bfc21103 chapter2

BFC21103Hydraulics

Chapter2.UniformFlowinOpenChannel

TanLaiWai,WanAfnizan&ZarinaMdAlilaiwai@uthm.edu.my

Updated:February2015

LearningOutcomes

Attheendofthischapter,studentsshouldbeableto:

i. Understandtheconceptofuniformflow

ii. CalculatenormalflowdepthinvariablechannelsectionsusingChezyandManningequations

iii. Determinethebesthydraulic/effectivesectionofopenchannel

BFC21103HydraulicsTanetal.(laiwai@uthm.edu.my)


OpenChannelFlow

ClassificationbasedonTime

ClassificationbasedonSpace

Steady Unsteady Uniform NonUniform

GVF RVF


Uniformflowisconsideredtobesteadyonly,sinceunsteadyuniformflowispracticallydoesnotexist.

Steadyuniformflowisrareinnaturalstreams,onlyhappensinprismaticchannels.

Weadopt/assumeuniformflowformostflowcomputationsbecauseuniformflowcalculationissimple,practicalandprovidesatisfactorysolution.


The132kmlongAllAmericanCanallinksCalifornia'sImperialValleytotheColoradoRiver.Thisnewconcretelinedsectionsavesabout3.8millionofwaterayearoveritsleakyearthenforerunner


TheconcretechannelofLosAngelesRiver(NGM,2010)


TheKlangRiver,KualaLumpur&Selangor


Inuniformflow,thenormaldepthyo occurswhendepthofwateristhesamealongthechannel.

Normaldepthyo impliesthatthewaterdepth,flowarea,wettedperimeter,velocityanddischargeateverysectionofthechannelareconstantwithinaprismaticchannel.

Thus,inuniformflow,theenergyline,watersurfaceandchannelbottomareparallel,i.e.theslopesareequalSf =Sw =So =S.

gV2

2

EnergylinegradientSf

yo

WatersurfaceSw

BottomslopeSo

2.1VelocityDistribution


Naturalchannel

0.840.820.800.760.700.620.48

Vmax

Vmax

0.52

0.500.450.400.35

yo

0.53

Rectangularchannel

Vaverage

0.6yo

Vmax0.2yo

V

y

yo

Velocitydistribution

Dependsonthegeometryofthechannelandwettedboundaryroughness


2.2ChezyandManningEquations

Twomostcommonequationsusedintheuniformflowcomputations:

1. Chezyformula

2. Manningformula

21

21

oSCRV =

21

321

oSRnV =

xo

xSRV constant=Thus,thegeneraluniformflowequation:

C =Chezyroughnesscoefficient

n =Manningroughnesscoefficient


611

Rn

C =DifferencebetweenChezyandManningformulae

Factorsdeterminingtheroughnessaresurfaceroughness,vegetation,channelirregularity,channelalignment,siltingandscouring,obstruction,sizeandshapeofchannel,stageanddischarge,seasonalchange,andsuspendedmaterialandbedload.


TheChezytwoassumptionsare:

1. Theforceresistingtheflowperunitareaofthechannelbedisproportionaltothesquareofthevelocity:

2. Theeffectivecomponentofthegravityforcecausingtheflowmustbeequaltothetotalforceofresistance.Thisisalsothebasicprincipleofuniformflowwhereuniformflowwillbedevelopediftheresistanceisbalancebythegravityforces:

DerivationofChezyequation

PLVkFf2=

sinALFg =


gV2

2

EnergylinegradientSf

yo

L

W

W sinW cos

Datum

WatersurfaceSw

BottomslopeSokV2PL

A

P


sin2 ALPLVk =oALSPLVk =2

oSPA

kV

=2

21

212

1

oSRkV

=

21

21

oSRCV = whereC =Chezycoefficient

1221 sin MMpFWp f =+ Sinceforuniformflow, 2121 and MMpp ==


Arectangularchannel2.0mwidecarrieswateratadepthof0.5 m.Thechannelislaidonaslopeof0.0004.TheChezycoefficientis73.6.Computethedischargeofthechannel.

