chapter 9 flow in open channel
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CHE 493: FLUID MECHANICS
Chapter 9: Flow in Open Channels
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Learning outcome
1. Explain the concept of uniform flow
2. Describe velocity transmission of a wave
3. Describe and calculate hydraulic jump phenomenon
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Introduction
• Open channel flow implies flow of liquids inchannels open to the atmosphere or in partiallyfilled conduits
• Characterized by the presence of a liquid-gasinterface called the free surface
• Most of natural flows encountered in practiceare open-channels flow
• Eg: Rivers, floods, draining of rainwater throughroofs, highways
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Uniform/varied flow• Uniform flow - if the flow depth (average velocity)
remains constant
• Encountered in long straight sections of channels withconstant slope and cross section – the liquidaccelerates until the head loss due equals theelevation drop – reaches terminal velocity – uniformflow is established
• Remains uniform as long as the slope, cross sectionand surface roughness of the channel remainunchanged
4
Uniform/varied flow
• Flow depth is called the normal depth –important characteristic for open-channelflows
• Non-uniform/varied flow - Flow depth varieswith distance in the flow direction
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Why are open-channel flows important?
• Many natural systems responsible for the transport of sediment are channelized, in both sub- aerial and subaqueous environments.
• Nearly all of the modeling performed on the entrainment and transport of sediment is either in open channels or in 1-D boundary layers.
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Uniform Flow in Channel
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Head loss = Elevation Loss
Flow depth = y
Average flow velocity = V
Bottom slope = S0 = tan α
During open channel, Head Loss = Elevation Drop
--------------- (1)
Since hL = S0L and Dh = 4Rh ------------------------- (2)
Sub (2) in (1): ------------------------ (3)
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• Rearrange (3), uniform flow velocity:
where
• Flow rate:
Chezy CoefficientAntoine Chezy (1718-1798)
hoc RSCAQ
Note: Determine using Moody chart,
open channel typically is
turbulent flow and fully develop.
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Gauckler and Manning made recommendations:
Where: a = dimensional constant = 1 m1/3/s
n = Manning coefficient (depends on roughness of the channel surface)
For uniform flow velocity & flow rate:
and 2/13/2
oc SRAn
aQ
h
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Mean value for Manning coefficient
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Types of Channels
Hydraulic radius
Circular channel
Rectangular channel
Trapezoidal channel
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Example 1
Water is flowing in a weedy excavated earthchannel of trapezoidal cross-section with a bottomwidth of 0.8m, trapezoid angle of 60˚ and abottom slope angle of 0.3˚. If the flow depth ismeasured to be 0.52 m, determine the flow rate ofwater through the channel. (Given n = 0.030)
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Specific energy
Consider flow of a liquid in a channel
Where:
y - flow depth
V - average velocity
Z – elevation of the bottom of channel at that location relative to some reference datum
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Total mechanical energy in terms of head:
• Not realistic representing true energy
• It can be realistic if the reference datum is taken to be the bottom of the channel so Z = 0
• Then, the total mechanical energy = Pressure + Dynamic Head
• This term is called specific energy, Es
- ------------------------ (1)
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• Consider flow in an open channel of constant width, b.
• Volume flowrate: .
• So, the average flow velocity
--------------- (2)
• Sub (2) into (1)
ybVVAQ c
yb
Q
A
QV
c
22
2
2 bgy
QyEs
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There is minimum specific energy Es,min required to support specific flow rate, Q
Therefore, Es cannot be below Es,min for a given Q
So,
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Critical flow depth
Critical velocity
To find character and flow, using Froude Number
Lc = Critical LengthFr < 1 = Subcritical or tranquil flowFr = 1 = Critical flowFr > 1 = Supercritical or rapid flow
3/1
2
2
gb
Qyc
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Example 2
Water is flowing steadily in a 0.65 m widerectangular open channel at a rate of 0.25 m3/s.If the flow depth is 0.15 m, determine
(a) The flow velocity and type of flow
(b) The alternate flow depth (Es1=Es2 ) if thecharacter of flow were to change
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Hydraulic jump
• It called rapidly varied flow (RVF) if the flow depthchanges markedly over a relatively short distance inthe flow direction.
• Occur when there is a sudden change in flow, such asan abrupt change in cross section.
• RVF is complicated—since there will be affect ofbackflow and flow separation.
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Hydraulic jump• In compressible flow, a liquid can accelerate from
subcritical to supercritical flow
• It can also decelerate from supercritical tosubcritical flow by undergoing a shock which isknown as hydraulic jump
• Hydraulic jump involves considerable mixing andagitation and thus significant amount of mechanicalenergy dissipation
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Hydraulic jump formed on a spillway model
for the Karnafuli Dam in East Pakistan.
Classification of hydraulic jumps: (a) Fr =
1.0 to 1.7:undular jumps; (b) Fr= 1.7 to
2.5:weak jump; (c) Fr= 2.5 to 4.5:
oscillating jump; (d) Fr=4.5 to 9.0:
steady jump; (e) Fr= 9.0: strong
jump.
Assumption from figure:
• Velocity is nearly constant across the channel at section 1 & 2 –therefore the momentum flux correction factors β1 = β2
• Pressure in the liquid varies hydrostatically, we consider gage pressure only since atmospheric pressure acts on all surfaces and its effect cancel out.
• The wall shear stress and associated losses negligible relative to the losses that occur during the hydraulic jump due to intense agitation.
• The channel is wide and horizontal
• No external or body forces23
y2
ρgy2ρgy1
y1
(1) (2)
hL
Energy line
Control
volume
v1 v2
Consider steady
flow through a
control volume
that encloses
the hydraulic
jump
x
• From momentum equation
• For channel width b
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• Substituting and simplifying:
• Eliminating V2 by from the continuity equation gives:
• Canceling factor y1 – y2 from both side and rearranging gives:
where
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• Therefore, depth ratio:
• The energy equation for this horizontal flow section can be expressed as:
• Noting that;
and
• The head loss associated with hydraulic jump is expressed as:
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• The specific energy of the liquid before the hydraulic jump is
• Then , the energy dissipation ratio:
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Example 3
Water is discharged into a 8 m wide rectangularhorizontal channel from a sluice gate is observed tohave undergone a hydraulic jump. The flow depth andvelocity before the jump are 0.8 m and 7 m/srespectively. Determine:
(a) The flow depth and the Froude number after thejump
(b) The head loss and the dissipation ratio
(c) The wasted power production potential due to thehydraulic jump
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