Manning's equation in some way that will make it dimensionally homogeneous.*'hich actually was the intent of Manning when he rejectedthe e q u a t i o nt h a t n o w b e a r sh i s n a m e .E q u r t i o n ' l 2 l i r n p l i e st h a t M a n n i n g ' sn c a n b e thought of as having dimensions of lcngth to the ; Po\\er, with K,, then having dimcnsionsof Lrl:/l The nondimensionalityof the equation,however,still is qucstionable. becauseK- u'ould have to take on a value of l.8l ftl/:/s compared to 1.0 ml/2/sin the SI systenlif lUanning'sn were to be convertedfrom ml/6 to ftl/6' Yen (1992b) suggestedthat this confusing siate of af-fairscould be alleviated by d e f i n i n gM a n n i n g ' se q u a l i o nt o b c
"-(+)(#)**:(f),c*
rA S)\
\9.,-/
This would allow the equationto be truly homogeneous in which n, : ngtt?|K,. n" from ftr/6to mr/6or vice versawith no correof converting capability with the unitsof so longasthedimensional in theequation, changeof coeflicients sponding values of Manning's current the However, consistent. variables remained other all = to n" in ftr/6by multiptyingthemby 32.2t/211.19 n wouldneedto be conyerted rn = 3.13.Yen( 1992a)converted 3.8I andto r" in mr/6by multiplyingthemby 9.81 of Manning'sn in thisway.In addition.he derivedvaluesof equivChow'stable"s natureof cur&,.for thesetables.Giventheestablished roughness, alentsand-grain rent valuesof Manning'sn, the useof the tablesfor n, is likely to be unpopular despiteits desirability.
4.18 CHANNEL PHOTOGRAPHS Thesephotographsare provided by courtesyof the U.S. GeologicalSun'ey and comefrom the work by Bames( 1967).For eachriver shown,the dischargeand the water surfaceprofile over severalcrosssectionswere measuredfor a flood event, and Manning's n was calculated from the equation of gradually varied flow in Chapter5. Figures4.18 to 4.32 give Manning'sn valuesfor maindescribed showsthe meachannelflow only (Bames1967).The captionfor eachphotograph n value. In somecases, Manning's sureddepthat the crosssectionalongwith the doesnot necn the Manning's and multrpleeventswith differentdepthsarc shown, inundated for in vegetation to changes essarilyremainconstant.This could be due flows, or changes in shallow differentdepths,effectsof large roughnesselements in bed forms with stage,which will be discussedin moredetail in Chapter 10.
FIGTJRE4.18 SaltCreekatRoca,Nebraska: z = 0.030;depth= 6.3ft. Bedconsists ofsandandclay.(U..r. GeologicalSurtey)
FIGTIR-E4.19 Rio ChamanearChamita,New Mexico: n = 0.032,0.036;depth = 3.5, 3.1 ft. Bed consisrs of sand and gravel. (U.5. Geological Survey) 143
FIGURE 4.20 = = Salt River below StewartMountai! Dam,Arizona:z 0 032; depth l 8 ft' Bedandbanksconsistofsmoothcobbles4tol0in.indiameter,ar.eragediameterabout6 in. A few bouldersare aslargeas l8 ir. in diamete' (U'S GeologicalSuney)
FIGURE 4.27 = = ft' WestFork Bitterroot River nearConneqMontana:n 0'036; depth 4 7 = Suney) = (U Geological mm S' 265 Bed is graveland boulders;dro l"l2 rsm''du 144
FTGURE4.22 MiddleForkVermilionRivernearDanville,Illinois;n : 0.037;deprh= 3.9ft. Bedis gravel andsmallcobbles.(U.5.Geological Sun'ey)
FIGURE4.23 Wenatchee RiveratPlain,Washington: z = 0.037;deptl: I l.l ft. Bedis boulde$:dso: 162 mm;dro= 320mm.(U.S Geological Suney\ 145
FIGURE4.24 EtowahRivernearDawsonville,Georgia:n=004t,0'039'0035;dePth=98'9'0'44ft' Survey) fallentreesin thereach(Il S' Geological Bedis sandandgravelwith several
FIGURE4.25 n= 0'043'0 041'0 039;depth= 92' 87'63 ft' Georgia: CreeknearMacon, Tobesolkee SuNe!) of sand'gravel,anda fewrockoutcrops(IJ'S Geological Bedconsists t46
FIGURE4.26 MiddleForkFlathead RivernearEssex, Montana: z = 0.041;depth: g.4ft. Bedconsists of boulders; dro: 142mm:d* = 285rnm.(LI.S.GeotogicalSuney)
FIGURE 4.27 BeaverCreeknearNewcastle,Wyoming:z : 0.043;depth = 9.0 ft. Bed is mostlysandand silt. (U.5. GeologicalSuney) t47
FIGURE4.2E lt : 0.045;dePth= 4.2ft. Bedconsissof sandand MurderCreeknearMonticello,Georgia: gravel.(U.S.GeologicalSuney'1
FIGURE 4.29 SouthFork ClearwaterRiver nearGrangeville,Idaho:n = 0.051;depth = 7 9 ft. Bed consistsof rock and boulders;d.n = 250 mm; d6o= 440 mm. (U.5. GeologicalSurvey) 148
FIGUR.E4.30 MissionCreeknearCashmere, Washington: z = 0.057;depth= 1.5fL Bedof angular-shaped (.U.5.Geological boulders aslargeas I ft in diameter. Survey'1
FIGURE4.31 Haw RivernearBenaja,NonhCarclina:n = 0.059;depth= -1.9fr. Bedis composed of coarsesandanda few outcrops. (U.S.Geological Suney) 149
150
CHAPTER {:
Uniform Flow
FIGURE4.32 RockCreeknearDarby,Montana: n = 0.075;depth= 3.1ft. Bedconsists ofboulders;4n= 220mm; d* = 415 mm. (U.5.GeologicalSurvey)
REFERENCES Ackers, P "Stage-Discharge Functions for Two-Stage Channels: The Impact of New Research."J. of the Inst. of Water and Environmental Managemint j, no. | (1993),
pp.524r.
Ande6on, A. G.; A. S. Paintal: and J. T. Davelport- TentatiyeDesign prccedure for Riprap_ Lined Channels, NCHRP Report 108. National Cooperative Highway Researchpro_ gram, Highway ResearchBoard,NationalResearchCouncil,Washingon,DC: 1970. Arcemenl G. J., and V. R. Schneider Gaidefor SelectingManning's RoughnessCoeficients for Natural Chaanels and Flood plains, Report No. FTIWA-TS-g4-204. FedeA Hish_ \ryayAdmioistration, U.S. Dept. ofTranspofiation, National Tecbnical Information Sirvice. Springfield VA: 1984. ASCE. Gravrry Sanitary Sewer Design and Construction, Manual No. 60. New york: ASCE, 1982. ASCE Task Force. "Friction Factors in Open Channels,progress Repon of tbe Task Force on Frietion Factors in Open Channelsof the Committee on Hydromechanics.',"/. Ilyd. Div., ASCE 89, no. HY2 (1963),pp. 97-143. Bames, H. H., Jr. RoughnessCharacteristicsof Natural Channels,U.S. Geological Survey WaterSupplyPaper1849.Washington,DC: covemment printing Office, 1967. Bathu$t, J. C. "Flos ResistanceEstimation in Mountain Rivers." J. Hydr Engry,, ASCE I I l. no.HY4 ( 1985).pp.6254J.
C H A p T E R4 : U n i f o r mF l o w
l5l
"Unstable Berlamont, J. E.. and N. Vanderstappen. TurbulentFlou.in OpenChannels.', "/. H I d D i l l . A S C E1 0 7 n . o .H Y 4 ( 1 9 8 1 )p, p . . 1 2 7 - . 1 9 . "Carr)ing Bhowmik.N. 6.. andM. Demissie. Capacityof Floodplains." J. H\,.1Diy.,ASCE 1 0 8n . o . 3 ( 1 9 8 2 )p. p . , 1 . 1532 . Biswas,A. K. Histor, of Hldn>log"-. publish_ Amsrerdanl, the Nerherlands: .r-onh-Holland i n g C o . .) 9 7 0 . Blodgett,J. C. RockRiprapDesignfor Protectionof StreanChannelsnear High*,ay Struclnfes,Vol. l. Hldraulic CharactcristicsoJOpenCfiarrrels,\\RI Repong6-412?.Den_ r e r : U S G e o l o g i i aSl u n e y .I 9 8 6 . Bousmar.D., and Y Zech."lvlomentumTransierfor practicalRo\\ Computationin CompoundChannels." "/.Htdr Engrg..ASCEI 25,no. 7 ( t999),pp. 696 J06. Camp,T. R. "Designofsewersro Facrlirare Flow. saradeWork,ii. no I ,1946r,OO,_tU. Chen,Y H., and G. K. Cotron.Designof RoadsideChannelswith Fletible Lirines. FHWA_ IP-87-7,Hldraulic Engineeriog Circular I5. FederalHigh*.ayAdminirrrirron,U.S. Dcpt.of Transponarion, Narional Technical Information Sen,ice,Springfield, VA: l9gg. Chou Vcn Te. Open ChunnelHtdrarrlics.New york: Mccraw-Hill. 1959. Colebrook.C. F. TurbulentFloq in Pipes,wirh panicularReferenceto theTransitionBetween the Smoothand RoughPip€t-aws."J. Inst.Ciu Eng.Iond. ttl (1939),pp. 133_56. Cotton,G. K. "HydraulicDesignof FloodControlChannels." In Htdraulic DesignHandhr,,o,(, ed. t arry W Mays.NewYork:McCrau Hill, 1999. Cowan,W L. "EsrimaringHydraulic RoughnessCoefficients."Agricultural Engineering 3'7,no.1 (t956), pp. 413,15. Cunge, J. A.: F. M. Holly, Jr.; and A. Vetwey. Practicol Aspectsof Conputational H)draulics.Marshfield, MA: PirmanPublishing tnc., 1980. Daily, J. W, and D. R. F. Harleman.F/uid D!-namics.Reading,MA: Addison_Wesley, 1966. Dickman,B. "LargeScaleRoughness in OpenChannelFlow,"M.S. thesis,Schoolof Civil Engineering, GeorgiaInsrituleof Technology, 1990. Dooge,JamesC. l. "The ManningFormulain Context.,,In ChannelFlow Resistance: Cen_ tennial of Manning's Fon,tula, ed. B. C. yen, pp. 136_g5. Linleton. CO: Warcr Resources Publications. 1992. French,R. H. Open-Channel Hldraalics. New york: McGraw-Hill. 19g5. Hager,W. H. Wast?v'ater H).draulics.Berlin: Springer-Verlag, 1999. Henderson,F. M. OpenChannelFlon New York:The Macmillan Co.. 1966. Hey, R. D. "Floq Resistancein Gravel-BedRivers,"J. Hrd. Dir.., ASCE 105, no. Hy4 ( 1979),pp. 36s 79. Kazemipour,A. K., and C. J. Ap€lt. "New Data on ShapeEffecs in SmoothRectangular Channels."./. Hydr Res.2O. no. 3 (1982),pp. 225-33. Kazemipour,A. K., andC. J. Apeit. "Shap€Effecrson Resistancero Uniform Flow in Open Channels."./. Hydr. Res.t'7, no.2 (1919), pp. 12941 . Keulegan, C. H. "Lawsof Turbutent Flow in OpenChannels." ./.of Res. of N.B.S.2l ( 193g), pp. 707-.+I . King, H. W Harulbookof Htdraulics, lst ed. New york: McGra*-Hill, l9lg. Knight, D. W., and J. D. Demetriou."Floodplainand Main ChaanelFlow Inreraction.""/. H,-dr Engrq.,ASCE109,no. 8 (1983),pp. toj3-92. Kouwen,N., T. E. Unny,and H. M. Hill. "Flow Rerardance in VegetatedChannel."J. 1rrig. anciDrainage Div., ASCEIR2 (1969),pp. 329-42. "Equivalent Krishnamunhy,M., and B. A. Christensen. Roughnessfor ShallowChannels." 1 1ly'dDir, ASCE98, no.HYl2 (t972), pp.225't-43. Kumar, S. 'An Analyrical Merhod for Computationof Rough Boundary Resisrance." In ChannelFlo* Resistance: Centennial ofManning'sFonnula,ed.B. C.yen,pp.241_5g. Littleton,CO: WarerResources Publications. 1992.
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- l : U n i f o r mF l o w CHAPTER
"General Algorithmfor RoughConduitResistrnce." Kunrar.S.,andJ. A. Roberson. J. Hll D i r , A S C E 1 0 6 .n o .H Y l l ( 1 9 8 0 )p, p . 1 7 , 1654 . Limerinos, J. T. "Determination of theI\,lanning Coefficient from \{easuredBedRoughness in NaturalChannels." U.S.CeologicalSurvey\\hter SupplyPaper1898-B.Washington,DC: Govemnrent PrintingOffice,1970. andPipes. Transactions Manning,R- "On the Flowof Waterin OpenChannels ofthe lrrstitutionof Civil Engineers of lreland2O(1889).pp. 166 95. of Riprap."J. Hldr Engrq..ASCE I I7, no.6 (l99l), pp. Maynord,S. I F'lowResistance 687 95. Moody. Lewis F. "Friction Factorsfor PipeFlow." Trans.AS.llE 611(^-ovemberl9-1,1). "CompositeRou8hness of NaturalChannels." Motayed,A. K., and M. Krishnamunhy. J. Hrr.i.Dnr. ASCE 106,no. HY6 ( 1980),pp. I I I l-16. "Discharge Ratiosin Smoothand RoughCompound Ivlyers,R. C., andJ. F. Ll ness. Channels."J. Fhdr Ergr:,ASCE l13. no.3 (1997),pp. l81 88. Myers.W R. C. "\{onrentumTransferin a CompoundChannel.'1 Hrdr Res.16,no. 2 ( 1 9 7 8 )p. p . I l 9 5 0 . "Flow Characteristics of PVC Se$er P)pe."J. of Sanitan Neale.L. C., and R. E. Price. Engrg.Dir:. ASCE 90. no. SAI (1964).pp. 109 29. Pezzinga. G. "VelocityDistribution in Compound ChannelFlous b1 r.\umerical Modeling." J. H)-dxEn|rg.. ASCE120,no. l0 (1994),pp. 117698. Pomeroy,R. D. "Flow Velocitiesin Small Sewers,"l l+Arcrn)llution ControlFed.. ASCE 3 9 ,n o . 9 ( 1 9 6 7 )p. p . 1 5 2 5 - + 8 . "Limit Slopein UniformFlow Rao.N. S. L., andK. Sridharan. Computations." 1 H-r,r/. Dir, ASCE96, no. HYI ( 1970).pp. 95 102. Roberson,J. A.. and C. T. Crowe.EngineeringFluid Me.ldnic.r.6th ed. Boston:Houghton Mifflin. 1997. Rouse,Hunter."CriticalAnalysisof Open-Channel Resistance-' J. H)-d.Dlr, ASCE91, no. HY4 ( 1965).pp.,l75-99. Rouse,Hunter."SomeParadoxes in theHistoryof Hydraulics.'J.HId. DiI. ASCE 106,no. HY6 ( 1980),pp. I077-8,1. Samuels. P G. "The H)drauljcsof Two StageChannels: Revier.r, of CurrentKnowledge.' HR PaperNo. {5. Wallingtbrd. England:HydraulicsResearch Limited.1989. Handbookof ChannelDesignfor Soil and llhter Corr.ndlior. Stillwater. SCS-TP-161. OK: U.S.Soil Consenation Service.1954. Shih,C. C., and N. S. Grigg."A Reconsideration of the H).draulicRadiusasa Geometric Quantity in Open ChannelHydraulics."Proc- l2th Congress,IAHR | (1967), pp.288-96. Shiono,K.. andD. W. Knight.'Turbulent Open-Channel Flows\\ilh VariableDepthAcross theChannel.'-/.Fluid Mech.222(1991).pp.617.16. Strickler, A. 'BeitriigezurFragederGeschwindigkeitsformel undder Rauhigkeitszahlen ftir Kdnaleund geschlossene Leitungen"(Conrributions Strcime, to the questionof flow roughnesscoefficients for rive.s. channels,and conduits). Mirr?ilug 16. Amt fi.ir Wasser$,irtschaft: Bem.Switzerland, 1923(in German). Sturm,T. W, and D. King. "ShapeEffectson Flow Resistancein HorseshoeConduits.""/. H,"dr Etryrg..ASCE ll,1.no. ll (1988),pp.1416-29. Sturm,T. W., andAftab Sadiq."WaterSurfaceProfilesin CompoundChanneluith MultipfeCitical Depths."1 H_r'dr Eagrg.,ASCE 122.no. 12 ( 1996),pp. 703-9. Thein,14."Experimental Investigation of Flow Resistance andVelocityDistributions in a Rectangular Channelwith LargeBed-roughness Elements." Ph-D.thesis.Ceorgialnsti, tuteof Technoloev. 1993.
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I53
Tracy,H. J.. andC. Nl. Lesler. Resistance Cocfficientandvelocily Distribution in Smoorh paper1592-A.\\'ashingRectangular Channel." U.S.Geological Suney,Water-Supply ton, DC: Government PrinlingOffice.1961. \Vark.J. B.. P C. Samucls, andD. A. Ervine. 'A Pracrical\lerhod of Estimarjng \relociry and Dischargein CompoundChannels." ln I ternationalConference on Ri|er Flood Hydraulics,ed.W. R. White.New York:JohnWiley & Sons,1990. White.F. l!,|.Fluid Meclnnics,Jrd ed.NewYork:McGraw-Hill.1999. White.F. M. llscousFluid Flott.NewYorkrN{ccraw-Hill.197i. Williams.G. P ManningFormula-A i!,lisnomer?" J. Htd. Dit.. ASCE 96. no. Hyl ( 1970).PP.193-99. "Flow Disrribution Wornrlealon. P R..andP Hadjipanos. in Con]pound Channels.'. "/.Hrdr E g r , A S C EI l l , n o . 2( 1 9 8 5 )p,p .3 5 7 , 6 1 . Wormleaton. P R.. andD. J. Merren.'An ImprovedMethodof Calculation of SleadyUni_ fornr Ffow in PrisnaticN,lainChannel/Flood PlainSecrions.J. Htdr Res.2g.no.2 I l q q o) . p P 1 5 7 7 l "Linear Wright,R. R.. andM. R. Carstens. NlomenrunFlux ro OverbankSections.,./. Flld D l y , A S C E9 6 ,n o . 9 ( 1 9 1 0 )p. p . l 7 8 l - 9 3 . Yen,B. C. "HydraulicResistance in OpenChannels.ln Clunnel Flow Resistance: Centennialof ltlatmhg'sFomtula,ed.B. C. Yen.Lifiieton,CO: WaterResources publications.1992a. Yen,B. C. "Dimensionally Homogeneous Manning'sFormula.'J.Hr.drEngrg..ASCE I lg, no. 9 ( 1992b).pp. 1326-32. Yen,C. L.. and D. E. Ovcnon."ShapeEffectson Resistance in Floodplain Channels.""/. H v r . rD. i l . A S C E9 q .n o . I I l a 7 J , .p p .2 t 9 - 3 8 . Zheleznvakov,C. Y Interactionof Channeland Floodplain Streams."proceedingsllth IAHR Conference,Paris,pp. 145,.18.Delfr, the Netherlands:Inremalional,Association for l{ydraulicResearch. 1971.
EXERCISES 4,1. Detennjnethenormaldepthandcriticaldepthin a trapezoidal channel$.irha bottom widthof .10ft. sideslopes of 3:l, anda bedslopeoi 0.0O1fuft. TheManning'sn value is 0.025and the discharge is 3,000cfs. ls rhe slopesreepor mild? Repearfor n = 0.012.Did thecriticaldeprhchange? Why or why not? 4.2. Compulenormalandcrilicaldepthsin a concrete cul\en (n = 0.015)with a diame, ler of 36 in. anda bedslopeof 0.002fVfr if rhedesjgndi\chargeis l5 cfs.ls rheslope steepor mild? Repeatfor S : 0.02 fr./ft. 4.3. For a discharge of 12.0mr/s,derermine the normalandcriricaldeprhsin a parabolic channelthathasa bank,fullwidthof l0 m anda bank-fujldeprhof 2.0 m. Thechan_ nel hasa slopeof 0.005andManning's11: 0.05. ;1.4.For the horseshoe conduitshapedefinedin Figure'1.1.deriverhe relarionships for AlAf. RlRt and QlQr. whereAr, Rr, and 01 represenrrhe full flow \,aluesof area, hydraulicradius,anddischarge, respectively. On the samegraph.plol rle relationshipstogetherwith thosefor a circularconduit.plotJ././on thevenicalaxis.Notethat for the horseshoe conduit,A/ = 0.8293dr andRr: Q ljlg /.
l5.l
C H A P T E R. l : U n i f o r m F l o w
4 . 5 . A d i v e r s i otnu n n ehl i s a h o r s e s h o e s h a p e r l i r h a d i a n t e t c r o f l l . g m . T h e s l o p e o f t h c runnelis 0.022,andir is linedwjlh gunire(|l : 0023). Find rhenormaldeDrhfor a discharge of 950 mr/s.Is thc slopesteepor mildl 4.6. A trapezoiditchannelhasvegetated banks\{ith \tanning.sa = 0.0-10 and a stable bolrom\rith Nlannings a : 0.02-5. The channelborrom\.\idthis l0 l.l and the side slopesare,l:l. Findthecomposite valueof lvlanning's n usingrhefourmethods given in rhischaprerif theflow deprhis 3.0 fr. ,1.7.Find the nomraldepthina l2 in. diameterpVC slonn sewer flowingat a dlscharge of L2 cfs if it hasa slopeof 0.001.Treatthepipe assmooth,andusetheChezyeou-arion.Verifyyour solutionwirh the graphicalsolutiongivenin rhe re\1.Whri is'rhe equivalent valueof Manning'sn for your solution?Wouldthevalueof rl be the same for otherpipediarleters l ,1.8.A circularPVC plastic(snrooth) stormsewerha-sa diameterof lg In. At the desien flow. it is inrended ro havea relarivedeprhof 0.9.Ar wharminimumslopecanir 6e laid so rhatthe velocityis ar least2 fr,/sat desienflow anddeposited solidswilt be scouredout by the designflowl Horr would rour rnswer foi rhe mininrumslope .h.rngefor I concrele\e\rer? ;1.9.Designa concrete sewerthathasa maximunr designdischarge of 1.0mr/sanda mini_ mum discharge of 0.2 mr/sif its slopeis 0.0Olg.Check rhevelocityfor self-cleansing. 4.10. Derermine the designdcprhof flow in a rrapezoidal roadsidedrainage ditch wirh a designdischarge of 3.75mr/sif rhedirchis lined with grasshavinga rerardance of classC. The slopeof theditchis 0.00,1andit hasa botrom$,idthoi 2.0m with side slopesof 3:1.Is thechannelstable? 4.11. A verywiderectangular channelis to be lined\rith a tall standof Bermudagrassto preventerosion.If thechannelslopeis 0.01fuft. determine the maximr_rm aliowable flow rateper unitof widthandvelocityfor channelstability. 4.12. Derivea relationship betweenthetrapezoidal channelsideslopeard theangleofrepose of thechannelripraplining suchthatfailureof the rock riprapoccurssimultaneously on rhe bedandbanks.Allow tie angleof reposeto r.an,between30. and42.. What is lhe mtnimumvalueof thesideslope.m: l. sotlat failurealwayswouldoccuron lhebedfirst? il.l3. Designa riprap-lined trapezoidal channelrhathasa capaciryof 1000cfs anda slope of 0.0005fr-lft.Crushedrock is ro be used and rhe channelbottom widrh is nor io exceedl5 fr. Determine theriprapsize,thesideslopes.andthedesigndepthof flow. ;1.14.A rectangular channelhasa widthof l0 ft anda \,lanning'sa valueof 0.020.Determinethechannelslopesuchthatuniformflow uill alwayihavea Froudenumberless thanor equalto 0.5 regardless of thedischarge. 4,15. A rect:rngular channelin a laborarory flumehasa widthof 1.25fr anda Manning,s /] of 0.017.To erodea sedimenr sample,rhe shearsrressneedsto be 0.15 lbs/fl. A supercritical uniformflow is desiredrlith the Froudenumberlessthanor equalto 1.5 t o a r o i dr o l lq a te . .
C l A p t E R 4 : U n i f o r mF l o w
155
(a) Calculatethe nraximumand minimumslopes to satisfythe FrouqenumDer cnterion. (b) Choosea slopein rhe rangedetemined from part (a) and calcularethe deprh anddischarge to achierethe desiredshearstress. 4.16. A mounlainsrreamhasboulderswith a mediansize (r./ro) of 0.50 ft. .Ihe streamcan beconsidered approximarel! rectangular in shapeandvefuwide,with a slopeof 0.01. You may assumethat k, = 2.d50. (a) For a dischargeof 7.0 cfs/ft, calculaterhe normal depthand critical depthand classifytheslopcas steepor mild. (r) Discusshow a Manning,sn thatis variablewift depthaffectsthecnricalslope andtheslopecla..rficarron. 4.17, Findthebesthydraulicsectionfor a trapezoidal channel.Expressrhewettedperimeterof a trapezoidal channelin termsof area,A, anddepth,r, ihen differcntiate p with respect to ), settingthe resultro zeroto showthatR = p with ,r/2.Also differentrate respect to the sideslope ratio.rr, rnd setthe resulrto zero Whatis rhebesrvalue of 't and what do you concludeis the besttrapezoidalshape? 4.18. A.comporind channelhassymmetricfloodprains, eachof whichis lo0 m wide with Manning's.r= 0.06,and a main channel,whichis trapezoidal witi a uottomwidth of l0 m, sideslopesof 1.5:1,anda bank_fulldepthof ).5 m. Ifthe ctrannet slopeis 0001 and the rotaldeprhis 3.7 m, computerhe uniformflo* Ji."hu.g" u.,ng tf," dividedchannelmethod,frsr with a venical inrerfaceborh with and wtthout wetred perimeterincludedfor the main channer,then with a diagonar interfacewrti wetted penmeter exctuded. 4,19, The power-lawvelocity disrributionin a very wide open channelin unilbrm flow is stven bv
u
( z\'
in whicha is rhepoinl velocity;ll. is rheshearvelocity;d is a constanr: z is the distanceabovelhe channelbed:k, is theequivalent sand-grain roughness; andm is the glvenIracttonal erponenlthal is constant. (a) Findthemeanvelocity,y, in termsof a.., andra, \4here!-., is rhemaximum vtloclly at i - )oi vu - depthofflow: andm - etponent power in taw. , {D) wnte theexpression for y from pan (a) in theform of a uniiormflow formura anddeducethe valueof the exponent ft thatis compatible with Manning,s equation. 4.20, Velocirydarahavebeenmeasuredat the cenrerlineof a tilring flume havinga bed of crushed.rockwirh d50= 0.060 ft. Considerthe Run 12 data"rhar forows, tor which 0 = 2..10cfs andS : 0.0O281. The elevations. a,,aregiven*i,f,..rp"o to rhe botrom of rhe flume on which rhe rockshavebeenlaid one-layer thick. ,Iire averagebed ilevationof the rocksis 0.055 fl abovethe flume bortom.iaking this etevalionas rhe origin of the logarithmicverociry disrribution,determinethe s-hearstressfrom the velocirydistributionand compareit with the valueobtained from the unifbrm flow rorTnuta. Lrlscuss theresulls
156
C H A P T E R4 : U n i f o r m F l o w t,. ft
0..1.10 0.368 0..t27 0.285 0.265 0.1,1,1 0.221 0.201 0.r82 0.170 0 .r 5 7 0.1.15 0 .[ ] 0.120 0.108 0.096 0.08? 0.079
vebcit), tys 2.19 ?.10 2.03 1.95 l.E7 1.83 1.80 t.69 L6,r !.58 !.56 r.,19 L35 t.30
r.15 L05 0.97 0.E8
the normal of yourchoice,thatcomputes 4.21. Writea computerprogram,in the language depthandcriticaldeplhin a circularchannelusingthe bisectionmethod.The input coefftcient. slope,and Manning'sroughness datashouldincludetheconduitdiameter, Illustrateyour p.ogramwith an exampleand verifythe resultsfor normal discharge. andcriticaldepth. or rock 4.22. Write a computerprogramIo designa trapezoidilchanneluith a vegetative aregiven.Allow the userto adjustthe ripraplining if the slopeanddesigndischarge andthechannelbottom vegetalretardance class,or therocksizeandangleof repose. widthandsideslopeinteraclively. 4,23. Using the computerprogramYcomp in AppendixB, find the nomlal and critical channelsectiongivenin a datafile.Thedischarge depthsfor thefollowingcompound Is thisa subcritical or superis 5000cfs,andtheslopeis 0.009.Plotthecross-section. criticalflow? "DUCKCR". 19..1 -480.796 -.140,788
-,{20,786 -305,784 -t't 5,182 ,95,780 -50,778 -30,176 -)\ 71t
2,172 17.172 20;114
C H \ P T t i ? . 1 . L l n t f . , n nF l o w
28.780 50.780 670.780 990.782 1070.784 I r20.786 l ? 6 0 . 81 0 , 9 5. . t . 2 8 , . 0 ' + , 6 7 0 . . 0l 8 ? .6 0 , . 0 5
157
CHAPTER 5 '*I,}:e:;1alnffig*:'ie*r
GraduallyVariedFlow
5.1 INTRODUCTION Gradually variedflowis a steady, nonuniform flowin whichthedepthvariation in the directionof motionis gradualenoughthatthetransverse pressure distribution canbe considered hydrostatic. Thisallo$s theflow to betreatedasonedimensional with no transverse pressure gradients otherthanthosecreatedby gravity.The methodsdeveloped in thischaptershouldnot be appliedto regionsof highlycurvilinear flow, suchascanbe foundin the vicinirv of an ogeespillwaycrest,for example, because the centripetal acceleration in cunilinearflow altersthe transverse Dressuredisributionsothatit no longeris h\ drostatic, andthepressure headno longer canbe represented by thedepthof flou. Even with the assunrptionof gradually variedflow, an exactsolutionfor the depthprofileexistsonly in thecaseof a u.ide,rectangular channel. The solutionof the equationof graduallyvariedflow in this caseis calledthe Bresse function, which providesusefulapproximations of watersurfaceprofilelengthssubjectto the assumptions of a very wide channeland a constantvalueof Chezy'sC. The solutionsto all otherproblems, in the past,wereobtainedgraphically or from tabulationsof the variedflow functionbasedon hydraulicexponents asdeveloped by Bakhmeteff(1932) andChow( 1959).Cunently,theuseof personal computers and the application of soundnumericaltechniques maketheseoldermethodsobsolete.
5.2 EQTTATION OF GRADUALLYVARIEDFLOW In addition to the basic gradually varied flow assumption.we further assumethat the flow occurs in a prismatic channel, or one riat is approximately so, and that the slope 159
160
CHAPTER 5 : c r a d u a l l yv a r i e dF l o w
of the encrgy grade line (EGL) can bc evaluatedfrom unifornr tlow formulas s irh uniforrn flow resisltncecoelicients. using the local depth as thoughthe llo\\ \\ere locally uniform. With respectto Figure 5.1. the total hcad at anv crossscctlon is
v1 H = a+ | + 0*
(-5.I)
in which I - channel bed elevarion;l - depth; and V = mean velocity. If this expressionfor H is differentiatedwith respectto _r,rhe coordinalein the flou direction. the following equationis obrained: dH
: d,,
s":-so*
dE
(5.2)
d,{
in whichS" is defincdas the slopeof thecnergygradeline: Sois thebedslope( = dJdt); andE is the specificenergy.Solvingfor dEldr givesthe firstform of rhc equationof graduallyvariedflo*: dE
:So-S.
(5.l.)
dt
dH
I
i
EGL
Bed
t ^ .
I I 1 FIGURE5.1 Definitionsketchfor graduallyvariedflow.
; UATUM I
2
C H A P T I ' R5 : G r a d u a l l\!' a r i e dF l o \ r '
l6l
It is apparentfrom this tbrn of the cquation that thc specilic encrgy can either increaseor decreasein the dou'nstreamdirection.depcndingon the relativemagnitudcsofthe bed slopeand the slopeofthe energ)'gradeline.Yen ( 1973) sho\r'edthat, in the gencralcase.S" is not the sameas the friction slope.t/ ( = r,,/7R) or thc encrgy dissipationgradient. Nevenheless.we hare no better rray of e\aluating this slope than uniforn flos fornulas suchas thoseof Manning or Chezy.It is incorrect.however,to mix the friction slope,which clcarly comesfrom a tnontetltuntanalysis'u'ith terms involving a. the kinetic elerg,! correction(Martin and Wiggert 1975). The secondform of the equationof graduallyvaried llow can be derivcd if it is - Fr recognized that dEldt : dEld-r'' dr'/dr and that. from Chapter ?. dEld.r' = | becomes Then' Equation 5.3 properly defined. provided that the Froude number is
d,r'
So
S"
(5.4)
d { l F r The dellnition of the Froudenuntberin Equation5.4 dcpendson the channelgeonetry. For a compound channel,it shouldbe the compoundchannel Froude number as defined in Chapter 2, while for a regular,prismatic channel, in which dald-r'is negligible,it assumesthe conventionalenergydefinition given by ap:8/gA1.
5.3 CLASSIFICATION OF WATERSURFACEPROI'ILES Equation 5.4 can be used to derive the expected shapesof water surface profiles for gradually varied flow on ntild. steep, and horizontal slopes. for example. It is imponant to identify theseshapesbefore running a water surfacc prollle program becausethe location of the control, where a unique relationship cxists between stageand discharge,and the direction of computation(upstreamor downstream) dependon this knowledge.ln effect,identificationof the control for a given profile amountsto specificationof the boundarycondition for the numerical solutionof a differentialequation. Equation 5.,1 provides the tool for determining whether or not the depth is increasingor decreasingin the downstreamdirection and also for determininglhe limiting depths very far downstreamand uPstreamior particular gradually varied flow prohles. In order to deducethe shapesof the profiles,it is sufficientto determine qualitatively the relative magnitudesof the terms on the riSht hand side of Equation5.4. For this purposeand in the numericalcomputationol graduallyvaried flow profiles, we assumethat the local value of the slope of the energygrade line, S", can be calculatedfrom Manning's equationusing the local value of depth as though the flow were uniform locally. Therefore, the following inequalities hold when cornparingthe magnitudeof the local depth ) at any point along the profile with normal depth ,ln:
_ r( l o :
S")So
(s.5 )
)>)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
(s.7 ) (s.8)
With the\e inequalitiesand Equation 5..1,the gradualll varied Uorv profile shapes can be drtcrmined, as shown in Figure 5.2. The l-lowprolrlcsshownin Figure 5.2 are designatedM, S, C, H, or A for mild, steep.critical, horizontal,and adverseslopes.respectively.The flow profiles are funher identifled numericallyas l. 2, or 3 counting from the largestdepth region do\l nward. basedon the two or three regionsdelineatedby the normal and critical depth lin.'s. Only two rcgions occur for critical, horizontal. and adverseslopes. bccausenormal depth does not exist in the latter t$o cases.while in the former case. normal depth equalscritical depth. Funhermore, all profiles are sketched assunringrhat flow is from left to right. It is important to keep in mind that the control alwals is downstreamfor subcritical flows and upstream for supercritical flo$ s. Hence, the directionof computationof subcritical profiles is upstreanr,and for supercriticalprofiles.it is downstream. From the inequalConsider the rnild slope in region I for which,r' ) ,r'o) _r'... ities in Equations-5.5through 5.8, we can conclude tlat d_r'/dx> 0 so that the Ml protrle aluays must have an increasingdepth in the downstreamdirection.As y approaches]0 in the upstreamdirection, dy/dr approacheszero asymptotically, while in the downstreamdirectiond_r/drapproachesS0,so that a horizontalasymptote exists. The Ml profile sometimesis called the backwater profle and exists "backs r,,herea reservoir up water" in the tributary stream flowing into it. In region ( _r'( _ro,S. > 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,, (<1.5) and sub\tilute it inlo the secondequa_ lron to solve for p. Then. substi(uteO and t.ointo the first equationand iterateuntil the result is 3.5 m (11.5 fl) tbr rhe heacl.Alternalirely, the secondcquationcan bc substi, tuted into the llrst and a nonlinearalgebraicequationsolver can be applied Solr ing by t n a l a n de r r o r .$ e f i r s lr r i a lg i \ e s r 0 = 3 . 0 m ( 9 . 8f 1 ) .O = l 0 8 m r / s ( 3 g l 0 c f s ) . a n d , y : 3.66 m ( 12.0ft). For lhe \econd trill. r,o: 2.5 m (8.2 ft). Q = 6.r.31rr7.1'r920cfs). antt ( l 0 . 0 f t ) . F o r t h e r h i r da n d f i n a l r r i a l . ) o = 2 . 8 7 m { 9 . { 2 f r ) . H:3.06m IOI mr/s e: (3565 cfs), and H = -].50n) ( I 1.5 fr), which gives rhe llnal anslr.er.The crjrical depth can be calculatcdro be y, = {?r/g)r/r = 2.18 m (7.15 fr). so the slope jDdeedis mild. The Froude nuntberof the uniform flo$ is 0 66. The l\ll profile \rsn\ from a depth of 2.5 m (8.2 ft) al the downstreamend of thc channeland approachesnormal deprh before it rcachesthe upstreamlake. since the chanrrelis very long. This also can be referredtcr as a hdraulitallt l
5.5
IVATER SURFACEPROFILE CON{PUTATION Thecompuration of watersurface profiles hasmanyimponant apptications in engi-
neeringpractice.A prismaticdrainagechannel,stomt sewer,or culven designedfor uniform flow ntay be checkedfor its performanceunder graduallyvaried flow con_ ditions. Floodplain mapping,which is the determinationof the extent of floodins for a flood of specifiedfrequency,requircswater surfaceprohle computation\ in ; naturalchannelof irregularand variablegeometry.slope,and roughness. The problem formulation in $'ater surfaceprofile computationsusually specifics a dcsign di:charge set bY fiequerrs).r,:urrsidcrrtions and requrrcstlre sclcition of channel roughness,slope, and geometry.In the case of a nirural channel. the channelroughness,slope,and geometryare measuredfor a seriesof reacheswithin which these parametersare relatively constant.With this information given. the mathematicalproblemis to solre the equationofgradually varied flo\,, roibrain the depth as a function of distancealong the channel,,r.- F(.r), subjectto a boundary condition establishedby the channelcontrol.The control can be a measuredstagedrschargerelation,nornal deprh. cntrcal depth,or a depth set by a hydraulic control structure. Two typesof methodscan be usedto solvethe equationof graduallyvaried flow in the form of either Equation5.-j or Equarion5.4. In the firsr type, rhe disranceis dcterminedfor a specifieddepth change.This approachcan be classifiede.rpliciland sometimes is called the direct step method, becausethe soiution is direct, requiring no iteration.Equation5..1.for example,can be reprcsentedsymbolicallyas d,r,/drf,r'), where/lv) is the nonlinear funcrion of r specifiedby rhe right hand side of Equation5.,1,in which both S. and F dependon the local depth r,. This is an ordi_ nary differentirl cqurrion for q hich rhe varilble:,can be ,epaiareda. .
d,f
,tt)l
(5.13)
168
CHAPTER 5 : G r a d u a l l\yi r i e d F l o *
'fhis
nr i l n i t e - d i l l e r c n c e . r p l r r , r r i e q u a t i o nc a n b e s o l r e db y n u m e r i c ailn t c g r i r t i o o a i n d c p t h I i s s p c c i f i c da n d t h e c o r r e s p o n d i r ) g i n e i t h e r c a s e . c h a n g e nationl c x p l i c i t l y T . h i s n t e a n s n o c o n t r o le \ i ! l \ o \ c r t h e p o s i l i o n s i n . r i s c o m p u t c d chan-ce qherc for depth are obtained.thich is no probstations. the solutions .r, or channel lem in a prisnratic channel.becausethe cross-sectionalpropcnies do not change u i t h d i s t a n c e . rI.n a n a t u r a cl h a n n e lo. n t h c o t h e rh a n d .c r o s s - i c c t i o n aplr o p e n i e s are determinedbelorchand at particu)arlocations.so thal a ditlercnt approachis r e q u i r e di.n r ' " h i c hd e p t hi s c o m p u t e da s a l u n c t i o no f s p c c i l r e dc h a n g e si n d i s t a n c e . In this case,the rariabl-'sarc scparatedas
(s.r.1)
d_r.= /(_r.) dr
and it appearsthat the numericalsolutionprocedurehas to bc-iteratire to conrpute the value of Jr for a specihedLr. becausethe unkno$n appcrrs on both sidesof rhe equation. If iteration is required.the approachis consideredinplicit. On thc other hand, a cfass of techniques,callcdpredi
5.6 DISTANCE DETERNIINED FROM DEPTH CHANGES Dir€ct Step Method ln principle,the directstepmethodcouldbe appliedto eitherEquation5.1 or 5..1 with theformer.Equation5.-3is placedin Inite diflbrence but usuallyis associated asdescribed dEld\ with a fbrwarddif-fcrence. approximating the derivative form by gradeline value of the slope of the energlA. and by taking the mean in Appendix (-r,+r u'hich t and the subscript i inererse -r,) in distance the step size Ir over The result is downstream direction. in the F
L I + I
Jo -
F
(5.15) J.
uhere S" is the arithmetic mean slope of the energy grade line bctuecn scction: i and i * I, with the slope evaluatedindividually from Manning's equiltionrt euch crosssection.The variablesE,-,, E,, and 5" on the riSht hand side of Equation-5.l-5 all are functions of the depth r'. The solution proceedsin a step$'iseflshion in Lr
C H . \ p r E R5 : C r r d u a l l l a r i e dF l o w
169
b1 assurningvaluesof dr.'pth_r'and thereforeva)uesof specificenergy.E. As Equation 5.l5 is *ritten. -r increasesin the do\\nstreamdirection.In gencral.upstrc,am c o m p u r a t i o n us r i l i z eE q u r t i o n5 . 1 5 r n u l t i p l i e db v l . s o t h a rr h e c u r r e n rv a l u eo f specific energvis subtractcdfrom the assumedr aiue in the upstreamdirectionand It becomes(.r, - .r,_,).rvhich is negative.Thcrefore, iI the equationis solved in the upstrean)dircction fbr an M2 profile, for erample, the contpuredvaluesof Jr should be negativefbr increasingvaluesof _1.Decrcasingvaluesof _r'shouldrcsult also in ncgativeyalucsof Ir for an Ml profile. For an M3 profile, which is supercrirical, incrc-asingvalues of depth in the do\\ nstreantdirection correspondto decrcasingvaluesof specificenergy.and Equation 5.l5 indicatespositivevaluesof \r since S, > du Although the direct srep method is the easiesrapproach.it requiresinrerpolation ro find the final deprhat the end of thc profrle in a channelof specifiedlength. Some care nrustbe takcn in specifyingstartingdeprhsand checkingfor depth limrts rn a computerprogram.In an M2 profile, for exantple.rhe staningdepth should be taken sJightlv grearerthan the computed crirical deprh. if it is the conrrol. becauseof the slight inaccuracyinherent in the numericrl evaluationof critical depth. ln addition, the M2 profile approachesnormal depth asyntptoticallyin the up\tream direction. so that some arbitrary stopping limit must be set, such as 99 perccnt of normal depth. E x A l t p L E s . 2 . A r r a p e z o i dcahla n n ehla sa b o t t o mw i d r hb. . o f 9 . 0m ( 2 6 . 2f i ) a n d a sideslope rarioof 2:L The Mannings n of rhe channelis 0.025.and it is laid on a slopeof 0.001.Il rhechannclendsin a freeoverfall,computethe warersurfaceprofile for a dischargeof -j0 mr/s. So/rltbt, First.norrnaldepthandcriticaldepthnlustbedetermined. Fromllannine.s equation.
+ 2r,,)l'l i-r',18.0
It.or :-'.v/i + l]r'
0.015 x 30 1 . 0x 0 . 0 0 t, ,
from whichr,,: t.75.1m(5.755fr). SettheFroudenunrber (OBl.r)/(V.SA:r) = I and solvefor criticaldepth:
r lr.(8.0+ 2r,)lr r l 8 . o+ , 1 t],r
= _
30
= o sq
v 9.81
from which t. = 1.03m (3.18fr). Therefore, rhis is a rnildslopeandwe arecompul, ing an M2 profilethathascriricaldeprhar rhe freeoverfallasrheboundary condition. The directstepmethodcanbe programmed as shownin AppendixB or solr.edin a spreadsheet. as sho$nin Table5-1.The valuesoi r areselected in the firstcolumn: andthe formulasfor determining the specificenergy.E, andslopeof theenergygrade line,S".for a givendepthare shownat the bonomof the spreadsheet. The arithmeric meanof S"(Seba.r) is compured in column7. andthechangein specificenergylE {Del f) il theupstream direcrion is shownin column8. Then,theequation of graduallyvar, ied flow in finitedifference form is solvedfor the distance steD.I(. as
r.6:E.0,1 s--s
lo 0or
=
r 6.o9f-03
0.028m (-0.092fr)
-
t
- E
o
y.n
a> *
ii i4=qa q f ? 5 5 n : ; 7 r t = - s
E
: - . L
3 : 170
t l i l
1 - .
> E
) l l a c = ( . ' . ' :
i -,
; : : : : I 1 : j r j ; lt ; r . t ; = s ; : i i l '
: = : : -rt rJ -:: _
l:
ci
cj
-- a f -d Jl J
3 tlj.rJ
= a? J --,: r
?oo r _ :!
* f i . =l = -j. ; -:. 2 r: :-:; ; ; ci ci .i .i .i : :
i;
- _ : _ ': : = 5q i --i -:- -:i :- _:i -:_
t; a =53
3 : i = "i : i : € i f : i i i f : 2 n i r n a : € ? 6 3 5 * ! c e F R E 5€ s € q f r; j ; s
;a23 -a- . 4-
Y.
-
.t
''
4. d b (. : t I = J x , t € 9? . - . -? j r,.1 r , r I " . q r ^ n . r o - . a j : r r , 6 € € o , 3 3 3 _
2 . i = _ I1: i: ! ? 7 . i i! ! M
E
: + a
!i i F
< -€ -4l , oi - d
l r t l l l
t7l
t12
C H A P T E R5 : C r a d u a l l yV . U i e dF I o w
Normal depth
M2
E
h
i
o
0
000
-€00
-40o
_600
-200
m UPstream, Distance T.ICURE5.4 profilecomputed by thedirectstepmethod M2 uatersurface
in the first step. Note that at leastthree significantfigures shouldbe retainedin AE to avoid large roundoff enors when the differences are small in comparison to the values of E. In the last column, lhe cumulativevaluesof lr are given' and theserepresentthe distance from the staning point to lhe point where the specified depth -! is reached After t}le first step, uniform incrementsin depth,\'.$ith \ increasingin the upstreamdirection. are utilized.The valuesof ] are stoppedal the tinite limn of 1.745m (5 725 ft)' *hich is 99.5 percentof normal depth.The length requiredto reachthis point is I27l m (,1170ft), which is the lengthrequiredfor this channelto be consideredhydraulically long, but that length varies.in general.The depth incrementscan be halved until the changein profile lengrhbecomesacceplablysmall-Alte.natively'smallerincrementsin depth can be used in regionsof rapidly changing depth, and larger incrementsmay be appropriate in regions of very Sradual depth changes. A portion of the computed M2 profile is shown in FiSure5.4.
Direct Numerical Integration The direct numericalintegrationmethodis applied to Equation5 4' which also can be solved by the direct step method, but in this case numerical integration is employed.In the integratedform, Equation5.4 b€comes
t' f' I a^-,,.,-,,=l
),
J,
l-F: [ ' g ( \ d' ). r ^ ^d'=l
Je
r,
(5.16)
).
The integrandon the right hand sideof Equation5. l6 is a functionof .r, glr'), which can be integratednumericallyto obtain a solution for !r, as shown in Figure 5 5'
C H \ P r E R 5 : C r a d u r l l yV a r i e dF l o w
r,"^ FIGURE5.5 Water surfaceprolile computationby direct numerical integration.
A varietyof numericalinregrationtechniquesare available.such as the trapezoidal rule and Sinpson's lrule, which are cornmonlyusedto find the cross-scctional area ofa naturalchanncl.lbr example.Sinrpson'srule is ofhighcr order in accuracythan the trapezoidalrule. which simply meansthat the sarnenumericalaccuracycan be achievedwith fewer integrationsteps.Application of the trapezoidalrule ro the right hand side of (5. l6) for a single step produces g[r',-r) + g(l'i)
To determinethc full lengthof a flow profile. (r, trapezoidalrule resultsin
(,r',., - l')
(5.171
.r0),multiple applicationof the n
l
el
g(,\'r)+s(,\',)+Z) L-xn-.r-0-,[
r = 1
(5.r8)
where L - profile lengthand tr_]: (-I,,-| - y) = uniform depthincrement.Because the global truncationerror in the mulriple applicationof the trapezoidalrule is of order (A_r)r,halving the depth incrementwill reducethe error in the profile length by a factorof l/4. By successivelyhalving the depthinterval,the relativechangein the profile length can be calculatedwith the processcontinuing until the relative error is lessthan some acceptablevalue.
C H \ P l l k 5 : C r r d t r ; l l rV u r r c rl ll ' r * f,. /
FRO}{ DISTANCECHANGES DEPTI{CONTPUTIlI) Thc second approachto thc solulion of thc eqtrctionof gradually raricd llow is e x a c t l yo p p o s i t ct o t h r t l r \ t . l n t h i sc l a s so f m c t h o d sd, c p t hc h a n g e sa r e d e t c n n i n e d for specifiedchangesin di\tancc. This solutionstrxtegyis ntore appropriatefor nar propertiesarc determincdby surreys at speural channelsin u hich cross-sectiontrl citlc locationsalong the channcl. but it can be used lbr artificial channtls as rvell' t o o b t a i na s o l u t i o nl b r d c p t ha s a l u n c t i o no f d i s t a n c e . I f E q u a t i o n5 . - 1i s i n t e g r a t e d lI Decomes
i ' s o- S " r , - r - r , = J , - f " . ,o . -
l'tor*
(s.l e)
Thc dilficultv with (5.l9 t is readily apparcntwhcn we recognizethat the integrand itsclf is a llnction ofthe unknown depth.-v An altcrnativeis to usethe Tlylor series expansionfoli,-, and drop all terms beyond the llrst derivativetem''
- r ' , r 1= , r ; + / ( - r ' , ) l . r
(5.20)
where/(,r) : clr'/dt, wbich can be evaluatedat point.r, from Equation 5.4 This method,known as Euler's nrcthod,sinply extendsthe slope of the solution curve for depth-vforward from .t, as a straightlinc to obtain the next estimateof) at ri*r' The termsdroppedfrom the Tay)orscriesexpansionmake the local truncationerror O(Arr) as discussedin Appendix A, while the global truncationerror (local plus propagated)for multiple stepsis O(Ir) (Chapraand Canale 1988) This is referred to as a first-order method. In gencral,it requiresvery small step sizes' and therefore considerablecomputationaleffon. to achieveacceptableaccuracy An improved Euler's method can be formulatedby evaluatingthe slopeof the function at both ri and ,r,- ', then applying the arithmeticmeanof the two slopeestimatesto move the solution forward However,becausethe slope cannot be evaluated at i + I , since -r is unkno* n there, the value of 1',*, is first predicted by the Euler methodto evalua(ethe sloPe/(ri* r) The valueof ,v,*, then is conectedusing this estimateof the slope in the determinationof the mean of the beginning and ending slopesover the interval.The resultingpredictor-conectorequations'known As the Heun method, or corrected Euler method, are
,rl-r : t, +/(,r;)Ar
+/(-Il-,)l Lf(v,) Ar
(s.21) (s.22)
valueof -v,*rin (5 21), zerois usedto identifythepredicted in whichthesuperscript (5 22)' This, refened given by formula the corector which then is substitutedinto class of solution part of a larger is method, to as a one-sleppredictor-corrector Equations5 21 is that apParent Also methods techniquesknown as Runge-Kutta
C H A p I E R 5 : G r e d u a l l y\ n r i e d F l o w
175
and 5.22 can be iterated back and forth to inpro,,e the .r,lution. Hou$er. the iter_ ative approach must be used with caution because rhe c,rror lctullly nlay grow rather rhan shrink (Beckett and Hun lg6T). lf Equation. _i.ll and -5.ll rr" not it".ated. the)'can be shown to bc a second{rde-r iunge,Kutta method (Chapra and Canale 1988) with a global enor that is O(Irr). EIAIIpLE 5.j. C o m p u t et h e M 2 p r o t i l co f E x a m p l e5 . 1 u s i n gt h ec r r r r e c t e d Euler melhod \r i(houl iterationand conrparethe results. Solulion. fhe solurionis accomplishedin the spreadsh:crsho*.n in Iable -5_2,using Equations5.2I and 5.:2. Firsl, the valuesof area (A I). hlijraulic radius(R l). and ropwidth (B I) are conputed for the rnitial \ tluc of depth ,,,,i,. h".,rr," (her are neeoerl1o calculalethe functionjr(l)
l' - sj f t' r r : t The valueof/Iyl) is givenin column6 Thc predlcledr.alue ofr. (,\f:pred)at theend of.the spatialintervalis givenin column7, compuredfronr Equerion 5.21usingrhe valueof/(r',) in column6. Coluntnsg. 9, and l0 are ne.ded lo computethe valueof f(y2:predJin columnI l. The corrected valueof _y.is compuredfrom Bquarion 5.22in column2 of the nextrow for a givenstepsizein i. anOr-t" pr*er.-begins again_ At a drsrance of l27l m (4170fr). the correcred Eutermerhodgires a deprhof 1.744m (5.722fl), whilctheairecrstepmethodyieldsa deprhof 1.7{-5m 15.72S frl at thesame location.This is a relarivedifference in depthof iess*an O.f f"r".nr. If rheinterval sizein depth_r, is halvedin rhedirectstepmethod,theresulringdepthrounds to | .744m (5.722fl) in agreemenr wirh rheconectedEulermethod...frit," f,.glnningof rn".o. putatlonin Table5-2,the stepsin the spatialcoordinite.t hare be-en takento be very smallbecause of thesteep,rapidlychangingslopeof the \12 *.atersurtace prol.ile near criticaldeprh. The
popular Runge-Kutta method is the founl_order method, which -most four equarions or sreps ro proceed from point i ro poinr i i L fhe equa_ :,e-1u-tle: tlons are recursive,in that each usesa value computed from the previousone.The method can be summiLrizedby
= )i * ,v,+r
flto,
* ro,+ 2kr+ e.,)]l.r
(5.23 )
in which
i<, = /(r" -v,)
(5.23a)
.f*,)
(5.23b)
* tAo r' )\
(5.23c)
ka=f(x,+ Ax,,r,a Axk.,)
(5.23d)
/
r, =/(x, + (. t =- / \ .( r , +A
n"
],rr x
T,r,
\
w
r ;
=
; a -:< -
-
II
< J
t16
c
a;E
.,.r Q
::
c
x
t
?
r
a :
t
t : -
4
i .2,1 :
=- - : l : l i 3 j : 3 a_2 1 = t_ i ". y 2 t- y. -=
- =: ; i 1 1 = : - !;, :i = v 4 ' : i .t; .rt !_ ;_ i_ z: a j : i i: Z z t i f, - . i, -.
l;i!!ri!:?5::551:1i:riii:?: -- -' , . 3 * .r . 991 r 9 :€ - - ... -'-- - -. .| 1 :;7,7' 7, i' Y;i, 9..9 3 f, t, i : S : : -
?
i = r . z) -= : = 4 . 7 r ; ! ' 2: ': :L=. a ', _ : 1? 3! : - : = : r 1 1 : : : 1 ti ;r i i i 3 ; ?_ s
i : :
: : z ! z 3 E I I 3 s l :
i:
=:
3 3 3 1 53 3 3 3 5 3
: : i : : r r : : ; ? i: ? ; = ; 5 i i + i E ; ; t ' c q ' v ' - ' e
rIi;19;:i:rl:55f$:i::llilir t r : : ' r ! t r-2 -- . - -c -r -: :- : ,cr -^ .
' ' " ::
i:, I X i n ri I E d E f R F,
liiil;qi;i;ig!:!!l!::ii:3a: i ; t_=' :7' :_?: i: jl: 7: :' i: !:i 2 € : l :F: f: ,! fi :: : :: !: F :| i| :i :r: : i: 5
f ; " T : t C : E r = F g s: *Ei F r 6 E F F E = E E , r , l | _
t1'1
178
C n A p r E R5 : G r a d u a l lVy a r i e dF l o u
The founh-order Runge Kutta rnelhodcan be applicd u,ith adaprive\rep rire control such that. at each srep.thc stcp size is first taken as a full srep and thcn taken as l$o half steps.Thc djffcrence between thc t*o esrimaresof depth is used to adjust lhe step sizc so lhat some specificdrclativeerror critcrion is rnet on a stepb y - s t e pb a s i s( C h a p r aa n d C a n a l e ,1 9 8 8 ) .I n g e n e r a lt.h e d i r e c ts r e pm c t h o do f i e n is sufficient for water surface prolile computation.but the founh-order Runge, Kutta metbodmay be useful u,herea high dcgrceof accuracyis required. An iteration prcredure 1or the sccond-orderpredictor-correctorntcthod of (5.21) and (5.21) has been proposedby Prasad(1970) for water surface profile con)putationin rirers. His procedurcis summarizedby the follo$ ing: l. Calculatef(r') for r - r'.:
.,.,_ " I l \"' | -
se-s"lr,) I
F.(,r,)
(5.24)
2. Set/(vi,r) =,f(.r',)as an initial guess. 3. Calculate.v,-, lbr a given Ir front
-lrll
.i^.a J
d\'/,lr
C"lrrrlqtc
=
fr '
-
tr
'
,)l L,f(r,)+ ,f(,u,-
( 5 . 2) s
a r' l t frnm
(5.26) 5. Check/(,r,-r) from step.l againstthe previousvalue and repeatstcps3 through 5 u n t i l l h e y a g r e eu i t h r na c e n a i ne n o r c r i t e r i o n . While this method does converge,numericalproblemscan arise when critical depth is approachedas in an M2 or M3 profile. When rhis happens,rhe denominator in/(),) approacheszero as Fr approaches1. Theseproblemscan be handledby using smallerstep sizes near the critical depth and startingand stoppingthe profile computationu ithin some finite intervalaway from critical depth. It also shouldbe apparentthat, for overbank flow. the compoundchannelFroude number shouldbe used in the equationof graduallyvaried flow. Otherwise,incorrect valuesof critical depth are accepted,and the resultingprofile is incorrectas well. Ex A v p L E s.l. Considerrhelakedischarge problemof Example5.1.exceprthat the rnild slope(S : 0.005)hasa lengthof 500 m ( 16.10fr), followed by a slopewith a valueof 0.02 and a lengthof 200 m (656 ft). as shownin Figure5.6. The Manning'sn cf thedo$'nstream channelis 0.030andirswidrhis 10.0m (32.8fr). \\'hicharerhesame valuesasfor the upstreamchannel.Sketchthe possiblewatersurfaceprohlesandcomputeoneof themfor a downstream lakelevelof 5.0 m ( 16.4ft) abovetheoutlelinven.
Curerrr
5: GraduatV l ya r i e dF l o w
ljg
110
E
105 =--CDL-=-=
c
.9 6
M2 :---..\
Hydrar Jlrc
:--\
S o =;.i(
tr 100
NI
sl \
o.o2 95 o
100
200
300
4oo
500
600
7oo
Distance,m (a)WaterSurlaceProfiles(O = 101 m3/s)
100 ao
E c.
LL
/t
E o E > 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,<. and geometrychange.1'hcrefore,water surface profiles in natural channelscan b: r'iewed as transitionprofiles betueen normal depths:that is, a collection of i\tl and M2 protiles on mild slopes.If thc normal depth at a specificlocationis desired as a downslreamboundarycondition, several \\ ater surfaceprofilescan be startedfrom funher do$ nstrean until asymptoticconvergenceto normal depth is achiered (seethe Ml and M2 profiles in Figure 5.2). In reality,when thc depth rcached by two backu,aterproliles is within a specified tolcrance.convergcnceis assumed.Davidian (198,1)suggeststhe use of trvo M2 profilesto deterrnineconvergence Cross sectionsfor water surface profile computationare selectcdto be represenlativeof the subreachesbenreen thenr.rs shown in Figure 5.9. Such locations as major breaksin bed profile, minimum and maximum cross-sectional area,abrupt changesin roughncssor shape.and control sectionssuch as free overfalls alrlays are selectedfor cross sections.Cross scctionsneedto be taken at shorterintenals in bends,expansions,low-gradientstreams,and where therc is rapid changein conveyance(Davidian 1984). Somecrosssectionsmay require subdivisionwhere thereare abrupttransr,erse changesin geometryor roughness.as in the caseof overbankflows. This must be done with care,however,or unexpectedresultsare obtained.[n general,if the ratio of overbankwidth to depthis greaterthan 5 or if the ratio of main channeldepth to overbankdepth exceeds2, subdir ision is recommended(Davidian 1984). The occurrenceof both supercriticaland subcriticaldepths in a river reach, referredto as a mi.red-Jlow' reginre.requiresspecialattentionin naturalchannels.In a prismatic channel in which a hldraulic jump is expected,as for example in a reach with an upstream supercritical and dow,nstreamsubcritical profile. the momentum function is computed for each profile and the intersectionof the two nromentumfunction profilesdeterminesthe locationof a hydraulicjump, as shown in Example 5.4. In a naturalchannel with a slope near the critical slope,however, finding the exact location of the jump is not possiblebecauseof the continuous variationin geomelricpropertiesof the crosssections.Instead.the HEC-RAS program computesa subcritical profile in the upstreamdirection, starting from the downstreamboundarycondition. tien computesa supercriticalprofile in the downstreamdirection.usuallybeginnins from critical depth.At eachcrosssectionwhere both a supercriticaland a subcritical solulion exist, the value of the momentum function is computedand the depth with the higheryalue accepted.If, for example, the subcriticaldepthhasthe higher valueof the momentumfunction,this meansthe jump would be subnrergedat this location and nrove upstream,so the subcritical depth would be accepted.At an\ cross sectionwhere the HEC-RAS program or "balance" WSPRO cannot the energy equation,the critical depth is taken as the solutionand computationsproceed.If the depthis critical for both supercriticaland subcrilicalprofiles at a given cross section,then it is likely to be a critical control section. Water surfaceprofiles computed using the Prasadmethod and the compound channelFroudenumber(Sturm. Skolds, and Blalock 1985)are illustratedin Figure
184
C H \ P r t R 5 : C m d u r l l y \ / i r r i e dF l o w
1 0'[\ 8
F t -1\ 2 0
water surlace /
| \ weeds & w,ll*
.l I
s e e o t ' n s sU l
1r\,''4'{"1 40 60 20 CrossSection1
Segrnenls
(."1
I
Scanered
/)r
4A 20 60 CrossSection2
2 -< sec\\oo CloY
(D
g
80 Fr
Subseclrons t-i i
1.000
1 ? &
I
Cross seclion 1O 3 8 6 F l a 2 0 Cross Seclron 3
FIGURE5.9 n usedin assigning segments. andsubsections sho*ingreaches, crosssection Hypothetical (Arcement 1984). andSchneider values 5.10 for a laboratorymodelstudy.The channelis a 21.3m (70 ft) long movablewereusedin the bedmodelof an alluvialriver A totalof eightrivercrosssections ( Water surfaceprom3/s I .20 cfs). of 0.0341 di scharge for a constant computaiions The sediequilibrium bed had approached the sediment after files weremeasured : 0 016' : (0.0108 n ft) Manning's mm and 3.3 with dro was uniform mentsize channelFroudenumberwasusedto calculatethecriticaldepthlor The compound
C H A P . t E R5 : C r a d u a l l yV a r i e dF l o w
t85
0.95
Crosssectionno. 13 12 14 0.90 l l
e i 9
11
10
0.85
9
8
7
O = 0.0341m3/s
0.80
_9
LU
0.75 0.70
0.6s 0
2
-
Bed B Critical
8 6 Station,m
4 - -
10
Computedsupercrilical
-
12
14
Computedsubcritical
^ Measured
FIGURE5.IO qatersurface profilesin a ri\er modcl(Slurm.Skolds'andBlalock andcomputed lvteasured "lVaterSurfacePruJi[esin (Sorrrce. T. ll. Srunn,D M. Skolds'and M E Blalttck. 1985). Cttnference'Htdraulics and Conpountl Channels. I'roc.of the ASCEHtd. Div Specialo^ ofASCE ) bY pentrissittn Reprotlucetl /98J ASCf. Age. O Hyt!rologt h theSntttllCttnrputer each cross sectionand to idenlity a panicular solution of the energy cquation ls supercriticalor subcritical.Both subcritical and supercriticalprofiles were compuied. as shown in Figure 5.10. At cross sections l0 and 13. critical depth was returned as the solutionfor both profiles becauseneithera subcriticalnor a suPercritical solution could bc found. The measureddepths also are in close agreement with the computedvaluesof critical depth at these two cross sections,indicating that they indeed are critical.At cross sections l2 and 9. just downstreamof cross both a supercriticaland a subcriticalsolutionexist sectionsI 3 and 10.respectively, This would indicatea weak hydraulicjump ralues are subcritical. but the rneasured cross sec(ionsl3 and I2 and bet\."een waves between standing or perhapssimpl! higher valueof the momentumfuncwith the the depth l0 and 9. Computationally, depthsat crosssectionsl2 and subcritical the supercritical tion is chosenbetween example. the subcriticalsolulf' for 1998) (U.S. of Engineers Corps Army and 9 jump would be then the function. momentum value of the tion has the higher of correctlypreThe importance moved upstream. and dro* neclout at that section an Incorothenrise. be apparent: should in this example depth dicting the critical can occur. u regime of the rong selection profile and the rect interpretationof EXAIlIPLE 5.5. (ADAPTED FRO\I U.S. AR\TY CORPS OF E}iGINEI]RS l998 ). Appl) the HEC-RAS program to Skillel Creek. as shown in Figure 5 l l, and
of 2000cfs (56 6 mlls; In lhe upper profilefor a discharge computethe oatersurface and a totalof 2500cli (70.8mr/s)in the ( Creek. mr/s)in Possum ."u.h. 500 "f, 1.1.2
186
CItApTER 5: GraduallyVa-riedFlo*
FIGURE 5.I I Streamlayoutschematic for HEC-RAS,Example5.5.
entirelower reachof Skillet Creek.Assumea subcriticalprotile and usea downstream boundaryconditionof the slope of the energygrade line equal to 0.0004at Slarion 2500.The difference betweenstationsindicatesreachlengthsin feerin Figure5.11. Manning'sn valuesare0.06in rhelefi floodplain,0.035 in themainchannel, andvary from 0.05 to 0.06 in the righl noodplain.(All cross-sectjon dataare not shown.) 'fhe Solutitn. schematic layourof the river systemshownin Figure5.1I is enrered graphicallyby the user,and the cross,sectiongeometrydata are enteredand edited inleractively.The usermust thenenterdischargedataand boundaryconditionsbefore computingthe profile.The computedwater surfaceprofrle,alongwith thecritical depth line,areshownin Figure5.l2 for themain srem,with thedisrance scaleindicatingdistanceupstreamof Station2500.The water surfaceprofile is computedup to Station
C H A P T E R5 : C r a d u a l l yV a r i e dF l o w
187
Example5.5 ExistingConditions
€ o
LIJ
500 .---r--
WS 50 yr
1000 1500 2000 MainChannelDistance,tt - - {- - Crit 50 yr
-e-
2500
Ground
t-tGURE5.r2 Computedwatersurfaceprofilefor HEC'RAS,Example5.5.
