water flow in open channels
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CHAPTER 6:
Water Flow in Open Channels
University of Palestine
Engineering Hydraulics
2 nd semester 2010-2011
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Introduction.
Type of Open Channels.
Types of Flow in Open Channels.
Flow Formulas in Open Channels.
Most Economical Section of Channels.
Energy Principles in Open Channel Flow.
Non-uniform Flow in Open Channels.
Hydraulic Jump.
Content
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Open channel hydraulics, a subject of great importance to
civil engineers, deals with flows having a free surface inchannels constructed for water supply, irrigation, drainage,and hydroelectric power generation; in sewers, culverts,and tunnels flowing partially full; and in natural streamsand rivers.
An open channel is a duct in which the liquid flows with afree surface.
This is in contrast with pipe flow in which the liquidcompletely fills the pipe and flow under pressure.
The flow in a pipe takes place due to difference of pressure(pressure gradient), whereas in open channel it is due tothe slope of the channel bed (i.e.; due to gravity).
Introduction
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It may be noted that the flow in a closed conduit is notnecessarily a pipe flow. It must be classified as openchannel flow if the liquid has a free surface.
for a pipe flow:
The hydraulic gradient line (HGL) is the sum of theelevation and the pressure head (connecting the watersurfaces in piezometers).
The energy gradient line (EGL) is the sum of the HGLand velocity head.
The amount of energy loss when the liquid flows fromsection 1 to section 2 is indicated by hL.
Introduction
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Pipe system
Introduction
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For open channel flow :
The hydraulic gradient line (HGL) corresponds to the water surface line (WSL); the free water surface issubjected to only atmospheric pressure which iscommonly referred to as the zero pressure reference inhydraulic engineering practice.
The energy gradient line (EGL) is the sum of the HGLand velocity head.
The amount of energy loss when the liquid flows fromsection 1 to section 2 is indicated by hL. For uniformflow in an open channel, this drop in the EGL is equalto the drop in the channel bed.
Introduction
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Introduction
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Based on their existence, an open channel can be naturalor artificial :
Natural channels such as streams, rivers, valleys , etc.These are generally irregular in shape, alignment androughness of the surface.
Artificial channels are built for some specific purpose,such as irrigation, water supply, wastewater, waterpower development, and rain collection channels.These are regular in shape and alignment with uniform
roughness of the boundary surface.
Type of Open Channels
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Type of Open Channels
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Type of Open Channels
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Based on their shape, an open channel can be prismatic ornon-prismatic:
Prismatic channels: a channel is said to be prismatic when the cross section is uniform and the bed slop isconstant.
)Non-prismatic channels: when either the cross sectionor the slope (or both) change, the channel is referred toas non-prismatic. It is obvious that only artificialchannel can be prismatic.
The most common shapes of prismatic channels arerectangular, parabolic, triangular, trapezoidal and circular.
Type of Open Channels
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Type of Open Channels
The most common shapes of prismatic channels arerectangular, parabolic, triangular, trapezoidal and circular.
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Types of Flow in Open Channels
The flow in an open channel can be classified into thefollowing types :
A).Uniform and non-uniform flow:
If for a given length of the channel, the velocity of flow,depth of flow, slope of the channel and cross-sectionremain constant, the flow is said to be uniform.
Otherwise it is said to be non-uniform.
Non-uniform flow is also called varied flow which can befurther classified as: Gradually varied flow (GVF) where the depth of the flow changes
gradually along the length of the channel.
Rapidly varied flow (RVF) where the depth of flow changessuddenly over a small length of the channel. For example, when
water flows over an overflow dam, there is a sudden rise (depth) of water at the toe of the dam, and a hydraulic jump forms.
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Types of Flow in Open Channels
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Uniform Flow Types of Flow in Open Channels
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Types of Flow in Open Channels
B). Steady and unsteady flow: :
The flow is steady when, at a particular section, the depthof the liquid and other parameters (such as velocity, area ofcross section, discharge) do not change with time. In anunsteady flow, the depth of flow and other parameterschange with time.
