OpenChannelHydraulics
i
.l
I
. l
TerrvW. Sturm
Georgia Ins'tut. oI Te.hnology
I
{:
I
I
i
{
lA Madison,Wl NewYork
Boston BunRidge,lL Dubuqu_e,
SanFranciscoSt.Louis BanEROkBogote Caracas KualaLumpur
Lisbon LondonMaddd MexicoCity Milan Montreal NewDelhi
SantiagoSeoul SingaporeSydney Taipei Toronto
t
t
-
"
--.-r.
J
1
McGraw-Hill Seriesin Water Resourcesand Environmental
Engineering
CoNsuLTtNcEDrroR
George Tchobanoglous, Univenity
of Caldornia,
Davis
Bailey and Ollist Biochemical Engineering Fundamentals
Bedsmin: WaterChemistry
Bishopt Pollution Prevention: Fundamentals ond practice
Canlert Environmcnral Impact Assessment
Cha etli Envirc^rnantal Protection
Chapran SurfaceWarer-Quality Modeting
Choq Maldment, ard May$ Applied Hydrology
Crites and Tchobanog)otrsi Snwll and Decentrclized WastewoterMonogerrun, Sysrems
Davis and Cornwelft Inrruduction to Errvimnnqatal Engineering
deNevers: Air PollutioaCon,rol Engineering
Eckenfefder: Industrial WaterPollurion Contml
Ewels, Ergas, Chang, and Schroeder: Bioretwdiation principles
LaGrega, Buckingham, and Evans: Hdzardous Wd$teManogenan,
Linsley, Franzini, Frtyberg, and Tchobanogtous: Water Resourcel.
and Engineeing
M&het
WaterSupplyand Sewage
Metceff & Fddy,Inc.i WastewaterEngineering: Collection and pumping olWastewater
M€tcalf & Fddy, Inc-i WastewoterEngineering: Treatment, Disposal, Reusc
Peary, Rowe, and Tchobanoglous: Environmental Engineering
Rittmann snd McCarTli Environmentol Biotechnology: principles and Applicotions
Rubin: Introduction to Engineering and the Environment
Sswyer, McCarty, and Partkin: Chemistry for Environmental Engineering
Slurmr Open Charnel Hydraulics
Tchobanoglous,.Thies€n, and Vigit: Integrated Solid WasteManagement:Engineering
Principlesand ManagementIssues
McGraw-Hill Higher Education
A Obnd
d lle Md;tH
il Co"tffi
U
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
l0 09 08 0 7 0 6 0 5 M 0 3 0 2
20 09 08 07 06 05 04 03 02 0l
UPE BJE
Ouote on Daqev: Maclcaq Norman. A Nver RunThrough/t. @l9?6, Univ€rsityofChicago Press
nieprinteawirh peurussionof Univeasityof Chicsgoftess
Library of Corgresr Cstalogirg-in-PublicatioD Dsts
Stutr\ Terry W.
Openchannelhydraulics/ T€rryW. Sturm.- I st ed.
p. cm.
Includesindcx
ISBN 047462,145-3
l. charrlels (Hydnulic engineering)2. Hydraulics. l. Title ll Series'
TC175.57742ml
627'.2!-4c21
[$\'.mhhe.com
WheB orderingahirtitle, useISBN047-12018G?
Printed in Singaporc
00{4898?
CP
Evntually, oll things nerge into one, and a river runs through it. The river was
cur b)' the xorld's great food and runs over rocksfrom the basenent of time. On
some of the rocks are tinteless raindrops, Under the rock are the lords, and
some of the u'ords are theirs.
I arn haunted by waters.
-Nomtrnanl\'laclean,A River Rlns Tltrouphlt
CONTENTS
Preface
xl
BasicPrinciples
l.l
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
LlO
lntroduction
Characteristics
of OpenChannelFlow
SolutionofOpenChannelFlowProblems
Purpose
HistoricalBackground
Definitions
BasicEquations
Surfacevs. FormResistance
DimensionalAnalysis
ComputerPrograms
I
I
I
)
J
)
6
lt
IJ
l7
Specific Energr
2l
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2l
23
26
28
Definitionof SpecificEnergy
SpecificEnergyDiagram
Choke
DischargeDiagram
ContractionsandExpansionswith Headloss
Critical Depthin Nonrectangular
Sections
OverbankFlow
Wein
JI
34
39
48
Sharp-Crcsted Recmngulor Notch Weir / Sharp-Crested
Triangulu NoEh Weir / Brcad-Crested Weir
2.9 EneryyEquationin a StratifiedFlow
))
Momentum
6l
3.1
3.2
3.3
3.4
3.5
3.6
6l
6l
74
78
8l
84
Introduction
HydraulicJump
StillingBasins
Surges
BridgePiers
SupercriticalTransitions
Design of Supercritical Con rcction / Designol
Supercritical &pansion
Conlents
vut
4
Uniform Flow
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
Introduction
AnalYsis
Dimensional
MomentumAnalysis
of theChezyandManningFormulas
Background
LogarithmicFormulafrom ModernFluid Mechanics
Discussionof FactorsAffecting/and n
Selectionof Manning'sn in NaturalChannels
Channelswith CompositeRoughness
Uniform Flow ComPutations
4.10 Pattly Full Flow in Smooth,CircularConduits
4.1I GravitySewerDesign
4.12 ComPoundChannels
4.13 Riprap-LinedChannels
4.14 Grass-LinedChannels
4.15 SlopeClassihcation
4.16 Best HydraulicSection
Manning'sFormula
4.17 DimensionallyHomogeneous
97
98
99
l0O
102
109
I 14
ll4
I 19
l2l
122
126
129
132
137
l4l
4.18 ChannelPhotograPhs
142
142
Gradually Varied Flow
t59
5.1
5.2
5.3
Introduction
Equationof GraduallyVariedFlow
Classificationof WaterSurfaceProfiles
5.4
5.5
Lake DischargeProblem
WaterSurfaceProfile Computation
DistanceDeterminedfrom DepthChanges
DirectStepMelhod / DirectNumeicalIntegration
5.6
5.7
5.8
DepthComputedfrom DistanceChanges
NaturalChannels
5.9 FloodwayEncroachmentAnalysis
5.10 Bress€Soludon
5.1I SpatiallyVariedFlow
159
159
l6l
165
t67
168
174
l8l
189
190
r92
Hydraulic Structurts
201
6.1 Introduction
6.2 Spillways
6.3 SpillwayAeration
zol
202
2to
lx
Contcnts,
6.4 SteppedSpillways
6.5 Culvens
hlet Control / Ou,lerContml / Road
Otenopping / Improvedlnlets
2t3
215
6.6 Bridges
HEC-2 and HEC-MS / HDS-I / USGSVtidh
Contraction Method / WSPROModel / IISPRO Input
Data / WSPROOutput Data
233
Governing Equations of UnsteadyFlow
26'l
7.1 Introduction
?.2 Derivationof Saint-Venant
Equations
267
269
Conlinuity Equatibn / Momentun Equatiorl
7.3
7.4
7.5
7.6
Transformation
to Characteristic
Form
Mathematical
Interpretation
of Characteristics
Initial andBoundary
Conditions
SimpleWave
274
277
219
282
Dom-Break Problcm
NumericalSolutionof the UnsteadyFlow Equations 295
8.1 Introduction
295
8.2 Methodof Characteristics
8.3 BoundaryConditions
8.4 ExplicitFiniteDifference
Methods
297
301
305
Lal Difasive Scheme/ kapfrog Scheme / Lax.Wendrof
Scheme / Predictor-Corrector Methods / Flux-Splining
Schemes/ Stability
8.5
Implicit Finite Difference Method
8.6
E.7
8.8
8.9
Comparison
of NumericalMethods
Shocks
Dam-BreakProblem
PracticalAspectsof RiverComputations
Simplified Methods of Flow Routlng
9.1 lntroduction
9.2 HydrologicRouting
313
319
3m
324
326
JJJ
JJJ
334
Resemoir Routing / River Routing
9.3 KinematicWaveRouting
9.4 Diftrsion Routing
9.5 Muskingum-CungeMethod
345
352
356
Contents
l0
Flow in Alluvial Channels
l0.l Introduction
Properties
10.2 Sediment
Pa icle Size/ PorticleShape/ PanicleSpecifc
Gravity/ BulkSpecifcWeight/ Fall Veloci4/ Arain
SizeDistribution
10.3 Initiationof Motion
10.4 Applicationto StableChannelDesign
10.5 Bed Forms
Relationships
10.6 Stage-Discharge
Method/ VanRijn'sMethod/ Karin'
Engelund's
KennedyMethod
Discharge
10.7 Sediment
Sediignl
Bed-loadDischorge/ Suspended
Discharge
Discherge/ TotalSediment
10.8 StreambedAdjustmentsand Scour
AggradationandDegradation/ BridgeContraction
Scour / Incal Scour/ ToralScour
AppendixA NumerlcalMethods
A.l Introduction
A.2 NonlinearAlgebraicEquations
Intenal HolvingMethod/ SecontMethod / Netton'
RaphsonMethod
A.3 Finite DifferenceApproximations
AppendixB Examplesof ComputerProgramsln
VisualBASIC
B.l YOYC Programfor Calculationof Normal andCritical
Depth in a TrapezoidalChannel
Y|YCFormCode/ Y|YCModuleCode
B.2 Ycomp Programfor FindingMultiple Critical
Depths in a CompoundChannel
YcompFormCode/ YconpModuleCode
B.3 WSP Programfor Water SurfaceProfrle Computation
WSPFormCode/ WSPModuleCode
Index
311
3'tI
372
380
388
389
396
404
457
457
458
463
467
467
469
476
483
PREFACE
The studyof open channelhydraulicsis a challengingand excitingendeavor
becai.rse
of the influenceof gravity on free surfaceflows. The position of the free
surfaceis not known a priori, andcounterintuitivephenomenacan occur from the
liewpoint of the first-time sludentof open channelflow. This book offers a study
of gravity flows staning from a firm foundarionin modern fluid mechanicsthat
includesboth experimentalresultsand numericalcomputationtechniques.
The
developmentof the subject matter proceedsfrom basic fundamentalsto selected
applicationswith numerousworked-outexamples.Experimentalresultsand their
comparisonwith theoryare usedthroughoutthe book to developan understanding
flow phenomena.Computationaltools range from spreadsheets
of free-surface
to
computerprogramsto solve moredifficult problems.Somecomputerprogramsafe
providedin Vsual BASIC, both as leamingtools and asexamplesto encouragethe
use of computationalmethods regardlessof the platform available in a very
dynamicenvironment.In addition, severalwell-known computerpackagesavailable in the public domain are demonstratedand discussedto inform userswith
respectto lhe methodologiesemployedand their limitations.
The basicequationsofcontinuity, energy,and momentumare derivedfor open
cbannelflow in the first chapter,from the viewpoint of both a finite control volume
and an infinitesimalcontrol volume,althoughthe completederivationof the general unsteadyform of the differential momentumequationis savedfor Chapter7.
Dimensionalanalysisis introducedin somedetail in the hrst chapterberauseof its
use throughout the book. This is followed by Chapters 2 and 3 on the specific
energyconceptand the momentumfunction.respectively,and their applicationsto
open channelflow problems.Designof open channelsfor uniform flow is examined in Chapter4 with a detailedconsiderationof the estimationof flow rcsistance.
