Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA
Transcript of Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA
Chapter 13: Open Channel Flow
Bryan PearceDepartment of Civil and Environmental Engineering
University of MaineFall 2009
Objectives
Understand how flow in open channels differs from flow in pipesLearn the different flow regimes in open channels and their characteristicsPredict if hydraulic jumps are to occur during flow, and calculate the fraction of energy dissipated during hydraulic jumpsLearn how flow rates in open channels are
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 2
Learn how flow rates in open channels are measured using sluice gates and weirs
Classification of Open-Channel Flows
Open-channel flows are characterized by the presence of a liquid-gas interface called the free surface.Natural flows: rivers, creeks, floods, etc.Human-made systems: fresh-water aqueducts, irrigation, sewers, drainage ditches, etc.
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 3
drainage ditches, etc.
Classification of Open-Channel Flows
In an open channel, Velocity is zero on bottom and sides of channel due to no-slip conditionVelocity is maximum at the midplane of the free surfaceIn most cases, velocity also varies in the streamwise directionTherefore, the flow is 3D
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 4
Therefore, the flow is 3DNevertheless, 1D approximation is made with good success for many practical problems.
Classification of Open-Channel Flows
Flow in open channels is also classified as being uniform or nonuniform, depending upon the depth y.
Uniform flow (UF) encountered in long straight sections where head loss due to friction is balanced by elevation drop.
Depth in UF is called normal depth y n
( “Normal refers to uniform”)
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 5
Classification of Open-Channel Flows
Obstructions cause the flow depth to vary.Rapidly varied flow (RVF) occurs over a short distance near the obstacle.Gradually varied flow (GVF) occurs over larger distances and usually connects UF and RVF.
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 6
Classification of Open-Channel Flows
Like pipe flow, OC flow can be laminar, transitional, or turbulent depending upon the value of the Reynolds number
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 7
Where ρ = density, µ = dynamic viscosity, ν = kinematic viscosityV = average velocityRh = Hydraulic Radius = A c/p
Ac = cross-section areaP = wetted perimeterNote that Hydraulic Diameter was defined in pipe flows as Dh = 4Ac/p = 4Rh (Dh is not 2Rh, BE Careful!)
Classification of Open-Channel Flows
The wetted perimeter does not include the free surface.Examples of Rh for common geometries shown in Figure at the left.
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 8
Froude Number and Wave Speed
OC flow is also classified by the Froude number
Resembles classification of compressible flow with respect to
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 9
compressible flow with respect to Mach number
Froude Number and Wave Speed
Critical depth yc occurs at Fr = 1
At low flow velocities (Fr < 1)
2
22 1
cc A
Q
gg
Vy ==
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 10
At low flow velocities (Fr < 1)Disturbance travels upstreamy > yc
At high flow velocities (Fr > 1)Disturbance travels downstreamy < yc
Froude Number and Wave Speed
Important parameter in study of OC flow is the wave speed c0, which is the speed at which a surface disturbance travels through the liquid.Derivation of c for shallow-water
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 11
Derivation of c0 for shallow-water Generate wave with plungerConsider control volume (CV) which moves with wave at c0
Froude Number and Wave Speed
Continuity equation (b = width)
Momentum equation
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 12
Momentum equation
b
Froude Number and Wave Speed
Combining the momentum and continuity relations and rearranging gives
For shallow water, where δy << y, (Wave speed c0 is only a function of depth, c0is both phase and group velocity.)
Shallow water is typically defined as the wavelengt h, λ λ λ λ > 20y
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 13
Shallow water is typically defined as the wavelengt h, λ λ λ λ > 20y
For “deep water” λ λ λ λ < 2y. C= λλλλ /T=gT/2π π π π (Phase velocity)
In between is “transitional. C= λλλλ /T=gT/2π[π[π[π[tanh(2 ππππy/λλλλ)] (Phase velocity)
Froude Number and Wave Speed
How long will it take a tsunami generated in the Aleutian Islands to reach Hawaii?
Assume: The
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 14
Andr1.movDart_04.swf
Assume: The average depth of Pacific is about 13,000 feet & it is about 4000 km.