C). Laminar and turbulent flow:
The flow in open channel can be either laminar orturbulent. In practice, however, the laminar flow occurs
very rarely. The engineer is concerned mainly with
turbulent flow. In the case of open channel Reynold’s number is defined as:
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Types of Flow in Open Channels
Recall that Reynold’s number is the measure of relative effects
of the inertia forces to viscous forces. 17
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Types of Flow in Open Channels
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Types of Flow in Open Channels
D). Sub-critical, critical, and supercritical flow:
The criterion used in this classification is what is known byFroude number, Fr, which is the measure of the relativeeffects of inertia forces to gravity force:
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Types of Flow in Open Channels
D). Sub-critical, critical, and supercritical flow:
The criterion used in this classification is what is known byFroude number, Fr, which is the measure of the relativeeffects of inertia forces to gravity force:
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Flow Formulas in Open Channels
In the case of steady-uniform flow in an open channel, thefollowing main features must be satisfied:
The water depth, water area, discharge, and the velocitydistribution at all sections throughout the entire channellength must remain constant, i.e.; Q , A , y , V remainconstant through the channel length.
The slope of the energy gradient line (S), the water surfaceslope (S ws), and the channel bed slope (S0) are equal.
S = S ws = S0
D Water Surface
T.E.L
channel bed
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l l h l
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Flow Formulas in Open Channels The depth of flow, y , is defined as the vertical distance
between the lowest point of the channel bed and the free
surface. The depth of flow section, D , is defined as the depth of
liquid at the section, measured normal to the direction offlow.
D
Water Surface
T.E.L
channel bed
Unless mentioned otherwise, the depth of flow and the depth of f low section will be assumed equal. For uniform flow the depth attains a constant valueknown as the normal depth, yn 22
l l h l
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S RC V h
1.The Chezy Formula(1769)
C is the Chezy coefficient (Chezy’s resistance factor), m 1/2/s, a dimensionalfactor which characterizes the resistance to flow .
Pwetted
Awetted Radiushydraulic
slopebed S
Rh
Flow Formulas in Open Channels
Many empirical formulas are used to describe the flow in openchannels
The Chezy formula is probably the first formula derived foruniform flow. It may be expressed in the following form
23
Fl F l i O Ch l
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2. The Manning Formula: (1895)
2/13/21S R
nV h
tCoefficienM
PwettedAwetted Radiushydraulic
anning n
slopebed S
Rh
Flow Formulas in Open Channels
where n = Manning’s coefficient for the channel roughness,m-1/3/s.
Substituting manning Eq. into Chezy Eq, we obtain the Manning’s formula for uniform flow:
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Fl F l i O Ch l
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Flow Formulas in Open Channels
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Fl F l i O Ch l
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3. The Strickler Formula:
Flow Formulas in Open Channels
where kstr = Strickler coefficient, m1/3/s ComparingManning formula and Strickler formulas, we can see that
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Fl F l i O Ch l
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Example 1
1 . 5
m
3.0m
2
1
open channel of width = 3m as shown, bed slope = 1:5000,d=1.5m find the flow rate using Manning equation, n=0.025.
sVAQ
V
P
A R
P
A
S Rn
V
h
h
/m84.49538.0
m/s538.05000
1927.0025.0
1
927.0708.9
9
9.70835.132
m95.1935.0
1
3
3
2
22
2
3
2
Flow Formulas in Open Channels
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Fl F l i O Ch l
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Example 2
open channel as shown, bed slope = 69:1584, find the flow rateusing Chezy equation, C=35.
Flow Formulas in Open Channels
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Fl F l i O Ch l
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sVAQ
V
P
A
R
P
A
S RC V
h
h
/m84.11352.1627.0
m/s7.01584
69.0917.035
917.018.177
52.162
m177.1804.552.28.166.38.115072.0
m52.16215072.06.32
52.272.08.1652.2
2
04.552.2
3
2222
2
Flow Formulas in Open Channels
Example 2 cont.