3450at the Crosstown lunction.Then,theen€rgyequationis appliedacrossthejunction. first from Station3450 to Starion0 on the tributaryand thenfrom St ion 3450ro Station3500on themainstem.A lengthof 50 ft wasspecified acrossthejunction.Both frictionlossesandminor losses(contraction andexpansion) areincludedin theenergy profilesin the main slem calculation. Oncethejunctionhasbeencrossed, theseparate andtributarycanproceed.For a subcritical flow splitin thedownstream direction,the programrequiresa trial and-error distribution offlow untiltheenergies calculated from just dou nstream thetwo branches of thejunctionareequal.Forsupercritical andmixed flow cases,seethe HEC-RASmanual(U.S.Army Corpsof Engineers 1998). Figure5.13illustratcs themostupstream crosssectionat Station5000on themain stem.The compulationdetermines only onecriticaldepth,which occursin the main (WS)indicates channel. andthe \aatersurfaceelevation overbank flooding.The output datafor this crosssectionaregivenin Table5-3-The watersurfaceelevationis 81.44 ft (24.82m) andthe vel(rity is 2.67ft/s(0.81m/s).The flow is splitinto main channel andoverbank contributions by takingtheratioof theconveyances of each\ubsection to thetotalconveyance andmultiplyingtimesthetotaldischarge. Themainchannelvelocity is approximately four timesgreaterthanthe overbankvelocities. The geometric propeniesof eachsubsection aregivenin the table,leadingto a valueof the kinetic enersvcorrection coefficienta : 2.10.
188
C H \ P T E R 5 : C r J d u n l l ) V i r r e dF l o w
Example5.5 ExistingCondilions Upstreamboundaryof SkilletCreekStation50 RS = 5000 95
c
:9 dB0 LU
70 100
150
200
2so
300
350
400
Slation,ft - - i- - Crit 50 yr ____.r_ WS 50 yr
-----o- Ground _____._Bank stalion
FIGURE 5.I3 Upslream crosssectionandcomputed watersurface elevation from tlEC-RAS.E\ample5.-5.
TA I} I, tJ 5,] HEC-RAS cross-sectionoutput table for the upstream end of Skillet Creek Plan:Erist River:SkillelCreek Reach:Upper Riv Star5000 Pro6le:50,,-r E.G.Elev(ft) 81.67 Elemenl l,efi OB Channel Vel Head(ft) 0.13 Wt. n Val. 0.060 0.0t5 WS. EIev(ft) 8l .-1{ ReachLrn. (i) 150.00 5(y0.0u Crir W.S.(f1) 16.21 FlowArealsq fr) 105.98 162..16 ts.G.Slope(f/fi) 0.000656 Area(\q li) 105.98 l6t lE 1000.(n Flowtcfs) 2.| :8 I568.tJ Q Totallcf!) T o p W i d t h( f r ) 131.87 TopWidrh(fr) 81..H 10.00 VelTotal(lL\) ,1.31 ).61 Av8.Virl.(lts) l.l? Max Chl Dplh (ft) I l..l-l Hldr. Depth(ft) 2.51 9.06 Con\.Total(ct\) 78102.3 Conv.(cf\) 9121.1 6lll7., LenglbWtd. (ft) 198..17 WexedPer.(fl) 8:.06 J5.6t) '/0.W Min Ch El (ft) Shrar(lbl\q lil 0.10 0.31 Alpha :.10 Srream P o w e(rl b / f rs ) 0.t: l.1l FrcrnLoss(tr) 0. ]2 Cum Volumetacre-fr) 6.58 tj.l C & E Loss(ft) 0.00 Cum SA (acres) l.lt !.18
RighO t B 0.053 550.00 180.65 r80.65 r90.58 I t0.,{{ L05 1.64 71-12.3 I I0.6.1 0.07 0.07 6.8,1 3.58
C H A P r I R - 5 : C r a d u a l l yV r r i e d F l o w
lE9
waler surlace Encroached
Letl encroachment station
Rightencroachmenl stalion
FI(;LRE 5.t,1 analysis. Fleod\ray encroachment
5.9 FLOODWAYENCROACHMENTANAI}'SIS planning andin floodinsurance for land-use Floodway boundaries areestablished studiesbasedon the amountof encroachntenton the floodplainthat can be allowed without exceedingsome specifiedregulatory increasein water surfaceelevation. Floo<Jwayboundariesand floodway encroachmentare illustratedin Figure 5.14. In the encroachedareas.all tloodway conveyanceis assumedto be lost. ln the United States,the 100-yearpeak flood dischargeis establishedas the baseflood for floodwal analysis.and the increasein the natural $'ater surfaceelevationcausedby floodway encroachmentcannotexceed I ft. Floodway analysisproceedsby first running a water surfaceprofile for the base flood undernaturalconditions.Then, encroachmentsof varyingamountsare added, accordingto certaincriteria, so as not to exceed the targetwater surfaceelevation increase.The resultingboundariesfrom the floodway analysisusuallyare the result of severalilerationsand may have to be adjustedfor undulationsfrom crosssection to crosssectionand for unreasonablelocationswhen comparedto existingland use and topography. Five separatemethodscrn be selectedin HEC-RAS to determinefloodway boundaries.Theseare summarizedhere (Hoggan 1997; U.S. Army Corps of Engineers 1998): I - In encroachmentmethod l. the exact locations and elevationsof the encroachments are specifiedin each floodplain, as shown in Figure 5.14. 2. Encroachment method 2 specifies a ltxed top width of the floodway that can be specifiedseparatelyfor each cross section.Each encroachmentstationis set at half the specifiedwidth, left and right of the channelcenterline. 3- Method 3 calculatesencroachmentstationsfor a specifiedpercentof reduction in conveyanceof the natural profile for each cross section.The conveyance reductionis appliedequally on each side of the cross section,but the conrputed encroachments are not allowed to infringe on the main channel.
190
C H A p tL R 5 : G r a d u a l lVl a r i e dF l o \
- 1 .T h e i n t e n t o n f lcthodJis1()specifvalargetfortheallo$ablcincreascinthenat_ ural watcr surfaceclcvation.The rerulting gain in convclancc in rhc f.loodwar i s t a k e nu p e q u a l l yo n r h c l e f t a n d r i g h r f l o o d p l a i n sA. s s h o u . ni n F i g u r e5 . l j . the increasein conveyanceJK - ,(. + K&. where K, lind Ko arc the blockcd on the lefr and righr fltxrdplainsand K. : KR : JK/?. - llnfellqccs 5. N'Icthod5 is an oprirnizationIechniquethat automatiiallv iicratesup to 20 times to achievethe targetwater surfaceele\ ationsfor all cros.sscctions.Both a tarcet water surfaceelevltion increaseand a target energy grade line clcvation irc specified.In each iteration.the entire \vatersurfaceprolile is conrputedfor a set of encroachments, and then the encroachmentsarc adjustedwherethe targetwas violated for the next iteration. Methods4 and 5 are n]ost useful to esrablishan inilial solution for the flood\\'ay boundarics.[n fact, they can be run u,ith severaldifferent rargetIncreases in $'atersurfaceelevations.The final determinationof the lloodway boundary usually is madewith rnethodl. which definesthe specificencroachments at eachcrosssec_ tion and allows engineering.judgmenr to be appliedto the tlnal adjusrments.
s.l0 BRESSE SOLUTION Only u.ndervery special assumptionsis an analytical solution to the equatlon of graduallyvaried flow possible.This solution was first obtainedby Bresse for very wide rectangularchannels.The solution approachwas extendedby Bakhmeteff and finally fully developedby Chow (t959) into a method called the hvdraulic e.rpotrcnt nrctlnd.lt is a numericalmethod in rhe fornr derelopedby Chow, but very a t e d i o u ro n e t h a t n o l o n g e ri r i n u s u . To obtain the Bressesolution,the equationof graduallyvaried flow is written in the form: /
c \
s ""\ { l - 1 )
d)
Sn/
d":
o.8 I - - '
(s.3 3)
gA'
Now if Manning'sequationis writtenin rermsof conveyance, K = e/Srn,theratio of S"/Soin Equation5.33becomes(KnlK)2,in which K^ is the uniiormflow convel anceandK i\ theconve)ance conerpondingro thc l;cal deprhr.. Funhermore Q'lg in Equation5.31canbe replaced b1,AjlB. for thecriricalcbnditionof Froude numbersquared equalto l. With thesesubstitutions, Equation5.33becomes d_y
dr
'.1'(?)'] A:/B"
l - - A'/B
(5.34)
CHAPTER - 5 : G r r d u a l l l ' V a r i eFdl o w
19l
The hydraulicexponenlassunptionsare madeat this point. \*reassumethrl the t\r'o ratio terms in thc nunerator and denontinatoron the right hand sidc of Equation 5.3-l can bc set cqual ro rhe ratio ol either the nornral or the critical depth to the local dcpth taken to a po* cr dcsignatedM or N:
9l dr
(+)"1 ' (+)"
'.[,
( 53 s )
For a rectangularchannel.it is easily shorvnthat M = 3. Howcver. the value of N is a constantintegeronlv for a wide. reciangularchannelusing the Chezy equation \l ilh constantC; and it. too. has the valueof 3. Under theseassumptions,the equation of graduallyraried florv can be integratedexactly to give the solution
s o-, ' ' - " o I r ( ; ) ' ] r ( " . )
(5.36 )
- l, given by in uhich d is a function of -ry'r'n
otut -
t . fa : - r r t l 6'"1 i, ri ]
I I l rI I tttt"nf2, A , - tJ]t .,,
(5.37)
in which A is an arbitrar_v constant.The valueof the constantis immaterialbecause the function is evaluatedbetweentwo points locateda distance(x, , ,r,) apan, and so the constantA cancels.The Bressevaried flow function d is shown graphically in Figure 5.15 for subcriticaland supercriticalprofiles.ln the casesof Ml, M2, 52, and 53 profilcs. the approachto normal depth is asymptoticas shown. The determinationof the downstreamboundarycondition for a subcriticalprofile in a natural channel with no critical control section requires an asymptotic method, as discussedpreviously.The computationis staned further oownsrream than the reachof interestand severaldepthsare tried successivelyto find an asymptotic depth as the do\\'nstreanr boundary condition for the reach of interest. The Bressemethod for a very wide channelcan be usedto answerthe questionof how far downstreamto stan the process,at lerrt in an approximlte manner The length of an M2 profile from 75 percentof normaldepthdownstreamto 97 percentof normal depth upstreamcan be shown from the Bressesolution to be given by (Davidian 1984)
jlJ^ 0 . 5 7- 0 . 7 9 F '
(M2curve)
( 5 . 38)
in which L is the requiredtotal conrputationlength:S0is the bed slope;,vnis the normal depth: and F is the Froude number of the uniform flow. In a similar fashion. the leirgth of an Ml profile from 125 percentof normal deprh downstreamto 103 percenlof normal depth upstreamis given by ljo .,0
0.86 - 0.64Fr
(Ml curve)
(s.39)
r9l
C H A P T F R 5 : G r a d u a l l yV a r i e dF l o *
2.O 1.5
't.0 s
0.5 0.0 -o.5
a
-1.0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . O 1. 2 1 . 4 1. 6 1. 8 2 . 0 Y/Yo
F I G U R E5 . I 5 withconstant ChezyC. variedflow iunctiond for verywidechannels Bresse
slopeof 0.00t, normaldepthof 3.0m (9.8 For example,a channel* ith an average ft), anda Froudenumberof 0.25wouldhavean M2 profilelengthof approximately 1560m (5120ft) whiletheN'lI profilelengthwouldbe greater,with a valueof 2460 m (8070ft).
5.11 SPATIALLY VARIED FLOW Spatiallyvariedflow is a graduallyvariedflow in which the dischargevariesin the flow direction due to either a lateral inflow or a lateral outflow The goveming confusioncan center equationin thesetwo casesis different.and considerable aroundwhich equationis appropriatefor a given case. For the caseof lateralinflow,suchasa sidechannelspillway.themomentum because theenergylossesare not well known,while equationis moreappropriate the lateralinflow momentumflux can be specified.lf it is assumedthatthe momentum correctionfactorB is approximatelyunity and that the inflow entersin a direcequamomentum to themainchannelflou thegeneralunsteady tion perpendicular tion (derivedin Chapter7) canbe simplifiedto dv dd
s"-s.
2 q' V
I F ,
gA
(s.40)
C H A P T E R 5 : G r a d u a l l yV a r i e dF l o w
t9l
in r"hich Sr: friction slope - r,/yR and 4. : lateral inllow rate per unit ofchanncl length. In the caseo[ the side channel spillu'ay, g. is a constant,such that the channel dischargeQ(.x) = qrr, where r : 0 at the upstreamcnd of the channel. BecauseQ variesrvith,r, Equation5.+0 has to be solred numericallyby specifying a r alue of -r and iteratingon _vin a stepwisefashion along the channel. The variationof B with r alsocomplicatesthe determinationof the critical section. Critical depth can occur at an)'point along the channel,with subcriticalflow upstreamand supercriticalflow dou nstreamof the critical section.lf it is assumed that critical depth occurs when the Froude number l' = I and the numeratorof (5..10)is zero, so that d-y/dr * 0, then the location of the critical section can be shown to be given by (Henderson.1966)
8qi t
,P r r ' L s-"; ; ]
l'
(5..11)
in which r. = location of critical section; 4L = lateral inf'low per unit channel length; 8: channeltop width; S0 = bed slope: P - wetted perimeteriand C = Chezy resistancecocfficient.Equation 5.,11is solved simultaneouslywith the crirerion that the Froudenumber is ecual to unitv at the critical section:
,. , Q'6)a, eAi
,
(s.42)
where Q(.r) : qrr. lf r, > 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 ) .
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e33E Eii4 i4n-a €E36 brgS
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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 <el,'.a 3.0
4.0
5.0
7 1,,
r
Squareedge
ul,as"
6.0
7.O 8.0
9.0
Q lAdo 5
FIGURE 6.14 platecomrgatedmetalconduits(FedInlet controlcurves-{ircular or ellipticalstructural Administration 1985). eralHighway
CHApTER6: HydraulicStructures 229 cunes for differententranceconditionsconstructedfrom Equations6.l2 and 6.I3 for a circular or elliptical structuralplate comrgated metal conduit.The maxinrunr incrcasein dischargeat HWld = 3.0 due to bcvcling is about 20 perccnt in contparisonto a thin edgeprojectinginlet. The next lcvel of improvementis the side'taperedinlet shown in Figure 6.15. The side-taperedinlct has an enlargedface sectionwith a 4: I to 6;l side taper as a transitionto the entranceto thc barrcl of the culvert. called the t/rroal.The floor of the taperedsectionhas the sameslopeas the barrelof the culvert,and the hcight of the face shouldnot exceedl. I times the height of thc barrel.The headwaterheight on the throat is greaterthan on the face due to the slopc of the taperedinlet. However.an increasedheadon the throat can be achievedby rotatingthe culvert about its dow streamend such that there is a fall from the naturalstreambedto the invert of the face. Thc sidc-tapcrcdinlet is designedby first calculatingthe head on the
Face section
LrS
Symmetrical wingwallflare anglesfrom '1 to 5" 90'
to6:1) Plan
FtcuRti 6.15 Side taperedinlet, no fall (FederalHighway Administration 198-5).
230
C H A p T E R6 : H y d r a u l iSc t r u c l u r e s
throat,Hly,, for a gir en designdischarge,culven size,and allowableheadu,aterelevationusing inlct conl-rolnomographsor cquationsdcvelopedfor this case.The elevation of the throat thcn is sct as the hcadwatcrelevationninus the head on Lhe throat.This may requirethc inclusjon of some fail in the throat below the normal streambedelevation.Then inlet control equationsor nomographsfor face control are used to obtain the minimum u idth of the face for the given head on the face, assuminga maximum increasein elevationof I ft from the throatto the face in\en. The face width is roundedup slightly to be conservative,so that control will be at the throat and not the face.Once the facc width is fixed, the length of the side taper is calculatedfrom a chosentaper ratio bctween,l:l and 6:1 (longitudinal:lateral ), and the actualelevationof the face can be determinedfrom the slopeof the barrel. If it is more than I ft higher than the throat,the calculationmust be repeatedwith a rrewface elevation. The final level ofinlet inprorcment is shownin Figure 6.16, which depicts the slopc-taperedinlet. In this inlet improvement,the entire lall is concentratedfrom the face invcn clevationat the nalural streambedelevationto the throat inven ele-
Bevel (optional)
Symmetrical wingwallflare anglesfrom '1 5" to 90'
r'tGURE6.t6 Slope'tapered inlet(Federal High*ayAdministration 1985).
-T_
l"
C A p t E R 6 : H y d r a u l i cS t r u c l u r e s l l l
vation dctcrmined for throat control. Separateface control nomographsor equarions lor thc slope-taperedinlet are used to find the nininrunr face width. The lall slopeis selectedto be in the rangebetween2:l and 3: | (horizontalto vertical),and the side taper remains in thc range of .1:l to 6:l to determine the length of the trpered section.The amountoffall should bc in the rangcbetwcen 0.25dand 1.5d. box culverlto carrya designdischarge E x A ]l Pt, D 6 . 2 . Designa concrete of 500 cfs ( 1,1.2 m1/s)rr ith an allo*ableheadwatcr of 10.0ft (3.05m) abovethe inlet inven. m) Iongandhasa slopeof 0.02.The do\\nstreamchannelis Theculvertis l0O ft (91-.1 wirh a bottomqidth of 20 ft (6.1 m), sideslopesof 2: I, r : 0.020,and rrapezoidal slopei = 0.001frft. S(,,1ll/ror.Slan by choosinga 6 ft (1.8 rn) by 6 ft (i.8 m) box culvenwith a square (lC-l). so that.from edgcin a head*all.Assumeinletcontrolwith theinletsnhmerged (6.I I ). theheadfor thedesigndischarge is
a'
500r
x -16rx C; 2g(CuA,,)' 6,1.,1
-r.00 c)
Then,irom Table6 3, for rr/D: 0, itssume a valueof HW/d : 2.Ofot vhich C, : 9.51 : : afldHW 3/0.51I I L5 ft (3.51 m). Repealwith HW'ld= l l.716 = 1.95,andC, = 0.505.so thatHltl = l l .8 ft (-1.60 m). This is acceplable agreement, so lor inletcon trol,thehead.dly, of I1.8 ft (3.60m) exceeds theallowablehead$ater. The nextslep couldbe to increasethesizeof theculven,blt it wouldbe cheap€rto beveltheedges. With l'l, : 0.012, the ilerationon C, from Table6-2 producesC, : 0.-55and illtz = headwater. 9.9ft (3.0m). $ hich isjusl les\thantheallowable On the otherhand,Equations 6.12 and 6.13 are somewhatmore accuratefor inlet control.The value of :500(36 x 6"t) = 5.67,so Equation6.l3 is applicable. Table6-3 givesc Ql(.Adat1 = 0 . 0 3 1 ,a1n d l ' : 0 . 8 2f o r a . 1 5 ' b e v ealn da 9 0 ' h e a d w a lS l . u b s t i t u t i ni ngt oE q u a t i o n 6.13resullsin HIV : 10.9ft (1.32m). This is slightlygreaterthanthe allowableheadwater.For a grealerfactorof safety,increase theculvertsizeto ? ft (2.I m) by 6 ft ( l 8 and m) highbut still usebeveleded8es.ln thiscase,Equation6.l3 remainsapplicable Hly: 9.3 ft (2.8 m). This mightbe an acceptable design,but we shouldalsocheckfor outletcontrol.In fact,from Manning'sequation, the normaldepthin theculvenfor Q : 5 0 0c f s( 1 , 1 . m 2 r / s ) , , :l 0 . 0 1 2S, : 0 . 0 2 a, n db = 7 . 0f t ( 2 . 1m ) i s 2 . 9 7f t ( 0 . 9 0 5m ) . = 5.41fr (1.65m). Consequenrly, rhis is a and crjricaldeprh'-,.: 1600n)1132.21tt1 steepslopcand inletcontrolis likcly to governunlessthereis a high tailwater from Manning'sequaThe tailwalerfor 0 : 500cfs ( 14.2mr/s)canbe calculated tion with rl : 0.02 andS = 0.001for the givendimensions of the downstream trapedepthof 3.83ft ( l.l7 m) abovetheoutletinven. zoidalchannel.The resultis a tailwater (y. + d)/2 : \5.11+ 6)12: 5.'1fr ( 1.7m). *hich is grearerthanthe tailwaCalculate ter deprhof 3.81 fr ( l.l7 m), so use5.7 ft ( 1.7m) in the fr-rllflow equation. Subsritut ing intoEquation6.l5 \\'iththefrictionlosstermevalualed by Equation6.17andK" = 0.2 for bevelededgesfromTable6'5, we have
Hw-s1 002,,00,(' o,. = 3 . 8f t ( 1 . 2m ) Clearlythe inlet controlheadof 9.3 fl (2.8m) is higher,andit \\'ill control.
232
CHAp'rER 6 : H y d r a u l iSc r r u c l u r e s
Whilethisis a perfccrlyacceprable design.ir is \\onhwhilclo explorerheeffccrof utilizingsidc-rapered andslopetapcredinlerson Lheonginrl 6 fl bj 6 fi box culvert designusingthe FHWA prograrnH\'8 (Federalt.lighu,ay Adminisrrarion 1996)..fhe progranrHYS allowsinteractive entryof culvenand inletdataanddo$,nstream chan, nel characterislics. It thencalculates thetail$'atermtingcurv-e anddcvelops a full per_ fomlancecune for lhe selected culven.It calculalescomplere*.atersurfaceprofiles \\Jrenrequirc'd andprovides graphicat screcnresultsandprinledourputlablesandfiles. To designa side-tapered inlel. as\umea larrral erprnrion of,t:l and specify bevclededges. Thenchoosea facewidrhlargerthanlheculvcn\ridlh.andtheprograrn computes thefacecontrolcurve aswell asthethroalcontrolperformance curve.Adjust the facewidthuntil thefacecontrolcune is belo$ the throatcontrolcurveso thatthe throatis the control,at leastfor p greaterthanor equalro the designdischarge. The pertonrancecurvefor the 6 fl by 6 ft culven rrirh sidc-tapered intethavinga face \\ idth,B/, of 9 fr (2.7m) is shownin Figure6. l7 in comparison wirhrheperfonnance curvesfor a squarcedgeandbeveled cdgeon the 6 ft by 6 ft culvcn.At thedesigndischargeof500cfsil,l.2rnr/s),rheheadfortheside,raperedinlet(SDT)is9.llfl(2.7g m), which is a 26 percentreducrionfrom rhe headof 12.39fr (,'1.7g m) for a squareedgeinlet.AIsoshownjn Figure6.17is theperformance cun.efor a slope_tapered rnlet ( S L T ) w i t h a f a lo l f 2 f t ( 0 . 6 1m ) , a f a l l s l o p eo f l : l . a n da f a c eq , i d r h of l2.0fr(3.66 m). The headat 500cfs ( 14.2mr/s)is 7.23fr (2.20m), or a .l2 percentreduclionfrom the headfor a square-edge inlet.It is apparent tharrhecutverrbanelcouldbe reduced in sizefurtherif a sidetapered or slopetaperedinlet wereused. Justbelowthc perfomrance cune for thc slope-tapcred inletin Figure6.l7 is the outletcontrolperformance curvefor the slope-tapered inlet design.We seethrt the out_
l 14
12 q
l
ConcreteBox Culvert 6 f t x 6 f t ; S = 0 . 0 2 ;L = 3 0 0 t t Allowable headwater
10
T
1:1 bevel
r ^
4 2 0
r
z
a -1
.?-,2
'1"
) '/'r' SDT-4:1,8/=9tl
S L T - 4 : ' 12,: 1 ,B / = 1 2 f t 100
,27
Square
200
300 400 Discharge,O, cfs
4 Outletcontrol Design discharge
t
500
FIGURE 6.I7 HY8 resultsshowingeffectof intproved inletson culren perfonnance curves.
600
C H A p t E R 6 : H y d r a u l i cS t r u c l u r c s 2 3 3 let control curve intersectsthe slope tapcred inler curvr'ar a dischargeslightly grearer lhan 600 cfs ( l7 Inr/s).For all di\chargesgrell!'r lhan rhe intcrscctionpoint. lhe cut\ crl is in outlel control.and the headrises.rra grcalerrate rhanfor inlet control.One design philosophyis to use a tapcredinlel * ilh a fall such that rhc intersectjonwith the oullel control curve occursexactly at the allorrableheadof l0 ft (3.05 m) qhere e is greatcr than the design value.This fully utilizes the inlet capacityof the cul\cn at rhe design head and providesa factor of saferl in culvert capacitt'.Alternatively.the culvert wilh inrprored inlet can be designed!\ith a fall such lhat rhc inlet controt curve rnlcrsecls exactly lhc poinl corresponding1()lhe designdischargeand allowableheadwater.This often is acceptablc.if some additionalheadwatercan be toleraledor if roado\ crtopprnB is allowed.The final possibledesign point is the inler\!-ctionof the inlet control curve u,ith thc designdischargeal the lo\\csl possiblehcad, $hich is limiled b1,the natural wlter surtaceeleYationin the streamupstreamof the culvc'rt.The final choiceof desisn pornt nrus{bc milde by thc engineerbascd on local conditionsand judgment
6.6 BRIDGES Theflowconstriction caused by bridgeopenings andbridgepiersgivesriseto both contraction ande:pansionenergylosses,wilh a resultingrisein watersurfaceelevationupstream of thebridgein conrparison to thatwhichwouldoccurwithoutthe bridge.This excesswatcrsurfaceelevationin the bridgcapproach crosssecrron, referredto asbechrater,is shownin Figure6. l8 ash'f.TypeI flow shownin Figure 6.18is definedfor subcriticalflow throughoutthe approach, bridge,and exit crosssections. In TypeII flow,the constrictionis so serereas lo producechoking and the occurrence of criticaldeprhin the bridgeopening.In TypeIIA flow, the llow dcplhdoesnot passthroughthe downstream criticaldeprh,so a hydraulic in the cascof TypeIIB flow, the flow downstream Jurnpdoesnot occur.However, of thebridgebeconres junrpformsirnmediatcly supercritical anda hydraulic tlownstreamof the bridge.Finally,Type III tlow. which is not shownin Figure6.18. occurswhen an approachsupercriticalflow remainssupercritical throughthe bridgeopening.In TypeI flow, rhe bridgebackwateris rhe resultof headlosses, includingthe approachfrictionloss,contractionloss.and expansion loss.In the caseof TypeII flow, the chokedcondition,additionalbackwater is causedby the upstream dcpthnecessary to increase the available specificenergyto rheminimum valuein thebridgeopening. Severaldiffercntmethodsare availablefor derermining the bridgebackwater. especially for TypeI flow,whichis rhemostcommon.Thesemethods arediscussed individuallyhereandincludeempirical,monentum.andenergyapproaches to the problem. HEC-2 and HEC-RAS In thenornrrdbridgeroutinein HEC-2(U.S.Army Corpsof Engineers l99l ) or the energymethodin HEC-RAS(U.S.Army Corpsof Engineers 1998).thegradually
234
C H A P T E R6 : H \ d r r u l i c S l r u c l u r e s
w.s.
??
fz > fzc Normalwalersudace
t tc
(a)TypeI Flow(subcrilrcal)
Criiicaldepth
I
ir.vu
16*
'
r-"
-- - --,,'
(b)TypellA Flow(passesthroughcritical)
W.S, jump Hydraulic Crilicaldepth Normalwatersurlace fo > fzc
(c)TypellB Flow(passesthroughcritical) r-tGURE 6.18 Flow througha bridgeopeninS(Bradley1978).
variedflow profile calculationsare continuedthroughthe bridee using the standard step method, as though the bridge opening were just anorherriver cross section. This nrethodusually is used when thereare no piers or rhe head loss causedby the piers is very small. The cross sectionsare locatedas shoun in Figure 6.19, numberedfor consistencyu ith other methodspresentedhere.Crosssections3 and 2 are locatedirnmediatelydownstreatnand upstreamof the bridge opening, respectively, at a distanceof only a few feet from the face of the bridge.The approachsection I in Figure 6.19 is in the region ofparallel flow beforeflow conrractionoccurs,while the exit section4 is locatedat a point where the flow hasreexpanded.Traditionally, the Corps of Engineershas rccommendedthat the lengrhof rhe contractionreach from crosssection I to 2 be takenas I times the averagclength of the side obstruction causedby the embankments(CR = 1). In addirion.the expansionreachlength from crosssection3 to.1 has beenrecornmended in the past to be 4 times the average length of the side obstruction(ER = 4). However.the Corps of Engineerscon-
C H,\PTER 6: H\draulic Structures
215
-o Contraction reach
Typicalllow ,, Iranstuon / pattern ,' // /'./
ldealizedflowtransilion pallen for one-dimensional modeling Expansionreach
+ I
-@
F I G U R E6 . I 9 Crosssectionlocations at a bridge(U.S.ArnryCorpsof Engineersl99g).
ducted a numerical study of the contractionand expansion reach lengrhsusing a two-dimensional numerical modcl (U.S. Army Corps of Engineers l99g). The resultsshowed that rhe Iengrhsrequiredfor expansionof the flow dependon rhe geontetriccontractionratio, the channelslope, and the ratio of overbankto main channelvaluesof Manning s rr, while contractionreachlengthsdependonly on the latter two variablcs.ln gcneral,contraclionreachlengthswere in ihe rangeof I to 2 times the averageobstrucrionlcngth.and expansionreach lengthsfcll in the range of I to 2.5 times the averageobsrructionlength. Best fits of the numencal results were obtained.but they are specificto the valuesof the independentvariablestested in the nunerical rnodel.which includedbritJpeopening lengrhsfrom 100 ro 500 ft (10.5 to I-52m), a floodplain width of 1000fr (305 nr). overbank Manning's n values lrom 0.0-1to 0.16, main channclManning'srr of 0.0.1.dischargesfron5,000 to 10.000 cfs (1.12 to 8-50nrr/s), and bed slopes from 0.00019 to 0.0019 (see U.S. Anry Corps of Engineers1998). Twc addirionalcrosssectionsare createdby HEC,RAS insidethe bridge open_ ing. Only the effectivellow areafrom I to C is usedin the cross-sectron DroDenles of cross section 3 as rvell as cross scction2. Standardstep flow profilei are com_ putcd through this total of six crosssectionswith friction lossesand expanslonand contractionlossescomputcdin the usual way.
236
CH^I,1L:R6: HydraulicSlructures
In the spccirilbridge nrethodin HEC-2. the program computcsa momentum balancebetwcenan upslreamcrosssectionand a sectionjust insidc thc bridge section and bctwccn the bridge sectionand a dorvnstreantscction.to determineif the flow is Type I or Type ll. If the flow is T1'peI. thcn the enlpiricalYarnell cquation (Hentlerson 1966) is used to detcrrlline thc changc in watcr surlace elevation' IH.
thc bridge oPening ,. thror.rgh
'{4n=r,,rl1r"+sFi - 0.6)(A + .l 5 A : )
r6.l9)
: pier in which 1., = downstrcanrdcpth: l'r : downstrcam Froude nunrber,K" lo 1 25 for a nose and tail coefllcicnt varying from 0.9 lor a picr with scmicircular = area due to obstluctcd arca ratio pier with square nose and tail; and A, sets the depth equal piers/totalunobstructcdarea.If the flow is Type II. then llEC-2 to critical depth in the bridge and deternincs the upstreamand downstrearndepths from a momentumbalance(Eichert and Peters 1970) In I]EC-RAS. the Yarnell method or the nlonel)tun nlcthodcan be chosenas the dcsiredbridge hydraulicsanalysismethod for Type I flow ln the momentum nlethod,the mon]entumcquationis written in threesteps:( I ) from just upstrearnof thc bridgeto a pointjust insidethe bridge.(2) throughthe bridgeopeningitself' and (3) from just inside the bridge cxit to a point just downstreamof the bridge. This provitlcsa solution for thc depth at the t\!o crosssectionsinsidethe bridge and the cross-sectionimmediatelyupstreamol the bridgc. The pier drag force is included i n s t e p( 1 ) . Dctection of Type II flow and calculation of the approachdepfh can also be accomplishcdusing a contbinationof the monlen(umand energyapproaches.First' the monrcntumcquationis written hctweenthc bridge sectionand do$'nstrcanlsection 4, with critical dcpth assumedin the bridge scction.Thc rcsult.which is given in tcrms ofthe width ratio r = b,lht thJl causeschoking. is (lJenderson1966)
'
(2 + r i r)rFl (l + lFl)r
(6.20)
- downin which b, = width of bridge opening:b, : exit channelwidth; and F. strcamvalue of the Froude number.The approachdepth is obtainedby writing the energyequationbetwecnthe approachsection I and the critical sectioninside the bridge with an appropliatcheadloss coefllcient.
H D S -I The FcdcralHiShwayAdnlinistrationde\ eloped an energymethodof bridge analys i s p u b l i s h e di n t h e t { y d r a u l i cD e s i g nS e r i e s( H D S - l : B r a d l e y1 9 7 8 ) l t w a s u s e d prior to the developmentof WSPRO. Refening to Figure 6 20' the cnergy equatron is applicd bet$'ecnsectionsI and 4 to obtain
o.V rr t ; -r.,-*'j'-,/,, S , . / , . ,r ' - ' l :
( 6 . 2| )
W.S.alongbank Sectlon
g " ' r { |r ? YlmatWS.
s-/. ^
Fto**
\'2 .,.1 v4 \ \ 2at
a i\ a"l
"o I
th
ni
Actuatw.S. onq
(a) Profileon StreamCenterline
Section
Q6
Qa
Section (d) Planal bridge
}.IGURE6.20 Normalcrossing: (Bradley Wingwall abutments 1978).
231
2 - 1 8 C H , r l t r - r 6 : l l ] d r x u l i cS l R r c l u r e s With respcctlo tl)c in which lr, is thc rotal cnergy loss bet$'ccllsectionsI and l ir, is just l-'ll'rncedby of portion nnrm"l *at., surface.thc unifornt-flow rcsislanc€ 21 6 ' we have Equation frorn the vcrtical fall in thc channclbotton so that a.Vi
tlLYi _
.
)-
),,
t
L
(6.22)
and can be in which h,, is tic additionalhead loss due to the bridge constriction expresscdin temrr ol'a tninor loss coclllcicnt. r(' delined b1
" '
d1V:1
h,
Y*
(6.23 )
t8
the flow area where {,2 - the mean vclocity in the contractedsection based on bclowthe.normalwalcrsurlaceinclusiveofthcareaoccupiedbybridgcpiers'Now for fi,,' Equaif (,r', 1o) is replacedby ft'f and [lquation6 23 is used to substitule becomes of continuity, the aid tion 6.22. with
, , i = r . + - " [ ( */4"\'lI": )' \A' ,/l2s
(6.r{)
the difference The secondterm on the righr hand side of Equation6 24 represents mucb smaller is generally term in velocity heads betweensectionsI and 4' This secondtcrm with the iteration by than the irrst term, anclEquation6.24 is solved water area gross A,' is the that note equal to zero in the first trial. tt is importantto i[thecontractedsectionmeasuredbelownorntalstage,andV,'isarefcrence = 1 0, and it is assumedthat ar - cr1' velocity equal to QlA"..The value ofa, K' To'calculate tn. Lo.k*ot"t, the vaiue of the minor head loss coefficient' field and laboratory from developed must be detcrmined.Valuesof K have been studies.and K is considcredto consistof additivecomponents
K*=Ktral(r"+aK.+aK,
(6.2s)
JKn = prercoefficienrlAK. = eccentncrty coefficient; in which K, : contraction coefficientFor simplicity'we consideronly a coefficientiandAK, - skewness or skewnesseffects-The valuesof K, no eccentricity normalbridge crossingwith Thecontraction 6 22' respectively 21 and 6 Figures from andAK- can"beobtairied dischargeratio openingthe bridge Mo, on depends lo"fn.i".nt K, in Figure 6.21 elevationin water surface normal the for 6'20 Figure givenas Q/Q and d-efinedin curvein lower the length' ft in 200 exceeding abutments itr" uppt*itt t .tion. For 22' the pier 6 ln Figure type' abutment of regardless figuie O.Zf is recommended to the co".fficienti. given asa functionof "/,the ratio of areaobstmctedby the Piers section 2 The al water surface normal the below waterway grossareaof ihe bridge value for the it is conected then and of "/, function as a hrst ialue of AK is determined of Mn to give \K, = 5Yo. USGSWidth ContractionMethod flow The USGS has an interestin bridgesfrom the viewpoint of using them as it a result' As stages downstream and measuringdevicesby measuringupsfeam
C H { P T I R 6 . H ) d r J U l r cS t r u c r L r r e \ 239
90'
45"
l nrfrm I l
2.4
2.O v
L
,.,1"*n.,lN 90' \ \.-\ /
to 200n
I
O
.,, 0.8
t \
All spillthrough __ or 45" and 60" WW abutmenlsover - _ 200 tt in lenglh
t
tTrilfnililf
l
90'Wingwall 45'Wingwall
30' \ J
\i: Spillthrough
\:
\r
0.4
S< \
0 0.1
o.2
0.3
o.4
0.5 Mo
0.6
0.7
0.8
0.9
1.0
I'IGURE 6.2I Backwatercoefftcient basecurves-subcritical flow (Bradlev1978).
der eloped an energy approach (Kindsvater and Carter 1955; Kindsvater,Carter, and Tracy 1958;Matthai l9?6) that utilizesa bridge dischargecoeflicient,C. Firsr, with referenceto the crosssectionlocationsin Figure 6.20, the energyequationis written betweencrosssections1 and 3, but with section 3 insidethe bridge,to obtain
o,Vt, 2r''
"'
a , V1 1,
I h\ | h'
h'
(6.26)
in u,hich h, = stage(water surfaceelevation)in the approachsectionl; ftr : stage in the bridge section3; l" : entrancehead lossl and hr: friction head loss from sectionsI to 3. If the entranceloss is expressedin termi of a minor Iosscoefficient as h" = K"(\)212g and if continuity for the bridge sectionis writrenas
Q = C,b)"3v3 16.21 ) : : rvhich in b bridgeopeninglength,and C. the contracrion coefficient, thcn Equation6.26canbe solvedfor V, andexpressed in termsof O usingEquation6.27 to glve Q:
CAI
[4;;;fl
(6.28)
In Equation6.28,thebridgedischarge coefficicnr,C, is definedby
C" \/"t
+ K-
(6.29)
!! dlh ol prernormal!o llow. tl
f , w " - t h - w oA on aaseo n n ln fl (n"z UVU >, Yt:
Helghloi p er erposed
LenehD)
tj
tN wph.2 =Tora protecred a r e ao l p r e r sn o r m a l t o Ilow.fl2
t-\
NormalCrossin9
Gross waler cross sectron in conslriclionbased on n o r m a l w a l esr u r l a c e ( u s ep r o j e c t e bd r i d g e € n g l hn o r m a l l ol l o w lor skewcrossrngs)
(A.2 83sed on Lengh D cos d,)
4L
Note: Swaybracingshould be rrcluded in wdth ol pile benls
JKp= AKo .o2
.04
.06
.10
.08
.12
J (a)
FIGURT6.22 for piers(Bradley1978). coefficient backwater lncremental
240
.14
.t6
.18
C H A p T E R6 : H \ d r a u l i cS t n r c t u r e s 2 4 1 andAl = lr, i , u h i l e . , 1 r: b r r . A s a n e x a m p l e v. r l u e so f t h e b d c l c ed i s c h a r s e coclficicnt are given in Figurc 6.2.1lor a Type I bridge consistingoirectangul-ar abutnrcntswith or wirhour * ingrvals. 'fhe basecoefficient c' is cleterminedfronr the upper graph and correctedfor the Froudc number and comer roundjng bv mul_ tiplying C' by lo and k, to get C. Curvesof rhis type have beendevelopeJfor rhree additionalbridge types,discussedin thc scctionon WSpRO. (For the conplete set o f c u r v e s s. e eM a t t h a i 1 9 7 6 :F r e n c h1 9 8 5 ;o r U . S .A r m y C o r p so f E n g i n e e r s1 9 9 g . The purposehcre is only ro show the conncctionbetwecn rhe USGS method and WSPRO.) Each hridge tvpc has jts own sct of correcrionfactors.In addirion. some corrcction factors are common to a)l four bridge types. such as thc concction for piers or piles as shown in Figure 6.2.1,and thcseare mulripliedrintesrhe basecoef_ ficient. The pile and pier adjustmenrfactors,Jepc,nd on thr. rrrio/ = ArlA,, where A, is the submergcdareaof the picrs projcctedonto the planeof crosssection3 and ,4r is the gross arca of cross scction -1,L/b : rxio of iburnrenr\\idrh in the flow directionto bridge opcnina lcngth, and |n : channelcontracrionratio. The value of the basedischargecoefficient c' is a function of the channelcontractionratio. ,r. which is defined as the obstructeddischargein the approacbchannelcross sectlon divided by the rotal discharge.and L/b.ln terms of HDS-1. nr = (l - M,) where M, is the unobstructeddischargeratio, defined as in HDS-l exceptthat it is evalu_ ated at the approachwater surfaceelevation.i in\tead of lt the normal water surr. face elcvation.To delcrminethe backwater.Equation6.2g can be solved for Aft. but this is only the drop in tvater surface from the approach to the bridge section. WSPRO, to be describednext, utilizesthis ponion of the energybalanceinvolving C but also the encrgy equation written from sections3 to 4.
WSPRO N{odel The USGS in cooperationwith FHWA devclopeda compurerprogram that com_ bines step backwater analysis with bridge backwatercalculations.The program, namcdWSPRO (Shcarmaner al. 1986),is recommendedby FHWA. It is c-ontiined in the,HYDRAIN suite of programs (Federal Highway Adminisrration 1996). WSPRO allows for prcssure flow through the bridge, embankmentovertopping, and flow rhrough mulriple bridge openings including culuerrs. Tne Liidg-e hydraulicsrely on the energy principle but have an improred techniquefor detei_ mjning approachflow lengrhs and an explicit considerationof an expansionloss coefficient.The flow lengrh improvcmentwas found necessarywhen the approach flow occurson very wide. heavily vegetatedfloodpiains. The crosssectionsnecessaryfor the WSpRO energyrnall,sisrre shown in Figure 6.25 for a single-openingbridge with or without spur dikes having a bridge openinglength ofb. Cross sectionsl, 3, and 4 are requiredfor a Type I flow anal-y_ sis, and they are referred to as the approach section, bridge sectiin, and,exit secIr.rn, respectively.In addition, cross section 3F, called the full vallet. section, is neededfor the water surfaceprofile cornputationwithout the prcsenceof the bridge contraction.Cross section 2 is uscd as a control point in Type Il flow but requiris no input data.Two more crosssectionsmust be defincd if spurdikes and a roadway
---; €-
withoutwinswalrs tro", (L_*,tn*,ng:L l 1.00 0.90
N\ t \ - \ \
0.80 0.70
-
:
l br
:
2.00 or greater :----------
\
S l a n d a ' dC o n d r r o n s\
1o = o ob . ! ! 1 \= o F=0.5
e-100
6-9'
J=O
t.....*-
0 . 8 00.60 0.40 o.20 0
\
__=0
0.60 0
' " " 1
= - 1.00
0.10 0.20 0.30 0.40 o.5o 0.60 0.70 o.8o 0.90 L00 ChannelContraction Ratio(m) (a) Base Coetficjentof Discharge
1. 1 0 ..* 1 0o 0.90 0
0.r0
0.20 0.30 o.4o o.5o 0.60 0.70 0.80 FroudeNumber(F)= Q/Ae{ifs NumberAdjustmenl Factor {b)Froude
1.20 1. 1 5
zA
\:' | .10 1.05
I
r
1.00 0
?
z
7
0.30 0.20
0.02 0.04 0.06 0.08 0.10 0.12 0.1 Rarioof CornerFiounding to Widthof Opening(f) (c)CornerRounding Adjustment Factor
FIGURE6.23 Bridgedischarge coefficient for TypeI bridgeopenings (Marthai1976). 242
m = 0.80 0.50 0.40
o o
N
N
o o
\\\
fi\
o E
I
- g
ii :
o
o i i
I\
R E E
@ 6
;;;;;; E
ilt l R 8 g J Q
I
IL L
I
c (
6 - 9
I
o F
-
!
< ^
I
t
( !
a
9
l \
^ . c
<
o
o -i
Ir
N
N
o
F'
r-i
L L
o
)
-i
!,
(r o =/rollrt I r-oci
io - r o tl
A
o o o o
,I
E
-t-
r
l|
fs-r nn rz1
S;
N\ NN '/,4
Ei
E
|
i---i
.E:
s
l
,i-
NNN,,
<-{
NJ UN T4
el
3l
t
l
I
I l;
tl
243
C H \ P T I R 6 . H _ \ d r J U l iSc l n r ( t r r r c \
4: Exit |z/
I
I b
b
(a)WithoutSpur Dikes
3: Bridgeopening
L*-- ! O
D
(b)WilhSpur Dikes FIGURE 6.25 WSPROcross-section locations fo. streamcrossing with a singlewatenlayopening(Shear_ manet al. 1986).
profile are specified.The approachsectionis locateda distanceof b upstreamof the upstreamface of the bridge,while the exit sectionis a distanceof b downstreamof the downstream face. The basic methodology for a single-openingbridge wirh no spur dikes and free-surfaceflow consistsof writing the energyequation,first betweencross sections I and 3 and then betweencrosssections3 and 4, as defined in Figure 6.25: ht = ht + h4 t h3:
honI
h11-21! h1p-t1
h u a ,+ h t 1 4 ) + h " -
h"r h,l
(6.30) (6.3r )
in which l, : the water surfaceelevationat cross sectionii h,.i - velocity head at : the friction headlossbctweencrosssectionsi andj: h,,,= normal. sectioni; h/ii_./)
Cfi,rPr rtr 6r Hydraulic Strucrures
245
,I'A B I-E 6.6
in \\'St)RO.,( = conYeyance and C = bridgedischarge Flnergylosscxprcssions coefficicnt Loss bet\retn seclionnumbers
Tl p€ of toss
I 2 (no spurdikes) I I 3 -1 3 -1
Friction ljriction Fricrion Lxpansion
S.!r..r
l:nergJlossequalion h t ' \ = L , , Q ) I Kt K t
ht 1 \t = I. .Q:tKi h] ,)': bQll\K, K4^) 1 , "- p t t p s A l t ' ] f i . .rj1 2p,(A./Ar) * .r. (Ar/Ar):l rhcrc a, = l/Crind Pr = l/C
Dala ift'n Shcr.nrln er 3l i1986).
water surfaceelevational crossscctionj: and r,, : exit headlossbet\\'eencrosssections J and '1.Equations6.30 and 6.3 | arc solvedby assuminginitiai trial elevations which are used to computc thc right hand sidcsof thc 1\r'oequations for lr, and 12.,, to obtain updatedvalueson the Ieft. Iterationis continueduntilthe changesin hr and h, are small. Energy loss expressionsnecdedin Equations6.-30and 6.ll are surnmarizcdin Table 6-6. F'iction loss calculations utilize the gconrctric mean convcyance betweenany two cross sections,and the flow lenSthfronr section I to 2 is the average length.L,,, as determinedby thc nrethoddevelopedby Schneideret al. (1977) and shown in Figure 6.26. The approachflow is dividcd into 20 streamtubesof equal conveyance,and the flow distanceof cach slrcamlubclronr the approachsection 10 the bridge is areraged for the calculationof the approachfriction loss.The length L,,0,is thc distancc from the bridge opening to lhe approachseclion where the flow is nearly one dinrcnsional.detcrnrinedas function of the geonretriccontraction ratio basedon potentialfltxv thcory (Schncideret al. 1977).Thc value of Loo,is equal to the bridge openinglength.b, at a geonrclriccontractionratio, lrlB = 0.12. lt L,,ptis Icssthan b as in Figure 6.26a.then the parallelstraight-linelengths of eachstreamlinefronr thc approachsectionto the dashedline at L.,",plus the converging straight-linelcngths to thc bridge opening are averagedto obtain L"". ln Figure 6.26b, L.fl is greater than b for a very severecontraction.In this case,a parabolaapproximating an r'quipotentialline is constructedfrom the edge of the water at the upstreamdistanceof b. Then, the parallelstreamlinesare extendedto intersectwith the parabolabcforc bcing turned to thc bridge opening, if the interscction point is downstreamof the dashedline locateda distanceof Loo,from the bridge opening.The approachhead loss also dependson the geometric meln conveyancesquaredfrom I to 2, dellned as thc product of (, and K,, rr hcrc Kl is the conveyalrceof the approach section.and K, is the minimum of the conveyanceat section3 (trr) or the conveyanceK,, defined as the conveyanceof the segmentof apprcachflow that can flow through the bridge opcning with no contraction. 'fhe friction loss through the bridge is basedon the conveyanceK, as shown in Table 6-6. The lcngth of the expansionreachused in the friction loss calculationis one bridge opcning lcngth. b. and so the bridge exit cross sectionlocation should
:16
C H A P T E Rb : H l J t r u l t cS t r u c l u r e s
,-
Approachseclion
a
f--
"u
f__
",
Cenlerline \
o (! t!
i
,
.on"o,*",".-/
f-
.*--l
(a) For Relatively Low Degreesof Contraction FIGURE 6.26 (Shearman WSPROdefinitionsketches of assumed streamlines et al. 1986).
not be changed. A separateexpansionhead loss computation is based on the approximate solution of the momentum,energy.and continuity cquationsfbr an abrupt expansion given by Henderson(1966) and discussed in Chaprer 2. It dependson the coefficientof dischargefor the bridge as developedby Matthai ( 1976).By comparingEquations6.28 and 6.30, it can be shown thatq1 = l/Cr.The USGS width contractionmethod is used to find the bridge dischargecoefllcient, which then appearsin the expansionhcad loss expression. Pressureflow through the bridge opening is assumedto occur when the depth just upstreamof the bridge openingexceedsl.l times the opening hydraulicdcpth. The flow then is calculatedas orifice flow with the dischargeproportionalto the
C H A p T E R 6 : H y d r a u l i cS t r u c t u r e s 2 1 j
f
o (5 c o
6 ul
(b) For Relatively High Degreesof Contraction FIGIIRE 6.26 (continued)
squareroot of the effectivehead.Unsubnrergedorifice flow is illustratedin Figure 6.27 with the orifice discharge,Q,, computed by
q"
c o e , . , , l z e 1-r ,z z r t , , . 1
(6.32)
in which A,"", = net open area in the bridge opening,and Z = hydraulic depth = ,4.,,"/b. Submergedorifice flow is treated similarly, with the head redefined as showr in Figure 6.28, and given by
Q.: CoA."",\/2sLh
(6.33 )
Totalenergyline
Flow ------>
Datum
FIGURE 6.27 et al. oriltceflow conlprlations(Shearman WSPROdefinilionsketchfor unsubmerged 1986).
Flow
Datum
FIGUR.E 6.2E (Shearman el al. 1986) orificeflow computations WSPROdcftnitionsketchfor submerged 2,18
CrAprER
6: Hydraulic Structures
2.19
TA BI,I' 6.7
conditions Rridgeflow classificalionaccordingto submergence Flo\\ throughbridg€op€ningonl,
Flo$ throughbridgt openingand oter road
Classl. Freesurfaceflos Class2. Orifice flo\r' orificeflou Class3. Submerged
Class-{. Fr!'erurflcc flo\r Class5. Orificeio\l Cl.r\!i6. S brnert!'doritice io$
TA tll. E 6'8
Bridge tlpe classi{ication T}'pe
I 1 l -l
Embanknlenls
Abutments
Wingwalls
\tflical SbpinS Sloprng Sloping
Ve(rcal Venical Sloping Venical
With or wrrhoul None None
coeflicientis 0.-5over a wide rangeof orifice flow, the discharge In unsubmerged orificecase. submerged YjZ, whlleit is equalto 0.8 for the the bridgeopeningsimultaneously flow through WSPROalso can consider rvithdischarge a weir discharge computed as which is overflow, with embankment classification of (see This leads to 6. l3). power Figure proportional to headto the] ln [reein Tablc 6-7. (Shcarman 1986), as shown et al. ilow classesI through6 surfaceflow.thereis no contactbelweenthc watersurfaceandthe low steelelcvawhile in girderis submerged, tion of the bridge.In orificeflow.only the upstream girders are submerged. downstream and orificeflow boththe upstrcam submerged in A totalof four differentbridgetypescanbe treatedby WSPROasdescribcd ( et al 1986) given are by Shearman Table6-8.Furtherdetails of WSPROresultswith severalothermodelsand field measureComparisons mentsof watersurfaceprofilesthroughseveralbridgesare given in Figure6.29 (Shearman HEC-2(N)andHEC-2(S)are the normaland et al. 1986).The methods an olderUSGSmcthod.\\'SPROcompares while E.131 is specialbridgeroutines. profiles.Maxinum errors are 0 3 ft tbr water surface very well with the observed ft and 0.6 ft for the higherand lowerdisand 0.4 Buckhomand CypressCreek, from WSPROand HEC-2 are The results on PoleyCreek. respectively, charges, well asfor the profilesupstream profile as Creek Cypress for thc entire comparable on PoleyCreek The water the low discharge Creek and of thebridgefor Buckhorn verl *'ell by HEC-2 reproduced is not bridge. however. surfaceprofile throughthe by WSPRO, values computed (1997) comparcd backwater Kaatzand James values for l3 backu ater with measured nethod Bradley HEC-Z,andthe modified
250
C H A p r l : R 6 i H l d r a u l i cS ( r u c t u r e s
) a
TL c
.9 UJ
U) (D
3
-
Observed
---+-- wsPRo - -o--.{.--{--
0
1s00
4500 3000 RiverDistance.tl
l]EC-2 (N) H E c - 2( S ) E431 6000
7500
(a) BuckhornCreek,near Shiloh,Alabama I'IGURE 6.29 Comparison of watersurfaceprofiles(Shearman et al. 1986).
flood eventsat nine bridgesin Louisiana,Alabama,and N,lississippi. The modified Bradley method used essentially$,as the HDS-l method given in this chapter, exceptthat tie contractionreachlength was taken ro be one bridge openinglength, as in WSPRO. The bridge opening lengthsvaried from 40 ro 130 m ( 130 to 430 ft) and the dischargecontractionratio, rn. definedin the USGS method,variedfrom 54 to 79 percent of total flow obstructed in the approach section. Both the normal bridge meth.rdand the specialbridge methodwere usedin HEC-2, in which rhe latter method simply is an applicationof the Yarnell equation to determinethe water surfacedrop through the bridge. The downstreamexpansion reach length for the HEC-2 methodswas taken to be one bridge opening lengrh. as in WSPRO,but the HEC-f recommendedvalue(4 times the averageobstrucrionlength)also was tried. When using the expansionreachlength of one opening length, the HEC-2 normal bridge method gave the most consistentresults,with computed backwatervalues
CHApTER 6 : H y d r a u l iSc t r u c t u r e s 2 5 1
I
u) o p co
b LL c
.9 I
LU (E
-
U) i6 =
-
--+--+-. -a- --{--
0
1000
2000 3000 RiverDislance,ft
Observed
wsPBo H E C _ 2( N ) HEC-2 (S) E431
4000
5000
(b) CypressCreek,near Downsville. Louisiana FIGURE 6.29 kontinuedt
showingan overall averageof 2 percentlessthan measuredbackwatervalues,while WSPRO gave computed valueswith an overall averageof 3l percentgreaterthan measuredbackwatervalues.The HEC-2 specialbridge method (yamell) and the modified Bradley method both gave consistentlylow valuesof computed backwater,which is not too surprising,since neither methodwas developeclfor bridges in wide, heavily vegetatedfloodplains.When the expansionrarioof 4:l was apptied in the HEC-2 normal bridge method, rhe overall averageof computedbackwater valueswas 36 percenthigher than measuredvaluesand the computedwater surface elevarionsdownstreamof the bridge uere significantly higher.It was concluded that, althoughthe WSPRO model gave backwatervaluesthat were somewhathigh, it provided an accuraterepresentationof the downstreamwater surfaceelevations and the water surfaceelevationsin the immediarevicinity of the bridge. In laboratoryexperimentsconducted at Georgia Tech. water surfaceprohles were measuredin a large compoundchannel (4.3 m (l:l fr) wide by lg.3 rn (60 fr)
t5l
CHAP] I]R
o : H y d r r u l i cS t n r c t u r c - s
240 A .'4,
239
.;P'.'
'2. --.v -;?'
) a
234
g 237 o -o
ii
E 236 IL
/A
i .9 ^^_9
'./.
3 234 !
4,600c
f
a 5 233 6
.l
T,V'
rtJ
3
t
-
--+-1,900
(D
231
m
- -o -.+.--{--
l Observed
wsPRo HEC,2 (N) H E C - 2( S ) E431
I I
800
1600 2400 3200 RiverDislance, tt
4000
(c)PoleyCreek,nearSanlord, Alabama l' lG U R\) 6.29 (cont i n ued)
long) for which the lvtanning'srr valueswere determinedin uniform flow experimentsto be 0.0155 and 0.019 in the floodplain and main channel,respectively,for compoundchannelflow. The compoundchannelwas asymmetricwith a floodplain width of 3.66 m ( 12.0ft) and a trapezoidalmain channelbank-full width of 0.55 m (1.8 ft). The measuredwater surface profiles for a bridge abutmentin place are cornparedwith WSPRO resultsin Figure 6.30. in which the total depthsrelativeto the botton of the main channel are given. The bank-full depth is 0.l5 m (0.5 fi). length for this caseis 4.1percentof the floodplain width The abutment/embankment (l,JB, = 11.141. Almost exact agreementis lound bet$een the WSPRO depth and the measureddcpth at the downslream face of the bridge. while WSPRO depths upstreamof the bridgc are approximately2 to 3 percenthigh. Measuredand computed velocity distributionsare superimposcdon the shapeof the compound channel at the bridge approachsection in Figure 6.3 l. The WSPRO velocrtresare com-
C H \ P r r R 6 : H \ d r i r u l i cS l r u c t u r e s t 5 _ l
. Measured -
Normaldepth
^ WSPRO
0.250 O=00gSm3/sS ; =00022 VerlrcalwallabutmenltLa/q = 0 44
0.225
E
o.2oo A o o
0.175
0.150 o
2
4
6
8 1 0
1 2
1 4
1 6
2 A
X Station.m Fr(;uRE 6.30 of measured dcpthsandWSPROcomputeddcpthsin a laboratory compound Conrparison 1998).(Soxr.?.Tern W. Stunnuru! !.ltlonisChrisochannel(Stumrand Chrisochoides elnitles.Ott-Ditertsiottalund T*o Dintensbtnl EstinutesoJAbutntentScourPrediction Vtriubles. In Trunsport ti)n Rescar<'hReconl 1617, TransportotionResearchBoard, Nlltional Reseor.'|1 Countil, Wushitgton,D-C., 1998.Reproducedbr pennissionofTransportotiotl Research Boord.)
puted fiorn the dischargesin each of 20 strcamtubesharing equal conveyance dividcd by the flow area of each streanltube.Relatively good agreementbetween velocitiesis sho* n both in the floodplain measuredand computeddepth-averaged WSPRO vclocitiesconrputedin this wav did not agree and main channel.["[owever, at all with measuredresultantvelocitiesnear the face of the abutnrent,where the flow was not one-dimensional(Sturm and Chrisochoides1998). A user's instructionmanual for WSPRO was developedby Shearman( 1990), and it shouldserveas a sourcefor more dctailedinformation on using the computer model. The applicationof the nethod is describedbriefly in the follo\\ ing seetions and an example is given.
WSPRO Input Data All the input data recordsfor WSPRO are identified by a two-lener code at the beginning of each record.Thesecodes,summarizedin Table 6-9. can be divided into four groups: titles,job parameters(optional),profile control data, and crosssection definitions.The record identificationcodes must appear in the first two columns of each input record.Data valuesare enteredin free format ('F as the first data record) and can be separatedby comnas or blanks. Default valuesof certain
C H A P T E R 6 : H v d r a u l i cS l r u c t u r e s
1r')
. Measuredvelocity ^ WSPROvelocity - i/easuredWSE - Bed 2.O , =00022 o = 0 0 8 5m 3 / s S Vertical wallabulmenl. La/q = A 44
E
5o o.e
9 f
r!
n c
€:,
Ll.i
9 ^" ."
0.8 co
b o o
" _
0.0
..1'
t a a ' a
ao a
LU
a 0.4 =
-\L 1.0
2.0 3.0 Transverse Station,m
4.0
0.0
FIGURT]6.3I velocityandWSPROcomputed \elq:ity in a Iaboratory Comparison of measured compound channel(Stumr and ChrisochoidesI998). (So&rce.'Terr\^llt. Sturfi oru1Artonis Chrisochoides.One-Dirrensionala d T)ro-Di ensionalEstimatesof AbuttrentScourPrediction Variables. In TransportationResearchReutrul 1647, TransportationResearchBoard, D.C., 1998.Reproducedbt permissionof TransNational ResearchCouncil,Washington, portttIiort Research Board.1 parameterscan be used by entcringan asteriskor doublc commas.The input data records are createdeasily using word-processingsoft*,are that has a text-file creation feature or within HYDRAIN (FederalHighway Administration 1996).The input and output data can be in either SI or English units. Profile control data consistsprimarily of Q, WS, SK, and EX records.The Q record allows a whole seriesof dischargesto be analyzedin a singlecomputerrun. The stafiing waler surface elevation can be specified directly for each Q with a WS record,or the critical water surfaceelevationwill be assumedifWS hasa valueless than critical, such as the lowest ground elevation. Alternatively,a slope of the energy grade line can be enteredon an SK record to obtain a starting water surface elevation by the slope-areamethod.The EX record is used to specify a computation in the downstreamdirection(supercritical)with a lalue of unity or an upstream (subcritical) computationwith a value of zero (default). The ER record ends the data input file. Cross'sectiondataconstitutethe bulk of the input data and includeground elevations and locations,roughnesscoefficients,and bridge and spur dike geometry. Header codes for cross-sectiondata are given in Table 6-9. The actual r--l coordinate data for each cross section are entered on GR records and must be referenced
C l l A P r E R 6 : H ) d r a u l i cS t r c t u r c s -TAI}LE
6.9
\\'SPROinput data rocords
T l . 1 : . I 3 ' A l p h i r n u n r e r il cr r l cd i t a t { ) . i d c n r r f i . r t i o no t o u r p u r J l { r r o r t ( t e r ! n c e s .l e \ t v a l u e \ . e t c . Jl inpurInd oullul conlrol prrantetcrs J l \ N c i a l t r b l rn g p a | a n t c t e r s I'nlilt .o tnl dnta Q Jir.harsc(s) lirr t()ijle conrpulalion(s) \ \ ' S s t a n i n gw r t e r s u r f l . e e l e ! a t i o n ( s ) S K - n c r 8 y g r i r d i e n l ( \ )f o r s l o p e - c o n \ e \ i t n c ec o I n p u l a r i o n E X e \ e c u l i o n i n \ r r u c r i o ni n d c o m p u t a t i o nd i r e c t i o n ( s ) \ nd of inpul (cnd of run) L R . i n d r c . r t ee Cn!i \cLtion L|?li itio,l Herder\ X S - - r e g u l a r \ a l l c y s e c l i o n( i n c l u d i n g a p p r o r c h s c c l i o n ) BR bndge-openinesection S D s p u rd i k e s e c l i o n X R - ( ) a d g f a d es e c t i o n CV {ol\'en section X T l e m p l a r cs c c t i o n C r o \ \ - \ e c t i o n i l g c o m c r r yd a t a C R . t . r c o o r d i n a r c so f g r o u n d p o i n t s i n a c r o s \ s e c r i o n( s o m e c x c e p t i o n sa l b r i d g e s ,s p u r d i k e s , r o a d s .c u l v e n s . a n d i n d a t a p r o p a g a t i o n ) R o u g h n e s sd a t a N - f o u s h n e s s c o e l f l c i e o t s( M a n n i n g s , l \ a l u e \ ) S A ( ( , ' , ' J r n . l ( . ,o l . L h , r ( a b r e r l p , , r n r r. n d 1 r . . . . e \ t r . n N D i e p t h b f e a k p o i n l sf o r v e n i c a l l a r i a l i o n o f N v a l u e s F I o \ r l e n g t hd a r a F L f l o w l e n g t h sa n d / o r f i c t i o n s l o p e a v e r a e i n gr e c h n i q u e B n d g e s e c r i o nd a t a ( M = r n a n d a l o r y iO : o p r i o n a l ) D€ri8lr ,ro./r (no GR da{a) B L - b d d g e l e n g l h . l o c a t i o n{ M ) B C b . i d g e d e c k p a r a m e t e r s( M ) A B - - a b u t r D e n l s l o p e s( N I . T y p € l ) C D b r i d g eo p e n i n gr y p e ( M ) PD pieror pile dala (O) K D { o n \ e y a n c e b r e a k p o i n t s( O )
Fi.ted gettmetn nod€ (requires CR dara) C D b n d g e o p e n i n gl } p e ( M ) A B - a b u l m e n t r o e e l e v a t i o n( M , T y p e 2 ) P D p i e r o r p i l e d a r a( O ) K D . o n v e y a n c e b r c a k p o i n r s( O )
A p p r o a c h s e c r i o nd a t a B P h o r i z o n t a ld a t u m c o r e c t i o n b e l r e e n b r j d g e a n d a p p r o a c hs e c l i o n s T e m p l a t eg e o m e t r yp r o p i g t l i o n G T r e p l a c e sG R d a t a \ I h e n p r o p a g a t i n gl e m p l a r es e c t i o ng e o m e t r y (nar,
Dir
f r u m S h e a r n , J n( l q 9 0 ) .
255
256
C H A P rE R 6 : l l l d r a u l i c S l r u c t u r e s
to a cornmondatlrnr-Roughncssdata are entercdon N rccordsand ntust con.espond to the subsectiondcllnitions civen by SA rccords.Thc SA record gives thr- rjghr hand boundaryas an.r r.aluefor each subsectiongoing from Iefr ro r.ighr.L'\ccpt fbr the last onc, rvhich is the limiting right boundary.The l\lanning'srr \ aluesrhen cor, r e s p o n dt o e a c hs u b s e c t i o nT.h c r t v a l u e sc a n v a r y u i t h e l e v l t i o nu i t h i n e a c hs u b scction by using an ND record. u'hich gives thc vcrtical brcakpoinrsfor rhe additional valuesof rr cnlcrcd on the N record. ln rvhat is called thc desigttnrode,specificbridgc parametcrscan be varjcd on the BL, BC, and AB recoriis.as sho*,n in Table6-9. Other bridge recordsdcfining bridge and cmbanklrentconfisuration(CD), pic'ror pile dara(PD), spur dikcs (SD), and road grades(XR) are discu\sedin more dctail in the user'srranual.Four bridge typcs arc possiblein thc desien mode.as shownpreviouslyin'lbble 6-8. In the fixed gcometryn)ode.thc bridgesectionis cntercdas a seriesof slationsand elcvationsas for naturalchannelsections.exceptthat the seclionInust be 'closcd" by reenlcring the first gcometricpoint at the ieft abutmcntas the last geornetricpoint. Data propagationis a r ery convenicntltature of WSPRO, * hich avoids reentcring data that do not change frorn onc crosssectionto the next. Data on N, ND, and SA records.for example.can be codedonly once and propagatedfrom one section to the ncxt until they change.A singlecrossscctiondefinedby GR recordsalso can be propagatcdby specifi,ing only rhe slope and longitudinaldistance to each succeedingscction or by dcfining a tcmplatesecrion(XT).