Applicationsincludethe designofchannelswith vegetativeandrock riprap linings,
and the design of storm and sanita4r sewers.Chapter 5, on gradually varied flow,
emphasizesmodem numerical solution techniques. The methodology for watersurface profile computation used in current computer prcgrams promulgated by
federal agenciesis discussed,and example problems are given. The design of
hydraulic structures,including spillways,culverts, and bridges,is the subjectof
Chapter 6. Accepted computer programs used in such design are introduced and
their methodologiesthoroughly explored.Chapters7, 8, and 9 developcunent
techniquesfor the solution of the one-dimensionalSaint-Venantequationsof
unsteadyflow and their simplifications. In Chapter 7, the Saint-Venantequations
are derived, and the method of characteristics is introduced for the simple wave
problem as a meansof understandingthe matbematical transformation of the governing equationsinto characteristic form. The numerical techniquesof explicit and
implicit finite differences and the numerical method of characteristicsare given in
Chapter 8, with applications to hydroelectric transientsin headracesand tailraces,
xl
Prcface
xii
co'rerssimplified
problem,and flood routingin rivers Chapter9
the dam-break
nrethod'
diffusion
method'
wave
of flow routingincludingrhe kin-ematic
methods
chanalluvial
of
subjecr
complcx
;;;';; lt;;;ki"s";-b,i,lg" nt"tttod Finrllv' the
explored
is
surface
free
adjusiable
nel flows thathavea movablebed as rvell asan
links amongsedimentdisinponant
the
emphasizcs
chapter
rrtls
r" ti "pli ioof
essentialto an.u.nderstanding
.n-gaiU"O forms,and flow "ti'tunt" that are
adjustchannel
alluvial
are
l0
op.n'.itunn"tnonuin rivers.Also coveredin Chapter
flow blockase
and shapeiand bed scouiin responseto the
J;;;';iilLrm,
causedbY bridgefoundations'
to supPlementthe text material The first is
The book includestwo appendices
techniquesthat can be used
, ".n"iui discussionof somi selectednumerical
someexamplecomputerprocontains
,rrt?r"rr"*-t.trebook. The secondappendix
;i noIilal and critical depth in prismaticchannels'
;l#ffi;';;;;;";;,t*
of waler-surfaceproirles These
channels'and computation.
i".i"oi"g .t.p"""h
aids for. more exrensrveprc.
as
leaming
;" written in visual BASIC
;;;;r".:
chapters'6n a website for the book' addiu,',i. "na-or*tal
!;ffi;;;;t;i;;;
exerciseson unsteadyflow comadvinced
ionJ proir^*. for solutionof the more
if necessaryin a dynamic
updated
be
can
""o,iJ", i"" be found, where they
;'f"':l"J#ilj?i,),.Ti;n,
r.*?;l;;;;;"i"
andnrsrundersraduales
roradvanced
is inrended
environmental
and
i" trt" g"**r fieldsof waterresources
sufficient
provide
9.
?
throush
Chapters
and
i'il"dsh 5
ch;ffi
coveringbothsteadyand
hydraulics
courserr openchannei
materialfor a semester
courseor a sengraduate
u*i*Oy no". fn" bookalsocanbe usedfor a first-year
andriverhydraulics'utilizingChapters
i", .i.tii"i-.."*" on hydraulicstructures
andexample
whichincludesseveralapplications
+,'s,'is, "iJ 10.n"s materia.l,
for
responsibilitv
; ur.rur ,o the fractitionerchargedwith the
ilii
#il;r,
culvert
spillwaydesignfor smallteseryoirs'
i""ft'lrks "t nt aplainmanagement'
of alluoi stabilityandflow resistance
investigaion
-J."*", a"tig" r,ir orainage,
applied
this
of
"stimatioriofbridgeiackwaterandstour.Because
il ;;;il;J
library
consultingengineer's
i*"r J,ii"'m"r. it shouldbe a uJefuladditionto a
flow'
ofopenchannel
'
u, " po.ti"al textbookon *re fundamentals
-ur-*"fi
to ard.intheunderstandproblems
example
worked-out
;;;h'rp";"ontains
form
Wherepostibt"'tolution'aregiven.indimensionless
ins of thetextmaterial.
and
problem
the
of
iniuiJu'"una.''ondingof thephvsics
llt#n"il ;;il;;
chapeach
of
the
end
of its solutionovera widerangeof variablesAt
thebehavior
in thechapteras
of the.material
thatinvolveapplication
;" ptesented
;;; ;;;;;i;
In some
material'
the text
""'U "t ti"J* "'-pf"tation of furtherrarnificationsof
by
presentation
and
i"do.utoryresultsaregivenfor datareduction
;;;#;;;i
verify text material'
studenb
"'""d; to experimentally
of it"ttctional andresearchmaterialsdevelopedover
out
bJ;;*to*n
in openchannelflow andsedcout'esequence
*J "i"A in a g'adoate
**r"iy"-.
thatI havetaughtat the
eduiation.course
i**io't .pon ^t *ell asii a continuing
of its uniquefocus on fundamentalsas
6'""r *iJi i*" of Technology'Becau-se
resultsas well as numericalanalvsis'this
ffiiuuJ *fi;;il'-*J"*pJtl-""ttl
ffitffi;;:
book shouldfill a niche b€tweenexhausiivehandbooksand purely academicuea_
tiseson ihe subjectof open channelhydraulics.
I am indebtedto more peoplethan t can enumerateherefor the completionof
..
this project.My initial motivationfor preparingfor an academiccareerin hydraul_
ics datesbackto a keynoteaddressthatI hearddeliveredby HunterRouse,who was
an accomplishedorator as well as writ€r, at a conferenceheld at the Universitv of
Iowa. The subjectwas the careersof famoushydrauliciansincluding rheir foiLles
as well as achievements.I later graduatedfrom the Universityof Iowa under the
lateJackKennedy,who was a continuinginspirationto a strugglingph.D. student.
I am much indebtedto the continuingencouragement
given by BJn C. yen at the
Universityof Illinois, where I receivedmy B.S. and M.S. degreesin Civil Engi_
neering,and Fiward R. Holley at the Universityof Texasat Austin over the course
of my careerC. SamuelMartin hasservedas mentorandcolleaguefor many years
at GeorgiaTech.The encouragement
and researchcollaborationof my coileague
Amit Aminharajahhas been invaluible. I owe much to the previous treatiseson
openchannelhydraulicsby Ven Te Chow and F. M. Henderson,as do many olher
authorsaswell as practirioners.Reviewcommentsby JohnnyMorris, Larry Mays,
and Ben C. Yen, and suggestionsby EdwardR. Holley have led to an improved
manuscript,although I bear the responsibility for any errors or shortcomings that
remain.I expressmy gratitudeto Mark Landersofthe USGSfor locating and providing copiesof the river slidesby Bames.
My students have been a continuing source of motivation for me to try rc
explain complex aspectsof open channelhydraulicswirh clarity. I have leamed
much from their curiosity and probing questionsabout the details of o;rn channel
flow phenomena.
Finally, I am forever indebtedto my wife, Candy,whosepadence,love, and
s-upportbrought me through this project, and to my grown children, Geofrrey,
Sarah,and Christy, through whoseeyesI continually seethe wodd anew.
Torhememoryol my brorherTim ( 1949-1998),
entlangdem sichewig windendenStrontdesI'ebens'
Reisegenosse
C]IIAPTI]R I
BasicPrinciples
t.l
INTRODUCT'ION
O p e nc h a n n ehl y d r a u l i ci s r h es r u d yo f r h ep h y s i c o
s f f l u i df l o w i n c o n re ya n c e si n
whjch the flowingfluid formsa freesurfaceand is drivenby gravity.The primary
cascof inrerestin thisbookis waterastheflowing fluid ha!ing an inlerfaceor free
surfaceformcd*,ith theantbientatmosphcre,
but lhe basicprinciplesalsoapplyto
othercasessuchasdensity-stratified
flows.Naturalopenchannels
includebrooks,
streams,rivers,and estuaries.
Anificial openchannelsare exemplifiedby storm
sewers,sanitarysewers,and culvens flowing partly full, as well as drainage
ditches,irrjgationcrnals,aqueducts,
andflooddiversionchannels.
Applications
of
opcnchanneihydraulics
rangefrom the designof anificial channels
for beneficial
purposes
suchasirrigation,drainage,
watersupply,and wastewater
conveyance
to
the analysisof floodingin naturalwaterwaysto delineatefloodplainsand assess
flood damagesfor a flood of spccifiedfrequency.Principlesof open channel
hydraulicsalsoareulilizedto dcscribethetransporland faleof environrnental
contan'linanls,
includingthosecarriedby sedinrcnts
in morion,as *ell as to predict
flood surgescausedby dambreaksor hurricanes.
1.2
CIIARACTERISTICS OF OPEN CHANNEL FLOW
Althoughthe basicprinciplesof fluid mechanicsare applicableto openchannel
flow, suchflow is considerably
morecomplexthanclosedconduitflow dueto the
frce surface.The relcvantforcescausingandresistingmotionandthe inefliamust
form a balancesuchthatthe frcc surfaccis a strca',rlinealongwhichthe pressure
is coDsl.lr)t
andequalto almospheric
pressure.
Tltis extradegrceof freedomin open
2
: B a s i cP r i n c i p l e s
C B P T T - :l R
chanacl flow means rhat the flow boundariesno longer are fixed by the conduit
geometry, as in closed conduit flow, but rather the free surfacead.justsitself to
.rccomnrodatethe gi\ en flow conditionr.
Another importantcharacteristicof open channel florv is the extremevariability encounteredin cross-scctionalshapeand roughness.Conditions range from a
circular gravity scwerflowing partly full to a natural river channelwith a floodplain
subject to overbank flow. Roughnessheights in the gravity sewer correspondto
those encounteredin closedconduit flow, while roughnesselementssuch as brush,
vegetation,and deadfallsin natural open channels make the roughnessextremely
difficult to quantify. Even in the case of the circular gravity sewer,resistlnce to
flou is complicatedby the changein cross-secrionalshapeas the depthchanges.In
allur ial channels,the boundaryitsclf is movable, giving rise to bed forms that pror ide a funher conlributiunto florr re.istrnce.
Becauseof the free surface,gravity is the driving force in open channeltlow.
The ratio of inertial to gravity forces in open channel flow is the most irnportanr
governing dinrensionlessparanreter.It is called the Froude number, deltned Lty
F: - Y*
(sD)'-
(l.l)
in which V is the meanvelocity,D is a lengthscalerelatedto depth.andg is gravIn someinstances
the Re1'nolds
numberalsois imponant,as
itationalacceleration.
in closedconduitflow, butoneof the few simplifications
in naturalopenchannels
is the existence
of a largeReynoldsnumberso that viscouseffectsassumeless
imponance.Flow resistance
in this casecan be dominatedby form resistance,
pressure
which is associated
with asymmetric
disfibutionsresulting
fromflow separation.The success
Manning's
equation
in characterizing
of
openchannelflow
resistance
in factdepends
on theexistence
of a Reynoldsnumberlargeenoughthat
the Manning'sresistance
factoris invariantu irh Reynoldsnumber.