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Fl F l i O Ch l
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Flow Formulas in Open Channels
Example 3: Group workThe cross section of an open channel is a trapezoid with a ottom
width of 4 m and side slopes 1:2, calculate the discharge if thedepth of water is 1.5 m and bed slope = 1/1600. Take Chezyconstant C = 50.
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M t E i l S ti f Ch l
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During the design stages of an open channel, the
channel cross-section, roughness and bottom slope are given.
The objective is to determine the flow velocity , depth
and flow rate, given any one of them. The design ofchannels involves selecting the channel shape andbed slope to convey a given flow rate with a givenflow depth. For a given discharge, slope and
roughness, the designer aims to minimize thecross-sectional area A in order to reduceconstruction costs
Most Economical Section of Channels
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M t E i l S ti f Ch l
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A section of a channel is said to be most economical
when the cost of construction of the channel isminimum.
But the cost of construction of a channel depends on
excavation and the lining. To keep the cost down or
minimum, the wetted perimeter, for a given discharge,
should be minimum.
This condition is utilized for determining the
dimensions of economical sections of different forms of
channels.
Most Economical Section of Channels
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M t E i l S ti f Ch l
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Most economical section is also called the best
section or most efficient section as the discharge,passing through a most economical section of channelfor a given cross sectional area A, slope of the bed S0and a resistance coefficient, is maximum.
Most Economical Section of Channels
Hence the discharge Q will be maximum when the wetted perimeter P is minimum.
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M t E i l S ti f Ch l
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The most ‘efficient’ cross-sectional shape is determined
for uniform flow conditions. Considering a givendischarge Q, the velocity V is maximum for theminimum cross-section A. According to the Manningequation the hydraulic diameter is then maximum.
It can be shown that:1. the wetted perimeter is also minimum,2. the semi-circle section (semi-circle having its
centre in the surface) is the best hydraulic section
Most Economical Section of Channels
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Most Economical Section of Channels
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Most Economical Rectangular Channel
Because the hydraulic radius is equal to the water crosssection area divided by the wetted parameter, Channelsection with the least wetted parameter is the besthydraulic section
Rectangular section
Most Economical Section of Channels
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Most Economical Section of Channels
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DBA BD2P
D
A 2DP
0dD
dP
222 2 02 D
D B
D
A
D
A
dD
dP
D
B 2
2
B D
Most Economical Rectangular ChannelMost Economical Section of Channels
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Most Economical Section of Channels
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D ) Dn(B A
212 n D B P
Dn
D
A B
212 n D )nD
D
A( P
0dD
dP
012 2
2 nn
D
A
dD
dP n
D
An12
2
2
D Dn Bn
D DnD)(Bn 2 12 2
2
2
Dn2Bn1D 2
or
Most Economical Section of ChannelsMost Economical Trapezoidal Channel
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Most Economical Section of Channels
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DOF
The best side slope for Trapezoidal section
3
1 k 60
0dk
dP
k
Most Economical Section of Channels
Other criteria for economic Trapezoidal section
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Most Economical Section of Channels
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Circular section
2sin8 4
22 d d A
d r P 2
d D 95.0154
Maximum Flow using Manning
Maximum Flow using Chezy
d D 94.0151
d D 81.075.128
Maximum Velocity using Manning or Chezy
Most Economical Section of Channels
Most Economical Circular Channel
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Most Economical Section of Channels
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Most Economical Section of Channels
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Most Economical Section of Channels
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Example 4Circular open channel as shown d=1.68m, bed slope = 1:5000, find theMax. f low rate & the Max. velocity using Chezy equation, C=70.
sVAQ
V
m P
A R
d P
d d A
S RC V
h
h
/m496.117.269.0
m/s69.05000
1485.070
485.05.4
17.2
m4.568.1180
154
m17.21542sin8
68.1
180154
4
68.12sin
84
3
22222
154Max. flow rate
Most Economical Section of Channels
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Most Economical Section of Channels
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m/s748.05000157.070
57.03775.3
93.1
m 378.368.1180
75.128
m93.175.1282sin
8
68.1
180
75.128
4
68.12sin
84
22222
V
m P
A R
d P
d d A
S RC V
h
h
75.128Max. Velocity
Most Economical Section of Channels
Example 4 cont.