Froude Number and Wave Speed
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 15
Indo_gl2.mov
Froude Number and Wave Speed
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 16
Discuss Refraction and Shoaling
Froude Number and Wave Speed
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 17
Specific Energy
Total mechanical energy of the liquid in a channel in terms of heads
z is the elevation heady is the gage pressure headV2/2g is the dynamic head
Taking the datum z=0 as the bottom of the channel, the specific energy E is
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 18
the channel, the specific energy Es is
Specific Energy
For a channel with constant width b,
Plot of Es vs. y for constant V and b
ybVVAQ c ===V&
22
2
2 ygb
QyEs +=
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 19
222 ygbyEs +=
2
2
2 :get weand
Q/bq case,lower use common to isit channelsr rectangulafor that Note
gy
qyEs +=
=
Specific Energy
This plot is very usefulEasy to see breakdown of Es into pressure (y) and dynamic (V2/2g) headEs → ∞ as y → 0Es → y for large yEs reaches a minimum called the critical point.
There is a minimum Es required to support the given flow rate.Noting that Vc = sqrt(gyc)
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 20
Noting that Vc = sqrt(gyc)
For a given Es > Es,min, there are two different depths, or alternating depths,which can occur for a fixed value of Es
A small change in Es near the critical point causes a large difference between alternate depths and may cause violent fluctuations in flow level. Operation near this point should be avoided.
1D steady continuity equation can be expressed as
1D steady energy equation between two stations
Continuity and Energy Equations
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 21
Head loss hL is expressed as in pipe flow, using the friction factor, and either the hydraulic diameter or radius
Continuity and Energy Equations
The change in elevation head can be written in terms of the bed slope α
Introducing the friction slope Sf
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 22
The energy equation can be written as
Uniform Flow in Channels
Uniform depth occurs when the flow depth (and thus the average flow velocity) remains constant
Common in long straight runs
Flow depth is called normal depth
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 23
Flow depth is called normal depth yn (as in regular not perp.)
Average flow velocity is called uniform-flow velocity V0
Uniform Flow in Channels
Uniform depth is maintained as long as the slope, cross-section, and surface roughness of the channel remain unchanged.During uniform flow, the terminal velocity is reached, and the head loss equals the elevation drop
We can the solve for velocity (or flow rate)
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 24
Where C is the Chezy coefficient. f is the friction factor determined from the Moody chart or the Colebrook equation
Uniform Flow in Channels
Antoine Chezy came up with this relationship in about 1769 when given the task of getting water to a palace near Paris.
Values of the “Chezy” Coefficient are tabulated.Many people, especially in Europe, use the Chezy formula.About one hundred years later an Irishman saw fit to mess with success (who wants a French formula?) and came up with “Manning’s” equation, by finding a formula for C in terms of a different fudge factor, n.
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 25
finding a formula for C in terms of a different fudge factor, n.
“American’s”, generally but not all, use Manning’s “n”.Values of Manning’s “n” as well as “C” are tabulated.
sftssmaWhere
SRn
aAorSR
n
aR
n
aC hhh
/49.1/)2808.3(/1
QV
3/13/13/1
2/1
0
3/22/1
0
3/2
0
6/1
===
===
Uniform Flow in Channels
sftorsmaRRn
aAorRR
n
ahh /49.1/1QV 3/13/12/1
03/22/1
03/2
0 ===
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 26
Water is flowing in a weedy excavated earth channel of trapezoidal cross section with a bottom width of 0.8 m, trapezoid angle of 60°, and a bed slope angle of 0.3 °, as shown. If the flow depth is measured to be 0.52 m, determine the flow rate of water through the channel. What would your answer be if the bed slope angle were 1°?
Note: from chart , n=0.03
Q=0.6 m3/s for 0.3°, & 1.1 m 3/s for 1.0°.Review Solution
13-6 Best Hydraulic Cross Sections
Best hydraulic cross section for an open channel is the one with the minimum wetted perimeter for a specified cross section (or maximum hydraulic radius Rh)
Also reflects economy of building structure with smallest perimeter
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 27
13-6 Best Hydraulic Cross Sections
Example: Rectangular ChannelCross section area, Ac = ybPerimeter, p = b + 2ySolve Ac for b and substitute
Taking derivative with respect to y
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 28
To find minimum, set derivative to zero
Best rectangular channel has a depth 1/2 of the width
13-6Best Hydraulic Cross Sections
Similar analysis can be performed for a trapezoidal channel yielding
For θ=90o this results in y=b/2 as before.
Plugging into the Rh formulas yields Rh=y/2 (for any θ, Rh=y/2 yields the best cross section).
Similarly, taking the derivative of p with respect to q, shows that the optimum angle is (a hexagon).
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 29
q, shows that the optimum angle is (a hexagon).
