Chapter 1

Hydraulics II Module 2011
Hydraulic and Water Resource Engineering Dep. AMIT 1
CHAPTER ONE
INTRODUCTION TO OPEN CHANNEL FLOW
Course objectives:- atthe endof the course studentswill be expectedto:
o Differentiate betweenopenchannel &pipe flows
o Understandfluidphenomenonapplicationtodesignvarioushydraulicstructures
o Acquaintlaminar&turbulentflows,boundarylayertheory,velocity-shearstress
relationship
? WHAT IS OPEN CHANNEL
Openchannel:- It maybe definedasapassage inwhichliquidflowswithitsuppersurface
exposedtothe atmosphere.e.g. :- curvets,spillways,andsimilarhumanmade structures
Differencesb/nthe flowinpipes&openchannel flow
Openchannel flow
 Is exposedtoatmosphericpressure.
 The cross-sectional areaof the flow is
variable.(thatdependsonmany
parametersof the flow)
 The force causingmotionisgravity.
Pipe Flow
 Is closedchannel
 The top surface iscoveredbysolid
boundary
 It isnot exposedtoatmospheric
Pressure.
Where HGL - Hydraulicgrade line (coincidewithwatersurface)
EGL - Energygrade line
Hf - headlossdue to friction
g
V
2
2
Y2
Fig 1(a) Pipe flow
Z2 Z1
Z2
Y1
Y1
Y2
Hf
HGL
EL
HGL
EL

V2
/2g- velocityhead
Types ofchannels
 Natural channels:These channelsnaturallyexistwithoutthe influence of humanbeings.
E.g. Rivers,streams,tidal estuaries,aqueducts.
 Artificial channels:Such channelsare formedbyman’sactivityforvarious
purposes.E.g.irrigationchannel,navigationchannel,sewerage channel,culverts,power
canal…… etc.
 Prismaticchannel:- channelswithconstantshape andslope.
 Non-prismaticchannels:- channelswithvaryingshape andslope.
 Open channel:-A channel withoutanycoverat the top.
 Closedchannel:-The channel havingacoverat the top.
ACTIVITY1.1
What isopenchannel?
What are the differenttypesof channel? Give example ineachcase.
1.1 Types of flow in open channel
The flow ina channel classifiedintothe followingtype,dependingonthe change inthe
depthof flowwithrespecttospace and time.
a) Steadyflow&Unsteadyflow
b) Uniformflow&Nonuniformflow
c) Steadyuniformflow&Unsteadyuniformflow
d) . Unsteadyuniformflow
Time as criteria
Steady flow & Unsteady flow
Whenthe flow characteristic(suchasdepthof flow,flow velocity andthe flow rate atany cross
section)donotchange withrespecttotime,the flow inachannel isto be steady.
Mathematically, 0

t
V


, 0

t
p


and 0

t
y


The flowissaidto be unsteadyflow whenthe flow parametervarywithtime.

Mathematically,
0

t
V


,
0

t
p


and
0

t
y


Space as a criterion
Uniformflow& Non uniformflow
Flowina channel issaidto be uniformif the depth,slope,cross-sectionandvelocityremain
constantovera givenlengthof the channel.
Mathematically,
0

s
V


, and
0

s
y


Flowinchannel issaidto be non-uniform(varied)whenthe channel depthvariescontinuously
fromone sectionto another.
Mathematically,
0

s
V


,and
0

s
y


Time and space as a criteria
Steady uniformflow: - The depthof flow doesnotchange duringtime interval andspace under
consideration.
Unsteadyuniform flow:- Thisis a flow inwhichthe depthisvaryingtime butnot withspace.
Unsteadynon uniform flow:- Is the flow inwhichthe depthisvaryingwithspace and time.
ACTIVITY 1.2
Explainbrieflythe following:
1. Steadyand Un steadyflow
2. Uniformand nonuniformflow
3. State the conditionunderwhichuniform andnonuniformflowsare produced.
1.2 Geometric elements of open channel section
Geometricelementsare propertiesof achannel sectionthatcan be definedentirelybythe
geometryof the sectionandthe depthof flow.The mostusedgeometricpropertiesinclude:
1. Depth of flow(y):itthe vertical distance fromthe lowestpointof the channel tothe free
surface.
2. Top width (T): itis the widthof channel sectionatfree surface.