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Most Economical Section of Channels
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Trapezoidal open channel as shown Q=10m3/s, velocity =1.5m/s,
for most economic section. find wetted parameter, and the bedslope n=0.014.
m D
D D D A
DkD B A
mV
Q A
B D
D B D
kD Bk D
78.1
667.6)2
36055.0(
667.65.1
10
6055.0
2
2322
31
2
21
2
2
2
Example 5
Most Economical Section of Channels
43
Most Economical Section of Channels
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m P
k D D P
k D B P
49.72
3178.12)78.1(6055.0
126055.0
12
2
2
2
To calculate bed Slope
6.1941:1
5.189.0014.0
1
89.0
49.7
667.6
m49.7
m667.6
1
3
2
2
3
2
S
S V
P
A R
P
A
S Rn
V
h
h
Example 5
cont.
Most Economical Section of Channels
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Most Economical Section of Channels
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Use the proper numerical method to calculate uniform water
depth flowing in a Trapezoidal open channel with B = 10 m, asshown Q=10m3/s if the bed slope 0.0016, n=0.014. k = 3/2. to aprecision 0.01 m, and with iterations not more than 15.Note: you may find out two roots to the equation.
1
ManningFrom
2
2
2
1
2/1
3/2
22
2/13/2
S
P
A
n A
Q
P
A R
DnD B P
nD B B D A
S RnV
h
h
Example 6:
Most Economical Section of Channels
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Most Economical Section of Channels
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sm A
QV
m A
m D
/326.07.30
10
7.30)28.2(2
328.210
28.2
22
Example 6 cont.
Most Economical Section of Channels
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Variation of flow and velocity with depth in circular pipes
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Variation of flow and velocity with depth in circular pipes
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Energy Principles in Open Channel Flow
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g
V y Z EnergyTotal
2
2
Energy Principles in Open Channel Flow
D
Water Surface
T.E.L
channel bed
Referring to the figure shown, the total energy of a flowingliquid per unit weight is given by
Where:
Z = height of the bottom of channelabove datum,
y = depth of liquid,V = mean velocity of flow.
If the channel bed is taken as the datum (as shown),then the total energy per unit weight will be.This energy is known as specific energy, Es. Specificenergy of a flowing liquid in a channel is defined asenergy per unit weight of the liquid measured from
the channel bed as datum 48
Energy Principles in Open Channel Flow
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Energy Principles in Open Channel Flow
D
Water Surface
T.E.L
channel bed
The specific energy of a flowing liquid can be re-written inthe form:
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Energy Principles in Open Channel Flow
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Energy Principles in Open Channel Flow
It is defined as the curve which shows the variation of specific
energy (Es ) with depth of f low y. It can be obtained as follows:
Let us consider a rectangular channel in which a constantdischarge is taking place.
Specific Energy Curve (rectangular channel)
But
g
V y E specific
2
2
2
2
2
A g
Q y E s Or
50
Energy Principles in Open Channel Flow
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Energy Principles in Open Channel Flow
Specific Energy Curve (rectangular channel)
g
V
2
2
H
G
E
The graph between specific energy (x-axis) and depth (yaxis)
may plotted.
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Energy Principles in Open Channel Flow
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Referring to the diagram above, the following features can be
observed:1. The depth of flow at point C is referred to as critical depth, yc. It
is defined as that depth of f low of liquid at which the specificenergy is minimum, E min , i.e.; E min @ yc . The flow thatcorresponds to this point is called critical flow (Fr = 1.0).
2. For values of Es greater than E min , there are two correspondingdepths. One depth is greater than the critical depth and the otheris smaller then the critical depth, for example ; Es1 @ y 1 and y 2 These two depths for a given specific energy are called thealternate depths.