1.3
SOLUTION OF OPEN CHANNEL FLO\Y PROBLEMS
The complexitiesofferedby open channelflow often can be dealtwith througha
combination
of theoryandexperiment,
asin otier branches
of fluidmechanics.
The
basicprinciplesof continuity,
energyconsenation,andforce-momentum
flux balancemustbe satisfied,
but we often mustreson to experiments
to completethe
solutionof theproblem.Theresultingrelationships
canbe quitecomplicated,
espegeometryis considered.
cially whenthe variabilityof thecross-sectional
past,the designof openchannels
ln the not-too-distant
wasachieved
with the
aid of numerous
nomographs
andgraphicalrelationships
because
of thenonlinearity of thc govemingequations
combinedwith complexgeometry.
More extensive
analysisof unsteadyflow problemsor gradually variedflow problemsassociated
with river floodplainsrequiredmainframecomputers.Presently,
the proliferationof
personalcomputersandengineeringworkstationshasprovidedmuchgreateraccessibility and flexibilityfor simpleas well as complexproblemsin openchannel
I 1 B a s i cP r i n c i p l c s l
CtlAPrF-R
\ \ ' i t hi m m c d i a l ef c c d b a c ko f r c s u l t s
h l t l r a u l i c s .P r o g r a m sl h a t a r e l r u l ) i n l c r a c t i ! e
T h e h 1 d r a u l i ec n p i n c ccr a n
i n t h e f o r n l o fs c r c c n g r a P h l c s r ' t n t ' \ - * r i r t c n u i t h e a s e
3 n d t h e i r i n l p l i c a t i o n si n a c o m p l c l e l )
i " r a . u i g " , "a * i d e a r r - a yo f d c s t g ns o l t t l i o n s
O n t h e . o t h e rh a n d 's u c h
r n o d ci n t h c n t o d c r nc n g i n c c r i n gr r o r t s t a t i o n
in(cractive
o
a p p l i c a t i o n sf a c c c p t e dp r o g r a r l st h a t
. l " r a o i u r a , o t n . , i t n e sI c r d st o n r i s i n f . r n t e d
10 flcrsonalcomptrlcrs
havc bccn lran\Poned from lltc tnainfrlmc
1.4
PURPOSE
n u n l c r i c atl c c h n i q u c sf o r t h e s o l u t i o n
T h c t h ! ' m eo f t h i s b o o k t s t o p r c s c n ln r c d c t n
n":^o:::l:',::.i:,ii",...,lli"l,ir'l'r"il'*r:::i:i;l:::'JT
"i'"p."'ii-".r
exPcnmclr
well as to en'lphaslze
transponin ailub1'sedinrent
;;;t. Tt; proii.. or u variablebed surfacccaused
of
aPplication
the
placcdon
.'ii .f,r"".it is (reatedas well ln addition'focusis
probflow
oinu,o -"tr"nitt ro the formulationof .pen channel
;;'';;;i:';i;:
modelsnow widely
ano limitationsof the nurnerical
lems.so thatthe assumptlons
andnumerexperimenta!'
of theoretical'
;;;il;;i" "r" rnadecleai The cornbination
become
has
that
to openchannelflow providesa synthesis
;;;h-",q;;plied
thehallmarkof modernfluid mechanics'
1.5
HIS'[ORICAL BACKGROUND
of hydraulics
relieson the excellenthisloricaltreatment
The followingdiscussjon
f
u
r
t
l
r
e dr e t a i l s '
f
o
r
i
s
r
e
f
c
r
r
c
d
r
e
a
d
e
r
t
h
e
"'v R o u s ea n dI n c e( 1 9 5 7 )t,o w h i c h
b
has
channels
opcn
in
\tater
of
cotrvcyance
the
;ro; irr" Jt*i "i.i"irization'
andMesopolamiasinigationfor thellSyptians
becnusedto mcetbasicneeds,sLrch
in theMiddle
U'1ry:lt
for
disposal
*istt
and
ans,watrr supplyfor theRonlans'
ln
transmission somecases'
discase
,og;t,;irh ,ttl'ail"strousresultsof waterborne
while in otbersnaturalriver channels
artificialopen channelswcre ton'trutttd'
\""'e r eu r i l i / e dt o c o n \ c y$ a t c rJ n d \ \ a s t e s '
throughcanals
ur.d u du* for waterdiversionandgravityflow
d;;;;pii^"t
developedcanalsto
Mesoporami.ans
to ,rirtnuu['iut", from the Nile River,andthe
rivers'but thereis no recordedevi.
transferwater from the Euphrat€sto the figris
principJesinvolved'The Chinese
i""i. "f""y ""a"tstanding of the theoreticalflow
for prorectionfrom.flooding several
are known to havedeviseda systemof dikes
of watersupplvpipesandbrickconduitsfor drainage
Evidence
,h;;;
;;;;;g"
of
valley The success
y;rs B.c.hasbcenfoundin int indut River
;;;;i000
onlv'
likelv the resultof experience
;;;; ;;tiy,;;;""tive hvdraulicworkswas
water from springsto distribution
transport
to
were used
Romanaqueducts
by masonry
supported
masonry.canals
rvcrereclangular,
The aqueducts
reservoirs.
in longitudinal slope.The
"r.rr.r, ""0 they conforned to the n,iural topography
areaof flow
is the cross-sectional
waterdischargein the aqueducts*ut 'neu'u"d
4
C H A p T T Rl : B a s i cP r i n c i p l e s
u,ith no regard for the r elocity or slopc producingrhc velocity.Alrhough rhe exis_
t e n c eo f a c o n s e r v a t i opnr i n c i p l ew a sr c c o g n i z e dt .h c c o n s c r v e d
q u a n t i t ro f v o l u m e
flux was misundcrstood.Yct, these itqueductsserved their .ngin".r,,,g purpore,
albeit inefficiently and uneconomicallyin nrotlernlerms.
The philosophicalapproachof the Grceks toward physical phenomcnawas
rcvived by the Scholasticismof the N4iddle.{ges, and it renrainedior L_eonardo
da
Vinci to introducethe exprcrimcnlal
rnethodin open channelflow during the Renais_
sance.konardo's prolific writings includedobservations
of the velocitv distribution
in riversand a correctunderstanding
ol the continuityprinciptein srcams with nar_
rowing width. Someearlt experimental
resultson pipe andchannelflo*, $ ere rcponcd
by Du Buat in 1816,but rhe cxperintcntalwork on canalsbegunby Darcl,ard com_
pletedby Bazinin the late l9th centurv,and Bazin'sexperimentson weirs.\\ cre unsur_
passedat the timc and remainan enduringlegacyto the experimentalapproach.
The problcm of o;xn channclflow resistancewas recognizedas imponant by
many engineersin the I Sth and l9rh centuries.The work of Chezy on fl oq, resisrrnce
beganin 1768,originaringfrom an engineeringproblemof sizing a canal to deliver
water from the Yvctte River to paris. The resistancecoefficieni attributed to him,
however,was introducedmuch laterbecausehis work dealt only with ratios of the
independentvariablesof slopeand hydraulicradiusto the l/2 power in a relationship
for velcrity ratiosin difrerentstreams.His work was not publisheduntil the lgth century.The Manning equationfor openchrnnel flow reiistarrce,
aboutwhich much will
be said in this book, has a complex historical development but was based on field
observations.Thc Irish engineerRoben Manning actually discarded the formula
becauseof its nonhomogeneityin fayor of a more complex one in l gg9. and Gauck_
ler in 1868preccdedManning in introducinga formulaof the type that no$. bears
the
nameof Manning.
The theoreticalapproach to open channel flow rcsts on the lirm foundation
built by Newton, Leibniz, Bernoulli. and Euler, as in other branches of fluid
mechanics;but one of its early fruits was the analyticalsolution of the equationof
graduallyvariedflow by Bressein I860 and the conect lormulation of thi momentum equationfor the hydraulic jump, which he attributedro the lg3g lecturenotes
of Belanger.In addition, Julius Weisbachextendedthe sharp_crested
u eir equation
in l84l to a form similar to that usedtoday.By the end oflhe lgth centurv.manv
of the elementsof the modem approachto open channelflow, which inclujes borir
theory and experiment,had beenestablished.
The work of Bakhmeteff, a Russianemigre to the United States.had oerhaos
the most imponant influence on the developmentof open channel hl,draulics in
the early 20th century. Of course, the foundations of modern fluid mechanics
(boundarylayer theory, turbulentvelocity and resistancelaws) were
beins laid bv
Prandtl and his students, including Blasius and von Kiirmiin, but Bakh-meteffis
contributionsdealt specifically with open channelflow. ln 1932, his book on the
subject was published, based on his earlier l9l2 notes developed in Russia
(Bakhmereff 1932). His book concentraredon .'varied flow" and introduced
the
notion of specific energy, still an imponant tool for the analysisof open channel
flow problems. In Germany at this time, rhe conributions of Rehbock to weir
flow also were proceeding,providing the basisfor many further weir expenmenrs
and weir formulas.
C l l A p r F RI : B a \ i cP r i n c r p l e s5
B y t h c n r i d - 2 0 t hc c n t u r yn. l i l n ) o l t h c g r i n s i n I n o w l e d g ei n o p c nc h l n n c l f l o w
h a d b c c n c o n s o l i d a t c da n d c r t c n t j c Liln t h c b o o k sb y R o u s e( 1 9 5 0 ) C
, how ( 1959),
a n d l J c n c j e r s o(n1 9 6 6 ) .i n r v h i c l cr r t r ' n s i v cr c f c r c n c ccsa n b c f o u n d .T h e s eb u l k s \ e t
t h e s t a g cf o r r p p l i c a t i o n so f n t o c l c nnr u n t c r i c aal n a l y s i st e c h n i q u cas n dc x p c r i n r c n I a l i n s t n r n r c n t a t i ol no p r o b l c o r so f r ' f . ' n c h l n n e l f l o * .
1.6
DI.]FIN IT'IONS
I n a s t c a d yo p en c h a n n c l l o w . t h c d c 1 l l ha n dr c l t , , - i t ya t a p x ) i n dl o n o t c h a n g ca s a
f u n c t i o no f t i m e . I n t b c n r o r eg c n e r i rcl a s co f u n s t c a d yf l o q . b o t h v c l m i t y a n dd c p l h
v a r y u i t h t i l r e . a s i n t h e c a s eo f t h e p r s s a g co f a f l o o d $ a r c i n a r i v e ra s s h o w ni n
Figure L la relative to a fired obscrrer on lhc rivcrbank.Thc changein rc'lrn--it;and
(a)
(c)
Rainfall
TITTTTTTTT|ITTT|N
Q
(d)
r
I
(e)
F I G U R I :T . T
T)pesof L,penchannelflow: (a) unstcidy;(b) stcady,
riniform:(c) steady.
gradutllyvaried
(GVF).tn,lsleady,rapidly!aried(RVF)i(d) uns{ead}'.
rapidll rrried: (e).paliall}varied.
6
C H A P T E Rl : B a s i cP r i n c i p l e s
deplh in a large river nlay occur so gradually and over such long distancesthat the
obsen er can seeonly a gradualrise rnd flll of rir er stlge. If rhe flood wave results
from a dam break,on the other hand,an abrupr changein depth and velocity and a
distinct u'avefront or surgemay be observed.In the former case,only ncarthe peak
of the flood wave could the florv be consideredapproxirlately steaoy,or quasl_
stead). allowing steadyflow analyses.