WSPRO Output Data The usercan specifyccnain rypesof dataoutput.but of more interestis a definition of the output variablesthat appearin thc computerprinloutsshownhcre.Thesedefinitions are summarizedin Table 6-10. In general,the output consistsof an echo of input dataand cross sectioncomputationsfor eachsucceedingcrosssectionfollowed by the water surface profile results. The bridge backwater is the differcnce between the constrictedand unconstrictedwater surfaceelevationsat the approachsection. Ex A \t p L E 6. i. A nomlal,single-opening bridgeis to be constructed ar the cross sectionshownin Figure6.32.whichshowsthesubsections androughnesses. The averagestreamslopein theviciniryof thebridgeis 0.00052ftlft.Thebridgeopeningbegins at Starion230 ft (70 m) and endsal Sration430 ft ( l3 | m) for a loralbridgeopening lenglhof200 ft (61m). It hasvefticaiaburmenrs (TypeI bridge)and andembankments a bridgedeckelelarionof 35.0ft ( I0.7 m) wirh a low chord(or low sreelelevation) of 32.0ft (9.75m). The bridsehasthreepierswirh a spacingof 50.0fl ( 15.2m) anda widthof 3.0 ft (0.91m) each.No overtopping is allowed.For a discharge of 20,000cfs (567mr/s).calculate the backwarer causedby thebridgeandthe meanvelocityat rhe bridgesectionusingWSPRO. Solulion. The input datarecordsare shownin Table6-l l. The specifieddischargeof 20.'J00 cfs (567 mr/s)is enreredin rheO record,andrheconcsponding sranrng\larer surfaceelevationis obtainedby the slope-area methodusingthe slopeof 0_00052 on the SK record.The exit crosssectionis locatedat station1000fl (305 m). and the groundpointsshoxn in Figure6.32are enteredin the GR records. This singlecross sectionis propagated upstream in rhisexample. Thebridgeopeningis 200 fr (61 m), so
C I H A p -Ll R 6 : H ; d r a u l i c S t r u c t u r e s 2 5 1 TA Bt.ti 6-10
\\'SPROdcfinitionsof output variirbles AI.PH AREA BETA BI-EN C CAVC CK CRWS DAVG DI\{AX EGL EK IjRR I"LEN FLOW FR# HAVG HF HO K KQ LEW LSfl. M(K) i\'1(Cl OTEL Pl!-D P/A Q REW SKEW SLEN SPLT SPRT SRD TYPE VAVG VMAX VEL VHD WLEN WSEL XLAB XRAB XLKQ XRKQ XMAX XMIN XSTW XSWP YNIA\ YMIN
\tl.{ir} headconecrionfaclor Crosqseetion area }lomenlun)cor.ection facrn Bridgt,opcninglcngrh Cctillcicntof discharle A\.rase wcircoefficient (0.0default) Conrracrion losscoetficienl Crirical\-!alcr-\urfaccele\alion A\.rage depthof fio$ o\cr road$ay l{a\inrum deprhof floq overroad$ay Energygradeline E\pansionlosscocfficienl (0.5default) Error in cnLygv/discherle balance Roq dinance FIoq cla\sillcalion code Froudenunbe. A\era8e lolal head Fricrion headloss }tinor headlosses(expansiocontracnon) Crosssectjon conveyance Con\eyance of Kq sccrion lrfr edgeof wateror lefi edgeof $eir Lo!\ \reelr,ubmcrgencr,
-So!r..r Darafrom Shearrnan (1990).
258
CHAp'l ER 6: Hydraulic Sttuctures
T{Bl-E 6-lr \\'SPRO input data file for Exanrple 6.3
S IO TI T: T]
Example6-3.-Normal BridgeCrossing LSEL = 32.0ft; No Ovenopping BedSlope= 0.iX1052; BridgeOpening(TypcI): X = 2l0lo,l30 f(; 3 piers Discharge 10000.
SK
XS GR CR GR GR GR GR
* N sA * Ir * BR
Slopefor slope-areamelhod 0.00052 EXIT seclioo Sectionrcference distance, skew(0),ek (0-5),ck (0.0) EXIT IOOO.
0..35.0 0.,28.0 1.10.,23.5200..21.5 230..21.0 250.20.5 280.20.4 300..20.0 3r0..19.0 330..10.0 360.,3.0 380,8.0 4 0 0 . 1 8 . 0 , 1 3 0 . . 2 1 . 04 5 0 . , 2 0 . 01 7 5 . 1 1 . 0 500..17.5 5,{0..I8.0 600..20.0 730.,?8.0 730.,35.0 brealpoints Subsection n valuesandsubsection 0.0.15 0.0.15 0.07 0.035 430. 200. 300. geomelric PropaBate datafromexit sectionto full valley FULV 1200. * * * 0 00os2 Createbridtesection BRCEI2OO,
*
low chord eleva(ion
BC
J2,O
'
ri8hl abut.sta. hndgelengd.left abr-tt..ta.. 200.210.430.
BL 0 ' PD0 * cD * XS EX ER
pier elev.,grosspier width, no. of piers 8.0.3.0,t10.0,3.0,110.0.6.0.220.4.6.0.220.4,9.0,1 bridgelype, bridge\^idth I 40.0 Approachsection APPR 1440.
C H \ P T L F o . H r d r : r u l iSc t r u l l u r e s 1 5 9
n=o.oas]o07 ] o o 3 si
ooos
o uJ 20
200
FIGURE 6.32 Example6.3,bridgecrosssection. both the full ralley and bridgesectionare locatedar slation1200fl (366 m) ar rhe downslreamface of the bridge.The approachsectionis one bridgeopeninglength or 200ft (61 m) upstream ofthe upsrream faceofthe bridge.As a result,rheapproach section is at Station1,140 takinginroaccounr thewidrhof rhebridgeof ,10.0ft ( 12.2m) as givenin lhe BL record. The bridgcrecord(BR) is givenin thedesignmode.in whichtheprogramcreates the bridgecrosssectionfrom rhe succceding records.The BC recordenlersthe low chordelevationof 32 lt (9.75 m), which is neededto determine if the flow is freesurfaceflow or orifice flow. The bridgelengrhof 200 fr (61 m) beginningar r : 230 ft (70 m) andendingar.r = 430 fr ( 131m) is givenin rheBL record.The elevarions at the bottomof thepiersandtheir cumulativewidthsare shownin thepD record.Finally, the CD recordindicatesa Typ€ I bridgeopeningand a .10fr ( 12.2m) bridgewidth- No roughness dataaregiven,so theseare propagatedfrom lhe downstreamstation. Sampleoutputis shownin Table6-12 for Q = 29.11p.r, (567mris).Inpur data echohasbeensuppressed for brevity.First, the water surfaceelevationsfor the unconstrictedflow are given at the exit, full valley,and approachsectionsto representthe waler surfaceprofile without the bridge in place.Theseresultsare followed by the watersurfaceelevationsal the bridge sectjonand the approachsectionfor constncled flow,or with thebridgein place.Thecrilicalwatersurface elevations andFroudenumbersat the bridgeandapproachsectionsbothindicatesubcriticalflow at thesesecrions. The backwater of L12 ft (0.34 n]) is obrained by subrracling rheunconsrricted warer surfaceelevationat the approachsection from the conespondingconstrictedvalue \29.37- 28.25).The bridgeopeningvelocityis 8.14fvs (2.54m./s),andthe approach velocityis 3.16ftls (0.96m/s).The flow is ClassI, or TypeI in rheourpur,which is freesurfaceflow throughthe bridgeopeningwithoutembankment overtopping,and the bridgedischargecoefficientis 0.7.15.
TABLE 6.T]
Output data for Fixanple6..1 sPRO*******++* l'ederalHigh$a] Administralion-U. S. GrologicslSurvey l\lodel for \latcr-SurfaccProfilc CoInputations. lnpu( tlniL!: English/ Oulput thits: English E X A \ I P L E6 . . ] . . N O R M A LB R I D C EC R O S S I N C BED SLOPE = 0.ql05li LSUL = 31.0I:f; NO OVERTOPPING BRIDCEOPENING(TYPEI), x = 130TO J-10l--T.3PIERS wsEI- vltD AREA SRDL a V ECEL I]F K FLE\ CR\\ S HO SIi AI-PHA FR # Se,rtioni EXIT HcadcrTypc: XS SRD: 1000.000
28.008 :E -r_i8 2 r .895
.150
Section:FULV HeaderType:FV SRD: 1200.000
?8.1t7 28.{66 2l.999
.3.{9 .r0.1 .000
10000 ft)o 1628 .101
551:.0t9 876n61 t0
I-EW REW I]RR .t00 7.r0.(xx)
1.70E
20000.000 5515.t02 200.000 .100 8775,17.80 200.0)0 7t0.000 1.6:6 .304 .0005 L108 .00.1
<<< The Preceding Daia Reflecr The Unconsrricted Profile >>>
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 <<
>>
Section:APPR HeadefType:AS SRD:1140.000
.t.19 .125 .000
<<>> <<>> WSEL EGEL CR\\'S Section: BRGE HeaderType:BR SRD: 1200.000 SpecificBridge Bridge Type I Pier/PileCode
VIID HF HO
27.135 1.941 29.,182 .189 22.508 .835 Information Flow Type I 0
WSEL EGEL CRWS
AREA K SF
Q V FR #
I-EW REW ERR
20000.000 1198.501 200.000 230.0.12 8 . 3 3 9 5 r0355.60 200.000 .130.058 .124 l.800 .001 P/A PFELEV
.7453
SRDL FLEN ALPHA
XRAB
32.000 200.000 230.000 .130.ffX)
.055
VHD HF HO
BLEN
a FR#
AREA K SF
SRDL FLEN ALPHA
LEW REW ERR
Se{tion:APPR 29.169 .25,r 20000.000 6338.016 200.000 .08,r HeaderType:.\S 29.623 .t' 73 Ll56 10-585,13.00220.202 ?30.0t6 SRD| 1140.m0 22.t24 .069 .242 .0005 1.6-t2 .00r ApproachSectionAPPR Flow ContractionInformation M(G) M(K) KQ XLKQ XRKQ OTEL .'726
260
.398
636363.5
242.723
112j39
29.369
C H A I , T E R6 r H y d r a u l i cS l r u c t u r e s 2 6 1
REFERENCES Bodhaine.G- L. "Measurcnrent of PeiikDischarges at Culrr'ns b1'lndirectMethods."ln of llatcr Resourccs Techniqrtes Inv'cstigatiorrs. Book 3. ChapterA3, Washington. DC: CorcmmcntPrintingOlfice.U.S.CeologicalSurver'.1976. Bradlel-. J. \. Hvlraulicsof BritlgeWaten+att(HD.t'//. ll\ draulicDesiSnSericsl. \\/ashington.DC: FcdcralIlighwayAdministration. U.S.Dept.of Transporration. 1978. Cassid1". J. J. "Designing SpillwayCrestsfur fligh HeadOperation." J. H;d. D/1. ASCE96. n o .H Y 3 ( 1 9 ? 0 )p. p . 7 ' 1 55 3 . Chanson. H. Ilvlroulic Design rf Sttpped Costades, Chuatels, Weirs.awl Spilluats. Oxford.England: Perganron, ElsevierScience,I99-1b. Chanson.H. 'Hydraulicsof SkinrmirrgFlow ovcr SteppedChannelsand Spillways. J. pp.,145-60. Hvdx Res.32.no.3 ( 199.1a), VenTe.OpenChunnelHtrlrarrlics. Chouvi NewYtrrk:McCra*-tlill, 1959. Christodoulou. G. C. EncrgyDissipation on StcppedSpill\\a)'s.'"L Hrr& frr3rg.,ASCE | 19.no. 5 ( l99l). pp.614 -19. de S. Pinto.N. l-. Cavitation andAcration."ln Advtnted Dam Engineering for Design. Construction. antlRehabrlitution. ed.R- B. Jansen. Ne\\ York:VanNostrand Reinhold, 1 9 8 8p. p . 6 2 03 4 . 'ComputerDeterminalion Eichcn.B. S.,andJ. Peters. of Flow ThroughBridges." "rtH,r'd. D i r . A S C E9 6 .n o .H Y 7 ( 1 9 7 0 )p, p . l 4 - 5 5{ 8 . Fcdcral llighway Administration. IIY-6 ElectronicCo,ttputerI'rogrun for llydraulic Anahsisrtf Cuherts.Washington. DC: Fcderallligh\\l! Administration. 1979. FederafHighway Adnrinistrrtion.Hydroulic Anal,-sisof Pipc-Arch und Ellipricol Shape CultertsUsingPrograntnuble Calculutors: Calcuktnr DesignSeries No.4. Washington. DC: FederalHighwayAdministration, 1982. FederalHighwayAdrninislrution. Hvlraulic Designof Culterts.ReponFHWA,IP-85-15, Hldraulic DcsignSeries5 (HDS-5).Washington. DC: FedcralHighwayAdnrinisrration. U.S.Dept.of Trrnsportation. I985. FederalHighwayAdn)inistration. User's Manualfor Hl'l)RAIN; l tegratedDr.rtuage Washington. DesignContpute r St.\tet (V(r.tion6.0).FHWA-SA-96-06,1. DC: Federal High* ay Adnrinistration. I996. French.R. Open'ChanneI Hvlrualrcs.Netr York:McGrau-Hill, 1985. Henderson. F. M. Opn ChannelFktw.NewYork:Macrnillan.1966. Kaatz.K. J., and W. P James."Analysisof Alternativesfor ConputingBackwaterat Bridges."J. Hrr.iirErgrg.,ASCE 123.no.9 ( 1997).pp.181-92. "DesignChanfor Predicting Keller,R. J.. andA. K. Rastogi. CrjticalPointon Spillways." J . H r r l D i r : , A S C E1 0 3 n. o .H Y l 2 1 l 9 1 ' l ) . p p .l 4 l 1 - 1 9 . Kindsvaler. C. E..andR. W Carter."TranquilFlowThroushOpen,Channei Constrictions." Transattitns ASCE120(19-5-5 ). Kindsvirter. C. 8.. R. W Carter.and H. J. Tracy.'Corrpuration of PeakDischarge at Con, tracrions.U.S.Ceological Survey,Circular28J.Washington, DC: Govemment Printi n g O f l r c c ,1 9 5 8 . lvlatthai.H. F 'Measuremenl of PeakDischarges at Width Contractions by lndirectMeth ods" ln Tethniques oJWdterRetources In\..stigutiuts. Book 3, ChapterA4. U.S.GeoloqicalSurvey. Washington. DC: Governnrent PrinringOffice.l9?6. Rajarrtnam. N. 'SkirnningFlow in StcppedSpillwavs." J. H*h Engrg., ASCE Il6, no.'1 ( 1 9 9 0 1p.p .5 8 7 9 l . Rcese.A. J.. andS. T. Mrynord. Designof Spillu,ayCrests."1 Hr./r Ezgrg.,ASCE ll3, no. J ( 1987).pp..17G90.
16l
C H A p IE R 6 : H y d r r u l i S c trucrures
Rice. C. 8., and K. C. Kadavy...ModelSrudyof a RollerCompacted CoDcrete Srepped Spillway."./. Hrzlr t,.q1g.,ASCE 122,no. 6 ( I 996), DD.2g2 gi _ Schneider. V R.. J. W Board,B. E. ColsLrn, F. N L:e, and L. A Druffcl...Cornputarion of Backwaterand Discharge at WidrhConst.ictions of HeavilyVegetatcd Floodplains.,, U.S.Ceolo!icalSurveyWater-Resources lnvesrigations 76-1i9, Washinlton. DC: Gov e f l ) n t e nPl n n r i n g O l f i c e ,I g 7 b . Shearman,J. O. Ltser'sl,lanualfor llSpRO A ContputerModelfor l+tter Surfa(eprolile Cortputations. ReponFIJWAIp-89 02?.Washington. DC: U.S.Dt,pr.of Transpona_ lion, Federall.lighwayAdninisrration, 1990. Shearman, J. O.. \V. H. Kirby,V R. Schneider, and ll. N. Flippo. Britlgel*ttentrlr.sAnah_ sis Model: Res.atdt Reporl ReportFHWA,RD g6/10g.Washington, DC: Federal HighwayAdministrar ion,U.S. Dept.of Transponation, 1996. ..Experimenr, Straub.L. G., andA. C. Andcrson. on Self AcrtreJFlou,in OpenChannels.,. Ir.rn.r.ASCE(1960),p. I25. ..One Sturm.T. W. andA. Chrisochoides. Dimensional andTwo Dimensional Estimates of AbutnrentScourPredictionVarirblcs."TrcnsportationResearchRer:ord1617.\Nash_ ington,DC: NationalResearch Council,December199g.rrD.lg_26. U.S.Army Corpsof Engineers. Hvlraulit DesigttCrilcn,r.Vicksburg.MS: U.S. Army Corpsof Engineers Waterq,ays Experiment Station,1970. U.S. Army Corpsof Engineers. HtdruulicDesignof Spillvats.EngineerntanualI I10,2_ J603.Vicksburg, MS: U.S.Army Corpsof Engineers.1990. U.S. Army Corpsof Engineers. HEC-2 Ltser,s Munual. Dayis,CA: U.S.Arrnv Coms of Engineers. HydrologicEngineering Center.lgg l . U.S. Army Corpsof Engineers.HEC-RASHydraulic RefcrenceMttnua[,Versiort2.2. Davis. CA: U.S.Army Corpsof Engineers, HydrologicEngincering Cenler,199g. U.S. Bureauof Reclamation. Designcf SnallDams,a lUaterResources Techtli.dl t,ublicdprintincOffice.lgg7. Iior, Srded.Denver:U.S.Government ..Ceneral Wood.L R.. P Ackers,an{lJ. Loveless. I\lerhr.xl for Critrcalpotnton SDillwavs.,. -/. Hr'lr Errgrg.. ASCE 109.no.2 ( 1983),pp. 108-t2
EXERCISES 6.1. A higholerflow spillwaywtthplHd > i.33 hasa maximumdischarge of 10.000cfs with a maxinlumheadof 20 ft. Delermine the designhead.spillwaycrestIength,and the minimumpressure on the spillway.plot the complctespillwaycrestshapefor a compound circularcurvein the upstream quadrantof rhecrest. 6.2. RepeatExample6.1 for an ellipticalapproachcrest,usinga dcsignprocedure lhar guarantees a mtnlmumpressure headof - l-5ft. plot the head-dischargc curve. 6.3, An ogeespillwayhasa cresrheighrof 50 fr abovethe roeanda nraximuntheadof 15 ft. A minimumpressureof 1.5 psi is allowed.The maximumdischargeis 16,000cfs. (.r)Delermine thecrestlengthof thespillu,ayassuming a conrpound crrcutarcurve for lhe upstream crestshape. Whatis the pressure at ther.r,srfor the maximum discharge? (b) If thespillwayis designed asa stepped spilluay.rvitheachsrep2 fr highby l.-5fr long.whatis theenergydjssiprlion in feetof waterat themaximumdischarge?
C H A p r r , R 6 : I l v d r a u l i cS t r u c t u r e s 2 6 3 6.4. An existing ogee spillway with an elliptical crest has a cresr height of 7.0 m rnd a crest length of 15.2 m. A nrinimunr gage pressureof Tero (atn)osphericprcssure) occurs at a head of 1,1.0nt. Whal maximum head and dischargeuould you recom rnendfor this spill*ay ) 6.5. A 0.91 m diametercom-rgaled metalpipe culven (l = 0.021) has a lengthof90 m and a slopeof0.0067. Thc entrancehasa squareedgein a hcadwall_At rhe designdischarge of 1.2 nrl/s, the lailwateris 0..15nt abovetie outlet in\en. Detennine the headon the culve11 at the dcsign discharge.Repcatthe calculatiotrfor headif rhe culven is conerele. 6.6. Show lhat Equalion 6.10a for a box culverl in inlet control uilh rhe entranceun\ubmcrgcd can be placed in a form in *hich Q is proportionalto rhe head,F1ly.to rhe 3/2 power. 6.7. A 3 ft by I ft concrete(n : 0.011)box culven has a slopeof 0.006 and a lcngrhof 250 fi. The eniranceis a squareedge in a headwall.Determinethe head on the culvert for a dischargeof 5{) cfs and a dischargeof 150cfs. Thc downstreanllail\-\,ater elevationis ().5 ft above the outlet intert for 50 cfs and 3 ft above the o!l/?/ i/llet at 150 cfs. 6.8, Designa box culven to carry a designdischargeof 600 cfs. The culven inven elevarion is 100 ft and the allowable head$ater elevarion is I 1,1ft. The pared roadway is 500 ft long atrdovcnops at I15 ft. The cul\en lengthis 200 ft wirh a slopeof 1.0percent.The following tailq ater elevationsapph up to the maximunrdischargeof 1000cfs: 8, cfs
:00 ,100 600 8m 1000
Tll', tt
t0) .1 102.6 !03.I 103.8 lol I
Preparea perfonnancecune for theculverldesignby handandcornparewith theresult.s of IJYS.Also useHY8 to prepare a perfonnance cur"'eif theslopeis 0.1percent. 6.9. A circularconcreteculverthasa diameterof 5 ft with a square-edged entrance in a headwall. The culvertis 500ft longwith a slopeof 0.005andan inletinvenelevarion of 100.0ft. The downstream channelis lrapezoidal with a bortomwidthof l0 ft, side slopesof2:1, slope= 0.005.and Manning'sa = 0.025.The pavedroadwayhasa constantelevationof 130ft wirh a lengthof 100ft anda widrhof 50 ft. The design discharge is 250 cfs andthemaximumdischarge is 500cfs.UseHY8 to consrruct and plot the performancecurve for thesedata.and comparethis \rith the perfonnance curve for a 5 fl diametercomrgatedsteelpipe. Also comparethis with the perfor, nlancecunvefo. the 5 ft diamelerconcrete culven* ith a side-tapered inlet. 6.10. Provethator : l/Cr in the WSPROmethodology where(rr : kinelicenergyflux correction coelicientat section3 andC: USCSbridgedischarge coefficient. 6.11, Apply the HDS-l melhodto the datagivenin Example6.3,and conrpare the back\vaterto thatoblainedfrom WSPRO. 6.12, Usingthe USGSwidth-contracrion curvesin Figures6.23and6.2,1,verifythe value of the bridgedischarge coefficient andthe discharge ralio tn (= M(K)) givenin the WSPROoutputfor Example6.3.
261
C H A p r r , R 6 : H ) . d r a u l iS c trucrurcs
6.13. Change rhe bridgerype ro Type 3 for thc WSpRO example (E\anrple 6.1), rnd dr,rer_ minc rhe backwarerfor p valuesrrnging from 5.00Olo 25.000 ct\. plot the resultsin a graph contpring lhe Type I and Type 3 bridses in this range of dischargesfor a bndge lcnSrhof 200 ft. 6,14. For a b dge lenglh of 200 fl rnd a Type I bridgc-.change the lo\\ chord elevarion in rie WSPRO cxantpleto:8 ft with a coDslilnrroad\\a\.ele\aljon of -tl fl. Allo\r overtopping to occur and detenninethc back*.aterfor the santt rangeol.dischargcsas in Exercise6. 13. Plot the resullsin conparison * irlr Erample 6.-j. Note: Thc XR heade. rccord is rcquiredto locatethe centcrlineof the road\ay folloxcd br CR recordsto give the road\\ay profile for overloppinganall sis. 6,15. For Exanplc 6.3. reducerhe bridge lengthro 150 ft (\r irh trr.obridgepiers).and inlro, duce a reliel bridge with a length of 50 ft al a loearion of your choice in one of the floodplains.Plot rhc resulrsfor bdckwatero\er rhe stnte range of dischargesls in Exercise6.I3 in conrparisonwjth the resultsfrom E\ample 6.-1. 6.16. Anallze the exisringbridge o\cr DLrckCreek using \\'SPRO or HEC,RAS (WSPRO option) and the following tableof cross-section dara.Use the fixed gconletrymode for the bridge secrion.The bridge is Type .l wirh a rr idrh of j0 fr, embrnkmentside slopes of l: l. enbankmenlelcvationof 790 fr, and wine* all angleof 30.. T}e tow chord elerarion is 788 ft. The designdischargeis 6950 cfs $ ith a watcr surfaceelevallon of 78'166 in the exit crosssecrionThe full va ey sectionshouldbe idcnricarro rhe bridge sectionover the bridgeopeningwidth and essenrialll.is the samea\ the e\rt secLronln both floodplains. (a) Determinethe backwaterfor the existing b.idge. (b) Design a new bridSelo replacethe old one so that the back*ater <0 is 25 lt. Duck Crcek cross sections Exit, Station 1000 Point I
2 3 1 5 6 1 8 9 IO ll t2 I] l,{ l5 t6 1i l8
Distance
Elevalion
150
192 780 780 '778
-20 l8 22 _15 50 210 600 E60 1005 1050 l 2 l]60 I 3r 0
1't3 1'72 172 780 780 1'79 '779
- r05 -'70 -28
780 78: 78,1 786 788 798
C H A P T E R6 : H ) d r a u l i cS t r u c t u r e s 265 Subsrclion I
) -l 4 5 6
X Subsection
2tt 35 :t0 600 860 l3l0
l\tanninq'sn
0.08 0 0.1 0.08 0.05 0.08 0.05
llridge,StationI100 Point I
2 l .l 5 6 1 Subs€ction
Distance
'71 -71 30 t5 25 X Subsection
Elevation 7E8 778.1 116 112.5 712.5 7'7 5 788 Nlanning'sa
0.04 Approach, Station 1230 Distance I
-.r80
2 l 1 5 6 '7
1-10 .120 - 305 - 175
8 9 l0 II t2 l3
t7 l8 l9
-30 -25 2 t7 20 ?8 50 670 990 1070 lD0 l:60
Subsection
X Subsection
l5
I
2 -l :1
28 670 t:60
Ele\ation 796 788 786 78.1 1'42 780 7?8 '7'l6 112 '7 72 780 780 780 '782 78.1 786 810 Nlannins'sz
0.l0 0.0.{ 0.08 0.05
266
C r { A p r E R6 : H l d r a u l i cS t r u c r u r e s
6.17. Apply HEC-RASro Eraniple6.3usingthcencrgy.monrentunr. yamelt and nrerhods. Setup the crosssecrionsin the schcrnatic layoJistaningat the up.irerm ,tationof 1450.Usea consranrslopeof 0.00052andaid rheappr-opriare ailount r,r att eteva_ ttonsglvenlbr stationlO00in Exarnple 6.j. Thcnin ihc geonrerric daraerlrlor,copy the crossseclionsdo!|nstreantadjustingthe elevafiond-own*ard accorJrnglo the slopeanddistancebet*eenstalions. Usevaluesof0.3 and0.1 for the erp:nsronand contractron losscoefficients. respectively. Eslablish slalionsat 1250.l2m. and 1000 to correspond with the\\'SpROseclions. Add a bridgeat station1225usi]']!lne geo_ metncdataedjtor.In the bridge/culvert editor,enterthedeckandroao\\a)data.pter data,and checkthe boxesfor all threcmethodsof compuration as *ett as rhe box choosingthe highestenergyanswerin rhe brrdgemoelelrng.rpproach uindow. Also enlera picr dragcoefficientof 2.0 anda yarnellprer.oati,.,.n, of 0.9_In the cross sectiondataeditor,addineffcctive flow areasat stationst 2SOana t iOOro $e lcfrand dght of lhc bridgeopeningspecified at elevarions abo,,.e the low choroet!-\ationbul belowthe_ top of the road*ay.In_lhesteadyfl()\! Llalamenu,enterthe or.cnargeot 20,000cfs and choosenormaldepthas the downstream controlurrn 3 slopeof 0.00052,Finally.choosesleadyflow analysis andclick rhecomputebrrron.Colnpore the resultswith WSpRO.andthenmakea secondrun with the exit sectronat station 800insteadof l0{D_Discussthercsurrs.
CHAPTER 7 },t.'.g:?45txed..I*drr-
GoverningEquationsof UnsteadyF'low
7.1 INTRODUCTION In unsteadyflou. r'elociliesand dcpthschangewilh time at any fixed spatialposi_ tron tn an open channel.Open channel flow in natural channelsalmost alwavs is unsteady,allhoush it often is analyzedin a quasi-steadystatefor channelde.ign or floodplain rnapping. Unstcadyflow in open channelsby nature is nonuniform as \\,ell as unstcad) bccauseof the free surface.Mathcmaticallv,this nteansthat the trvo dcpendentflow variables(e.g..velocity and depth or dischargeand depth) are functionsof both dislancealong the channeland time for one-dintensionalaoolica_ tions. Problem formulalion requirestwo partial differcntial equationsrepreienring the continuity and ntomcntumprinciplesin the two unknown dependcntvariables. (The differential fonn of the cnergycquationcould be uscd in cascswhere the flow variablesare continuous.bul the ntontcntumcquation is required where they are discontinuous.as in surgesor tidal bores.)The full differential forms of the two governing equations are cnlled the Saint-Uenantequutiottsor the dvnamic u.ave equatiott.s. Onlt in rathersevcresimplificationsofthe governingequalionsare ana_ Iytical solutionsarailable lor unstcadyflow. 1'his situationhas led to the extensive developmentof appropriatenurncricaltechniquesfor the solution of the govenring equations.Scveralof theserechniqueswill be exploredin the next chaprer. Unsteadyflou. problemsarisein hydrauJiccngineeringin a varietyof settings, ranging from *aves formed in irrigation channelsby gate operarionor in hydioelectric plant headraccsand lailracesby turbine operation to natural flood waves and dam-breaksurgesin rivers.Thc types of wavesconsideredin theseslruauons are calledlrarislalon war,erbecauseof their continuousmo\ement along the channel as cpposed to periodic or oscillatory ocean waves, which are not considered here. In addition. only shallou water wavesare considered,in which water move_ ment occurs o\c'r the full depth and venicai velocity and accelerationcan be neglectedto allo\\ the usc of one-dimensionalfornrsnf the govcrningequations.In 261
26!
Crr\r'r F
-.
C o r e r n i r rLg( l u . r t r , ' n \ o fnI . r : " J r f l o *
V-c<1--+ V+c
(a) Subcritical Flow
-r-" l----+ V+ c
(b) Supercritical Flow
F I G U R E7 . I \\'are plopngirlion in subcritical andsupcrcritical tlo\\.
all the wavc problcnrsconsidercd,thc pur-poseof obtainingthe solutionof the govem jn8 equations(referrcdto as rurrtirg in the context of llood wavcs)is to describe the flow velocity and depth as functionsof spaceand 1in'rc.ln other words.thc spatial shapeand temporaldevclopmenlof thc translalorywave are sought. A nrorefomral dcfinition of thc tanslaton rlar,eclescribes it as a disturbance mor ing in the longitudinaldirectionthat gives rise to changcsin discharge,velocity. with an absolutespced,dcsignatedby dr/dr, which and depthwith time. Il propagates is the sum of thc nreanu,atcrvclocity, V, and the wave celeritv \rith respectto still !rarer, c. as illustratedin Figure 7.1, with positive y in the positiver direcrion. Becausethc tvavecan Inovein both upstreamand downstrcantdireclions,its absolute speedis given by V 1 c. The celerityof a long rvaveof small amplitudeis given by (-gr)ra. in which,r'is dcpth, so thc valucsof dr/dt dcpendon the Froudc number,F, dr'fincd by V/c. In subcriticalflow in which V { t and l' < l. dVdr hastwo possible r aluesgiven by V * c in the downstrermdireclion and V - c in the upstrearndirection. as shownin Figure7. la. On the othcr hand. thc t$,o possiblewaveproplgation (). are both jn the downspeedsin supcrcriticalflow, givcn by (V + c) and (V streanrdircction,becauseV ) c and I' > I . as illustratedin Figure7. ib. Thc physicrl propeny of two possiblervare propagationspcedsis particularto hl perbolic panial dilferentialequations,which have the mathenraticalprope(y of tso possiblecharacteristicdirectionsor paths along which discontinuitiesin the deri!atives travcl.Thc conncctionbetweenthe physical and mathematicalproperties of the Saint Venantequationsallows them to assumea simplerfbrm tn characteristic coordinatesassociatedwith the path of two ntoving observerstravcling at the speedsof (y :l c). As a result. u,e first derive the Saint Venantcquationsand then begin thc study of unsleadyllow. uith a translbrnrationof the equarionsto characterislicfornl to provide a deeperundcrstandingof the physicsof wave propasation.ts well as thc initial and boundaryconditions necessaryto solve the Saint\.'nanl equations.The characleristicequationsarc sirlplified for the caseof a "simple rr 3v9." with no gravitv or friction cffccts. and applicd to sluice gate operation prob)emsas a lcarning tool for undcrstandingthe characteristicform. In thc ncxt chapter,which covcrs numcrical solution techniquesfor thc govcrningequrlions. $e alsoapply finite difftrence techniquesto the \olution ofthe governingequations in characteristicform. which has come to be called the nethod of tharactertsrtts.
C . r r , r I t rr <7 ( ; o \ c r n i n gE q u a t i o nosl L l n s r c a dl rvl o w
269
l r a d d i l i o nu . c c o n s i d c rc x l l i c i t a n d u r r p l i c i tl l n i t cd i l - f c r c n ct c c h n i q u easp p l i e r rJo l h e u n t l l | n \ t o D c d S l i n t , V ' c n a nctq U ; r i , r ) , a 1n \ d d i s c u s s{ h c a d \ a n l a g c sa n d d i s a d vilntascsol elch ntcthod.r\pplicatiorr.,rrrcluclclhc problcntso1 h;droelcctricpo*.er load acceptanccard rcjcction.drnr frreaks.and llood routing.
7.2 DF]RI\A'IIONOF SAINT-\'ENANl'I'QUATIONS AlthoLrghthc govcrnjn-ccquationsof continuity and mon)!'ntunlcln be dcrived in lununrbcrof ways, we uppll' a conlrol r olume oI snall but llnite lcngth, Ir. that is reducedto zelo lcngrh in rhc limir ro obrain lhc llnal diiTcrentialequarion.The der_ i \ a t i o n sn r a k et h c t i r l k r s i n ga s s u r n p l i o n1sY * j o i c h I 9 7 5 :C h a u d h r y1 9 9 3 ) : l. Thc shallowwater approxinralionsapply so that vcrlictl accclcralionsare negli_ gible, rcsulting in a vertical pressurc disrributionlhrt is hydrostatic;and the depth-t, is sntall contplrcd to the \\'avelengthso that the wave celcrity c = (.g,r')"1. 2. The channcl bottont slope is sntall. so that cosl 0 in thc hydrostaticprcssure force fonrulation is approximatelvunity. and sind - rand - So,the channelbed slopc,where d is the angle of thc channelbed reiativcto thc horizontal. J. The channclbed is stable,so that lhe bcd elevatjonsdo not changewirh time. 4. The flow can be rcpresentedas one-dimensiontilwith (a) a honzontal water surfaceacrossany cross scction such that trilnsvcrscvelocitiesare ncgligible and (b) an averageboundaryshcar stressthat can bc appJiedto the whole cross section. 5. The frictional bed rcsjsttrnceis the santc in unstcadvllow as in steadyflow, so that the Manning or Chezy cquationscan be uscd to craluatelhe nean bound_ ary shearstress. Additional simplifying assumptionsmade subscquenrlymay be true in only certain rnstances.The momentum flux correction factor. B, for exantple. will not be assumedto be unity at first becauseit can be significantin river overbankflows.
Continuity Equation First, considerthe continuity equation.which will be derivedfronl the control vol_ ume of height equal to the depth, ), and length, A.r, as shown in Figure 7.2. As in the derivationof continuity in Chaprer l, which used the Reynoldstranspontheorem, the basic statenlentof volume conservationfor an incompressiblefluid flow_ ing through the conrrol volumc is Ner Volume Out - - Changein Sloragein the time interval,Ar. This can be expressedas aO .- -\rlr rrr
.711.rJr
}
AA ,- ll dt
(1.t)
270
C H ^ P I E R 7 r G o v r ' r n i r gE q u r l i o n so f U n s l c J d vF l o \ , , Lateral inflow, gllx
fttti\ilfl r I
*-----
j t, Q + Q/nx)Jx \d
Protile
CrossSection
F I G U R E7 . 2 Conirolvoluinefor dcrivarjon ol unstcirdy conlinuitycqurlion.
in which4. : lateralinflow rateperunitlengthofchannelandA - cross_sectronal areaof flow. Dividingby A,rtrlandtakingboththe controlvolumelensthandthe time intervalto zero,the continuityequalionis
aA ao i t '
(7.2)
a t - q L
Substituting d,,1: Bd.r.fromFigure7.2,jn whichB : channclrop widrhat rhefree \ u r f a c eL,o l i n u i t vb e c o m e \
a) *
=,t, uP
(7 . 1 )
By definitionof the dischargeas 0 - AV, in which V = rneancross-sectional veloc_ ity in the flow direction (r), rhe dQlar term in (7.3) can be wrirten as A(Ayldr) + V(dA/6x),using the product rule. However,the term involving d,4/Armust be eval_ uatedcarefully becauseA can vary u irh borh depth. and distance,r, if the chan_r. nel width is changing:
AA
aAI
ar
,. 1,
dr
"ax
( 1. 4 )
wherethe first term on *re right handsideof (7.4)represents the derivativeof A with respcctto.r while holding-r'con\ranr. For pri.rnaiiccharrnels, rhis termgoes to zero.Finally,with thesesubstirurions for Ael6x and then dAldr, and dividing throughby thetop width.B, thecontinuityequationreducesto
!d ,t * u duf t _ oa xa v , v l l B dr I,
4r B
( 7. 5 )
( - -t l ^ p T E R 7 : C o r e r n i n gE q u i l i o n so f U n \ l c ' a d y FIow
2jl
Laleralinflow,qr.\x
i l \ t t i l tr
r
Dl-,
* { VL
\________y_/
\\ ^ .1,.//
Fpt = fh6A
,-v \\
Centroid //
w =yAJx profile
CrossSection
I.'IGURFJ7.3 Controlvoiumefor derivation of unsteady monlenlumequalion. in which D - A/B = hydraulicdepth.For a prismaticchannelwith no lateralinflow, the founi temt on the left hand side as well as the right hand side go to zero.Furthermore,if a rectangular crossscctionis considered,the continuityequationbecomes
a)'
da
(7.6\
in which 4 : V_r'- flow rate per unit of width. In this form, it is evidentthat temporal changesin depth at a point must be balancedby a longitudinalgradientin flow rate per unit of width.
Momentum Equation The momentum equationis derivedwith referenceto Figure 7.3, in which the forces acting on ttte control volume are shown. Pressure,gravity, and shearforces are con_ sidered,and these must balancethe time rate of change of momentum inside the control volume and the net momentumflux out of the control volume.In the.xdirection, which is takento be the flow direction,the momentum equationcan be written
4 , + 4 , - r , . =,qf f
- pqlrru. cos{ *.*]o' . * ll ou:aef* (1.1\
in which Fo, : pressureforce componentin the _rdirection; F", = gravity force componentin ther direction;F,, : shearforceconlponentin thi x direction;u, : point vefocityin ther direction;q, = lateralinflow per unit of lcngrhin the flow direction;andu. = velocityof lateralinflow inclinedat angleg to ther direction.
2 ' 7 2 C H A p - I F r7R: C o v e i r i n gE q u a l i o nosl L . n s t c a dl rl -l o \ r ' Expressionscan be derelopcd for each of lhe force Ierrns.Assurninea hydros t a t i cp r c s \ u f cd i s t r i b u t i o nt .h e p r e s s u r ef o r c e .F r , : F , , L 1 , , . a n d i s g i v e nb y
r,,, -
iJ .r-t
.ir
(7/r,..1)-\.r
7I
r, \
l\
(7.8)
in which l. : vcnical distancebelow lhe fiee surfhceto the centroidof thc llow cross-sectional arealA = cross-scctionalarca on which thc lbrce lcts; ancl,1lr,.: Idl''L\(r) Ilb(n) dn, which representsthe llrst ntomentof lhc areaabout the frcc surfacewith b : local width ofthe crosssectionat height rl lron) thc bottorr 0fthe channcl.Notc that the pressurefbrcc contribution arising liom a chrngc in erosss e c t i o n aal r e ad u c t o a n c x p a n d i n go r c o n t r a c t i n gn o n p r i s m l t i cc h a n n c li s j u s t b a l ancedby the componcntof prcssureforce on thc channrl banksin thc floq, direc t i o n ( l - i g g e t t1 9 7 5 ;C u n g e ,H o l J y .a n dV e r u c y l 9 8 0 ) . C o n s c q u e n t l tyh, c e v a l u a t i o n of the derivativeshown on the far righl hand sidc of Equrtion 7.8 ignorcsthe variation in channelwidth with .r anclcomes only from the intcgraldcllnition of A,, and the Lcibniz rule. The gravity force componenl in the.t directionis given by
F., = 7A -\,rJx
( 1. 9 J
i n w h i c hS n: b e d s l o p e: t a n 0 .w h i c hh a sb c e nu s e dt o a p p r o x i r r a tsci n d f o r smallvaluesof slope.Finally,thc boundaryshearforcein the.r directroncan be expressed as (7.l0) in which 2,, - mean boundaryshearstrcss:and P = boundlry wctted pcrimeter. On the momentunrflux sideof thc momeotum equation.the net convectiveflux of nromcntumout of lhe control volume can be $,ritten
- o ppv',r -\, r,'rolr, .' f i 'rr drLJ4 J
(r rr)
velocity. with B = momentum tlux correctjonfactor and y : meancross-scctional The time rate of changeof momentum inside the control volume for an incomDressiblefluid becomes , l t t
d
. I I p u , , t AJ r = p ; l v A l J t o{t I I oI
( 7. 1 2 )
Equations 1.8 to'7.12 into Equation7.7.dividingby pAr, andletSubstituting ting A-{go to zeroresultsin
#. *('uI).
= ua{rr.o)ro{ros1)t 4,u,c o s0
(7.13)
in whi;h Q : AVISr- friction slope = ro/(yR); R = hydraulicradius - A/P; and g. = lateral inflow per unit of length with velocity, uL, and at an angle of { with respectto the r direction.In ofder from left to right, the termson the left-handside of Equation7.l3 come from: ( I ) the time rate of changcof momentum inside the
C H 1 p T [ R 7 : C o v c r n i nE g q u a t i o nosf U n s l c a dFy l o *
2j3
c o n t f o l\ o l u m e . ( 2 ) t h c n c t l D o n t e n t u nf 'ltu x o u t o f t h c c o n t r o lv o l u m e .a n d 1 j ) l h c nct plcssuretorce in the.r ciirt'ction.On the right-handside. \\,e havc the contdbut i o n so f : ( l ) t h c g r a v i t yf o r c c . ( 2 ) t h e b o u n d a r ys h t a rf o r c e .a n d ( 3 ) t h c m o m c n l u r n f l u x o f t h e l a t e r a il n f l o r v a, l l i n t h e . rd i r c c t i o n E . c l u a t i o7n. 1 3r t ' p r c s c n ttsh e n t o r n e n tlrm cquationin conserr,alionlbrlrr fbr a prisnatic channel.This simply mcans that, if the terntson the right hand sidc of (7.13) go to zero, thc force plus ntomentum flux tenls on the lcft hand side oI the equationare conserved;and this rral' be the most appropriatefbrnr in * hich to apply sonle nuntcricalsolution schemes. Equation 7.13 somclimes is placcd in reduccd lbrm by applying the product mle of differentiation.subsrirutingfor 4,,1/rrfront continuity.and dividing through by cross-sectional area,A, to \ ield
# * r t u - r ) r , 1 I + ( Br ) v r 4 +e v . f + s + -
s,5
q'
. s . t'
Funhermore,the monrenlunt equation oftcn is gitcn for the case of F ABlilr - 0 for prismaticchannels:
AV
*
(1.14)
, u , c r ) . d -t Y )
aI' aV * ,t;; * -s g(si,- sr) + cos6 y) ar:I ! Qr,
I and
( 1. t5 )
It is intcrestingro note thar rhe two lateral inflow tcrms on the right hand side of ( 7 . 1 5 )i n c l u d ec o n t r i b u t i o nfsr o m b o r ht h ee , r n r e c t i v m r ' o n t c u t u Df l u x a n d t h e l o c a l changein rnomentum,respectively.The convectiveterm goes to zero if the lateral inflot\.is at right anglesto the main flow ($ :0), bul rhe local contriburionremains unless4,_: 0. Ifthe lateral inflow is zcro, and (7.l5) is rearrangcdas fbllows. the contribution of the variousterms in the monentunr equationwith respectto differcnt types of flow can be identified: J , = J n"
'
I
3y VaV ldV : - - dx g'ir 8ar
s l e 3 0 vu. n r l o r mr i o w I \ t e a J ) .B r u d u a l l y\ r r i e d l l u w u n . t e a d yg. r a d u a l l )v r r i e dl l o w
(7.l6) ] I
The steady,uniform flow case sirnply mcansthat ru : 7RSo.as derivedpreviously in Chapter 4. The momentum equation for steady,gradually varied flow can be derived in a more familiar forn by startingwith Equations7.2 and 7.13 $,irh the lime derivativeterms set to zero. The result in terms of d-r/dr, with dBldr = 0, is given by (Yen and Wenzel 1970)
d1 dr
so
St+
;QL t - F 2
c o s $- 2 P v ) ( 1. t 1 )
2'l1
C H A p r r , R7 : G o \ e r n i n E g q L r a t i oonfsU n s t c a dFvl o w
in which Fj - B\/)l(gD\ = momenlunr fornt of the Froude nunrberi and D = h y d r a u l i cd e p t h = A / 8 . E q u a t i o n7 . l 7 c a n b e u s e df o r s p a t i a l ) r ' r a r i c dl l o w o r f o r 'lhc steady.graduallr r aried flow with no lrteml inflow. diflbrcnccsbetwecn(7. l7) and the encrgy fonn of the cquationfor gradually varied flou. derir,edin Chapter5 l i e n o t o n l ) i n t h e d i f l e r e n td c f i n i t i o no f t h c F r o u d en u n b e r ( s i r h B i n s t e a do f r r l but also in the friction slopcS' which is defined as inl7R and replacesthe siopeof the cnergy grade iine S" in the energycquirticu.As a practicalnrartcr,both S.and S" are evaluatedbl, the Manning or Chczy equarions.but (he) havc differcnr Llcfinit i o n s( Y e n I 9 7 3) . The choice of depcndcntvariablesnay dcpend on rhe numerical technique applied to solve thc Saint-Venantequations.[n the prefcrred conscrvationform. u'ith dischargeQ and depth r. as rhc dependentvariablcs.Equarions'/.2 and 7.13 would be appropriatefor the continuity and n'tomenluntequalions,respcctivel),. Anothcr commonly usedform is the reduccdform of the continuity and momentunt equationswith r,elocityV and depthr,as dependentvariables.as given by Equations 7 . 5 a n d 7 . 1 5 .N u m e r i c a lt e c h n i q u easr e c l i s c u s s ei d n t h e n e x tc h a p t e r .
7.3 TRANSFORI\TATION TO CHARAC'TERISTIC FORNI The transformationto the characteristicform of the pair of partial differentialequations given by (7.5) and (7.l5) allows them ro be replacedby four ordinary differential equationsin the,r-rplane(r representsthe flow dircction and r is time). Much sinrpler,ordinan,differential equatjonsmust be satisfiedalong two inherentcharacteristicdircctions or paths in the.r-l plane in the charactcrislicform. Although numericalanalysisby thc methodof characteristics has fallen out of favor because of the difficulties involved in the supercriticalcasewith the formation of surges,it hasthe advantageof being more accurateand lendinga deeperunderslandingof the physicsof shallow water wave problernsas well as the mathematicsof requiredinitial and boundary conditions.In addition,the method of characteristicsis essential in some explicit finite difference techniques,specifically for the evaluation of boundaryconditions.Finally, the methodof characteristicsis useful for explaining kinematic wave rouring in Chapter9. There are t\r'o methods of arriving at a characteristic form: (l) taking a linear combination of dle momentum and corrtiluity equations and rearranging the terms and (2) performing a matrix analysis that relies on the fundamental mathematical meaning of the characteristicform. We begin with the first approach becauseof irs simplicity.We assumea prismaticchannelwithout Iateralinflow for the samereason. The momentum equation(Equation7.15) wirh the foregoing simplificationsis multiplied altemately by the quantity x(.Dlg)tt2and added ro rhe conrinuity equation (Equation7.5) to give two new equations,the solutionof which is the sameas the original pair- The quantity D is thc hydraulic depth given by A/B for a general
CuAprr-tR7t GolerningEqualions of Unstcadv Flow
2:'5
nonreclangularcross s!.ctionr',hereA is the cross-\cctjonalarea anclB is lhe top r r i d t h . T h e r e s u l t i n gt * ' o n e w e q u a l i o n se a s i l ya r c s h o * , nt o b e g i v c nb y
(v -' c) ., : c(so t ) l,t.1 v+ c ; 1 ] r + ;l;.' lv , l
lj,.,'.,*.],. i [*.(v
.tf
( 7 l. 8 )
= (7.re) ]u <s, .t)
i n * h i c h c = ( g D ) t t 1: w a v e c c l e r i t yi n a n o n r e c l a n g u J ca hr a n n e l a. s s h o w n l n C h a p t e r2 . O f p a r t i c u l a ri n t e r e sitn ( 7 . l 8 ) a n d ( ? . 1 9 )a r c t h e t w o o p e r a l o r a sppear_ ing in bracketson the lcft hand sidesofthese t\\o equations.First the operalorsare applied to dcpth _r.and lhen to velocity V in both equations,and rhcy differ only in t h e m u l t i p l i c ro n r h e . rd e r i v a t i v ew. h i c h i s g i v e n b y ( V + c ) i n ( 7 . 1 g )a n d ( V _ c ) in (7.19). This particularoperatorcan bc recognizedas rhe total or matcrialdcriy_ atire D/Dt found elservherein fluid ntechanicsoperatingon the density,p, in the continuity equationor on the velocity vector to give the accelerationin the equa_ tions of motion. In general,if a function/varies \\,jth both position..jr,and rime, r, the total derivativeis given by the chain rule ro be
Df
af
af dx
dt
dr dt
= * f
D,
(7.20)
Equation 7.20 can be interpretedto define the toral time rate of changeof the func_ tion / as seenby an observermoving through the fluid with spceddr/dr, with the first term on the right hand side of (7.20) giving the local changein/wirh rime and the secondterm reprcsentingthe convectivechange inl Applying this inrerpreta_ tion to (7.18) and (7.19),it can be said rhar Equation 7.lg iian oidinary differential equation that must be satisfiedalong a parh in the x-l plane describedby an observer nroving with the speed (V + c), whiJe Equation 7.19 nlust be satisfied along a path describedby a secondobscrveru jrh sper<J1V - c). Mathematically, the pair of goveming partial differcntial equationshas been transformedinto four ordinary differentialequationsthat have the same solution as the original system:
alongC I : CI: alongC2: C2:
/ o ' \ - c / o v \ - c(So- Sr) \D,/,
t \Di
dr CT
= (y + c)
_ t/ o _ r \I +
c /D Y \| | _
\ D r , / , g \ D r/ ,
dr dr
:(v
c)
(7.21a) (7.2tb)
: c(56 - 57)
(.7.Zlc) (7.2td)
in *hich the subscriptsI and 2 refer to the two total derivativeoperatorsdefined in (7.18) and (7.19) with two differenr
.'_h
C|l\ltlt,-.
C o \ , r f l r n !l r q . - r i t r . ,o, fnl.r r . t c . , t l lr- 1 *,
FIGT]RE7.4 Characterislics in the.r-rplancLlcfiningthe solutionsurfrcefor dcpthI = lt, r).
(y r'), respectively. The lnmily of charlcteristicsdefinedby ( 7.I Ib). along which (7.21a) nrusl bc satisfied.are designatedCl characteristics, rvhich have also been referrcdto asfrn|urd clnructeristicsor ytsitirc
C , \ p r E R7 : G o r e n r i n E g c l u a r i oonfsU n s t e a dFv ) o w
2.7j
N u m e r i c a ls o l u t i o nt e c h n i q u e lsi r r E q u a t i o n s7 . 2 1 a r e d e r e l o p e di n t h c n c x t chapter Thc assuntptionsof no lateral inl'low and a prisntaticchannel need not be n'rade. Thc rciaxationofthesc assunrptions sintply produccsailciitionalsourccternts o n t h e r i g h t h a n ds i d e so f E q u a r i o n is . 2 l a a u l 7 . 2 l c . i \ t r h e o r h e rc n d o f t h e c o n r p l e x i t y s p c c t r u mE , q u a t i o n s7 . 2 1 t a k eo n a s i n p l c r l b r n r f o r t h e s p e c i a lc a s eo f a reclangularprisrnaticchannel. Bccausec = (g).)r,l for this case. i1 can be shown that dry'd/- (2.clg)d<'/dt.from u'hich it follows lhat rhc characlcri\rrccquilrions leduceto
alongcl: CI:
I D r y+ 2 . ) ' l [-.''.' o, .1, t{t,,
'tr)
-dr, _ = 1 V + c ) OI
a l o n gC 2 : C2:
I D ( Y - 2 c )l
^-r.i dr dI
:(v
c)
= s(s,, s,)
' ( 7. 2 2 a ) (1.22b) (7.22c) (7.22d)
From Equations7.22. ir is clear rhat rhe firnction subjectto time variarionsis (y + 2c) along thc Cl characteristicsand (y 2c) along the C2 characreristics.This sugSests thc inleresringcase,althoughnol very practical.of the right hand sidesof ('7.22a)ancJ(7.22c) becoming zero so thar (y + 2.) and (V , 2c) become consranr along the Cl and C2 characterisrics. respectively.Such a sirnplificationforms the b a s i so f t h e s i n r p l e* a v e p r o b l e n tf o r u h i e h r n r l l r r c . r l\ r ) l u t i o n sc \ i s t . T h c s i m p l e wave problem is cxplored in morc detail Iater in this chapter. T h e p h 1 ' s i c l lc o n n c c t i o nb c t \ \ e e nc h l r r c r c r i s t r cd i r e c t i o n sa n d p a t h so f w a v e propagatronnorv should be clear. The n]oventcnt of clementary waves both upstreamand downstreamfrom a di\lurhilnce u ith ,,peeLJr rV + c) and (y - c) in subcriticalflow delineatcspathsalong which the charactcristiccquationsare \atis, f i e d .T h e c o m p l c t es o l u l i o nd e s c n b e ra s u r f l c c i t b o r el h e . r - l p l u n e t h a t g i v e st h e valuesofdcpth and vclocity for all_rand I, as illusrratcdin Figure 7..1for rhe depth. The propagationof wavesboth upstreamand downstrcamis limited to subcrit_ ical flow, whilc in supercriticalllow thc absolutespeedsof (V + c) and (V - c) result in downstreamtravel only as shown in Figure ?.-5,in which both the charac_ terislicsare inclined doq'nstream.
7.4 MATHEN,TATICAIIn\TERPRETATION OF CHARACTERISTICS As mentionedprcviouslv,a secondapproachfor transformingthe governing equa_ tions into characteristicfornr is a nratrixanalysisthat arisesfrom the mathematical intcrpretationof characteristjcs. Characteristics are definedmathematicallvas Daths
278
C l r { p r c R 7 : G o l c r n i n gE q u a t i o n rs. r fL n s t e a d yF l o , : t
1
A
B
A
(a)Subcritical Flow
B
(b) Sup€fcritical Flow
I-IGURE 7.5 Characteriitjcs in subcritical andsupcrcritical flow.
in the r-r piane. along which discontinuitiesin the firsr, and higher_order cleriva_ tives of the dependentvariablespropagate.physically. such disJonlrnurtiescorre_ spond to propagationof infinitesimallysmall u,avedisrurbancesin the limit. To translatethc ideaofdiscontinuitiesin derivalivesinto characlensttcform for the simplest case.the continuity and momentum equationsfor a prismatic rectan_ gular channel without lateralinflow are rvrittenin matrix lbnn as
v 0
r,l I .r',
t- J] g
l
d r 0 0 d l
o I - s7)
l_ ls(so
t ,l - i
Y,l
L
|
d, I
(7.23)
or' l
in which rhe subscriptsin the column vector on the leti hand side denorepartial derivativeswith respectto time, 1, and distance.r. The secondtwo equauonsln (7.23\ are simply rhe definitionsof rhe total differentialsof dcpth. r.. and velocity, v lfa unique solutionforthe dcrivativesexists,then from cramer'srule, lhedeterminant of the coefficientmatrix in (7.23) must be nonzero.Therefore.a condition fbr the solution to be indeterminant(and lor the derjvati\cs to be discontinuous) is that the deterntinantof thc cocfficient matrix is exactl] zero. Settingthc delcrmi_ n r n t t o z e r o r e . u l t si n d\' dt
-vt\6:v:tt
(7.24)
which describes the characteristicdirections. However. there is no solutton of ( 7 . 2 3 )u n l e s st h e d e l e r n i n a n to f t h e c o c f f i c i e n m t a t r i x $ i l h o n e c o l u m nr c p l a c e d b y t h e r i g h t h a n ds i d ev c c t o ro f ( 7 . 2 3 )a l s oi s z e r . , i n u h i c h c a s e .t h e s o r u t i o n has the indeterminareform 0/0 basedon Cramer's rule (Lai l9g6). Sellinq this deter-
C H ^ p r r , R7 : G o v e r n i nE g q u a r i o norf U n s t e a dFYI o w
219
r n i n a n tt o z c r o r e s u l t si n t h c c h a r a c t e r i s t iecq u a t i o n st h l t n r u s lb c s a t i s f i e da l o n g thechrractcristics Dy
.8,*;
c Dl'
o
=c(.Sn-S)
(1.2s)
in q,hich the total derivativesD/Dr lppear and are dcfincd along rhe Cl characteri s t i c w i t h t h e p l u s s i g n a n d a l o n g t h e C 2 c h a r a c t c r i s r iwc i t h t h c n t i n u ss i g n , a s before.Now thc transfornationof varirbles from I to c in (7.25) yiekls the charact e r i s t i ce q u a t i o n isn t h e f o n n g i v e np r e ri o u s l yb y E q u a t i o n s7 . 2 2 .
/.:
INITIAL AND BOUNDARYCOn'DITIONS 1'hc dcpcndenceof the solution10the characterisriccquationson initial conditions is illustratedin Figure 7.6. 1'hesolution for dcpth and velocityat the intcrscctionof Cl and C2 charactcristics at point Pdepcnds on knouledge ofthe initial conditions at A and B, as wcll as on all points bct*ecn A and B. As observerI proceedsfrom
Observer
Domainof !-IGURE 7.6 Domainof dependence andranseof influencedefinedbv characteristics in the.{ t plane.
t80
q q u r t i o n \o l U n s t c a d }F l o w C r { A p l E R 7 : C o v e r n i n -E
point A. C2 characteristicsemanatingfrom thc interval;18 continuously intersecl the path and alter the dcpth and \.elocity.In thc same$ av,0bserrer 2 reccires inforntation from the Cl characteristicsoriginating frtrm interval .{B unlil mecting o b s e r y c rI a t p o j n t P . A s a r c s u l t .t h e s o l u l i o na t P d c p c n d so n t h c i n i t i a lc o n d i t i o n s along the intervalAB. which is callcd thc intervl of dt,pcndcrir.c. In reality. an inf-rnite nunrberof characteristicscontinuouslyinterscctAP and BP so that the region ABP is called the donrait of dapertdencrlor point P. Fronr a differenrpoinr of Yicw. a s i n g l ep o i n t C o n t h e . r a x i s h a s i n i t i a lc o n t l i t i o n sr h a ti n l l u e n c er h e r e g i o nC e R becauseall the C I characterisrics coming from rhe lcft of CQ and all the C2 characteristicsintersectingCR from thc righl are influencedby rhe iniriat values at C. For this reason,thc region CBR is called rhe rorge oJ inllucnce. As a conscquenceof r"arc propagationin characteristicdirections.both initial conditionsand boundaryconditionsnrustbc specifiedcarefull),.Thc scneral rulc is t h a t ( h e n u n r b e ro f i n i t i a la n d b o u n d a r yc.o n d i t i o n sn u s t c o i n c i d cu i r h t h e n u m b e r ofcharacteristicscnteringatt:0forall-\oratbounclaries.r=0andr:Lforall time. as shown in Figure 7.7 (Liggett and Cunge 1975:Cunge, Hollr,. and Verwel, 1980).For the ini(ial condirions,we scc in Figure 7.7a lhar t\\o conditionsrnustbe specifiedat point A to derermincthe initial slopcsof the C I and C2 characteristics as given by (7.24). Wirh refercnceto Figure 7.6, the modific:uion of the initial slopesat A and B comes from pairs of initial dara spccilied on .48 unril lhe two characteristicsintersectar point P. At this point. the characrcristic.or compatibility. equations(7.25).are solvcd simultaneouslyfor the depcndenrvariablesar p. As this process marches fonvard in time, the solutions at subsequentintersection points depend less on the initial conditionsand more on infomrarion carricd by characteristics conring from the boundariesand hcnceon the boundaryconditions.
; I
c1
c2
, / | \ c'l I c2
;4C /;, c 1/ ' C 2 "
I
I
c1 c2 (a) SubcriticalFlow
(b) Supercritica] Flow
FIGUR,E 7.7 Spccificationof boundary conditions and initial cL]ndirionsiD subcrilicaland supercritical flow (after Cunge. Ilolly, and VeNey 1980). (.t.xrc?. Fisure used cott']est of Io\a Institut. of Hydraulic Rcsearch.)
C s r r r r t 7 : C o v c r n r nEgq u a r i o nosf U n s t e a dF! l o \ \ . l g l l n s u b c r i t i c afll o * , a s s h o u n i n F i t u r c 7 . 7 a .o . c c h a r a c l c r i s r icca r r i c si n l b r r . a t i o n upstrean at thc do\\ nslreanrbounclarv, .r = 1-.and onlv one charlctcristiclransrnits infornrationdo$ nslrcan into the solution domuin jrorn thc upstrclnr Dounoarl,at - r : 0 . l n o t h e r r r o r d s .o n l y o n ! ' b o u n d a r yc o n d j l i o ns h o u l db e s p c c i f i e da t b o r ht h e u p s t r e a ma n d d o r v n s l r e a rbno u n d l r i c sr n s u t r e r i t i r . f. lrol u . 8 1 , 9 6 1 1 1 ; 1t5N1o. b o u n d , ary conditionsnust be spccifiedat the upslrelnrboundaryrir cn bv .t : 0 fbr super_ c r i t i c a lf l o w l s s h o r v ni n F i g u r e7 . 7 b ,u h i l e n o b o u n c l a rcyo n d i r i o n sa r e s p e c i f i e d ar t h e d o w n s t r c a mb o u n d a r yf o r t h i s c a s e . T h e f a c t t h a t i n i t j a lc o n d i t i o n sh a v el c s sa n d l c s si n f l u e n c ea s t i r l r cp r o g r c s s e s m e a n st h t . i n s o m es i l u a t i o n s u c ha s t i d a l l ' l o w si n r ' s t u d r i c st h . c i n i t i a lc o n d i t i o n s neednot be knos n very accuratcly.so long as a startuppcriod is usedun(il thc solul i o n b c c o m e sd e p e n d e not n l y o n b o u n d a r lc o n d i t i o n sI.n r a p i d t r a n s l e n t ss.u c ha s t-rccurin hydroelectrictailracesor bcaclraccs. on the other hand. the initjal condit i o n s m u s t b e k n o u n v e r y r v e l l .b e c a u s et h c y u i l l i n f l u c n c et h c e a r l y p a n o f r h e s o l u t i o nw . h i c h i s r e r y i n t p o n l n ti n t h c a n a l y s i so l t h e t f a n s i e n t tsh a to c c u r .l n a d d i t i o n ' i f I i t l l c o r n o f r i c t i o nc x i s t s ,r h c i ' i r i a l c o n d i t i o n sc a n c o n r i n u e( o b c r e f l e c t e d from upstreamlnd downstreamboundariesfor a very lonr time. as the transient oscillltes about sonrc steadystate The initial and boundaryconditionsrhat are spccifledmusr bc indcpendentof one another.Specifyingboth the Ialue of the depth and its derivarivewjth time. for example. as initial conditionsdoes not satisty the condition of independe,nce nor does the specificationof both depth,!. and delit.r. becauscthey are relatedby the continuity equation.In gcneral.a stageor dischargehydrograph.or some relation betweenstageand dischargeas given by a rating curvc. can be speciliedas singlc boundary condirionsin subcriticalflorv.A rating curvc should nor be speciliedas an upstreantboundary condition.hou,ever.bccauseof the fcedbackbet\r.ccndeDth and dischargeas tine progresses(Cunge,tlollv. and Vcrqel, l9g0). A linal consequenceof characterislicsto be discussedhas tremcnckrusjnflu_ ence on some of thc nuncrical solution techniqucsdescribedin the nexr cnarrcr. By referring to Figure 7.8, we see that the chrracterisricsdeflnc a nttural coordi_ nate system which limits the size of rhe time step that can bc taken in a finite dif_ f-erence approximation.lf we attenpt to approxintatethe timc derivati\eo\cr x trme step AI > At. \r'e arc seekinga solution outsjdethe domain of dependenceestab_ lished by the characteristics(Liggett and Cunge 1975).The result is instabiliry in the numericalsolution, in which a snrallperturbationgrou.s $ ithout bound uniil it s w a m p st h e t r u e s o l u t i o n T . h e C o u r a n tc o n d i t i o n w . h i c h l i m i r s r h c t i m e s t e ps u c h that the numerical solutionstaysinsidethe domain of depcndencc.can oc sratedas
.
l
1r <
.
r
v : c
( 1. 2 6 )
Altematively, the Courant number C, can be deflned as lhe rario of aclual ware ' e l o c i i y t o n u m e r i c aw l a v ev e l o c i l y w , i t h t h e r e s u l t h a tr h c s l a b i l i t ' c o n d i t i o na. l s o c a l l e dt h e C o u r a n t - F r i e d r i c h - Lye $( C F L t c u n d i t i , , nb. e c o n r e 1 r 4 5 i n 1 11 9 7 5 , a
Y 1 c __,
\. \,
<<1
( 7. ) 1)
t8l
C H A p T E R7 : G o v e r n i n gE q u a t i o n os l U n \ t e a d vF l o w
V+c
FIGURE 7.E Limitationson thetimestepimposedby theCou.anlcondition.
Becausethe vclocity and wave celerity continuously changewith time, the time step must be adjustcdconstanrlyduring the nunterical solution processto avoid instability.
7.6 SINIPLE WAVE A simple wave is definedto be a wave for which So = St: 0, with an initial condition of constantdepth and velocity and with the water extendingto infinity in at least one direction.While neglectinggravity and friction forces may not be yery realistic, the simple wave assumptionis useful for illustratingthe solution of an unsteadyflow problem in the characteristicplane. Equarions7.22a and j.22c, the characteristicequationsfor a rectangularchannel, assumea panicularly sinrple form when the right hand side goes to zero. The result is
alongCl:
cl: alongC2:
Y+ 2c: consrant
(7.28a)
|o l: v * .