3. If the flow depth y > yc , the flow is said to be sub-critical (Fr < 1.0). In this case Es increases as y increases.
4. If the flow depth y < yc , the flow is said to be super-critical (Fr > 1.0). In this case Es increases as y increases.
Energy Principles in Open Channel Flow
Specific Energy Curve (rectangular channel)
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Energy Principles in Open Channel Flow
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Froude Number (Fr) h
r D g
V F
T
A
WidthSurfaceWater
Area)(WettedFlowof Areah D
T
T
Flow Fr
Sub-critical1 > Fr
Critical1 = Fr
Supercritical1 < Fr
g A
T Q F r 3
22
Energy Principles in Open Channel Flow
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Energy Principles in Open Channel Flow
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Critical Flow
Subcritical
Super criticalcritical
Energy Principles in Open Channel Flow
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Energy Principles in Open Channel Flow
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Rectangular Channel
31
2
g
q y
C
h
r D g
V F 1
At critical Flow D g
V
D g
V F h
r For rectangular section
3
1
2
2
g B
Q yc
q=Q/B
Energy Principles in Open Channel Flow
a) Critical depth, yc , is defined as that depth of flow of liquid at
which the specific energy is minimum, Emin,
b) Critical velocity, Vc , is the velocity of flow at critical depth.
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Energy Principles in Open Channel Flow
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Rectangular Channel
Energy Principles in Open Channel Flow
c) Critical, Sub-critical, and Super-critical Flows:Critical f low is defined as the flow at which the specific energyis minimum or the flow that corresponds to critical depth. Referto point C in above figure, E min @ yc .
and therefore for critical flow Fr = 1.0
If the depth flow y > yc , the flow is said to be sub-critical. In this case Esincreases as y increases. For this type of flow, Fr < 1.0 .
If the depth flow y < yc , the flow is said to be super-critical. In this caseEs increases as y decreases. For this type of flow, Fr > 1.0 .
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Energy Principles in Open Channel Flow
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Rectangular Channel
gy p p
d) Minimum Specific Energy in terms of critical depth: At (Emin , yc ) ,
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Energy Principles in Open Channel Flow
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Other Sections
)2(2
)53(
c
ccc
Dn B
D Dn B E
sin
)2sin2(
16)cos1(
2 d d E c
cc D E
4
5
Trapezoidal section
Circular section
Triangle section
cc D E
2
3 Rectangular section
at critical flow Fr =1 where: 13
2
2 g A
T Q F r
gy p p
58
Energy Principles in Open Channel Flow
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Example 1
13
2
g A
T Q
1
81.93
14.2
3
124.233.1
23
2
3
2
cc
c
cc
c
D D
D
g DnD B
nD BQ
m31.0c D
Determine the critical depth if the flow is 1.33m3/s. thechannel width is 2.4m
gy p p
59
Energy Principles in Open Channel Flow
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Example 2
4481.9
253
2
3
1
2
3
1
2
23
1
2
2
B B B g B
Q yc
0.0062
D
0.016
125
2
P
AR
SR n
1V
32
c
32
B D
B
B D
B D B D
cc
c
c
Rectangular channel , Q=25m3/s, bed slope =0.006,determine the channel width with critical flow usingmanning n=0.016
gy p p
60
Energy Principles in Open Channel Flow
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m B
B
B
B
B B
B B
B B
3
0.006
8
4
0.016
1
4
25
0.006
42
4
0.016
1
4
25
32
35
31
32
32
32
32
Example 2 cont.
gy p p
61
Non-uniform Flow in Open Channels:
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p
Non-uniform flow is a flow for which the depth of flow is aried. This
varied flow can be either Gradually varied flow (GVF) or Rapidly varied flow (RVF).
Such situations occur when control structures are used in the
channel or when any obstruction is found in the channel
Such situations may also occur at the free discharges and when a
sharp change in the channel slope takes place.
The most important elements, in non-uniform flow, that will be
studied in this sectionare:
Classification of channel-bed slopes. Classification of water surface profiles. The dynamic equation of gradually varied flow.
