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Rectangular Weir and Hydraulic Jumps 1
Florida Institute of Technology
CVE 3033 Lab 9
Lab #9: Rectangular Weir and Hydraulic Jumps
Objectives
1.
2.
3.
4.
Calculate flow over a rectangular weir
Simulate a hydraulic jump in the water channel
Take measurements of flow depths at the specified locations
Calculate the critical depths, Froude numbers, head losses, energy losses and flowrate in the
jump
Introduction
The hydraulic jump is a natural phenomenon that occurs when shallow and rapid flow, called
supercritical, is forced to change to a deep and slower flow, called subcritical, by an obstruction to
the flow. This abrupt change in flow condition is accompanied by considerable turbulence and loss
of energy. This phenomenon is governed by the Froude number which balances the inertial forces
in a fluid against the gravitational forces. The Froude number is defined as follows:
𝑣
𝐹=
√𝑔𝑦
Where F = Froude’s number
v = velocity of flow
y = depth of flow
g = gravitational acceleration
Therefore,
Supercritical (shallow and rapid) flow: F > 1
Critical Flow: F = 1
Subcritical Flow: F < 1
Table 1: Froude numbers and their associated flow conditions are defined below
< 1.0
No jump since flow is already sub-critical
1.0 to 1.7
An undular jump, with about 5% energy dissipation
1.7 to 2.5
A weak jump with 5% to 15% energy dissipation
2.5 to 4.5
Unstable, oscillating jump, with 15% to 45% energy dissipation
4.5 to 9.0
Stable, steady jump with 45% to 70% energy dissipation
> 9.0
Rough, strong jump with 70% to 85% energy dissipation
Diagrammatic representations of these jumps may be seen in Figure 1.
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Rectangular Weir and Hydraulic Jumps 2
Figure 1: Classification of Hydraulic Jumps
The hydraulic jump may be illustrated with the aid of a specific energy diagram (SED) as shown
in Figure 2. The flow enters the jump at supercritical velocity, 𝑣1 , and depth 𝑦1 , that has a specific
energy of 𝐸 = 𝑦1 +
𝑣1 2
. The kinetic energy term,
2𝑔
𝑣1 2
2𝑔
, is predominant. As the depth of flow
increases through the jump, the specific energy decreases. Flow leaves the jump area at subcritical
velocity with the potential energy predominant.
Figure 2: Specific Energy Diagram (SED)
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Rectangular Weir and Hydraulic Jumps 3
Nomenclature
Table 2: List of Symbols and Description
Parameter
Nom.
Units
yc
ft
y1
ft
y2
ft
Depth taken at subcritical section of flow; look for where the
flow has calmed after the jump
Weir/channel
Width
bw
ft
Width of weir is taken to be the same as the width of the channel
Weir Height
hw
ft
Height of weir (req'd if upstream height is recorded as the route
to get critical depth)
Cross-sectional
Area
A1
(ft2)
Cross-sectional Area at depth y1, supercritical region.
A2
(ft2)
Cross-sectional Area at depth y2, subcritical region.
Gravitational
Constant
g
(ft/s2)
Acceleration due to gravity = 32.2 (ft/s2)
Flowrate
Qc
(ft3/s)
Flowrate calculated using depth yc
Flowrate
Q1
(ft3/s)
Flowrate calculated using depth y1
Velocity
v1
(ft/s)
Velocity; recall Continuity Eqn.: Q = vA, manipulate so that the
velocity can be had to calculate Froude's Number
F1 or
Fr
Dless
Froude Numbers give an indication of the type of and other
characteristics associated with the jump.
hL
(ft)
Conjugate Depth
y2/y1
Exp. (ft)
Conjugate depth after the hydraulic jump; experimental value.
Conjugate Depth
y2/y1
Theo. (ft)
Conjugate depth after the hydraulic jump; theoretical value
calculated by applying the momentum eqn. to the hydraulic
jump.