GivenB =2.0m,y =0.5m,So =0.0004andC =73.6

A =By =1m2,P =B +2y =3m,R =1/3m

B

y

oRSACQ =

0004.031

6.731 =Q

/sm850.0 3=Q

Activity2.1


Water flows in a triangular channelwith side slope 1.5(H) : 1(V),bottom slope 0.0002 andChezy coefficient of 67.4. The depth offlow is 2.0m. Find the flow rate and average velocity. Based onFroudenumber,determinethestateofflow.

zy1

Giveny =2.0m,z =1.5,So =0.0002andC =67.4

A =zy2 =6m2,P =2y =7.211m,R =A/P =0.832m,D =A/T =6/2zy =1m

Activity2.2

oRSCV =0002.0832.04.67 =V

m/s869.0=V


AVQ =869.06=Q

/sm217.5 3=Q

gDV=Fr

181.9869.0

Fr =flowlsubcritica277.0Fr =


ChezyresistancefactorC

ThefollowingtwoequationscanbeusedtodetermineChezycoefficient:

1. GanguilletKutter

2. Bazin

Rn

S

nSC

o

o

++

++=

00155.0231

100155.023

Rm

C+

=1

87

n =Kuttercoefficient

m =Bazincoefficient


Table2.1aValuesofManningroughnesscoefficientn

Surfacecharacteristics Rangeofn

(a)Linedchannelswithstraightalignment

Concrete

i.formed,nofinish 0.013 0.017

ii.trowelfinish 0.011 0.015

iii.floatfinish 0.013 0.015

iv.gunite,goodsection 0.016 0.019

v.gunite,wavysection 0.018 0.022

Concretebottom,floatfinish,sidesasindicated

i.dressedstoneinmortar 0.015 0.017

ii.randomstoneinmortar 0.017 0.020

iii.cementrubblemasonry 0.020 0.025

iv.cementrubblemasonry,plastered 0.016 0.020

v.dryrubble(riprap) 0.020 0.030

Tile 0.016 0.018

Brick 0.014 0.017


Table2.1bValuesofManningroughnesscoefficientn


Sewers(concrete,asbestoscement,vitrifiedclaypipes)

0.012 0.015

Asphalt

i.smooth 0.013

ii.rough 0.016

Concretelined,excavatedrock

i.goodsection 0.017 0.020

ii.irregularsection 0.022 0.027

Laboratoryflumessmoothmetalbed,glassorperspexsides

0.009 0.010


Manningroughnesscoefficientn=0.020 0.022




Manningroughnesscoefficientn=0.020



(b)Unlined,nonerodiblechannels

Earth,straightanduniform

i.clean,recentlycompleted 0.016 0.020

ii.clean,afterweathering 0.018 0.025

iii.gravel,uniformsection,clean 0.022 0.030

iv.withshortgrass,fewweeds 0.022 0.033

Channelswithweedsandbrush,uncut

i.denseweeds,highasflowdepth 0.050 0.120

ii.cleanbottom,brushonsides 0.040 0.080

iii.denseweedsoraquaticplantsindeepchannels

0.030 0.035

iv.grass,someweeds 0.025 0.033

Rock 0.025 0.045

Table2.1cValuesofManningroughnesscoefficientn



(c)Naturalchannels

Smoothnaturalearthchannels,freefromgrowth,littlecurvature

0.020

Earthchannels,considerablycoveredwithsmallgrowth

0.035

Mountainstreamsincleanloosecobbles,riverswithvariablesectionwithsomevegetationonthebanks

0.040 0.050

Riverswithfairlystraightalignment,obstructedbysmalltrees,verylittleunderbrush

0.060 0.075

Riverswithirregularalignmentandcrosssection,coveredwithgrowthofvirgintimberandoccasionalpatchesofbushesandsmalltrees

0.125

Table2.1dValuesofManningroughnesscoefficientn


Grassedswale

Table2.2ValuesofManningroughnesscoefficientforgrassedswale

Surfacecover

Manningn

Shortgrass 0.030 0.035

Tallgrass 0.035 0.050


Table2.3ProposedvaluesofBazincoefficientm

Descriptionofchannel Bazincoefficientm

Verysmoothcementofplanedwood 0.11

Unplanedwood,concrete,orbrick 0.21

Ashlar,rubblemasonry,orpoorbrickwork 0.83

Earthchannelsinperfectcondition 1.54

Earthchannelsinordinarycondition 2.36

Earthchannelsinroughcondition 3.17


Calculatethevelocityanddischarge inatrapezoidalchannelhavingabottomwidthof20m,sideslopes1(H):2(V),andadepthofwater6m.GivenKutter'sn =0.015andSo =0.005.