Spatialvariationsin velocityand dcpth in the flow directionare disringuishedbv
the tcrms uniform nnd nonunifornt. In a uniform flow. the mean crosi-sectional
velmity and dcpth are constantin the flow direction. as shown in Figure l.lb. This
flou conditionis difficult to createin lhe laboraroryamdrarcly occursin the field, but
oflen js usedas the basisfor opcn channcldesign.It requircsthe existenceof a chan_
nel of unifornrgcometryand slopein the flow direcrion;that is. a prismaricchannel.
The nonunifornrflow conditioncan be divided inro tuo rypes:gridLrallyvariecland
rapidly varied.Gradually varied flow is nonuniform flow. but the curvatureof the
free surfaceand ofthe accotnpanyingstreamlinesis so slightthatthe transvcrsepres,
sure distributionat any stationalong the flou can bc approximatedlr hl,tlrosiarrc.
This assumptionallows the flow to be treated\{ith one-dimensionalfonns of the
gor eming diffcrentialequationsin which we are concemedwith variationof the flow
lariables in the flow directiononly. Fortunatel\'.most river flows can be feated in
this ntanner.Rapidly variedflow, on the other hand, is not amenableto this approach
and often requircsapplicationof the monentum equationin control volume ftrm as
in the hydraulicjurnp or a two dimensionalformulationof the gor eming differential
equationsas in the highly cunilinear flow over a spillway crest.Examplesof gradu_
allv raried and rapidly variedflow are shown in Figures l.lc and I.ld.
Spatiallyvaricd flow really is a classof nonuniform flow but oues irs nonuni_
formiry to Iariation in the flow dischargein the directionof motion as well as to an
inrbalanceof gravity and resistingforces.Examplesof spatiallyvariedflow include
side channelspillwaysard continuousrainfall additionsto gutter flow, as snown rn
Figure 1.1e.
1.7
BASIC EQUATIONS
Thebasicequations
of fluidmechanics
areapplied
to openchannel
flowwrrnsome
modificationsdue to the f'reesurface.These equationsare the continuity.momen_
tum. and energyequations,which can be derived directly from the Rcynoldstrans_
pon theorem applied to a fixed control volume as shown in Figure 1.2a. The
Retnolds transporttheorem is derived in mrnr elementaryfluid mccnanrcsrextbooks (Robersonand Crowe 1997:White 1999) and is civen bv
d
dr
B
d
I
dr
t
b p d Y + I b p ( V . nd
)a
I
(1 . 2 )
inuhich8:systemproperty:/:time;
b : r h e i n l e n s i v ev a l u eo f B p e r u n i t r n a s s
nr, dB/dm. p = fluid density;V : volume of rhe control volume (ca); V : veloc,
C l l ^ f l t - R l : B : l s i cP r i n c l p l c \
Control
(b)
(a)
F - o il
r
t
i
sEc.c-c
c <-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 - fo@.s..r'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<lgrarn roughnesshcighti and a = constant.
(r') Find the ratio of the maximum relocity. *hich occursar the free surface*.hcre
: = the depth,]0, to the mean velocity for a very wide channel.
(b) Calculatethe valuesof the kinetic energyconec(ionco€fficienta and the
momcntufilflux correctioncoefficientp for a very wide channel.
1.9, An alteniativeexpressionfor the !elocjty djstributionin fully rough, turbulenl flow is
S i t e nb y r h e I o g a r i t h n r rdci . L n b u t i o n
r.,
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<c
(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<lbe formedin a trapezoidal
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
n)
r
N
o
o
rO
o
(o
O
@
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-<l
charges:andthebestperformancewasobtainedforthediagonalinterface'A
<D
regressionequationwas proposedfor estimationof the index as a function of
floodplain depth' and
floodplain,
and
channel
main
between
veiocity difference
methodfor
calculation
a
discharge
proposed
(1993)
also
floodpiainwidth. Ackers
adjustdischarge
a
suggestedHe
data
Wallingford
the
using
co-pound channels
conthe
full-channel
of
the
ratio
as
defined
coherence,
pends
on
de
meni factorthat
of
weighting
perimeter
with
unit
a
single
as
treated
(with
channel
the
veyance
the
subby
summing
calculated
conYeyance
total
to
the
boundaryfriction factors)
Four differentzones were definedas a function of relative
sectionconveyances.
depthlratio oi floodplainto total depth)with a differentempiricalequationfor dislimchige a justmentfor eachzone.In both methods,theregressionequationsare
laboratory'
in
the
observed
variables
experimental
of
ited io thi range
An altemitiveappioachto obtainingthe dischargedistributionhasbeenthe use
Ervine
of numericalanalysisto solvethe govemingequationsWark,Samuels'and
equaNavier-Stokes
(
lhe
depth-averaged
used
l99l
Knight
and
)
( 1990)andShiono
lateral
distrithe
for
solve
to
channel
prismatic
a
in
flow
uniform
iion, ior rt"udy
of
bution of veloiity. Their approachrequiresspecifyingthe lateraldistribution
threeto
the
closuremodel
t-e
turbulence
(
a
applied
1094)
Pezzinga
eddyviscosity.
secdimensionali3l) Navier-stofesequationsfor steady,uniform flow to Predict
ondarycurrentsandthe lateralveloiity distribution'He showedthatusingthe diag-
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
<r)
a
42
Angleof Repose,d, degrees
(b) Recommended
Channels
Side Slopesof Trapezoidal
FIGURE 4.I3
Angle of reposeand recommendedchannelside slopesfor rock rjprap (Andersonet al.
r970).
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<l/6.Theroolsare.\,/h:l098and0.0ll5.frorn$hich_v,=3.29
m ( 10.8 ft) and (1.03.15
m {0.1 l3 li). The correspondirrgdischargescome from lhe rela_
tionship bcllveenllow rirteper unil of width q and r., for rectangularchannels:
./. \{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.
152
- 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.
C H A p T E R. 1 : U n i f o r mF l o l r
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<I
(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<tngchannel. We explorehow long this realh, is in the next section.
This exanrplenegiecledthe approachlelocity head and the channelentranceloss. but
thesecan be addedeasily \4ithout changingthe sotutionprocedure.
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<tor-t offedor tnetltods.thrt esscntially are explicit also can be applied to the problen posed in this \\'a)'.r,'ith the
depth unknown at specifiedlocationsalong the channel.
Regardlessof the nunerical solutiontechniquechosenfor solutionof thc cquation of graduallyvaried flow, \re will assumethat the slopeof the energygrlde line,
5", can be evaluatedfrom Manning's or Chezy's equationusing the l@-alvalue of
depth. Essentially,the flow is assumedto behaveas though it rverelocally uniform
for the purposesof evaluatingthe slopeof the energygrade line. Effectsof nonunicoefficient.but the condition of gradually
formity can be lumped into the resistance
varied flow still must be satisf-red.
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
<D
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 ) .
=.?
g".9
r.E
6 Z
5-
3-
E
)
-
:
; f
g
i
:
a
3
a
. O
c
!
t
J
a
-
-
t
t
:
f
!
t
t
i
!
;
-:
E
6
=i
d
.
.&,
t
t
i
l
=i
5
.
z
;
a
.
l
5
&&.
t
.
;
a
V
!
a
a
z
!
i
2
i
;
4
:
V
=
a
&.
;
A
i
:
.i.
221
-t r .: .-r r: 2
.t c !i 9 r 1
I I r
- - 3 - -
q. q
I 9 -
!
r---,:
r: c r ir-c
r:-r
- - a i - 3 - . -
rt ar
t
i
-
-
c -
-
-
! 3 ; EE EEE
e E - re e ? c v r e c q ! v rq n n q o q q "- cEc g
o - o c o o c c )
-.c
- . 0
tl
:
- '2
a=?oti
qdfl*:
.izi.!a
i-
- E4ei4*
; r o r f . =r : : . !
z9 9== ;* i+ ! =; * i - i
!x;Ee;
a
= + v =_i" 76 ::qyi , .
9
e : - = Y q E = . 9 ! E
- i d E i ; : : Z - . - . i i
- i o > r 5 " > r 6 - - 9
3" .9
6 i
o u t
e33E
Eii4
i4n-a
€E36
brgS
I= ji -= ad $. ::u Sn :u :u at: i -= . *= ,E 1a . .- ,-. .. E= ,-11=3- +. 1!: . :+: +- :- :uIEGgEdg
Ei l uzzE zzE EgX; ' L ' i"!:E
i :i €a ;EEE9 9 ;
;
i
A
;
i
E
Z
)
;
;
Eei i i Ff r =3i ' ;
s ; E 3 , r3is
.a
>
-
o
E
:
U
U
U
€
€
I
G
g
9
:
F
o
;
;
Z
a
:
o
F
222
c
3
E
>
o
F
k
-
E
E
0 -
9
E
=
-
9
;
h
k
o .
9
t
.s
:
r!
A
--!
A
a
C HAprl-tR6: t.lydraulic
Structures 223
Walersurface(W.S.)
w.s.
FIGURE 6.I2
Typesof outletconlrol(FederalHighwayAdministration
1985).
Outlet Control
Typesof outletcontrolare shorvnin Figure6.12.Flow condition(a) is the classic
full-pipeflou in whichpressure
flow occursthroughout
the banel.In flow condition (b), the outletis submergcdbut the inlet is unsubmerged
for low valuesof
headwater
because
of the flow contraction
at the inlet.The outletis unsubmereed
in flow condition(c),but theculvenstill flowsfull dueto a hish headwater.
ln tlow
221
E R 6 : H l d r a u l i cS t r r t c t u r e s
CHAPT
partly full ncar
coldition (d), the ou(let not onl)' is unsubtncrged'thc barrel flo\\ s
thcoutletandpassestlrroughcriticaldepththcre.I-.inally,inflorrcondition(e).both
flow that is subthe inlet and outlet are unsubtnergcdand we havc open channcl
while
critical on a mild slope Flow conditions(a) and (b) are USGS llow Typc:1'
(e)
for
conditions(c) and (d) can be consideredUSGS flow Type 6 Flo$ condition
uhether the
on
depcnding
or
3'
2
Type
flow
USGS
is
cither
flow
op"n.h"nn.i
than crilical
downstrcamcontrol is critical depth(M2 profile) or a tailwatcr Ereatcr
(a)' (b) (c)'
conditions
llow
next'
shown
As
respectivcly
profile),
(M
or
M2
I
depth
condition (d)'
,nl (i) ,tt can be rreatcd as full flow with sonte adjustnent for
with submerged
Hence,we refer to theseflow t1'peshere as OC-l for outlet control
so it is clasinlet'
unsubnrerged
has
an
hand'
(e),
other
the
on
condition
inlet. Flow
sified OC-2.