E1
(ft)
Energy entering jump; Specific Energy calculated at depth, y1
E2
(ft)
Energy entering jump; Specific Energy calculated at depth, y2
Ec
(ft)
Energy entering jump; Specific Energy calculated at depth, yc
%Eloss
(%)
Difference/loss in energy as a result of the jump; energy
dissipated from jump:= %𝐸𝑙𝑜𝑠𝑠 = ℎ𝐿 ⁄𝐸1 × 100
Qalt
(ft3/s)
Critical Depth
Sequent Depth
Froude Number
Headloss
Specific Energy
Energy Loss
Flowrate
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Description
Depth taken from top of weir (approximately middle) to surface
of water
Depth taken at supercritical section of flow; usually lowest
elevation of the free surface wrt channel bottom
Note Bernoulli's Equation
Flowrate calculated using equation 11.
Rectangular Weir and Hydraulic Jumps 4
Hydraulic jumps dissipate a large amount of energy in open channel flows. This makes them very
useful in canal, dam and spillway designs. However, assistance is sometimes needed to make the
hydraulic jumps occur at the desired locations near spillways. This can be done by increasing the
surface roughness, adding a baffle wall, or sloping the basin floor. As shown above, the best design
range for the Froude number is between 4.5 and 9.0. In this range, a well-balanced steady jump
will occur with a large amount of energy dissipation. A jump with a Froude number between 2.5
and 4.5 is the worst design case since a jump in this range will create large waves that could cause
structural damage.
Theory
1. When F =1, then 𝐹 =
𝑣
can be rewritten as
√𝑔𝑦
𝑣2
𝑦 = 2𝑔
Eqn. 1
2. Assuming that the specific energy of the flow system is constant, by Bernoulli’s equation:
𝑇𝑜𝑡𝑎𝑙 𝐻𝑒𝑎𝑑 (𝑆𝑢𝑝𝑒𝑟𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙) = 𝑇𝑜𝑡𝑎𝑙 𝐻𝑒𝑎𝑑 (𝑆𝑢𝑏𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙)
3. Therefore,
𝑃1
𝛾
𝑣2
+ 2𝑔1 + 𝑧1 =
𝑝2
𝛾
𝑣2
+ 2𝑔2 + 𝑧2 + ℎ𝐿
Eqn. 2
Where P = pressure, 𝛾 = unit weight of the fluid, z = elevation from a datum.
4. From Figure 2 above,
𝑦1 =
𝑃1
𝛾
+ 𝑧1
𝑦2 =
and
𝑝2
𝛾
+ 𝑧2
5. And substituting the above into Equation 2, we can define a new quantity, which is the total
head with respect to a local datum:
𝑣2
𝑣2
𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 = 𝑦1 + 2𝑔1 = 𝑦2 + 2𝑔2 + ℎ𝐿
Eqn. 3
6. Taking the left side of the Eqn. 3 above,
𝑣2
𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 = 𝐸𝑡 = 𝑦1 + 2𝑔1
Eqn. 4
7. From conservation of mass, the continuity eqn. may be expressed as:
𝑄1
𝑄1 2
2
𝑣1 =
𝑡ℎ𝑒𝑛, 𝑣1 = 2
𝐴1
𝐴1
8. And letting A1 = 𝑏𝑦1 for a rectangular open channel where b = bw.
Substituting all of this into Eqn. 4 yields
𝑄2
𝐸𝑡 = 𝑦1 + 2𝑔𝑦
1
𝑄2
9. Letting K = 2𝑔𝑏
𝑤
2
2𝑏 2
𝑤
Eqn. 4a
, we get
𝐾
𝐸𝑡 = 𝑦1 + 𝑦
1
2
Eqn. 5
10. If Q, g and bw are constants, then K is a constant, and if we plot Eqn. 5 on a y vs. Et graph,
we will get the curve in Figure 3.