3. Stage (h):is the elevationorvertical distanceof the free surface above a datum.
4. Wettedperimeter(p):itisthe lengthof the channel boundarywhichisincontact with
water.
5. Wettedarea (A):is the cross-sectional areaof the flow normal tothe directionof flow.
6. Hydraulic radius(hydraulicmean depth)(R) : itis the ratioof wettedareato itswetted
perimeter
R=
P
A
7. Hydraulic depth(D):the ratio of wettedareato the top width,
D=
T
A
8. Sectionfactor (Z):is the productof the wettedareaand the two-thirdpowerof the
hydraulicradius
Z=A D =A
T
A
=
2
1
3








T
A
=A 3
2
R
9. Conveyance (K) :
Q=VA………………………….V= 2
1
3
2
1
S
R
n
Q=A 2
1
3
2
1
S
R
n
=A 3
2
R 2
1
1
S
n
=K 2
1
S S= bedslope
K=
n
1
A 3
2
R n= Mannings constant
= CA R c= Chezy’sconstant

Where S0- bedslope of channel
Sw- Water surface slope
S- Slope of EGL
W – Weightof water
0 – Shearforce
L- Lengthof channel
Uniformflowisthe resultof exactbalance betweenthe gravityandfrictionforce
Wsin= o
 .P.L…………………………….(1)
A L sin= o
 .P.L
But sin = hf/L= S, solvingfor o
 ,
o
 = S
R
S
P
A
.
. 
  ………………………………… (2)
Where - unitweightof the water
The shear stressisassumedproportional tothe square of the meanvelocity,
or o= kV
2
…………………………………..……..(3)
Therefore,kv2
=RS
V2
= RS
k

,
L
W
Wsinө
S0
Sw
Y0
S
0
X
Hf (Z)
Fig. 1.2

Let 2
C
k


-constant(b/c&k- are constant)
.
RS
C
V  ……………………………………………….... (4)
Thisis the Chezy –formula
C= chezycoefficient(chezy’sresistancefactor)
V= Average velocityof flow
ManningFormula
V= 2
1
3
2
0
1
S
R
n
………………………………………………(5)
 The bestas well asmostwidelyusedformulaforuniformlyforuniformflow.
n- isthe roughnesscoefficient.
A relationbetweenthe Chezy’sCandManning’sn maybe obtainedbycomparingeqn(4) & (5)
n
R
C
6
1
 …………………………………………..(6)
 The value of n ranges from0.009 (forsmoothstraightsurfaces) to0.22 (for very
dense floodplainforests).
? What is hydraulic efficiency channel (most economical channel) means
 A channel sectionissaidtobe efficient(economical)if itgivesthe maximumdischarge
for the givenshape,areaandroughness.
1.3 Most economical channel section
Most economical rectangular channel section
Let B and Y be the base widthanddepthof flow respectively
A=BY………………………….(i)
P=B=2Y……………………….(ii)
From eqn.(i),B=A/Y
Substitutingin(ii) P=A/Y+2Y………….(iii)