Flow conditions (a)' (b)' and (c) (Type OC- 1) all are govcmed by thc cnerg)'
\ \ r i l t c n f r o n t t h c h e r d u u l c rl o t h c t a i l \ \ J l c r :
c(ruf,lion
Hw--7w-sor+ (r . r,. th) #
( 6 r. 5 )
=
slopelL =
in which IW: tail\\'aterdepthrelativeto the outlet inverti S0 culven
=
factor:
friction
culven length; K" - entranie loss coefficient;/ Darcy-Weisbach
culven
=
and
culvert cross-scctionalareai
0
R = lull-fl-ow hydraulic radius;A
form
in
the
writtcn
and
reananged
be
can
equation
This
discharge.
u-^
2g(Hw-Iw+s/-)
( 6 .l 6 )
on therighthandsideof Equation6 16is
in thenumerator
The termin parentheses
of theheadin elevations
it
is
thedifference
because
H"u,
calfedtheeiectit'e heur1,
(H
6 12)can
Figure
in
H.r
based
on
nomographs
waterandt;lwater. Outletcontrol
emphabe
It
should
1985)Administration
(Federal
Highway
be foundin HDS-5
6 16
6
15
and
in
Equations
appears
slope
culvert
why
the
sizedthattheonly reason
of
the
culven
the
invert
to
relative
HW,
head.
the
of the d.finition of
is because
througha
inlet.As long as the effectiveheadis the same,the full-flow discharge
slope'
the
barrel
of
regardless
the
same
will
be
culvertof specifiedlength
is written in termsof ManTtre treaOloss tenn in Equation6.l6 sometimes
in whichp/4R is replaced
equation,
ning'sequationinsteadof theDarcy-Weisbach
as follows:
L
'4R
2gn2L
t,: R4 r
(6.17)
in which n : ManninS'srt valuefor full flow, andK^ I 0 for SI unitsand l'49
for Englishunits,as in Chapter4. Typicalvaluesof Manning'sn for culvertsare
shownin Table6-.4.
C l l ^ p T r ' R 6 : llydftrulic Stnlcturcs
225
1tBt_E 6-.1
Rccomnrended
I\Ianning's n values for selected conduit-s
Tl pe ofconduit
\\'all and joinl description
\lanning's a
Con.r.re pipe
CclodJoinls,\mrxnh $i{lls
CoodjLrinrs.
rou!h walls
Poor.joints.
roush!\ all\
0 . 0 1 l. 0 . 0 t 1
0 . 0 1 .01. 0 r 6
0 . 0 r 60 . 0 1 7
Con.rr'lebox
GoodJoinrs.\m(xnhijnishcdralls
Poo.joinrs.rough.unlini\hed\r alls
0.012
{)) 0l5
0 . 0 r 10 . 0 1 8
('()rrusrredmelalpipesandboxes.
lnnular corRrgrtions
2 i by 1 in. corrueations
6 b) I In.corru-qalions
5 b) I in. corrugarions
I by I rn.conxlalions
plrte
6 b) I in. slrlrc{ural
plate
9 b1 l I in. structural
0.02711.021
0.0250.022
0.0260.025
0.02rJ
0.02?
0.015--().013
0.0170.033
( i r r r u - ! 3 t e d m e t a l p i p € s .h e l i c a l
2 i by j in. corrugarions.
2,1in.
plite ridth
0.0t2 11.02.1
at ll in. spacing.
] by.,lin. recesses
g{x loinls
0 . 0 1 20 . 0r 3
c o r r u ! a l i o n s . f u l 1c i r c u l a r f l o r
Spiral rib nlelal pip€
i - a r . . e D . r a i r o m F e d e r aHl i C h w a A
y d m i n i s r r a r i o( 1
n9 8 5 ) .
Valuesof the entranceloss coefficientfor outlet control are given in Table6-5.
The value of K" for a squareedge in a hcadwallis 0.5, while for bevelededgesand
the groove end of concretepipe culverts,K" = 0.2. On box culvertswith a square
edge. a small reductionin K" to a value of 0.4 is obtainedfor wingwalls at an angle
of 30'-75' from the centerlineof the banel; otherwise,wingwalls have either no
effect for concretepipcs or a detrimentaleffect if constructedparallelto the sides
of a box culven.
The flow condition(d) in Figure 6. l2 actually requirescomputationof the subcritical flow profile from the outlet to the point where it intcrsectsthe crown of the
culvert. Numerousbackwatercalculationsby the FHWA, however,led to a simpler
procedure for manual calculations.A full-flow hydraulic grade line is assumedto
cnd at the outlet at a point halfway betweenthe crirical depth and the crown of the
culvert. Cy.+ d)/2, and is extendedto the inlet as though full flow prevailedthrough
the entire length of the culven. Then the full-flow equation,Equation6.15, can be
used to calculatethe head-discharge
relationwith flV replaccdby (),. + d)/2. If the
tailwater is higher than (,". + Al2, then the actual tailwater depth is taken as rhe
value of ZW. ln computerprogramssuch as HY8 (FederalHighway Administration
1996). the water surfaceprofile for condition (d) is computed until it reachesthe
crou,n of the pipe, after which full-flow calculationsare made.Thus, it is given a
special lype 7 in additionto USGS Types I through 6, which are used in the program. Since it is a mixture of OC-l and OC-2, as defined here, it shouldbe given
its ou n designationof OC-3 in HY8.
226
CBAPTLR 6: Ilydraulic Structures
'I'ABLE
6.5
llntrancelosscocfficienf,s:
Outlct control,full or partly full entranceheadloss,where
/v,\
H ' , = K' \I 2 s il
T]-peof \lructrrreand dc\ign of eotrance
Pipe,concrete
Projectingfrom fill, socketend (grooveend)
Projecting
from fill, squarecul end
Head$allor headwallandwing$alls
Socketendof pipe(grooveend)
Square
edge
(radius= ll d)
Rounded
Mitered lo confonn to fill slope
End seclionconforminglo fill slope
Bevelededges,33.?'or 45' bevels
Side-or slope-tapered
inlet
Pipe,or pipe arch.corrugatedmetal
Projecting
from 6ll (no headwall)
Ileadwallor headwalllrndwingwalls,
squareedge
Miteredto conform to fill slope.plved or unpavedslope
End sectionconforminB10fill slope
Belelededges,33.7' or.l5" bevels
Side or slope'tapered
inlet
Box. reinforcedconcrete
(no wing' alls)
lleadwallparallelto embankment
Squareedgedon threeedges
Roundedon tbreeedgesto radiusof .r barreldimcnsion,or b€veled
edgeson threesides
Wingwallsal30'-75' to barrel
Squareedgedat crown
Crown edgeroundedIo radiusof * barretdimension,or be!eled
top edge
Wingwallat l0'-25'to banel
Squareedgedat crown
Wingwallsparallel(extension
of sides)
Squareedgedat crown
Side or slope-tapered
inlet
Co€mcient i'.
0.2
0.5
0.2
0.5
0.2
0.'l
0.5
0.2
0.2
0.9
0.5
0.1
0.5
o.2
o.2
0.5
0.2
0.4
0.2
0.5
0.'7
0.2
Sdlr"r Data irom FederalIiiShway Administrarion( 1985).
Outletcontrolcondition(e) (TypeOC-2)in Figure6.l2 requiresthecompuration of a graduallyvariedflow profile from the outlet proceedingupstreamto the
culvertixlet.This will be eitheran M2 or an Ml profile.Ar rheinlet,the velociry
headandentrancelossesfrom Table6-5 are addedto tie inlet flow depthto obtain
the upstream
headwater,
HW. Theflow profileis computed
in HY8 usingthedirect
stepmcthod.
C I r A Pr E R 6 : I l y d r a u l i cS t r u c t u r c s 2 2 1
Cn= krC,
3 . 10
d. 3.00
Gravel
2.90
0.'16 0.20 0.24 0.28 0.32
1.00
HWt/Ll
(a) DischargeCoeflicientfor HW./Lr> 0.15
3 . 10
3.00
Paved,,,/
2 . 9 0/
d-2.80
2.70
2.60
2.50
0
\
090
c ravel
0.80
I
Paved
\\
0.70
0.60
/o'i*'
2.o 3.0 4.0
HWr,t\
tot HW/L,< O.15
(b)Discharge
Coefficienl
1.0
0.50
0.6
0.7
0.8 0.9
ht/Hwr
'1.0
(c) SubmergenceFactor
6.13
I.'IGURD
(Federal
l9E5).
HighwayAdministration
for roadwayovertopping
coefficients
Discharge
Road Overtopping
When the roadway overtops,the roadway embankmentbehaveslike a broadweir is writfor a broad-crested
weir,asshownin Figure6.13.Theequation
crested
case
as
ten for this
(6.18)
Q = C,L(Hw,)t'
in which Q : overtoppingdischargein cubicfeetper second;C. = weir discharge
coefflcient;L = length of roadwaycrestin feet; andHW, - headon the roadway
crestin feet. Figure6. l3a givesthe dischargecoefficientfor deepovertopping,and
Figure 6.13b showsits value for shallowovenopping The correctionfactor tr in
of the weir by the tailwaterAn iterativeprocedure
Figure6.13cis for submergence
128
C r i ^ p r r R 6 : l l y t l r ' l l u lS
i ct r L t c t u r c s
h a s t o b c c n r p l o y c d( o d c t e r n ) i n et h c d i v i s i o n o l t l o \ \ b c t \ \ ' e e nt h e c u l r e n l n d
c r n b a n k r D eo
n vt c r f l o w .D i l l c r c n th c a d u a l c rc l c v a t i o n sa r e l s s u m c du n l i l t h c s u m o l '
flow
thc culvcn
and crubanknrentovcrllow cquals thc spccilicd ciischargc.
Improved Inlets
W h c n a c u l | c r t i s i n o u t l e tc o n t r o l .o n l y m i n i n r a li r n p r o r c n l c n t cs a n b c r n l d c 1 o
incrcasethe dischargcfor a given headwalcrelcvation. Bevcling of lhc cnlrunce
reduccsthc cntrancchcad loss.but the barel friction loss is likelv lo be the doninant head loss.Thc barrel friction Iosscan be rcducedb1 using culrcrts ftrbricated
from materialshaving lowcr valucsof N{anning'srt. but this becomesan ecL)lonric
i s s u e .O n t h e o t h e rh a n d .a c u l \ ' c r tt h l t i s i n i n l e t c o n t r o l i s a n r e n a b lteo c o n s r d e r ablc inrprorcmentin perltrrnranceby dcsign changesto thc inlct itsclf.
The purposc of inrprovedinlets is flrst to reduce the Ilow cootraction.which
increasesthe effectivc flow arca as well as decreasesthe head loss that occurs in
severecontractions.In addition,improved inlcts can include a/n//. or depression.
that increasesthe head on the throat of the barrcl. uhere thc control section is
l o c a t e df,o r t h e . a m e h c r d u l t e r e l e v r l i ' r n .