Hydraulic jumps as examples of rapidly varied flow. 62
Non-uniform Flow in Open Channels:
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p
63
Non-uniform Flow in Open Channels:
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pBed Slopes-Classification of Channel
The slope of the channel bed can be classified as:
1) Critical Slope: the bottom slope of the channel is equal to the criticalslope. In this case S0 = Sc or yn = yc .
2) Mild Slope: the bottom slope of the channel is less than the criticalslope. In this case S0 < Sc or yn > yc .
3) Steep Slope: the bottom slope of the channel is greater than thecritical slope. In this case S0 > Sc or yn < yc .4) Horizontal Slope: the bottom slope of the channel is equal to zero(horizontal bed). In this case S0 = 0.0 .
5) Adverse Slope: the bottom slope of the channel rises in the direction
of the flow (slope is opposite to direction of flow). In this case S0 =negative .The first letter of each slope type sometimes is used to indicate theslope of the bed. So the above slopes are abbreviated as C, M, S, H, and
A, respectively. 64
Non-uniform Flow in Open Channels:
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pBed Slopes-Classification of Channel
65
Non-uniform Flow in Open Channels:
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pClassification of Flow Profiles (water surface profiles):
66
Non-uniform Flow in Open Channels:
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pClassification of Flow Profiles (water surface profiles):
67
Non-uniform Flow in Open Channels:
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Classification of Flow Profiles (water surface profiles):
68
Non-uniform Flow in Open Channels:
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Classification of Flow Profiles (water surface profiles):
69
Non-uniform Flow in Open Channels:
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Classification of Flow Profiles (water surface profiles):
70
Hydraulic Jump
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A hydraulic jump occurs when flow changes from a supercriticalflow (unstable) to a sub-critical flow (stable). There is a sudden
rise in water level at the point where the hydraulic jump occurs.Rollers (eddies) of turbulent water form at this point. Theserollers cause dissipation of energy.
71
Hydraulic Jump
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General Expression for Hydraulic Jump:In the analysis of hydraulic jumps, the following assumptions are
made:(1) The length of hydraulic jump is small. Consequently, the loss
of head due to friction is negligible.
(2) The flow is uniform and pressure distribution is due tohydrostatic before and after the jump.
(3) The slope of the bed of the channel is very small, so that the
component of the weight of the fluid in the direction of the flowis neglected.
72
Hydraulic Jump
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Hydraulic Jump in Rectangular Channels
But for Rectangular section
73
Hydraulic Jump
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Hydraulic Jump in Rectangular Channels
74
Hydraulic Jump
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Hydraulic Jump in Rectangular Channels
75
Hydraulic Jump
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Hydraulic Jump in Rectangular Channels
76
Hydraulic Jump
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77
Hydraulic Jump
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78
Hydraulic Jump
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Example 1 A 3-m wide rectangular channel carries 15 m 3/s of water at a 0.7 m
depth before entering a jump. Compute the downstrem water depthand the critical depth
72.27.081.9
14.7
/14.77.0
5
366.181.9
5
/s.mm53
15
1
1
1
1
1
3
2
3
gd
V
F
smd
qV
md
q
r
c
md
d
365.2
1)72.2(812
1
7.0
2
22
79
Hydraulic Jump
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Example 2
dn = Depth can calculated from manning equation
d1=dn
d2
80
Hydraulic Jump
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calsupercritiisflowthe142.108.181.9
63.4
/63.43
15
08.1
004.032
3
01.0
1
3
15
322
3
1
1
11
1
1
1
32
32
gd
V F
smd
V
md d D
D
D
D
D B D P
D BD A
S Rn A
Q
r
n
h
d1=dn
d2
a)
b) md
d
7.1
1)42.1(812
1
08.1
2
22
81
Hydraulic Jump
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c)
d1=dn
d2
m g g
E
smd
V
032.0294.27.1
263.408.1
/94.27.13
15
3
15
22
2
2
82
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