For maximumQ,P- is minimum.
0
)
2
/
(
0 


 Y
Y
A
dY
d
dY
dp
0
2 



Y
A
Y
B
Y
A *
2 2



So,B=2Y (orY=B/2)
Thus the rectangularchannel ismostefficientandeconomical whenthe depthof waterisone
half of the widthof the channel andthe discharge flow will be maximum.
EXAMPLE -1
1 .A rectangularchannel isto be dug inthe rocky portionof a soil.Finditsmosteconomical
cross-sectionif itstoconvey12 m3
/sof waterwithan average velocityof 3 m/s.Take chezy
constantC=50
Given
Q=12 m3
/s
V=3 m/s
C=50
Solution
The geometricrelationsforoptimumdischargethrougharectangularchannel are
Then
WhenB,Y andR are base width,depthof flow andhydraulicradiusrespectively
Now
From thisequationsolvefordepthof flow
Therefore base widthof flow
Hydraulicradius,
Also chezyformula
Hence
Mosteconomicaltrapezoidalchannelsection

But for mosteconomicsection
EXAMPLE-2
An irrigationchannel of trapezoidal sectionhasside slope,m=2and carries a discharge of
15m3
/s on a longitudinalslope of 1in5000. The channel isto be linedforwhichthe value of
frictioncoefficientinManning’sformulaisn=0.012. Findthe dimensionof the mosteconomic
sectionof the channel.
GIVEN
Side slope m=2
Discharge Q=15m3
/s
Longitudinal slopeS=1:5000
Manning´scoefficientn=0.012
SOLUTION

ACTIVITY1.3
What do youmeanby mosteconomical sectionof anopenchannel?How isit determined?
What are the conditionsforthe rectangularchannel of bestsection?
Showthat the hydraulicmeandepthof a trapezoidal
1.4 Specific energy
? What is specific energy
 Specificenergyisthe energy perunitweightof flowingliquidabove the channel
bottom.
For any cross section,shape,the specificenergy( E) at a particularsectionisdefinedasthe
energyheadtothe channel bedasdatum.Thus,
g
V
Y
E
2
2


 ……………………………………………..(1)
( - iskineticenergycorrectionfactor 1)
ET1
ET2
Datum
EGL
Z1
Z2

Fig1.3 SpecificEnergyata particularsection
For a rectangularchannel,the value of flow perunitwidthisQ/B=q,andaverage velocity
Y
q
BY
qB
A
Q
V 


Therefore eqn(1) becomes:
2
2
2
2
2 gy
q
y
g
y
q
y
E 








 …………………………………… (2)
g
q
Y
y
E
2
)
(
2
2

 (Forthe case of constantq)………………………… (3)
A plotof E VsY isa hyperbolalike withasymptotes(E-Y) =0i.e.E=Y and y=0. Such a curve is
knownas specificenergydiagram.
Y
Sub critical
section
Super critical
E
Y1
Yc
Y2
Ec E0
Specific Energy diagram

For a particularq, we see there are twopossible valuesof Yfora givenvalue of E.These are
knownas Alternativedepths(fore.g.Y1 & Y2 on fig.above)
 The two alternative depthsrepresenttwototallydifferentflow regimesslow &deepon
the upperlimpof the curve (subcritical flow) &fastand shallow onthe lowerlimbof
the curve.(supercritical flow)
? What is critical depth
 Depthof flowatwhich specificenergyis minimumis calledcritical depth.
The velocityof flowatcritical depthisknownas critical velocity.
For example,arelationforcritical depthinawide rectangularchannel canbe foundby
differentiationEof eqn.2withrespecttoY to findthe value of Y for whichE isa minimum.
3
2
1
gy
q
dY
dE

 …………………………………………….. (4)
AndwhenE isa minimumY=Ycand 0

dy
dE , so that
3
2
3
2
1
0 c
c
gy
q
gY
q



 ………………………………. (5)
Substitutingq=vy= VC*Yc, gives
c
c gy
V 
2
c
c
c
y
q
gy
V 

 ……………………………………….. (6)
It may be expressedas:
3
1
2
2










g
q
g
V
y c
c ……………………………………….. (7)
From eqn(7) ,
2
2
2
c
c y
g
V
 hence,
c
c
c
c
c
c y
y
y
g
V
y
E
E
2
3
2
1
2
2
min 