At the first level of inlet improvement,the inlet edges can be beveled.The
dcgreeof inrprovementcan be seen in Figure 6.1,1,which is a set of inlet control
3.0
Mtitercd\//
i
Thin edgeproiectinsf
I
2.0
i -
'1.0
a
z
7
'1.0 2.0
z7
4 <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-
<l<
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,
<lc'arr"n
Flo* conlraction
ratio(conlcyance,
Geomelricconrraclion
ra(io(*idlh)
Road olenoppinSele.r'ation
Pieror pilecode
Pier arearatio
Discharge
Riehtedgeof ll'aleror dghredgeof *eir
Skeq of crosssection
Sraight-lincdislance
Stasnalionpoinr,lefr
Sragnarion
poinr,righr
Sec(ionreference
disrance
Bridge openinglype
A\erage velociry
Marimum velocity
Velocity
Velcriry head
\\'eir len8lh
\\hter,surfaceelevalion
Abulmentstation,left llx
Abulrncntstalion.righl toe
L-efiedgeof Kq secrion
Righredgeof Kq secrion
}faximum stationin crosssection
V r n r m u m. r a r r o inn c r o l r5 c c l r o n
Cross-secrion
lop widrh
Cross,sectionwelredpenmeler
Var rmumelevarion
rn . ro\s .r, rion
N{rnrmum
ele!l,nn rn.ro\\ ,(.rion
-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
<<<ThePreceding
DaraRetleclThe Unconstricled"
Profile>>>
Section:APPR
HeadefType:AS
SRD:1140.000
.t.19
.125
.000
<<<TheFollowingDataReflectThe ConstrictedProflle>>>
<<<Beginning
Bridge/Culven
Hydraulic
Conrpulations
>>>
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 <peedsof moving observirs,(y + c) and
.'_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 <lurut'tcritri( r. The C2 characteristics,also knonn as bac'ktardor rrcgotivecharucteristics.
are defined by (7.21d).
a l o n gw h i c h ( 7 . 2 1 c )m u s tb e s a t i s f i e d .
The two familicsof C 1 and C2 characteristics
are shownin the -r,r plane in Figure 7.,1for a caseof subcriticalflo*. ObserverA, bcginningat point A, follows the
Cl characteristicpath to meet obsencr B, who follou'edthe C2 parh,at point P. Ar
point P, both observersnrustseethe samevaluesof depthand vclocity,even though
they got there by differentpaths and experienceddiffercnt ratesof changein their
initial valuesof depth and velocity. which they picked up ar rhe srartingpoints. A
and B. The solution for the valuesof depth,_r'",and velocity,V". at point P comes
from the simultaneoussolutionof (7.21a) and (7.21c)ar rhe posiiionr" and time 1".
determinedby (7.21b) and (7.21d). This prrrcesscan be rcpcatedlbr each pair of
C I and C2 characteristics
emanatingfrom the r axis,along u'hich initial conditions
are specified,to determinethe solutionsfor velocity and depth as well as positions
of all points P at the first setof intersectionsor time lcvels.Thesesolutionsbecome
the initial conditionsfor the next time levelsuntil the solutionis defrnedat all interior points in the r'l plane.The boundary conditionscomplele the solution for the
cntire x-t plane. ln essence,the characteristicgrid is a curvilincar coordinatesystem built as pan of thc solution processto define points whcre depth and velociry
can be obtainedin a simultaneoussolution of all four equalionsgiven by (7.21).
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<ttt. Both the physicaland chilrlclcristicpllncs are showninFigureT.l0.The .r
coordinatehts bc-encho\cn positi\e in lhc direclionol th.- ad\incinq negatirewave.The
inirial rrlue ofdr/d/ (= lr,, * c,,) :
i + 16.05 l-j 05 frls (3.9g nr/s).shown as rhe
slopcofthellrstClcharitctcrisliclhalselafatcslheloneofquietflomthenclau\cwale
regron Additional Cl charlcrcrjsticselltanitlefrom the I axis a( 0..5lu intenals *ith
slope\ givc'nb)' t7.l-la). in *hich r: is rpccilredby rhe boLrndarl,condition
cxpressedby
(7.15r. Along cachol thcscchuritclerislics,
borh thc deplh and \clocilv are conslant.wlth
the deplh, r'. spt'cificdby lhc boundary conditrcn and \elocitv. y. delelnlined fron]
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<ithecharacteristic
plane.
7.6. W-aterinitiallyflo\\s at steadystateundera sluicegate.The upsrream
flow depthis
3.0 m and rhedownsrream
flow depd is 0.30m. The stuicegite is raisedabruprly,
completelyfreeof the flowing *'ater.
(d) Find thedepthanddischarge
at rle gateafterthe gatehasbeenraised.
(b) Dctemtinerheheightandspeedof thesurge.
Stetch your resultson rhe physicalplaneand rhe characterjstic
plane.Neglecrbed
slopeand resistance
effects.
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
<t)
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<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<o: nrediangrain diameter;and I0 = depth of uniform flo*. (Note that a value of c, : 0.03,1in English units contmonly is used for
the Stricklcr constant,rs discussedin Chapter4.)
lf the grain size is such that the flow is not fully rough turbulent,then the \ alue
of r-,. is obtainedfrom the Shields diagram and substitutedjnto a Keulegan-t)'pe
equationfor velocity derived br Einstein(1950) and given by
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
<z
(10.32a)
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
<oooo8
i
I
Lrr:"16','l
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<o]or
X 1.65x 32.2
= 0.046ft/s (0.014n/s). The shearvelocity is
x 0.000885105
,,:\Gy,S
= 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<iand Wolman (1957, 1960) and Lane (1957).The significanceof the relationship in Figure 10.22is that any engineeringchangesthat result in a change in
slope;62n causemajor changesin planform of the stream.
Characteristicwidths and dcpths of alluvial streamshave been relatedto the
mean annual dischargeto the power 0.5 for uidth and 0.4 for depth by Leopold
and Maddock (1953). However, in a study of 36 stablealluvirl rivers in the Greal
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
<D
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.
--
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,
C H \ P n R I 0 : F l o q i n A l l u r r r lC h : r n n c l s
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
Prlyat€ Sub c[ilc.lcu1.t._Cll.ct
(]
0 . t-rtQ.T,€.rt
a0 r trt8o.?€xt
call Ycoq,(DI8!(), Er.Ev(), xSUaO, RxO, r{p, tqsnB, e, 90, irf, rc1, yC2, e&, Qq)
tr.tYE.Tsxt . lolrlat (tr, .lff.000.,
If ICl < 0llt.n
. .Do€3 Not Elst.
txttcl.rerct
1136
- Forrlat(rc1, .11t.000.)
txttcl.lext
Enil rf
rfIC2<0tbeE
! .Do€a llot ttL6tr
tntlc2.lart
tI!a
tr3tyc2.lcrt , toraAt (tc2, .lfl.o00.)
E d I f
tr.tQlJ,t6r.t . lorD.t (0!, '|11..)
tr.tQu.aGxt . lord.t (0O, .fff.')
ldd
S{b
Prlvat.
suD @illagut_cl tck ( )
Dlt! f, Aa Intogar
Dlr rtrlltrla
lr Strhg, strxlt Ar gtrlBg
cdbFlle . ghorot oa
. citbFlto . "t tef|!.
BtllllarD
' rr l:b.a Elt Sub
Il rtrlllcrn
r sttlllal!
txtltEara.Iaxt
ODc! .trtll6lD
tor latrut rr 11
lanrt 11, 3tgg, rP, pgUS
lorR.1!oID
4'72
B: Examples of Computer Programs in Visual BASIC
AppENDlx
tBt ut 11, DIs!(x),
lldat
EGv(x)
f,
lorx.l!9Nsol
ragut 11, xs[B(x], Rn(x)
x€xt x
clorc ,.
r atrxS
txtlg.td.t
lad glrb
Prlv.to 8u! @dlt lt-cllck
lad
EDd Aub
()
YcompModuleCode
ODttor holtclt
Dh Drsr(1001, Etlrf(100), xstB(20), RI(20)
Ycl la 5l!91., Yc2 l. slsglr'
Dl! O ,rr strgl.,
aO r! slagl.
Dlr Qu rr 8IBg1., Qr, A! thgt.,
Dh
ll8ttB la
Int€q€r'
f,P It
Yll L
alagle
llteg.l
xltltBt),
EdO. lfP, N8I'B, 0, 80, YIr tc1, YC2, Ql,' Qu)
ILEyO,
&.rb Icoq,(DISIO.
glDgl.,
glagla,
Aa slngl.
Emr.B.rl|x:rol,l.
E{^X
Ar
Di! ttr(Ix As
ws!4t It slagl.
w93 l! ghgl€'
ltg2 It stngl.,
n81 ls 3lug1.,
Di.n xg lr 8lngl6,
tr gbgl.,
FRot DEl'qx 13 slagl.
rR(rvDll
Dt! lRoItDa Is ahgla,
DlrrtAr la glDgla
E1 ts ghg1a,
rr ghgl.,
EralrT2 lr 81DgI.,
DLt! rrf,Rtll
Dl!
aI Ir
lDtag.r'
D{s
ftfR
rt
DLD rtrc
EUIX As lutsg.r,
l!t.g.r,
As gtrhg
KLf! A.
cott.t O.32.21
>. EIln(rP)
tf tlrv(l)
E|rx r !I,lv(1)
Ttoa
Inte96-,
'flail
f, la
|!xier!
IIIBAtrR
Int.g6l,
KRT A!
Iateger,
.lovetlon
!l!c
! rI
&il
f,l(If
!o!
If
r llllt(rP)
rf
. 1r Eu! ! ltlV(Xl{I{)
dde|,l
x . I fo [P 'fha
< EIIU It a
t!EV{f)
Eanf r llEv(l)
'
Xlltl
Eid
lLnt
!o!
It
U,TAttrx.t-1
f€xt
il
cbrt r.l
X
1 !o fSItB rfl[d
xAgD(,t) > DIST(II(II)
r il
tR|!8ll[f,
lnd
lB lal[
tf
x
it .
lrit
atavatloa
lor
tf
rubs.ctlo!
rh6!
rulb€r
of
braL3
la
IRTalllf,
Iatege!
lE
latagcr
AppEriDrx B: Examples
ofComputerprogramsin VisualBASIC
:. !o nP 'ftnal
polDt Dueb€r
. xsgB(ILTaATK) tih.n
KL!.X
X .
8o.
If
of
l€lt
4"t3
blrnt
DISI(X)
lxlt
aor
Enal rf
tl€xt
for
X
X .
If
DIsI(r)
1 To llP ,flad
b.*
DolDt auDbe! of light
! xSttB(lRlBr.nX) I|!d DISI(X) . DIa?(X + 1) rhoD
K R I ' K + 1
lxlt
tot
Elg.If
Dlgl(f,)
. x.$ta(IiTarNK)
Th€D
f,Rf.X
l:.lt
lor
lnd
If
l{erct x
If
EI,EV(RLI)
<! ELEV(KRT) ITe!
EI,EVBITRTOI,IJ . ELEV(XIT)
!1r.
ELEVBINKESIJ! i
ELW(XRA)
Ead Il
tfsl
r EIII
Call
+ 0.01r
ra2:
ti8€ctlon(wgl,
EUlIr
rS2,
DIS!(),
rt!€
.ttORXl
.
lrrEv0 r rgnl0,
calcul.tr
lorral
RtrO, rp,
degth
NSsr, e,
SO, srrc,
rs3 )
rr.rg3-El{rr
wa '
EIIVBAI|KFIIII.: !t!c
tnOoDEl . rc(ngr DIstO,
rCRIt.
i
8!tv(),
'calculat6
xsgBo,
crltlcel
BtfO,
d€ptb(r)
q, s0,
lrp, !rs!!,
srrc)
Qgi0/raosDEl
ItQTQETh€r
. -1r IC1 . -1 'calculat€
EICRIIl
uDD€r lrc
w81 ! EIJEVEIITTIIIJ!
t rcRIT.