(7.28b)
Y - 2c = constant
(.7.28c)
C2: *=u-. OI
(7.28d)
which statesthat V + 2c is a constantalong the Cl characteristics and V - 2c is a constantalong the C2 characreristics. The constantvaluesin generalare different
C H A f T E R 7 : C o r e r n i n gE q u a l i o n so f U n s t c a d yF l o w
283
FIGURE 7.9 StraightlineC I characteristics for the simplewave.
fbr each characteristicand are calied the Riema n inrori.utts (Abbott 197-5).How_ evcr, the simplificationis even more powerful becauseit can be shown that y and c are individually constanlalong each Cl characteristic.all of which are srrajght lines, and tha( the C2 charactcristicsdegencrateinto a constantvalue of V 2c everyw,here in the r-t plane (Stoker 1957:Henderson1966). With rcferenccto Figure 7.9 and following the proof by Stoker( 1957),the initial C I characteristic, C!, is shou n at the boundaryof a constantdepth region. It is a straightline inasmuchas dr/dt - Vo + cu,where Vnand coarc the initial constant valuesof vclocity and wave celerity, respectivcly,as rcquircd by rhe conditions of the simple wave problem.The constantdepth regionextendsto the right of the initial cl characteristic. where the initial flow is undisturbed;this reaion is called the a o n eo f q u i e t , w i t h i n w h i c h b o t h C I a r r dC l c h r r a c t e r i s r i cr sr e , r i a i g h t l i n e s .e a c h with the santeconstanrvalue of depth and velocity.Wc extendr*o i2 .haru.te.istics front the initial C I characrcrisricto a secondC I characteristic. Cl , as shou,nin Figure 7.9. By definition of rhc characteristicequationslor rhe C2 charactensrics. we nrust have V - 2c - constant.so that Vp
2cr:V^
Vp - 2cp = V5
2r*
e.Zg)
2.,
(7.30)
but, by definition of the initial condition. ue also musl have l,/ = Vo andc, w i t h t h e r e : u l rl h a { 7 . 2 9 jr r r d 1 7 . J 0 1r i n r p l i f i r o Vr - 2ca = V, - 2c5
co,
( 7 . r3)
28.1
C H , \ p r E R 7 : G o \ . n i n g E q u a t i o nos l L l n \ l c a d )F, l o w
I n r d d i t i o n .a l o n gt h e s e - - o n dC l l c h u r a c t c r i s t i V c . + 2 c : c o n s l a no tr l,'* * 2c*:
V. 1 2r'.
(7.32)
A f t e r a d d i n ga n d s u b t r a i t i n g( 7 . - 3 1a) o d ( 7 . 3 2 1r. v e r e a d i l vc a n s h o w t h a t V * = V . t h a tt l r cs e c o n dC l c h a r a c t e r i s t a i cl s oi s a n dc ^ = c . , w h i c h l c a d . t o t h cc o n c l u s i o n a straight linc rlong \\hi.'h thc velocity and rvalc cclcrity, or dcpth. are constant. G e n t - r a l i z a t i ooni ( 7 . 2 9t . t 7 . 1 0 ) .a n d ( 7 . 3 1) t o a n y C I a n d C 2 c h i i r a c t e r i s t incc a n s . h e C 2 c h a r a c t e r i s t i ct hs e m , t h a t y - 2 . i s a c o n s t a n cr v c r y r r h e r ci n t h e . r / p l a n e T . c c a u s ew h c r ca s i n g l eC f c h a r a c t e r i s t i c s e l v c sa r c c u r v c di n s t e a do I s t r a i g hIti n e s b crosscsdillcrcnl Cl char?ctcri\lics.there must be diffcrcnt vllues of rclocity and dr-pth.as at R and P, for examplc; so a different sJopcis givcn by V c. The C2 characteristicsin f;rct nr, longcr iirc nccdcdin thc sinplc rvavcproblem il V - 2c i s c o n s t a r te v c r y w h e r er d t h e rt h a n . j u sot n i n d i v j d u a C l 2 charac(eristrcs. 1'hc regionof Cl characteristics a(ljlcentto thc initial Cl characteristicand the tore rcgiort.\/clocitics and zone of quict in Figure ?.9 is refcrr.cdto a\ the ri/r?/r1c wave celeritjesarc dcternrinc'dcomplctelyin this rcgion by the fact that y 2c constlnt e\ery\vhcreand tha( the slopcsof thc Cl charactcristicsare given by dr/dl = V * r'. If V and c are the velocity and wave celerity at any point in the simplc wave region.and if V,,and .0 arc the constantinitial conditions,the conrplctc solution fbr yand c at specillc locationsand timcs dcfined by the slopcsofthe Cl chara c t e r i s t i ciss e i \ e n b v V d.-
dr
2c-Vj
V*c=Vo
2co 2 c o *3 c
(7.33) (?.34a)
if the wave celerity.r' {or dcpth), is specifiedas a boundarycondition on the right h a n ds i d co f i 7 . 3 4 a ) .o r b 1
-dr= l i + c = 3 y d t 2 .
Y,, :+c" 2 "
(7 . 3 4 b )
if the velocity V is specified as a boundarycondition.Thus. boundary condition. expressedat,r - 0 in Figure 7.9 for all time determinethe slopesof the characteristics along which both c and y are indiridually constant.Obscrverslear.ingfrom .r = 0 carry r.,",ith them unique valuesof depth and velocily that can be locatedat a n y s u b s e q u e nt itm e i n t h c . r - l p l a n e . Thc sinrplc $,ave rcsion is applicablcto ncgativewavcs.u hich are formed by a smaller depth propagating into a region of larger depth. Becausea decreasing . e s i n l p l ew a r c r e s i o nc o n d e p t hr e s u l t si n a s m a l l e rr a l u e o l d r / d t f r o m ( 7 . 3 ' 1 a )t h sistsof divcrgingcharactcristicsin the.r-t planc.A positivcw,avc,on thc other hand, results in conlerging characteristics.which eventually inlersect and can form a surge for u'hich the as\unltion of infinitesimal wave disturbancesis no longer valid. A diffcrent sct of characlcristicswould bc rcquircdupstreamand downstream of the surge,acrossuhich there is an cnergy loss. In this casc. the surge can be treatedby the application ol the continuity and momentum equationsto a finite conlrol volume that has hccn made stalionary.as describedin Chapter 3. Numerical solutiontcchniqr,rtsl()r surgcsarc discussedin thc next chapter.
C H . \ i , f t , R 7 : C o r c | n i n gf : q u . i r i o n o s f [ , n s t e a d \F l o w
:g5
7 . t . T h c i n i l i t l l l o \ \ ' c o n d i r i o ni ns a n e \ l u a r vt r e E\\\tpLt: - g i r e nb 1 a r c l o c i t y 1 1 )= - 1l v s ( 0 . 9 1n / s ) , r n dd e p ( h! 0 - l l l l ( l . l n r ) .a s \ h o q n i n F i q u r e7 . I 0 . T h e b o u n d iir) condrtion ilt thc nlouth of the estultr) whcre _r = i) i\ !i\en by /rt I cosl
\ 6
n\
i
2 /
( 1 c , r 0- < I = l )
(7.3-5)
in $hich I i\ lime in hoursand J,r is thc dcplh in licr ar rhc left hand boundary.Finclthe deprhprolrlert/=Ihr. Soluti
10
l- 3 hr
3 fvs
20
,i,
ro I
6. 5 9
lz.oo 7.48 8.00
5
10
15 x, mtles
FIGURE 7.IO Simplcu ar e solutionof estuaryproblem.
186
C H A p r r , R 7 : C o v c r n i n gE q u n l i o n so f L n s t c r d \ I - l o w (7.13)T . h e i n t e r s e c t i oonf c a c h C l c h a r a c r c ' r i s r\ i\cr r h t h r ,t i n t el i n c o l t = ] hr dctcr_ mines the.t position of lhc depth and velercrtlitssocialr-d $ith thnl characlcfislic.and thus the depthprofilc as wcll as lhe velcrir] along the dr,lrh prollle arc delerntin!-dFor c x a r n p l et.h c c h a r a c l e r i , i t ihca tb e g i n si l t I = l h r h r \ r d c l l h o l T 0li (l.l nl) $ith. = 1 5 . 0f l / s ( . 1 . 5 8n / s ) f r o m 1 7 . . 1 5a)n d a s l o ; ^ -r t r / d r = ( t x 1 6 . 0 5 )+ ( 3 x 1 5 . 0 ) I = 9 . 9 0 f t l s 1 3 . 0 2n V s l l i o m ( 7 . 3 . 1 a )l l.s r e i o c i r ) t , -i i l x t 6 . 0 5 )+ ( 2 x 1 5 . 0 1 - - 5 . l 0 f t t s ( - l . 5 5 r r / s ) f l o ( 7 . - l l ) .T h t ' i n r e r s e c t i o\nv i r hl h c r i ) e l i n e = r, 3 hr is locatedar-r = (d!:/dr)x (tt t) :9.9O x I x j60{l/5180 13.5 nri (I 7 trn). So Jr a localjonof ll.5 nri (21.7 km) upstreamof lhe c'rruar\rlxrurh.the depth is 7.0 fl (2.I m) and the velocity is 5. l0 fr,/s( L55 ds) ar / : 3 hr.
Dam-Break Problem As anotherapplicarionof the mcthod of characterislicsapplicd to the simple u,ave, we consider next lhe suddcn removal of a r,cnical platc behind u,hich a known depth of water is at rest.The simple-wavesolutior]of this problcrn.rvhich Stoker (1957) refened to as rhe breakingofa dam. is oversintplifiedin comparisonto the solution of a realisticdam break discussedin more dettil in the next chaDter.How_ cver. it illustratesrhe applicationof a velocity boundaryconclitionand rh., formr_ tion of a surgein a submergeddownstreamrivcr bed. and providesfurther insight into thc unsteadydevelopmentof a negarivewave as interpretedby the method of characteristics. The next two exantplesare prescntedfollowing more closely the practical approachof Henderson(1966), u,ho rclaredthe dam-breakproblem to sluice gate operation and hydroelectric load acceptancein a headrace,than the mathematicaltreatnlentby Stoker ( 1957). E X A I U p L E ; . 2 . A v e n i c ap l l a l ei s f i x e da r t i m e/ . - 0 a t . {= 0 w i t ha c o n s t a n d te D t h of waterupstream equalto 1owhilethechanneldo\\'nslream ofthe gateis dry,asshown in Figure7.11.The waterupstream of rhe plateiniliallvis at rest.Ar r > 0, the plate suddenlyis accelerated to the left to a conslantspeedyp.Determine the simplewave profilefor thjscaseandalsofor thecaseof the plalebeingrenroved instanraneousty. Solution. The physicaland characterislicplanesareshownin FiAure7.1L The zone of quiet,denotedRegionI. is esrablr\hed on rhe righrhrnd sideof rhecharacreristic planeby a characteristic havingan inverseslopeofcoconesponding lo theinitialdepth )b sincetheinitialvelocityis zero.On the lefr bouDdary, whichis moving,thecharac_ teristicpath is describedas a straightline beginningat the origin \\,ithan inverseslope of - ye.Because the wateris in contactu il}| the mo\ing plale,it nlusthavea constant velocityequalto that of the plate.As a result,a constantdepthregronrs creareo y upstream of theplatebecause 2c must be a constant alongthepathof the plate. whichformstheleftboundaryin thccharacreristic plane.Because bothyand. areconstant,dr/dralsois a constant, so thatthe characterislics areparallellinesin RegionIII in FiSure7.1L In betweenthe zoneof quiet on the right and theconstantdepthregionon the left, the characterislics fonn a fan shapein RegionII dueto the decreasing valuesof &/dr = 3r: - 2cooccasioned valuesof depth.Fort harl,:terisirc by decrea\ing A0 in Region II, fo. example,the inverseslopeof the characteristicis fixed. The deprnls constant along the characterisric and dereminedfrom (7.3Ja)ro be.; : (l/3) (dr/dr)o + (2/3)co.The velocity,too, is constantalong the characteristic and equalto (2co - )c,r)
CH rpr I,R 7: CiovcrningEquetion\ oI ['nst!-ed] F-lo$'
,
I I
I
187
----'-i--s---
FIGURE 7.I I Sinple wave solution of vertical plate renlolal al consfanl speed lo left with a reservoir behindit (afterHenderson1966).(S.xrr.e:OPEN CHANNEL FLOIf bt llenlerson, Q 1966. Reprinted bt pernissiotl of Prcntice Hall, Inc., Upper Saddle Rivr
NJ.\
from (7.33).Sol\ ing for lhe depth profile in Region Il is onl)' a matterof fixing a series of valuesof dr/dl and determining the -r positionsof thc inlersectionsof a fixed time line, r : r,, wilh the characteristics. Then associatedwith each charactensticis a constantdepth and velocity, *hich can be cxlculatedfrom (7.3.1a)and (7.33), respectively. In Region III, we must be careful to avoid an impossiblesituationwhen specifying the constantplate vclocity. Vn.For example,along characteristicBBr in the con' stantdepthregion.the q a\'ecelerityfrom (7.33)is r, : co - Vnl2,which cannot be negative.The limiting case is c, = 0. for which fn = 2c,,.Hencc. we must have yp < 2c0 f o r l h e ! o n . l u n l d e p l hr e E r o nl o e \ i . t . The limiting caseof co = 0 is interestingbecauseit can be seento have a leading featheredge of the advancing *ave. which mo\es downstreamat a speed of zco, as shown in Figure 7.12. In fact. for this case,the plate can be eliminatedand inraginedto be removedinstantaneously becauseit has no real influence.For this reason,the situation shown in Figure 7.l2 has come to be known as the dam-breakproblem but could also apply to the abrupt raisingof a sluicegare.Note thrt the fan shapedRegion II has in Figure 7.12-Furt-hermore, the time axis has cxpandedand Region III has disappeared itself becomea characterislicalong which velocity and depth are constanl.It is easily shown from (7.34a) that. since d/dr : 0 for this characteristic.xe must have c : (2/3)cn or l = (4/9)rn = constaDtat -r : 0. Likewise. from (7.34b). t\e see that the constant
288
C H A p T E R ? : C o v e m i n gl l q u r t i o n so f U n s t e a d Y FIow
t^^
t'tGURE7.t2 Simplewavesolution of theinstantaneous dam-break problem *,itha drydownstream chan_
nef (afterHcnderson1966).(Soarce.OpEN CHANNELf'LOW b\ Hentlerson, @ 1966. Reprintedb'- pemission o;fPrentice-Hdl,Inc., LtpperSatltlleRn,er ,VJ.) velocilyat r = 0 is -(2/3)co.The resulrjs a constanr discharge per unirof widrh,q, at theorigin,whichcanbe shownto bc a maximum,givenby R _ qmu =;)0vg)o
(?36)
The coDstantdischargeoccursbecauseI y | = . at the origin u.ith the water verocrry equaland oppositeto the wavecelerity. FiDally,the waveprofilecanbe deduced from setringdr/dr = r/r in (i.34a).since thecharacteristics all issuefrom theorigin.The resultfor anylime tj is
u
: rn6
2\..4;
t.'7.3'l )
which can be seen to be a parabolatangenl10 the channel bed at the leading edge of the wave. ExA M pL E 7.-1. If the inirial condirion in rhe dam-breakproblem of Exanrple7.2 includesa submergeddownstreamchannelwith depth = ),! and no velocity,while Ihe upstreamdepth remainsconstantat,yo,find the wave profile at any time ,r if the plale suddenlyis removed,or a sluice gate suddenlyis raised,at r : 0.
C H . r t l r , a 7 : G o r c r n i n g E q u a l i o r )o\ f L l 0 \ t e a d )F l o w
r89
Y3+Ca
F]GURE 7.I3 Sirnple wave solurionof the instantaneous dam-break problem with a submergeddown_ sueam channel (after Henderson 1966). (Source. Opt,N CHANNEL FLOW bt.Henderson, A, 1966. Reprinted bt pernission of Prentite-Hall. Int., Llpper Soddle Rirea NJ.)
.So14fibr. Wilh reference ro Figure7.13.the simplewaveprofilecannorsimplyintersectlhe conslanldo\,\,nslream watersurface*.ith depih = ,),rbecausethis would cause a discontinuily in thevelocity.Thediscontinuit)canbe resolved only by theformation of a surgewith a speedV, to the left, as shownin rhe figure(Henderson 1966).The intersection of thesintplewaveprofilewith the backof the surgeresultsin a constant deplhrcgionandparallelcharacteristics. While l/ 2. still mustbe a constant for the simplewave,the surgemustbe analyzedseparately by applvingthe momentumand continuityequalions afterthe surgehasbcenmadestationary. asdiscussed in Chapter 3 andshownin Figure7.14.Theunknownvaluesin theproblemarenow q, )r. and y,. First.theconlinuityequation canbe writtenwi*l reference to Fieure7.1,1as
1,,1.=(4-t,).,:
(7.38)
Se.-'ond, themomentum equalionalsocanbe v,,nnenfor the stationary surgein Figure 7 . 1 4t o y i e l d
v, Vgr.,
f l.r'3/.r', ')] L2 .Yr\,t .
' (1.39t
290
C H A P T L R 7 : G o r e m i n gF , q u a i i o nosf U n s t e e J yF l o $
'
l
F-
v3
-Vs-Vz ys+
FIGURE 7.I4 Making thc surgein the dam-break problem stationary for momentum and continuity analysrs. in whichthe negative squareroot ha\ beentakento agreewith the signconvention in Figure7.13$at hasbothy, and4 in rhenegative r direclion.Finally,rhesimplewave equation belweenpointsA andB in Figure7.13mustbe satisfied so that
vj: 2^,Gi - 2^vEa
(7..10)
In principle.Equations 7.38through7.40canbe solvedfor thc unknownvaluesof V,. to presentthe solutionin dimensionless ,Il, and y,. Ho$ever.it is instructjve form, as in Figure7.15.Equation7.38is dirided by co : (g_r1)1/2 and solvedfor yrl.!. In rhe sameway,Equation7.40 is divided by coand solvedfor _y.,/_yo, keepingin mind that V, is inherentlynegativein the sign convention. Equation7.39 is solvedas a quadratic equationfor,r',/ro.The nondimensionalized solutionsfor yrlca,yr^r. and r!/)o from (7.38),(7.39),and (?.40),respectively, now canbe plorredas a functionof V1c.,,as shownin Figure7.15.For a giveninitial ratio of ].r/_)0, all threeunknownscan be determineddirectlyfrom the graphin Figure7.15.Nore thar.as _r,.,A,o approaches l. V. approachesco for a small wave disturbance.Also. V, is of the samesign as V, and alwaysis smallerin magnilude. As draun in Figure7.| 3, it is apparent thatpointsB andD bothmovedownstream with thedistance betweenthemgraduallyincreasing. Howcver,pointB couldbe posi, tionedto the right of the-r axis and move to the right in the upsrreamdirection.Under thesecircumstances, theconstant depthregionextends acrossthe.taxisandsubmerges the constantdepthof (4/9).ro thatorherwisewouldoccur,with the resulttharrhe dis, chargeat the originis smallerrhanrhe maximumvalucgivenby (7.36).Tle ljmiring case,ofcourse,is for pointB to lie exactlyon the.t axis,so that yt = -(2/3)coandr.l : (4/9)yo. We canshowtharthis limiringconditioncorresponds to !.,4,n= 0.138.lf 0.138,thenpointB nrovesto theright,$hile it movesto the left for 1./ro )r/)'oexc€eds < 0.138.For the lattercase.the Froudenumber,as seenby a stationary obsenerat x = 0, hasa valueof unity sothatthe depthmustbe criticalandthedischarge a maxr-
E q u a t j o no s f L J n s t e a dFyl o w C H A p T E R7 : C o r c r n i n - q
291
1.0 18
0.9
fz/fq'
0.8
l
Yz/Yt
o.7 0.6
'"
Va/Ca
14
)/'
Ya/Yo
10 0.4
.--
0.3
o.2 0.1
8
Vs/ca
6 4
X
2
0.0
0
'10 Vrlc4
I'IGURE 7.I5 Variationof surgespeed.depthbehindthesurge,andvelocitybr:hindthe surgewith theiniratio. tial submergence theflow to theleft of thepointr : 0 is supercritical nrum.Underthesecircunstances, asseenby an observer behindthesurgeasseenby a slationary observer andsubcritical nrovingat the speedof thesurge. generally;however,it illustrates lhe Tle simple$ave solutionis not applicable methodof characteristics in graphicalform. This will be usefulin inte.p.etingthe of the simplewaveno resultsof thenextchapter. in wbichthesimplifyingassumptions longerare made and numericalsolutionsof the govemingequationsin characteristic form are sought.
REFERENCES ' Abbott,M. B. "Method of Characteristics. In UnsteadyFlov in Open Channels,ed. K. PubMahmoodandV Ye\jevich,vol. I, pp.63-88.Fon Collins,CO: water Resources lications,1975. 1993. Chaudhn,M. Hanif.Open-Chaanel F/ow Englewood Cliffs,NJ: Prentice-Hall, Cunge,J. A., F. M. Holly, Jr., and A. yer'Ney.Practical Aspectsof ConlputationalRiter Lirnited,1980.(Reprinted by IowaInstituteof Hydroulics. London:PilmanPublishing HydraulicResearch. IowaCity,lA, 199,1.) Henderson,F M. Open ChannelFIow.New York: Macmillan, 1966. Lai, Chintu. "Numerical Modelingof UnsteadyOpen-ChannelFlowi' Advancesin Hydrovol. 14,pp. l6l-333. NewYork:AcademicPress,1986. sclence,
292
C H \ p T I R 7 : C o r c r n i n gL q u t l i o n \ o f U n \ l c i ] d yF l o \ l
LiSgett.J. A. Brsic Equltions of Un\tead) Flo$. ht Ltn:tttJt f'lotr in OpLttChttnnt,ls. ctl K. \{ahnrood rnd V. Y.\Jc!ich. vol. l. pp 29-61. Forr Collins. CO: \\'arcr Rcsourcr\ P u b l i c a t i o n s1. 9 7 5 . LigSetl.J. A.. and J. A. C unge. \unterical \lcrhods ol Solution ol lhe Llnstead].1--lo!\Equa t i o n s _ l n u a v e o r l r . F l . ) \i n o r c h a n n L , l s . e dK.. l v l a h m o o da n d V . \ t r j e v i c h . r o i . l . pp. 89-18:. liorl Collins. CO: \\hrer Resourccspublicarion\. 1975. Stokcr.J. J. \l'oter l\l^r,s. New Ytrrk:John \\'iler & Son\. 19.57 Ycn, Ben C. Opcn Channt'l Flow Equrljons Rcvisircd. J. Enqrg.ltltL.h.Dil. ASCE 99. no. Elrl-5{ 1973).pp. 979 t009. Ycn. Ben C.. and ll. G. \\tnzel, Jr. Drnanric Equarion: for Sleed,vSprrirll), \rrricd Flo*,.. " L H l l D l l . A S C E 9 6 . n o . l { ) , 3 ( t 9 7 0 r .p p . 8 0 1 - 1 , 1 . Yeljcrich. V lntroduclion. Lnsteath Fktrt in Open Chunnel:. ed. K. l\.lahmoodanrj V. Ycrjer ich, \ol. l. pp. I-1.1. Fon Collins. CO: \\,atcr Rcsourcespuhlications.1g75.
EXERCISES 7.1. Starting$ith Equalion7.11,deriveEquation?.1.1. 7.2. Derire Equations 7.18and7.19.thecharacteristic equations. 7.3. In the estuaryproblemgilen as Exanrple7.1,determineulgebraicallt. (not graphi_ cally) from the sinrple*ave merhodthetiDrein hoursrequiredfor rhedcprtrio drop to 6.50 ft al a distanceupstream ofr = 25,000ft. plot on the santeaxesrhc deplh hydrographs atr = 0 andr = 25.000fr. 7.4. Waterinitially is al rcstupstrcamand downstrcant of a sluicegate.which is completel) ciosedin a rectangular channei.The upstreamdepthis 3.0 m andthe down_ streanrdepthis 1.0m. Thegatesuddenlyis openedconrplctely_ Dctermine lhe spced of the surgc,rhedepthandvelocirlbehindthesurge,andthespeedof lengrhening of the constantdeplhregion.Showyour resuhsin bothIhe physicalplaneandthecharacteristic plane. 7,5. Waterflowsin a reclangular channelundera sluicegate.Theupslream depthis 3.0m andthedownstream depfi is 1.0m for a steadyflow.If thegateis slamnred shut.com putethe heightandspeedof rhesurgeupsrream of the gareandrhedeplhjust downstreamofthe gate.Linderwhatcondilions wouldthedepthdownstreamof ihegatego to zero?Showyour resultsin bothr-hephysicalplanean
C,r\o-lR
-
C . ' \ c n . - , i . q r u t r ' n " t, n f .tr.JrIl ,r
t9l
7 . 7 . P r o ! et h a lt h c I r m r t r n gc o n d r r r oonf \ r / \ - 0 l : l 'r thc 'rrnpleuare dam break prob letu deleflnrne)!{hcther the aonstint dapti : n r p o r nIt r n F r g u r e? l J t n t o r e s upstreamor do* nstrcam 7.E. ln the srnple l are dam breakproblcm. denr e i- :rprerrronfor tie dischargcpcr unll o f * r J t h a t r = 0 . q o . a s a f u n c ( r o no f t h e r a : , r f I n r r l adl e p t i s .r o / r , . \ o n d i m e n sionalrzr91,as qt/(,)t) and plor lhe re\ulLs. 7,9. Det.nnine (he mir\imum pos\iblc heighr of tl.: ,;rge. I,rr - Ir). in Er.ample?.3 in lemrs of lu. and the valuc of r.r/ru for *hich rt :r:urs. 7.10, The headracefor a turbine is a long rectangula.i.anal that feedswater from a lake to the Iu.binepenst()cks.Supposethat the designc:iharge for a turbineis 68 ml/s. The canal *idth rs ll m. and the canal slope rs ver., ,:all !rrrh an a\e.a8erlater depth in the canal of 1.5 m \ai0r no flow. If the turbrr rs broughton line suddenly (load whal will be lhe depth of no,, ar thc Jownstream acceptance). end of thc canal whcre lhe penstockinlet is located?Ho\r long u ill rr :ie for the negative*ale lo reachlhc rescr\oir rI the canal is 2 km long? 7.1l. Suppox that lhe lurbine in Exercise7.l0 is oprlr:ng at steadystateat a dischargeof 68 mr/s. and the conespondinS normal depth oi !.low in the canal is 3.0 m. lf the turbine is shut down suddenly(load rejection). * h.::r ill be lheheightof $e sur8eat the downstreamend of the canal? 7.11. lf a dam ' ilh a maximum *ater depth of 54 ft fails abruptly.estimar the time required for the surge to reach a communily 5 t.lt:do$ nsream of tie dam. \l'hat will the surgeherghrbel Inilially. the do\anstrearnr-.rerhas a ncgligible\elocity and a 'hat faclorsaller vour estrmate!.L1d deDlhof 5 ft. \ in ,*hatdirection?
CHAPTER 8
NumericalSolutionof the Unsteady
FlowEquations
8.1 INTRODUCTION Severalnumericaltechniqueshave been developedto solve the panial differential equationsof unsteadyflow. Some of thesetechniquesare more applicablein specific typesof engineeringproblems than in others.The purposeof this chapter and the next one is to introduce the engineerto the more commonlv used techniques, especiallythosefound in well,known commercialcodes,as well as ro assistin identifying the most appropriatetechniquefor a given problem. This chaprerconcentrateson solving the govcming equationsdevelopedin Chapter ? with no major simplifications.Chapter 9 considerssimplified forms of the gorerning equarions and correspondingnumerical solution techniquesthat often are employedin some types of flood routing problems. The methodsdevelopedin this chapterdependon approximationsof the derivatives in the goveming equations, either in characteristicform or in the original partial differential form. The major difference between the two approachesis tiat the derivatives are approximated on the characteristic grid along rhe characteristics themselvesin the caseof the characteristicform of the equationsor on a fixed rectangularr-l grid in the caseof the original panial differentialequations.The former case is refened to as the method of characte ristics,while the laner includes both explicit and implicit finite difference methods.Explicit finire differencemerhods advancethe solution to the end of the time step at a single grid node, using an explicit tunction of the dependentvariablesalready determinedfor several grid nodes at the beginning of the time step. Implicit methods, on rhe other hand, approximatethe derivativesusing valuesof the dependentvariablesboth at the beginningof the time step, where they are krown, and at the end of the time step
296
Cnrlttl
8: \LrnlericalSolutionof the Unsteady Flo\\'Equarions
fcrrmore than one grid node,u'herethe dependentvariablcsare unknown. In the latter case,a systemof simultaneouseqLrations must be solvedto adlance the solution by one time step. The method of characteristicsgenerallyis utilized only in special cases,often as a check on some other nrethod.However,it frequentlyis used in explicit finite differencetcchniquesfor a more accurateapproxinlationof boundary conditions. The explicit finite differencemethodis usedin problenrsof rapid transientssuchas those in the headraceor tailraceof hydroelectricturbine operarions.A nunrberof readily available codes such as BRANCH (Schaffranek,Baltzer. and Goldberg l98l), FLDWAV (Fread and l-ewis 1995),and UNET (U.S. Army Corps of Engineers 1995) inrplement the implicit finite differencenrethodto solve problemsof flood routing and dam brcaks. ln contrastto finite differencemethods.finite clement methods approximate the solution rather than the differentialequrtions by using polynonial shapefunctions that dcpend on the unknown nodal valuesof the dependentvariablcs.In rhe Galerkin approach,the residualsbetweenthe approximatesolution and the exact solution are minimized by integratingthe product of weighting functions and the residualover the solution domain of finite elementsand setting the result to zero. The finite element method has bcen applied to problems of discontinuousopen channelflow causedby surges(Katopedesand Wu 1986).Becauseit is not usedas extensivelyas finite differencemethodsin widely availablenumericalcodesfor the solution of the one-dinensionalunsteadyflow equations,it is nor discussedhere. In finite differencetechniques,the continuousgoverningequarionsare approximated and transformedinto discretedifferenceequationsllhercfore, it is essential that methodsbe chosen for which the truncationerror in the approximationof $e equationgoes to zero as the time step,At, and spatialstep..&. approachzero.Such a condition is refened to as colrsiste/rc)(Ames 1969).Without consistency.extraneous terms that are incompatiblewith the original differential equationsmay be introduced.Consistencyalone,however,may not guaranteethe ultimate goal of the finite differenceapproximationof the governingequations,which is for the solution to approachthe true solution as Ar, Ar -J 0; that is, converge.If small errors such as roundoff errors grow during the numericalsolution process.then the solution becomesunstable,so that the enor grows without bound, swamping the true solution and preventing convergence. The pitfall of instability is rhat a perfecrly reasonablefinite differenceapproximationcan lead to garbagefor a solution.The remedy may be to place limits on the discretizationsize,but in some casesit may be necessaryto switch to a completelydifferentapproximationscheme.Such difficulties can cause unacceptablemistakesthrough uninforned use of commercial codes or casualprogramming of seeminglystraightforwardfinite differencetechniques,as is illustratedin this chapter. Application of numerical techniquesin hydraulicshas become commonplace as computershave become more powerful. Desktopcomputersthat are as fast as mainframecomputersof a only a decadeago are very affordable.In addition.cumberso:ne batch processing and text-basedoutput have been replaced by userfriendly program interfacesand colorful graphics.In this environment of readily accessiblecomputationalhydraulics,the importanceof proper calibration and ver-
CHApTER8: Nunrerical Solutionofthe Unsteady Flow Equations 29'/ ification of numerical models of unsteadyflow cannot be overenrphasized. Application of any of the numerical techniquesdiscussedin the next two chapters rcquirescomparisonsof computedresuitswith measurements in the field or laboratory and appropriateadjusttnentof calibrationfactorsthat have physical validity. The model also should be verified with entirely different data sets from thoseused in the calibrationprocedure.It is insufficientto acceptnumericalnrodel resultson the basisof qualitativesimilarity with what might be expectedto occur. Engineers must be demandingof numericalmethodsand neverignorethe crucial link between numericalanalysisand laboratoryand field data.
8.2 I\{ETHODOF CHARACI'ERISTICS As discussed in Chapter7, the transfonnation of the equations of unsteadyflow into characteristic form givesrise to two familiesof characteristics that form a kind of naturalcoordinate s)'stemthatis partof thesolution.In thenumericalmethodof characteristics, a numericalsolutionof thc tlansformed equations is soughtalongthe characteristic directions, Cl andC2.With reference to Figure8.1,*'e seeka solution for velocityy anddepth) at pointP in thecharacteristic planeasa functionof vadablesat L and R. For simplicity,the rectangular the valuesof the dependent
ty
XL
xp XR
X
FIGURE 8.1 Numericalsolutionof the govemingequationsin characteristicform on the characteristic gnd.
298
Flow tquations Cs,qpret 8: NunrericalSolutionofthe Unsteady
channel case is considcred so that the characteristicequalions to be solved are Equations 7 .22 of Chapter 1 . The equations can be wrilten in integrated form as Vp+2cp=Vt*2cy
.l,'
8(So
Sr) dt
xp-xL+f'{u*.o ) , Vp- 2cp: Vt z.^+ x p : , r R+
s/)dr f s(So
[,_'tr,
oo,
(8.1) (8.2) (8.3) (8.4)
in which c : (.gr-)'o.Nowby evaluating the integralsapproximatelyusingthe rule,Equations8.1to 8.4 in discreteform become trapezoidal V r i 2 c r : V r - r 2 c , + j ( r " - r r ) [ g ( S ,- S 7 p )+ g ( S o- S 7 r ) J ( 8 . 5 ) (8.6) xc - xt : ! Q, - tr)(V, * cp 't V1 r c1) V p - 2 c p : V ^ - 2 c a+ j ( r " - r * ) l g ( S s- S y " )+ g ( S o- S 7 n ) l ( 8 . 7 ) Xr - xn : L,(tp - t;(Vp - c, I V^ - c^)
(8.8)
The discreteforms given in Equations8.5 to 8.8 alsocould havebeenobtained from a forwarddifferenceapproximation of thederivativcs, asdescribed in Appendix A, and takingthe meanvalueof the integrandbetweenpointsZ and P and pointsR andP. In an1'case, the resultis four nonlinearalgebraicequations in the values Vr, unknown x", tr, andc", whichhaveto be solvedby tteratron. One methodof iterationbeginsby seningV" andr" cqualto yL andc., respectively,on theright handsideof (8.6) andequalto V^ andc^, respectively, on theright handsideof (8.8)and solvingfor,t" andt". Then,by setting+p - S/Lon theright handsideof (8.5)and S/p= S/non therighthandsideof (8.7)andusingtheinitial valueof t" just computedin the previousstep,(8.5)and(8.7)canbe solvedfor V" andcr. Thereafter,the new valuesof V" andc" aresubstituted on the right handsides of (8.6)and(8.8)to obtainnew valuesof .r" andt"; then(8.5)and (8.7)aresolved for thenextvaluesof V" andc" in iterativefashion.LiggettandCunge(1975)suggestedan iterationmethodwith improvedconvergence properties thatdefinestwo residualfunctionsfrom (8.5)and(8.7)thataredrivento zeroby Newton-Raphson Iterauon. Regardless of the iterativeprocedureused,the solutionis obtainedon the :-r grid at irregularintervalsdetermined as shownin Figure by thecharacteristics, properties if theyareknownonly at fixed 8.I . Thisrequiresinterpolation of channel grid intervalsof x. In addition,the final solutionmustbe interpolated if the water surfaceprofile is desiredat a specifiedtime or if a stagehydrographis requiredat a specifiedlocation,for example.This inconvenience is overcomeby the Hartree method,alsocalled the methodof specrfedtime intervals.
CHApTER 8: Numerical Solution of lhe UnsteadvFlow Eouations
299
^{
A R
I t
c
l ,\x
FIGURE 8.2 Numericalsolutionby the methodof specified time intervals on thecharactedstic grid overgrid. laid on a rectangular
In the Hanree method, fixed time intervals and uniform spatial intervalsare specified,and the solution at point P is projectedbackwardin time to points R and S, as shown in Figure 8.2. In this case,the generalnonrectangularcross sectionis considered,so that the equationsof interest are (7.21a-d). The finitc difference approximationsof the derivativesalong the characteristicsare substituted,and the resultingdiscreteequationsare given by
vp- vR+fi {-t"- r^) = s(s,- .s/^)Ar rp - xn = (V* + c^)Ar V, - V, -.q {,," - .rr) : g(Se- S75)Ar rc-rs:(Vr-cr)Ar
(8.e) (8.10) (8.1l) (8.12)
To avoidileration,tie valuesof {, andwavecelerity,c, in (8.9)and(8.I 1) areevaluatedat pointsR and5, wheretheyareknown,asaretherighthandsidesof (8.10) and(8.12).Strictlyspeaking,the Hartreemethodusesa second-order approximation in which the mean valuesof Sr,c, and y 1 c betweenpointsR and P and in the finitedifference approximation. The betweenpointsS and P are substituted first-ordermethodsuggested by Wylie and Streeter(1978)is presented herefor
300
CHApTER8: NumericalSolutionofthe Unsteady Flow Eqtrations
simplicity. In either case,the valuesof the dependentvariablesat R and S havc to be determinedby interpolationbeforethey are known. [f linear interpolationis utiIized, for the velocity at point R with referenceto Figure 8.2, *'e have
Vc-Vn Vc-V^
- r(V* + c*)
Ax
in which r - Al/i1!r and Equation8.10 is used to substitutefor (r" ilar intcrpolation,the value of cRcan be obtainedfrom
= r(V^ + c^)
(8.l3) r^). In a sirn-
( 8 .l 4 )
If we solve for V^ and c^ from (8. l3) and (8. l4), we have
vR
V, L r(
Vrc^ + crV^)
| + r(V,-
V^|
cr- c^)
cc*rV*(co-cc) cI
| + r(cr-
( 8 .l 5 ) ( 8 .l 6 )
c^)
In the same manner,the valuesof V" and c" can be obtainedfrom V, I r\crV" VJ
ctVc)
l+r(-Vr*Vu*cg-c6)
gqjj.Yr(.. _, ,,,)
(8.r7) (8.18)
l*r(c6-c6)
These interpolationsassune subcriticalflow, as shown in Figure 8.2, and would have to be rederived for supercritical flow. Now, with the dependent variables known at R and S, we can subtract(8.1I ) from (8.9) and solve for _1"to give
r I + f(v" + .".rl-f "t'": (." * .r) .r'"c, frsc*
v ) (S/R- ^ , . l l .(8.19) .ts)Arl.l^ . ^ .
Then,it followsfrom (8.9)thatyp is givenby Vp=Vn
/
St \
l D
-
(^8
\'D
\
/
l-
( S , "- S e ) g J r
(8.20)
Equations with the interpola(ion 8.19and 8.20,together equations, canbe usedto solveexpiicitlyfor y and,yat all interiorpointsof ther-r plane.beginningwith the initialconditions specified on the-raxis.At therightandleft boundaries for subcritical flow, (8.9)and (8.1l) aresolvedsimultaneously with the boundaryconditions. The tirnestep,however,mustbe chosensuchtha(theCourantconditionis satisfied for all grid pointsat a giventimelevelunlessthemodifications suggested by GoldbergandWylie ( 1983)for an implicittimelineinterpolation are implemenred.
CH -\pTt,R 1l: Nunre.rcalSolution of thc Lnsteady flou Eouations
Headrace
301
:----+ o(l)
Fieservoir
Turbine Ho- constant
o= o0(r)
tL
!++
Pftrl-J, I'IGURE8.3
Boundary conditionsfor the hydroclectricturbine Ioad acceptanceproblem.
8.3 BOUNDARY CONDITIONS As an initial illustration of the applicationof boundary conditions, considerthe hydroelectricturbine load acceptance/rejection problem shown in Figure 8.3. The reservoirat the upstreamend supplieswater to a headracechannel,which in turn conveysthe dischargeto the turbine shown schematicallyat the downstreamend althoughin reality it is at the bottom of the pensrocks.When the turbine is brought on-line in a relativelyshorttime, a netrtive wavepropagatesupstreamfrom the turbine and then is reflectedback as a positivewave.This is the load acceptanceproblem. In contrast,the load rejectionproblem occurs \\,henthe turbine is shut down in a finite but shon time interval,causinga surgeto propagateupstream.In either case, the boundary condition set by the reservoir ar the left hand boundary is the maintenanceof a constantvalueof headH = H^ in the reservoir.At the downstream boundary,the turbine dischargeis specifiedas a function of time. 0 : O0(r). Considerfirst the upstreamboundarycondition at r : 0, illustratedin the characteristicplane in Figure 8.3 bclow the headraceentrance.Onll the backward(C2) characteristicfrom 5 to P is of interest,and the equationto be satisfiedalong that characteristicis Equation 8. I l, which can be rearrangedto give P|o Y" - -. = V, -:
SYc
g ( S r 5- S e ) A r- K ,
(8 . 21 )
in which the right handside,designated r(, for convenience, is knownfrom the solutionat the previoustime stepand the interpolationequationsfor point S given by (8.17)and(8.18).The boundaryconditionis appliedas a conditionof energy
302
C s a p r E a 8 : N u m e r i c aSl o l L l t i oonf l h c U n s t e a dFyl o wE q u a t i o n s
conservationat lhc enlranceto the reservoirwhere,ftrr simplicity, the entrlncc Ioss coefficientis neglected: r/l
(8.22)
Hr=),+;
Equation 8.22 is soived for -\'pand substitutcdinlo (8.21), which is rearrangedto give a quadraticequation in V,, that is solvedby the quadraticfornula:
u":arf' *
(8 . 2 3)
At eachtime step,(8.23)givesthc valueof V" at -t : 0 and (8 2 | ) or (8 22) prov lgl u eo l r ' " . d u c e rt h ee o r r c s p o n d i n is shownin Figtheforward(C| ) charactcristic boundary, At thedownstream : xL The equagiven at x is by ure 8.3 wherethe boundarycondition Q QoQ) (8.9), as which can be reananged is tion to be satisfiedalongthecharacteristic
v,*T
= u^*
?
- so)ar = Kn s(srr
(8.24)
asK* andobtainedfromtheinterwhere,thistime,therighthandsideis designated theboundaryconpolationEquations8.t5 and8.16.Fromthecontinuityequation, be stated as ditioncan
(82s)
Qolt) : VpAp
on the unknowndepth, areaof flow tbatdepends in whichA, is the cross-sectional given shape. On solving(8 24) channel parameters for the yo, andon thegeometric equationin Ip. (8.25), nonlinear algebraic the result is a into for Vpandsubstituting the Newtonat eYery time step, be applied condition must Sincethis boundary by function f defined with the for its solution, Raphsontechniqueis chosen
- o"(-T . Fop): co(,)
(^)
(8.26)
iterationat eachtimestepis givenby Thenthe Newton-Raphson . , r + l - - . - ,-r .tP .tP
r(Y*') l..,b,!)
(8.27)
the valueofyp at the ,hh iterationlk + I des,t indicates in whichthe superscript (t + I ignatesthe valueof )p at the )th iteration;andf is the first derivativeof the (8.26) for evaluated functionF in )p at the kh iteration.Equation8 27 is iteratedat negligiblysmallchangein y", afterwhich V" can is some eachtime stepuntil there be solvedfrom Equation8.25. Additional boundaryconditionsare shownin Figure 8 4 Illustratedin Figure 8.4ais the specificationof stage,,, or depth,-r,asa functionof time at the upstream
CHApTER 8: Numerical Solution of the UnsteadvFloq' Eouations
303
P1o oP2
O = lo9p2l Ratingcurve (a) RiverFloodRouting
Mainstem
YP1=YP2=YPz Qp1 + Qp2 = Qp3
(b) RiverJunclion
Qp1= e.l3')Cd(2dltzLlyn - P)Y Qpt = Qpz Qpt+
-_-_;-ep2 P2
P'l
(c)Weir FIGUREE.4 conditions for unsteady Additionaltypesof boundary flow.
endof a river reach.In this instance,Equation8.I I is solvedfor V", giventhe value of yp at the boundary.In a typical flood routing problem,the upstreamboundary conditioncould be of this type or, altematively,it could consistof the specification of O(l) as the inflow hydrographto a river reach.The downstreamboundarycondition in a flood routing problemalsomight be a stageor dischargehydrograph,but relationshipas illustratedat the downin somecases,it could be a depth-discharge streamendof the river reachin Figure8.4aandgivenby ^ - | r, - r/-. \ U-^PvP-Jo\lPl
(8.28)
in which/o(v") is a specifiedfunctiondeterminedby a gaugingstation,a weir, or some other control that could include uniform flow. Equation 8.24 for the right
304
Soltltionof lhe UnsleadyFlow Equatlons CHAP'tER8: ..-umcrical
hand boundaryis multiplicd by A" and rearrangedto producea funclion, F, that can be solved for -r'"utilizing Ncwton-Raphsoniteration:
++ r0,")=/.,(r,")
A,Kr
(829)
Tbe valucof V" followsfrom (8.28t. maybe required,asillustrated conditions irtcrnalboundary In sorncsituations, junction formedby two tributaries shows a in Figures8.,1band 8.4c.Figure8.4b in energy is lostandthedifferences river. If no significant flowingintoa mainstcm conditions bccome internal boundary velocityheadaresmall,the -\'Pt Aptvpt+
=
Ap2Vpl-
,) Pl
:
,rPl
AplVp3
(8.30) (8.31)
reprewith unknownvaluesof r'"', -r'".,,lra,Vrr,Vrr, and Vn,.The thrceequations with two forward(CI ) charsentedby (8.30)and (8.31) aresolvedsimultaneously I and2 andoneback$ard(C2) characwrittenfor tributaries acteristicequations teristic equation\r ritten for the main stem. The two forward characteristic pointsfor the two interpolation in termsof two separate are expressed equations iteration. canbe solvedby Newton-Raphson The systemof equations tributaries. For a weir or spillway,as shownin Figure8.4c,thereare t$'o grid points,Pr by a negligiblysmalldistancewith unknownraluesof -t'",,V",, andP2,separated yp, just upstreamanddownstream of the weir.The solutionfor thisinterand )F2, equation theforwardcharacteristic conditions, nal boundaryrequirestwo boundary of the equationdownstream upstreamof the weir andthe backwardcharacteristic arewrittenas flow. Theboundaryconditions weir for subcritical
g ", = i cot/zst(_y",- p),', Qn-Qcz=AptVpz
(8.32 ) (8.33)
in which P and L : heightandcrestlengthof weir and C, : dischargecoeffiat R and S haveto be written in termsof two cient.The interpolationequations velocityand depthare determinedin the for which and P2. points, Pr separate step. previoustlme Both the boundaryconditionsand the characteristicequationsfor the interior They also in this sectionas first-orderapproximations grid pointsare expressed by Liggett and as described approximations, as second-order expressed could be for the should be used (19?5), order approximation case the same but in any Cunge grid points. for the interior as conditions boundary for subcritical flow.If in Figure8.4 aredescribed All theboundaryconditions the flow is supercritical,both unknownsmustbe specifiedon the upstreamboundary, while no boundaryconditionsare specifiedat the downstreamboundary,as equations for theCl andC2 characexplainedin Chapter7. Thetwo characteristic boundary' at thedownstream teristicsaresolvedsimultaneously as Boundaryconditionsare specifiedusing the methodof characteristics the characteristics and numerical method of for both the chapter in this described explicit finite difference methodto be discussednext. Otherwise, instability or of the yariablesat the boundariescan result.In the implicit finite overspecification
CHApTER8: NumcricalSolutionof the Un-.teady Flow Equations 305 difference mcthod, both inlcrnal and cxtemal boundar) conditionssimpl;,bccorne thc additional compatibilit)'equationsnecessaryto sol\e the matrix r'cluationsat each time step, as explainedlater in this chapter.
8.4 EXPI,ICIT FINITE DIFFERENCEMETHODS Althoughexplicitfinitedilfercnce techniques arerelatirely simpleto program, they with instabilitythatgo beyondthe satisfacarc fraughtwith difficultiesassociated considerthe cornputational molecule tion of the Courantcondition.As anexample, moleculedefinesthegrid pointsused illustratedin Figure8.5a.Theconrputational approximltions of thc dcrivatires for a particularnumerical in the finiredifference The grid pointsareidentifiedby thesubscripts i andsuperscripts t to indischeme. respectively. For example, catespatialintervalsandtimeintervals, 1f-r is thediswherex cretevalueof thedepthat a distance of (iAr) from the left handboundary, : 0, if uniformspatialintervalsareused,andat a time that is (k + I ) time steps from the initial time of t : 0, but the time stepsmay be nonuniformdue to the requirements of the Courantcondition.The timeand spacederivatives in theorigon thisgrid andwithinthecominalpanialdifferential equations areapproximated putationalmolecule,which is appliedrepeatedly for all the interiorpointsat any giventime step. For the computational moleculeillustratedin Figure 8.5a,an unstablefinite are approximated as differenceschemeresultsif the V andy derivatives
av At
':
a v _ v : + t - v ; _ t. Ax 2Ar
a
d,r'_ -rl- ' - -"1 t l t
(E.34)
d-v_ )f-r - _vf-r Ax 2-\-r
(8.3s)
wherethe dependentvariablesy and) in Figure8.5 are represented by the general aresubstituted into thereduced functionl Ifthesefinitedifference approximations (7.5)and(7.l5), we canshowthat form of thecontinuityandmomentum equations, (LiggenandCunge1975).In somecases, the resultingsolutionusuallyis unstable stability can be achievedby artificially increasingthe friction terms,but in general it is betterto avoidthisscheme.
Lax Diffusive Scheme With nrinormodification, theunstable scheme canbe madestablein theLax diffumoleculeshownin Figure8.5bno longerusesthe sivescheme.The computational point (i, l) in the evaluationof the time derivativebut someweightedaverageof the
306
CHAPTER 8:
NumericalSolutionof the UnsteadyFlow Equations
(k + 1).\t klt
(l- 1)Ax lAx (l+ 1)Ax
(t- 1)Ax llx
(a) UnstableScheme
(b) Lax DitfusiveScheme
Ax (k+ 1)Al
(kr 1)lt
(k + 112)Jt - - t -
(k- 1)ar
klt
( l - 1 ) A x t A x ( t+ 1 ) " r x
Ax
--tLt/zl
kAt
(l+ 1)Ax
+112 .1t2
11i Itf_, ( l - 1 ) . l xi l A x l ( i + 1 ) A x (i- 1/2)Lx (i+ 112)Lx
(c) LeapfrogScheme
(d) Lax-WendroffScheme
FIGURB 8.5 Computational molecules for someexplicitfinitedifference schemes.
solution at adjacentgrid pointsat the lrh time level. Using the generalfunction,/, to representthe dependentvariables,the derivativesbecome
ar ri.' lxri* 7 c:.,* r:-,t) At
At
af, d.r
2L,
(8.36) (8.37)
CHAp-tER8: NumericalSolutionoflhe Unsteady Flow Equarions 3O]' in which X is a weighting factor betrleen 0 and l. For ,1,= 1, wc recoverrne unstable schemeiwhile for X = 0, we har,ca pure diffusive schemecalled the Lax diffusive scheme,which is stableso long as the Courantcondition is satisfied. If the finite differenceapproximationssuggestedby (g.36)and (g.37) with x : 0 are substitutedinto the rcduced fclrm of the continuitv and momenlumequations as given by (7.5) and (7. I5) for a prismatic channelwithout lateralinflow, the result is two differcnceequationsthat can be solved explicitly for depth and velocity at the grid point (i, t + l) with the "free variables"or cocfficientsevaluatedas the meanof the valueson either side of the grid point (i, t): \t t l'/ r' . r _ |, r \ . ^ r* ,r . r : 2 1 . \ ,r f. ., .vri. 'r ) ' : _f a. . \{ -
-
'_
r/l
I
]r,.* 7tr;.,-r',,)
L)rul., - ul r vir,= Ivi., + v! ) -*
( 8 . 38 )
(4j1-)rvi-, - v1_,; + st'so ,.,[qd.
+ (si)f ' 2
(8.3e)
in which S, = QlQl/K, and K - channelconveyance. The absolutevaluesiqn applicdto O in rhedefinitionof the fricrionslopeS,ensures rhepropersignl-orrie shearforcefor flowswith changingdirecrions. Liglett andCungi 11975)showthar the Lax schemeis not consistent, sincethe finite difference approximation intro_ ducesdiffusivetermsthat should nor appear.but it is stableprovidedthat the Courantconditionis satisfied andaccurare so longas(It]:/Jl is smallenoughthat thediffusivetermsdo not influencethe solution. The l-ax diffusive schemecan also be appliedto the preferrcdconservatron form of theconrinuityandmomenrumequations asgivenby (7.2) and(7.13) for a prismaticchannel.The differenceequauonsare
e!.'- Iai-,+Al.,)- ftrcft, - oi-,1
(8.40)
- (* . *r"), o!"=;@:, +et.,)#J+ * ,o4):., ,] +a,rdi.,d +l , , ) \
2
/
( 8 . 4 1)
in whichd - gA(S0- S/).The sourceterm d hasbeenevaluated as the meanof the valuesat poinrs(i - I . ,t) a-nd(t + l, t) assuggested by TerzidisandStrelkoff (1970)andChaudhry(1993).The valuesof andA aredetermined at eachtime e step,from which the valuesof velocity, V, and depth,)., can be calculatedfor the given channelgeometry,and the valuesof Ah. follow from its definition for the givenprismaticchannelshape(seeTable3-l,) for usein thenextrimesrep.
108
8 ; N u n r e r i c aSlo l u t i o on f t h e U n s t e a dI l' l o wF q u a t i o n s CHAPTER
Leapfrog Scheme scheme' Anothcr explicit nlcthod that has been used extcnsivelyis thc leapfrog gcnof the ln ternls 8 5c in Figure shou'n rvhich has t'hccomputarionalnrolccule by e!aluated are eral function,f, thc timc and spacederivatives
at t:'ti' 2lr
At
;tfI:. -f: ,
'
2l.t
ar
(8 . 1 )2
lf the finite and any coefficientsare e\aluated at (i' t) (l-iSgettand Cunge I975) consetvltion into the (8'12) substitLttcd are clifferenceapproximationsgiven by resultform of the continuity and momentumequationsas for the l-ax schenle'the given by ing clifferencecquationsfor Q and A are
A r r l- A r - r- i l t o l - '
(8.43)
Qi ,)
e : . ': e : ' ' * l ( * + s A l , . ) ' . , ( * - ' * ' . ) , , 1 + z r r o i
(8.44)
for the sourccterm expression in whichd - pA(Sn- S) asbeforeAn alternative (i, 't l) in a weighted can be deueloie,Jusing'thegrid points(i' k + l) and fashionso that implicit-explicit
( 0 1 '- +I o i
di='At[s,#
2
')l )
(8.4s)
with (LiggettandCunge1975) ln comparison in which K : channelconveyance the Lax diffusivescheme,the leapfrogschemeis of secondorderratherthanfirst thatis' no diforder,providedthatAx andAt ari uniform,andit is nondissipative: useof necessitates This wave front' a of smearing cause terms numerical fusionlike 5omeartificialdampingtcrmsto simulatesleePwa\e fronts' Lax-Wendroff Scheme The Lax-Wendroffschemeis developeddirectly from a Taylor's seriesexpansion with thecontinuityandmomentumequatrons in the time directionin combination havebeenwntten equations point, the governing to this form.Up in conservation vectorform: the in write them to is convenient sometimes but it out separately,
au*s=srul
(8.46)
in which l A l u - ll t^t lI ;
l
| . F ( u :) lI 0 - '
o1_
,rr]'
(84?) sru)= lrers.os/)]
CHApTER 8: NunrericalSolution of rhc UnsteadyFlow Equations
309
The Taylor's seriesexpansion for Ur* I is develo;rcd tround the known values of Ut as rr{+l
ru r ,r - r. _ fa u l ' j l
I L dI j,
: r r i a : ul r :
l _ , L,tl- ),
(8 . 1 8)
in which all terms bcyond the second-orderterm are droppcd.Valuesfor the first and secondtime derivativesthen are expressedin rermsof F(U) and i(s derivatives, using the original equationsgiven by (8.46). Finally, finirc differenceapproximations are substitutedfor the r derivativcsof F (seeAmes 1969).The resultingdifferenceschemecan be simplified and is cqui\ alent to a two-stepmerhod(l-iggett and Cunge 1975;Abbott and Basco 1989) in which the Lax diffusive schemeis used in the first half of the timc stcp at (* + l)fr and rhen rhe leapfrog method is applied in the secondhalf of the tinte slcp. The computationalnolecule is shown in Figure 8.5d, jn rvhich thc circles represenrrhe grid points involvcd in the first stageand the x symbolsidentify the computationalpoints in the secondstageof the scheme. Applying the Lax,Wendroff scheme to the continuity and momentum equa(ions in conservationform resultsin first-stagedifferenceequationsgiven by
Aiiii:=:@:,,* ol)-
., - o:) za,A(of
(8.49)
o i ; t i : : : @ : .+, o : ) , - f |[ ( * * r o r , ) : .-, ( # " * " ) , ] AT \ Q , - t + Q , ) 1
1
(8 . 5 0 )
Equations8.-19and 8.50 are applied a second rime to obtain valuesof A and Q at the grid point (/ - i, [ + i.). as shown in Figure 8.-5d.In rhe secondsrageof the scheme.the valuesdcternrinedat the half rime step are utilized in a leapfrogtype of evaluation.as given by
A l ' :' A l - f O i : 1 , o ' -l r l ; ) f
( 8 . 1s )
- (#. *,.): o:''=Qi *"[(#-'*,.):_', ;] +JI
(d1lL+ : : d 1 l ,ir)
(8.52)
The Lax-Wendrofftwo-stepschemeis of second order and dissipative(diffusive) for shoder$ave componentsonly. so that it has been usedto model moving shocks (surges),as is discussedlater in this chapter.This propeny of the method tendsto smooth thc $ avy water surfacebehind rhe surge.However,fbr a hydrauiicjump in steadyflow, instabilitiescan (rccurat the jump, so that some additionaldissipation is neededthrougha "dissipativeinterface"(Abbott and Basco 1989)or an artificial r iscosity(Cunge.Holly. and Verwey 1980).
310
CHApI ER 8: n'umericalSolution of the UnsleadyFlo\-\Equations
Predictor-Corrector Ntethods For unstcadyflow problenrsinvolving regionsof both supercriticaland subcritical flow that move with time (mixed-flo$ regintesor transcriticalflow), computing can introduceseverenunrericaldifficulties.The predictorthroughthe discontinuities involYe a two-step contputationlt eachtime step in which there methods corrector is first a forward sweep in the spatial direction to carry the influenceof upstream boundaryconditronsin the predictorstep followed by a back\\'ardsweepin the correctorstepthat propagatesthe effect of downstreamboundaryconditions.The MacCormack schenre(Fenncrnaand Chaudhry 1986)is a good exampleof this classof ncthods in which the two-stcpcomputations,with referenceto the vector form of the equationsin (8..16).are givcn by
u i - ui
- - . ( F ) - , Fi) + llsi
u ;: u i
fltol-Ff ,)+lrsf
( 8 . 5 3)
I t
(8.5'1)
in which the 2 superscriptrcfers to the valucs ol lhe variablesconpuled in the predictor step and the c superscriptrefers to the valuesdeterminedin lhe corrector step. Note that the spatial deri\atives use only two grid points and they are computedas forward differencesin the predictorstepand backwarddiffcrcncesin the correctorstep.The order of the pr€dictor and corrrec(orstepscan bc revcrsed at every other time step, but Chaudhry (1993) suggcststhat the predictor step shouldbe in the directionof the advancingwave front. At the end of the predictorcoffector steps,the solution is taken as the mean of lhe prcdicted and corrected values:
u i - ' - I ( u 1+'u , )
(8 . 5 5 1
Fennemaand Chaudhry( 1986) and Carcia-Navarro.Alcrudo. and Saviron ( 199?) appliedthe MacCormackschemeto transcriticall1ow in opcn channels.albeit with differentadaptiveartificial viscosity schemesto dissipateoscillationsat surgediscontinuities.These dissipationmethods are consideredadaptivebccausethey are appliedonly in regionswhere the water-surfacegradientsbecomelarge. Meselhe, Sotiropoulos,and Holly (1997) introduced a predictor-conector schemederivedby replacingthe panial derivativesin the governingcquationswith T a y l o r s e r i e . a p p r o x i m u t i o n\r' e n l e r c dl r o u n d t h c S r i d p o i n t r i + l t l o r . p r t i a l derivativesand (t + i.) for the time derivatives.Their MESH schenreusesonly two points for evaluationof the spatialderivativesand allows for inrplicit evaluationof the source tcrms. It also employs artifrcial dissipation terms in the predictorcorrectorcquations.Sinrulationsof choked flow over a channelbottom hurnp followed by a hydraulicjump as well as a jump on a steepslope downstreamof a slope break agreedwell with analytical solutions.They also showed satisfactor) performanceof the numericalscheme for supercrilicalflow on a steepslope followed by a hydraulicjump upstreamof a weir locatedmidway along the channel.
C H A p T E R8 ; N u n r e r i c S a lo l u t i o on f t h e U n s t e a dFyl o * E q u a t i o n s 3 l l passageto supercriticalflow downstreamof the weir, and anotherhydraulic junrp downstreamof the weir, which moved upstreamwith time due to a rising tailwater ievel. For further detail on thesepredictor-cofiectormethods,refer to the original papers.
Flux-Splitting Schemes Flux-splitring schemestake advantageof the characterisricdirectionsof the governing equations.The vector fonn of the equations,given by (8.46),can be rewritten in the form
lV*af:srur
(8.s6)
where the matrix A is the Jacobianof l-(U), wbich is refened to as theflux vector becauseits componcntsconsistof the massflux and the force plus momentum flux per unit of density.The Jacobianmatrix is given by
AF dU
1,, tn
r l
l -o-' - - 2- '0 1 A ' A ]
(8.57)
ascanbe verifiedby rhereaderin theExercises. We canshowthattheeigenvalues, I, of the matrixA in factaretheslopesof thet$ o characreristic directions givenby y 1 c by settingdetlA AI] : 0, in whichI r. rheunit matrixwith diagonalvaluesof onesandzeroesfor all othcrelements (seetheExercises to thischapter). The diffcrenceevaluationof the flux, AF, can be evaluatedapproximately as AAU. whichcanbc splitintopositiveandnegative pans,corresponding to thelocalcharacteristicdirections.Then spacederivativesinvolvingthe positiveand negative componentsof A are evaluatcdby backwardand forward finite differencesto preservethe directionalpropenies of signalpropagation in subcriticalandsupercritical flow. Fennemaand Chaudhry(1987)haveappliedthe Beamand Warming scheme,which is of this type,to rhe dam-break problemdescribed in Chapter7. The flux splitting schemehas been modified and improvedfurther by Jha, Akiyama,and Ura (1994,1995).A variationof it hasbeenintroduced by Jin and Fread(1997)into the NationalWeatherServicecompurerprogramFLDWAVfor regionsof mixedflow (supercritical andsubcritical)nearthe criticalstate,with a movinginterfacebetweenthem.Suchsituationsarisein rapiddam breakswith largedifferences betweenupstream anddownstream depths.
Stability A completediscussion of stabilityis beyondthescopeof thisintroductlon to numerical methodsfor the unsteadyopen chanrielflow equations.Stabilityanalyses
312
C H A p T E R8 : N u m e r i c aSl o l u l i o n o f t h U e n s ( e a dFyl o $ E q u a t i o n s
involve substitutionof Fourierseriesterms for the solution into the finite differcnce schenreand detenrriningwhetherthe penurbationsincreasein amplitudewith timc (instability).Such classicstabilityanalysestypically are applied to Iinearizedforms of the equations.so that nonlinearinstabilitiescan bc found onJy throughnumerical experimcntation.Sufllce it to say that, for any explicit scheme.the Courantcondition must bc satisfiedfor stabilitv: \r <
Ar V ! c l
(8.s8)
The Courant condition seemsto imply that Ar can be increasedto keep the tirne stepsfrom becoming too small. Horvever,the Koren condition for explicit nrethods. which results from the explicit treatmentof the friction slope evaluation.placesa l i m i t o n t h e s p a t i a ls t e p s i z e a s w e l l ( H u a n g a n d S o n g 1 9 8 5 ) .U s i n g n u m e r i c a l cxperiments,lluang and Song show that the Koren condition is applicableto the method of characteristicsas well as to exDlicit schemes.The Koren condition is srven Dv
At<
Vl -rrr;
I
8so F,, "yo
(8.s9)
in which Fo - Froudenumberof initial steady,uniformflow of velocity,Vo,on whicha disturbance is supcrimposed andSo: channelbedslope.lf theKorenconditionis combinedwith theCourantconditionsothattheCourantnumberis exactlv I, thena maximumstepsize,A-rmar, is givenby SeA.r.o"
- (V' + rq - r)1r+ r',,;
(ri.60)
in which 1,0- hydraulic depth of uniform flow. Becausethis limitation on A.r can become somewhat restrictiveat small values of the Froude number. Huang and Song (1985) suggestedseveralsemi-implicitmethodsfor evaluationofSrthat ease t h i sc o n s t r a i n t . While severalother explicit schemeshavebeen used successfully.the onesthat have been presentedprovide a sufficient illustration of applicationsin unsteady open channelflow. lt is not advisableto extendan explicit schemeto the evaluarion of the boundary conditionsbecauseol ambiguitiesand redundanciesthat can occur. The method of characteristics is better suitedfor the boundary condirionsin combination with the explicit scheme for interior grid points. In general. explicit schemesmay seem easierto program than other merhods.but the combinationof characteristics-based boundary conditions,the need for anificial dissipation,and the stabjlity constraintson explicit methodsmake them more demandingto implement than may first appear.The applicationof the explicit method is limited to relatively shon-duration transients,such as occur in hrdroelectric turbine or sluice gateoperations.for example,becauseof the limitation on the time stepimposedby
C H A p ' t I R8 : N u r r ] e r i cSaol l u t i o on f t h eU n s t e n dFyl o wE q u a t i o n s - l l l the Couranl condition. Explicit merhodsordinarily are not applicd to flood routing problcms in large rivers, which more oftcn are lreared by inrplicit methods. as dcscribedin the next section,becauseof their nrorefavorablest;bility propenres. E x A \ t p L E 8 . t . A h y d r o e l e c t tr ui cr b i n ei n c r e a s ei tss l o a dl i n e a r l yf r o m0 t o 1 0 0 0 cfs (28.3nrr/s)in 60 sec.The headrace channelis rrapezoidal wirh a lengthof 5000ft ( 1 5 2 0m ) . a b o r t o mw i d r ho i 2 0 . 0f r ( 6 . 1 0m ) . s i d es l o p e so f 1 . 5 : 1M , a n n i n g .as = 0.015.anda bedslopeof0.0002.Compurethedeplhhydrographs at.r/l = 0.0.0.1.0.,1. 0.6,0.8. and 1.0.usingthe nerhodof characrerisrics with specitied lime inrer\alsand the l-ax diffusivescheme. So/l/ion. The channellengthis dividedinto50 spatialinlervals, andlne ume sleprs selected so thattheCourantnumberis = I for all grid nodesat the curent trmelevel. The Lax diffusivemethodis appliedto the conscrvation form of ihe go\erninsequa tions.The merhodof characterisrics is usedfor theboundarycondirionifor borhrnirh_ ods.The upstream boundaryis a rcservoirfor whjchthe energvequation ts writtentor flow from the lakeinto theentrance of the headrace channcl.The downstream boundary condirionis a discharge hydrograph with a linearincrease in turbinedischarge fiom zeroto tic steadystatevaluein a specifiedtime.which is 60 secin this example. At hme t = 0, the waterin theheadrace is at restwith the same$.atersurface elevalion as the resenoir. resulrsareshownin Figure8.6afor theLax diffusirenrethod. andFigureg.6b _ .The for the rnethodof characteristics. A veryrapiddecrease in depthis observed at thetur_ binesduringthe stanupperiod,thenwe seea moregradualdecrease to the nunimum depth.This is followedby a gradualapproachto the sready_state depthar bolh the upstreamand downstreamboundaries. The solutionsare nearly indistinguishable exceprar rheminimumdeprhregionat theturbine(x/L = 1.0).The rrinimumdeerhfor theLax schemeis 4.67ft ( L42 m), $,hileit is ,1.94ft ( L5 I m) for thenrcthodoi char_ acferistics. Therealso is a very slightwideningor smcaringof rhe minimumdepth regionb! rheLax scheme, duero diffusion.whichmay accountfor theslightlysmailer mininrumdepth.
8.5 IMPLICIT FINITE DIFFERENCEMETHOD The implicit methodutilizesmore than one grid value of the dependentvariablesat the forward time in the computationrlmolecule,a! shown in Figure g.7. ln Figure 8.7, the computationalmoleculeis a.,box" used in the preissmannmethod(Cunse. Holly, and Verwey 1980).The spatialderivativesare found as weightedaverages'of the first-order difference approximationsat the two time levels with a vadtble weighting factor,0, while the time derivativesdepcndon the differencein the arirh_ metic averageof the grid values at each time level (or a weighting factor of 1). Specifically,for any function/ the spatialand time derivarive,or" *.,,r.n u, df
dr
,/f[+.r-
f{*r\ + /l AI
0)ui-,- f)
( 8 . 6 )1
10
x/L = 0.0
N v
.e.
fxtt =t .o
!