For this angle (θ=60o ), the best flow depth is (half a hexagon):
13-6 Best Hydraulic Cross Sections
Water is to be transported at a rate of 2 m3/s in uniform flow in an asphalt lined open channel. The bottom slope is 0.001. Determine the dimensions of the best cross section if the channel is (a) rectangular and (b) trapezoidal.Note: n=0.016 from table of experimental values.
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 30
13-6 Best Hydraulic Cross Sections
Water is to be transported at a rate of 2 m3/s in uniform flow in an asphalt lined open channel. The bottom slope is 0.001. Determine the dimensions of the best cross section if the channel is (a) rectangular and (b) trapezoidal.Note: n=0.016
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 31
13-7 Gradually Varied Flow
In GVF, y and V vary slowly, and the free surface is stableIn contrast to uniform flow, Sf ≠ S0. Now, flow depth reflects the dynamic balance between gravity, shear force, and inertial effectsTo derive how how the depth varies with x, consider the total head
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 32
13-7 Gradually Varied Flow
Take the derivative of H
Slope dH/dx of the energy line is equal to negative of the friction slope
dx
dV
g
V
dx
dy
dx
dz
dx
dH b ++=
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 33
Bed slope has been defined
Inserting both S0 and Sf gives
13-7 Gradually Varied Flow
Introducing continuity equation, which can be written as
Differentiating with respect to x gives
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 34
Substitute dV/dx back into equation from previous slide, and using definition of the Froude number gives a relationship for the rate of change of depth
13-7 Gradually Varied Flow
This result is important. It permits classification of liquid surface profiles as a function of Fr, S0, Sf, and initial conditions.
Bed slope S0 is classified asSteep : yn < yc
Critical : yn = yc
Mild : yn > yc
Horizontal : S0 = 0Adverse : S < 0
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 35
Adverse : S0 < 0
Initial depth is given a number1 : y > yn
2 : yn < y < yc
3 : y < yc
13-7 Gradually Varied Flow
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 36
13-7 Gradually Varied Flow
Typical OC system involves several sections of different slopes, with transitionsOverall surface profile is made up of individual profiles described on previous slides
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 37
13-8 Rapidly Varied Flow and Hydraulic Jump
Flow is called rapidly varied flow (RVF) if the flow depth has a large change over a short distance
Sluice gatesWeirsWaterfallsAbrupt changes in cross section
Often characterized by significant 3D and transient effects
BackflowsSeparations
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 38
13-8 Rapidly Varied Flow and Hydraulic Jump
Consider the CV surrounding the hydraulic jumpAssumptions
1. V is constant at sections (1) and (2), and β1 and β2 ≈ 1
2. P = ρgy3. τw is negligible relative to the losses that
occur during the hydraulic jump4. Channel is wide and horizontal5. No external body forces other than gravity
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 39
13-8 Rapidly Varied Flow and Hydraulic Jump
Continuity equation
X momentum equation
Substituting and simplifying
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 40
Quadratic equation for y2 / y1
13-8 Rapidly Varied Flow and Hydraulic Jump
Solving the quadratic equation and keeping only the positive root leads to the depth ratio
Energy equation for this section can be written as
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 41
Head loss associated with hydraulic jump
13-8 Rapidly Varied Flow and Hydraulic Jump
Often, hydraulic jumps are avoided because they dissipate valuable energyHowever, in some cases, the energy must be dissipated so that it doesn’t cause damage
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 42
damageA measure of performance of a hydraulic jump is its fraction of energy dissipation, or energy dissipation ratio
13-8 Rapidly Varied Flow and Hydraulic Jump
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 43
13-9 Flow Control and Measurement
Flow rate in pipes and ducts is controlled by various kinds of valvesIn OC flows, flow rate is controlled by partially blocking the channel.
Weir : liquid flows over deviceUnderflow gate : liquid flows under device
These devices can be used to control the flow rate, and to measure it.
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 44
13-9 Flow Control and Measurement Underflow Gate
Underflow gates are located at the bottom of a wall, dam, or open channelOutflow can be either free or drownedIn free outflow, downstream flow is supercriticalIn the drowned outflow, the liquid jet undergoes a hydraulic jump. Downstream flow is subcritical.
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 45
Free outflowDrowned outflow
13-9 Flow Control and Measurement Underflow Gate
Ct coefficien discharge Add
22
and VV :assume andfriction Ignore
22
2. and 1 pointsbetween BernoulliApply
121
22
1221
22
22
21
11
gyVyg
Vyy
g
Vy
P
g
Vy
P
=→=
<<<<
++=++γγ
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 46
gate. theof bottom theofheight thea and
width,channel theis b where2CQ flow,for or
2C
Ct coefficien discharge Add
1d
1d2
d
)(
gyab
gyV
contractavenaandfriction
==
Note that Cd for free outflows is between 0.5 and 0.6.