WA2 r Elrtx! rt!€
C.11 Blsoctloa(wsl,
wS2, DrsTO, Er.E.'{), xsgl0,
Rr(},
S0, 3trc, ws3)
r WS3r fC2 . llJcRIr2 - EIII
ELRIIz
rcrlculatc
lor!!
trc
ll,CRIa2 . -1! yC2 . -1 ,Dolrlbly
so lrpDor yc
+ 0.01: Wa2 . ltitVBNfKEt'&:
r81 . Efir
| TCRIT.
str!
llre
C.ll
alsectlo!(fa1,
S0, rtrc,
Drsr0,
t!'JrVO, xgltE(),
tr(},
rp,
!SUa. e, _
Wg3)
t Wg3: ICl
EI€RIfl
rdl
rS2,
Np, nsOB, Q, _
: ELRITI
- ltcr
Il
11 r ILIVllNKn
DEII.IY r 0.01
1.!
- EriII)
(ELavBNxlIrIJIr
r{S I lLEvBrrNIFttLL
. 2l . DELItt!
. rcRrt.
rttc
ERottDEr lC (nS, DISIO, Et,Ev(), xglrE0, Bt[(], [p, xsrrB, e. gO, strc)
If IROIID! < fROgDll Tb.n
uDto! lrc ritb
yc
QIJ r Qg ,Caac I-odly
!o urltlDle
'o! Carc llr-oa1y
Elt
Sub
loy€t !rc rLtb !o ErLtlDl.
Enit rf
,fiEal tb€ DlxLEre lrolral€ DuDb€!
FROSDE|IX - IROUDE1
For IIER '
.
1To
100
.calcutato
yc
!+
_
AppENDIx B: Examplesof Computerhoglams in visu.l BASIC
414
t11
+ DIIIAI
rb€! llxtt lor
11 > EqI
t rcRu'
t{g r !1! ttrc
DISTO, ll,EVO,
n{oODE r lc(r8,
ll
ll
Il
XSIBO,
tJ|O,
NP, IssB,
0,
s0,3trc)
lROnDl > lRottDE|ll( .l
IROIIDEI{II( r fBOoDf
. El
tfs|}ql
lad
rf
xext
IaER
QIt / (EROqDlutX / rROuDBl)
Ql.'
utx)ar yc rith
Er1tj.pl.
I! O > Og Th.r E.tt sub 'caBc I-only
l"c
rfQ>QtTb.a
Wa1 ' WS||lx; W82. EtlX 'Ca3. tl-both
lore!
||ltd Ellp€r yc
BtrC ' rCRIlr
cetl
als€ctloa{wg1,
Ed.t
sl|b
'C$c
rll-only
1or€r yc rltb
-1
lI,CaII2 ! -1: IC2.
!1so
ElErrO,
xssa0,
R!O,
rP,
xasB, 0, _
- EIII
r tlE3: lC2 . BLRITZ
ELR.ITI
hit
t{92, DrsTO,
ws!)
ao,3trc,
yc
erltl.Dlc
rf
EDit aub
8ub als€ctlod(ltgl,
r|s2, DISTO,
Dh
rs
awsl lt
Dh
I la
l.
alEgl.,
ElvO,
xSuBO,
F$s2 ,u
atngl.,
an().
Slnglc,
Q, 80, rt!C, !qa3)
sIDg1., !2 l! gt!916
lIP, EttE,
ms3
t
l!t€g.!
Coa.t !R.0.0011
wg . wS1
lrl81 . !c(ws, DIaT(), EIJEVO. xsgB(),
Wg " WS2
ELEVO ' xsItBO,
ma2 ! FC{WS, DISIO,
It !W81 r rlag2 > 0 jlAor f![lcoq,.Prllt
!lP, llgDB, Q, 80, !trc)
- 1
RNO, trP. xSItB, q, S0, stac)
'l l[O ROO4!
I l':
Edt
Alt.b
- 1
RxO,
t o l I . 1 ! o 5 0
(FEl + ngz, l2
YlS3.
wg . na3
ws3 . Fc(Is, DIsr(), lLlv(}, rsuBo, RnO, !lP' llguB' Q, E0, .ttc) - 1
tZ.rflAlrFfg3
Illl<0lt6a
rS2 r rS3
llrel!!Z>0llhea
fl81 : rS3
Elra
gub
Bslt
Edd II
Ia lbs(ws2
Nar.t
- Itgll
< lR rh€!
t!.lt
sub
I
.lRiOR
tllTcoqr.E
gub
lst
tir.actlon
Fc(Ka,
CRIIERIOI
IOf
gaTISlIlD.
lDd
Dt!
AsuB(2o)
Drglo,
Ar glagle,
Elavo,
Psu8(2o)
x8n8o,
rr
RNo,
8laglc,
np,
lsua,
cgua(2o)
rr
Q, 90, rtrc)
siDsl.
rr
ghgl.
BAslc
APPENDIx B: Exarnplesof ComputerProgramsin visual
Db
Dt!
Dt!
Dt!
D!!
Dl!
Pr t! strgl'
R'r lt glnglc'
g1ng1''
a!
8lng1'
63
82 tr
DP lt 8bt1'
Dx tr SlDgl''
It abtrl''
It| rt thgl'
DPD! ls glngl"
8lBg1"
Ar at!91'
Dl!!tr
t! ghglc'
Trntf
Tlinxl lr glDgl"
:ntcg'r
L A!
cl!
!T l'! slngla,
AI A! glBg1.,
s1 rs 8lIrgl',
tr,Pa l. slagl.,
DY
12 at slngla,
Yl t! 8lEglc,
ghgl.,
At
B
l!
P
r l| slDgla,
R lr
il lr
slugl.,
c tt
8IDgh,
lrt6gci,
f, la
tDt.g.t,
Coart O ' 32.21
AT r O: Br . Ot c ! '
8l . Or82 'Or 83.
f,!
glstl''
0! Rt ' 0t Pr ' 0t IIPEI ' 0
a'
0: DPDI'0
0! P'O:
Ol t'
IJ . 1 To f,sug
' Olt PasE(L) ' Ot: cSUa(t') ' 0l
lrltB(!)
t
ll.rt
itcruSt'ctlo!
6vrluatloat
J t 1 rb€gtn g€@atrlc
Yl . wS - ll,Ev(l)
lot
no'
! o ! x . 2 r o N P
Y2!rE-tl,EV(K)
DxrDrgr(x)-Drsr(x-1)
Dt ' Y2 Elto Dl r Y1
Y2 t 11 tt€!
' 12
It DI . 12 ttsa lN ' tl EIs€ Yll
i
1
2
)
(
Y
1
<
0
l
t
a
It
DA.Dr.DY/{DX-If,l
l , t + 0.5 i !8 r Df
rf
DP.sqr(Dx'Dx+DrrDI)
P . l l D P
I t E . U
ll.clt
D?Dt.DpDr+DP/Dl
(Y1 +Y2) >0fb.l
( f 1 + t ? )
a : r + o . 5 . D A r
- 12) '
Dp. gc!(Dx r DI + (11
(f1 - !2))
P ' P + D P
8 . 8 + D X
If fU . 0 ttdl
DPDI ! DPDI + DP / lrlf
Elal rf
11.12
'..rrrufit
If
ttlo9€rtl€3
ta rubs.ctlo!
DtST(f,) . xgttB(it) tb.!
It l ' 0 lb.tt
lf
GDd ot
i'tch6d
subsoctlon
ra!B(J).01
PS!B(J) . 0l
csgB(J).01
Elaa
t.rbs(t)rE.Jtb3(B)
R ' h / P
R ^ (2r / 3l) / 8f,(,t)
1.a9. r.
' c
lSuS(,I) . A! PEIts(.t) ' P: CSEB(it)
r
+
B
B
I
+tr 8.I
It.lI
c.
c | ! . q f r C ! P I T P T + P
9 1, g 1 + ( C / l ) a 3 ' ( 3 t ' B - 2 1
8 2 . 3 2 + C ^ 3 / l ^ 2
( 5 1r B - 2 t
S 3 ' A 3 + C / r '
Edd
If
4'75
'
R ' D I D Y )
i R ' D P D I )
AppENDlx B : Examplesof ComputerProgramsin Visual BASIC
4'16
J ' i r +
1 . 0 !
!!d
Next
If
1
P '
0 : 8 . 0 !
D P D .I 0
:
rf
x
tbe[
rT.0
EtLt
ttllctloa
R T . I T / P I
A I P E A ' 4 2 / ( C { ' ! / A T ^ 2 )
TEsrlrAT^ 2 r s1 /el^3
(2 i Ar r aT / cT r 3 - Ar ^ 2 r s3 / CT ^ {)
T8rr:t. 82.
' t:Eltlll + aERt|2
D tI
^ 2 . B ! / ( O . AT ^ !,
lEAt|1, N,PEA. 0
Ar ^ 2)
A'
T E n r ! 2 . Q ^ 2 i D A T , D IY l Z .
'cRrrr
:rben
rt strc .
- TERlil)
!C ' 8qr(I8ru1
il|oRltr
.
!he!
stEC
lls.If
. ! c . Q /
( c T ' s 0 ^
( 1 /
2 ))
EAd If
!t!d
functlor
Data File
rCAlr, e.3 rtrrfl
of cross BoctloD,
'crountl Doiats ! atatlo!,
DuDb€l of
61e?ation
0,110
a,106
60a,106
510,:.00
582,100
588,106
1288,106
L292,rto
6 0 { , . 0 8 , 688 , .03 , L292 , .OA
gEould DotEt!,
lbar ot subsactlot
(fro
loft to rlght Iooklng dlowattr€al
't6crtlo!
,t6.Dnlac'.
of
rt lsoctloaa:
a oa tb€
left
stltLoa,
.LaL
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
>
a
b
3
o
rl
E
I!
EE
I
I E
EH
! t !I ! t !
E
'
EEH
EE EH
a rtt
.-r
{
lr>
>
a
b
3
o
rl
E
I!
EE
I
I E
EH
! t !I ! t !
E
'
EEH
EE EH
a rtt
.-r
{
lr>
in VisualBASIC
Programs
of Computer
B: Examples
4'18 ApPENDIX
dependingon the profrle classificationin order to avoid startingon the wrong profil;. The aALCULAIE PROFILEbuttonis clicked,and the profile type is classified and shown.The profile is calculatedin Sub PROFILE using the direct step
merhodby dividing the depthintervalappropriateto the particularProfiletype into
The specificenergyandslopeofthe energygradeline arecalculated
30 increments.
in Sub SPENERGY.Inter?olationis appliedto obtain the depth at the end of the
specifiedlengthof channel.The channelstationsincreasein the downstreamdirecof whetherthecomputationproceedsin the upstreamdirection(subti,onregardless
criticaDor downstreamdirection(supercritical).If normaldepthor critical depthis
approachedwithin 0.1 percentbeforethe specifiedchannellength is reacbed'the
cbmputationis stopped.The resultsfor depth,YP; velocity,VP; and momentum
function,MB are printed at eachstep,XP, in the RESUUIS picture box The output canbe directedto a printerwith the PRINT button.The programaccommodates
horizontal,mild, and steePslopes
WSP Form Code
ODtloa EC,ltctt
Dh O rs shgl..