 ……………… (8)

And min
3
2 E
yc  ……………………………………………………………..(9)
From eqn.(7):
3
max c
gy
q  ……………………………………….………….(10)
For nonrectangularcross sectionthe specificenergyeqn.
2
2
2gA
Q
y
E 
 …………………………………………………….. (11)
[V=Q/A]
To findthe critical depth,
dy
dA
gA
Q
dy
dE
3
2
1
 ………………………………………………….. (12)
Fromfig1.3 (b) dA = dy*T (at Yc, T= Tc)
Therefore the above equationbecomes:
1
3
2
max

c
c
gA
T
Q
…………………………………………………………….. (13)
The critical depthmustsatisfythisequation
From eqn.(13)
c
c
T
gA
Q
3
2
 and substitute in eqn.(11) then
c
c
c
c
T
A
y
E
2

 …………………………………………………………..(14)
eqn.(13) can be solvedbytrial & error forirregularsectionbyplotting 3
2
)
(
gA
T
Q
y
f  and
critical depthoccurs forthe value of y whichmakesf(y)=1
What are sub critical, critical, and super critical flow?
 Subcritical flow:-whenthe depthof flow inachannel isgreaterthanthe critical
depth(Yc) inthiscase Fr <1
 Critical flowisone inwhichspecificenergyisminimum.A few correspondingtocritical
depthalsoknownas critical flow.
 Supercritical flow:-whenthe depthof flow inachannel islessthancritical depth(Yc) in
thiscase Fr>1.

If specificenergycurve forQ- constantis redraw alongside asecondcurve of depthagainst
discharge forconstantE, will showthe variationof discharge withdepth.
Y
yc
q qmax
For a givenconstantdischarge fig
i) The specificenergycurve hasa minimumvalue Ecat pointC witha corresponding
depthYc knownas critical depth.
ii) For any othervalue of E there are two possible depthof flow knownasalternative
depthone of whichistermedsubcritical (y>Yc) and the othersupercritical (Y<Yc).
a) For a given constantspecificenergy( fig.1.5(b))
i) the depthdischarge curve showsthatdischarge isa maximumat the critical
depth
ii) For all otherdischargesthere are twopossible depthof flow ( sub- &super
critical) forany particularvalue of E,
Fromeqn.(13) above if we substitute
Q= AV (continuityequation),we get
1
3
2

gA
T
Q
1
1
2
3
2
2



gA
T
V
gA
T
V
A
butA/T = D ( Hydraulicdepth),then[ D=Y forrectangularsection)
gy
V
gy
V


 1
2
……………………………(*)

1
gy
V
Froude numberatcritical state.

gy
V
F  ……………………………………….(**)
Thus, i) F= 1critical flow
ii) F< 1 subcritical flowType equationhere.
iii)F>1Supercritical
ACTIVITY1.4
What isspecificenergy andspecificenergycurve?
What do youunderstandbycritical depthof an openchannel whenthe flow initisnotuniform?
Examples
1. For constant specificenergyof ,calculate the maximumdischargethatmay occur in
a rectangular channel 5m width.
Given
Solution
For constant specific energy discharge is maximum
2. Most efficient rectangular channel, which is laid on a bottom slope of 0.0064, is to carry
20m3
/s of water.Determine the widthof the channel whenthe flow isincritical condition. Take
n=0.015.
Given
Solution

1.5 Hydraulic jump
What is hydraulic jump?
 A flowphenomenonwhich occurswhensupercritical flow hasitsvelocityreducedtosub
critical.There issuddenrise inwaterlevel atthe pointwhere hydraulicjumpoccurs
e.g (Rapidlyvariedflow).
Hydraulicjumponhorizontal bedfollowingoveraspillway
Where V1-velocitybeforejump
V2 –velocityafterjump
Y1 –waterdepth before jump
Y2 –waterdepthafterjump
Lj –lengthof jump
Y2
V2
Y1
V1
Lj