_
10
x/L=O,1 x / L = 0 . 2 ,0 . 4 ,0 . 6 , 0 . 8
20
30
40
Time,min (a) Lax DitfusiveMethod 10
A8
i o o .
FS*
xlL = O.0
v
,-e.
/xrt = t.o
2
_x/L=0,.1 x l L = O . 2 , 0 . 40,. 6 ,0 . 8 0 10
20 Time,min
30
40
(b) Methodof Characteristics FIGURE 8.6 Depth hydrographsbetweenthe reservoir(i/L : 0.0) and the turbine (-ta : 1.0) for load acceptance. 3t4
CHApTER 8: Numerical Solutionofthe UnsteadyRow Eouations
315
(k + 1).\t
ll
ktt
lAx
(i + l)Ax
TIGURE8.7 Prcissmann implicitscheme.
af f l - ' + , f l ; i ) f l + , f 1 . , ) At
2^t
(8.62)
while the evaluationof the coefficientsin the governingequationsis given by
o(fi.' + fi:i) + (l - 0xlf +/l- r)
(8.63)
Thesefinite differcnceapproximations areappliedto the continuityandmomentum equations in theconservation (7.2) and(7.13\for prismatic form of Equations channelswithout lateralinflow.In the vectorform of Equation8.46,this can be writtenas
u f - r + u l i i - u l - u f . ,+ 2 + l d' ( F i i r-, F l * , )+ ( l - 0 ) ( F , F f) l A-r - a 4 o ( s f - s, +l i i )+ ( l - o x s+l s f - , ) l (8.e) in which the vectorswerepreviously definedby (8.47).Theseequarions are nonlinear,especiallyin theevaluation of Sr,whichdepends on thedependent variables K. In addition,it is much easierto work with Q andA as well as the conveyance stageZ anddischarge variables in a naturalriver.Therefore, the Q asthedependent simplersystemof governing equations derivedfrom (7.3)and (7.13)is givenby
dz ao B- + ::0 At
AQ
A (Q,\
r-r\;/
dx
A7 ool +sA:+8A--0 ox A-
(8.65) (8.66)
316
C H A p T E R8 : N u m e r i c lSl o l u t i oonf l h e U n s t e i r dFyl o wE q u : l i o n s
in which Z : stage : :, 4 ,r and :6 = bcd elevation,is used nrorc often in thc irnplicit mcthod. al(houghthe systenlis not stricrlyconservatire. This usuallyis satisfactoryas Iong as thc implicit methodapplied to this form of rhe equationsis not uscd to modcl ven stcepwave fronts.where conservationof mass and nlonlenluul must be observednrorestrictly (Cunge,Holly, and Verwey 1980t. Nore rhat thc bed slope,5n.does not appearexplicitly in (8.66),bccauseit has been incorporatedinto dZld-r,sinceS,,: -6:r/itx where:, = bed elevation.If the implicil approximrrions o f ( 8 . 6 1 ) ,( 8 . 6 2 ) ,a n d ( 8 . 6 3 )a r e a p p l i e dt o E q u a t i o t ' r8s. 6 5 a n d 8 . 6 6 .t h e r e s u l r i n g algebraicdif'ferenceequationsare
E ( z l - 1+ z f : l t
z:
zit;)
+ : l r i a { o i i , 'o f ' ' )+ ( l o ) ( o l -. , o l ) l= o ; @ i ' ' + o i i-io i " o i - , )
(n.67 )
-(*),.].,' .*{,[(+): ,,[(?), (*),]] +!
- zf-')+ (t o)(z:,, - zl)) gA:.e1zi;l
.' lf:-flrn:.' n:.'t+er:; er:i'1) + 9 r 9 o : o i t +o i ,o , -f , ) )
(8.68)
in which the coefllcientswith an overbarare cvaluatedaccordingro Equation8.63. Equations 8.67 and 8.68 form a pair of nonlinear algcbraic ecluationswith four unknown values at the forward time level. By extension,the conputational molecule will yield 2(N l) equationswith 2N unknowns as ir is applied repeatedly with overlappingacrossthe grid in the ,r direction,where the rotal nunlberof computationalpoints is N and the number of reachesis (N I ). The remainingtwo equationsmust come from the boundaryconditions,and the slstem of equations has to be solved simultaneously. Solution of (8.67) and (8.68) is accomplishcdby the Nellon Raphsontechnique for multiple variables.If we dellne rhe left hand sidesof (8.67)and (8.68) as G and H, respectively,the equationsfor eachapplicationof the computationalmolecule can be written as
G , ( 2 "Q " Z , , r Q , * , ) : 0 H , ( 2 "Q " Z , * , . Q , . t ) = 0
(8.69a) (8.69b)
w h e r ei : l , 2 . . . . . N I f o r ( N - l ) r e a c h e sT. h e s u p e r s c r i p tosn r h e d e p e n d e n t variablesare omitted for conveniencebecausethey all are (t - l): that is, we seek
C H A p r r , R 8 : N u n r e r i c aSl o l u t i o no f t h e U n s t e { d } ' F l o $E q u a t i o n s 3 t ' 7
t h e s o l u t i o nI i r r t h c s cl b u r u n k n o r v n sa t t h e ( l - l ) t h t i r l e l c v c l i n t e r m s o f t h e kntrs n valucsof the clependcntvariablesat the (th time Ievel.Each nonlinerr Iunction has a subscriptbecluse the known valuesare diifcrent for each applicltion of the cquations.Thc tuo additionalequationsnr'r'dcdfronr the boundaryconditions al:o can bc cxprcsscdas flnctions set to zcro. For cxample.ln upstrean specified sta-.^ehydrograph and a do$nstream specified slage-dischargerelationship are gl\ en Dy Zj(t) = g
(8.69c)
B,:2"-f(Q\):o
(8.69d)
Bt = Zt
in u hich 2,,(l) is the specifiedstageas a function of time and the stage-discharge refationship.or ratin!:curye. is givcn by Z : ftQt. T h e g e n e r asl o l u t i o no f t h e s y s t e mo f n o n l i n c a r c q u a t i o nrse p r e s e n t ebdy ( 8 . 6 9 ) can be obtainedusing Ncwton's iteration method. Thc solution begins u,ith cstimates of the unknown valuesof Z and O that u ill result generallyin the right hand sidesof the systemof equationsin (8.69) being nonzero.or in other words, having residuals.At the rth iteration,this can be expressedas
Blz'1.Qi\ - Bi
(8.70a)
c , ( z ' iQ . ' i .z ' i r , .Q i _ , ) : G ' i
(8.70b)
H , ( Z iQ , ' iZ , i-,): Hi , " ' * ,Q
(8 . 7 0 c )
B,\(Zft'Q\) : 8""
(8.70d)
in $hich I - 1,2,...,N I a n d t h e s u p e r s c r i prt r e f e r st o t h e p r e s e n vt a l u e so f the unknownsand the functions at the nth iteration. Note that the residualson the right hand sidesof (8.70) simply are the evaluationsof the functionswith the nth estimatesof the unknowns.To obtain the (d + I )th estin)atesof the unknouns, the functions are expandedin a Taylor serieswhile retainingonly the first derivative terms. For example.for the ith continuity function, G,. we have
aG: . ac: aG'; c i ' ' = c : - - . : J Z , ' . ^ [ ' Q .' ^ - ) 2 , . , dl,. dZ, AQ, ,
aG: ^ ^- lO,.' dQ,.
18.71)
,
in which AZ, = (z)^'t Qi)^: lQr : (.Q)"-t - (Q)^: L2,,, - (Zi*)"'t ( 2 , - , ) " , a n dA O , * , = ( O , * , ) ' * ' ( 0 , * r ) ' . E q u a t i o n so f t h e f o r m o f ( 8 . 7 1 )c a n b e u ritten for each of the original nonlinear equations in (8.69). Then. as in the NewtonRaphson techniquefor a function ofone variable, we set (G)'*r and all sirnilar functions to zero to obtain the root. The result can be reanangedas
aBi u',t''*
aBi -Bi a...'JQ'-
acl aG: aG: dc: =d L , \ 2 , . =o v: , L o , - o_L , . J Z , _ , + d. v^, _ 1 0 , . , t
t
ci
r812a) t8.72bl
318
C H A p T E R 8 : N u n r c r i c l lS o l u t i o no f r h e L i n \ t c a d )f,: l o $ E q L r a l i o n s
aH"' . __ ^,'J7
dH,,' at!: ., t .V,. ;1. 0 . . , . ; ' \ 1 t,/.,. 1
dH, . . . ^, ,'L 1 , - \ 0 . . . .
'.'..0,1 ,r, (llli rc,., tV,,
d/, \
Hi
rx.71c1
di (x72J)
in whichi = 1.2.....Nl. EqLrarion 8 s. 7 2r e p r c s e nat j i n e a rs y s r e no l e q u a t i o n st h a t c a n b e p l a c e di n n r a r r i xf o r m a s I E I { ' \ _ r } = { D ) i n w h i c h {lr} - uec_ tor of changesin the unknownsat each iteration: {b } : I.ectorof negativeresitlu_ a l s ; a n d I E ] = n a t l i x o f d e r i v a t i v c bs a n d c da l o n g t h e d i a g o n a lu , i r ha m a x i n u m width of four cler.cnts.This bandedproperty alows fbr more efficient sorutionof the systemof equations.The system is solved rcpearedlyunlil the changesin the u n k n o w nv a l u e sb c c o m ea c c e p t a b lsym a l l . An ad\antagcof the inrplicit method conrparedro the mclhod of characterisrics and the explicit methodis its inherentstability r{,ithouthaving ro satisfythe Courant limitation of small time steps.Stability of a numericalschemeoccurswhen srnall pedurbationsin the solutiondo not grou exponcntiallywirh rinre.It is derermincd mathematicallyby substitutinga Fourierseriesrepresentation of the finite dilference solution at the grid points into lhe differenceequarionsand determiningthe conditions under which the enor in the sorution grows with time. The Fourier stabirity analysis,often attributedto von Ncumann (Strelkoff 1970),generaltyis appliedto a simplerIinearizcdsetof equationswith the assumptionthat the resultsalsoare applicableto the more complex nonlinearsystem.Numericalexperimcnr gcnerallyionfirm the validity of this approach.l_iggertand Cunge ( 1975)show for a linearized form of the goveming equations,that the condition for stability of the preissmann schemedependson the weightingfactor 0. If 0 = ], then thc solutionis not damped wjth time nor does it grow with time. \ hile for 0 < 1 rhe solurrongrows with time and always is unstable.For 0 > ]. the solution always is stablebuisome dampins occurs.l is temptingthento usea valueofd = j, but becauseofdifferencesberweei the numericalwave celerityand the actual wave celerity,srnallundesirableoscilla_ tions in the solutioncan occur,althoughthey do not grow with time. For this reason, a slightly largerIalue ofd is neededto damp out the oscillations.As a practicalmaG ter, Liggett and Cunge ( 1975)recommend0.6 < 0 < I .0. Samuelsand Skeels( 1990)included both the convectiveterm and the friction slopeterm in their stabilityanalysisand showed analyticallythat g > j is required for numericalstabilityin agreementwith previousinvestigators;howevir, they also showedthat the absolutevalue of the Vedernikov number,V, must be less than or equal to unity, where V is definedby
v:sAqtF b R d A
(8.73)
in whicha : exponent on thehydraulicradiusandb - exponent on thevelocityin the unifomtflow evaluation of thefrictionslope;A - crois-sectional areaof flow; R : hydraulicradius;andF : Froudenumberof the flow.The Vedernikov num_ ber adscsin stabilityanalyses of steady.uniformflow in openchannels (Liggett and Cunge1975;Chow 1959).When the Vedemikovnumberexceedsunitv,ioll wavesform.The roll wavesarea seriesof transverse ridgesof high voniciiy that occur in supercriticalflow (Mayer 1957).The roll wavesCanbreakandresemblea
C H A p T I R8 : N u n r e r i c S a lo ] U t i o n ol hf e[ J n s t e a dFyI o \ .E \ quations 319 successionofmoringhldraulicjumps.ForIulllrorrgh.turbLrlcntflou,,,h=2,and u s i n gt h e M a n n i n ge q u a t i o na, = . 1 s. o r h u tl i r r 3 , " r , u i t J ec h l n n e l .t h e V e d e r n i k o v s t a b i l i t yl i m i r r e d u c e sr o ! - < I . 5 . W h a l t h e a n r l y s i sb y S a m u e l sa n d S k e e l ss h o r v s is that the Prcissnrannschemc musr sa(isfynot only 0 > 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
o
2
_ -
x t L= o , 1 x l L = 0 - 2 , 0 . 40, . 6 ,0 . 8
0 Time,min (a) Lax DiffusiveMethod
'12
10 I
o
2
_ -
x t L =0 , 1 x / L = 0 . 2 , 0 . 40, . 6 ,0 . 8
0 Time,min (b) Methodof Characteristics FICURE 8.9 Depth hldrographsbetweenthe reservoir(i/L = 0.0) and the turbine (;/L = r.0) for road rejection.
323
324
C H { p ' r E R 8 : N u n r e r i c aSl o l u t i o n o lt h c U n s l e a d \F l o \ \ E q u a r i o n \ br a morc grtdual incrersein depth due to lhe incnia ofthe llo$ins *att-r Tlre positive rr avc is rcflcctcdback from thc reservoirwith lo\r er deplhs.ard it is applrent lhat a relali\'ely lo g tirnc is requiredfor the \Iater to come contplelelyto rest.as reflections conlinue back and forth along the headracc.The solutionsby the tuo melhodsare eren closer in agreementthan in Extimple8.l. The only differcncesare a slishrl) more gradual rise and a very slight roundingal thc peak of thc deprh hydrographsat iDlemtediare points llong the channelfor the Lax scheme.The marimunt depth for the Lax scheme i s l 0 3 1 f t ( 3 . 1 5m ) . n h i l e i t i s 1 0 . 3 6f l ( 3 . 1 6m ) f o r t h e n l e t h o do f c h i r a c r c r i s r i c s .
8.8 DAIII.BREAK PROBLEM Severallargc dam failuresin the United State\.including the Tcton Dam failure on the Teton River in ldrho in 1976,have led to dam safety progranrsin many states and the need to predict the peak dischargeand time of travel of dam-breachflood waves. In the dam-breakproblem. the routing of shocks in the downstreamri\er channel dcpends greatly on the hydrograph created at the dam, which in rum depends on the time of failure and the geonretry of the breach.The National Weather Service combined an implicit flood routing technique (Preissmann method) with a paramcterization of the breach geometry to generate the ourflow hydrograph resulting from a dam break and route it downstream in the program FLDWAV (Fread and Lewis 1988), which combines the formerly used programs DAMBRK and DWOPER (Chow, Maidment, and Mays 1988).The dam breach geometry is trapezoidalin shape,as shown in Figure 8.10 and given by Freadand Harbaugh (1973) and Fread(1988):
br: b",
m(ha- h)
(8.7+)
Dam
! _
_ _ _ _ /
\___________
FIGURE 8.10 Definition of embankment dam breach Darameters.
/
CHApt ER 8: NumericalSolutionof theUnsteady Flow Equatjons 325 in which b, : final bottom width of the trapczoidi b,, = averagebreach width: nr : side slopeof the breach(horizontal:venical):fta = elevationof rhe top of the dami and i, = final elevationof rhe bottom of rhe breach.Triangularatrd rectangular breachshapesalso can be simulatcd with br: 0 and ra = 0, respcctively.The instantaneous elevation,lrr,, of the bottom of the breachis given by hr,,: h,t
(ht
_,i(:)
(0=r
(8.75)
in which ,l?,= elevationof the top of the dan;' h, = final elevationof rhe bottom of the brcach(taken to bc the bottom of the dam unlessthere is an erosion.retarding layer);I = time from the beginningof the breach;7 = total faiiure time; and p = I to 4, with the linear rate usually assunred.Likewise, thc instantaneous value of t h e b ( ) l l o mu i d t h .b . . r r l 'r h c b r e a c hi s
, - r(:)'
(8 . 7 6)
in whichb, is thefinalbottomwidth of thebreach.Estimates of the failuretime,r, andaverage brcachwidth,6,,,,are neededto completethedescription of the timevaryinggeometry of rhebrcach.Froehlich( 1987)sratisrically zt3embank_ analyzed mentdam failuresfor damsrangingin heightfrorn l2 ro 285ft (3.7ro 87 m) and proposed thefollowingrelationships: ,* : 0.47&o(Y*)o:5
(.8.71)
r* : 79(V*)ot1
( 8 . 78 )
in which b* = b,,/lt; /t,, = I.4 if rhe failure rnodeis ovenoppingantl tn : 1.6 1g the failure mode is not overtopping;V* : V,lH): Ha = height ofdam: y" = volu m eo f w a t e r i nr e s e r v o iar t t i n t co f f a i l u r e ;a n dr * - r ( g / H 1 \ 0 5E. q u a t i o n 8 s . ? ?a n d 8.78 have coefficientsof deternrinarionof 0.559 and 0.913, respectively.If the height of the dam. H,r,is nor equal ro rhe height of the breach,(&d f), then H, is replacedby (ft, l/) in the definirions of the dimensionlessvariablei. In a subsequent study, Froehlich ( 1995) recommendeda side slope ratio m : l.,l for overtopping and ra : 0.9 otherwise. The outflow hydrographfrom the breacheddam is computedeither by level pool reservoirrouting (see Chapter 9) or dynamic routing by the inrplicit numerical model with the breached dam outflow as an jnternal boundary condition betweenthc upstreamreservoirreach and the downstreamriver reach.Level oool routing is used for wide, flat reserroir surfaceswith gradualchangesin u,arersurfaceelevation,while dynanricrouting is neededfor narrow valleyswith significanr water surfaceslope in the reservoir.The outflow relationshipfor the breachutilizes the head dischargerelationshipfbr a trapezoidalbroad-crested weir given by Qh = C,K,[3.] b,(h"
hh)ts + 2.45n(h,,. h).t]
(8.79)
in which Q, : breachoutflow in cubic feet per sccond;C,, = approachvelocirycorrectionfactori K, : weir submergencecorTectionfactor:b, = instantaneous bottom
326
CIIApTER8: NumericalSolutionofthe Unsteady FIowEquations
width of the breachin feet (Equation8.?6); ft" = elevationof the water surfaccin feet: and ,r, - instantaneous elevationof the bottom of the breach(Equationg.75). The FLDWAV model deals with transcriticalflow and shocks or surses bv two different methods (Fread and Lewis 1988; Jin and Fread 1997). The first method is an approximale approachusing the inrplicit method in which the entire riyer reach is divided into supercriticaland subcritical subreachesat each time step. For supercriticalsubrcaches,two upstreamboundary conditions are applied that consisrof the dischargefrorn the next upstreamsubreachand critical denth. For subcritical subreaches,rhe downstreamboundary condition is crirical deoth and the upstream boundary condition is the dischargc from the next subreich upstrcam.The position of surgesis adjusteduntil thc shock compatibi)ity equations are satisfiedbefore moving to the next time step. In the second method, an explicit, characteristics-based schemeis available,as describedpre\.iously,to be used in contbination *ith the implicit schemc for different reachesduring the samc routing. For transcriticalor mixed-flow subreaches,the explicit schemc is applied alongside the implicit method for subreachesu,here niar_critical flow does not occur. To further simplify the dam-breakproblem and provide quicker forecastsof dam-breakflood waves. dimensionlesssolutionshavi been develoDed:for example, Sakkasand Strelkoff (1976) for inslrnraneousfailuresand the NWS simplified dam-breakmethod for gradual embankmentfailures (Wetmore and Fread l9g3). These methodsare basedon a large numberof routed hydrographsfor typical val_ ues of the independentvariables,with the resultspresentedin dimensionlessform. Wurbs (1987) tested a nuntber of dam-breachflood wave models. includins simplified models,with measuredfield data and concludedthat a dynamic routing model providesmaximum accuracyalthough none of the rnethodscould be con_ sidered highly accumte becauseof uncertaintiesin the breach devclopmcntwith timc, rapid changesin dorvnstreamchannelgcometry,lack of one-dinensionrl flow conditions,and loss of flow volume.Another contributingfactor to inaccurucyis that rnost dam-break flood waves exceed stagesexperiencedfor any historical floods so rhat calibration of parameterssuch as Manning's a is not possible. Regardlessof thesedifficulties,dam-breakflood wavepropagationcan be modeled to provide reasonableestimatesof the consequences of a catastrophicdam failure.
E.9 PRACTICAL ASPECTS OF RIVERCON,IPUTATIONS Riversseldomareprismaticandfurtherexperience an abruptchangern crosssec_ tion as the flow transitions betweenbank-fullflow and overbankflow. The main channelmaymeanderacrossthe floodplainandconsistof numerous branches and loops.Underthesetryingcircumstances, theone-dimcnsional l-lowassumplons are severelystrained. As long astheflow remainsin themainchannelor the flow com_ pletelyinundates the floodplainfollowingthe generaldirectionof the valley,onedimensional flow is a reasonable assumption. In the transitionbetweenthesetwo
CHAprr,R8: Nurncrical Solutionof the UnsteadyFlow Equations 32'1 cxtrcmes,it is a questionof how nruch lateraldrop in the \\ ater surfacecan be tolerated as the flow moves into the floodplain on the rising side of the hydrograph and returnsto the main channelon the falling side. sontetimesonly partially. Severalanifices hare been devisedfor thc rolc of floodplain sroragein flood wave propagation.One possibilityis to includean inactiveareaof flow in the floodp l a i n i n t h e c o n t i n u i t ) ' e q u a t i ow n h i l e u s i n g t h e a c t i v e\ \ i d t h i n t h e m o m c n t u m eqLration.In the continuity equation,the tirne derivativebecornesd(A + A0)/Al,in which z1orepresentsthe inactive flow area. In this way rhe storagecffects of the floodplain are takcn into accountin an cd lroc manner,but considcrubleskill on the pan of the modclcr is necessaryto dcsignateinactiveflo* areas.Another approach is to treatthe main channelof the river with one-dimensionalmethodsbut with storage pocketsat specificnodescoming off the main channel (Cunge,Holly, and Verwey 1980).Then the challengebecomescorrectly modeling the exchangeof flow between the main sten and the storagepockets,usually by anificial weirs. If the storageareasare linked. lhen a kind of two-dimensionalnetwork of loops can be generated.and the SainrVenantcquationsmay be simplified in thc storagereaches by neglectingthe inertial tenns (seeChapter9). For meanderingchannelswith flow in the floodplains.the flow path in the main chrnnel may be longer than in the floodplains,and the device of a conveyanceweighted reach length can be used.in which an averagelength is basedon the relative magnitudesof the conveyancesof the left and right floodplainsand the main channel.In some instances.as in the flow through multiple bridge openings,onedimensional methodssimply no longer may suffice, and two-dimensional,depthaveragedmodels nray be required,dependingon the purposeof the hydraulic modeling effon. Calibrationand verificationof unsteadyflow models are essentialto gain confidence in their use. The selectionof a panicular one-dimensionalmodel, whether the dynamic form or sonresimplified form as describedin the next chapter,is an imponant consideration.Considerabletime and effort are requiredfor calibration and verification, so a simplified model should not be used if engineeringriver works or extensivefloodplainand channelalterationsare expectedin the future that w o u l d r c q u i r ee x t e n s i v e recalibration. The calibration of a one-dimensionalmodel often begins with selectionof Manning's n valuesbasedon past experienceand running a steady-flowmodel to verify previouslymeasuredpeak stages.This shouldbe done for the entirerangeof dischargesexpected to be encounteredin the unsteady flow model. Once the steady-flow values of the resistancecoefficient have been established,rhen the unsteadyflow model is implemented,with further tweaking of the resistancecoefficients to reproducemeasuredflood hydrographs.The initial condition for the unsteadymodel can be the steady,graduallyvaried flow computation,but running the unsteadymodel during a startupor warmup period by maintainingsteadyflow may be necessaryto dampenany initial instabilities. Stagehydrographsrarherthan dischargehydrographsare bestfor calibrationof the unsteadymodel becauseof the uncertaintyof the stage-discharge relationship. The stage-discharge relationship,or rating curve, often is looped with higher dischargesoccurring on the rising limb of the hydrographthan on the recessionlinrb.
328
Crrprrn
Flow Equarions 8: NumericalSolutionof lhe Unsteady
downstreamstagehydrographis a prelcrleddownstreamboundUse of a rneasured ary condition,exceptthat such a hydrographmay not be availablefor future modeling runs to determinethe effectsof changesin the river Therefore,it is useful to relationshipthat has been measuredover a wide range of have a stage-discharge discharges.The downstrcamboundary should not bc subject to significantbackwatcr effectscausedby a reservoir,for cxamplc. Either the dou nstreamboundary should be moved downstrealnall the way to the dam or moved upstreamout of the backwater jnfluence. The establishmentof a steady-state,single-valuedrating curve at the downstreanrboundaryessentiallycausesreflectionof wavcs upstream "free-flow ' condition.This can work only il the that otherwisewould not occur in a not to influence the river reachof interest.One boundaryis far enoughdownstream offered by FLDWAV for the dorvnstream boundary condition is a of the options with looped rating curve using Manning's cquation S, detcrminedfrom computed the monlentum equation for the last two the implicit finite differencesolution of grid points. spatial The deviation of a looped rating cune for unsteadyflow from the singlevalued, steady-flowrelationshipis influenccd by the rate of rise of the discharge hydrograph,roughnesscoefficients,and channel slope among other factors.The more rapid the hydrographrise, the greater the deviation from the steady-state curve. For channelbed slopesin excessof approximately0.00| , loops usually do not occur, while thcy always exist for slopes less than 0.0001 (Cunge, Holly. and Verwey 1980).IncreasingManning's n during the calibrationto reducethe computed flood peak also may causea widening of the looped rating cur\e at il panicular crosssection. Compoundchannelsectionsare particularly challengingduring the calibration process,becausetbe wave celerity is drastically reduced in the transition from bank-full to overbankflow due to the abrupt increasein area.The wave celerity reachesa minimum at relatively shallow depths on the floodplain, thcn begins to increaseagain.The wave celerity thereforeis very sensitiveto both flooded valley width and elevationoi the main channel banks at which floodplain inundation begins.Calibrationmay requirecheckingtheseparticulargeometricdata very carefully rather than adjusting Manning's n alone to accommodatelarge differences betweenobservedand computedpeak travel times.Also, the locationof crosssections may be deficientin that the chosencross-sectionlocationis not reprcsentalive of the river subreachof interest,especiallythe floodplain width. If calibrationdifficultiesoccur. then severalsourcesof errorsshouldbe examined. The size of the time step or distancestep may be too large.Aside from srability considerationsin explicit methods,the size of the time step should be small enoughto adequatelydiscretizethe boundary conditionssuch as tidal variationsor flood hydrographs.The distancestepdependson the slopeof the water surfaceand the desiredaccuracy.Jin and Fread( 1997) recommenda distancestepfor the FLDWAV ;nodel selectedby
T.M
(8 . 8 0 )
CH,lpTt_Rg: NumericalSolutionofthe Unsteady FIo\rEquations -llg in which ?",= rise time of the inflow hydrograph:Cr, = bult wave celerity of the pcak discharge;and,M = constant value, ,e.on,,n"nded to be about 20 for the implicit scheme.Other sourcesof en.or may be or,ersintplihc-ation of the basic cquatrons(seeChapter9 for iimitarions),inadequateor inaJcurareflood stagedara, and insufficientlydetailedtopographicdara. Topographic and hydrauljc data needs for calibrating and verifying a , dynamic flow routing nrodcl includc detailecleleration data, jescriprions of r eg_ etation and other roughnesselements, bridge geometry, fl;;i;;" boundaries, and stage hydrographsat several along the river. Existing topographic .locations maps may be insufficient to establish variationsln topographynecessrtalrngspor aerial or ground surveys to ausm
.":,:"',:L?:iTfr"Jl,l ;;;;ii;;"",;;;i;,i"; il;:.;;#ili ::il:i:: :::ffi,.J may requireas buirtpransor e'en groundtruth me.surements. The exrstence
of peak slagemeasurenrents alone may requireestablishment of severalgauging stationsto obtaincontprehensive. consistent datavery .u.ff in tfr. projecr,er en if it turnsout not to be datafor a
rhen,ore dara rhar u,.uuu'utrJ.ut"# iiili iiii;#ll,ir: T,ffi"J""il::i:;
unsteadyfl ow variations. For a moredetaileddiscussionof theapplication of unsteadyflow modelsto riversalongwirh casestudies,refer to Crnge,ffrffy, unJ V".*.'y itCgOl.
REFERENCES Abbott, M B ' and D. R. Basco.conlputationarFruid Dtrntnics. New york: Johnwirey & Sons.1989. Amein,M., and C. S. Fang...ImplicirFloodRoutingin NaturalChannels.,, J. Hyd. Dir.., ASCE96, no.Hyl2 (1970),pp. 2.181_2500. Ames,William F Nnnericat Metiids for partial Diferenriat Etyratiorr. New york: Bames andNoble,1969. Chaudhry,M. H. Open-Channel Flon: EnglewoodCliffs. NJ: prenrice,Hail,1993. Chow,VenTe. Open-Channel H.vlrarlics. New york: V.Cru*_Hifi, iS59.' ch"Y:.Y:l D. R. Maidment;and L.w.Mays. Appliedai'a-Le.r'N.; J!, v-i: MccrawHi . 1988. Cunq,e,, J. A., F M. HoIy, Jr, and A. yerwey. practical Aspects o! ContputattonalR^.er H,-drautics. London:pitmanpubtishingLimirea,rgtO lifeprinLJif io*" rnrtttrr" of HydraulicResearch, IowaCity, IA, 1994.) Fennema, R. J., and M. H. Chauihry...ExplicitNumericalSchemes for UnsteadyFree_ SurfaceFlowswirh Shocks."tvdre..R"so urcesRes.22,".. i-i-iidSif, pp. rgz:_:0. ..simulation Fennema, R. J..andM. H. Chaudhry. of One Oi."J"J fjl.-sreak Flo*.s.,, J. Hydr Res.25.no. I r tS87),;p a | _) L Fread.D. L. the NWSDAt4BRKMi)el: TheoreticatBackground/lJser Docutnentation.Silver Sp,ing,MD: HydrologicResearch Laboratory, NJrlonatWeartrer _ iervice, l9gg. ..Transient F""d:P .::ld]. E. HarbaL_rgh. Simularionof n."act J eini Dams.,,"/. I/r.d f Div.,ASCE99, no. Hyl (19i3), pp. 139_54.
330
C H A p T E R8 : N u m e r i c aSl o l u r i oonf r h eU n s r e a dFyl o wE q u a l i o n s
Fread,D. I-., andJ. M. Lewis...FI_DWAV: A Ceneralizcd Flood RouringN{odel.,. ln prrrceedingsofthe1988Natio al Conference on Hydrtulic Erlgi eering, ed.S. R. Abt and J. cessler,pp. 668-73.ColoradoSprings, CO: ASCE, I988. Fread,D. L., andJ. \1. Lewis. Iie NWSFl DIVAVModel euick Ilser,s Guile. Silvcr Spring, MD: HydrologicalResearch Laboratory, NationalWearherScrvice.I995. Froehlich, D. C. 'Embankment-DamBrcach parameters_" In proceetlngs oJ tlp IggT National Conferenceon Htdraulic Etrgineering,ed. R. M. Ragan, pp. 5?0_75. W i l l i a m s b u rVgA . . A S C E .t 9 8 ? Froehlich,D. C. "Embankment Dam Breachparameters Revisited.,, In proceedings of the First Intentational Conferenceon WaterResourcesEngineerhg, ed. W H. Espey,jr. andP G. Combs,pp. 887-91,SanAntonio.TX, ASCE. 1995. GarciaNavarro,P1 F Alcrudo;andJ. M. Saviron...1-DOpenChannelFIow Simulation Using TVD-\tccormack Scheme.,' J. Htdr Engrg.. ASCE lt8, no. l0 (1992), pp. 1359,'72. Goldberg, D. 8., andE. B. Wylie.,.Charactcrisrics MethodUsingTime_Line lnterpolations.., J. Hldr Engrg..ASCE109,no.5 (t983),pp. 6?0_83. Henderson, F.M. OpenChannelFlov..Newyork: Macmillan.1966. Huang,J., and C. C. S. Song...srabiliryof DynamicFlood RoulingSchemes.,. 1 i/r-r& E n g r g .A, S C E l l l , n o . l 2 ( 1 9 8 5 )p, p . 1 . 1 9 7 _ 1 5 0 5 . Jha,A. K., J. Akivama,and M. Ura. ..First_and Second_Order FIux DifferenceSplitting Schemesfor Dam-Breakproblem.",/. H,vdr Engrg.,ASCE l2l, no. l2 (1995), pp.877-84. Jha,A. K., J. Akiyama,and M. Ura.,,ModelingUnstea<Jy Open_Channel Flows_Modi_ ficationro Bean andWarmingScheme." J. Hydr Engrg, ASCE 120,no. 4 (1994), pp.46l-76. Jin, M., and D. L. Fread.',Dynamic FloodRourjngwith Explicirand ImplicitNumerical SolutionSchemes." "/.H)dt Engrg.,ASCE123,no. 3 (1t97), pp. 166_73. ,, Katopedes, N., and C. T. Wu. ..ExplicitCompuration of Disconrinuous ChannelFlow.,,J, H1-drEngrg.,ASCE I12, no.6 (t986),pp. 456_75. "Nurnerical Lai, C. Modelingof UnsrcadyOpen-channel Flow...ln AJtancesin UF[roscience,vol.ll, pp. l6l-333. Newyork: Acadenticpress.19g6. Liggett.J.A., andJ. A. Cunge...Numerical MerhodsofSolutionoflhe Unsleady FlowEqua_ tion{' In UnsreadyFlow in OpenChannels,ed.K. MahmoodandV yevjevich,voi. I, pp. 89 182.Fon Collins,CO: WaterResources publications, 1975. ,.Open-Channel Manin,C. S., and E G. DeFazio. SurgeSimulationby DigitalComputer,, J. Hr"d.Di\'., ASCE 95, no. Hy6 ( t969), pp. 2M9_10. Manin.C. S.,andJ. J. Zovne.',FiniteDifference Simulationof Borepropagatron.,, J. F1)d Dnr, ASCE97. no. HY7 (1971),pp.993_1007. Mayel P C. W. 'A Srudyof Roll WavesandStugFlowsin InclinedOpenChannels.,, ph.D. thesis,CornellUniversity,1957. Meselhe, E. A.; F Sotiropoulos; andF M. Holly,Jr...Numerical SimulationofTranscrirical Flow in OpenChannels." "/. Hydr Engrg.,ASCEt23,no.9(1997), pp.114 83. _. Price, R. K. 'A Comparisonof Four NumericalMerhodsfor Flood nouiing.,,f Hyd. Dit., ASCE 100,no. HY7 ( 1974),pp.8?9-99. Sakkas,J. G., and T. Srrelkofi .Dimensionless Solurionof Dam_Break FloodWaves.,,J. Hyd Dir,.,ASCE 102,no. HY2 (1976),pp. t7l_84. Samt'els,P C., and C. p Skeels..,stabiliryLimirs for preissmann,s Scheme.,, J. Ilyrlir Ergrg.,ASCE I t6, no. 8 (t990),pp.997_t012. .A Schaffranek, R. W.: R. A. Baltzer:andD. E. Goldberg. Model for Simularron ol FIow in Singufarand lnterconnecledChannels.', ChaplerC3 in Techniquesoflvater Resources In,estiqationsof the U.S.CeologicalSaney. Washington,Di: Govemnrentprinting Office,l98l.
C l l A p r E R8 : N u m e r i c aSl o l u t i o on f r h e U n s l e a dFy l o wE q u a r i o n s 3 3 1 Slrelkoff.T. NumericalSolutionof Saint-Venanr Equalions."J. H,-d.Div., ASCE 96, n o .H Y I ( 1 9 7 0 )p, p . 2 2 35 2 . Terzidis,G., andT. Srrelkoff.'Computarion of Open-Channel SurgesandShocks.', I Hvd D n r , A S C E9 6 .n o .U Y l 2 ( 1 9 ? 0 )p. p . 2 5 8 l - 2 6 t 0 . "One dimensional U.S.Anny Corpsof Engineers. Unsread). Flow Througha Full Nelwork of OpenChannels." ln LINETUser'sMoual. Davis.CA: U.S.Army Cor?sof Engi_ neers.HydrologicEngincering Center,1995. Wetnrore,J. N.. and D. L. Frcad.The NIUSSinplifed Dtun,BreakModel Executiv,e Brief. SilverSpring.MD: NarioDal WeatherSen'ice.Officeof Ily,drology, 19g3. "Dan)-Breach Wurbs.R. A. FloodWaveModels.'j ffrdr Engrg.,ASCE ll3, no. I (19g7). pp.2946. Wylie.E. B.. andV. L. Streeler. Fluid Transients. Newyork: lvlcGraw-Hill. I97g.
EXERCISES 8.1. Write out the completedifference equations for an internalboundaryconditionof a weir What if the weir werein a submerged condition? 8.2. Whatcauseslhe diffusivebehaviorin theLax diffusivemethod? 8,3. Derivethe Lax-Wendroff methodusingthe Taylor'sseriesexpansion for ,(l + A0 abouta(t).Hinl: Writethesecondderivative of a with respectto r in termsof deriva_ iivesof F usinglhe governing equarions andsubsrirure A,lU : AF. 8.4. Verifythedefinirionof A givenby (8.57)by takingthe Jacobian of F (aF/aU). 8.5. Showthattheeigenvalucs. i, ofA aregivenby y +. and y - cby lakingthedcrerminantof (A ll). E.6. Rederive theLax diffusiveschemeusingthemethodof treatingthesourcetermgrven by (8..15). 8,7. Apply the simplewavemerhodof Chapter7 ro Example8.l. Calculate thenrinimum depthat the turbineandcompareit with the computerresulrs. Whatis rnemaxrmum discharge thatcanbe suppliedto lhe turbjneby thenegative wave? 8.8. Apply the momentum andcontinuilyequations in finitevoiunreform (shockcompat_ ibility equations) to rhesurgedeveloped in Exampleg.2 andcomparethe resulrsto the numericalresults. 8.9. For an eanhendamrharis 80 fi in heighrwirh a volumeof waterin srorage of 50,000 ac-ft.estimarerhe time of failureand the averagebreachwidth.What will be the heightandbotromwidlh of rhebreachal onethirdrhetime to failureI 8.10. Usethe computerprogramCHAR (on the website)to routea triangular dam,breach hydrograph wirh a peakdischarge of 100,000 cfs at a rimero peakof I hr anda base time of I hr in a downstream reclangular prismaticchannelhavinga widthof 200ft, a slopeof 0.0003.and Manning'srr = 0.025.The initial discharge in the channelis
332
CHApTER8: NumericalSolutionof rheUnsready FIow Equations 2500cis, andit is l0 mi long.\lhar will be rhe pcakdischarge ar rne oownstream boundaryandhow nruchtime will it take for thc pealidischargeto ilrrive?
8.11. Apply theconrpurer programCI{AR (on rhewebsire)for a hvdroelectric loadaccep_ tanceproblemin a trapezoidal headrace havinga bottomwidthof l0 rn, sideslopes of 0.-5:1, Manning'sa of 0.016.and a bedslopeof 0.0001.The sready_stare lurbrne discharge is 40 nrr/s,whichis brouehton line in 2 min. The headrace is 3 km lone. Repcdtfor a .lopeof 0.0004. 8.12. Apply the computerprogramCHAR (on the website)for the samecondiriorls as in Excrcise8.1I exceptfor the loadrejectionproblem. 8.13. Apply the computerprogranrLAX (on rhe wcbsite)for lhe sameconditionsas in Exercise 8.11andcomparethe resuhswith thosefrom CHAR. 8.14. In theloadacceptance problem.shou thatthenegative wavecannotsupplythestead! stateturbinedischargeif the correspondingFroudenumberof the unjform flow. F,.exceeds 0.319.Setthe steadystatedjscharge per unitofchannelwidth,Vy,r,equatro the unit maximumdischarge for $e negativewave,whichdepends on the ups[eam heador specificenergy,80, if the slope is snrall.Then solvefor Fn. E.15.A steadydischarge of 8000cfs is released by hydroelectric turbinesat a ctamrntothe downstream riverat full load.The discharge is increased linearlyfrom theminimum releaserareof 700 cfs to 8000cfs in 20 minutes,held steadyat g0O0cfs for 2 hr, and broughtback down to 700 cfs linearlyin 20 ninutes.The river crosssectionis approximalely rrapezoidal in shapeu ith a bortomwidthof 300ft and l: I sidesloDes. Thebedslopeis 0.0005ftlft, andManning'srr: 0.035.Whatwill be therimelo Deali andthepeal discharge at a locarion25,000ft downsrream of rhedaml UserheiomputerprogramLAX or CHAR. 8.16. In the dam-break problenrof Exercise8.10,repearrhecompurarion usingCHAR or LAX, but for a Manning'sn : 0 05 instead of 0.025.plot on thesameaxesthe max_ imumstageasa functionof distancedownstream of thedamfor bothvaluesof Mann t n Ps n ,
CHAPTER 9
SimplifiedMethodsof Flow Rouring
9.1 INTRODUCTION While in the previouschapter.rhe full dynamic equationsof continuity and momentum werc solved numericallr,.this chapterprescntssimplified methods in which one or more ternlsof the gor erning cquationsare neglccted.Thesesimplified nreth_ ods are presentedin the context of flow routing problerns,where they most often are used.Bylorl rorirlag.* e refer to the trackingin time and spaceof a wave char_ rcteristic such as the peak dischargeor stageas it moves along the flow path but superimposedon the physical flow itself. FIow routing problems range from the routing of a flood or dam-breaksurgein a river to rouring runoff from a parkng lot or upland watershedto generatea runoff hydrograph.The generalsolution sought in the flow rourineprobJemis the disrributionof dischargeor stagewith time at the downstreamend of a rivcr reachtthat is, the outflow hydrograph,given the inflow hydrographat the upstreamend of the reach and the streamgeometry,slope, and roughness.In particular,the translationand attenuationof the peak dischargeor stagewith respectto time most often are of interest. While flow routing metiods can be classifiedin a number of ways, one of the most importantdistinctionsis betwecnhtdrologic routing tnd hvlraurlicrouting.In hydrologicor storageroutinc. rhe momentumequationis ignoredaltogetherand the one-dimensionalcontinuir) equationis integratedspatiallyin the flow direcrion so that it becomesa Iumped svstem spatially,with no yariationof parameters,within the resultingcontrol volume. Hydraulic rouling is a disrributedsystemmethod that detenninesthe flow as a function of both spaceand time (Chow, Maidment. and Mays 1988). The continuity equation in hydrologic routing simplifies to the storageequalion civcn as
d5 : I - O dt
( e .)l 333
ll,1
C u A p r r , R9 : S i n r p l i f i eNdl e l h o dosf F l o \ rR o u r i n g
in which S : storagein thc reach(control volumc); / = inflos late Io thc reach; and O - outflo* rale from thc rcach.An additionalcquation is requirecito solve for the outllow in Er;uation9. l. and it is pr.ovidedby a known functional rclarion_ ship betweenstorageand thc inllow and outflow; thar i\, S = /(/, O). In contrast, h y d r a u l i cr o u t i n g i n c l u d e st h e f u l l o n c - d i m e n s i o n aul n . s t c a d lc, o n t i n u i r yc q u a t i o n and all or part of thc.ntomentuntcquation,as lbllows:
do Ax
aA at
(9.2)
AV ;tV ;rr + Y + 9 . - 8 ( S o 5 / ) =0 (tt OX
., t
t---
llinemaric -] clilTusion
l--
(9.3 )
-l d;namic
l---
As shown in Equarion 9.3, dynamic routing includesall terms in the momenrun] equation,while diffusion routing neglectsthe inenia terms (lcrat and convective acceleration),and kinematic routing includesonly the gravity and flow resistance terms.In termsof spatialvariarion,all the hydraulicrouting methodscan be considereddistributedmodelsbur applicableonly underconditionsfor u'hich the neglected termsare small relativeto thc remainingteflns.The previouschaptcrconsideredthe caseof dynamic routing, while this chaptertreatssinrplifiedrouting methods,borh lumped and distribured. Finite difference numericaltcchniquesare describedfor the sinrplified meth_ ods of flow routing in this chapter.Problems of stabilityand numericaidiffusion are consideredbecausethey arejust as importantas in the previouschapter,when solving the dynamic routing equarions.Solutionsof the simplified equationsalso are comparedwith solutionsof the dynamic equationsso that the conditionsof appli_ cability of the simplified merhodscan be identified.In addition. we show that the kinematic routing method can be recastin terms of the method of characteristics with a result analogousto the simple wave probJem,treatedin Chapter 7. Finally, we seethat a hybrid method,the Muskingum-Cungemethod,can be developedby matchingnumericaldiffusion and physicaldiffusion termsin the routing equations.
9.2 HYDROLOGIC ROUTING With referenceto the control volume shown in Figure 9.1, the one-dimensional continuity equation given by (9.2) can be integratedalong the flow path from the inflow sectionat r, to the outflow sectionat,r. as follows:
[\ do
t
"
dr+
IT
dr=0
(9.4)
C H A p r I R 9 : S i r n p l l t i e dM e r h o d so f F ] o w R o u t i D g
335
Wedgestorage,S,v
Q,= t--'---->
Prismstorage,Sp
s = s , " +s p
+Qo=O
xa
r-IGURE9.I Inflow.outflow,andstoragein hydrologicroutingrhrough a riverreach. The first inrcgral in (9.4) becomesthe difference bctwcen the dischargeevaruated at.r, and r,, (0. - q, or simply the difference betweenourflow and in ow rates lor thc conrrol vorumc. By applicationof the rribniz rure,the time derivativecan b e b r o u g h to u r r i d cr h e . e i o n d i n r e g r asl o r h r t ( g . + ) bccomes
O.
Q,+
*t'
Adr:0
(e.5)
Nou'recognizing theintegralin rhe.thirdterrnasthe storage volume,J, andreplac_ ing theoutflowrare,O,, wirh O andtheinllow rate, 0,, J,i l, *";"ver thesror_ ageequationgivenby (9.l). In theMuskingummethodof riverrouting,rhc second relationshtp _ (in addition to the.storage. equation) requiredto.solverhe-routing p.oUI.rnir rrppfiedby a linear relationship betweenstorageanda u.eighted funiiion of inflow'andoutflow:
s : d [ x /+ ( 1 x ) o l
(e.6)
i n w h i c h d = t i m ec o n s t a nXt ;= w e i g h t i n fga c r o r( < l ) ; = S s t o r a g e, f;= i n f l o w = outflowrate.Equation9.6 somerimes is justifiedphysicallyby argu_ Jate:,and 9 rng_that channelstorage consists of prismsrorageandweOge ;;;r;", * illustrated in Figure9.1; thus,it dependson boih in no* an-doutflo*.-F'irrn"rl".ug. i. ,t u, po.tion of storage associared wirh a sreadyuniformflow piofil"'O"f.nj.n, onry on ouu flow, whilewedgestorageis theremainingstorage trat occursiu.ingrn. un.t.uOy rise and.fattof stage,so it dependson thJ differ"ence b",;;;;;;ii;* and ourflow for the,riverre.ach. Alternativeiy,it can be argued,f,utlquution l.i'o srmplya conceptualmodelwith the parameter0 indicitive or trr" t."n.iution'of. the inflow hydrographand the parameterX considereda weigbting ru"ioi r.iut"o to storage and attenuation of the floodpeak,with both0 and-Xto i. O"r..rin"O by calibra_ tion. If thetimeconsrant, 0, indeedis,heldconsranrforuff Ji*fr_g*, then Equation 9.6 establishes a linearrelationship bet\r,eenstorageand weightedflow in jl': co€qcient of proportionatityian u. inte.p.ete'J"; ;. ;;u" havel rime l:h.t"c'lr In the n\ er reachas will be iustifiedlater. As will be discussed subseouently, rhe valueof the MuskingumX generally falls.jntherangeof 0.0 to 0.5 (althoughnot alwa),s). these timiti on rherangeof X valuesestablish two differentbehavrors u ith respecrto thepropagation of a flood wave.For X : 0, Equation9.6 simplifiesro rhe srorage*;ii;;;o lbr a linear
336
C H A p T E R9 : S i m p l i f i e dM e r h o d so f F I o w R o u r i n g
Outfiow,
(a) ReservoirRouting(X = 0.0)
(b) ldealizedBiverRouting(X = O.S)
FIGURFJ9.2 Routingof hydrographs with puresloraSe(a) and pure translation(b).
reservoilseveralof whicb sometimesare usedin seriesfor catchmentrouting. Reservoir routingcanbe interpreted as a specialcaseof Muskingumriverrouting, in whichthe storagedepends only on outflow.So long as the reservorr watersur_ facecanbe assumed to be approximately horizontal. otherwiscknownaslet,elpool routing,boththe storageandthe outflowdependso)elyon therescrvoirwiltersurfaceelevationandtherefore on eachother As the reservoirinflow increases with time anda portiongoesinto reservoirstorage, theoutflowis reduced, as shownin Figure9.2a.The resultis a peak outflow arrenuated in comparisonto the peak inflow,whichis preciselythepurposeof a floodconrrolreservoir. It is instructi\e to notethatthe peakoutflowoccursat the intersection of the inflow andoutflo$, hydrographs. This is because the maximumstoraseoccursat the sametlme as the maximumoutflow whendS/dr: 0 and therefore/ = O. Reservoirroutingprovides the limitingcaseof purestoragewirh associated attenuarion andspreading in time of the outflowhydrograph which is sometimes referredto asdiffusion. At the otherextreme,X - 0.5, the Muskingumroutingtechniqueweights equallythe inflow andoutflowin the storagerelationship. The result,as is shown subsequently, is puretranslation of the inflow hydrograph, in whicheachdischarge is delayedin timeby thewavetraveltime in thereachdetermined by d. In thisspecial case,theinflowhydrograph theoretically underg<xs no attenuation or changein shape, asshownin Figure9.2b.Mosr riversbehave betueenthetwo extrcmes given in Figure9.2andexhibitbothdiffusionandtranslation of theinflow hydrograph. Reservoir Routing Equations 9.1and9.6canbe solvednumericallyusinga finitedifference technique. If a forward differenceis taken for the time derivari\€ in Equation9.1 with the meanvaluesof 1 andO evaluated oYerthe time intervalA/, the followingresults:
C r i A p r E R 9 : S i m p l i t i e dM e t h o d so f F l o w R o u t i n g
:sr-s,111111111111111 \!!: -9tj!. .1r
2
2
331
(e.7 )
in which the subscriprsI and 2 ref'erto the beginningand end of rhe rinre inrerval, analogousto the indicesI and,t + | for the currcntand subsequenllime levelsused in Chapter8. In the storageindicarionrnc'thod,ls ir sonrerimesis called, Equation 9.7 is nultiplied by 2 and rearrangedto l,ield
25,
25, tr+O,=1,*17*t-O,
(9.8)
Becauseboth storageand outflow are functionsof resenoir stage,a separaterelationship can be developedbetweenthc left hand side of Equation 9.8 and outflow O.. Then the right hand side of Equation9.8 is computedin subsequenrrirne steps to determinethe lcft hand sidc, frorn which O, fbllows, basedon the seDaraterelationship of 2SlAr + O Ys. O. EXAI\tPLE 9.1. A smallwatersupplylakehasa normalpool elevationgivenby Z = 0.0 m with an emergency spillwaycrestelevation givenby Z : 0.50m ( 1.6,4 ft). TheemerSency spill\\ay is a broad-crested weirq,itha discharge coefficient C, : 0.848 anda crestlengh of 20 m (66 fr). The elevarion storage-ourflow relarionship is given in Table9- 1. If thelakelevelinitiallyis ir rhenoonalpool,rourerheinflow hydrograph givenin Table9-2 throughthe spillwayanddetemrine rhepeakourflow. Solution. First.the rouringinrenallr = 0.5 hr is chosensuchrhatrherisingsideof the inflow hydrographis adequarely discretjzed. Then,in Table9-1, calculations are shownfor the quantiry2sll/ + O to be usedin the routingwjth careful attentionbeing paidto usingconsistent unils,whicharecubicmetersper secondin this example. The routingtable(Table9-2) is developed basedon Equation9.8.To srant}lerouting,the flrst outflow valueis setto zeroandthe corresponding valueof 2sllt + O is placedin
TABI-E 9.I
El€\'ation-storage-outnowrelationshipsof Example9.1 Z, m
0.0 tr.J
1.0 1.5 2.0 3.0 3.5 4.0 4.5 5.0
.!, m3
+06 6.90E "l.-l2E+ 06 +06 8.588 9.,198+06 I .o:lE+ 07 L l.lE+07 L2.lE+07 L35E+07 L,l6E+07 1.58E+07 1.708+0?
o, mr/s
LsldJ+ o
0.0 0.0 10.2 28.9 53.1 8t.8 I t4.3 150.2 r89.3 231.3 2'7 6.0
7668 8579 9546 I0569 | 1642 t216'7 t3942 t5166 | 6440 t/"763 19t36
338
CHApTER 9: Simplified Methodsof Flow Ror.:rjne
TABLE 9.2 Storage-indication method of flov, routing of Example 9.1 Time, hr
0.0 0.5 t.0 1.5 2.O t.t
3.0 3.5 4.O 1.5 5.0 )_J
6.0 6.5 7.0 8.0 8.5 9.0 9.5 10.0 10.5 I1.0 It.5 12.0 t2.5 t3.0 13.5 14.0 14.5 15.0 t5.5 16.0 16.5 t't.o 17.5 t8.0 18.5 19.0 19.5 20.0
zS/At- O 0.0 79.2 212.5 320.5 38 1 . 9 400.0 386.I 352.3 308.,1 261.1 2t6.5 r75.6 140.1 I10.2 85.7 65.9 50.3 38.I 28.6 2t.4 15.9 It.7 8.6 6.3
t.8 l.l 0.9 o.'7 0.5 0.3 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0
1668 '/718 8039 8512 9260 10003 10722 r1362 I i 897 t 2 ll 9 t2632 12846 12971 t3038 13044 r3009 t2912 12852 t2716 12631 r2 5 1 0 12386 12261 t2t3'7 12016 | 1898 l r785 I 1675 I 1570 1r469 lt3'72 |219 lll90 llt05 I1023 t 0945 r0871 t0799 I0731 10666 t0601
2S/At + o 7668 77.18 8039 857: 9?15 100.12 r0789 I r.160 i2023 1216'l t2t91 i-r024 r3 1 6 2 t3221 13234 13196 l3l25 r 3030 12918 t2796 12668 t2531 12,106 12216 121.18 t2024 I I 90.1 l I789 l1678 512 ll471 I1373 I | 280 lll9l lll05 11024 10945 i08?l 10199 l0?31 10666
0, mls 0.0 0.0 0.0 0.0 '|.3 19.3 33.9 .t9.0 62.8 71.1 82.6 88.9 92.1 94.5 94.7 93.1 91.7 89.r 86.0 82.6 '79.3 75.9 '72.6 69.3 66.0 62.8 59.8 56.9 54.0 51.5 19.2 47.l 45.0 42.9 41.0 39.2 3'1.4 35.7 34.1 32.6 3 l .t
thetablecorresponding to theinitialconditiorof normalpoolelevarion (Z = 0), which is telow thespillwaycresr.Thevalueof 2SlAr_ O follows from,i. ,ufii ln *. zSla, + O columnminustwicetheoutflowat rhesamevalue "irirn..'fir"", ilr"iS.Si. solvedfor 2SlAr+ O in rhenexrtimesrepandinrerpolated in fuUi"'i-llo oUt"_ tt " corresponding outflowvalue.The outflowremainsat zerouniil JeIakJ-tevelrises
C r l A p r L R 9 i S i m p l j f i e dl l e r h o d so f F l o w Rouring
339
E
o
15
ri"]J r','
20
I.'IGURE9.3 Inflowandoutflowhydrographs for reservoir routing. Exampte 9.1.
abovethe spillway crest,after which finite values of outflow are obtarnedfrom the inte.poration in Tabre9-r. This processis continuedun,ii ,r," o*no,u tals to some small rarue.The peakoutflow rare-which occurs at rhe poini oiinr.ir..t,on *,tr, tr," inflow hydrograph,is 9.1.7m3/s(33,10cfs) , , = 7.6 ;; ;;';,iur'ii.n ..ou."a r.orn rhepe{k inflow rateof 400 nrr/s( 14,r 00 cfsr at l : z.s rr. rt"-r".rrtio." .r,o*n in nig,
River Routing
riverrouring. Equation 9.6.canbeemptoyed roevatuare J.2andSr
f::Yl-r\llt:r $ r t h t h e r c s u l t ss u b s t i t u t eidn t o { 9 . 7 r r o y i e l d
0 l x ( r 2 r- ) + ( l _ x x o , - o r) l :
]1,'
* r ) _ ( o ,+ o , ) l I . g )
Collectingtermsin (9.9)andsolvingfor Or, we have Ot : Colz+ CJt + C?Ot in which the coefficientsCo,C,, and Cr, called routing
-
"
-0X + 0.5Ar 0-0X+0.5Lt
(9.10)
coeficients,are definedby (9.ll)
C t { , \ p r E R 9 r S i n r p l i l i c d\ l e t h o d s o f F I o w R o u t i n g
+ (--' . = , . .0 X 0 . 5 1 t 0 {rx + 0.-51/ 0 - 0X 0.51t
-
0
dX + t).-5lt
(9.12) (9.13)
Becausethe dcnominaloris the sanrcfor all thc routing coefllcients.we readilr sec that (C0 + Cr + Cr) - l. Thus. at leastconccptually.the routing coefficientscan be viewed as wcighting factors applied to the intlows al thc beginningand end of the time inlervaland to the outflorv at the bcginningof thc linrc intcrval to solle for the outflo$' at thc cnd of the time interval.The rouling cquationas deflnedby (9. 10) can be applied repcltcdly to obtain the outflorv hytlrographat the end of the rivcr reach.given the inf'low hydrograph that cn(ersthe reach. In thc contextof the finite differencenotationuscd in Chaptcr8. Equrtion 9. l0 can be reu ritten as
Q l ; , -' c o Q i ' ' ' c Q : ' c . . Q : .
(9.1,1)
representing only one reachand in which the cornputationalmolecule is rectangular, a singletime step.Note from (9. 1,1)that, for pure translationover the time interval At, the valuesof C, and C. should equalzero so that Cr = | and Of- i = Qf. These conditionsare satisfiedin Equations9. | | through9.13 for the specialcase of X = 0.5 and At : 0, as statedpreviously.The lattcr rcquiremcntof At = d is equivalent to specifyinga value of unity for the Courantnumber,providedthat A can be interpretedas the wave travel time over the reachlenSthIr as is provenlater. The choiceof the time step. -\t, and the spatialinterval.,Lr, are imponant. and somegenerallimits must be considcred.First, Jt shouldbe chosensuchthat the rising limb of the hydrographis approxirnatedrdcquatelyby a seriesol straightlines. which usually rcquiresAt < /5. where lp is the time to peak of the inflow h1'drograph. Second,it would sccm that negativevaluesof the routing coefficients are counterintuitiveif they indeed representweighting valuesfor the inflow and outflow (Miller and Cunge 1975). However,Ponceand Theurer (1982) showed from numericalexperimentsthat it is necessaryonly for Co > 0, while C, and C. can be negativewithout affectingthe accuracyof the routing (definedas avoidanceof negative outflows). Requiring Co > 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
xl | 12.2R'
I{ : 5.75u..Ios L- t:-l
(10.19)
in whichu," : criticalvalueof the shearvelocity= [r.. (SG - l)gdro]or;R' = causedby of form roughness independent hydraulicradiusdueto gmin roughness, ripplesand dunes(to be discussedin the next section)l.r = a correctionfactor for smoothandtransitionalturbulentflow, which is equalto unity for fully rough turbuto d65, whichEinsteinequated roughness, sard-grain lentflow; andt, - equivalent the65 percentfiner grainsize.The correctionfactor,r, is a functionof ,t/6, asshown in Figure10.7,where6 : viscoussublayerthickness: 11.6vlu'. andal : shear
1.8
1.4
FlIry rolrgn
1.0
T
0.8 ; mf,o 0.6 0.4 0.1
10
1
100
ksl6 FIGURE 10.7 meanvelocityin smoothandtransition Einsteinvelocityconectionfactor.-!, for calculating turbulentflow (Einstein1950).
386
C H A p T E R l 0 : R o \ i i n A l l u v i a lC h a n n e l s
veJocitydue only to grain or surfaceroughness: 1gfi'Ss)o5.Coarse sedimentshavc no bed forms so the hydraulicradiusR = R', and furthemrore,r= I .0 for fully rough turbulentflow, with the resuIt that Equation10.I 9 reducesto Equation 10.I 7 for sedimentscoarseenoughto fall in the fully rough turbulentregime. The relationshipsfor critical velocity in Equalions10.| 7, 10.I 8, and 10.l9 can bc placed in dimensionlessforr in terms of a critical value of the sedimentnumber, N,", as defined by (Carstens1966)
v
( 10.20)
vGF r)s4,
Neill (1967)hasdoneextensive experiments on "first displacement" of uniforrnly gradedgraveland proposeda bestfit relationship as shownin Figure 10.8and siYenbv /J
\
0:0
Ni = z so(.,3)
(10.21)
in which d" - geometricmeandiameterand yu - depthof uniform flow. As reportedby Pagdn-Oniz(1991),Parolaobtainedsimilarexperimental resultsfor uniformflow over a gravelbed whenutilizingNeill'scriterionof hrst displacein Figure10.8areEquations ment.Shownfor comparison 10.17and 10.18in terms d5s= d; andtheStricklerconstantc" : 0.034 of N,. (with r-" : 0.045;k, = 2d5o;
7'c = 0 045 cn = 0.034(EN)
z .-;---'-
/
.;-.tF.\
---t-r
1
0.01
0.1 dsotfo
1
FIGURE1O.E (datafrom Neill 1967). Criticalsediment numberfor initiationof motionof coarse sediment
C l i A p r c R l 0 i F l o $ i n A l l u \ i a lC h a n n e r s 3 8 7
in Englishunits).For drol.r,u > 0.I, Manning'sn beginsro vary rvithdepthas the roughness elemcntsbecomelargerclativeto thedcpthasdi\cussedin Chapter4. In this zone,ivlanning'sequariontendsto overestintate rhe cdricalvelocity;wbile Keulegan'sand Neill'scquationsunderestimate it and so are on the conservative side.Manning'sequationprovidesa rnoreconservative e\rinrateif c- = 0.039in E n g l i s hu n i t s . E X A l t p t , E 1 0 . 2 . F i n d t h e c r i t i c a ls h c a rs t r e s sa n d c n r i c a l v e l o c i t yf o r a m e d i u m sand uith dru = 0.3 rnm (9.8 x l0 { fr) and a medium gravel \\.jth d.n : l0 mm (0.0328 fr) for a uniforn flow depth of warer (20"C) of l.O m (3.28 fr). So/llnbn. First calculalethe dimensionlesssedimentnumber. /.. for both sediment sizes.For sand with a specificgravity of 2.65 and warer u irh a viscosiryof I X 10-6 mr/s ( 1.08 x l0 5 ftr/s), .1. is determinedby
r ',' =[ (Ls c ; t ) q -d lll : LI
l/l
= 7. 5 9
A similarcalculation for the gravelyieldsd. = 253.Then, from Figure10.6,r.,: 0.041 for the sandand0.045for the gravelwith the former in the transitionalrurbulent rangeand thelatterin thefully .oughturbulentrange.The corresponding valueof critical shearstressfor the sandis i, - (y, - f)dsor,,:
1.65x 9810 x 0.OOO3 x 0.041
: 0.20N/m' (0.0042lbs/ftr) andfor the gravelit is 7.28N/nr2or Pa(0.152lbs/frr). To fi nd thecrilicai velocityfor thesand,useEquation10.l9 wirhx determinedfrom Figure10.7.Assumetharno bedformsexisrar initiationof motion,sorhatR, = R.Take k , : 2 d \ o : 0 . 0 0 0 6 m ( 0 . 0 0 2f t ) a n d5 : 1 1 . 6v / u " , : 1 1 . 6x t 0 6 te . 2 } t r c m ) \ n = 8.20 x l0 a m (2.69x l0 r fr).Then,t/6 : 0.?3,andfrom Figur€10.7,.r= 1.57so thatthe criticalvelocityis calculated from Equarion10.l9 as /r | 1 . 2'."2 v I^ r v-- : 5 . ? 5 . / l o-sLl I
k, J
vp
( x - 5 . ? 5x ( 0 . 2 0 / 1 0 0. 0- ).' ' l o-g[l{ 1 2 . 2 1\ .. 00 \ 1 . 5 7l) 2ftls) 0.0006 r-aa L I=037m/s(1 For the Bravel,useEquation10.18(Manning)with c, : 6.9414and : l.O for SI 4 unitsto obtain K
-
{ - rV(sc - r)r..dl"'rj6 =
1.0 x ( 1 . 6 5x 0 . 0 4 5 ) rxi ' ( 0 . 0 1 ) r xi 3 ( 1 . 0 ) r /=6 t . 4 2 m l s 0n4t4
or 4.66ft/s. Forcomparison, thereadercanconfirmthatthe criticalvelocityfor the for thesamevalueof r.. is I .3?m/s(4.50fVs) travelfrom Equationt0.17(Keulegan) andfromEquation 10.21, it is 1.01m/s(3.31ft/s).ThelanervaluefromNeill'sresults is considerably moreconservative rhaneitherEquation 10.l7or 10.l8for thisvalueof dr/yo: O.O33.