Cd for various outflows can be picked off chart of experimental values.
13-9 Flow Control and Measurement Underflow Gate
Es remains constant for idealized gates with negligible frictional effectsEs decreases for real gatesDownstream is supercritical for free outflow (No Friction) (2a)Downstream is supercritical for free outflow (With Friction – Real) (2b)Downstream is subcritical for drowned outflow (2c)
Schematic of flow depth-specificenergy diagram for flow throughunderflow gates
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 47
13-9 Flow Control and Measurement Free Outflow
Water discharged through a sluice gate undergoes a hydraulic jump. Find y2 and V2. Assume conservation of energy between 1 and 2, or E1=E2.
y1 = 8 ft
Sluicegate
y2
y3a =1 ft
Hydraulicjump
For free outflow, we only need the depth ratio y1/a to determine the discharge coefficient. The corresponding discharge coefficient is found from Fig. 13-38 to be Cd= 0.58. Then the discharge rate becomes:
8ft 1
ft 81 ==a
y
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 48
= 0.58. Then the discharge rate becomes:
/sft 13.16 3=== ft) 8)(ft/s (32.22ft) ft)(1 1(58.02 21gybaCdQ
ft 042.8ft)] ft)(8 (1)[ft/s 2(32.2
)/sft (13.16ft 8
)(22 22
23
21
2
1
21
11 =+=+=+=byg
yg
VyEs
Q
122
2
2
22
22 )(22 ss Ebyg
yg
VyE =+=+= Q
ft 042.8)]ft)( (1)[ft/s 2(32.2
)/sft (13.162
22
23
2 =+y
y
13-9 Flow Control and Measurement Free Outflow
y1 = 8 ft
Sluicegate
y2
y3a =1 ft
Hydraulicjump
8ft 1
ft 81 ==a
y
ft 042.8)]ft)( (1)[ft/s 2(32.2
)/sft (13.162
22
23
2 =+y
y
Solve: y2=0.601 feet
ft/s 21.9====ft) ft)(0.601 (1
/sft 16.13 3
22 byA
Vc
QQ 97.4ft) )(0.601ft/s (32.2
ft/s 9.21Fr
22
22 ===
gy
V
Examine flow after the jump.
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 49
ft 3.94=
×++−=
++−= 22
223 97.4811ft) 601.0(5.0Fr8115.0 yy
ft/s 3.34=== )ft/s 9.21(ft 3.94
ft 601.02
3
23 V
y
yV
Examine flow after the jump.
ft 93.3)ft/s 2(32.2
ft/s) (3.34-ft/s) (21.9ft) (3.94-ft) 601.0(
2 2
2223
22
32 =+=−
+−=g
VVyyhL
Find the difference in energy before and after the jump to calculate headloss.
13-9 Flow Control and Measurement Example 13-8 Drowned Outflow
Water is released from a 3-m-deep reservoir into a 6-m-wide open channel through a sluice gate with a 0.25-m-high opening at the channel bottom. The flow depth after all turbulence subsides is measured to be 1.5 m. Determine the rate of discharge,Q.
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 50
13-9 Flow Control and Measurement Overflow Gates – Bumps and Weirs
:is change theand
2
ERemember
22
.elevations bottom downstream and upstream theare
& note andfriction Ignore
2
s
22
22
21
11
21
g
Vy
g
Vyz
g
Vyz
zz
bb
bb
+=
++=++
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 51
figure. in theshown curve on theliemust Qconstant afor depths flow All
2E
b, idth,constant wFor EEE
:is change theand
22
2
s
2211
12s2s1s
ygb
Qy
VbyVbyQ
zzz bbb
+=
==
∆=−=−=∆
13-9 Flow Control and Measurement Bump
height". bump"or given afor y find want to weand
Vknown ,known b,constant with channelr rectangula a have We
22
11
bzy
∆cal.supercriti one and lsubcritica one
solutions real twohas yfor equation cubic The 2
When the bump gets high enough we reach critical flow. We cannot increase the flow without increasing the energy. If we continue to raise the bump the flow will “Choke”.