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Rectangular Weir and Hydraulic Jumps 5
Figure 3: y vs. Et graph
Determining head loss:
11. From Equation 3:
𝑣2
𝑣2
𝑦1 + 2𝑔1 = 𝑦2 + 2𝑔2 + ℎ𝐿
12. Therefore, headloss
ℎ𝐿 = (𝑦1 − 𝑦2 ) +
𝑄
𝑄
1
2
(𝑣1 2 −𝑣2 2 )
Eqn. 6
2𝑔
13. Now, 𝑣1 = 𝐴 𝑎𝑛𝑑 𝑣2 = 𝐴 , and substituting these values into Eqn. 6
(𝑄 2 )
ℎ𝐿 = (𝑦1 − 𝑦2 ) + [
2𝑔
1
1
1
2
× [(𝐴 2 ) − (𝐴 2 )]]
Eqn. 7
14. Predicting 𝑦2 given 𝑦1 :
Applying the momentum equation to the hydraulic jump:
𝑦2 =
15. Taking 𝐹1 =
𝑣1
√𝑔𝑦1
= √𝑔𝑏
𝑄2
𝑤
2
𝑦1 3
𝑦1
2
8𝑄 2
× [√1 + 𝑔𝑏
𝑤
2
𝑦1 3
− 1]
Eqn. 8
, Eqn. 8 can be rewritten as:
𝑦2 =
𝑦1
2
× [√1 + 8𝐹1 2 − 1]
Eqn. 9
16. For rectangular channels:
3
𝑞2
𝑦𝑐 = √2𝑔
where q = Q/bw
17. Simplifying this equation, we get:
𝑞2
𝑦𝑐 =
=≫ 𝑞 = √𝑦𝑐 3 𝑔
𝑔
18. Now substituting for q – Q/bw yields:
𝑄 = 𝑏𝑤 √𝑦𝑐 3 𝑔
3
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Eqn. 10
Rectangular Weir and Hydraulic Jumps 6
19. Determining Q given 𝑦2 and 𝑦1 for a rectangular channel with no slope:
From Eqn. 9, solve for Fr
𝑦2 1
= [√1 + 8𝐹1 2 − 1]
𝑦1 2
20. Use Fr to yield velocity, v using
𝑣
𝐹𝑟 =
√𝑔𝑦1
21. Now, Solve for Q:
𝑄 = 𝑣𝑏𝑤 𝑦1
Eqn. 11
Apparatus
• Flow Demonstration Channel as shown in Figure 2
• Rectangular weir
• Water
• Screw driver
• Stopwatch. Ruler
The flow demonstration channel unit consists of a transparent flume, a headtank with adjustable
undershot gate, a movable tailgate, a reservoir, a circulating pump, a flow meter and flow control
valves. A motorized jacking system adjust the slope of the channel bed. The complete assembly
is mounted on a castered frame for easy portability. All of the wetted parts of the equipment are
made of non-corrosive materials. The only utility required for operation is electric service.
Figure 4: Flow Demonstration Channel
Water is allowed to flow over the weir and later hindered by a trap door. Rulers will be used to
measure the depths for the supercritical and subcritical flows. The trapdoor at the end of the flow
tank acts as a barrier to the flow and thereby results in the hydraulic jump. In this experiment,
when the trapdoor is lifted, the flow on the trapdoor is faster than the horizontal flow. The
horizontal flow at the down-end of the tank is suddenly slowed considerably as compared to the
up-end of the tank. The hydraulic jump will then be simulated as shown:
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Rectangular Weir and Hydraulic Jumps 7
Figure 5: Simulated Hydraulic Jump: as more flow is slowed than sped-up, the hydraulic jump region would travel
upstream very quickly (like the crest of a wave).
Procedure
1. Adjust the slope indicator to the zero mark.
2. Ensure that all the dials are off.
3. Plug in the apparatus.
4. Level the apparatus.
5. Ensure that the tailgate horizontal by adjusting knob near the control panel.
6. Measure and record the dimensions of the rectangular weir on data sheet provided.
7. Place the weir in the channel and attach with screws.
8. Open the small and large orifice knobs slightly.
9. Turn on the pump.
10. Close the small orifice knob.
11. Slowly open the large orifice knob to increase flow in the channel.
12. Set the flow of the water to a certain level. Next, lift the trapdoor up to a sufficient angle
to cause a hydraulic jump.
13. Measure the different levels 𝑦1 and 𝑦2 of the hydraulic jump as it rolls back through the
length of the flow tank. Measure the critical depth 𝑦𝑐 . Record these values. Use your Data
Sheet and the Nomenclature Table as a guide.