C H A p T E Rl 0 : F l o w i n A l l u v i r l C h a n n e l '
r0.4 CI{ANNELDESIGN TO S'TABLE APPLICATION engionce the critical shearstressfor initiation of motion is elaluated,an ubvlou\ The neeringapplicatiortoccursin stablechanneldesign,as discusscdin Chapter'l that the nraxis ro choosca rock-riprap linirrg of sufficientsize Oesigriphiiosoptry at the design flow dcrs not exccedthe critical shcar s(ress' itress sh."r i*uii te,l barks' r. : r,, : ln the simplest problem,which is a very wide channeI with 'stable, channel linlf vr'^ S^. in whicir ],, is the nornraldepth and 5o is thc bed slope the of the Shields inj rnureriatis coaiseenoughto be in the fulll rough turbulentregion is prramelcr diigram. which is the uslralcase'then the critical value of the Shiclds diamcter' grain to the ".Jn.tunt "nd th".ritical shearstressis directly proponional = ro = 7.,(7, 7)dto Thercfore' setting r, For r-" = 0.045, it follows that 7. resultsin dso = 13.5r',,50
( 10.22 )
the dePthor in water'From(10.22)ue canseethatincreasing for quartzsediment of thechanfor stability largerrock-riprapsize a proPonionately the sioperequires slopethe and channel nel bei. On itreothe;h;nd, for a givennati\e sedimcntsize thechanrequires flow depthmustbe limitedto a specificntaximumvalue,rvhich bed maintaining while largerdesignflows nel widih to be largerto accommodate stability. too wideto achievebedstalf i canalin theexistingnativesedimentbecomes usedasa riprapliningof a narrower bility at thertesignflow,thena largersediment, for ease The channelshapeusuallyis chosento betrapezoidal is required. channel, as well bank the sloping andstabilityof the rock-ripraplining on of constructio;, stress on shear the critical an issue.In general.the valueof ason thebedbecomes addiof the bed because thebanksof thechannelis not the sameason thechannel channelThis tionalforceof gravityactingdown the side slopeof a trapezoidal initiationof causing force.in gravityforce.Jrnpon.ntasiiststhe hydrod-vnamic coincides force' Fo' drag irotlon.tn ttresimplestcase,asshownin Figure109' the down has components with the directionof motion andthe submergedweight' %' motion point ofincipient the the sideslopeand perpendicularto the side slope At on thechannelbanl, tireratio of the forcesparallelto the bankto the forcesnormal
T'IGURE10.9 bank. Stabilityof a particleon a channel
C I A p r 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
to the bank, must equal the tangenl of the anglc of internal friction or angle of reooscof the bank material.With rcfcrenceto Figure 10.9'this is written as Iat\ 0 :
\a4l (u,;t"df
(1 0 . ? 3 )
W,cosd
The tlragforceis thecriticalshearstresson thebankor wall' 7:' timesthe surface into (10 23) andsolvingfor thecriticalshcarstress areaofa grain,A,. Substituting on the wall gives
w
f
r 'l = - c o 5 t , t l n d , / l A, V
onto
( r0.2.1)
, tan-O
Whenappliedto the bed.tand = 0 andcos0 = l, so thatEquation10.24implies r. on the bedis equalto (W,/A,)tan S. If we thentaketheratioof the shearstress on the waff to that on the bed, thich is called the tractiveforce ratio, K,. lhere results K '.
rt' 'f.
f
= c o s 9 . 'l \
t^ to
L tan-@
f-;t
./l v
\rn @
(1 0 . 2 5 )
identity. Thelaststepon therighthandsideof ( 10.25)is theresultof a trigonometric stabilityof thesideslope,soK. < l. Fora given 0 < d for gravitational By definition, sideslopeangle,0, andangleof repose,S, whichdependson thegrainsizeandanguby multiplyingthecritical on the sideslopeis obtaincd larity,thecriticalshearstress by thetractiveforceratio,K,. It remains on thebedfromShields'diagram shearstress only to comparethe ntaximumshearstresson the side slope'which was given by with thecrilical shearstresson thesideslopeto determine Lane( 1955a)as0.757r',,Sn. thatledto Equation10.22'theresultis stability. Settingthcnrequal.asin theanalysis d56 =
l 0 .t O
_roSo
(10.26)
From ( 10.26),we can seethat the flatter is the side slope,the largerthe value of K. and the smailerthe minimum sedimentsizethal will be stableon the side slope-The stabilityof the bed also must be checked;and for this purpose,Lane (1955a) gave the maximum shearstresson the bed as 0.977-vfr' Design of a channelrock riprap lining !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!n this way is referred to as lhe Bureau of Reclanation procedure More for riprap design are given in Chapter4. methods detailed
10.5 BED FORMS Bed forms are irregularitiesin an alluvial channelbed with respectto a flat bed that are higher than the sedimentsize itself. The three main types of bed forms are nppies.dunes,and antidunes,each \\ ith a diffcrent physicalorigin. Ripplesand dunes ire caffedlorler rri ine bedfornts. becausethey occur generallyin subcriticalflow'
390 CHApTeR l 0 : F l o wi n A l l u v i aCl h a n n e l s whileantidunes existeithernearor in supercritical flow.As t-tre discharge or Froude numberincreases,a transitionzoneforms betweenthe lower regime ripples and dunesandtheupperregime,whichconsists of a flat bedwith sedimenr uansponoi antidunes.The transitionzonecan consistof severalbed form typ€s occurringin differentparrsof rhe bed simultaneously. Spccifically,ripplesor junes andflat bed canoccurtogetherduringtransition. Sketchesof severalbed forms are shownin Figure 10.10.Ripplesare approximatelyriangularin shapewith a long,flat upstream slopefollowedby unit*pt steepslopeapproximatelyequalto the angleof reposeof the sediment.HowevJq ripplessometimes can be nearlysinusoidalin shape.Ripplesand dunesmove slowlydownstream at a velocitymuchlessthantheflow velocityaserostonoccurs on the upstreamslopewith depositionon the downstream slope.Rippleshave amplitudes of approximately 3 cm (0.I ft) andwavelengths on the orderof 30 cm ( I ft); andtheyusuall),occuronly in sandswith grainsizessmallerthan0.6 mm (0.002ft). Ripplescan occuron the upstreamslopesof dunes,*,hich are much largerin amplitudeandwavelength.
Typicalripple pattern
Planebed
<>i Duneswith ripplesuperposed
Antidunestandingwave
Dunes
Antidunebreakingwave
rii\
Pool Ciut€ Washed-outdunes
Chutes and pools
FIGURE IO.IO Formsof bedroughness in an alluvialchannel (SimonsandRichardson 1966).
C H A P T E Rl 0 r F l o v ,, r ,A l l u v i r lC h a n n e l s 3 9 1 W h c r e a sr i p p l e sa r e t h c r c \ u l t o f t h c g r o * t h o f a n y , . r r : r l d l i s c o n t i n u i t oy n t h e bedcausedbvdcfornrationofthebcd,dunestendtob(,,lJtcdlothelargestscale turbulene t d d i e si n t h e l l o w $ i t h a h e i g h ro n t h e o r d c r , , 1 t h c f l o u ' d e p t h .I n b o t h c a s e sa, l t e r n a t i n gr c g i o n so f s c o u ra n d d e p o s i l i o n; 1 1 cg 1 , . ; r t cidn t h e f l o w d i r e c t i o n t h a t p r o d u c eg r o u t h o f t h e b e d f o r n r st o s o m er c l u r i v c l y. t r b l e s h a p eY. a l i n ( 1 9 7 2 ) sho$s thal the wavelengthof dunesmust be relltcd to It,,r dcpth, sincethe largest c d d y s i z e sa r e d c p t hd e p e n d e n D t . u n e so c c u ra l h i g h c rl l r , w v c l o c i t i e st h r n r i p p J e s , but thcy are sinrilarin shapewith a gentle,slighrly convcr upstreamslopefollowed by an abrupt drop at the anglc of repose.Duncs ma1'be rrl sufllcientheightlo cause surlacewar es. but theseare of nuch snrallerantolitudcthan the dunesand are out of phasewith the dunes.Ripplescan be t\\,odimensional.rr ith prrallel crcststransvcrse to the flow direction,or can exist as a three-dimcnsionalanay of individual r i d g c sa n d r a l l e y s ; w h i l e d u n e st e n d t o b c t h r e c - d i m c n s i o n a e lx c e p t ,p o s s i b l y i,n narlow laboratory flunes. As the flow velocity increasesto cause ripples just beyondthe thresholdof motion and thcn dunesat higher r elocities.sedirnenttransport increases.With funher increasesin velocity or stream power, the dunes are washedout to form a plane bed with sedimenttranspon. In contrastto ripples and dunes.antidunesare not causedby either bed deformation or disturbancedue to the largest-scaleturbulenteddies,but rather by the standingsurfacerravesthat occur u'henthe Froudenunrber is nearunity (Kennedy 1963;Yalin 1972).The alternatingregionsof scourand depositionin the flow direction due to the surtacewares createantidunesin phase with the surfacewaves, which can become breakingwaves.Antidunes can move upstreamor downstream or remain stationary.Kennedy (1963) shows that the uavelength of antidunes dependson the Froude numberof the flou as givcn by
.\'o
8)o
( 10.27\
- flow depth;and V: flou lelocity. Sedimenttransin which A = rvavelength;,r'o pon continues to increaseas the bed passesthrough transition to flat bed and antidunes. Severalother bed form types havebeenclassified(Vanoni 1977).Bars are bed fbrms having a triangular longitudinalprofile, like dunes. but are of a scalecomparableto the channel\.!idthand depth.Point bars occur on the insideof meander bcnds, while alternatingbars occur in relatively straight river sectionsas the thalweg undulatesfrom one bank to the other. Chutes and pools, as shown in Figure 10.10. consist of large depositson which supercriticalflow forms a chute that servesto connectdeep pools. Becausebed forms dependon flow conditionsin the river, they generatea variable form roughnessdue to flow separationin the lee of the bed form with attendant separationeddiesand turbulentenergydissipation.This has led to the idea of separationof the total bed shearsrressinto a ponion that can be attributedto form roughness(ril) and the remainderdue to surfaceor grain roughness(r;). Assuming linear superpositionof the shearstresscomponents.this is written as
( 10.28)
391
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
10 i
-
o Standingwavesor antadunes I
Transition zone 5
€
t
(r
/
.
.9
7,
(! I
Upperregime (planebed, waves, slanding antidunes)
regrme LOWer (dunes) a
1.0 a a
0.5 '1.0
'10 Velocity(fvs)
FIGURE IO.I1 ofhydraulicradiusto velocityfor Rit GrandeRivernearBemalillo,New MexRelationship ico (Nordin1964).
where ro is the averageboundary shear stress in uniform flow given by 7Rnlo, in which Rn is hydraulicradiusand Sois the channelslope.Such a separationhasbeen relationshipsfor alluvial found to be necessaryto correctlypredict stage-discharge channels,as describedin the next section.The existenceof a changingflow resistancedue to variablebed forms as river dischargeincreasesgiYesrise to discontinrating curves,as sho*n in Figure l0.l I for the Rio Grande uous stage-discharge River, for example.ln the lower regime, the tlos resistanceis high, but as the dischargeor velocity continuesto increase,the ripples and dunesare washedout in a transitionzone to producea plane bed with lower resistance.albeit with a larger value of sedimenttransport.Thus, for a given slope,more than one depth-velocity
CHApTERl0: Flow in Allurial Channels
2.O
A
A
A
1.0 0.8 0.6
^
T
3 0.04 c E lt)
0.02 0.01 0.008 0.006
t
a
r
.
'
-
oi
? 3E
-
"
'
;
:
I
.
-
.
'
S
-
l
.t ;.
ri
o
g"'. -; Riooles o \
:-
^b gA '
'
r
.,'{ r. '}./
r.
"'
fa
: o -Q . a ' " / F r a r
a
0.001 0
-
..t .8..t - [. !. o
0.004 0.002
-
8O
rransrlron fl
-.'-
a
)io.{
J
s.;h'iff.:^", .!
o.2 0.1 0.08 0.06
A
A2
s" $ --:; -",.*r!a!-i? i^
o.4
-i
andtn ^ Anlidunes
^fp"fi
393
^
o o . a ^
Ripple Transition Dune Antidune Flat
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (mm) MedianFallDiameter
FIGUREIO.I2 Prediction of bedform typefrom sediment fall diameter andsreampou,er(Simonsand Richardson 1966). combination is possibleastheroughness andconesponding sediment transponrate change(VanoniandBrooks1957;Kennedy1963). Numerous attempts havebeenmadeto identifybedformsasa functionof flow properties. andsedirnent The Simons-Richardson diagram( 1966)shownin Figure 10.12plots streampower,definedas the productof meanbed shearstressand velocity,againstsedimentsize to identifyregionsof occurrence of variousbed forms.Figure t0.12 is basedon extensiveflume data collectedby Simonsand
C H . \ p r E Rl 0 : F l o w i n A l l u v i aC l hannels Richardson,and reponed by Guy, Simons, and Richardson(1966). on rircr dara from the Rio Grande at severalsrationsand the Middle Loup Rirer at Dunning, Nebraska,and on irrigation canal data from Indja and Pakistan.Ripples occur at low valuesof streampower for fine sands,whilc lower regintedunes transitioninto antidunesand flat bed as thc streampower increases.Simons and Senrurk(1977) report that the diagram performswell on small naturalstreams.bur on rhe l\,lississippi River, it predictsflat beds for casesthat have been obscrvedto be dunes. Van Rijn (198.1cyproposcd a bed form classificationsystem based on the djmensionlessgrain diameter.r/,, and a shear-stress parameterrelated to scdiment transponand given by f = Q'./r,, - l), in which zl = value of Shields parameter for the grain shear stressand r-. : critical value of Shields' parameter.As shown in Figure 10.13.Van Rijn suggesredrhat ripplespredominarer,,hend, < l0 and I < 3. Dunes fall in all other partsofthc region I < 15. Transition is defined by l5 < T < 25, and upper regime bcd forrns occur for T ) 25. From laboratory and field data, including sorneof thc same data used by Simons and Richardson. Van Rijn also developeda direct predictorfor the height of dunes, _\, given by
i =o"(<';"( l - e o i r ) ( 2 5 - r )
( 1 02 9 )
is flow depth;d.ois mediangrainsize;and I is the sedimenttranspon in which-r,o variabledefinedpreviously. Clearly,if thevalueof I exceeds 25 in ( 10.29),thena flat bedis prcdictedrrith A = 0. The wavelength of thedunes,\ *as givenby Van '100 Upperregime t"q)
Transilion
10
E (E
Dunes
Dunes
Bipples
Dunes
3
o
0.1 1
10 Dimensionless Particle Diameter, d.
100
FIGURE10.I3 Prediction of typeof bedfonnfromtranspon parameler, panicle I anddimensionless diameter.d. (VanRijn l98ac).(Source: L. C. trn Rijn."SedinentTtunsport III: BetlFonnsand AlluyialRoughness," J. Htdr Engrg.,A 1984,ASCE. Reproduced bt pennission of ASCE.)
l hannels 395 C H A p tr , R l 0 : F l o \ ri n . { l l u v i a C Ri.jnto be 7.-jr,,.which is in good agreenrcntwith Yalin's ( 1972) theorcticalvalue of ,\ : 2r.r',,.Julien (1995) suggestedrhar rhc van Rijn classificationalso suffers from poor prediction of the upper regimc for vcry large rirers, such as the Mississippi, for u hich dune-coveredbcds hare bccn observedfor f valueswell in excess of 25. This appearsto be due 1o the fact rhat lhc Froude nunrberapproachesunity for Z approrintarelyequai ro 25 in laboratoryexperimenrs.whereasin large sand_ bed rivers. the Froude nurnberis considerablyless than I atT = 25. As a result. Julicn and Klaassen(1995) proposcddropping rhe dependenceof bed-form height on 7 and deternined. front field data for severallarge rirers. that bed-form height can be sivcn bv
-o-= r., (',n)o' )o
( 10.30)
\.to ./
while bed-forrl length is approximatelyA = 6.25r'0. Another difficulty in bed forrn prediction is caused by water rempcrarure effects.As discusscdby Shen,Mellema, and Harrison ( 1978),thc Missouri River experiencesa changefrom dune-coveredbed to flat bed from summerto wlnter as the temperaturedecreases at rhe samevaluesof dischargeand slope.Obviously,the stream po\\'er in the Simons-Richardsondiagram would remain the same evcn though the bed fornr changeswith temperature. Brownlie (1983) studi:d the transition regime and suggestedthat it can be delineatedb1' the value of the grain Froude number or sediment number. N- = y/[(SG I.)gd5pJ050, and the ratio of grain diameterto viscoussublayerthickness, d.o/6, where 6= 11.6 vlu',. For slopcsgreaterthan 0.006. Brownlie found that all the bed forms were in the upper regime.while for slopes less than 0.006 he suggesledthe following relationshipsfbr rhe lower limit of rhe upper regimebasedon both flunre and river data:
. N , ,ng N;
+ o.rsr; 002.r6e roef + o.s:sr(nsf )'
,o,
f
. , I l0.l la)
r o g $ : I o gt . 2 s
( r 0 . 3l b )
o
where Nf - t.74 S r/r and S = slope. For the upper limit of the lower regrme, B r o w n l i ep r o p o . e dl h c h c \ t - f i 1e q u t t i o n s N.
r. o s g = - o z o : o + 0 0 i 0 z d,,, 6 r .*sofo ] 3 /o ( rd.,, . e \''f )
i,I.
f
r o g [ = I o g o . s fo, !!! - 2 3
( r0.32b)
396
C H A P T E Rl 0 : F l o w i o A l l u v i a l C h a n n e l s
10 N j = 1 . 7 4 Sr / 3 Lowerlimitof upperllow regime
Transition Transition Upperlimitof lowerflowregime
0.1
1 d5o/6
10
I'IGURE IO.I4 of bed form transitionzone from lower regimeto upperregime(Brownlie Delineation "Flow Deprh in Sand'BedChannels"'J' Hldt E't|rg' 1983).(Soarcerll. R. Bronnlie. b!-pennisskmofASCE) O 198J,ASCE.Reproduced Theserelationshipsfor the transitionzone ate shown in Figure l0 l4' and we can see that the variible ,1,0/6,the ratio of grain size to viscous sublayerthickness' reflectsthe viscousinfluencenear the bcd and thus indicatesa temperrtured!'lendencc for the bed forns.
10.6 RELATIONSHIPS STAGE-DISCHARGE Perhapsthe most fundamentaldifferencebetweenalluvial channelflows with movable bedsand rigid-bedchannelsis the effect of variablebed forms on flow resistance and thereforeon the stage-dischargerelationship lt cannot be emphasized enoughthat the Manning's rr valuesof Chapter4 no longer are applicablein alluexperiencingactivesedimenttransport,becauseof the accompanylng vial clhannels the rising side of large flood hydrographs.dunesformed at low During forms. bed washerl out to prorluce a flat bed at the flood peak, which buffers be may discharges large variationsin stagewith discharge relationshiPsfor Many nlethodshave beenproposedto predict stage-discharge is referred The reader here. presented are number limited a but only sireams alluvial to Vanoni( l9?7) and Biownlie ( I983 ) for a more completetreatment Einsteinand Barbarossa( 1952)were thc first to separateflow resistanceinto form and surface (grain) resistancein alluvial channels.as indicatedby Equation l0 28 The actual
L l l A p T h R l 0 : F l o * , i n A l l u t r aC l hanncls ,l()7
scparation.f srrearstressin{o rbrnr and surfaceconrponr'nts uas lchiercd througrr rhc defrnitionof 1\\o rddiri\ c co^p.nents of the hl iriauric ratlius. r{' duc t. surfrir.r. r c s i s t a n cacn d R " d u e l o f i r n r rr e s i s r r n c er .u c h r h l r t h r ,r , n l l h l d r t u l c r a d i u sR = / i , + R " . T h e v a l u co l ' R ' w a s d c r e r m i n e dl - r o n rl b r m u l a sl o r f l ; \ \ r c s i \ t i t n c eI n n g l ( l bed channels.whire R" camc from the 'bar rcsistancecurve" rcritting v/rr1.in *hich V : mcan flow relociry and rrl' : shearvelocity clucro bcj lbrnts = (strfS)re. to thc Einstein sc{.linrent trtnspon paranr!'tert!' = (y, - 7)r1.r/(7R,S),$hich cssc . t i a l l y i s t h e i n v e r s eo f t h e S h i c l d sp a r a r n e t cur s i n gr / , , a s i h e r c p r J s c n t a t i r c grairr size. The physical reasoningbehind the bar resistancecune w,s tased on rrrc inlcrred rclationship betwecn the rate of sedincni transpon and the bed fonll topographyand. thus. the form resistanccof ripplesand dunes.Followtng t.inslcll and Barbarossa.others presentedmethodsfor separatingthe friclion lactor or thc slopeof the encrgy gradc line inlo lornt and grain resistrncecontDonenls.
Engelund's Nlethod The method proposedby Engelund (1966, 196?) divided rhe slope of the energy grade line into two componentsas S - 5' + S", in which S, is the grain roughness slope and S" is rhe additional slope due lo form drag on thc bed forms. The value of S" is expressedin terntsof an cxpansionloss due to the separationzone down_ \ l r e a mo f r i p p l e sa n d d u n e r .E n g c l u r r da p p l i e dt w o r i m i l a r i r yh l p o r h e s e sg i v e n a s follows: (l) In two dynamically similar streams,the Shieldsparameter11 (due to grain resistance)has equal values and (2), in two dynamicallysimilar streams,the expansionloss is the same fraction of the rotal energy loss.The latter hypothesis can be shown to imply thar, for rhc two dynamicallysimilar srreams, /ifi : /:/r, in which/, and, are rhe roral fricrion facrorsfor srreams1 and 2, and/i and/! ire the grain resistancefriction factors.Front the dcfinition of the friction factor. this is eouivalentto
1.2
T,t
( 10.33)
However,accordingto the first similarityprinciple,the valuesof rl on the right handsideof ( 10.33)areequal;therefore, somustthevaluesof r. be equal.This can be truein generalonly if z. is a functionof ri alone.Engelund plottedhume results summarized by Guy.Simons,andRichardson ( 1966)according to thisconclusion, asshownin Figure10.15.It is evidentfrom thedatathatseparate curvesfor lower regimeand upperregimebed forms exist wirh a rransirionbetweenthem. and apparent discontinuous stage-discharge relationships canoccurin alluvialstreams. In fact,Engelund(1967)showedthattheuseofFigure10.l5produced closeagreementwith the measured stage-discharge relationfor the Rio Grandegivenin Figu r el 0 . l 1 . The lowerregimecurvein Figure10.15hasan empiricallyfittedcun,egiven by Engelund(1967)as
r', = 0.06+ 0.4r?
(10.34a)
C H A p I E R l 0 : F l o wi n A l l u r i r lC h a n n c l s
198
8.0 6.0 4.O 2.O 1.0 d
0.8
G:i
0.6 0.4 o.2
| =O.O6+O.4t?
0.1 0.
FIGURE IO.I5 prediction(Engelundand Hansen1967; rl vs. r. diagramfor stage-discharge Engelund's 1966).(Source:F.Engelund Engefund196?,usingdatafrom Cu;. Simons,andRichardson "H\draulic Resistance ofAlluviol Streams,"J Hvd. Div.,Q 1967,ASCE Repro' Closureto ducedbv pennissiotlof ASCE.)
whilefor the upperregimecurve. for rl(1 r. - r', r . = ( 1 . 1 2 r5' - r 8 - 0 . 4 2 5 ) - r / r 8 f o r r l )
(10.34b) I
(10.34c)
Equation10.34bwas givenby Engelund(1967),while Equation10.34cwasproposedby Brownlie(1983)asan empiricalfit of thedata. methodrequiresthecalculationof thevelocity The applicationof Engelund's relationshipfor fully rough turbulent flow given by from a Keulegan-type Enselundas
v
R' , = 6 + 5 . 7 5 l o g 2dos
( r0.15)
in which V - meanflow velocitylR' = hydraulicradiusdue to Srainroughness; Note that the equivalentsandgrain and al : grain shearvelocity= (gR'S)r/2. of roughness, t,, is nken to be 2d6sin Equation10.35.Implicit in the application the Engelundmethodis a switch from dividing the slopeof the energygradeline To crecomponents. to dividingthehydraulicradiusintoformandgrainresistance from and y is computed a valueof R' is assumed relationship. atea depth-velocity
C H { p r E R l 0 : F l o * i n A l l u r i a lC h a n n e l s 3 9 9
( 1 0 . 3 5 )n h i l e r l i s c a l c u l r r eads R ' . ! / [ ( S G- I ) / { ] . T h c n .f r o n rF i g u r e 1 0 . 1 5o r Equation10.3.1, rhc valucof r. is obtainedfrorn ilhich lhc totalhydraulicradius, R . i s c a l c u i a t eads r.(SG- l)z/*
( 10.36) basedon the definitionof r,. For a verywide channel.rhc hydraulicradiusis taken equalto thedeprh,r,n,whichis a commonassunrption in alluvialrivcrs.Then,from contrnuit)'.4 = V.r'0. so thal llte depthdischargerelationshipalsocan be determincd.If q is givenandborhV andr,oareunknown,thenireraiionis requiredon R, u n t i lc o n t i n u i t iys s a t i s f i e d . For thetransitionregionof a discontinuous ratingcurve.suchastheoneshown i n F i g u r e l 0 . l l , B r o w n l i e l l g 8 3 ) s u g g e s t e d e x t e n d i n g a h o r il zi noenatcarl o sfsr o m thedepthat theupperlimit of thelowerregimeto theupperregimecurvefor grad_ uallyincreasingdischarges. For graduallydecreasing diicharg=es, a horizontalline wouldexrendfrom the lowerlimit of the upperregimeto thtlower regrme curve. Alternarively, an average couldbe takenof upperind lowerregimedepthsin the transitionregion,but ultimatelymoreneedsto be knownabouttie dynamics of the transitionitserfrecognizing thestochastic natureandthree-dimensionality ofthe bed form formation.
Van Rijn's Method An alternative approach for obtainingdepth_discharge relationships waspresented by 'ranRijn ( 198.1c). He usedthepredictetl heightoi the bedform ro intera formresistance componentof the equivalent sand-giainroughness suchthatt, : k, + tj'. The valueof lj = 3dr, whichis an averagevaluetakenfron a wlde variation in laboratoryandfielddatabetweenI and lOdno (vanRijn l9g2),is substitured inro the Keuleganequationto definerl, thegrainshearvelocity:
,
v l2R ) . / ) l o s-
( r0.37)
Ja gct
in which R : rotarhydraulicradius.This is a somewhatdifferent definitionof al than in Engelund'smethod.but the.finalvelociryfor a giren deprh is computed f r o mK e u l e g a ne' sq u a t i ount i l i z i n gr h er o t a lv a l u eo f k . a r i da , :
t2R _ v = >.l)ur tog:;- _
Jq% + k
(r0.38)
in wh.ich a. = (gRS)O5and k,' for the form roughnessis calculated from the bed lorm herght _\ and sleepnessA"/I a\
(J:
l.1A(l - e-25r/.\)
( 10.39)
400
l hannels C H A P T E Rl 0 : F l o wi n A l l u v i 0C
The applicationof the methodwhen dischargepcr unit of width r/ is givcn and both depth and velocity arc unknown is as follo\ s: l. Estimatea value of the hydraulic radius. R = r',,. 2. CalculateV = q/,r6. 3. Solve for al from Equation 10 37. ,1. CafculateT = u',1/ui..- I and the dimensionlessgrain dianreter,d.' 5. CalculateA from Equation 10.29and I = 7.3,r'0. 6. Detemrineiil' from Equation 10.39and velocity' V, frorn Equation 10 38' with k,=k"+k:' 7. Calculatea new depth.1u - qlV, and repeat,staning from step-j' - 0 in the oriSinal van Rijn If the value of T > 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
sothar... - 0.047from Figr-rre t0.6and,T: r',/r", l:O.52t0.04:' - I = 10.1. The heightof thedunesis obtained from Equarion10.29as
.t
(l-eorr)(25-T) orrr&)" \)0,/
^l
0.
, / 0.000852 \o -I 1.0 / \
-.n'
t u t ) r -2 5I 0 . l -; 6.29
so rhar.1 = 0.20 x 1.0 : 0.20fr (0.061m). Havingrheduneheighrandwith the wavelength,,l = 7.lfo, theequivalent sand-grain roughness heightdueto the bed forms can be estimatedfrom Equation10.39as r ? = l . 1 A ( 1- e - : s r / r )= l . l x 0 . 2 0x ( l _ e i - 2 5 x 0 ? o r r ) = 0.109ft (0.033m) Finalll, tie velocitycan be obrainedfrom F_{luation10.3gbasedon rhe totai shear velocit!: -
3d,()+ k"'
I2 < 1.0 = 5.15V32.2x 1.0x 0.0OtloelT 3x0.00157+0.109
) =r * n r "
or.!.6a n/s, The resulrfor dischargeper unir *'idrh is : V1o= 2.09 frrls (0.194 4 m'/s ). \r'hlchrequiresa seconditerationwith a largervalueof depth.For yo = 1.3f1 (0.40m), rhe trial valueof velocityis 2.31 tus (0.7M n/s) and al =d.lt+ fvs (0.0347rnis).Then r'. : O.28'7andT = 5.1l. This givesa duneheighrA = 0.291fr and t" = 0.171fr (0.0521m). Finally, the velocity is 2.29 ft/s (0.7bm.rs),wtrictris
l hannels C H , \ p r E R l 0 : F I o wi n A l l u v i a C verv close to the initial vllue. so the solutionby the vrn Rijn nrethodisr'0 = l.l0 fl {0..10m) and V = 2.-10ftrs (0.70 r s). 3. Karin-KtnneLlyMethoJ. First calcuhle the value of lhe Shields parameterlbr an i l r r u n r d. d c l ' r hu f I . 1 f r , 0 l 0 m r t o g i r e
'' =
\os 6ci - I)ri-
r
L3 0.001 = 0.925 t.65x 0.000852
rvhich is lcss than l.l and therefore in thc Io*er regime.The reliilive dunc hcighr fbllows from Equdtion l0..ll into \\hich the value ofr" has bct'n substituted'
t . o o , - : : r f" : r s ) , r . , . , ( 0 " ' ) ' f
,
,
\
1
.
. 70ef " el5 )' \
l
'
l
/
\
1
,
/
o ott .)' n..,t so.:-,f \
l
'
l
Therefbrc,the rclalivcralue of the friction factoris obtainedfrom Equation 10.,11as
I: ,/o
r 0 . i 2 7= 4 . 1 2 , . , 0* s . s : f : r . : o+ 8 . 9 2 -ro
Finally.thevelocitycomesfrom substituting intoEquiltion10..13 to give
{sc - Dsd,
- - b 6 8 i/ ( r J . ' r r r )
""
o . * , 0 \ 1v 4 1 2 ' h ' - r 0 . 5
= 2.23ftls (0.68m./s).The disso that y: 10.5x 0.65 x 32.2 x 0.0008-52)05 chargeperunitof widththenis 2.90ftrls(0.269rnr/s),\r,hichis close.but an addditionalilerationyields)o : 1.33ft (0.'11n))and V = 2.26ftls (0.69nts). The resultsof the lan Rijn methodandthe Karim,Kennedy nrethodare vinually identical,while the Engelundnrethodgivesa depthand velocilyboth of which are withinabout8 percentoi thevaluesfrom lheothertwo methods.
10.7 SEDIMENTDISCHARGE Theprediction of totalsediment discharge in an alluvialstream is an important aspectof river engineeringwith applicationsfrom the assessmentof changes in streamsedimentregime due to urbanizationto the evaluationof long-term bridge scour.This sectionfocuseson the bed-materialdischarge;that is, the ponion of the sedimentdischargeconsistingof grain sizesfound in the streambedas opposedto wasb load, which is defined as the fine sedimentresulting from erosion of the watershed. Two distinct approachesare taken to the problem of determining total bedmaterirl discharge.The first was pioneeredby Einstein(1950), in which total bedmaterialdischargeis divided into bed-loaddischargeand suspended-load discharge and summedto estimatetotal sediment discharge.The bed load is that portion of the sedimentcarriednear the bed by the physicalprocesses of intermittentrolling.
C H A p r I R l 0 : F l o u ' i n A l l u v i a lC h a n n e l s
s l i d i n g .a n d s a l t a t i o n( h o p p i n g ) o f i n d i v i d u agl r a i n sa t v a r i o u sr a n d o ml c r a t i o n si n the bed, so that tbe sedimentremainsin contactwith the bed a large perccntageof the time. Suspendedload, on the other hand,is composedof sedimentpanicles that arc lifted into the body of t}le flow by turbulence,where they remain and are transponed downstream.An equilibriunr distributionof suspendedsedimentconcentration developsas a result of the balancebetweenturbulentdiffusion of the grains upward and gravitationalsettlingof the grainsdownward.The sedinrentconcentration near the bed as determinedby the bed-loaddischargeis the essentiallink to estimalionof suspendedload dischargebecauseit providesthe boundary condition for the verticaldistributionof suspendedsedinlentconcentration. ln general, the opposing forces of turbulent suspensionand gravity are reflectedby the dimensionlessratio u,l\\, in which u, is shearvelocity and r, is the sedinrentfall vclocity. Bcd load is the dominant transponmechanismfor tl-/wr { 0.4, and suspendedload is the primary contributorto sedinrcntload for lJx', > 2.5 (Julien 1995).In betweenthesetwo linrits,nrixed load occurs,with componentsof both bed load and suspendedload. The secondapproachto determinationof total sedimentdischargeis to directly rclate the total rate of transpon to hydraulic variablcssuch as depth, r'elocity, and slopeand to sedimentproperties.This methoddependson largedatabasesof flume and field data to be applicableto a wide variety of situations,and the best-fit relavariablesfor the same rcason. tionshipoften is presentedin termsof dinrensionless In either approach,issues of water temperature,the effcct of fine scdiment, bed roughness,armoring,and the inherentdifficulties of nreasuringtotal sedimentdischarge can cause significant devialionsbetween estimatesand measurementsof total sedimcnt dischargeas demonstratedby Nakato (1990). Nevenheless,such estinratesol sedimcntdischargemust be madefor engineeringpurposes.This often inrolves the useof severaldifferent formulasdctcrminedto bc applicableto the situation of interestand relianceon engineeringjudgment to make the final estimate. This section presentsa few selectedformulas fbr estimating sediment discharge and lirnitcd comparisonswith field measurcmcnts.For a more complete treatment,refer to the referencesat the end of this chapter.The transpon formulas are presentedin terms of the volumetric transport rate of sedinrentper unit of streamwidth, g, with a subscriptof b for bed load, s fbr suspendedload. and t for total load.The sedimcnttransportrate also can be expressedin terms of dry weight of sedimenttransportedper unit of width and time as the symbol g u,ith the same subscripts,so that 8, = 1,qo lor bed-loaddischarge,for example. Thus 4r, for example,has dimensionsof t:/f (ftr/s or ml/s;. while go has dimensionsof F/I/L (lbs/s/ftor N/s/m). In the English systenr.lhe weight rate of transponrr ill be used, but in the SI systema mass transponrate traditionallyis used.The nrasstranspon rateper unit of channelwidth can be obtainedb1 dividing the correspondingweight rate by gravitational accclerationto obtain dimcnsions of M/T/L (slugs/s/ft or kg/s/m). The sedimenttransportrate for the full streamwidth is obtained as the productof transponrate per unit of width and streamwidth, and the synrbolsQ and 6 are utilizcd for this purposefor volumetricand weight ratesof transpon, respectively.with the appropriatesubscriptto indicatebed load (b), suspendedload (s), or total load (l).
406
C f l A p l 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
Bcd-Load Discharge Bed-loadformulas tend to be empirical by necessity.becauseof the complexity of the physics of movement of individual grains by rolling, sliding, and saltation. Early views of bed-loadtransport,such as that of DuBo) s ( 1879),assumedthat the bed load moved in sliding bed layers having a linear relocirl distribution.which led to a useful ranspon formula depcndalthough incorrect in theory,ncvertheless wilh coeficients determined by experimentsby Straub(Brown on shear stress ent (1971) formula can be deduced from a power shows that the same Graf 1950). and imposition of the bound\\,ith respect to the bed shear stress expansion series ary conditionsof no transportrate for ro = r. and ro - 0. where ro is the bed shear stress.The resulting bed-loadtransportformula is given by LD8 Yb
( r0..16)
.r'rrr0\/0 (dio,l-'
in which dro : thc median grain size in ntm: r0 - the bed shear stressin lbs/ftr; z" : the critical shearstressalso in lbs/ftl; 4b = the volumetric sedimenttransport rate per unit of width in ft2/s;and Cr, : 0.17 for this setof units. If the shearstresses are expressedin N/m: and 4, is in mr/s with d50inmm, then Co, = 6.9 x l0 6 Tbe most importantcontributionof the DuBoys formula is the relationbetweensediment transponrate and an excessbed shearstresswith respectto the critical value. Several other bed load transport formulas are of the DuBoys type. These include the formulas of Shields(1936) and Shoklitsch (Graf l97l), althoughthe latter is expressedin terms of a differencebetweenthe actual water dischargeper unit of width and the water dischargeper unil width at incipient motion. The Meyer-Peterand Miiller formula ( 19.18)is the result of laboratory e\perimentsat ETH (Eidg, ncissischeTechnischeHochschule)in Zurich. Suitzerland. for sediment sizesfrom 5 to 28.6 rnm. It can be placedin the dimensionlcssform
6t
Qt'
- 8 . 0( r .
0 . 0 4 7 ) rr
( 10.47)
in which d, : the sedimentsize; SG = the specific gravity of the sedimentiand r. : the Shields parameter.The value of 0.0.17can be interpretedas the critical valueof Shields' parameter,r.... Equation 10.47appliesto the caseof no bed forms in coarsesediment. Einstein( 19.12)introducedthe conceptof probability to the bed-loadtransport phenomenon,which resultedin a bed-loadformula dependenton shearstressrather than excessshear stress.[n other words, very small transPon rateswere predicted at shearstressesless than the critical value. Einsteinassumedthat individual sediment grains move in finite stepsof length L, proportionalto the grain size.Then, for a bed areaof unit width and lengthL,, the bed-loadtransportrate is lakenas the productof the numberof grainsin this bed area,the volume of a grain,and the proba b i l i t yt h a ta g r a i n w i l l b e m o v e dp e r u n i t t i m e ,p , T h e n u m b c r o f g r a i n s i n t h e b e d areais L,/C rdi, where C, is a constantof proportionalityand d, is the Srain\ize. \o the volumetric transDonrate is
C H A p r E R I 0 : F l o w i n A l l u v i a lC h a n n e l s
L,
a a=
(r0.'+8)
*C:d;P'
the volumeof an indilidual grainandL, = Cud,,in which shere Crrl] reprcsents p,' is confor the steplength The probability, Co is a const"ntof proportionality which Einscale, time by a characteristic by multiplying \ Jnedto a trueprobability a height through grain to fall a for rcquired the time or asd,/h7, steinexpressed for Lr andp, d,, since|'7is the fatl velocitySubstituting equaltoits own dianrcter, resultsin C. at = ,:,Cod'*10
( 10.49)
in which the probabilityp - pdlu t Finally. Einstein assunredthat the probability, p. was a funciion of the ratio of thc lift force on a grain. takento be proportionalto (7. ;odl, to the submergedweight of the grain. which was assumedproportionalto - y)d:r, so that
'n
ro
.t
'10,-
= l l - l
I
t)d,l
( r0.50)
2", which is seento be theShieldsparameter parameter dimensionless Thc resulting equathe Rubey by was expressed r'r, velocity, The fall l/ry'. to as Einsteinrefened ( t i o n 1 9 3 3 a) s wt= F'
(+r
)sa,
( r0 . 5 1 )
in which
EJ
36
v r - d : - di
( 10.52)
sedimentdiameterdefinedpreviouslyCombining and d. is the dimensionless in formulaexpressed resultsin a bed-loaddischarge 10.51 through Equations10.49 variables as dimensionless termsof Qn
Qt
: fQl')
( 10.5 3)
in which
( 10.54) in thefollowingdisthatwill becomeapparent and@r"is definedhere(for reasons rate suchthat transPort sediment dimcnsionless the Einstcin-Brown cussioij as (191'l) andthe ETH of Gilben = data laboratory plotted the Einstein 6u 6/F.. gravel, and coal (0 3 < sand, for 1948) and Mi.iller (see Meyer-Peter zirlcn aita
, 1 0 8 C u , r p ' r r n l 0 : F l o wi n A l l u v i a C l hannels
10 \.
B r o w n :A E B= 4 0 | 1l a ) 3
\
0.'l
n
\
g
0.01
0 :4 6 5 d e e= e - 0 3 9 1 u Einstern
--..-
-,
'
-=l
0.001
0.0001 o
2
4
6
8 1 0
1 2 1 4 1 6 18 A = 1lt.
20
24
FIGURE IO.I6 Einstein-Brownbedloadtransportformula (adapledfrom Brown 1950) (Soarce;FiSrtrc usedcourles\ of Iot|a I stituteof H)droulic Reseorch.) and derived an expod5o< 28.6 mm) in terms of thesedimensionlessparanreters given by It is nential relationbetweenthem. rer'y' ( 1 0 . 5 5)a 0.4656 ER- e-o However,in the chapterin Engineering H,tdraulicswritten by Brown (1950) and editedby Rouse,the samedata were fitted by the equation
r,,-*(;)
=+o(r.)r
(10.55b)
havecometo be knownastheEinsteintakentogether, and10.55b, 10.55a Equations for ry'> 5'-5 10.55aapplicable with Equation formula, Brown bed-loadtransport ( The Einstein10.16. in Figure as shown ry' 5.5, for applicable andEquation10.55b bed forms with no appreciable sediments for uniform was derived Brownformula is more then it forrns exist. lf bed conditions. field but often is appliedto other rf" because defined as stress' grain shear of the in tcrms to expressf appropriate transpon(Graf l97l t. for sediment the grlin shearstressis primarilyresponsible into whatnow is called relation bed-load (t950) his developed further iinrtein the lift force \r as probability. for expression ln the theEinstainbed-loadfurtctiott. to thehydraulic grain contribution using the velocity the shear in termsof evaluated corand a pressure hiding factor a ty''. ln addition, of radius.R', in the definition sediment with data for agreement better to obtain were introduced rectionfactor mixturesin whichlargergrainstendto sheltersmallergrainsin thebedfrom transandMiiller (1948)formulaalsohasa moregenera!form- in port.The Meyer-Peter
C t l A p r E Rl 0 : F l o wi n A l l u v i a C l hannels ,109 rr hich only the contributionof grain resistanceto the slopeof the energy grrde line is includedfor thc caseof significantbcd forms. Chien ( l9-56)shows that the ntore generalform of the Mel er-Peterand Miiller fbrmula can be modified and exprcssed in terms of the Einstein variablesas
6 . = - - J-: V(SC
f1 o.,rrl"
l).c,/],, Ia
l
( 10.56)
in which r,l' : (SG - I )d5olR'S.Chicn showscloseagreementbetweenthis expression of the Meyer-Peterand Mtiller fornrulaand the 1950 Einstcin bed-loadfunction. which usesd.,5as the representative sedimentsize.For bolh lbrnrulas,only the grain shearstressis used in the definition of fu' : 11r'-. Van Rijn ( 1984a)took a different approachto the estimationof bed-loaddischargeby modeling the trajectoriesof saltatingbed panicies.assuntedto be spherical in shape.He defined the bed-loadlranspon rate as 4, : ar6rcr. in which ri, is (he particle vclocity: 6b is the saltationheight; and c, is the bed)oadconccnrrarion. The equationsof motion for an individual saltatingspherewerc solYedwith a turbulent lift coefficientof 20, and an equivalentsand-grainroughnessheight of two or threcgrain diameters,which was usedin the logarithrnicvelocity distributionfor the fluid. These two parameterswere obtainedby calibration of the model with neasured trajeclories.Publishedvaluesof the drag coefficientwere used, and the initial velocitiesof the panicles were assumedto be 2u.. The maxintum thickness of the bed layer correspondingto the maximum saltationheight was approximately l0 grain dianreters,but the saltationheight varied with the transpon parameter,f, and the dimensionlessgrain diameter, d,, introduced previously in van Rijn's methodfor depth-dischargeprediction.From a set of computedtrajectory data for particlesfrom 0.2 to 2.0 mm in diameterand rr* valuesvarying from 0.02 to 0.1.1 m/s (0.066 to 0.46 ftls), the saltationheight and panicle velocity $ere correlated with the transponparameter,I and the dirnensionless grain diameter,d,. The values for bed load concentrationc, were obtainedfrom measuremcntsof bed-load dischargein flumes and the correspondingcomputed valucs of rr, and 6ri that is, c, = q,/(u63). The resultsfor c, also were conelatedwith ?.and d. and combined with regressionrelationsfor l, and 6o to obtain a bed-loadtransportfomrula given by * = vb
-40
\,{sc - Ds,ii
= n n . .' 7
d9'
( 1 0 . 5)7
The value of ?'in Equation 10.57dependson ai., which is calcuiatedfrom Equation 10.37.In a verificationdata set including flume data and Iimited field data.the proposedbed-loaddischargeformula was found to prcdict the measuredbed-load dischargewithin a factor of 2 for 77 percentof the data points.which was comparable to the variability in the data itself. The bed-load formulas presentedhere are limited to predictionsof bed-load dischargewhen bed load is the dominantmode of transportor to the predictionof the bed-loadcontributionto the total sedimentdischarge.They are not intcndedfor predictionsof total sedimentdischargein caseswhere both bed load and suspended load are significantcomponentsof the total.
410
C H A p T E R I 0 : F l o w i n A l l u v i a lC h a n n e l s
SuspendedSediment Discharge In steady,uniform turbulentflow in a stream,turbulentvelocity fluctuationsin the vertical direction uanspon sedimentpanicles upward. If the vertical velocity fluctuation is w', and the turbulentfluctuationin sedimentconcentrationis c', then a positivecorrelation betweenc' and r'' leadsto a mean turbulent flux of sediment per unit of area eiren by '1,'c' as shown in Figure 10.17.The positive correlation results from posirire (upward) valuesof rv' bringing parcels of fluid with higher sedimentconcenmtion (+c') with it since the suspendedsedimentconcenrration decreasesin the upward direction.For an equilibrium sedimenl concentrrtionprofile in the vertical having no changesin the flow direction,the only other sediment flux, as shown in Figure 10.17,is due to the gravitationalsettling of sedimentparticles given by x;C for a unit area,in which w7 is the fall velocity of the sediment and C is the time-averagedconcentrationat a point in the vertical. Now the turbulent flux is assumedto behavelike a Fickian diffusion processwith a turbulentsediment diffusion coefficientof €, so that
( 10.s8) Equatingtheturbulentflux to thegravitational settlingflux resultsin thefollowing equationthatgovemstheconcentration distributionC(;): differential (10.59)
€,-;+rrlL=u
Thisequationimmediately is integrable by separation of variablesif e, is a constant with respectto depth.The resultis an exponential distributiongivenby C C"
- "rl "-ol-f,r.
(10.60) wc
+ I I
l
l
t
l
l
I
I
+ wrC FIGURE IO.I7 Suspended sedimentflux balance.
l
C H A p T E R I 0 : F - l o wi n A l l u v i a lC h a n n c r s 4 l l
in rvhich C, - suspendedsedimentconcentrationat z : a. Unfonunately,e, is not a constantin alluvial channelflows, particularlynear the bed,where the turbulence characteristics are changingwith distanceabo',ethe bed.The distributionofe" with the vertical coordinatez is deducedbasedon the vertical disrributionof turbulent eddy viscosity,e, definedby du
(10.61)
o:
in whichr andll represent thepointshearstressandlongitudinal velocity,respectively,at anydistance z abovethebed,asshownin Figure10.18.First,we assume that €, = p€, wherep is a coefficientof proponionality.Second,we can show,from the Navier-Stokes equations, thatthe verticalshearstressdistribution in a steady, uniformflow in an openchannelis linear,as shownin Figure10.18andgivenby (,vo- :) (t0.62) ,-ro n in which ro : the shearstressat the bedandI'n : the depthof uniform flow. From the Prandtl-von Karmanvelocitydefectlaw,givenpreviously asEquation4.13,we can showthat du,u. dz Kz
(10.63)
in which r : von Karman'sconstant, havinga valueof 0.4 for clearfluids.Substitutinger : B€ into Equation10.61,alongwith Equations10.62and 10.63,and solvingfor e, gives
e, = pxu.
z ,
( 10.64)
which is a parabolic with a maximumvalueof the sediment distribution diffusion coefficientat mid-depth.Equation10.64can be substituted into the differential equation( 10.59)so that it can be integratedto produce
c a l& l(vo z) , c"=l Ltb-r)]
z
( 10.65)
U
Yo * \ro
I'IGUREIO.IE Shealstress andvelocitydistributions in steady, uniformturbulent flow.
Jl2
C H A p I 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
r o6
o.2
0.6
0.4
0.8
c/c. F I G U R EI 0 . 1 9 Rousesolutionlbr venicaldistribution of surpendedsedimentconce ntration(Vanoni1977). (Source:U A. Uanoni,ed. SetlinentatioaErtgineering,A 1977,ASCE.Reprotlucedb"-per,nissbnof ASCE.) in which C" = the referenceconcentrationat the distance: : a abovethe bed and R": w/(Bru.). Equation 10.65was derired by Rouse( 1937),and Ru is referred to as the Rorlre ruarlrer (Vanoni 1977; Julien 199-5).Equation 10.65 is plotted in Figure 10.l9 for differentvaluesof Ru Thc \ erticalcoordinateis dcfinedas (: - n)/ (,r'u- a) in which, arbitrarily,a : 0.05-r',,. As the value of Rn decreases,which would correspondlo a finer sedimentfor the same flow conditionsin the stream (a.), the concentrationdisrribution becomes more uniform. On the other hand. coarser sedimentparticlescorrespondingto larger values of Ro result in a suspendedsedimentconcentrationdistributioncarried in the lower ponion of the flow. The Rousesolutiongiven by Equation 10.65hasbeencomparedfavorablywith measuredsuspendedsedimentconcentrationdistributionsfrom rivers and flumes (Vanoni, 1977).Its applicationto measuredsuspendedsandconcenrrationsfor the Rio Grandeat Bernalillo. New Mexico (Nordin and Beverage1965),is illustrated in Figure 10.20at a single vertical location in rhe streamcross section.The conccntrationis pldted on the horizontalscale vs. the rariable (_r;- :)/; on the vertical scaleusing log-log axes.From Equation 10.65.ue seethat the Rousesolution should plot rs a straightline on log-log axes with an inverseslope of Rn as shown in Figure 10.20.Where there is a wide variation in the size distribution,Equation 10.6-5can be applied to separatesize fractions. Valuesof Ru can be obtainedfrom measuredconccntrationprofiles as shown in Figurc 10.20.but the rcsultsdo not alwa\ s agreeu ith predicredvaluesof Rn : x',/(Bxu.) with B : 1.0 and r = 0.4. The ron Karman constantcan vary from its
C H ^ p r r -R I 0 : F l o w i n A l l u \ i a l C h a n n e l s
{t.1
10 Rio GrandeRiveral Bernalillo, NN,1 f SectionA-2, 612'53.ya = 2.53ll / I
ti
1
a " - o z6z
i'' L
0.'1 100
1,000 c, ppm
10,000
FIGURE IO.20 Measuredsuspcnded scdimcntconcentration in Rio GrandeRiver and determination of Rousenumber.Ro.
clear-watervalue of 0.4 to a value of 0.2 at high concentrationsof suspendedsediment as shown by Vanoni (1953) and Vanoni and Brooks (1957) from laboratory experiments.Einsteinand Chien ( 1954)presenteda method for predictingthe von Karman constantin terms of the ratio of the power requiredto suspendsedimentto the rale of doing rvork by the boundaryresistanceforce. In addirion,the value of B ordinarily is taken to be unit). but flume and river data (Chien 1956) sho$ that it can vary from I to 1.5as Ro becomcslarger(>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
= v4r2
x . | 5 0 ' o 0 0 1 ?: 0 . 2 9 6f t s ( 0 . 0 9 0 m 2 /sl
Notethat a"/n, = 2.3 sothatthesediment discharge is mostll suspended load.
120
C t A p l l , R l 0 : f l o * i n A l l u v i i rCl h a r r n c l > |. Van Rijn's Method. The value of I is needed.and it dcpcndson rrl. As jn Erlrn, ple 10.3,al is obtainedfrom Kculegans equation using the nreasuredvelociry and k', = 3dn:
v 5.75tog
l2h l,/,,..
1 . 57 = 0.171fr/s (0.0524m/'s) ll x 1.6 5 15 lor: ^ i;.rr,,s;
rl = !:/l(SC - l)8d5J = 0.172rl(1.65 x 32.2 x 0.000885) Then,by definirion. = 0 . 6 3T . h er e s u l t i nvga l u eo f T = r i l ; . , I = 0 . 6 1 / 0 . 0 , -1 5l : l 3 . 0 . a n d l h e bed-load discharge from (10.57)becomes
- l;s"rL { au= o.o-srr.(sc = 0.053 l . b 5 , 3 2 . 2 ' 0 . 0 U 0 8 8 5' -' ; , ,
l l A l l
000t25tr:,
or l.l6 x l0-1 mr/s.For the suspended sedimentdischarge, ratuesofB. \. J\. a, andC, areneeded. Forthisexample.EquationI0.73givesa relatively smallcorrectionto drofor the effectivegrain size. so the valueof dro is used.The lalue of p comesfrom Equation10.69:
I u ,] : I o .r 2 9 l : , - t - 2 [ * J = r ' 2 1 0 2 q 6- ]I r 8 and thenfrom the definitionof Ro,we have u? R-'. =
-
PKu.
ol2g = 0790 1.38x 0( x 02%
Thereference concentration, C,, is calculaled from Equation10.71,in whichthereferencelevelis takenashalf the duneheightfrom Equation10.29to givea : 0.1I ft (0.034m). The valueof Coasa volumetricconcentration from (10.71)is d.n Tt5 o.ooo885 li.or' c , - 0 . 0 1 5- 0.015< 0.0032 dg3 0.l l 6t8,, Now the correctjonto \
follows from F4uation 10.70: 'l0. l:e 8 _ -" "s1f 0 0 .:so 10 l f 0 0 0 3 2l ^ . _ o , <
f x , - 0 8i c ARo:2.51 1;l
[ 0 . 6 sI
Lu.- LGI
so that R6 = & + A& = 0.79 + 0.15 : 0.94.The integrationfactor,1/.to calculatethesuspended sediment discharge comesfrom Equation10.73b:
l o l q- f o l ' ,
t;l
t;l
| - : .l t e | f- l 2 - R l l L
}ol
I o . l lro "
I o . ll I '
r
Ll.6 -t l . 6- t _ -l :ntK6 I nrrloq _ : : : l Il [12 0.941 L
L O l
Finally,the suspended sedimentdischargeis givenby
q, = IrVysC.: 0.166x 3.57x 1.6x 0.0032 = 0.00303 fi' /s (2.82x l0-1m:/s)
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
421
discharge, q,. is thesumof thebed-load andsuspended-load The totalsediment disandequalto (0.0o125+ 0.00303)= 0.00428frrls(3.98x 10 4 m:/s.).concharges vcrted !o tons/day.8t = 't,qt: 2-65 ><62.1 X 0.00428 x 86,400/20q) : 30.6 tons/dar(28.000kg/day) and C, : 106(t Jy)kl,/q) : 106x 2.65 x 0.00428/5.71= 1990ppm. Yang'sMethod. First,t}|e crilicalvelocityfrom Equalion10.76is needed,since udr,/v = O.296x 0.000885i 1.08x l0 '= 24.3< 70, so | ) ( Y' - r '' 1 L log(r.d..,z) - 0.12q ),
I -I 0b6 0.06 l
|
2 . s -1 \) - . : - - ^ ^ . | 0 . b 6 0 . 1 2 f8r l s ( 0 . 1 0 m I log(2].1) u.ub r
= 0.0470andV.S/x : 0.328x 0.0017/0.129 : Then,VS/r7: 3.57x 0.0017/0.129 = variables requiredareu.lwr = 0.29610.129 0.00432.The othertwo indepcndcnt x l0 5 : 10.6.SubstitutinS 2.30 andu,tlrolv = 0.1?9 x 0.000885/1.08 direcrly into Equation10.75,we have logC, : 5.435- 0.286log(10.6)- 0.a57log(2.30) + [1.799- 0.a09log(10.6)- 0.31alog(2.30 )][log(0.0470 0.00432)]
andthenC, : 1.740ppm. variablesarerequiredfor the total 3. Karin-Kennedys Metfiod. Threedimensionless sediment discharge computation:
v
3.5'7
= 16.5
v4.os, .r:: r o.osrrxs
\./isc rlql-
0.296 0.0.16
= l.l5
\(sc - | )sd.u \,G5 Lrr2 x orloos8s d.,,
1.6 - l,iOR 0.000885
Substituting directlyinro Equation10.77,we have tog - ---!
V(sc
. :
| )gdl,
-2.2'79+ 2.972log(16.5)+ | .060log(| 6.5 lo8(I . 15 ) ) + 0 . 2 9 9l o g (1 8 0 8 l)o g ( 1 . l 5 ): 1 . 4 7 7
Takingrheantilogandsol\ing,we haveq, = 0.00575ftr/s(5.34x l0 l mr/s).Con= 2,670ppm. On the veningro concentration. C, = 106x 2.65 x 0.00575/5.71 otherhand.if we usethe Karimpowerfonrula(EquationI0.?8).we have
\4sc
tlq4l,
, ito.5lra' (t.J0)ra' ls.5 - 0.00t.ro
wilh theresullthatq, = 0.00371ftr/s(3..17x l0 r mr/s)andC, = 1,740ppm.
C H A p T E R I 0 : F l o w i n A l l u v i a lC h a n n e r s
No conclusionscan be drawn about the accur.acyof the methods in ExamDle 10.4 based on a single data point for one river. The Niobrara Rivcr data are i n c l u d e di n t h e c o n r r o ld a r a s e t o f3 , l l d a t ap o i n t su s e db y K a r i m ( 1 9 9 8 )t o t e s th i s method as well as Yang's method, for which the mean normaiized errors are 45 percent and 63 percent.respectively.Funhermore,note that the measuredveloc_ ity and depth Ialues are used in rhe sedimentdischargepredictions, our are predicted rather lhan mcasuredin the generalcase.
10.8 STREATTBED ADJUSTMENTSAND SCOUR Thc sedimenttransportrelationshipsdevclopedin previoussectionsof this chapter assumedequilibrium sedimenttransportconditions,for which the sedimenttransport rate into a rir er reach was consideredidenticalto the sedimentrransDonrate out of the reachwith no net aggradation,degradation,or scour of the bed wlthin the reach.The bed itself was consideredmovablewirh bed forms, but on averase.rne bed was assumednot to be undergoingsignificantchangesin elevationon ai engineeringtime scale.which ntay be on the order of severalyears. In the short term, however,sedinrentstorage(plus or minus)compensates for inrbalancein the inflow and outflow sedimentdischargesfor a river reach.Under these circumstances. the independentvariablesare the streamslope and water discharge,in addirionto rhe sedimentproperties,and the dependentvariablesare the depth, velocity.and sediment discharge,$ hich are intenelaled.The bed forms adjust thenrselvesto provide a roughnessconsistentwith the depth and velocity necessaryto cany the equilibrium sedimentdischarge.On the other hand,thcre may be no depth-velocitycombination for the given watcr dischargeand slope to cary the equilibrium sedinent discharge,so that in the short,term,Iocal scour and deposition may occur, albeit without alteringthe streamslopeover a long reach(Kennedy and Brooks 1965). On a much longer time scale,on the order of hundredsof years,the water discharge and sedimentdischargebecomethe independentvariables:and the stream width, slope,and streamplanform adjustthemselvesso as just to be able to transport the water and sedimentdischargedeliveredto the upstreamend of the stream reach.This is Mackin's ( 1948)conceptof rhe "gradedstream."If, for example,the sedimcntdischargeto a streamreachover many years is too large for the streamto transport,some sedimentwill deposit,steepeningthe reach,or the meanderlength or streamwidth *'ill change,so that the streamequilibrium is restored. ln this section.applicationsof theseconceptsare consideredfor the important engineeringproblem of bridge scour Both long-tern.rand short-termchannelbed adjustmentsas uell as the scour causedby bridge obstructionscan undermine bridge foundations.with possiblefailure and loss of Iife. First, long-termchannel aggradationand degradalionare discussed,then contractionscour causedby the restrictedbridge opening is analyzed.Finally, local scour causedby bridge piers and abutmentsis considered.
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
423
Aggradation and Degradation Long-tenn aggradationand degradationof an aliur ial streamcan occur at a proposed or existing bridge site. In addition to changesjn bed elevationthat can be in the fomr of either scour or fill, the strean planform can shift laterallyaway from the designed bridge opening and cause local scour around the abutmentsand embankments.Some brief discussionof different rypes of alluvial streamswith rcspect to planform is nccdedto understandthe rarious geontorphicchangesthat can occur in responselo human activitjes such as building dams and bridges to cross the stream. Alluvial streamscan be classifiedas straighr. meandering.or braided, with transitional forms betueen each type. The sinuosity of a stream,defined as the strcam Jengthdivided by thc valley length, is used to distinguishbetweenstraight and meanderingstreans.ln general,a strcamis consideredto be meanderingil the sinuosity exceedsa valueof L5. Even straightstreamscan havean oscillatingthalweg at low stagesas the flow moves from one bank to the other around sandbars. ln meanderingstreams,the oscillatingthalweg initiatesstreambankerosionand the formation of a continuousseriesof bendsconnectedby crossings,as shown in Figure 10.21.Erosionof the outsideof a bendcarriessedimentto the insideof the next
(a) MeanderingChannel
==---r--===:= -
:'(
n
/
/-\
\
-)
Lowflow
_t r-1,--.
(b) BraidedChannel FIGURE 10.2I Schematic of meandering andbraidedchannels.
424
l hannels C H A p rt R l 0 : F l o wi n A l l u v i a C
bcnd downstreamwhere it is depositedas a point bar. [n addition,bccauseof the centripetalaccelerationassociatedwith the turning of thc flow through the bend. a transversepressuregradientmanifestedby a sloping water surfacetoward the center of the radius of curvaturedevelops.The result is a secondarycurrent with a transversevelocity componenttoward the inside of the bend at the streanlbottom and a returncirculationat the free surfacetoward the outsideof the bend leadingto a helical flow throughthe bend. Deeper pools devclop at the outsideof the bends. and they are connectedby shallow crossingsfrom one pool to the next. Meander loops can migratedownstreamas well as laterallyand forn cutoffs acrossthe neck leaving behind oxbow lakes. The rate of longitudinal and lateral migration of a nreanderdcpendson the erodibility of the sedimentsencountered,which can be In a study of 50 different nreanderingrivers, Leopold, Wolquite heterogeneous. (1964) found that the ralio of meanderradiusto streamwidth varman, and Miller to 4.3 with a median value of 2.7. The actualcauseof meanderinghas ied from 1.5 various factors, including heterogeneityof bank sediments to been attributed (Petersen1986),secondarycunents (Tanner 1960),and the needfor a streamslope to be flatter than the valley slope to cary a lower sedimentload than was available during the developmentofthe valley slope (Chang I988). Schumm ( l97l), on the other hand, arguedthat a cbange in the type of sedimentload carded by a stream accounts for meandering bchavior and the associatedincreasein sinuosity. A changefrom sandcarricd predominantlyas bed load to wash load transponedonly in suspensionwould result in an excessslopeof the energygradeline which mighr only be dissipatedby the decreasein slope associatedwith an increasein sinuositl. For larger streamslopesand increasedbed load. the alluvial streamtakeson a braided form that consistsof multiple channels around nunlerous sandbars.as shown in Figure 10.21.The channels are connectedin a network to form a wide shallow belt that is unstablewith unpredictableratesof lateral mjgration. As the sandbarsgrow and form islandsthat are large relativeto streamwidth, the braidcd streambecomesan anabranchedstream with somcwhatmore permanentchannels that can carry a substantial ponion of the total flow. Qualitatively,streamsinuosity first increaseswith slope and levels off before decreasingwith funher increasesin slope for a characteristicdischarge,which usually is taken to be the bank-full dischargewith a returnperiod of l-2 years.Such a relationship has been confirmed for several natural streams. as reponed by Schumm, Mosley, and Weaver (1987). For increasingvaluesof slope,the straight channeltransitionsinto a meanderingthalwegchannelwith large valuesof sinuosity and then into a combinationof meanderingand braidedforrns until the channel beconrescompletely braided with low sinuosity.Changesin the planfornl of the streamcan be analyzedwith the help of Figure 10.22,which showsdividing lines or thresholdsfor distinguishing meandering and braided streams as given by Leopol
1E-02
x
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
rOrO man
1E-03
N
Se0.25= O.O1O
Transitional < a
_
- 0.060 SO0.aa
Lane
1E-04 Meandering
1E-05 1E+Oz
,125
1E+03
1E+04
f-------.....-
SOo2s= O.OO17
1E+05
1E+06
ni<.h.r^a.1.
FIGUREI0.22 Changes in planformof streams with streamslopeat a givencharacteristic discharge (Richardson andDavis1995).
Plainsof the UnitedStatesand in the RiverinePlainof New SouthWales.Australia,Schumm( 1969)sho*'edthatthestreamwidthanddepthalsowcrefunctions of the percentsilt-clay,7cSC,in the sedimentformingthechannelbedand banks. The channelwidth-depthratio and slnuositywerefoundto be influencedprimarily by ToSCwith little effectof the meanannualdischarge. The largcris rhe percentageof silt-clayin the sediment,the smallerrhe width-depthratio and the largerthe sinuosity. Analysisof long-termchanges in thestreammorphology canbe achieved qualitativelywith theaid of the approximate relationship proposed by Lane( 1955b): QS
-
Q 'dyt
( r0.79)
in which Q = waterdischa-rge; S - energyslope;p, : totalscdimentdischarge; : anddro mediansedimentsize.In thechanneldownstream of a dam or sandand gravelminingoperation, for example,thesediment supplyis cut off sothatthereis a decrease in sediment discharge, whichis balanced by a decrease in slopefor the samewaterdischarge andsedimentsize.The decrease in slopeis accomplished by degradation of the channelbottom beginningfrom the dam and moving in the downstream directionwith the largestscouranddropoccurringin thechannelbottomjust downstream of thedam.As shownin Figure10.22,a decrease in slopecan causea changein streamplanform from braidedto meandering,for example.On the otherhand,a morerealisticanalysiswould indicatethat decreases in Q also occurdueto flow regulation by thedam,andthesediment sizemayincrease dueto
126
C H A p T E Rl 0 : F l o \ ri'n A l l u \ , i aCl h a n n e l s
streanrbedarmoring. in which the largersizesof the size distributionare left behind dcpendin the degradationprocess.In this case,the slopemay increaseor decrease, ing on the relative magnitudesof the othcr changes,but degradationlinlited by botll a r n o r i n g a n d r e d u c t i o n si n 0 i s a c o m m o nr e s u l t( L a g a s s ee l a l . l 9 9 l ) . S c h u m m ' s analysis( 1969) further showedthat long-term river metamorphosiscan result from changesin water dischargeand type of sedimentload. Again. using the construction of a dam as an exanrple,decrcasesin both water dischargeand bed-nraterial load (prinarill sand)can result in decreasesin width and width-depthratio while sinuosityincreases. Quantitatire analysisof long and short-lermchangesin streammorphologycan be accomplishedwith a numerical solution of thc sediment continuity equalion (Exner equation) given by
B(t-r,,)*-*:,
( 10.80)
in which B : streamwidth; p,, = porosityof the sedimentbedl :b : bcd elevation; ,{ = longitudinal distancealong the streamtand O/ = total volumetricsedimentdischarge. The equation can be solved simultaneouslyuith the one-dimensional unsteadyflo\^'equationsas describedin Chapter8, or if the changesin bed elevation are slow comparedto the time scaleof the changesin water surfaceelevations, a quasi-steadl'approach can be enrployed. In this approach, Equation 10.80 is solved and the sedimentbed elevationsare updatedfor the current quasi-steady, gradually varied flow profile. The change in bed elevation is assuned to be the points within the specifiedmo\ able-bedwidth. Then the sameat all cross-sectional watcr surfaceprofile is recomputedwith the new bed eler ations,using the standard step method for the current quasi-steadywater discharge.The sedimentand flow in this uncoupledfashionat eachtime stepto deterequationsare solvedalternateJy mine the developmentof bed elevationchanges.A sedimenttransponrelationship is requiredfor the solution of Equation 10.80,and the roughnesscoefficienthas to be specified.This is the basicapproachusedby the U.S. Corps of Engineers( 1995) program HEC-6, which also accountsfor bed arnoring using the mcthod proposed b y G e s s l e r( 1 9 7 0 ) . Chang ( 1982, 1984)proposeda similar water and sedimentrouting procedure, except that stream width changesare accountedfor by minimizing the stream power per unit of length,70S. This is equivalentto adjustingthe width of adjacent cross sectionsuntil OS approachesa constantvalue along the strean't.If p is relatively constantalong the stream.the resultis to minimize the variationin the energy gradient,S, in the streamwisedirection.In general,increasingthe width at a cross section correspondswith larger values of S and vice \ersa. A weighted average energygradientof adjacentcrosssectionsis computed;and if the actualenergygradient is higher (lower), channelwidth at this crosssection is decreascd(increased) to decrease(increase)the energy gradient.Once the width adjustmenthas been area is appliedto the bed. made,the remaining changein sedimentcross-sectional For deposition.the bed is allowed to build up in horizontal layers,while scour is applied according to the distributionof the excessshearstresswith respectto critical shearstressacrossthe section.Chang ( | 985. 1986) applied his water and sed-
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
427
iment routing model (FLUVIAL-12) wirh widrh adjusrntcntand simplified nrodeling of bank erosiondue to strean cun alure to define thresholdsfor different planfomrs of rivers from meandcringto braided. Thc water and sedinrentrouting model IALLUVIAL (Holly, Yang,and Karim 198,1)is a one-dinrensional model developedto predict long-term degradationof the lvtissouriRiver.Ratherthan specifying the value of Manning's l, the sediment dischargcrelationshipand the friction-factorrelationshipare coupledand solved at cach tinre step to ntodel bcd form changesand their interactionwith rhe flow and sedimenttransport(Karinr and Kennedy | 981, 1990).In the first stageof the time step. the water surfaceprofile is obtained fronr a quasi-steady, simultaneoussolution of the cnergy and continuity equationsas well as the sedimentdischargeand friction-factorrelationships.In the secondsrage,the sedimentcontinuity equation is solvedby an implicit finite-differenceapproximationto updatethe bed clevations uniformly. Bed arrnoringproceduresand the option of specifying a known bank erosionrate are includedin the rnodel. Severalother numericalmodels of aggradation-degradation have been developed. but all are linrited to varying degreesby an inconlpleteknowledge of the mechanicsof bank erosionand width adjustment.Kovacs and Parker( 199,1)provided some insight by developinga vectorial bed-loadformulation that takes into accountthe particlenrovenrenton steep,noncohesivebanks,as influencedby grat' ity as well as fluid shear.They applied their bed-loadformulation along with the sedimentcontinuity equationand the montentumecluationutilizing a simple algebraic turbulcnceclosure model for steady.unifbrm flow. The initial trapezoidal channel evolved inlo an equilibrium cross-sectionshapeconsisringof a flat bed near the centralpart of the channel that connectedsmoothly to a curr'ing,concare bank having a slopethat approachedthe angle of repose.Comparisonswith exp€rimental measurcmcnts showedgood agreement. Severalother nrodelsof width adjustmenthave been reviewedby the ASCE Task Committeeon River Width Adjusrmenr(ASCE 1998).Problemsof a variety of different bank failure mechanisms.unknown shear stressdistributionsin the near-bankzone,limited understandingof the erosionbehaviorof cohcsivesediment banks,lack of data on the longitudinalextent of massfailuresof the bank, and the significanceof overbankflows indicate that much rentainsto be leamedabout the mechanicsof bank erosionand width adjustment.The compurationaltools presently availablefor predictingwidth adjusrmentsare approximateat best.In spite of this. evaluationof a bridge-crossing site should include as much qualitativeand quantitative informationas possibleon the current stateof equilibrium of the streamor lack thereof,and possibleconsequences of the constructionof a bridge crossing.