(The water level upstream must rise to increase the energy.)V
yzV
y
y
VyVVyVy
zg
Vyz
g
Vy bb E
22E
22
21
2
1122211
2s2
22
22
21
1s1
++∆=+
=→=
∆+=++∆=+=
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 52
02
)E(
2E
2E
2
21
212
22s132
22
21
21
22s1
22
21
21
22s1
21
1
=+∆−−
+=∆−
++∆==+
g
Vyyzy
gy
Vyyz
gy
Vyyz
g
Vy
b
b
b
to increase the energy.)
Draw bump and EGL
g
Vyz
g
Vy b 22
222
11 ++∆=+
13-9 Flow Control and Measurement Broad-Crested Weir
Flow over a sufficiently high obstruction in an open channel is always criticalWhen placed intentionally in an open channel to measure the flow rate, they are called weirs
22
221 ++=++
g
VPy
g
VPH c
wcw
g
VHbg
g
VHbg
)2
(3
2CQ
(Ideal) )2
(3
2Q
2/32
1
2/3
2/1broadwd,Weir-BC
2/32
1
2/3
2/1Weir-BC
+
=
+
=
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 53
widthchannel AQ
)2
(3
22
3
222
weir over the critical is flow The22
2/32/1
cWeir
2
1
21
21
2
====
+=
=+→+=+
=
b
ybggybyV
g
VHy
y
g
VH
g
gyy
g
VH
gyVgg
cccc
c
ccc
cc
wPH
gHbg
/1
65.0C
)2
(3
CQ
broadwd,
broadwd,Weir-BC
+=
+
=
Can sometimes neglect V1.
For Lw<2H the flow may not reach Vc.
For Lw>12H the flow may be < Vc.
The best weir has 2H<Lw>12H .
wd=weir discharge coefficient - empirical
Can you examine these limits in the lab?
13-9 Flow Control and Measurement Sharp-Crested
Vertical plate placed in a channel that forces the liquid to flow through an opening or over the weir to measure the flow rateUpstream flow is subcritical and becomes critical as it approaches the weirLiquid discharges as a supercritical flow stream that resembles a free jet
22 uV 2
thenH2
and P If2
1w <<>>
g
VH
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 54
])2
()2
[()2(3
22Q
limits.) assigningin ion approximat (NoteQ. find toHh to0h from u Integrate
222
22
2/32
12/32
12/121
Hh
0h
Ideal
2
212
22
21
22
21
g
V
g
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uh
g
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g
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+−+=++
∫=
=
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,
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2/32/1Ideal
≤+=
=
=
wwrecwd
recwd
P
H
P
HC
HgbC
Hgb
13-9 Flow Control and Measurement V-Notch
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 55
58.0)20145 & 0.2m(HFor
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22
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)V neglectedagain have we(Hered)complicate more is (Integral
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13-9 Flow Control and Measurement Bump – Example 13-9
Water flowing in a wide horizontal open channel encounters a 15-cm-high bump at the bottom of the channel. If the flow depth is 0.80 m and the velocity is 1.2 m/s before the bump, determine if the water surface isdepressed over the bump
channel.r rectangula afor :NOTE
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 56
3/12
2
3222/322
)(
)(
channel.r rectangula afor :NOTE
gb
Qy
gybygbQ
gybyVAQ
c
cc
ccccc
=
==
==
13-9 Flow Control and Measurement Bump – Example 13-9
2
1122211
22
212
2
221
11
VVVV and
2723.015.0
873.0/81.9*2
)/2.1(80.0
2
y
yyy
g
VymmEE
msm
smm
g
VyE
ss
s
=→=+==−=
=+=+=
V 22 y
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 57
We get three solutions:0.59m, 0.36m, and -0.22m. How do we know that 0.59m is the solution?
0047.0723.0
047.064.0*073.0
*2
V723.0
23
222
2
22
221
2
22
222
1
=+−
+=+=+=
yy
yy
y
mmy
yg
yym
13-9 Flow Control and Measurement Sharp-Crested Weir Example 13-10
The flow rate of water in a 5-m-wide horizontal open channel is being measured with a 0.60-m-high sharp-crested rectangular weir of equal width. If the water depth upstream is 1.5 m, determine the flow rate of water
Evaluate assumption of V12/2g<<H
Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 58
0.077m is 8.6 percent of the weir head, which is significant. When the upstream velocity head is considered, the flow rate becomes 10.2 m3 /s, which is about 10 percent higher than the value determined. Therefore, it is good practice to consider the upstream velocity head unless the weir height Pw is very large relative to the weir head H.