zy1

B

Activity2.3

GivenB =20m,y =6.0m,z =0.5,So =0.005andn =0.015

A =By +zy2 =138m2,

P =B +2y =33.42m,

R =A/P =4.13m


Rn

S

nSC

o

o

++

++=

00155.0231

100155.023

GanguilletKutter

13.4015.0

005.000155.0

231

015.01

005.000155.0

23

++++

=C

769.76=C


oRSCV =Chezyvelocity005.013.4769.76 =V

m/s03.11=V

AVQ =Discharge03.11138=Q

/sm14.1522 3=Q


FindtheequivalentBazincoefficientm forthequestion inActivity2.3andcomparetheChezycoefficientsobtainedfromKuttern &Bazinm.

AssumethatforconcretewithKuttern =0.015,Bazinm =0.21

Rm

C+

=1

87Bazin

KnownA =138m2,P =33.42m,R =4.13m

13.421.0

1

87

+=C

Kutter)Ganguillet(from76.769Bazin)(from852.78 =C

Activity2.4


A trapezoidal channel is 10.0mwide and has a side slope of1.5(H):1(V).Thebedslope is0.0003.Thechannel is linedwithsmooth concrete n = 0.012. Compute the mean velocity anddischargeforadepthofflowof3.0m.

zy1

B

GivenB =10m,y =3.0m,z =1.5,So =0.0003andn =0.012

A =By +zy2 =43.5m2,

P =B +2y =20.817m,

R =A/P =2.090m

21 z+

Activity2.5


21

321

oSRnV =Manningvelocity

21

32

0003.0090.2012.01 =V

m/s359.2=V

AVQ =Discharge359.25.43 =

/sm625.102 3=


InthechannelofExample2.5,findthebottomslopenecessarytocarryonly50m3/softhedischargeatadepthof3.0m.

Activity2.6

GivenB =10m,y =3.0m,z =1.5andn =0.012

andA =43.5m2,P =20.817m,R =2.090m

21

321

oSARnQ =Manningdischarge

21

32

09.25.43012.01

50 oS=

0000712.0=oS51012.7 =oS


A triangularchannelwithanapexangleof75 carriesa flowof1.2m3/satadepthof0.80m. Ifthebedslope is0.009,findtheroughnesscoefficientC andn ofthechannel.

Activity2.7

Giveny =0.80m,So =0.009, =75,andQ =1.2m3/s

andA =zy2 =0.491m2,P =2y =2.017m,

R =A/P =0.2435m

21 z+z

y1 75

=2

tan

z

=275

tano

0.767=


21

321

oSARnQ =UsingManningequation

21

32

009.02435.0491.01

2.1 =n

0151.0=n

21

21

oSCARQ =UsingChezyequation21

21

009.02435.0491.02.1 =C197.52=C


A trapezoidal channel of bottom width 25 m and side slope2.5(H):1(V)carriesadischargeof450m3/swithanormaldepthof3.5m.Theelevationsatthebeginningandendofthechannelare685 m and 650 m, respectively. Determine the length of thechannelifn =0.02.

GivenB =25m,z =2.5,yo =3.5,n =0.02,andQ =450m3/s

zy1

B

A =By +zy2 =118.125m2

P =B +2y =43.848m21 z+R =A/P=2.694m

Activity2.8


21

321

oSARnQ =Manningequation,

21

32

694.2125.11802.01

450 oS=

00155.0=oS

Ho L

zS

=

( )HL650685

00155.0=

m13.22601=HL

Manningequation,


2.3Conveyance

ConveyanceK ofachannelsectionisameasureofthecarryingcapacityofthechannelsectionperunitlongitudinalslope.ItisdirectlyproportionaltodischargeQ.

1. Chezyformula

2. Manningformula

21

21

oSCARQ =

21

321

oSARnQ =

21

CARK =

321

ARn

K =


SectionfactorZ intheManningformulaisAR2/3,whichisafunctionofthedepthofflow.

InManningformula 21

321

oSARnQ =

Therefore,21

32

oS

QnAR =

SectionfactorAR2/3 isnormallyusedtocomputethenormaldepthyowhenthedischargeQ,bottomslopeSo andManningroughnesscoefficientn areprovided.

Computationofyo couldbethrougheitherdirecttrialanderrorcomputation,basedongraph,orthroughprovideddesignchart.