Di! Y0 It glDglc,
DL! stiPa
Dl.[
J L
l3
glDgla,
s L
Yc .trs Ehgla,
D ls 8lng1"
b as glsglc,
rcom la
xt tr alsglr,
A!
giaglc
3tllaqr
llP l,
I!t6gGr,
Illt69.l
DL! rP(100), YP(100), v"(100),
rc(100)
gob ddcalculato-cltch(
Erlvlt.
0 - V.I (ttrtQ.I€xt )
I
s.
i
siagl'
vr1(txt8,r.xt)
val(t:.t8.!6xt)
)
a ' val ( trt!.Ier.t
v l l (t x t 8 8 . I € : . t )
!.
C.U Y0!C(Q, S, b, !, a' i0, IC)
rt y0 . 10001 It.D
- rdlocr lot 6tirt'
txtlo.f.rt
b.
Elaa
.
tr.tYo.r.xt
rld
t*trc.T6rt
laal
torDrt
(YO, '|ff.O00')
tl
. fo.Drt
(iC,
'ttf.000')
8ub
()
gub @d!Ro!It !-cllct
Pllert.
ItJ . Yal (txtx!.T.rt)
)
Ycottl . vr1 ( tr.ttc€Fr.|Iaxt
g, b, & n, r0'
Paolnt(o,
crlt
b.tPaolM.ltnt
'
t€,
t tcR..ultt.c1r
t,lcRcrult!.Pllrt
Prlst
Dlcaalult!.
lorit.lTonP
rrr'
YclotE' .trPR'
n",
x"o'
l,
cuft'
ltrPR
'
l,
tt',
' r,!t',
'v,
ftlt',
'l!cd.
YP(),
vPo,
llPo)
B:
APPENDIx
Examples of Computer Programs in Visual BASIC
fll*|.0')
xr(J) " ror"at(xP(,t),'
'ff|.000")
YP(J) ! lot&rt(YP(.t),
't*|.00')
w ( , t ) - F o r o l l ( \ t P( . t ) ,
'}tltl.0')
ltP{,t) t rotllt(nP{J),
xP(.t),
plcRosultr.arht
rlP(J)
w('t),
rP(J),
iI
[6r.t
Edit 9rr!
gub cDdlrilrt-Cl
iclr ( )
.
'
A
r
r
.
a
l
'
Prlator.loatllarla
Prlvato
Prliter.lortsize
Print6r.CulteotY
?rlDt6r.Prht
'
12
. 1500
'rtAaER ggxllc8
gtrc(3o),
PRoFrLa - rt
strPR
Prlnt€!.?!ht
Prtator.Prlnt
lrlnt€r.Prlit
sDc(20),
'D."
!t
SDc(20),
' D="
Di !eb,
'Q, cf,, ",
rb, ft 'i,
labt
'SloD6Ert
Qt sDc(s)t
b
PilBtsl.Erirt
Y0 ' lorn t(YO, '**1.000')
YC . lo'.Dat (aC, 't**.000')
It g.0Ih€0
Prht.r.PElnt
sF(20),
'y0 do€s not exltt'
spc(20),
' Yo'ft
!1sc
?rlntet.Prht
anil
"r;
Y0
tf
Prlnter,Prht
SPc(20)'
' Ic,ft
SDc(20),
'x,
:'r
YC
Prlatc!.Priat
Prht€r.P:iiat
lor 't r 1 lo
iY,
ft"
ft',
rv,
ftl'i'
'u6.
llP
x!(,t)
- ronat(x!(J),'
lP(,t)
'
lorEet(rP(t).
ff**1.0')
'||t.000')
'fl*.00')
E ror&at(vP(.t),
't*l*|.0')
uP(it) ' toxrat(uP(J),
rP(J},
sDc{20), P(J),
Prlntor.Prllt
w('I)
vl(.t)'
l@(J)
l|. xt t
Prht6!
laa
. lddDoc
8ub
Prlv.t.
Eub @dExtt-clich
()
lEd
Enil Sub
WSP Module Code
Eqtltcit
Dls O ,rs gl!g1.,
Dti ! l! Stagt.,
optloa
s lr
EiDgta,
b ls
Y0 As sin91.,
g, b, !, B, Y0' lC)
Sub IoIC(Q,
Y2 rr sldglc'
As
slagl.,
Dt! Yl
stDg16.
rC lr
ER lt
9lug1.
ghgla
!
lr
Ehgla
l.r
g
479
AppENDrx B: Examplesof Computerprogramsin Visual BASIC
480
D{4
N? l.
IDteg6!
Yl r 0.0001
12.100
!R.0.0001
f , l r 1
CaII
alslgfrdt(Y1,
xF,
rR.
O,
S,
!,
!,
yC)
Y2, n!,
!R,
e,
g, b, a,
!,
y0)
Y2,
b,
M . 0 f b . !
Y0.10001
txLt
.
Aub
Ead If
11 ! 0.0001
Y2 t
100
||!r2
ClU
lull
BISEeuor(Yl,
gub
gub PROllLl(Q,
yC, rrJ, ycolrT, srrpR, np. p{),
yp0,
V?O, UpO)
gEoL l!
glDg1.
Si-agl.€, ILIU As shgl.,
! Ar glngla,
D!! Dx t, alag1.,
Dr ts lthglc,
I la glngh,
SlCIJl l! giagb,
!1 L gtagl.
Dt.[ aSGtJ2 r! St!gI.,
Et r! Shgl.,
SrCr,BrR l! gtEgl.,
vFJ t
Sln l.,
lr| rs gttrgl.
DI! I ta lBt.gc!,
ng tr Intcg.r
rl ICOIr . 0 tL.!
YCOTST. tC
'cl.sslfy
lf IC < y0 lb.a
Dllil llot€
Drofll€!
lf Yc:ottl > y0 fh6a
.rrpR r rl|1.. XtrGU r -1r tI,lI| , 1.001 r yO
llr.If
YCo!TT < yC ftb.!
atrPR r rx3.. Igr@f:
1: Yllrx r 0.999 r Yc
lk.
atlpR . rtor. xstet r -1r yutt r 0.999 . y0
Dia lgtcw
8, b, !,
n, r0,
Ar
If
YCIIII . lC tLar
tCOlaI .
1.001
r yC
Elal It
El!a
fl
> yC thd! ,clrsslfy
.tb6D .lo9€ Diofi1€s
. .glr,
XSICtr . -1! YIlIl| r 1.001 . trc
YCON! < t0 Tt.u
.EIPR r rg3.! Xgtcf. 1t ILII!:0.999.
f0
tCOm
rtrlR
lla.If
11.!
.rrpt r .82rr xgtqf. lt ILrx. 1.001 . y0
rf !CO!|l . yC !t r YCONT. 0.999 r yC
lld
lad
Il
Il
Il
a r
,clarllfy
0 lb€!
ff
ICUTT
borkortrl
!1o9. DrofII.,
>. YC taaB
.11.. lgr@t.
-1, II.t
.rrpR.
r yCOltT + 3l
yCOlrT
. tC t!b.! ICorlT . 1.001 r tC
If
!bc
rtrpR
lad
&rd If
nS . 30
.
.t2'r
If
.lDltl.llz.
vrrlabt.a
xglef
.
1: fllrt
r
0.999 r yc
r yC
APPENDIXB: Examplesof ComPuterProgramsin \4sual BASIC
l P ! N a + 1
-
(YL
Dl
r
rf
xsrclr .
.
r
Ycolrl)
/
NS
xP(1) .
1 !bs!
= !I,
0 Et66 x!(1)
YCO$I
9PENZRG{(Y, Q, S, b, r, [, E, sEcl, vEc,, Etll
. vEl: l@(1) . Ell
! Y! w(1)
i"(1)
a1 - Er sEell ' SlGl,
fot t - 2 [.o lls + 1 ,alla6ct st6D n€tbod 1o<at
c.II
t . I + D t (
srENEasr(r,
w(I)
Y?(I) .Ir
!2 r E: SlG'.2 i
'
sEGl,BrR.0.5
uI
(12 - 11)
-
vEl,
Q, E, b, r, n, E, sEc!,
. vELt l{P(I) ! Fll
c.ll
Ex)
SECIr
( 8 E G I J 1+ s l o l . 2 )
/ (S - sEGLalR)
rP(l)'xr(I-1)+D:(
E1 . !2: SEell . SEel.2
1€rrtb
for igcclfiait
> x! lhen 'tate4)o1atlol
rf xr(r)
- xL) / Dx
- Dr r (:e(r)
. r?(tl
Ir(r)
' :q,
xr(I)
llP'I
l!:.it
tad
rf
lor
rf
'l!t.4)olatlo!
< Ol rllatl
xr(I)
.Peclfi.d
fot
l.agtb
I?(r) . r?(r) - Dr. x?(r) / Dt(
. 0t
,19111
trP.I
tn lt
lad
For
tf
t6rt
t
Ead aub
Sub AIAECTIo!(Y1,
Db
r.r1 Ir
DL!
I ll
Slag1.,
12,
rlt
Nrltxc,
l!
lR.
8lng16.
Q' g, b' r, !, Y3)
lZ A! thg1.
!T3 L St!91€,
ht6g.r
t l(Yl,
fil'llc,
Q, 8. b, D' a)
!lro[C, Q, 8, b, i, D)
tZY? . !(I2,
rl rI1 r rY2 > 0 l'beB Ertt gub
rI1
lor l.1le
t0
y 3 . ( 1 1 + y 2 l l 2
. !(Y3, rllNc,
Q, g, b, t, a)
r
l
t
3
lz.lT1
sub
rf lz ' 0 Then E.it
Yl . f3
It !z < 0 ltea t2 ' Y3 lIa.
rf rbs( (Y2 - 11) / 13) < EA It€a tsit
Fn
llarat
lhd
8ub
I
a$b
FuDctlon
!,
DLi I
R ls
F(Y, [!m|C, Q, I, b,
Ar gl.agla, P ts thgl!,
. ( b+ n . Y )
r . I
D) A!
Sllglc
slagl.,
T L
glDgl.
481
482
AppENDtx
B:
Examples of Computer
2 r r r
P . b +
programs in Visual
BASIC
g q r ( +
1 r ^ 2 )
R ' r / p
! r b + 2 . ! . y
IfIFOXCrtrtc!
t'
Q - sqr(32.21 r I
r,
0 -
^ 1,5 I r
^ O.i
!1.!
&tit
(1.{96 /
a)
. f, r R ^ (2
/
3) I I
^ lL /
If
l![al lurctloD
Erb SPEtERqr(r, 0. A, b, r. B, E, gtt J, VEIJ, nr)
p & ahgl.,
DbIAr
tl!t16,
R L
ais91.
Dl.a llt rr At!grl.,
O .ls Etrgla
O . 32,2r Xr r 1.t85
t . r . ( b + ! r y )
r y l g q ! ( 1 + r . 2 )
P - b r 2
R . l / D
&d gub
v!!.0/r
I ! |. fb r r ^ 2 I Z r ! . r r
3 / 3+Q ^ 2 / (c.f,)
r . r r 0 ' 2 | l2 . o r l . 2l
8XlL. ! ^ 2 a Q ^ 2 / (x!. : . l ^ 2 r R ^ (at / 3r))
Zl
Index
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,
t09-t4
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