Bridg€ Contraction Scour The rccelerrtionof the flow causedby a bridgecontractioncan leadto scour in the bridge opening that extendsacrossthe entire contractedchannel.The contraction can arisefrom a narrowingof the main channelas well as blockageof flow on the floodplain,if the abutnrents are at the banksof the main channeland overbankflow
428
C H A p T E R l 0 : F l o u i n A l l u v i a lC h a n n c r s
is occurring.If the abutmentsare setback from the edge of the nrain channel,contractionscour can occur on the floodplainin the setbackareaas well as in the ntain channel.Relief bridges on the floodplain or oyer a secondarystream in the overbank areaalso can causecontractionscour. The type of contractionscour can be either clear water or live bed. In clearin the approachcross-sectionupstream water scour,the velocitiesand shearstresses of the bridge are insu,llcient to initiate sedimentmotion, so no sedimenttranspon is conring into the contractedarea.In this case,scour continuesin the contracted section until the enlargemcnt of the cross-sectionis such that thd velocity approachesthe critical velocity and no additionalsedimentcan be transportedout of the contraction.This is the equilibrium condition that is approachedasymptotically in time. I-ivc-bed scour,on the other hand, occurs when sediment is being transportedirto the contractionflom upstream.Scourcontinuesuntil the sediment dischargeout of the contractedsectionis equal to thc sedimentdischargeinto the sectionfrom upstream.at which time equilibrium conditionshave been rcached. Laursen( 1958a, I 960) developedexpressionsfor both live-bedand clear-water contractionscour,assumingthat the contractionis long so that the approachflow and the contractednow both can be considereduniform. The live-bed caseis consideredfirst with referenceto Figure 10.23,which shows the limiting caseof the contractedsectionformed by the abutmentsset at the banksof the main channelin an idealizedsketch.The approachmain channelwidth is 8', and the nrain channel in the contractedsection has a width of 8,. The approachchanneldischargeis Q", and the overbankdischargeis po. From the continuity equation,it must be true that
(a) Plan
I
(b)Prolile FIGUREI0.2.1 Scour in an idealizedlong contraction(Laursen 1958a).
C t l A p I . l : Rl 0 : F l o * i n A l l u v i a lC h a n n e t s 1 2 9
the (otafdischargein thc contractedsectionO/ = Q, + Q,t.ln addition.equilibriur-l]l ol rhe live-bed scour proccssis reachedwhcn scdinrentcontinuity is satistled;that is. assunringsedimcnttransportoccursonly in thc rnain channel,we have
C',Q,- C':Q,
(10.81)
in which C,, .= lhc nrcansedimentconcentralionin the approachsectionand C," : thc mern sedinrentconcenlrationin the contractedsection.L-!urscnappliedhis total s e d i n r c ndt i s c h a r g ef b r m u l al L a u r s c n1 9 5 8 b )g i v c nb y
- ),t,"; '"1+]'"(;
c , . r n *: ( l
(10.82)
in which C, is the tolal sedirncnlcortcentrationin partsper million (ppm) by weight; d.,, : mcdian grain size:r'o - depth of uniform flow; r,! - grain shearstressir. : critical shcar stress;and/(a./rr;) is a specifiedgraphical function of the ratio of shearvelocitl'. u.. to fall velocity, b'r, which Laurscndetermined from laboratory data. The ratio of grain shearstressto critical shear stress is evaluatedfrom the Manning and Stricklerequationsand from the critical value of the Shieldsparameter,rr.., to yield v'
t )gri"dio'
( 10.83)
in which c, : the constantin the Stricklerequationtg : gravitationalacceleration; - t t r . M a n n i n g e q u a t i o nc o n s t a n t= 1 . 4 9i n E n g l i s hu n i t s a n d 1 . 0 i n S I u n i t s : 4 V : mean velocity: and SG : specificgrar ity of the sediments,with all other variables defined as in the previousequation.Laursenapplied English units and used r.. : 0.039. SG = 2.65.and c. = 0.034 in English units (0.041 in Sl units) to give
li= 7,
v' I 20,yA,rdilr
t r n p 1 , , 7l r l
\
(10.84)
which is specific to English units.Furthermore,the shearr elocity also is evaluated from Manning's equationto give
n^uGo
a.=Vey"S=----:K,8v6o
( 10.85)
Then,assumingthat r'oh, )) |, andthatf(uJwr) is a pouer function: ko@Jw1\', thesedimenttransport formulafor C, (Equation10.82)is substituted into Equation 10.81alongwith Equations10.84and 10.85to produce
v, )'r
/ o , \ " ' / 8 , \ 9 l - r / , r ,\ | | f l - ' " t - . l - ' - " \Q./
\Br/
\nt/
( 10.86)
The valuesof a are the exponentin the power fit to the graphicalfunctionof z*/r.r,, 12l.andaa n dh a v et h ev a l u e a s : 0 . 2 5f o r u J w , ( 0 . 5 ;a : I f o r 0 . 5 < u . / w , 2.25 for uJv, ) 2. Theserangesin a,/u7correspond to trirnsponmodesof mostly load,respectively. bedload,mixed load,andmostlysuspended
, 1 3 0 C H A p rE R l 0 : F l o wi n A l l u v i a C l hannel. 'l The ratio of a valuesis assumcdto bc close to unity and so is neglectcd. hr:n special cascsof Ecluation10.86can be identilicd. For an ttvcrhlnk contructionin u h i c h B , = 8 . , t h e r e s u l tf o r l i v e b e d c o n t r a c l i o ns c o u ri s ,\'r
( Q r'\n'
.tr
\O,,/
( r0.87)
* h i l e f o r a n a i n c h a n n e cl o n t r a c t i o ni n n h i c h Q r = 0,, the equationfor contraction scour becomes
l r
(10.88)
\ B: /
in u hich p, has valuesof 0.59 (bed load). 0.64 (ntixed load.).or 0.69 (suspended load). Finally, Laursenassunredthat. at the end of scour.both the changein velocit) head and the friction loss tiom section I to section 2 were small. so that the energy equationreclucesto -v,/,r'- d,,l)t + l. in which d,. = depthof contraction scour as shown in Figure 10.23.[t is interestingto observethat live-bedcontraction of the mode scour for the overbankcontraction,as given by ( 10.87).is independent only, the ntode of sediwhile for main-channel contraction of sedimenttranport, pt. in the exponent ment transportmakessome difference The clear-watercontractionscour formula also can be derived from the long contractiontheory as describedby Laursen( 1963) for relief bridge scour.Following a simplification of the derivation as presented in HEC-18 (Richardsonand Davis 1995).the value of z0 is set equal to i. at the contractedsection(2) at equilibrium when the sedimenttransponrate out of the contractedsectionapproaches zero. Then Equation 10.83is solvedfor depth -r, and divided by depth,r.rto yield
-)r )r
( d', \ ,)r
+ , ) =
/ . i s\ ' \^';/
'f
a,
i r " . ( S G- r )s,rlrdior.l
( 1 0 . 8 )9
section;\'r = depthbeforescour in ivhich-y,- depthafterscourin thecontracted = q. : QlBi B, : contracted scour depthl contraction in rhe contractionl4. : : grainsize;andSG = rnedian sediment g gravitational acceleration; dro width: in front of the squarebracketshas gravity The coefficient of the sediment. specific choice of theStricklerconunits, depending on the value in or English the same S[ form. For cn in nondimensional 10.89 is expressed stant,c,l and so Equation (c:glK:)3n: in SI units, for example, units or 0.0414 in English n/dl[6 9.934, valuescan by Laursen, but other was taken equal to 0-039 value of 2." 0.174.The i n t o E q u a t i o ) n 0 . 8 9 . be substituted andDavis1995)for theapplicais providedin HEC-18(Richardson Guidance first step is to determineif live-bedor equations. The scour tion of the conhaction velocitieswith the critical approach is occurring by comparing clear-waterscour velocity,which can be determinedas describedin a previoussection.If thereis an overr;ankcontraction,heavy vegetationon the floodplain may preventsediment transportandso the casemay be oneof clear-waterscoureventhoughthe sediment itself hasa criticalvelocitylessthanthe floodplain velocity,basedon sedimentsize alone.This often is the casefor relief bridgeson the floodplain.Significantback*'ater causedby the bridgecan reducevelocitiesupstreamso that what otherwise
C H A p 'ItR l 0 : F l o u i n A l l u v i a C l hannels -t_]l may have becn lile-bed conditionscan be changedto cleitr-$aterscour in the con_ traction.Furthennore,if the value of u,/tr'ris very large.the incoming sedintentdischargeis likely to be rvashedthrough thc contractionas suspendedloid only, and so in reality'this is a caseof clear-waterscourbecausethereis no interactionbet$een the sedimentbeing scouredout of the contractionand the inconringsedimentload. For overbankcontractionsor maln chrnnel contrucritrns u,irh no setbackof the abutntcntsfron] rhe banksofthe main channel,the applicarionofeither the live_bed or clear-waterscour equationsis relatively straightfon'ard.For significantsetback orstances,separatecoDtractionscour conlputationsshould be ntaclefor the main channeland the setbackoverbank areas,with the flow distributionbetweenmaln channel and overbank area in the bridge contractionesrimatedby WSpRO, for example.lf thc setbankdistanceis lessrhanthreeto fir,eflorv depths,it is likely that contractionscour and Iocal abutmentscour occur simultaneousiyand are not inde_ pcndent(Richardsonand Davis 1995).This case will be consideredfunher in the discussionof abutmentscour
Local Scour I-ocal scour around bridge piers and abutmentsis causedby obsrructionand sepa_ ration of the flow w,ithattendantgenerationof a systemof vortices.There is a stagnation line on the front of the pier with decreasingpressuredownward due to the lower velocitiesnear the bed.This causesa downflow directedtoward the bed near the front of the obstruction that separates and rolls up into a horseshoe vonex wrappedaroundthe baseof the pier In addition,thereare \r,akevorticesin the seD_ arationzone.This systemof vonices fluidizesthe bed and carriesthe sediment our of the separationzone to create a highly localized scour hole adiacent to the ubsrruction.This is illuslrarcdfor a bridge pier in Figure 10.24.whici shows borh the horseshoeand $ake vonices.
Wake vonex ..-------->
-
_=-J \_-, ^
\-/
n
Horseshoe vortex
FIGUREI0.24 Schematic representarion of scourarounda bridgepier(Richardson andDavis1995).
432
CH'\PTER
l0: F'lowin AlluvialChannels scour Maximumclear-watet scourdepth Equilibrium
!
i
Live-bedscour .9 o-
Clear-waterscour
Time FIGUREI0.25 andDavis1995). of pierscourwithtimc(Richardson of development Illustrarion
hither clear-wateror live-bedlocal scourcan occurjusl as for contraction scour.The maindifferenceis in thetimescalerequiredto reachequilibriumscour' bedmaterial,it takeslongerto scourtendsto occurin coarser clear-water Because to equiasshownin Figure10.25for a bridgepier.The approach reachequilibrium, to occur scourdepthis consi,iered so maximumclear-water libriumis asymptotic. are negligiblysmall.Live-bedscour when furtherchangesin the bed elevations occursmore rapidlyas shownin Figure1025 and tendsto oscillatearoundthe of bed formsthroughthe scourhole.The equiequilibriumdepthdue to passage libriumlive-bedscourdepthfor piersis only aboutl0 percentlessthanthe maxiandKaraki1969) Scourdepthis shown scour(Shen,Schneidcr, mum clear-water as a functionof approachvelocityin Figure10.26with the criticalr elocitydividing clear-waterfrom live-bedscour.A peakis shownat the critical I elocity with an into the scour beginsto be transponed in scourdepthassediment abruptdecrease againto a second'louer Peakthat is the scourdepthincreases hole.Thereafter, with planingout of thebedforms(Raudkivi1986) associated Pier scour flow variables'fluid propScourdepthat a pier is a functionof piergeometry' erties,andsedimentproPenies: d , : f J K , , K r , b , V , . v gy , p , t L , ( p ,- p ) , d 5 s . o s )
(10.90)
in which K. : pier shapefactor= 1.0for cylindricalpiers;Ku - pier alignment dept}t:8= gravitavelocity;.vr- approach factor;b : pierwidth: Vr = approach p density; : fluid viscosity; p fluid density;p, sediment tionalacceleration; = : deviationof sediment geometric standard d.s mediansedimentsize;anda, variables and carryingout the siie distribution.Choosingp, V', andb asrepeating analysisresultsin dimensional
CHApTER l0: Flo\r inAlluvial Channels 4i3
.c o-
o a o (t)
Clear'waler Live-bed scour scout
vc Velocity FIGURX I0.26 lllustrationof clear,warer andlive-bedscourat a pier (afterRaudkivil9g6).(Source: A. !. Raudkivi. "Fu,lctiohal Trendsof Scourttt Bridge piers,"J. Htdr Engrg.,@ 19g6, ASCE. Reproducedbt pennissionof ASCE.)
d. I ;o= f : l Kl . K *
+*+T'il'",)
(loer)
CombiningrheFroudenumber,V,/t61,, )05,w jth {p, - p;/p andy,/droresultsin the sedimentnunrberN,: yrll(Sc - l)gd,,]105. whichcanbe replaced-by Vr/u,,from theShieldsdiagramin theabsence of viscouseffects(".c : constant). Funhermore, it is apparent from theKeulcgan equationwrittenfor criticalvelocitythat)r/40 can be expressedin termsof V.,/r-.,in which V. is rhecritical velocity.Finally,ihj ratio of Vtlu.. to V./a..just giyesa sedimentmobilityparameterV,/V..Thus,an alternativechoicefor the relativesubmerged densityis rhe sedimentmobilitypammeter ln addition,yrldro can be replacedby b/40. With thesesubstitutions and neglecting viscouseffects,theresultis
d.
V, v, h ] ' f ^Il r ' x * r', ' a ' ' b r G y ,y' ; , , ' " ' l
(10.92)
Mostpier scourequations canbe placedin thisform,but someof rie independent variables are neglected in all ofthem (Ettema,Melville,andBarkdollt99ti). The pier scourequationrecommended by the FederalHighwayAdministration in HEC-18(Richardson and Davis 1995)is the ColoradoStateUniversity(CSU) formula(Richardson, Simons,andJulien1990)givenby d.
. = 2 0 K , K o K b K/ ,r\. b\ )
015
F! r:
( r0.93)
-13-1 C H A p T E R l 0 : F l o w i n A l l u v i a l C h a n n e l s
in shich (, : piershapefactor;K, = pierskesncssfactor;K, : correction fictor for bcd condition;4 = beOarnroringfactor; r', = approachdepth directly up:treamof the pier: b - pier width; and F, = approachflorv Froudenunrber. Equation10.93is basedon laboratory dataandreconrmended for bothlive-bedand clea-r-water scour.The valueof K, : 1.0for roundnose.cylindrical,andgroupsof cllindrical piers,rrhile it hasthe valueof l.l for square-nose piersand 0.9 for piers.The skervness sharp-nose fhctoris expressed as a functionof the angleof attack,0, of the flow directionrelatjveto the lon-gitudinal axisof thepier: /
I
\Lrb5
K 6 - { c os 0+fs nd) \ D , t
(10.9,1)
in s hichLn = lengthof thepierandb = width of pier.The rraximumvalueof Lolb is taliento be 12,evenif theactualvalucexceedsI 2. The valueof K" = 1.0for 0 : 0.0. but it canbe significantly differentfrom unir1,. For L,/b = 4, for example,and piersshouldbe alignedwith rheflow directiondur0 : 30",K = 2.0.Therefore, ing floodconditions. For attackanglesgreaterthan5", K, dominates K,, which is takento be I .0 for thiscase.The valueof K, reflectsthepresence or absence of bed forms andso is relatedto maximumclear-r'atervs. live-bedscour.The valueof Kr : l.l for clear-water scourandfor live-bedscourwith planebed,antidunes, and smalldunes(0.6 < A < 3.0m). For duneheightsA from 3.0to 9.0 m, Kb - l.l to 1.2,whilefor A greaterthan9.0 m, K, - 1.3.Finally,thearmoringconectionfactor is definedby ,(" = [1.0
0 . 8 9 ( t . 0- y F ) : ] 0 5
(10.95)
in which V, : (Vt - V,)/(Vcn- V), Vt = approachvelocity in metersper second (m/s); V.- = critical velocity for d* bed rnaterial size in m-/s;and y, - approach velocity in m./swhen sedimentgrains begin to move at the pier The value of V, in m./s is calculated from f, 'r
-w.lFJt
10051 ,
I
L o )
%so
(10.96)
where V.ro: critical velocityfor drobedmaterialsizein m,/s,The factorK, applies only for dro> 60 mm.It hasa minimumvalueof 0.7 anda maximumvalueof 1.0 r h e n V ^> 1 . 0 . pier scourin the laboratory LaursenandToch( 1956)measured for conditions of live-bedscouraroundcylindricalpierswith a subcritical approach flow andbedload transponof sediment.They arguedthat neitherthe approachvelocity nor the sedinrent sizeaffected theirresultsfor depthof scourbecause a changein eitherone simply causeda proportionalchangein sedimenttransportrateboth into andout of the scourholeto setup a newequilibriumin transponratewith essentially thesame scourdepth.The resultingpier-scour formulaas givenby Jain( l98l ) is
f : "'[]]"'
(10.97)
The experimentscoveredthe rangeof I = \'t/b < 4.5, and dro from 0.44 to 2.25 mm (mediumto verycoarsesand).
C H A p T E Rl 0 i F ' l o wi n A l l u l i a lC h a n n e l s . l - 3 5 Jain (198l) proposeda formula for maxinturnclear-waterscour around crlin_ drical piers rhat includesan effect of sedimentsize.In the dimcnsionalurlysis of Equation 10.92,VrlV, = 1at maximum clear-waterscour so that the Froude number F, - F. = V1(glr)u5,which is rhe critical value of the Froude number calcuIatedfrom the crirical velocity evalualcdfrom Keulegans equationand the Shields diagram. as describedpreviously.The resulting fornula is based on the experi_ mental data ol Shen.Schneider,and Karaki (1969) and Chabcrt and Eneeldinser ( 1 9 5 6 ) .I t i s g i v e nb y
f ='*h]"' F:-
( r0.98)
The exponenton -r,,/bis the sameas for the LaursenandTochformula.The ranse in _v,/b of thedatavariedfrom0.7 ro 7.0,whilethenreansediment sizcsof thedaia werebetween0.24 and3.0 mm Equation10.98providesan upperenvelopefor thedata. Jain and Fischer(1980)invesrigated live-bedpier scouraroundcylindrical piersat high velocities. They measured the scourdepthsaroundpiersin a flume usingthreadsplacedvenicallyin the sedimentbed prior to scour.At the end of scour,the threadswereexcavated and lhe scourdepthwasmeasured at the eleva_ tion at which the threadswerebent over.This procedurewas intendedto avoid the biascausedby panialinfillingof the scourhole whcnthe experimental flow was stopped. The resultingscourformulais similarto EquationI0.9g,exceptthat the scourdepthis relatedto theexcessFroudenumber(F, - F-), because theformula appliesonly to the live-bedcase.The resultsshoweda slightdecrease rn scour depthaftermaximumclear-water scourfollowedby increases in scourdepthwith rncreases in (Fr F").The live-bedscourformulais
*:,'[]1"
lF, - F"lo"
(10.9e)
which providesan envelopeof the data.Most of the datahad,y,lb valuesof eirher I or 2 with threedatapointsin the rangeof 4 to 5. Sedimentsizesvariedfrom 0.25 to 2.5 mm. Melville and Sutherland (1988)and Melville (1997)developed an empirical pier scourequationbasedon a large numberof laboratoryexperimentsat the Uni_ versityofAuckland,NewZealand.It hasthe form d,b
K,KeK rKrKdK"
( 10.100)
in which ,(, andK, arethe shapeand skewnesscorrectionfactorsas before:rK, = expressionfor effect of flow intensity; (, : expressionfor effect of flow depth: K, = expression for effecrof sedimenrsize:andK" er,pression for effectof ied_ iment gradation.Raudkiviand Enema (1983) showedthat, for clear-waterscour, sedimentgradationcauseda large reductionin scourdepth due to armorine for o, ) I.3. However. MelvilleandSutherland { 1988)presented a methodfor accJunting for sedimentgradationeffectsby definingan armorvelocity y > % at which live-bedscourbegins.The value of V" is calculatedas 0.8 V.. in which'4, is the
. l - 1 6 C H A P T E Rl 0 : F l o \ i n A l l L r v i aCl h a n n c l s critical vclocity of the coarscstarntor size given by d","-/18, where dn',. is sotrte reprcsentativemaxinrurngrain size in the sedimentmixlure' Then, the flow intensity expression,K,, is evaluatedfrom v
- , *[
yr
(v,, v,.
rl
v, - lv" v,) < v, v\ (.v"- v) > 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
<25
a50 h : > )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,) _. <- I
v, - (v" v.) > v,
l
(10.l07b)
in which V, - arnrorvelocitydefinedin the sameway as for piers;4 = critical r,elocity;and V, = velocityin the bridgeapproachsection.The depthadjustment by Equations10.103,exceptthat factor,Kr, is the sameas for piers,as expressed shapefacby the abutmentlength,L,. The abutment the pier width,b, is replaced and0.75for wing-wallabutto be 1.0for venical-wallabutments tor is assumed valuesof K, - 0.6,0.-5,and0 45 for areassigned abutments ments.Spill-through s f s h a p ef a c t o r l yh. e s ev a l u e o 0 . 5 : l ( H : D , l : 1 , a n d 1 . 5 : ls i d es l o p e sr,e s p e c t i v e T for whichL/)t < 10.Shapceffectswerefoundto applyonly to shorterabutmcnts, be unimportantfor longerabutments,so that K, : 1.0for L/.i' > 25. For abutnrent \\'assuggested: a linearinterpolation lengthsbetweenthesetwo extremes, / 1.. \ K l - K , + 0 . 6 6 7 ( -1 K . ) ( 0 . 1 - j - l /
Lj .< 2 5 for l 0 < _1t
(10.108) andKf is the interpothe shapefactorfor shon abutments. in which K, represents Values K, for flow alignmentand of length abutments. latedvaluefor intermediate floodplain aregiven channel from th€ protrude into the main that Ko for abutments ( 1 9 9 7 ) . by Melville datasetfbr liveanalysisto a laboratory Froehlich( 1989)apptieda regression produce the relationship to investigators from several bedabutmentscour
4 = z.zt *, x,K,lLlo"utu,
(r 0 . 1 0 9 )
in which d, : local abutmentscour dePth;.r', = approachflow depth; K, : abutment shapefactor;Kd = skewnessfactor;L, : abutmentlength;and F, - approach flow Froude number The value of 1.0 added lo the right-handside of Equation 10.109is a factor of safety.Froehlichcalculatedthe approachFroudenumberbased on an averagevelocity and depth in the area obstructedby the embankmentand abutment in the approachflow cross section. All the experimentalrcsults in the regressionanalysiscame from experimentsin rectangularflumes. Richardsonand Richardson( 1998)arguedthat experimentalresultsfbr rectangular flumes that dependon abutment length as an independentvariable do not accuratelyreflect the abutmentscour processfor contpoundchannels,which have a nonuniform velocity and dischargedistribution acrossthe channel.Sturm and Janjua (1994) demonstratedthat a dischargecontractionratio. M, representsthe redistributionof flow between main channel and floodplain as the flow Passes
C H A p ' rE R l 0 t F I o w i n A l l u v i a l C h a n n e l s
439
Oobstt
A
t_ (a) Plan
Watersurface
---
Tl o.s
-----
Floodplain
(b) SectionA-A
FIGURE10.27 Definition sketch for idealized abutment (Sturm1999b). scourin a compound channel throughthebridgecontraction. As shownin FigureI0.27,thedischarge contraction ratio,M, is definedby Q
-
Q.r"t
( 1 0 . lr0 )
in whichQoo.,,- obstructed floodplaindischarge overa lengthequalto the abutment length projectedonto the approachcrosssectiontand Q : 1661discharge throughthebridgeopeningfor an abutment on onesideonly,asin Figure10.27,or Q : total dischargefrom the outer edgeof the floodplainro the cenrerlineof the mainchannelfor abutmentson both sidesof the main channel.The variablefl -rVf) wasproposedby Kindsvaterand Carter( 1954)to characterize the effectof a bridge on flow obstructionto measuredischarge,and it is usedin the FHWA/USGSprogramWSPROto determinebridgebackwater(seeChapter6). Sturmand Janjua (1994)showedthatM is approximately equalto the ratioof discharges per unit of width in the approachand conrractedfloodplain areas,qrylqp,for an abutmentthat termindtes on lhe floodplain. With reference to Figure 10.27,the idealizedlongcontraction scouris formulatedfirst,followedby equatingthelocalabutment scourto somemultiplierof the contraction scourasoriginallyproposed by Laursen( 1963).In two differentcompoundchannelgeometries, SrurmandSadiq(1996)andSturm(1999a,1999b)have
l hannels 4 . 1 0 C H A P T E Rl 0 : F l o q ' i nA l l u v i aC sho$n that this approach to the problem results in a clcar-water abutment scour equationgiYenby
*-"'r,l#A-o+]
+r
( l 0 . ll l )
in *,hich d, - local clear-waterabutmentscour;-v/0= floodPlain depth for uncontractedflow: K, : abutmentshapefactor;q/r - approachdischargeper unit width in the floodplain : Vr':rl, M = dischargecontractionratiol Vo. : critical velocity in the floodplain at the unconsricted depth.\i0 for setbackabutmentsand critical velocity in the main channel at the unconstricteddepth in the main channel for bankline abutments.The factor of I on the right hand side of Equation l0.l I I is a factor of safety. If the approtch floodplain velocity V71exceedsthe critical value vr.. then v/l is set equal to yr. for maximum clear-waterscour.The shapelactor K, = I for venical-wall abutments,while for spill-throughabutments,it is given by
0.67 t x, -_ t.5z _ gA ,
for 0.6? '< € < 1.2
( 1 0 .rl2 )
where ( - qtt/(Mvk)rd, and K, = 1.0 for ( > 1.2 as the contraction effect becomesmorb important than the abutmentshape.Equation l0.l I I is compared with the experimentaldata for an asymmetriccompound channel having a floodplain width of 3.66 m and a main channelwidth of 0.55 m in Figure 10.28,which
. Verticalwall o Spill{hrough Best fit
1 e. .
^
\ ' B' o t h-
'1.0
1.5 = qnl(MV6sfrc) I
FIGURE10.28 (Sturm1999b). channels for compound Abutment scourrelationship
oo
C H A p T E Rl O : F l o wi n A l l u v i aC l hannels .1.11 sho*s that d,/_r;shas a nraximumvalue of 10.The i valr.refor the best-fitequation withour a factor of safetyis 0.86 with a standardenor of 0.74 in r/.Aru. E x A \ r p L E 1 0 . s . A b r i d g ew i t h a 2 2 8 . 6m ( 7 5 0 f r ) o p c n i n gl e n g r hs p a n sB u r d c l l Creek.\r hichhasa drainage a-rea of 97I kmr (375 mir). The exit andbridgecrosssectionsareshownin Figure10.29with threesubseclions andthcircorresponding values of Manning'sa. The slopeof the streamreachat the bridgesiteis constant andequal to 0.001rdm. The bridgehasa deckelevation of 6.7i m (22.0fr) anda borromchord ele\arionof 5.,19m (18.0fr). It is a Type3 bridgetseeChaprer6) wirh 2:l aburnrent and embankment slopes.and it is perpendicular lo the flow direction(no skew).The topsof the left andrightspill throughabutments are at X stationsof 281.9 m (925fi) and510.5m ( 1675ft), andtheabutnrents aresetbackfrom thebanksof themainchannel.Therearesix cylindricalbridgepiers.each*ith a widthof 1.52m (5.00ft). The sedimenthasa mediangraindiarneter, d5o,of 2.0 mm (6.56 x l0 r fl). Esrimaterhe clear-\r'ater scourandpierscourfor the l0O 1r designflood,\\ hichhasa peak abutment discharge of 397mr/s(1.1,000 cfs). Solulion. The FHWL,-SGS program WSPRO. describedin Chapter6, is run ro obtainthehydraulicvariables neededin thescourpredictionformulas, althoughHECRAS couldalsobe used.The programactuallyis run twice, first to obtainthe watersurface elevationsior both the unconstrictedand constrictedflows at the approachcross sectionand,second, with theHP 2 dalarecordsto computethe velocitydistribution in the approachsection for the unconstrict€d(undisturbed)water surfaceelevationof 4.038m (13.25ft) andtheconstricted watersurfaceelevation of ,1.157m (13.6,1 ft). The scourparameters lhen are determinedfrom the WSPROresults.Calculations are madefor the left abutment,which hasa length.L,. of 233 m (76.1ft). Fromthe compuredvelocitydistributionfor theconstricted flow, the blockeddischargein rheapproach
Burdell Creek D.A.= 971km2;Q100- 397m3/s n=O.Q32
|n
0 . 0 4 2|
W.S.elev. at bridge
200
400 Transverse Station,m
FIGUREI0.29 Bridgecross-sections for Example 10.5.
n=0032
112
l hannels C H A p T E Rl 0 : F l o wi n A l l u v i aC area sectionfor the left abutmentis 39.I m'/s ( I -180cfs) w ith a blockedcross'sectional of 106.8mr ( I 150ftr). The dischargefrom tie left edgeof walerin the approachcross of the nain channelis 210 mr/s(7411cfs).Then,the valueof sectionto thecenterline = M = (.210- 19.ly2l0 : 0.81.Now we can calculateV11: Q1lA1r= 39.1/106.8 = : ( 8 1 . 5 0f t ) .I n a s i m i l a r w a y , 0 . 3 6 6 m , i s ( l . 2 0 f t / s ) a n d ! tA t =r l L , 1 0 6 . 8 / 2 3 3 0 . ' 1 5m crosssectionto be 0.35?m ( L l7 fl). the valueof rro is foundfor theunconstricted inlo by substituting The criticalvelocitiesfor coarsesedimentsare determined approach sectionandfor a equation). For rhe constricted Equation10.17(Keulegan's floodDlain deDthof 0.458m. we have
vtu = 5
'75x
( r2.tx0.458 )
(0.0,1s )(2.65 I ) ( 9 . 8 1 ) ( 0 . 0 0 2x ) I o 8 2 x 0 0 0 2
: 0.69m/s (2.3ftls) sizeand hasbeentdien to be 0.045for this sedinrent in whichthe Shieldsparameter = 2rl*. BecauseV11< Vyt,,it is apparent that sand-grain roughness,t, the equivalent we haveclear-waterscour In a similar manner,the valueof V;q for an unconstricted floodplaindepthof 0.357m (1.17ft) is 0.67 n/s (2.2ft/s). To cornputethe scour depth for the setbackabutments,substituteinto Equation l0.l l l to obnin
(0.r66x0.157 ) -d:. 8 . 1 4 x 0 . 6 3 xI 1 ( 0 . 8)1( 0 . 6x70 . 3 5 7 )
o + ]+ r o = : +
in whichtheshapefactorX, = 0.63from Equation10.I12andthesafetyfacto.of 1.0 hasbeenincluded.Finally,$e left abutmentscourdepthis 3.4 x 0.357: 1.2m (3.9 ft). In general,this calculationwould be repeatedfor the right abutment,but this examsymmetric ple hasan es.entially crosssection. Next, considerthe scouraroundthe bridge piers and usethe largestflow depthin the crosssection,assumingthat the thal$eg might migratelaterally.The WSPRO of 3.80 m (12.5ft) in thebridgesection,which resultsgive a watersurfaceelevation corresponds to a maxinrumdepthof 2.63 m (8.63ft). The maximumvelocityin the bridgesectionis 1.68m,/s(5.51 fi/s). The resultingvalueof the pier approachFroude into the CSU pier numberis V,/(8y,)o5: 1.68(9.81x 2.63)05: 0.33.Substituting scourformulaand recallingthat the pier $ idth , = 1.52 m (5.0 ft), we have f ! . 1 0r 5
d,: bx2.oK,KeKbK : l. t o)
Fl" f ) ^1']0J5
- l s 2 2 \ ( r 0 )r(O x r r x l 0 ) l ;;;
1 0 3 3 1- 02 ". s m
(or 8.2 ft), in which all the corection factors havethe valueof I exceptthe bed corequation gives rection,which is takento be l . l for clear-* ater scour The Laursen-Toch a oierscourdepthof
I r',lor
d , = b xr 3 5 L t ]
It61lor
1 . 5 2 / . ' l . l s|.: : : | L r . ) rI
- 2 . 4 m ( 7f.rq)
Total Scour in HEC-18(Richardson andDavis1995)thatdegradation, conIt is recommended produce pier be added to a conservative total tractionscour,andabutmentor scour
CHAPTERl0: F-lowin AlluvialChannels 443 scourestimate.For setbackabutments.contractionscourhas to be calculatedseparately for the setbackareaand the main channelin the bridge section.Another conservativedesignsuggestionis to use the calculatedmaximum scour depth at a pier in the main channelfor a pier in the setbackarea as well, assuminglateral migration of the main channelinto the setbackarea.For banklineabutments,contraction scour and abutmentscour occur simultaneouslyratherthan independenrly,so that adding abutmentscour and contractionscour for this casemay be overly conservative. If scourcalculationsindicatethat foundationdepthsare excessivelylarge,then such as rock riprap protectionand guide banks can be used scourcountenneasures ( s e eL a g a s s e t a l . l 9 9 l ) .
REFERENCES Ackers.P, andW. R. White. Sedinrent Transpon:New ApproachandAnalysis."J. H_rd D n : ,A S C E9 9 ,n o .H Y I I ( 1 9 7 3 )p, p . 2 0 4 l - 6 0 . Allen, J. R. L. "Conlputational Methodsfor DuneTime Lag: Calculations Using Stein's Rulefor DuneHeight;'Sedirnentan Geol.20.no.3(1978).pp. 165 2I6. Alonso.C. V "Selecting a Formulato EstimateSediment TransponCapacityin NonvegetaredChannels." ln CRELlts, ed.W G. Knisel,Chapter5. pp.,126-439. Conservarion Research ReportNo.26. U.S.Depanment ofAgricuhure,1980. "Relationships ASCETaskCommittee. BetweenMorphologyof SmallStreamsand SedimentYields."J. H,vdDir:. ASCE 108.no. HYI I (1982),pp. 132865. "RiverWidthAdjustment. ASCETaskForceon River\\idth Adjustment. II: Modeling.""/. H y t h E n g r gA. S C El 2 J . n o . 9 ( 1 9 9 8 )p. p . 9 0 3 - 9 1 7 . Bagnold,R. A. Ar Approachto the SedimentTronsportProblentfron General Physics.Prof. Paper422-1.Washington. DC: U. S. CcologicalSuney, 1966. Banon. J. K., and P Lin. A Studt of the SedintentTrattsportin Alluvial Cra|n"ls. Report No. 55JRB2.Fon Collins.CO: ColoradoAgriculturalandMechanical College,D€pt. of Civil Engineering, 1955. Bechteler,W, and M. Vetter."The Computationof Total SedinrentTranspon in Vie$ of Changedlnput Parameten.'In Proceedingsof the ASCElntemational Swnposiurn on SedimentTransportModeling,Neu Orleans,LA, 1989,pp. 548-53. Biglari.B., andI W. Sturm."NumericalModelingof Flow AroundBridgeAbutmenrs in CompoundChannel." J. Hrdr. Engrg.,ASCE 124.no. 2 (1998),pp. 156-61. Brown. C. B. "SedimentTrarsponation."In Engileeritlg Hydraulics,ed. H. Rouse,pp. 769 857.IowaInstituteof HydraulicResearch. IowaCity.lowa, 1950. Brownlie,W R. "Flow Depth in Sand-Bed Channels." J. Htdr Engrg..ASCE 109,no. ? ( 1983).pp. 959 90. Brownfie,W R. Prediction oJ Flou Depth and SedimentDischargein Open Chatnels. W. M. KeckLaboratory. ReportNo. KH-R-43A.Pasadena: CaliforniaInstituteofTechn o l o g y1. 9 8 1 . Buffington, J. M. "The Legendof A. F. Shields." J. H '-dr.Engrg.. ASCE 125.no. 4 ( 1999), pp.376-87. M. R. "Similarily[-a\\'sofLocalizedScour"J. H|dr. Di\'.,ASCE92, no.3 (1966), Carstens. pp. l3-36. Casey.H. J. "Uber Geschiebebewegung." Versuchsanstah Mitteilwtgender Preussischen .fiir Wasserbau wrd Schiflbarr Iin German].Heft l9 (1935). Chaben,J., andP Engeldinger.EtudedesAfouillementsAutourdesPilesde Po,rr.r.Chatou, France:Laboratorie Nationald'HYdraulioue. 1956.
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411
l hannels l 0 : F l o q i n A l l u v i aC CITAPTER
NewYork:JohnWilet & Sons'1988 h RiverEngineerinS' Chang,H. H. F/lvidI Processes J. H\d. Div ' ASCE 108.no Erodible Channels Model for Chang,H. H. Matbernatical H Y 5 ( 1 9 8 2 )p, p .6 7 8 - 8 9 . "Modelingof RiverChannelChanges"J. H.tdr.Dtgrg-,ASCE I l0' no 2 Chang,H. H. ( 1 9 8 , 1p) ,p . 1 5 7 - ? 2 . "River ChannelChanSes: Adiustments of Equilibrium"J- Htdr Engrg, Chang,H. H. 3 5 5 . A S C EI 1 2 ,n o . 1 ( 1 9 8 6 )P. P 4 . "River l\'lorphologyandThresholds." J H-r'drEngr3 ASCE I I l. no 3 ( 1985), Chang.H. H. p p .5 0 3 - 1 9 . "The PresentStatusof Researchon Sedin'lent Transpon."Iransactions[ASCE] Chien.N. l 2 l , n o . 2 8 2 4( 1 9 5 6 )p. p .8 3 3 - 6 8 . Colby.B. R., andC. H- Henlbree-Colrpt.zrtionofTotalSetlimentDischarge,\'iobrara River near Cotly, Nebrasla. Water Supply Paper 1357.Washington,DC: U S. GEological Survey,1955. "Flume Sludiesof the Sediment TransferCoetllcient."llhter Resources Colcnan.N. L. Resmrch6. no. 3 ( I970). Dawdy, D. R. Depth'Discharge Relationsof Alluvial Streants-Discontituous Raling DC: U S. GeologicalSurvey,l96l' Caner. WaterSupplyPaperl'198-C.Washington. "Effect of AdsorbedNatDennett,K. E., T. W. Sturm,A. Amirtharajah,andT Mahmood ural OrganicMatter on the Erosionof KaoliniteSediments"Waler Environment no. 3 ( 1998).pp. 268-75 Research.7O, "Le Rohneet les Rivieresa Lit Affouillable" Annalesdes Pont et Chaussees, P DuBoys, S e r i e5s , n o . l 8 ( 1 8 7 9 )P. P .l 4 l - 9 5 "Formulasfor theTransponation of BedLoad.' Transactions IASCE]107' Einstein,H. A. ( p p . 5 6 l 7 3 . no.2l40 1942), 'The Bed Load Functionfor Sedimenl in OPenChannels" Transponation Einstein.H. A. Service.1950. DC: USDA,Soil Consen'ation Bulletin1026.washington, Technical "River-Channel Iransactioas Roughness'" IASCE]ll7, H. A., andN. Barbarossa. Einstein. ( 1 9 5 2 ) p , p . ll2l-46. n o .2 5 2 8 Einstein,H. A., and N. Chien.Efrectsof Heat'ySedinentCo certlrationnear the Bed on Vcloci4,and Sedine t Distribution.MRD SeriesRepon No. 8. Universityof CaliforOmaha. nia at Berkeleyand MissouriRiverDivision,U.S.Army Corpsof Engineers, Neb..1955. L)od Einstein.H. A.. andN . Chien.SecondApprotination to the Soluliottof the Suspended Ifieon. MRD SeriesReponNo. 2. Universityof Califomiaat BerkeleyandMissouri Omaha'Neb.,1954 RiverDivision,U.S.Army Corpsof Engineers, "Hydraulic Streams." J. H\d Dit:, ASCE 92' no HY2 Resistance ofAlluvial Engelund,F. ( 1966).pp. -l l5-26. " "HydraulicResislance of AlluvialStreams J. i/rd' Dirr' ASCE93, F. Closureto Engelund, n o .H Y 7 ( 1 9 6 7 )p. P . 2 8 79 6 . Engelund.F., and E. Hansen.A Monogrophon SedimentTransportto '\lluvial Strea s' TeknikVorlag,1967 Copenhagen: " "ScaleEffectin Pier-Scour ExperimentsJ' Enema.R., B. W. Mel!ille. andB. Barkdoll no.6 (1998),pp.639-42. Htdr- Engrq.,ASCE 12,1, "Anallsis of On'SiteMeasurements of of Scourat Piers."ln Proceedings D. C. Froehfich, and J Gessler, ed. S. R. Abt EngineeriaS, on Hytlraulic Conferente 1988 Ntttional tlv ASCE,1988 pp.534-39.ColoradoSprings. 'Local ScouratBridgeAbutments."[n Proceedings of National Conference Froehlich.D. C. ASCE' 1989 pp. 18, New Orleans. l3 M. A Pons, ed. on HtdraulicEngineering. " "Beginningand Ceasingof Sediment ed.H.W River Mechonics, Motion in J. Gcssler, Shen.pp. 7-l to 7-22.Fon Collins.CO: ColoradoStateUniversity,l9?0'
C H A p T E Rl 0 : F l o \ \i n A l l u \ i r l C h a n n e l s 4 4 5 oj Dtbris bt Rtourirgllizrl'r Prof.Paper\o. 86.Wa\hittSlon. Gilr^Ln,G. K. Tntntportatbn S u r D C : U . S .C e o l o g i c a l v e yl ,9 l - 1 . Craf. W. H. Ittdraulicsof Selinett Tr tlsport.New York:\'lcCratr'-Hill.1971. Datafrorr Guy. ll. P, D. B. Simons.and E. V Rich&dson.Sununttn4 Allu\iotChttnttel I. U. S. \iVashinglon. DCI U.S.Ceologil9564l.Prot. Paper.162 Flune E.rperinents, cal Survey.1966. of MissouriRivr Prttgnosis Holli, F. M. Jr.,J. C. Yang.andM. F Karinr.Con4tuter'Bused IIHR Repon No. 281. Bed Degr,ttluion-RrJinenttntof Con4tutatioul Pro<'edures. 198J. of H)dra lic Research. IowaCily: IowaInstitute Midtlle Laup of Se'dilet Transpondtiotr. Hubbell,D. W.. andD. Q. Matejka.lnvestigation.r \\rashinglon. DCr U.S.CeoRivetat Du ing,Nebtrska.WalerSupplyPaper1.176. logicalSurvey.)959. 'MaximumClear-Water ScourAroundCircularPiers.l Hld. Dir, ASCE 107, Jain.S. C. n o .H Y 5 ( 1 9 8 1 )p. p .6 l l - 1 6 . /. F/)r1. Jain.S. C.. andE. E. Fischer.ScourAroundBridgePiersat High Flor Velocilies." Dn:. ASCE 106.no.HY I I ( 1980),pp. 1827-12. EquationsUsingField Data.' J. Hydr Engrg., of Pier-Scour Johnson,P A. "Comparison A S C El 2 l , n o . 8 ( 1 9 9 5 )p. p . 6 2 6 - 2 9 . Equations for BridgePierandAbutmentScour."In of Prediclion Jones.J. S. "Comparison Record950, SecondBridge EngineeringConference,vol. 2. TronsportationResearch Research Board,1983. Washington. DC: Transportation l.?rror.N-ewYork: CambridgeUniversityPress,1995. Julien,P Y Erosionantl Sedimed "Sand-Dune Geonretryof LargeRi\ersDuringFloods.'"/. Julien.P Y, andG. J. Klaassen. H y d r E n g r g .A. S C El 2 l , n o . 9 ( 1 9 9 5 )p, p . 6 5 76 1 . and HydraulicResistance in All!\ ial-Channel Rows." "/. Karim, F. "Bed Configuration Hvdr Engrg.,ASCE l2l, no. I (1995),pp. l5-25. Karim, F. "Bed-FormGeometryin Sand-BedFlows." J. Htdr Engrg.,ASCE 125,no. l2 ( 1999),pp. 1253-61. "Bed Material DischargePredictionfor Nonuniform Bed Sediments."",1.Ilr'dr Karim, Fr. no.6 ( 1988).pp.59?-604. Engrg.,ASCE 12,1. Predictorsfor SednnentDisclnrges and Karim, F., and J. F. Kennedy.Corltputer-Based ReponNo. 2'12.Io\ a City: IowaInstiluteof Streams.IIHR FrictionFactorsof AIlLtvial of lowa. l98l. HydraulicResearch, University "Menu of CoupledVelocityandSedimentDischargeRelaKarim. M. F., andJ. F. Kennedy. tionsfor Rivers." J. Htdr Engrg.,ASCE I16, no. 8 ( 1990).pp. 978-96. Channels."./. F/. M?cr., of DunesandAntidunesin Erodible-Bed Kennedy, J. n "Mechanics 16,no.4 (1963),pp.52144. WavesandAntidunesin Alluvial Channels." ReponNo. KH-RKennedy, J. F. "Stationary 2. Pasadena: W M. Keck Laboratory,Califomia Inslituteof Technology,1961. Kennedy,J. F. "The Albert ShieldsStory."J. Htdr Engrg.,ASCE l2l, no. tl (1995), pp.766-72. "l-aboratoryStudy of an Alluvial Streamat ConstantDisKennedy,J. F., andN. H. Brooks. Misc. of the FederalInteragetrcySedimentation Conferetrce, charge."In Proceedings DC: AgriculturatResrarchService,U.S. Publ. No.970, pp. 320-30.Washington, 1q65. Depanmcnt of Agriculture, Flow in OpenChannels." Research PaperRPI l5l. J. Keulegan, C. H. "Lawsof Turbulent of Researchof NationolBureauof Starulards( 1938). "Tranquil Flow ThroughOpen-Channel Constrictions." Kindsvater,C. 8., andR. w. Carter (1954), pp. 955 1005. Transactions IASCEI Kovacs,A., andG. Parker.'A New Vecto.ialBedloadFomrulationandIts Applicationto the Time Evolutionof StraightRiverChannels;' J. F l. Mech.261 (1994),pp. 153 83.
. 1 4 6 C H A p T F Rl 0 : F I o wi n A l l u v i a C l hannels Kramer.H- "SandMixturesandSandMovementin Fluvill Models."Iitrs.(.liozj IASCE] 1 0 0 .n o . 1 9 0 9( 1 9 3 5 )p. p . 7 9 8 - 8 3 8 . Lagassc. P F-..J. D. Schall,F Johnson,E. V. Richardson. J. R. Richardson, andF. Chanp. StreaDt Stabilind HighuoyStructures. ReponNo. HEC-20.Washinlton, DC: Federii H r ! h ! r y A d m i n i \ l r J t i oU n . S .D e f a r l m c norf T r a n . p o n . r r i o| 9n 9. l . pier-Scour I-anders, M. N., andD. S. Mueller.Evaluarion of Sclected Equarions UsingField Data.pp. 186,95.Trettspoatio Research Record1-t2J.Washington. DC: Narional Research Council.1996. Lane,E. W A Stadl,o/ theShopeof ChtnnelsFonnedby Nuuru! Steans Flctt+.ing in Erodible Muterial.MRD SerjesReportNo. 9. \{issouriRiverDivision,U.S.ArnryComsof Engrneers. Om.rha.Ncb..195?. Lane,E. W "Designof SrableChannels."TransactitnsIASCEI 120,no. 2776 fl955a). pp. 1234-19. Lane.E. W. "The Importance of FluvialGeomorphology pro.. in HydraulicEngineering..' A S C E8 1 ,n o .7 9 5( 1 9 5 5 b )p.p . l - t 7 . Lane.E. W., andV. A . Koelzer.Densibof SedinentsDepositedin Resenoirs.ReponNo. 9 of a Studyof MerhodsUsedin Measurement and Analysisof SedimentLoadsin Streams. Sr.Paul,MN: U.S.Army Corpsof Engineers. 1953. Laursen,E. M. 'An Analysis of Relief Bridge Scour.""/. I/ld Div., ASCE g9, no. Hy3 ( 1 9 6 3 )p, p .9 3 - I 1 7 . Laursen,E. M. "Scourat BridgeCrossings."Iowa HighwayResearchBoardBulletin No. g. IowaCity: IowaInstituteof HydraulicResearch, Universityof Iowa,l95ga. Laursen,E. M. "Scourat BridgeCrossings;'J. Hyd. Div., ASCE 86, no. Hy2 0960t. pp.3954. Laursen,E. M. "The Total SedimentLoad of Srreams."J. H!.d.Div., ASCE 54, no. Hy I ( 1958b),pp. l-36. Laursen,E. M., and A. Toch.ScourAround Bridge piers and Abutments.lowaHighway Research Board,BulletinNo. 4. IowaCiry: IowaInsrirute of HydraulicResearch, Uni_ versityof [owa,1956. Leopold,L. B., andT. MaddockJr."The HydraulicCeometryof StreamChannelsandSome Physiographiclmplications."USCS Professionalpaper 252. Washingron,DC: U.S. GovemmeniPrintingOffice, 1953. I-eopold,L. B., and M. G. Wolman. River Channel pauems: Braitled,Mearulering,aruJ SrraiSit. USGSProfessional Paper282-B. Washington,DC: U.S. Govemmcntprinting Office,1957. Leopold,L. B., and M. G. Wolman."River Meanders."6eologicalSocietyof AmericaBulletin, 7 | (1960\. tropold, L. G., M. G. Wolman,andJ. P Miller. (1964).Fluviat prccessesin Geomorphology.SanFrancisco:W. H. Freeman,l964. Mackin, H. H. "Conceptof t}e GradedRiver:' Geologica!Societyof AmericaBulletin 59 ( 1948)pp.463-512. Mantz, P A. "lncipient Transportof Fine Grains and Flates by Fluids-Exrended Shields Diagram."I Hyd Diu, ASCE,103,no. Hy6 (1977),pp.601_15. McKnown, J. S., and J. Malaika. "Effect of pafticle Shapeon Senling Velocity at L,ow ReynofdsNumbers."Trunsactions Unionl 3l ( 1950),pp. 74_g2. lAmencanGeophysical Mefville, B. W. "Local Scour at Bridge Aburments."J_Hydr Engrg.,ASCE l18, no. 4 ( 1 9 9 2 )p, p .6 1 5 , 3 1 . Melville, B. W. "Pier and AbutmentScour: InregratedApproach." J. Hydr Engrg., ASCE 123,no.2 (1997),pp. 125-35. Melville, B. W., andA. J, Sutherland."Design Method for local Scourat Bridgepiers..'./. Hydx Engrg.,ASCEI14, no. l0 (1988),pp. 1210-26.
l hannels 11'l C H A p T H Rl 0 : F l o \ ri'n A l l u v i a C 'Forrnulas for Bed-l-oad Trtnspon."Paperat thc Second lvlelerPett'r.E., and R. Muller. for HldraulicResearch. Stockholm.19J8. Association !lccting of International FormLllirs.'JH\dr Engrg.,ASCE I16. Nakato.T. Tcstsof SelectedSedirrcnt-Transpon no.I 11990).pp. -.162-79. Neill,C. R. "Me:rnVelocitvCritcrionfor Scourt)f CoarseUniformBed-Material.'In Pro' ceetlittgsof the 7\tclftlt Congressof the htenwtional Assoc.for Hrdr ResearcltFon C o l l i n sC. O . 1 9 6 7 . LlndSedinenl Trausport,Rio Grarule Near Nordin, C. F., jr. AspectsoJ FLot Resistance \4hshington, DC: U.S. GeologiBcrntlilkt,Ner Meri.o. WaterSupplyPaper1.198-H. cal Survey,196.1. Nordin, C. F.. Jr., and J. P Bererage.SeditftN Tnittsportin the Rio GrandeN?* Me-\ico. Washinglon. DC: U.S.Geological Survey.1965. Prof.Paper'162-F. Pagin-Oniz,J. E. Stabilin*of RockRiprupfor Protectionat the Toeof Abutt ents Located Washington.DC: Federal at the Floodpluit. PublicationNo. FHWA-RD-g1-057. I991. of Transponation. HighwayAdninistration.U.S.Dcpartment 1986, Cliffs.NJ: Prentice-Hall, Petersen. M. S. RiterEtrgheering.Englewood "FunclionalTrendsof Scourat Bridg€Piers."J. Hydr Engrg..ASCE I12, Raudkivi.A. J. n o . I ( 1 9 8 6 )p, p . l - 1 3 . "Clear-walerScourat CylindricalPiers;'J. Hydr Engrg., Raudkivi.A. J.. andR. Ettema. ASCE 109.no. 3 ( 1983).pp. 339-50. Richardson, E. V., and S. R. Davis.EvaluatingScourat Btidges.Repon No. HEC-18. U.S. Depanntent of TransponaWashington, DC: FederalHighwayAdministration, tion. 1995. of PierandAbutmentScou.:InteRichardson. E. V. and J. R. Richajdson.Discussion gratedApproach.'by B.W Melville."J. Hvdr Engrg.,ASCE 12,1,no. 7 (1998),pp. 111,'72. Repon Richardson. E. V, D. B. Simons,andP Julien.f/igfixdlj in theRiverEnvironmenl. No. FHWA-HI-90-016.washington,DC: FederalHighway Adnlinistration,U.S. 1990. Depanment of Transponation, Transactions Rouse,H. "ModernConceptionsof the Mechanicsof Fluid Turbulence." IASCEI 102.no. 1965(1937),pp.463-543. Rouse,H. An Anahsis of SedinrentTra sportationin Light of Fluid Turbulence.SCS-T of Agriculture, DC: Soil Conservation Service,U.S. Department P-25.Washington, 1939. Rubey,W. W. "SettlingVelocitiesof Gravel,Sand,and Silt Panicles."AmericanJoumal of Scien.e,5th Series,25.no. 148(1933),pp. 325-38. Schlichting,H. Bowrdary-ktter Theorr'.New York: McGraw-Hill, 1968. In RiverMechanics, Schumm,S. A. "Fluvial Geomorphology:The HistoricalPerspective." Vol. I, ed.H. W. Shen.pp.,1.l-4.30.Fon Collins.CO: H. W. Shen,l9? l. J. Httl. Div., ASCE 95. no. HYI (1969),pp. Schumm,S. A. "River Metamorphosis." Schumm.S. A., M. P Mosley, and W E. Weaver.Btperi rc tal Fluvial Geotnorphology. NewYork:JohnWiley & Sons,1987. "Local ScourAround Bridge Piers."J llyd Shen,H. W., V R. Schneider.and S. Karaki. (1969), pp. 1919-40. Div.,ASCE95, no. HY6 "Temperature andMissouriRiverStages Shen,H. W., W J. Mellema.andA. S. Harrison. NearOmaha.""/.Htd. Dit.,ASCE lM, no. HY1 (1978),pp. l-20. Shields,A. Application of Sinilarin Principlesand TurbulenceResearchto Bed-Iaad Mot'enenL trans.W. P On and J. C. van Uchelen.HydrodynamicsLaboratoryPubl. No. l6?. Pasadena: USDA. Soil ConservationServiceCoope.ativel-aboratory,Cali1936. fomia Instituteof Technology,
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Simons,D. B.. and E. V Richardson. Resistance to Ikt* ittAllutial C/rarrnelr. Prof.paper ,122-J. wrshinglon.DC: U.S.Geological Survey,t966. Simons,D. 8.. and F. Senturk.S.d/melllTransportTechnologl.. Fon Collins,CO: Water Resources Publications. 1977. Sowers,C- F. Intro(lucton Soililechanics urulFountlations.GeotechnicalEngheerin1,4th e d .N e u Y o r l : M u c M i l l r n I, 9 7 0 . Sturm,T. W- 'AbutmentScourin Conrpound Channels.'ln Strca,nStabilir\* and Scourat HighxavBridges.ed. E. V Richardson pp. {-13,56.ASCE, 1999a. andP F. Lagasse, Sturm,T. W. ,l6nrarentScourStudiesfor CotnpoundChannels.Repon No. FHWA,RD-99156.Washington. DC: Federall{ighwayAdministrarion, U.S. Depanmenr of Transp o n a t i o nl.9 9 q b . Sturm,I WW.and A. Chrisochoides. One-Dinensional and Two DimensionalEslinates ofAbutnrentScour Prediction Varitfules, pp. l8-26. TransponationResearchRecord 16,17. Washington, DC: Transponation Research Board,NationalResearch Council. 1998. Sturm,T. W-. and N. S. Janjua."ClearWaterScourAroundAbutmentsin Floodplains." "/. Hydr Engrg.,ASCE 120,no.8 (1994),pp.956-12. Sturm.T. W. and Aftab Sadiq.Cleor-lVaterScourAround Bridge AbutttentsIJnderBackwdter Conditions, pp. 196-202.TransponationResearchRecord I523. Washington, DC: Transponation Research Board,NationalResearch Council.1996 Tanner, W. F "HelicalFlow,a Possible Causeof Meandering." "/.Geophys.Res.65(1960), pp.993-95. U.S. Army Corps of Engineers.Scourand Depositionin Ri'ers and Resenoirs(HEC-6). Davis,CA: HydrologicEngineering Center,U.S.Army Corpsof Engineers (1993). U.S. lnteragencyConrmitteeon Water Resources.So'r,eFundanentals of particle Siie Analysis,A Studyof MethodsUsedin Measurentent erndAnahsisof SedinentLoadsin Slreams.ReportNo. 12.Subconlmittee on Sedimentation, *aler Resources Council. Washington, DC: Govemmenr PrintingOffice,I957. U.S.WaterwaysExperimentSrarion.S/!d,yof Rit'erbedlllaterial and their lJsewith Special Referenceto the ltx.er MississippiRiye,i Paper 17.Vicksburg,MS: U.S. Waterways ExperimentStation,1935. Vanoni,V A. "Some Effectsof Suspended Sedimenron Flow Characteristics." In proceedings,Fifth Htdraulic Conference,pp. 137 58. Builetin 3-1,lowa City: University of lowa. 1953. Vanoni,V A. "Transponarionof Suspended Sedimentby Water" Ira,sacrions [ASCE] I I l, n o . 2 2 6 7( 1 9 4 6 )p, p . 6 7 1 3 3 . Vanoni,V A., and N. H. Brooks.laborutory Studiesof Roughnessand Suspended load of Alluvial Streams.SedimenrarionLaboratoryRepon No. E68. pasadena:California lnstituteof Technology, 1957. Vanoni,Vito A., ed. SedinentationEngineering.New York: ASCE TaskCommitteefor the Preparation of theManualon Sedimenrarion, ASCE, 1977. van Rijn, L. C. "EquivalentRoughnessof Alluvial Bed;' J. Hydr €agrg., ASCE 108,no. H Y l 0 ( 1 9 8 2 )p, p . 1 2 1 5 , 1 8 . van Rijn, L. C. "SedimentTransporr,Part I: Bed Load Transpon."J. H,-dr Engrg., ASCE I 1 0 ,n o . l 0 ( 1 9 8 4 a )p,p . l 4 3 l - 5 6 . van Rijn, L. C. "SedimenrTranspon,Part Il: SuspendedLoad Transpon."J. Hydr Engrg., A S C El l 0 , n o . 1 1( 1 9 8 4 b )p,p . 1 6 1 3 - 4 1 . van Rijn, L. C. "SedimentTransponIII: Bed Formsand Alluvial Roughness." I ll1dr Engrg.,ASCE 110,no. l2 (1984c),pp. 1733-54. White, F. M. VscousFluid Flow. New York: McGraw-Hill. 1974.
l hannels 149 C H A P T E Rl 0 : F l o wi n A l l u v i aC willianls, G. P. Hunre E\p( nents ctnthe Transportof a CoarseStnd Prof. Paper562-8 Survey.1967. DC: U.S.Geological \\'ashington, Prcss.1972 Yalin, M. S. Meclraaicsof SedintentTransporr' \ew York: PerSamon 'lnceptionof SedimentTranspoft"J Httl Dnr, ASCE 105. Yalin,M. S.,andE. Karrhan. n o .H Y l l ( 1 9 7 9 )p, p . 1 , 1 3 3 1 3 . Transpon.'J.Hyd Dir' ASCE99. no. HYl0 Yang.C. T. IncipientNlotionandSediment ( 1 9 7 3 )p, p . 1 6 7 91- 7 O 1 . Yang.C. T. Sedihert Trdnsportl'heory and Proctice.Ncw York: McGraw-llill. 1996. Transpon." J Hrd Dirr,ASCE98'no HYl0 andSedilnent Yang,C.T."Unit StreamPorver ( 1972),pp. 1805-26.
EXERCISES equation(10-10)for fall velocityof naturalsandsand 10.1. DerivetheEngelund-Hansen for Cp (= 2'1lRe+ l5). Then plot the gravelsfrom theirproposedrelationship equationfor fall velocity (cn/s) vs. panicle diameter0nm) for Engelund-Hansen sandto gravelsizerangesin waterat 20'C andcomparewith theplotof calculated fluid (on the samegraph) fall velc'cityof quanl spheresin the samesedimentation 10.2. DeriveRubey'sequationfrom his proposedrelationshipfor CD(: 24lRe + 2) Plot the resultsfor fall velocity for sand to gravel size rangesand comparewith the equationon the sameSraph resultsfrom the Engelund-Hansen 10.3, A river sandhasa sievediameterof 0.3 mm. Find the fall velocityat 20"C usingtwo plotequation (l) useFigures10.2and 10.3.(2) usetheEngelund-Hansen methods: ted in Exercise10.1. 10,4. A river sandhasa measuredfall velociq in waterof l0 cm/sat 20"C. Find the sedimentationdiameter. 10.5. PIot the resultsofthe sieveanalysisgiven belowon lognormalpaper.Find dro,, drr.", dro,d. ando r. Derivea formula for d65andd$ in termsof ds andd8usingthe di\tribulion: lognormal lheorelrcal Sievediameler,mm
0.,195 0.,117 0.351 0.295 0.246 0.208 0.1?5 0.147 0.124 0.104 0.088 0.0?4 Pan
Grums retained 0.36 L52 4.32 ?.40 8.80 '7.M 4.56 2.88 L52 0.80 0.44 0.20 0.16
450
C H A p T E Rl 0 : F l o wi n A l l u r i a l C h a n n e l s
10.6. Fora slopeof 0.002in a wide stream. calculate thedepthal whichsedjment morion beginsfor a finegravelwirh d< = 3.3mnt. 10.7. Bank-fulldeprhfor a srreamin $ell-rounded alluviufrwirh dsn= t2 mm is 1.5m. Thestreamslopeis 0.0005.The thannelrhapeir approxrmarelirrapezoidal wirh 2: I sideslopesanda botom widrhof l0 m. Calculate ihe bank_fuli discnarge anddererminewhetherthebedandbarl'r aresrablear bank-fullcondirions. 10.8. A roadside drainage ditchhaslo carrya discharge of 100cfs.It hasa bortomwidrh of 5 fr andsideslopesof 2: t.-Tle dirchi5 ro be lrned* ith locallyavailable grarel, *hich hasdro: 25 n1m.At \\ hal marimum:lopecanrhechannel beconstructed for srabiliryof rhebedandbanks:r 10.9. Derermine thecriticalvelocil\ of a uniformflow of water(20"C)wrrha deprh of 3 tt overa bedof very coarsesandwith a mediansizeof 2.0 nm using Keulegan,s equarion.Repeatfor a laboraron,unifonn flow deprhof 0.3 ft. Ler k, : 2.5d5; 10.10.Calculate rhecriticalvelociriesin Exercise10.9usingManning.s equahonwith the Stricklerequationfor Manning'srr.Forcoarse.edrm-enn andiully roughturbulent flow' de.ivea generalrelationship for thecriticarsedim.ntnunrbe.us a functionof d50lf0,whereJ0 is flow depth.and the Shields'parameterusingManning,s equarion wirh Strickler'sequarionfor r_ The critical sedimenrnumberis defined as
^'-=--- I--
V(SC I )sd,o = in whichV" c.ilicalveloci! ard SG = specificgraviryof thesediment. l0.ll.
The RepubJican Riveris flou.rnoar a deprhof 3.0 ft uirh a relociryof 7 ftls for a slopeof0.00l?. The sedimenrsizesrre 4o = 0ll mm, do, = 0.39mm, andd, : 0.59 mm. Derenninethe bed form type uiing the Sinons-lichardson diagramand the van Rijn djagram.
