Ch9 Non-Uniform Flow in Open Channels
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Transcript of Ch9 Non-Uniform Flow in Open Channels
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Chapter 9Non-uniform Flow in Open Channels
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Overview
9.1 Critical , Sub-critical and super critical flow
9.2 Specific energy
9.3 Gradually Varied Flow
9.4 Hydraulic jump
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NON-UNIFORM FLOW IN OPEN CHANNELS
• Definition: By non-uniform flow, we mean that the velocity varies at each section of the channel.
• Velocities vary • Non-uniform flow can be caused by
i) Differences in depth of channel and ii) Differences in width of channel. iii) Differences in the nature of bed iv) Differences in slope of channel and v) Obstruction in the direction of flow.
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Non-uniform Flow In Open ChannelsContd.
• In the non-uniform flow, the Energy Line is not parallel to the bed of the channel.
• The study of non-uniform flow is primarily concerned with the analysis of Surface profiles and Energy Gradients.
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Hydraulic characteristics of the non-uniform flow in open channels: The bed slope i, water surface gradient Jp and the hydraulic slope J are not equal, i.e. show as fig.
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9.1 Critical , Sub-critical and super critical flow
Sub-critical Flow: If the small surface wave can propagate upstream as well as downstream, it will lead to a backwater zone of great distances before the obstacles. This flow is called a sub-critical flow, in which the velocity of flow is less than that of the wave propagation, namely, V<C.
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C C
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C- v C+ v
Sub-critical Flow
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Supercritical Flow
The small surface wave will be formed when the flow in the open channel is disturbed. If the wave can only propagate downstream, and can't propagate against the flow, the backwater zone is formed only around the obstacles. This kind of flow in open channels is called supercritical flow, in which the velocity of flow is greater than that of the wave propagation, namely, v>c.
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C + v
v > C
Supercritical Flow
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Critical Flow
If the velocity of the small surface wave propagating upstream is zero, which is just the critical situation to distinct the supercritical flow and the sub-critical flow, this kind of flow is called the critical flow, namely, V=C .
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2 C
v= C
Critical Flow
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9.2 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 Es is
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Specific Energy
In a channel with constant discharge, Q2211 VAVAQ ==
2
2
2gAQyEs +=g
VyEs 2
2
+= where A=f(y)
Consider rectangular channel (A = By) and Q = qB
2
2
2gyqyEs +=
A
B
y
q is the discharge per unit width of channel
How many possible depths given a specific energy? _____2
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Specific Energy Curve
• For a channel with constant width b,
• Plot of Es vs. y for constant V and b
ybVVAc ==Q
22
2
2 ygbQyEs +=
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Specific Energy Curve
For a given flow, if
Es < Emin No solution is possible
Es = Emin Flow is critical, y = yc, V = Vc Fr = 1
E s > Emin , V < Vc :
Flow is subcritical, y > yc , Fr < 1 , disturbances can propagate upstream as well as downstream
Es > Emin , V > Vc :
Flow is supercritical, y < yc , Fr > 1, disturbances can only propagate downstream
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Froude Number
• This is a Dimensionless Ratio Characterizing Open Channel Flow.
• Froude Number, gh
VF r =
forcegravity forceinertial
Kinetic energyPotential energy
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Froude Number and Wave Speed
• Critical depth yc occurs at Fr = 1
• At low flow velocities (Fr < 1)• Disturbance travels upstream• y > yc
• At high flow velocities (Fr > 1)• Disturbance travels downstream• y < yc
2
22
cc gA
Qg
Vy ==
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Multiple Choices
When the flow in open channel is supercritical flow:
A. Fr>1B. h>hcC. v<vcD.
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Judgement
The specific energy must increase with the increase of water depth.
Your answer: True false
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9.3 Gradually Varied Flow
• In GVF, y and V vary slowly, and the free surface is stable
• In contrast to uniform flow, Sf ≠S0. Now, flow depth reflects the dynamic balance between gravity, shear force, and inertial effects
• To derive how how the depth varies with x, consider the total head
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Gradually Varied Flow
• Take the derivative of H
• Slope dH/dx of the energy line is equal to negative of the friction slope
• Bed slope has been defined
• Inserting both S0 and Sf gives
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Gradually Varied Flow
• Introducing continuity equation, which can be written as
• Differentiating with respect to x gives
• 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
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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 as• Steep : yn < yc• Critical : yn = yc• Mild : yn > yc• Horizontal : S0 = 0• Adverse : S0 < 0
• Initial depth is given a number• 1 : y > yn• 2 : yn < y < yc• 3 : y < yc
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Gradually Varied Flow
• 12 distinct configurations for surface profiles in GVF.
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Gradually Varied Flow
• Typical OC system involves several sections of different slopes, with transitions
• Overall surface profile is made up of individual profiles described on previous slides
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9.4 Hydraulic Jump
• Flow is called rapidly varied flow (RVF) if the flow depth has a large change over a short distance• Sluice gates• Weirs• Waterfalls• Abrupt changes in cross
section• Often characterized by
significant 3D and transient effects• Backflows• Separations
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Hydraulic Jump
• Used for energy dissipation• Occurs when flow transitions from supercritical to
subcritical• base of spillway• Steep slope to mild slope
• We would like to know depth of water downstream from jump as well as the location of the jump
• Which equation, Energy or Momentum?
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Fig. Flow under a sluice gate accelerates from subcritical to critical to supercritical and then jumps back to subcritical flow
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subcriticalflow
supercritical
A hydraulic jump in a flume is illustrated below.
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Hydraulic jump in the laboratory.
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Itaipu dam spillway withhydraulic jump.
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Hydraulic jump variations: (a) jump caused by a change in channel slope,(b) submerged jump.
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CONJUGATE DEPTH RELATION FOR THE HYDRAULIC JUMP
The illustrated channel below carries constant water discharge Q in a wide, rectangular channel of constant width B. The flow makes the transition from supercritical to subcritical through a hydraulic jump.
Momentum balance in the illustrated control volume is considered (width B out of the page).
hydraulic jump
Fr > 1Fr < 1
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CONJUGATE DEPTH RELATION FOR THE HYDRAULIC JUMP
• Continuity equation
• momentum equation
• Substituting and simplifying
Quadratic equation for y2/y1
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CONJUGATE DEPTH RELATION FOR THE 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
• Head loss associated with hydraulic jump
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Hydraulic Jump:Energy Loss and Length
No general theoretical solution
Experiments show
26 yL = 14.5 13Fr< <
äLength of jump
äEnergy Loss
2
2
2gy
qyE +=
LhEE += 21
significant energy loss (to turbulence) in jump
( )
21
312
4 yy
yyhL
−=algebra
for
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Hydraulic Jump
• Often, hydraulic jumps are avoided because they dissipate valuable energy
• However, in some cases, the energy must be dissipated so that it doesn’t cause damage
• A measure of performance of a hydraulic jump is its fraction of energy dissipation, or energy dissipation ratio
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Hydraulic Jump
• Experimental studies indicate that hydraulic jumps can be classified into 5 categories, depending upon the upstream Fr