Purposes of hydraulic jump:-
i) To increase the waterlevel onthe d/sof the hydraulicstructures
ii) To reduce the netup liftforce byincreasingthe downwardforce due tothe
increaseddepthof water,
iii) To increase the discharge froma sluice gate byincreasingthe effective headcausing
flow,
iv) For aerationof drinkingwater
v) For removingairpocketsina pipe line
vi) Reduce downstreamerosion
vii) Veryuseful &effective formixingfluids
 Analysis of hydraulic jump
Assumptions
a. The lengthof the hydraulicjumpissmall,consequently,the lossof headdue tofriction
isnegligible,
b. The channel ishorizontal asit has a verysmall longitudinal slope. The weight
componentinthe directionof flow isnegligible.
c. The portionof channel inwhichthe hydraulicjumpoccursistakenas a control volume
& it isassumedthe justbefore &afterthe control volume,the flow isuniform&
pressure distributionishydrostatic.
Let usconsidera small reachof a channel inwhichthe hydraulicjumpoccurs.
The momentumof waterpassingthroughsection(1) perunittime isgivenas:
1
1
1
QV
g
rQV
t
p


 ……………………………………….(i)
Momentumat section(2) perunittime is:
2
2
2
QV
g
rQV
t
p


 ………………………………………….(ii)
Rate of change of momentumb/nsection1& 2
)
( 1
2 V
V
Q
t
P



 ……………………………………….(iii)

The net force inthe directionof flow =F1-F2 ………………..(iv)
2
2
2
1
1
1 , Y
A
F
Y
A
F 
 

 2
1 &Y
Y are the centerof pressure atsection(1) & (2)
Therefore F1-F2 =M=Q (V2-V1)
)
( 1
2
2
2
1
1 V
V
g
Q
Y
A
Y
A 




 ……………………………………(v)
From continuityeqn.Q=A*V, V= Q/A,so
)
.........(
..........
..........
..........
..........
1
1
1
2
2
2
2
1
1
1
2
2
2
1
1
iv
A
A
g
Q
Y
A
Y
A
A
Q
A
Q
g
Q
Y
A
Y
A





 


















Rearrangingthiseqn.:
2
1
2
2
2
2
1
1
1
2
M
M
Y
A
gA
Q
Y
A
gA
Q















= Constant.…………… (vii)
M1and M2 are the specificforcesatsection(1) & (2) indicatesthatthese forcesare equal
before &afterthe jump.
Y1= initial depth
Y2 = sequentdepth
Hydraulicjumpina rectangularchannel
A1=By1 the sectionhasuniformwidth(B)
A2= By2

2
,
2
2
2
1
1
Y
Y
Y
Y 

Nowfromeqn.(Vii) above:
)
..(
..........
..........
..........
..........
..........
..........
2
2
2
*
2
2
2
2
2
2
1
1
2
2
2
2
1
1
1
2
viii
By
Bgy
Q
By
gBy
Q
y
By
gBy
Q
y
By
gBy
Q


















Flowperunitwidthof q= Q/B Q=qB, theneqn.(viii)becomes
2
2
2
2
2
2
2
2
1
1
2
2
By
Bgy
B
q
By
Bgy
B
q



2
1
1 2
1
2
2
2
1
2
y
y
y
y
g
q 







 ………………………………… (.ix)
 
 
1
2
2
1
2
2
2
1
2
2
y
y
y
y
y
y
g
q



)
.........(
..........
..........
..........
..........
..........
0
2
)
........(
..........
..........
..........
..........
..........
).........
(
2
2
2
2
1
2
1
2
2
1
2
1
2
xi
g
q
y
y
y
y
x
y
y
y
y
g
q