10,12, The MissouriRiver at Omahaflows at a deprhof l0.l fr wirh a velocity of 4.4g h.ls and a slopeof 0.000155in summer(f = 7l"F). At approximately the samedis_ chargeof 32,000cfs, it flows ar a depthof 9.1 ft with a velocityoi 5.49 ftls und a slopeof0.00016in winter1f = 4l.F). The measured bedsediment sizesared5o= 0.199mm andd*, : 0.286mm in summer,while in u,inrer,dro = 0.224mm and d$ -- 0.306mm. predicrthe bed form rype for borhca... usini Figure 10.14and explainyour results.(Use the \'an Rijn methodto estimareal.) 10.13. A ? mi reachof lhe Chattahoochee Riverdownst.eamof the Buford dam ts subject to a maxrmumdischargeof 8.000 cfs due to hydropowerreleasesand a minimum dischargeof 550 cfs. The averagewidrh of rhe .ive; is l g0 fi, and it can be consideredvery wide. The bed sedimentis coarsesandwith dro = 1.0mm and d6J= L2 mm. The averagestopeis 0.00031ft/fr. (a) Calculatethe velociryand deprh at low andhigh flow usingthe KarimKennedymerhodand the Engelundmerhod. (b) Predictthe dominanrbed form ar low and high flow using the Simons_ Richardsondiagramand rhe r.anfujn diag."-. Whut u.. ih. b"d lorm dimen_ sionsat low and hieh flo*.?
C H A p T s Rl 0 : F I o wi n A l l u v i a C l hannels 451 10,14,Calculate the velocityandflow deprhfor the Republican Riverusingrhe measured per unit widlh of2l ftr/s as givenin Exercisel0.ll. Use rhe van Rijn discharge melhodand the Karim,Kennedymethod. 10.15.The Rio Crandefuver at Bemalillo,New Mexicohasthefollowinscharacreristics: Deplh range: Slope: Temperaturet Bedsediment:
_ro= 0.6 5.0 fl S = 0.(X1086 fr,ift a: 60'F dr5 = 0.24mm /so = 0.29 mm d u ,: 0 . 3 5 m m deo = 0.53 mm
Use a spreadsheet to preparea depth-velocitycuwe using three methods:( l) Engelund, (2) van Rijn, and (3) Karim-Kennedy.Plor the resulrson log-log scalesalong with the measured data from the table given below. Compare the results and discuss. Rio Grande River near Bernalillo, New Mexico, Sec. A2
Velocity,rus r1.06 5.96 5.09 3.11 2.84 2.65 1.66 3.58 3 . 1I 3.t2 2.34 2.23
r.98 I.9I 2.05 1.94 2.31 2.51 2.90 1.44 t.?0 1.40 2.01 t.97 1.85 1.76 1.89 t.76
Depth, ft
2.46 3.63 3.79 3.49 2.'t6 2.69 2.15 |.25 2.66 2.56 2.18 2.14 t.44 t.l8 t.23 t.29 |.32 t.42 1.50 1.29 1.12 1.48 0.19 t.o'l L70 1.03 t.3l l .t 0 l 70
Velocity,lus
2.09 |.62 t.l'l 1.30 3.'73 5.00 5.l7 .1.31 5.86 5.56 6.91 6.88 '7.82 't.7| 6.92 6.27 6.10 5.06 6.50 3.04 2.71 2.89 3.16 3.99 6.01 2.88 2.00
Deplh, ft
1.94 0.93 0.52 0.5E 3.06 3.25 3.01 3.68 4.46 4.1I 4.80 4.34 3.43 2.61 2.96 3.4{) 1.84 2.56 3.44 2.61 3.t2 2.33 3.22 I.54 l.m
solEdr Nordin andBcverasc(1965).
10.16. Calculatethe bed load dischargeper unit width in a wide gravel-bedstreamwith a slopeof 0.005and a flow depthof 1.0m. The medianbed sedimentsize is 20 mm.
452
l hannels C H A P T E Rl 0 : F l o wi n A l l u v i a C formula.Givethe andMiillerfonnulaandthe Einstein-Bto\\'n Uselhe Meyer-Peter (I metriclon : L00Okg). resultsin mr/sandin metrictons/day.
perunitwidthin a canalconstructed in coarsesand thebedloaddischarge 10.17. Calculate (dro: 1.0mm) at a slopeof 0.001.whichis flowingat a depthof 1.0m anda velocformula. Usethe vanRijn bedloadfonnulaand the Einstein-Brown ity of 1.5rn-/s. Discusstheresults. sandconcen10.1E.The followingtableSivesthe velocitydistribulionand the suspended tration distributionfor the size fraction betweenthe 0.07.1mrn and the 0.1O4mm sievesizesior venicalC-3 on the MissouriRiverat Omahaon \ovember'1,1952. On this day,theslopeof thestreamwas0.00012,thedepthwas7.8ft, thewidthwas 800 ft, the watertenperaturewas.15"F,and the flow u as approximatelyuniform. profileon log-log (d) Plot the velocityon semi-logscalesandtheconcentration analysisto obtainlhe best-fitstraightlines. scales.Do a regression (r) Computefrom thedataandthegraphsthe follou'ingquanlities: l,l. : Y= K: / = : \ g, :
shearvelocity, meanvelocity, von Karmanconstant, friction factor, Darcy-Weisbach Rousenumber. per unit width in lbVft/s. discharge suspended sediment
z. distanceabovebottom (ft)
a, velocity (ftls)
0.1 0.9 t.2 1.4 1.1 2.2 2.7 2.9
4.10 4.50 4.64 1.17 4.E3 5.t2 5.30 5.,10
3.4
5.42 5.50 5.60 5.60 5.70 5.95
C, conc€ntration (mg/L)
4ll 380 305 2E) 2'71 238 2t1 ls6
4.8 5.8 6.8 7.8
; l,t8 ll0
10.19. The MississippiRiver at ArkansasCity is very wide and has a bed sedimentwith dro : 0.30 mm anddr = 0.60 rnm.For a flow depthof 45 ft, thewatersurfaceslope is measurcdto be 0.0001,and the watertemperatureis 80"F. The meanvelocityis 3.0 fus, and the sedimentconcentrationat a locationof 5 ft abovethe channelbotfom is 1,000mg/L. (a) What is the dominantmodeof sedimenttranspon? (D) Calculatethe middepthsuspended sedimentconcentration. (c) If the watertemperature were40oFwhile all other factorsremainedthe same, would the suspended sedimentdischargechange?Explain.
C t i A p T E Rl 0 : F l o wi n A l l u v i aC l hannels 453 10.20. A reachof Peachtree Creek hasa slope of 0.0005.and a sandbed wiih d\o = 0.5 *ater discharge per unit of mm, du, : 0.6 mm, anddro: 1.0mm. The measured wrclthd = Il.) lf/s-
(a) Estinratc V/hatis the thedepthandvelocityusingthe Engelund merhod. equivalent valueof Manning'sn and how doesit compare with the Strickler value?Explain. (b) Determinethe bcd form type. and its heightand wavelengthfrom van Rijn's in pan (a). method.Usethevalueof 7l determined (c) What is the modeof sedimenttransport;that is, is it primarilybed load, mixed load,or suspended load? (d) Calculatethesuspended in ppmat a distanceof I ft sedimentconcentration abovethe bed usinga referenceconcentration,C,, and the methodofogyof vanRijn. (?) Calculate thetotalloadfrom Karim'smethod.Yang'snrethod, Engelund's method,andKarim-Kennedy s methodin flr/r andppmby weight. (, Calculatethe b€d load lransportrale in ft:/s usingthe Meyer-Peterand Miiller formula.first usingr, and then rl. Which result is the correctone and why? 10.21,Calculatethe total sedimentdischargein the NiobraraRiverusingthe Engelundgi\'enin Example10.4. Hansenformulafor theconditions 10,22. For the givenwaterdischarge in the NiobraraRiverin Example10.4,calculate the depth and velocity using the Engelund method and the Karim-Kennedymethod. Thendctcrminethe sensitivityof the calculatedsedimentdischargeto the depthand velocityusingoneof thesediment discharge formulasin theexample. 10.23.The NiobraraRiverin Nebraska hasthe followinesediment sizes: Bedsedinlent:
dro = 0.27mm d65: 0.34mm r/*=048mm
For eachof the measurcddatapoints in the table given below,apply the van Rijn method.Yangformula,Karim-Kennedyformula,andKaim formulato calculatethe total sedimentdischargeper unit of width g, in lbVsec/ftfor comparisonwith meaper unit width (convenfrom C, in ppm). Use a spreadsuredsedimentdischarges sh€etor write a computerprogram.For each methodplot a graphof measuredvs. calculated sediment discharges anddiscusstheresults. Niobrara River Depth, ft | .11 1.53 l.{,1 L6{) |.62 t.89 1.5? t.4l l.5l
lelocily, fts
3.20 2.26 3.57 l . 6r 2.15 l.t3 2.41
Slope
0.001,139 0.0013.{-1 0.001250 0.00170r 0.00170.1 0.00r?99 0.001269 0.001288 0.001t8E
Conc., ppm
t6 r0 970 I 1.10 1890 t'770 1780 780 '7 910
Viscosity,ftlls
t.6lE-5 0.99E-5 t.0EE-5 r.64E-5 t.JJE-)
1.34E,5 t.t3E-5 r.00E-5 1.03E-5 lcontinued)
'15,1 CHApTER l0: Flow in AlluvialChannels
Niobrara River (coltiraed) Depth, ft 1..1.1
r.5l t.56 L7l L38 L67 l.70 1.56 L6l 1.,{I r.l8 t.5.1 t..12 L36 L62
velocity, f/s
2.52 _r.21 3.07 3.70 I 3.r 2.95 3..{2 2.t5 2..{5 - J )
2.l9 2.05 2.20 3.33 3 . lt 3.61
Slope
0 . 0 0r17 { 0.001420 0.0011101 0.001685 0.001553 0.001250 0.001,177 0.001250 0.00r287 0.001 136 0.00r2s0 0 . 0 0 1t 2 0.001 136 0.001609 0.00 t610 0.00t667
Conc..ppm 1000 1780 1.190 1900 l7l0 893 1820 154 91.1 503 392 .129 ?36 1520 1660 2200
I'iscosity,ftrA l.l6E-5 l.2tE-5 t.l9E,5 r . 2 5 E5 t.82E-5 L768,5 l.76E-5 L08E,5 0.95E5 t.05E-5 0.888-5 r.0tE-5 L l9E-5 L79E-5 1.728,5 r.55E-5
S o l r . e rC o l b ya n dH e n b r e e( 1 9 5 5r : K a J i ma n dK e n n € d (y1 9 8t ) .
10.24.Discusstheexpected channeladjustmenrs in a sand-bed streamtharis in equilibrium asa resultof thefollowingchanges: (a) Sand-dredging in a localizedreachupsrream of a b.idge, (b) Rapidresidential development adjacent to the streamin an urbanarea, (c) Excavation of a wide. channelin a bridgecrosssectionto reduceoacx\\.arer, (d) Cutoff of a meanderbend. (e) Channelinproventent by removalofbrushon thc banksanddeadfallsin the channel. 10.25. Developthecomplete derirationof theequalion for live-bedcontraction scourgiven by EquationI0.86. 10.26. Considerthe caseof a ven long contractionwith overbankflow in the approach channelanda constant mainchannelwidth.Thebridgeabutmenrexlendto theedge of the mainchannel. Assumethatthereis no changein thechannelroughness from the approachthroughthe contraction and thatthe sedimentis coarsewithoutbed formswith bed-load transpononly in themainchannelandno sediment transponin thefloodplain.For the lile-bedcase,derivean expression for thecontraction scour usingrheMeyer-Peter and Viiller bed-loadformula. 10.27,Calculate the live-bedcontraction scourdepthsfor banklineabulmenls in o\erbank flow asa functionof thepercentof totalflow in themainchannel. whichvariesfrom 50 ro 90 percent.Assumea constantmain-channel width and an apDroach flow depthof I0 ft in the mainchannel.Plol rheresults. 10.28.Estimate the maximumscourdeptharoundcylindricalpiersin a riverbedif rhebed sediment hasa mediansizeof 1.0mm. andthebridgepiersare 1.2m in diameter. The approach flowjust upsrream of thepiershasa depthof 3.0 m anda velocilyof ( l) CSU,(2) Laursen L0 nts. UserhefollowinSformulas: andTcrh,(3)Jajn.(J) Mel-
C H A p T E Rl 0 : F I o wi n A l l u v i a lC h a n n e l s 4 5 5 ville and Surherland, and(5) Froehlich. Whatvalueof scourdeprh\{.ouldyou reporr to thegeolechnical engineer designing thepier foundationsJ 10.29. Find the abu,mentscour depth and pier scour depth for Example10.5 using Melville'sformulaandFroehlich's forntula. 10.30.The following field daraare giver for pier dianrerer, b: approachvelocity,V,: approach depth,-r',;andmediansedimenl diameter, d5o.Calculatethemaximurn pier scourdeprhfor a cylindricalpierhavingno skewusingrheequalions of Froehlich; MelvilleandSurherland; Jain,LaursenandToch;andCSU.Conrpare with thenreasuredscourdepths,4. River
,, ft
Jr',,ft
Vl, ftls
Red Oahu Knit
26.9 4.9 5.0
I,l.I
2.0 7.8 1.6
t0.2 1.9
d56,mm
0.060 20.0 0.5
Nleasured 4, ft I.t.l t.0
10.31. Write a computerprogramwith inte.activeinput that compuresthe depthandveloc_ ity in an alluvialriverfor a givendischarge per unit width. Implementall methods givenin rhischapter.
A P P E N D I XA
NumericalMethods
A.l INTRODUCTION Nume.icaltechniques areme&odsandalgorithmsfor obtainingapproximatesolu_ tions of algebraicand differentialequarionsthat can be programmedin computer lan!'uage. Suchalgorirhms musrbr efllcienrand accurare. fhe efliciencyof an algorirhmreferlro it havinga55matta numberof logicatstepsande\ecuri;nrime aspossible. Efficienc,atcocanenrarlminimi/ingcomputer memoryrequiremrnr\. The efficiencyof the programlhar aclualizesrhe algorithmalso is importantand program.dcveloped ha\ resuledin ,t.rcruredprogramming conceprs. rn *rucrured modLle,.whi.h dvoidindiscriminatr bjrnching.are ea.ierfor rhe u\er lo re3d. understand, andmodify,if necessaryTbe accuracyof a numericalalgorithmis essential.Roundoffenor or trunca tron enor canbecomeso largeas to "swamp"the numericalsolutionand male it numericallyunstableError analysiscan providesomehelp in idenrifyinginstability problems,but improvingaccuracysometimesis a matterof experiencJ obtained from numericalexperiments with a panicularalgorithm. Numericalmelhodsthatareirerativein naturemustsarisfysomeexpectation of con!ergenceto$ad lhe rue solurion.Somerime5 a llade-offmuy be madebetween efficiencyandaccuracy. The mosrefficientalgorithmmay div€rgein somekindsof problems.Thereis no hopeof accuracyin fiis situation,which is describedmore aptly in termsof lhe reliabilityof the algorithm. This appendixprovides only a few numerical techniquesthat are used throughoutlbs text.For a morecompletediscussionofnumericaltechniques. refer to the list of referencesfollowing this appendix.The numericaltechniquesDre.entedhereare givena, procedures {'ubprogramsrfor usern standardnoduler of Visual BASIC. The procedures are easilytranslaledto orherlanguages.
457
458
APPENDIxA: Nume.icalMe$Gls
4.2 NONLINEARALGEBRAICEQUATIONS Nonlinearalgebraicequationsare encounrered ofren in openchannelhydraulics. Forexample,the problemsofcritical andnormaldepthdetermination rcquireeitber a Lrial-and-enor or graphicalsolulion wirhoura numerical,olver. Nonlinearatseb|aic equalionscanbe placedin rhe forrn FLtl = O. rheroor of which is rhe sojution of the equation.Considerthe following algebraicequationin the unknownt,, for example,in which a and . ar€ f,osiriveconsrants: (A.I) This equationcanbe rearranged in fie form
r ( ) )= - r r + c : 0 )
(,{.2)
which can be solvedgraphicallyby finding the point ar which the function F(].) crossestbe) axisor. in otherwords,by findingthe zeroof thefunclion.This equalion actuallyhast\ro roo(s.which can complicarelhe rool search.ll represents $e equationfor ahematedeptbsassociated with a known specificenergyin a rectan, gular channel.In this case,knowledge of de critical depth, which separatesthe two rools.aidsthe solutionprocess.cene.aily, suchknowledgeof rhephysicalbasisof an equation,or at leastof its graph.canbe ol greatassistance in choosingan appropriatealgorithm. Three methodsfor solving such equationsin rhe form F(1,) - 0, in which I, is the root, are given in this section.There are severaladditionalmethods,but rhe methodschosenillusrrateth€ trade,offberweenefficiencvandreliabilitv_
Interval Halving Method The first methodis not very sophisricaredbut very reliable. Calledthe intenal hal'ring ot bisectionmethod,it easily is illustratedby the high-lownumberguessing game.For example,a professorrhinls ofa numberberweenI and l0O,asksa student1oguessthe number,and then providesfeedbackto the studenras to wherher the guesswas high or low. This informationprovided!o the srudentbrackersthe range within which the unknown numb€r lies. The srraighlA student then halves the interval after receiving the "highlow" feedbackfor each guessand converges rapidlytowardthe final numberin thjs way. The generalstrategyof inrervalhalvingthenis to dividean interval,in which a root of the equarionis known to occ4 into equalpans.which halfof the inrerval tberoot actuallyis in canbe dererminedby comparingthe signofthe funcrion, 4 at the midpointwith the sign at the left boundaryof the interval.The intervalis
A P P E N D TAX: N u m d c a l M e l h o d s 4 5 9
Flv)
Ya Y3 (a)IntervalHalvingMelhod
(b)SecanrMerhod
fly)
(c) NeMon-Baphson Method FIGUREA.I Graphical depiction of merhods for sohrtion ofnonlinear a-lgebraic equarions.
halvedagain,andthis processis continueduntil rheroot is app.oached as shownin FigureA.la. The generalprocedureis summarized as follows: l. Form a function. F, from the equation to be solved such that the root of FCy)= 0 is the solution. 2. Find rhe value of the function at the left boundary of the inierval from yr to )2 and storeit in FYl. 3. Halvethe intervaltretween)r and)2 by compuringy - Ot + hy2.
460
A: NumericalMebods APPENDIX
4- Evaluatelhe functionF(l) and storeit in FY. 5. Comparethe signsof FYI andFY by multiplving.[f fiey bavethe samesign,] lf theyareopposilein sign ) is $e newrightboundary' is the new left bor.tndary. some e.ror critenonrs met until 6. Repqt steps3-5 The absoluieenor ERABSin the inlervalhalvingmedrodis r€ducedbyhalfin each iterationso that it is givenbY -- -^ = Yr-)l tRAtts --
(A.3)
in which n = numberof iterationsand (y2 )r) = the originalintervalsize The error can b€ controlledby choosingthe numberof itemlions.but an approxmate The dif_ rclativeerrorcriterionmay be moreusefulfor stopPing$e computationsls recent estlmaie ferencebetw€en$e currenteslimateof lhe root and the mosl value falls When this r€lativeerrot dividedby the latrcrtoesiablishan approximale are stopp€d. below a specil'iedvalue,the comPutations The advantageof the interval halving methodis t}lat ii always bracketsthe root and (provideda singleroot existsin the originalinterval)-This avoidsdivergence methgood as the other two makesit very reliable.Itsefficiency,however,is not as it usuatlyrequiresmore iterat;onsto achievethe same odspresentedhere,because in Figu.eA.2a in VisualBASIC code Th; intervalhalvingmethodis presented asa procedurelhatcanbe placed1na standardcodemodule.This procedurecanbe invoied from a form module,which alsohandlesthe choresofdata input,printing output,and graphingthe function,if desired.The Procedurerequiresinput values forihe endsof $e originalintewalYt andY2. An addilionalprocedurethat eval_ uaresrh€ functionF()) mustalsobc supplied.Note that only onefunctionalevalu' ation is requiredin eachiteraiionandthat its sign is comparedwith lh€ sign ofthe functionalvalueat thelefl boundarythroughthecomputalionof FZ. Firsi the value of FZ is checkedto seeif it is zero.A valueof zero for FZ obviouslymeansthat "exactly,"which may seemunlikely.but this canhappenif the root hasbeenfound the error criterionis so restrictivethat the significantfigure limit of the computet thenthe func_ If FZ is negative, computations. hasbeenreachedin two successive endpoint righl hand tion has crossedthe y axis andthe midpointbecomeslhe new midpoint that the (Y2 = Y3). A positivevalueof FZ, on the otherhand' signifies should becomethe new left hand endpoint(Yl : Y3). The algorithm works with t Somecaremu\l be or dectealing equall)well if thefunclioni1 increa\ing exerciied.however,in definingthe function,so tha! it do€snol havea singulanly at either endpoint. be writtenin casetheerrorcrilerionis not metin thespecAn errormessagecan is to increase ified numberofiterations(I : 50). In this event,all thal is necessary can number ofiterations th€ maximomnumberofilerations.In fact. lhe ma\imum problem ofinterest be madetargeenoughthatthis will neverhappenfor a specilic so that exit from th€ FOR-NEXT loop alwaysoccurs ihmugh the relativeenor check qith the specifiedvalueof ER. If no root existson the chosenintewal' the towardthe right handendpointand may evensatisfythe erlor algorirhmconverges
APPENDIX A: Numerical Methods
461
ION METHOD***** Sub BISECTION {Y1. Y2, ER, Y]) FY2 As singl-e, Din FY1 As single, Din I As Integer FV']
=
F{V]
FY3 As single,
)
FY2 = F{Y2) rf FY1 i FY2 > 0 Then Exir sub F o r I = 1 t o 5 0 y3 = {y1 +y2) /2 FY3 : F (Y3) FZ = PY1 * FY3 If Fz = 0 Then Exit Sub If FZ < 0 Then Y2 = Y3 Else Y1 = Y3 If Abs (lv2 - vI) / Y3) < ER Then Exit Next I End Sub (b)
air**sEcA].IT
Fz As Single
sub
!.tETHoD****i
Sub SECANT (Y1, Y2, ER, Y3) FY3 As Single Dim FYL As Sinq1e, FV2 As Single, Dim I As Integer FY1 = P (Y1) FY2 = F(Y2) l o r 1 = 1 c o i ( , Y3=Y2+F,Y2* l \ 2 - Y 1 ) , / ( F Y 1- F Y 2 ) FY3 = F(Y3) < ER Then Exit. Sub If Abs (lY3 - v2)/v3) FY1 = PY2 Y 2 = Y 3 FY2 = FY3 NexE I End sub (c)
* *a a *NEWroN-RAPHSON
Sub NET^II (Y1, ER, Y2) g'Y2 As Sing1e, Ditn FY1 As Sinqle, FPR1 As Single, Ditn f As Ineeger FY1 = F (Y1) FPR1 = FPR (YL ) F o r I = 1 t o 5 0 Y2 = Y1 -FY1/FPR1 FY2 = F(Y2) FPR2 = FPR (Y2 ) < ER Then ExiE Sub If Abs (lY2 - Yll/l'2l FY1 = FY2 FPR1 = FPR2 Next
I
FIGURE 4.2 VisualBASICprocedures for nonlinearalgebr.icequationsolve$.
FPR2 As Single
A: Numerical Methods 462 APPENDTx criterion.This eventualityis avoidedby checkingprior to enteringrheloop to determineif a root existson the intenal and sendingan enor message if it doesnot.
SecantMethod The secondalgorithmto be discussedis calledthe secantmethod.As shownio FigureA.lb, two startingvaluesof the root,y, andyr, areguessed. Thena straightline is drawnthroughthepointsLyr,fg,,)l and[f2, F(rz)]. Its intersectionwith the y axis is the nextguessfor the root,)'1.Algebraically,y, is evaluatedfrom the equationof the straishtline: !3
-
F(v)(v,- vt) )2
r'(vr)- r(v,)
(A.4)
Then,y, andy, becomethe next two guesses for the root.This is continueduntil an errorcriterionis met. The Visual BASIC codefor the secantmethodis givenas a procedurein Figure A.zb. It is similar in severalrespectsto the intervalhalving program,but note thatthe two initial estimatesfor the root do not haveto bracketthe true valueof the root. The evaluationof the next estimateof the root usesEquationA.4, and only one functionalevaluationoccursfor eachiteration.Both the roots and theL fitnctional valuesare updatedafter the enor checkto preparefor the next iteration. The secantmethod convergesmuch more rapidly than the interval halving method.Its disadvantage is that it may divergefor sgme functionsbecauseit is open-ended in the sensethatthe root doesnot haveto be bracketed. An exampleof the divergenceof the secantmethodis shownin FigureA.3a.
(a) SecantMethod FIGURE A3 Divergence of numericalmethods.
(b) Newton-Raphson Method
APPENDTx A: Numerical Methods 463 Newton-RaphsonMethod The final methodto be consideredis the Newton-Raphsonmethod,which really is a refinement of the secantmethodasshownin FigureA.lc. In this method,the tangentto the curve given by F(v) is extendedto the :},axis to give the next guess for the root. This requiresthat the derivativeof F(y) be determined.From theTaylor seriesexpansionfor F(y), we can seethat, if higher-ordertems are neglected, we can expressthe value of the function at y2 in termsof the value and its derivative at v,: F(y,) -F(.y,) +
dF( v, )
-yr)
(4.5)
d;(),
in which termsbeyond the first derivativeterm havebeendropped.Now, we are seekingthe intersectionwith they axisat which F(v2): 0, so,for the nextestimate of the root, we have I::)r-
1t(v' ) ^ r Url
(A.6)
in which f'(v,) is the first derivativeevaluatedat )1. The Newton-Raphsontechnique is very powerful becauseof its fast convergence and capacityfor extension to multiple-variableproblems,but it requiresa functionfor which the derivativecan be evaluated.The procedurefor the Newton-Raphson methodis shown in Figure A.2c. It is nearlyidenticalto that for the secantmethodexceptfor the estimateof the next root by EquationA.6. Note that the SUB procedurerefers not only to a functionalevaluationF(Y) but alsoto an evaluationof the derivativeof the function FPR(Y). The main disadvantage of the Newton-Raphson methodis that it can be divergentundersomecircumstances as illustratedin FigureA.3b.
A.3 FINITE DIFFERENCEAPPROXIMATIONS In this textbook,the finite differencemethodis usedto solvethe equationof gradually varied flow, which is a first-orderordinary differential equation,and the Sainlvenant equations,which form a pair of nonlinearhyperbolicpanial differential equations.In both cases,the derivativesare approximatedover a small interval by finite differences.If we seekan approximationofthe derivatived1y'd-r, for example, we beginby writing theTaylor'sseriesexpansionfor the valueofy at grid point i + I in termsof the valueat point l, as shownin FigureA.4. The result is t-
!i+t = li + f'(x,)(x;,1- x,) + f"(x,) =}1z
-
-\2
*
(A.7)
in which y = /(x) and the primes refer to derivatives.All terms including and beyond the secondderivative term can be referred to as higher order terms (HOT)
464
APPENDIXA: NumedcalMerhods
v Tangent
y=n4
fi+t Yi
Forwardditference Centralditference Backwardditference
Yi-1
xi_1
Xi
Xi+t
FIGTJRE4.4 Finite differenceapproximations of theFirst Derivativeat -r,.
that are O(ri+ I - :i)2, wNch means that they are "on the order of' A-r2,where A-r : (.r,*, - .r,). If we neglectall HOTs beyond the first derivativeterm, there will be a truncation error that is O(A.x)z,so that halving the interval size A; quartersthe truncationenor. Writing the truncationerror in this way and solving for the first derivative results in
vj:-]--L + o(A'x) f'(x) =
(A.8)
If we drop the truncation error to approximatethe derivative by l l r , l 4 -
!i+l
)i
,i+ I
^i
(A.e)
then this is refened to as a frst-order approrimation of the derivative, since the truncation error is proportional to Ar to the first flower. This approximationof the derivative also is called aforward diference, becauseit utilizes data at points j and i + 1 in Figwe A.4 to estimatethe derivative. In the samemannerasfor the forwardTaylor's seriesexpansion,we can write a backuard expansionfory,_, in termsofy, as li-r:
!i
-
f ' ( r , ) ( r , x , - 1 )+ f ' ( x , ) G '
-l'-t)'
- ...
(A.10)
AppENDrx A: Numerical Methods 465 Again, truncatingthe termsbeyondthe first derivativeterm andsolvingfor the first derivative,we have ti r {.r,t - -
ti-l
^i
^i-l
(4.r1)
This is called the backward difference, andit too is first order. Now ifthe backwardTaylor'sseries(EquationA.l0) is subtractedfrom the for_ ward series(EquationA.7) and we solve for the first derivative,the resultis
f'@ :
Y'"'r
+ o(ax)'
:'-'
(4.r2)
in which A-r : (.ri+r - .rJ = (J, - _ri_l).The secondderivativetermscancel,and the third-ordertermswhen divided by Ar result in higher-ordertermsthat are of order (Ar)2. Then, when the higher-orderterms a.redropped,the second-order approximationof the first derivativebecomes ti+t r t,xt - _--
tt-, 't
-
(A.13)
Second-order approximationssometimesare usedto obtaina moreaccurarerepre_ sentationof the first derivative.EquationA.l3 is a centraldiferencerepresentation of the first derivative. Forward, backward, and central difference approximationsof the first deriva_ tive are showngraphicallyin Figure A.4. It is obviousthat the centraldifference representation givesa slopeor derivativecloserto the true value. Although illustratedfor ordinary derivatives,the samederivativeapproxima_ tions can be madefor the panial derivativesin the Saint-Venantequations.A firstorderapproximationof the time derivative,for examDle,becomes
ay- y!*' - y! at Lt
(A.14)
in whichAt is thetime interval,andthe superscripts referto valuesevaluated at timesof (k + l)Ar andlA, arthespatialgridpointlocatedat-r,.
REFERENCES Abbott, M. B., and D. R. Basco. Computational Ftuid.Dynamics. UK: Longman Scientific & Technical,and New York: JohnWiley & Sons,1989. Ames,W. F. Numerical Methodsfor panial Dffirential Equation.r.New york: Bames and Noble, 1969. Beckett,R., ard J. Hurt. Numerical Calculationsand Atgorithms.New york: MccrawHi , 1967. Chapra, S. C., and R. P, Canale. Numerical Methodsfor Engineers. New york: McGraw_ Hill. 1988.
A P P E N DI X B
Examplesof ComputerPrograms in VisualBASIC
B.l YOYCPROGRAMFOR CALCULATIONOF NORMALAND CRITICAL DEPTHIN A TRAPEZOIDALCHANNEL The Visual BASIC/azn is illustratedin FigureB.l. Tlteform code extractsvalues enteredin the text boxes and assignsthem to variablenameswhen the CALCU_ LATE button is clicked. The channel parametersare e = discharye in cfs; S = channelslopein ftlft; b : channelbottomwidth in ft; m = channelsidesloneratio asrun-to-rise;and n = Manning'sroughnesscoefticient.The form code thencalls a subprocedure Y0YC in the modulecode.The modulecalculatesnormal and critical depthin a trapezoidalchannelusingthe bisectionmethod.The resultsare then placedin the outputtext boxes.In the modulecode,the functionF evaluateseither the critical depth function or the normal depth function depending on whether NFUNC has the value I or 2, respectively.The BISECTION subprocedureis iden_ tical to thatshownin FigureA.2a,exceptthat additionalvariablejhaveto be passed to BISECTION in the parameterlist. The program input includes rectangular and triangularchannelshapesby specifyingm = 0 or b = 0, respectively. Y0YC Form Code q)tlon lbq)uclt Dt! Q ,u thgl., a Ir glDgl., b ra Sl.!gla, B lr DtD Y0 lr Stagl., r'c lr ghgL. gub (batcrtcrht._c1l.ct Prl$t. Q. Va1(txtQ.Iut) s r val(txta.Iar.t) b r vr1 (txtB.l!.x! ) ! r ve1 ( txt!,'Itr.t )
()
8hg1..
! t
gln i.
468
AppENDrx B: Examples ofComputerPrograms in MsualBASIC
SALCULATE
FIGURE8.1 VisualBASIC form for determiningnormalandcriticaldeptbin a hapezoidalchannelusing theprogramYoYC. r r va1 ( trtaa.I.:.t ) c.ll YoYC(Q,a, b, !, n, IO, tc) txtAo.r.6rt' . loEDat (t0, .fif.000.) r lollrt (IC, .ffl.000,) txtrc.I.xE ed eub gub .dtExlt_Cltcto Prlvat. laa lnal Sub
YOYCModule Code ODttoB lirl)ll.clt Dr.D 0 Ir Dta ! t
5L!91.,
g r.
stEgla,
f0
t!
gl,rg1€. Etlgla,
b a! SbgI., r As glDgl. yC l5 SlDgl.
APPENDIXB: Examplesof ComPuterProgramsin Visual BASIC sllb YOIC(Q,I, b, !, n, Y0, aC) Y2 l! ahglc, lR It Dlr Y1 l! shgl., tr Inl.g.r Dh f,! Yl . 0.0001
469
gi!91'
Y?.100 8R.0.0001 r l r 1 12, NF, lR, 0. 8, b, r, call BrsEctror(ll, yl . 0,0001 Y2.100 l|1"2 C.U. BIAEC"IIOI(II' 12, t{1, lR, Q' 8, b' r, E[it &rb
D' rc)
a, I0)
aob Brgre!ro!l(41, Y2, f,Fsxc, !r, q' 8, b' r, !, 13) glttgl' Dts r!1 As 8lngl., try2 ts atigle, !Y3 tt siaglc' lz As Db r Ir r!t.g.r tll . !(Y1, trrum, 0, g, b, D, E) Fr2 ' !(Y2, ltllm, Q, 8, b, !' a) If rT1 ' Fr2 > 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.
Ycomp Form Code Otrtlo[ &rDu.clt DtD DIg?(1001, ELEV(100),xsgB(20r, Rn(20) Db Q ls glnglo, tC1 rr 81.n916, yc2 ti glag16, t! Dh Qr Aa gtagl., 0g & gl!gl., a0 lr 8tagt. Dt! EP A! IDtogar, ltSItB lt lat.g.r
As glaglc
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S{b
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4'72
B: Examples of Computer Programs in Visual BASIC
AppENDlx
tBt ut 11, DIs!(x), lldat
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AppENDIx B: Examplesof Computerhoglams in visu.l BASIC
414
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BAslc APPENDIx B: Exarnplesof ComputerProgramsin visual Db Dt! Dt! Dt! D!! Dl!
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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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Abutments effectson spillways, 204 scourformulas,437-42 wingwall,236-38 Adverseslope,163 Aeraiionof spillways,210-13 Aerationramps,212-13 Aggndation of smambeds,423-21 Algorithms,457 Alluvial streamclasses,423-24 AmericanCeophysicalUnion (AGU) sedimentscale,372,373 Angleof repose,l3l Antidues, 390,391 Approachsection,241,244 Aqueducts,g Afiificial opeochannels,I Asphalt-lircd channels,Manning'sn for, I 16 Attenuation,3498 Backwardcharacteristics,276 Backwarddifrerence,465 Backwater 233 in bridgeapproaches, calculatingfor normalflow beneath bidges,238,239,240 WSPROmodeling,241-53, 256-60 Backwaterprofile, 162 Ba.ffteblocks,72-73, 74, 76
B. A.. 4 Bakhmeteff. curve,397 Bar resistance 391 Bars,in riverbedsediments, BeaverCre€k(Newcastle,WY), 14? Bed forms effectson stagedischarg€ 39904 relationships, factorsaffectingformation, 389-96 BedJoaddischarge,4O4-5,406-9 Bemoulliequation,lo Besthydraulicsection,for uniform flows, 141-42 Bevels,at culven inlels, 219-20,22V29 Bisectionmethod,45&52 Bottom features.SeeChannelbottom features Boundaryconditions,for unsteadyflows, 280-81.284.301-5 Boundaryshear,?2-73 Box culverts designexample,231-33 entrancelosscoefficients,226 inlet control desigDequations,221 Manning'sz for, 225 423,424 Braidedchannels, Bmssconduits,Mandng'sa for, I 15 Bressefunction,159 Bressesolution, I 90-92 Mandng'sn for, I 16 Bricklined channels, Brickwork conduits,Manning'sa for, I 15 Bridg€ piers amlysisof drag,16-17 dimensional
483
484
Ir*DEX
Bridgepiers-Cont. dischargeco€fllcientadjustment factors.243 effectson spilhaays,204 incremental co€ftlcients. 240 backwater local scouraround,431-37 momentum equation for,8l-84 Bridges contractionscour,427-31 HDS-I analyses.236-38 HEC-RASandHEC-2analyses.233-36 as hydraulicstructures, 201,233 method,238 4l USGSwidthcontraction WSPROmodeling,241-53 Bridgesection,241,244 weirs,discharge over.52-55 Broad-crested Brushyfloodplains.Manning'sn for, I l8 BuckinghamPi theorem,14-15 Built-upchannels, Manning's a for, l16 Bulk specificweightof sediments, 374 procedure Bureauof Reclamation for riprap design,389 stillingbasin Bureauof Reclamation designs,7t-77
Characteristic form,214-'77 mathematical Characterislics. inte.pretation, 277-19 Chezyformula,100,120,l2l-22 Choke in channelwidthcontractions, 29, 30.31,32 relationto specificenetgy,26-28 in sup€rcritical contractions, 87-89 Christodoulou curve,2l4 Chuteblocks.74,75, ?6,7? Chutes in riverbedsediments, 390,391 spilh'ay.74-78 Circularchannels criticaldepthcomputations, 36, 37138 momenlumfunctionfot.64,6'148 Circularconduits,flow resistance in, lll.l12 Circularculvens,inletcontroldesign equations, 221 Clay conduits,Manning'sr for. I 15 specificgravity,373 Clay sediments, Clear-waterscour,428,430,431,432 Closedconduits, Manning's n for, ll5 Coherence. 128 flow models,327-28 Compositeflow profiles,for graduallyvaried Calibrationof unsteady Canals,3 flo*'s, 164-65 valuesof Manning'sa Castiron cooduits,Manning'sz for, I 15 Compositeroughness, for, 11,1-19 Cavitation aerationto prcvent,210-13 Compoundchannels spillwayformsto prevent,202-10 abutmentscourin,438-42 Cavitationsafetycurves,208,209 calibratingunsteady flow modelsfor,328 Celerity.SeeWavecelerity findingcriticaldepths,469-76 Cemenlconduits,Manning'sD for, 115 Froudenumberfor, 39,4O47 CemenGlined chaonels, Manning's uniformflows in, 126-29 n for, 116 molecules, Computational 305,306 Centraldifference,465 Computerprograms Chainrule,275 advances in,296 Cbalnel bottom features.Seealso for designingop€nchannelsystems,2-3 Roughness;Scour;Sediments FLDWAV324-26,328 bedforms,389-96 HEC-2.233.236,249-51 effecton openchannelturbulent HEC-RAS,183,185-88,189, flows, 105-9 233-35,236 effecs on specificeneryy,2l-22,55-56 HY8, 232 Channelcontractioo ratios,242 VisualBASIC, 17 Channelphotographs, 142-50 WSB 180,47G82 WSPRO,241-53,441,442 Chanoelshape.S€eako Contractions; Expansions YoYC, 38,467-69 effectson flow resistance, 110-13 Ycomp,45-47,46u6 georneticelements for, 36 Computingth,rough, for shocks,321 momennrmfunctionsfor,64 Coocreteconduits specificenergyand,34-38 dischargecapacityexample,125-26 asvariable,2 entrancelosscoemcients,225 Channelwidthcontractions. SeeCon&actions Manning's, for, I15, 225
INDEX Concrete-lined channels,I 16, 140-11 Consistencyin finite differeoce techniques,296 Constant-parameter method, 362_63, 364 Continuity equalion applyrng to open channelflow problems,8, I I derivation for unsreadyflows, 269_71, 274-'15 in hydrologicrouting,333_34 reanangingfor diffusion routing technique,352-53 for rectangularchanoels,gO for supercriticalcontractions,g6 Contractions at bridges(ree Bridges) dischargediagramsfor, 28_31 with headloss,3l-34 reducingin culverts,228 of supercriricalflow, 8G89 Controlvolumes applying Reynotdstransport theorem,6-9 for derivationof unsleadycontinuitr equarion,269-70 for derivation of unsteadymomenfum equation,27l,2j2 Conveyance,divided-channelmerhod. 127_29 Comer rounding, 242 CorrecredEuler method, 174 Conugated metal conduits entranceloss coefficients,226 ManninS's|tfor, l l5, 225 Courantcondirion,28 1, 305,3 t 2 Creeping morion, 376 Crestpressures in \pillways,204,:05 Criticalboundary.l8l Criticaldeprh in composireflow profiles,164. 165 rn compoundchannels,40-17. 469_76 In nonrectangularchannels,34_39, 46'749 relalon to normal depth in uniform flows. I 19,21 In spatially varied flows, 193 in specificenergydiagram.2.1.15. j0 Criticaldischarge.138 Criticalsedimentnumber,386 Critical shearstress.Seealso Shear stress in bed-loadtransportformulas.406 computationsfor sediment-lined channels, 380-81 criteria for sediment,linedchannels, 382_83 variabilityin naturalchannels. 3gg g9
485
Criticalslope calcutatingjn lake dischargeproblem, 166 graduallyraried flow profileshape,I63 lor uniform flows. 137_i8. 139 Critical sr€p heighr, 26, 27 Critical velocity for initiationof motionin sediments, 38.1_85,386.387 for self-cleansingin sewers,123_26 Cross-sectionalshape.SeeChanncl shape Crosssecrions, in bridgeopenings.235 !ullrvatedareas,Mannings n for, ll8 Lu lverls designexample,23l_33 functionol 201 generaldesignissues,215-l 7 inletconrrolfiows.215,216,21j_22 inlet improvements,22g-33 ouUetconrrolflows,2|5, Zl6,223_26 roaduay overroppingwitlt. 227_2g _ Lumutariledisrriburionfunction.379 Cylinders,dimensionalanalysisof drag, l5-l7.See a/so Bridge piers
Dam-breakproblem as sintplewavesolution,296_91 unsteadyfl ow computations for.324-26 Dams, early uses.3 Darcy-Weisbachequarioo,9g Darcy-Weisbach/ in Chezy equatjonfor transitionalano smooth turbulentflows, l20 factorsaffecting in open channel flow,
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field measurements,l0g_9 relaiionship ro Mannins's,t.I05_7 Da Vinci. Leonardo.4 Degradationof streambed,s, 423_27 Degreeof retardance,vegetativelining classes.133 DensimetricFroudenumber,for stratified flows.56 Denlation, 74, 77 Depth. Seea/so Critical depth determiningdistancefrom chanses. 168-73 specificenergyand, 23_26 WSPRO computarionsin lab channel,
251-s3 Designhead,for spillways,202_3 Designmode,of WSpRO. 256 Developed,inflow jumps,72 Diffusionof reservoirflows.336
486
INDEX
Diffusionroutingtechnique.352-56 Dikes,earlyuses,3 Dimensional analysis applyingto openchannelflow problems, l3-17 of uniformfloq 98-99 Dimensional homogeneity, 142 Directnumericalintegrationmerhod,t72-73 Directstepmethod,16148, 168,12 Discharge coefficients with culvertinlet control,219-20 for ellipticalcrestspillways.204-9 Kindsvater-Carter fo.mula..19-50,5l for normalflow beneathbridges,239-41 for ogeespillways,202-4 for roadwayovertopping, 227 Discharge diagrams, specificenergy and,28-3I Discharge measurement with weirs,48-55 Distancechanges, computingdepth from, 174-80 Divided-channel methodof definingtoral conyeyance,12'7-29 Domainof dependenc e, 2'19.28O quadrantof spillwaycrest,204 Downstream Drag.Se€Resistance Dunes factorscausing,394-95 featu.esol 390,391 aslowerregimebedforms,389 Dynamicforerunner, 348 Dynamicrouting,325 Dynamicwaveequations, 267 Earlycivilizations, op€nchannelsystemuse in,3-5 Eanhenchannels, Manning'sa for, 116-17 Effectivehead,224 Einsteinbedloadfuncrion,408 Einstein-Brown bedloadtransportformula, 401-8,419 Einstein'ssuspeoded-load t-ansport method.4l4 Einsteinvelocitycorrectionfactorfor alluvial channels, 385-86 Ellipticalcrestspillways,204-9 Ellipticalculvens,inlet conrol design equations, 222 Encroachment, 189-90 Endsills,74,75,76, 77 Energyequation applyingto openchannelflow problems, l0-l l, 2l in stratifiedflows,55-56
Energygradeline,21,22, 182 Energylossexpressions, 245 Ene.gymethods of bridgeanalysis,23gl Engelund-Hansen formulafor lotal sedtmenr discharge, 416_17, 419 Engelund's method,for stage-discharge refationships, 397-99. 4A2-3 Entmncefosscoefficients,225,2?6 Equationof graduallyvariedflo\r; 159-61 ERABS,460 Estuaryproblem,285-86 EtowahRiver(Dawsonville, GA), 146 Euler'sequation,for incompressible, frictionlessfluid, l0 Euler'smethod,of solvingfor deprh,174 Excavated eanhenchannels, Manning'sn for, 116-17 Exit section,of flow b€neath bridges,241, 244 Exnerequation,426 Expansions withheadloss,3l-34 of sup€rcritical flow,89-91 Experimental method,4 Explicitfinitedifferencemethods compared to implicirrnethods, 319-20 for unsready flow computations. 295, 296,305-13 Extended Bemoulliequation,l0 Fall,in culverts,228 Fall velocity,of sediments, 37,1-78 Finitedifferencemethods explicitmethodsfor unsteadyflows, 295-96,30s-13 explicitversus implicit,319-20 generalapplication, 463-65 implicitmethodsfor unsteadyflows, 295-96, 3r3-t9 for reservoirrouting computations, 336-39 Finiteelementmethods, for unsteadyflow computations, 296 FirsForderapproximations, 464 First-ordermethods,174 FLDWAVprogram,324-26.328 Flooding effectsat mainchannel/floodplain intefiace,\21-28 kinematicwavebehavior, 348-52 roadwayovertopping discharges, 227_28 specificenergycomputations, 39-47 unsteady flow computations for, 326-29 (.reeaho Unsteady flow equadons)
INDEX Floodplains. Seea/soCompound channels effectson floodwavepropagation, 327 encroacbment analyses, 189-90 flow versusmainchannelflow, 127-29 Manning'sa fo\ 1l'7-18,126-27 mapping, 167 Floodroutingproblem,303-4 Floodwayboundaryanalysis,189-90 Flow profiles,16l-65.Seeako B.idges Flow resistance. .teeResistance Flow routing basisfor simplifiedmethods, 333-34 defined,333 diffusionrouringtechnique, 352-56 kinematicwavetechnique, 345-52 Muskingum-Cunge method,35G{6 FLUVIAL-12,427 Flux-splining schemes, 3l I Fllx vector,3l I Formcode,467 Formresistance, l2-13. SeealsoResistance Forwardcharacteristics, 276 Forwarddifference,464 Fourierstabilityanalysis, 318 Founh-order Runge/Kutra method,175-78 FrictionfactoN basedon logarithmicvelocity distriburion,104-9 of flow beneath bridges,24546 openchannelconditions affecting,109-14 Froelichabutment scourformula,438 Froelichpierscourformuta,43G37 Froudenumbef basicequation, 2 for compound channels, 39,40-47 criticaldeprhand,25 with hydraulicjumps,68 for stratifiedflows,56 usein slop€classification, 139-41 Full-pipeflow in culvens,223 Full valleysection,241,244 Galerkinapproach, 296 Geometricelements, for channelcrcss sections, 36 Glassconduits,Manning'sn for, I 15 Gradedstream,422 Graduallyvariedflow Bressesolution,190-92 computingdepthfrom distance changes,174-80 equationof, 159-61 featuresof, 6, 159
48.1
floodwayencroachment anaiysis,189_90 lale discharge problem,165-67 in natural channels, l8l-88 spariallyvariedflows, 192,95 watersurfaceprofile classification, I6l-65 watersudaceprofilecomputation, 167-68 Grainsize distributions in riverbed sedimenrs, 378-80 effectson sedimentmotion,383-86 Grass-lined channels, uniformflow computations, i 32-37 Gravel-bed streams, frictionfactor in, 107-9 Gravel-bottomed linedchannels, Manning'sa for, I l6 Gravity, 2, 17 Hartreemethod,298-300 Haw River(Benaja,NC), 149 HDS-I methodof bridgeanalysis,236-38 Headlosses, 3l-34 HEC-2program,233,236,249-51 HEC-RASprogram for bridgeflow constriction compurarions, 233,235,236 for graduallyvariedflow problems, 183,185-88,189 Heunmethod,174 Horseshoe-shaped conduits,flow resistance i n ,l l l , 11 2 ,I l 3 Horseshoe vortices,431 HY8 program,232 HYDRAIN suite,241 Hydraulicallylong channels,167 Hydraulicdata,for calibratingunsteadyflow models,329 Hydraulicexponentmethod,190-92 Hydraulichead,78 Hydraulicjumps energylossesin, 6l-62 momentum equationfor, 62-73 with sluicegares,164,165 in stillingbasins.?4-78 Hydraulic routing,333 Hydraulicstructures bridges(reeBridges) culvens(.re?Culverts) introductionto, 201 spillwayaeration,210-13 spillwayflow compuration, 202-10 stepped spillways,213-15 Hydroelectric turbineexample,3 l3
488
INDEX
Hydroelectric turbine load acceptance problem,30l-2 Hydrologic routing hydraulic routing venus, 333 introduction to simplified equations,
33+36 reservoirrouting,335-36,336-39 riverrouting,335-36,319-44 TALLWIAL,42'I Implicitfinite differencemethods comparedto explicitmethods, 319-20 for unsteadyflow computations, 295-96,3t3-t9 lncremental backwatercoefficients, 240 Initial conditionsfor unsteady flows,280,281 Initiationof motion,380-87 Inletcontrolin culverts, 215,216,217-22 Inletimprovement, 228-33 Intervalhalvingmethod,458-62 lntervalof dependence, 280 Inviscidfloq 12, 13. 14 Iterationprocedures intervalhalvingmethod,458-62 Newton-Raphson technique, 463 secantmethod,462 for second-order predictor-corrector method,178 for unsteadyflow compurations. 296 JainandFischerpier scolr formula,435,437 Karim-Kennedy method of predictingstage-discharge relationships, 400-402,404 total dischargecomputations, 4t't-18,4t9.421 Keulegan equarion,105,107,108 KindsvaterandCarterformula,49-50,5l Kinematicwavetechniqueof flow routing, 345-52,356-59 Kleitz-Seddon pinciple, 349-50 Korencondition,312 Lakedischarge problem,165-67 Laminarseparation,12 Lateralinllows, 192-93,2'l |, 2'13 LateBl outflows,19,L95 LaurseoandTochpier scour formula,434, 437 Law of the wall, 103
Lax diffusirescherne.305-7,114,322-24 Lax-Wendroff scheme.308-9,322 L€apfrogscheme,3O8.322 Least-squares methods,341-42 Levelpool routing,325,336 Limit slope,138-41 Linedchannels, Manning's lt for, 116 Live-bedscour.428-30,431,432 Loadacceptance problern,301-2 Load.ejectionproblem,301 Localscour,43l-32 Logarithmicorcrlap iayer,103 Logarithmicvelocir]'distributions in pipes andopenchannels,102-9 Loopedratin-ecurves.328 Lowerregimebedforms,389-90 Luciteconduis.Manning'sn for, I l5 Maccormackschern€,310 Main channeUfl oodplaininte.facefl ow computations. 127-28 Manningequation appliedto openrrap€zoidal channels,105-7 development of. -1,l0O-102 dimensional homogeneity and,142 normaldeprhcompurations, 120 reanangingfor kinematicwave technique.345 usein slopeclassification, 137-38 Manning's n for channelsof compositeroughness, I | ,1-l9 for compoundchannels,126-27 convcnedfo. dirnensional homogeneity, l.l2 conectionfactorsfor naturalchannels, I 14 for culvens.224.225 development of, 102 effectof bedformson,401 factorsaffecringin openchannel flow. 109-l4 field measurements, 109 relationship to Darcy-Weisbachl105-7 for vegetative channellinings.134-36 Masonry-lined channels,Manning'sa for, I l6 Massdensiryof sediments, 373 Masstmnsponrateof sediments, 405 Meandering channels.423-24 Meanslopeof energrgradeline, 182 Melvilleabutmentscourformula,43?-38 MelvilleandSurherland pier scour formula-435-36,437 M E S Hs c h e m e . 3 l O
INDEX Metal-linedchannels, Manning'sn for, I 16 Methodof characteristics defined,268 depthhydrograph output,314 versus,365-66 Muskingum-Cungeresults 295,f96, for unsteadyflow computations, 29'l-3N,322-24 Methodof specifiedtime intervals,298-300 andMiiller formula,408-9.419 Meyer-Peter MiddleForkFlatheadRiver(Essex,MT). 147 MiddleForkVermilionRiver(Danville, IL). 145 Mild slope defining,137 graduallyvariedflow profileshape,162. 163,164,165 WA), 149 MissionCreek(Cashmere, Mi[ed-flow regime,183 Modularlimit for tailwaterheightover weirs,55 Modulecode,467 Momentumequation applyingto openchannelflow problems,I l, 6l with,236 bridgeflow computations for bridgepiers,81-84 flows, derivationfor unsteady 271-'14,2'14-75 formsof,9-10 for hydraulicjumps,6l-73 ignoringin hydrologicrouting,333 for diffusionrouting rearranging technique, 352 for stillingbasins,74-78 transitions, 84-91 sup€rcritical for surges,78-81 for uniformflows,99-100 Momentumfuoction,63 Monoclinalwave,320,348-52,355-56 Moodydiagnm,97, 98-99, 106 MurderCreek(Monticello,GA), 148 method,342-44,356-46 Muskingum-Cunge Muskingummethod,335,336, 339-44,35641 Nappeflow,2l3 Naturalopenchannels examples,I graduallyvariedflows in, 181-88 Manning'sn for, 117 Negativecharacteristics,2l6,284 technique Newton-Raphson appliedto unsteadynows,302,316,3l? 463 basicequations.
{89
divergence,462 graphicaldepiction,459 visualBASICcode,461 Nonlinearalgebraic equations, 458-63 Nonrectangularchannels,sp€cific energy,34-38 Nonuniformflows,6 Nonuniformities, flow resistance from, lll-12 Normaldepth calculating for uniformflows,I 19-21 defined,97 usein slopeclassification, 137,140 Nlmericaldiffusion,359-61 Numericalmethods,457 463-65 finitedifference approximations, equations, 458-63 nonlinearalgebraic Ogeespillways, 202,204,205,210 predictor-corrector One-step method,174 Openchannelflow basicequadons.6-ll characteristics of, l-2 dimensional analysis,13-17 solvingproblemsin, 2-3 suriaceversusform resistance, ll-13 typesof,5-6 Openchannel hydraulics, 1,3-5 Orifices bridgesas.2499 culvensas,217,219 Outflowhydrographs, for dam-breal problem,325-26 Outletcontrolin culverts,215, 216.223-26 Overbankflow,specificenergy computations, 39-47 Ovenundischarge,349 Oxbowlakes,424 Parabolic channels, 36,64, 67 ParticleReynolds number,381 Particleshapeof sediments, 372-73 Paniclesizeof sediments, 372 Performance curves ofculvens,2l5-I7 Phi index,128 Piers.Se?Bridgepiers Pipeflow formulas.Seeako Culverts extended to openchannelflow, 102-9 g.avitysewerdesign,122-26 Planebed,390 Planformof sueams, 423-27 Pools,in riverbedsediments, 390,391
490
INDEX
276 Positivecharacteristics, hasadmethod,183-85 methods,168,31G-lI Predictor-conector Preissmann method, 313,315,319,392 ZlT,223 flow in culvefts, Pressure kism storage,335 PVC pipe capacityexafiple, | 24-25 discharge uniformflows in, l2l-22 68,69 Radialjumps, 212-13 aeration, Ramps, Rangeof influence,279,280 Rapidfloq 25, 26 Rapidlyvariedflow,6 channels Rectangular continuityequationfor, 80 36 c.iticaldepthcompuiations, in, I l0-l I flow resistance hydraulicexponentmethodfor gradually variedflow computation,190-92 limit slopefor uniformflows, I38-41 functionfor, &, 6'7,68-70 momentum turbulentflow in, 105 49, 202 Rehbockrelationship, Reservoirs weightin, 374 bulk sedimenr hydrologicroutingmelhods, 335*36,336-39 Resistance channelshapeand,I 10-13 3?5-?6 in fall velocitycomputations, of rock riprap,129-30 typesof, I l-13 Resistance coefficient,97-98 Reynoldsnumber criticalboundary,381,382 of;fandn on, 109-10 dep€ndence fall velocityand,375-76 in openchannelflow,2 Reynoldstransporttheorem.6-9 283 Riemanninvariants. Rio Chama(Chamita,NM), 143 Ripples factorscausing,394 features of, 390,391 aslow€rregimebedforms.389 channels Riprap-lined functionof, 129 stable,lesign,388-89 129-32 uniformflow computations. Riveroverbankflow,specificenergy computations, 39-47 Riverreachcontrolvolume.7
Rivers bedforms,389 96 hydrologicroutingmethods, 33s-36,33944 initiarionof sedimentmotion,380-87 practicalaspects of flow computations, 326-29 (s€esediment sediment discharge discharge) 372-80 sedimentpropenies, stablechanneldesign,388-89 relationships. 39G404 stage-discharge 227,249 Roadwayovenopping, RockCreek(Darby,MT), 150 R o l lw a v e sI ,1 3 , 3 1 8 - 1 9 Romanaqueducts, 3-4 Roughness effecton Darcy-Weisbach /, I l0 effecton hydraulicjumps,72 effecton openchannelturbulent flows, 105-9 effecton pipeflows, 104-5 effecton uniformflows,98-99 asopenchannelvariable,2, 97 of riverbedforms,39l-92 and.383-84 Shieldsparameter varyingwithinchannels,I l,l-19 Roundoffenor,457 412-13 Rousenumber, Routing,268 Routingcoefficients,33940, 351,363 Routingtables,337,338 Rubblemasonryconduits,Manning'sn for, I 15 Runge-Kutta methods,l?4-78 equations, 267,269J 4 Saint-Venant SaltCreek(Roca,NE), 143 SahRiver(AZ), l,l4 pipes,friction roughened Sand-grain factorin, 104,105 sanitarysewers,115,122-26 Scour abutment, 437-42 from bridgecontractions, 427-31 local,43l-32 pier,431,432-31 total,44243 Secantmethod,459,461,462 465 Second-order approximations, predictor-collector Second-order method,178 Sedimentation diameter, 372 426 Sediment continuityequation,
INDEX
Sedimentdis.harge 404-5 basicapproaches, bed load- -1O1-5.40G9 load,405,410-16 suspended 416 22 tolrl dischatgecomputations. Sedimentnumber.56 seedfuoScour Sediments. 423-2t' anddegradalion' aggradadon 80 372 proFenies, basic bedformsin. 389-96 channeldesignstabilityand,188-89 (J?eSediment dischargecomputations discha+e) initiationof motion,380-87 396-404 relationships, stage-dis.harge 210-lI Self-aerationSelf-cleansitgvelocitiesin sewers'123-26 Sequentdep$ iatios ior hydraulicjumpsin variouschannel shapes.63-70 74-78,79 in stillingbasins, Sewers,designingfor PartlYfull flo\r s. 122-26 Shapefacto( for sandgrains'372-13 weirs,discharge Sharp-cresrcd 48-52 measurcments, Shearstress in bed-loadtransportformulas'406 for initiationof sediment motion.380-83 at main channel/floodplain interface,127-28 in ripraplinedchannels,129-30 riverbed formsand,391-92 388-89 variabilityin naturalchannels, classes'132,134-36 of vegetalretardance Shieldsdiagram,381,383-84 381,382,383-84 Shieldsparameter. Shockcapturing,322 Shockfitting. 321 Shocks,320-24.326 weirs,discharge Short-crested 53, 55 measurements, inlets,229-30,232 Side-tap€red Sieveanallsis,379-80 Sievediameter,372 425 Silt-claype.centage, diagram.393-94 chardson Simons-fu Simpleware region,284 Simpleu ares of, 282-84 characleristics dam-breakproblem,28G9I estuarlFoblem,285-86 l/3 rule,173 Simpson's 423-2? Sinuosityof streams,
491
SkilletCreeke\ample,185-88 Skimmingtlo$. ll3-14 Slopeclassification for graduall!{ariedflows,162-65 I for uniformflo\rs, 137--'t inlets,230-33 Slope-tapered 122 Slugging, sluicegates,1fi. 165 friction factors,104,105.106 Smooth-pipe SouthForkClearwateiRiver(Crangeville, ID), l.{8 34Ml Spatialintenals.sel€cting. Spatiallyvariedflow, 192-95 Specificenergy 4 Bakhmeteffs explanation, chokeand.2G28, 29,30 with head andexpansions contractions loss,3l-34 defining.2l -23 diagrams,28-3I discharge momentumfunctionand,70-71 channels, 34-38 for nonreclangular overbankflow, 39-47 overweirs.48-55 in stratifiedflows,55-56 Specificenergydiagrams,2l,23-26 373-74 Specificgrality of sediments, 373 Specificweightof sediments, Spillways aeration,2l0-13 functionof. 201 202-10 computations, head-discharge hydraulichead,78 stepped,213-15 stillingbasinswith,7'1-78 methodof flow storage-indication routing,337-39 for, 303,304 unsteadyflow computations Spurdikes,24-1 3l l-12 analyses, Stability ratingcunes,392-93 Stage-discharge 39G404 relationships, Stage-discharge 327-28 Stagehydrographs, Standardfall diameter,378 Standardstepmethod,182 Standingware fronts,84-86,87 Steelconduits,Manning'sn for, 115 Steepslope.t62, 163,164,165 Stillingbasins.?,1-78,79 Stokes'law,376 Stokes'range.376 Storageequation,in hYdrologic routing.333-34,335 method,337,338 Storage-indication Stormsewers.122-26
492
INDEX
Stratifiedflows,55-56 armoring,426 Srreambed Streanlinecontrolvolume,7, l0 Streamtube conrol volume,7, 9, l0 Subcridcalflow basicfeatures, 25-26 gadually varied,183-85 specifyingunsteady flow boundary conditions, 280-81 Eanslatorywavepropagation in, 268 Subdivisionof channelcrosssections,183 specificweightof sediments, 374 Submerged Submergence effecton bridgeflows,247-49 effecton inletcontrolof culvertflow' 220 from kinematicwave Subsidence, absence computations, 3498 flow Supercritical 25,26 basicfeatures, chokingmodesfor conractionsin, 31, 32 designingfor contractions ol 8G89 designingfor expansions of, 89-91 graduallyvaried,183-85 flow boundary specifyingunsteady 280-81 conditions, transitiondesignchallenges in, 84-86 translatorywavepropagation in, 268 weirs,49, 5O-51 Suppressed Surfaceprofiles.SeeWatersurfaceprofiles Surfaceresistaace, ll-13. Seealso Resistance Surges from convergence of positivewaves,284 in dam-break problems, 289-91,326 examples, 78 momentumequations for, 78-81 uith monoclinalwaves,350-51 405,410-16 Suspended-load discharge, Tail$ ate.height modularlimit overweiN,55 in stillingbasins,74-78,79 Taylor'sseriesexpansion, 308-9 Temperatureeffectson river bedforms, 395 Theo.eticalapproach, 4 Throatof culvens,229-31 fime steps io numericalsolutionsto unsteadyflow problems,281-82,328-29 selectingfor flow routing computations, 34Hl Tobesofkee Creek(Macon,GA), 146 Topographicdata,for calibratingunsteady flow models,329
Tractive force ratio, 389 Tranquil flou 25-26 Transition zones,in riveabed sediments,390. 395 Translatory waves,26?-68 Trapezoidal channels critical depth computarions,
3r3'/, 38, 46'149 momentumfunctionfo.. 63,ff sideslopesfor riprap,l3l turbulentflow in, 105 Trapezoidalrule, l7l Tree-gro*'nfloodplains, l'lanning'sn for, I 18 Triangularchannels critical depthconrputations, 36 momentumfunctionfor. 64, 67 Triangularweirs,5l-52 Turbulentflow friction factorin, lM-9 Shieldsparameter and.383-84 Turtrulentseparation, I2 Type I flow aroundbridgepiers,81,82-83 depthcalculations for, 236 dischargecoefficients,242 throughbridgeopenings, 233,234 TypeII flou 8l-82,233.236 1}pe ILA flou 233,234 Type[B flow,233,234 Type m flou 233 TypeII stillingbasins,74,77 TypeIII srillingbasins,74, 76 Typery stillingbasins,74, 75 Underdesigning spillwaycrests,204 Uniform flows applications, 97-98 besthydraulicsection,l4l-42 channelphotographs, 142-50 Chery andManningformulas,100-102 in compoundchannels, 126-29 defined,6 dimensionalanalysis,98-99 factorsaffecting/andn, 109-14 in grass-lined channels, 132*37 momentumanalysis, 99-100 normaldepthcomputations, I 19-21 in parrlyfull, smooth,circulat conduits, l2l-22 pipe flow formulasextended to open channel,102-9 in riprap-linedchannels, 129-32 seu,erdesignfor, 122-26 slopeclassification,l3?-41
INDEX Unit streamF|ower, 417 Unsteadiness effects,I l3 Unsteadyflow equarions basicapproaches, 26749, 295-91 boundary condilions in, 284,301-5 chamcteristic fonn computations, 274-77 dam-breakprobJem. 324-26 derivationof Saint-Venant equations, 269-74 explicitfinitedifference me$ods,305-13 implicit finitedifferencemerhod,3 l 3-19 ini!ial andboundaryconditions,279-82 mathematical interpretation of cnaraclensocs, z//-/y merhodof characteristics, 297-300 numericalmethodscompared, 3 I 9-20 for rivers,326-29 shocks,320-24 simplewave,282-91 Upstreamquadrant of spillwaycrest.204 USGSTypeI floq 217,218 USCSType4 flow,223,224 USGSType5 flow,217,218 USGSType6 floq 223,224 USGSType7 flow,225 USGSwidth contraction merhod,238-41 VaoRijn classification sysrem,394-95 VanRijn's merhod of computingsuspended-load transport.4l4l6 of predictingstage-discharge relationships, 399-400,403-4 total discharge computations, 420-21 Variable-parameter merhodof MuskingumCungetechnique, 362-63,3g Vedemikov number. 318-l9 Vegetative linings,132,133,134-36 Velocitydefectlaw 104 Venruris.33, 34 Verificationof unsteady flow models,327-28 VisualBASIC, 17,461-82 Visualobseryation method,of measuring shearstress on rediments, 182-8.1 V-norchweirs,5l-52 Volumeflux, 8-9, l0
493
Wakevonices,43| Washed-out dunes,390 Washload,404 Watersurfaceprofiles classifying, l6l-65 computations for. 167-68, 169-73,| 82-88 VisualBASIC program,180,4?6-82 Wateru'ays Experimenr Stationspillwar shape,203 Wavecelerity in compoundchannels, 328 in kinematicwaverechnique, 346, 34'7 ,348,351 of monoclinaiwave,3.19,350-51,355-56 Wavefrontsin sup€rcrirical flow,8,+-86,87 Wedgestorage,335 Weirs culvertsactingas.2l?, 218-19 dischargemeasurements over.48-55 Iareraloutflowsfrom, 194-95 roadwayembankments as,22'l-28 on spillways,202 unsteadyflow computations for, 303,304 Wenarchee River(Plain,WA), 145 WestFork BitterrootRiver(Conner,MT), l,f4 Widthcontracrionmethodof bridge analysis,238-41 Wing*all abutmenrs, 236-38 Woodconduits,Manning'sn for, I 15 Wood-linedchannels. Manning'sn for, I l6 Wroughlironconduirs,Manning'sn for, I l5 WSPprogram,| 80,47G82 WSPROprogram,44l, ,142 featuresandperformance, 241-53 inputdata,253-56 outputdata,256-60 Y0YC program,38, 467-69 Yangmethod, 417,419,421 Yamellequation,236 Ycompprogram,45-47.469-76 Zoneof quiet,in simplewave,283