14. Repeat steps 12 and 13 for 4 more trials of varying flows (such that 𝑦𝑐 is observed).
Fall 2021
Rectangular Weir and Hydraulic Jumps 8
Florida Institute of Technology
CVE 3033 Lab 9
Lab #9: Rectangular Weir and Hydraulic Jumps
DATA SHEET
Name: _____________________________
Date Performed: ______________________
Lab Partners: ___________________________________________________________________
Table 3: Lab Data
Width of weir, bw
Acceleration due to Gravity, g
(ft)
(ft/s2)
yc
Trial
y1
y2
(in)
(ft)
(in)
(ft)
(in)
(ft)
A1
(ft2)
A2
(ft2)
Q1
(ft3/s)
v1
(ft/s)
F1
hL
(ft)
E1
(ft)
E2
(ft)
Ec
(ft)
%Eloss
%
Qalt
(ft3/s)
1
2
3
4
5
6
7
Trial
1
2
3
4
5
6
7
Trial
1
2
3
4
5
Fall 2021
y2/y1
y2/y2
Exp. (ft)
Theo. (ft)
Rectangular Weir and Hydraulic Jumps 9
Calculations/Sketch:
Fall 2021
Rectangular Weir and Hydraulic Jumps 1
Lab Report Format and Guidelines
Section
Description
Points
Points
Available
Awarded
□
Executive
Summary
Introduction
Procedure/
Equipment
Analysis/
Results
Fall 2021
Provide a summary (250 words minimum) of this lab,
which includes only the key points of your procedure,
analyses and conclusions.
□ What were the major steps in your data collection?
□ Include any problems encountered during your lab that
compromised the data collection.
□ What were the results once you analyzed your data?
□ You must include specific data and results as part of
this executive summary.
□ What are your major conclusions?
□ Spend some extra time to organize and edit the
information you wish to leave with your freshman
reader. The executive summary should be written last.
Use MS Word’s “Word Count” tool and include your
word count at the end of your Executive Summary.
□ What were the goals of the lab?
□ Give some introductory material on open flow in
channels and the purpose of hydraulic jumps to
familiarize your reader with the topic.
□ Include a picture and description of the equipment used
to perform this experiment. Label if necessary.
□ Briefly provide a list or paragraph of the steps
performed during the lab.
Prepare an Excel table for each set of your results. Sample
calculations for each equation should be handwritten (or
typed) and included in the Appendix. Don’t forget your
units! Also, please label all tables, figures and/or charts!
Calculations and Analysis of Data (20 pts):
□ Calculate the flowrate, Qc, at depth 𝑦𝑐 , using Eqn.___.
□ Calculate velocity, v1 using Eqn. ___.
□ Calculate and tabulate Froude’s Number (F1) using
𝑦1 values
□ Calculate and tabulate the headlosses of the various
trials using Eqn. ___
□ Calculate and tabulate the y2/y1 ratios for each trial.
□ Using the values of Froude’s Number (F1) calculated in
Step 1, obtain the theoretical y2/y1 ratios for each trial
using Eqn. ___.
□ Calculate and tabulate specific energy (E1) using
𝑦1 values, etc.; Eqn. ___.
□ Calculate and tabulate percent energy loss using
equation: = %𝐸𝑙𝑜𝑠𝑠 = ℎ𝐿 ⁄𝐸1 × 100
□ Calculate the flowrate, Qalt, using Eqn.___. Should be
the same as Q1 in step 1.
10
5
0
50
Rectangular Weir and Hydraulic Jumps 2
Discussion
Conclusion
Graphs of Data (15 pts)
□ Specific Energy Diagram: Plot 𝑦1 , 𝑦2 and 𝐸𝑡 values for
each trial on a y vs. 𝑬𝒕 graph using arithmetic scale and
compare the graph with the theoretical graph. Show the
critical depth.
□ Plot hL vs. Q for each trial. Add an appropriate
trendline and show the equation.
□ Attach the original data sheet from your lab. Be sure
that all work is shown including formulas and units of
measurements.
□ Discuss the classification of flow in an open channel:
uniform flow, non-uniform flow; classify the hydraulic
jump. Include if possible a sketch of the flow observed
in the lab and label accordingly.
□ Discuss the importance of hydraulic jumps citing a realworld application.
□ Do the values of Froude’s Number make sense with
what was seen in the lab? Did the Froude’s Numbers
follow the same trend as shown in Figure 1? Identify
each trial with a type of jump.
□ What is the importance of an SED? Do the Et values
make sense?
□ Is the appearance of the specific energy diagram as
expected? Explain.
□ Explain any variation in yc.
□ Discuss hL as a function of Q.
□ What sources of error did you encounter in the lab and
what magnitude of effect did it have on the results?