Thisis quadraticeqn.& the solutionisgivenas

)
........(
..........
..........
..........
..........
2
2
2
)
)(
.(
..........
..........
..........
..........
..........
2
2
2
2
2
2
1
1
2
2
2
2
2
2
1
b
gy
q
y
y
y
a
xii
gy
q
y
y
y




















)
)(
...(
..........
..........
..........
..........
..........
8
1
1
(
2
)
..(
..........
..........
..........
..........
).........
8
1
1
(
2
3
1
2
1
2
3
2
2
2
1
d
xii
gy
q
y
y
c
gy
q
y
y








The ratio of conjugate depths;
)
(
..........
..........
..........
..........
..........
8
1
1
(
2
1
)
)(
.......(
..........
..........
..........
..........
8
1
1
(
2
1
3
1
2
1
2
3
2
2
2
1
f
gy
q
y
y
e
xii
gy
q
y
y








3
2
2
2
2
2
2
1
1
1 ,
gy
q
gy
y
q
gy
V
F
gy
V
F 



)
..(
..........
..........
..........
..........
8
1
1
(
2
1
)
.....(
..........
).........
8
1
1
(
2
1
2
1
1
2
2
2
2
1
h
F
y
y
g
F
y
y
Therefore








Energy dissipation in a Hydraulic Jump
The head losshl.f causedby the jumpisthe drop inenergyfromsection(1) to (2) or:
hlf= E = E1 - E2
a
g
V
y
g
V
y )
1
.........(
..........
..........
..........
2
2
2
2
2
2
1
1 



















)
......(
..........
..........
..........
2
2 2
2
2
2
2
1
2
1 b
gy
q
y
gy
q
y 




















From eqn.(x) substituting: )
(
2
2
1
2
1
2
y
y
y
y
g
q

 intothiseqn.& byrearranging:
  )
2
......(
..........
..........
..........
..........
4 2
1
3
1
2
y
y
y
y
E
hlf




Therefore powerlost= Q hlf (kw)…………………(3)
ACTIVITY1.5
What ismean byhydraulicjumpinopenchannel andhow itoccurs?
Types of Hydraulic jump
Hydraulicjumpsare classifiedaccordingtothe upstream Froude numberanddepthratio.
F1 Y2/y1 Classification
<1 1 Jumpimpossible
1-1.7 1-2 Undularjump(standingwave)
1.7-2.5 2-3.1 Weakjump
2.5-4.5 3.1-5.9 Oscillatingjump
4.5-9.0 5.9-12 Steadyjump(45-70% energyloss)
>9.0 >12 Strongor choppingjump(=85% energyloss)

Examples
A 3∙6m wide rectangularchannel conveys of waterwithavelocityof .
a. Is there a conditionforhydraulicjump occur?If so calculate the height,length and
strengthof the jump.
b. What islossof energy?
Given
Solution
a.
b. Loss of energyforrectangularchannel

Exercises
1. A rectangular channel which is laid on a bottom slope of 0.0064 is to carry 20m3
/s of water.
Determine the width of the channel when the flow is in critical condition. Take C=66
2.An irrigationcanal of trapezoidal sectionhavingside slope 2in3 isto carry a flow of 10m3
/s on
a longitudinal slope of 1in5000. The canal is linedforwhichthe value of frictional coefficient in
Manning’s formula is n=0.012. Find the dimension of the most economical section
3. Determine the side slopeof the mosthydraulicallyefficienttriangularsection. .Show that the
head loss in a hydraulic jump formed in a rectangular channel may be expressed as
ΔE= (V1 –V2)3
/ [2g (V1 +V2)]
4. A rectangular channel there occurs a jump corresponding to Froude number (F=2.5).
Determine the critical depth and head loss in terms of the initial depth y1.
5. A trapezoidal channel havingbottomwidth10mand side slope 2:1(H:V) carries a discharge of
100m3
/s. Findthe depthconjugate tothe initial depthof 1mbefore the jump.Also determine
the loss of energy in the jump.

Chapter 1

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