2.4SectionFactor


Activity2.9

A trapezoidalchannel5.0mwideandhavinga side slopeof1.5(H) :1(V)islaidonaslopeof0.00035.Theroughnesscoefficientn =0.015.Findthenormaldepthforadischargeof20m3/sthroughthischannel.

GivenB =5.0m,z =1.5,So =0.00035,n =0.015,andQ =20m3/s

zy1

B

A =By +zy2 =5y +1.5y2

P =B +2y =5+2y21 z+ 25.3( )( )yyyPAR 25.325 5.15

2

++==


ArrangingManningequationasafunctionofsectionfactor,

21

32

oS

QnAR =

21

321

oSARnQ =Manningequation,

( )21

32

22

00035.0

015.02025.3255.15

5.15=

+++

o

oooo y

yyyy

( )( ) 036.1625.325

5.15

32

35

2

=+

+o

oo

y

yy

Therefore,yo =1.820m

yo (m)( )( )32

35

2

25.325

5.15

o

oo

y

yy

++

Bytrialanderror:

1

2

1.8

1.820

5.391

19.159

15.706

16.035


Graphically,

( )( ) 036.1625.325

5.15

32

35

2

=+

+o

oo

y

yy

yo (m)( )( )32

35

2

25.325

5.15

o

oo

y

yy

++

1

2

1.5

1.7

5.391

19.159

11.198

14.115

1.8

1.9

15.706

17.387 0

0.5

1

1.5

2

2.5

0 5 10 15 20 25

AR 2/3

y

o

(

m

)

yo =1.82m

16.036

Therefore,yo =1.820m


DesignChartisavailable,

Circular

Rectangular(z =0)

B

38

32

38

32

and

od

AR

B

AR

ody

By

and

0.2194

0.37


036.1632

=AR

38

38

32

5

036.16=B

AR2194.0=

37.0=By

537.0 =yTherefore,yo =1.85m

Atthexaxis,

Intersectingatz =1.5oftrapezoidalchannelgives


DesignchartforlinedopendrainfromUrbanStormwaterManagementManualforMalaysia(DepartmentofIrrigationandDrainage,2000)


Activity2.10

A concretelined trapezoidal channelwithn = 0.015 is tohave asideslopeof1(H):1(V).Thebottomslopeistobe0.0004.Findthebottom width of the channel necessary to carry 100 m3/s ofdischargeatanormaldepthof2.50m.

zy1

B

Givenyo =2.5m,z =1,So =0.0004,n =0.015,andQ =100m3/s

A =By +zy2 =2.5B +6.25

P =B +2y =B +7.07121 z+( )( )071.7

25.65.2++==

BB

PA

R


Manningequationasafunctionofsectionfactor,

21

32

oS

QnAR =

( )21

32

0004.0

015.0100071.725.65.2

25.65.2=

+++

BB

B

( )( ) 75071.7

25.65.2

32

35

=++

B

B

Bytrialanderror,B =16.33m


Activity2.11

Waterflowsuniformlyat10m3/sinarectangularchannelwithabasewidthof6.0m,channelslopeof0.0001andManning'scoefficientn =0.013.Usingtrialanderrormethod,findthenormaldepth.

B

y

GivenQ=10m3/s,B =6.0m,So =0.0001andn =0.013

A =By =6y

P =B +2y =6+2y

( )yy

R += 33


21

32

oS

QnAR =

21

32

0001.0

013.01033

6=

+ oo

o yy

y

167.233 3

2

=

+ oo

o yy

y

Bytrialanderror,yo =1.942m


Asewerpipeof2.0mdiameterislaidonaslopeof0.0004withn =0.014.Findthedepthofflowwhenthedischargeis2m3/s.

2rD

yo

( ) sin228

2

DAreaA =

PerimeterP = D

Activity2.12


21

32

oS

QnAR =Manningequation:

21

32

0004.0

014.02=AR

38

38

32

2

4.1=D

AR

2205.038

32

=D

AR

Fordesignchart:

6.0=Dyo

26.0 =oy =1.20mIntersectingatcircularsectiongives


DesignChart:

Circular

Rectangular(z =0)

B

38

32

38

32

and

od

AR

B

AR

ody

By

and

0.2205

0.6


SimplificationforWide RectangularChannel

Widechannel: 02.0

bfc21103 chapter2

uniform nonuniformgvf

xoxsrv constant

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