□ Briefly summarize the highlights of the lab and discuss
their significance in the real world
□ What did you learn from conducting this lab?
□ Give at least one recommendation to improve the
accuracy of this lab.
Total Points
Final Lab 9 Grade
Fall 2021
30
5
100
Note: Reformat to clearly distinguish equation numbers, nomenclature, their names and units.
Lab 9: Rectangular Weir and Hydraulic Jumps - Results
Width of weir, bw
Height of weir, hw
0.149
N/A
(m)
(m)
9.81
(m/s2)
Acceleration due to Gravity, g
yc
Trial
1
2
3
4
5
Trial
1
2
3
4
5
1
2
3
4
5
y1
y2
(cm)
(m)
(cm)
(m)
(cm)
(m)
measured converted measured converted measured converted
4.5
0.045
1
0.010
6.8
0.068
4.7
0.047
1.2
0.012
6.1
0.061
4.5
0.045
1.4
0.014
7.2
0.072
4.6
0.046
1.3
0.013
7.25
0.073
4.25
0.043
0.75
0.008
7.2
0.072
A1
A2
2
2
Qc
v1
F1
h L-1
3
(m /s)
(m )
(m )
(m/s)
(m)
calculated calculated calculated calculated calculated calculated
0.001
0.010
0.0045
2.990
9.55
0.39
0.002
0.009
0.0048
2.660
7.75
0.30
0.002
0.011
0.0045
2.136
5.76
0.17
0.002
0.011
0.0046
2.377
6.66
0.22
0.001
0.011
0.0041
3.659
13.49
0.61
y 2 /y 1
Trial
11/1/2021
y 2 /y1
E1
E2
Ec
%Eloss
Qalt
(m3/s)
Exp. (m) Theo. (m)
(m)
(m)
(m)
%
calculated calculated calculated calculated calculated calculated calculated
6.800
13.500
0.466
0.078
0.068
83%
0.004
5.083
10.962
0.372
0.075
0.071
80%
0.005
5.143
8.150
0.246
0.081
0.068
67%
0.004
5.577
9.413
0.301
0.082
0.069
73%
0.005
9.600
19.077
0.690
0.079
0.064
88%
0.004
Parameter
Nom.
Units
Critical Depth
yc
m
Description
Depth taken from top of weir (approximately middle) to
surface of water
Depth taken at supercritical section of flow; usually lowest
elevation of the free surface wrt channel bottom
y1
m
y2
m
Weir/channel
Width
bw
m
Weir Height
hw
m
A1
(m )
Cross-sectional Area at depth y1, supercritical region.
A2
(m2)
Cross-sectional Area at depth y2, subcritical region.
Gravitational
Constant
g
(m/s )
Acceleration due to gravity = 32.2 (ft/s2)
Flowrate
Qc
(m3/s)
Flowrate calculated using depth yc
Velocity
v1
(m/s)
Velocity; recall Continuity Eqn.: Q = vA, manipulate so
that the velocity can be had to calculate Froude's Number
Froude
Number
F1 or Fr
Dless
Froude Numbers give an indication of the type of and other
characteristics associated with the jump.
Headloss
hL
(m)
Conjugate
Depth
y 2 /y 1
Conjugate
Depth
y 2 /y 2
Sequent
Depth
Crosssectional Area
E1
Specific
Energy
Energy Loss
Flowrate
E2
Depth taken at subcritical section of flow; look for where
the flow has calmed amer the jump
Width of weir is taken to be the same as the width of the
channel
Height of weir (req'd if upstream height is recorded as the
route to get critical depth)
2
2
Note Bernoulli's Equation
Conjugate depth amer the hydraulic jump; experimental
Exp. (m) value.
Conjugate depth amer the hydraulic jump; theoretical value
calculated by applying the momentum eqn. to the hydraulic
Theo. (m) jump.
Energy entering jump; Specifc Energy calculated at depth,
(m)
y1
Energy leaving jump; Specifc Energy calculated at depth,
(m)
y2
Ec
(m)
Specifc Energy calculated at critical depth, yc
%Eloss
(%)
Difference/loss in energy as a result of the jump; energy
dissipated from jump:= (HL/E1) * 100
Qalt
(m3/s)
Flowrate calculated using equation 11.
Graphs of y vs. E
Has to be done per trial