CEE 370
Fall 2015
Laboratory #3 Open Channel Flow
Objective:
The objective of this experiment is to measure the flow of fluid through open channels using
a V-notch weir and a hydraulic jump.
Introduction:
Flow of water in a conduit is of two types: open-channel flow and pipe flow. These types of
flows are similar but differ in one important aspect: open channel flow must have a free
surface which is exposed to atmospheric pressure whereas pipe flow has no free surface
since it if flowing full. Open channel flows are driven by gravity alone, and the pressure
gradient at the atmospheric interface is negligible. Thus the basic force balance in an open
channel is between gravity and friction.
Figure 1. A sketch of open channel flow
Reynold’s number of open channel flow:
As in pipe flow, three states of flow are possible, laminar, turbulent and transitional. The
effects of viscosity relative to inertial forces of the flow determine the state of flow. The
relative effects of velocity to inertia can be represented by the Reynold’s number which is a
dimensionless number derived through dimensional analysis of the flow. The Reynold’s
number for open channel flow is given by:
𝑅𝑅 =
𝑣𝑅ℎ
Eq. 1
𝜈
Where, Re is the Reynold’s number; v is the mean flow velocity (in m/s); Rh is a
characteristic length (the characteristic length for open channel flow is the hydraulic radius)
(in m), and ν is the kinematic viscosity of the fluid (m2/s).
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Rh is calculated using the cross sectional area (A) divided by the wetted perimeter (P) where
b is the width of the channel, and y is the depth of water in the channel (Eqs. 2 and 3).
𝐴
𝑅ℎ = 𝑃
Eq. 2
𝑃 = 𝑏 + 2𝑦
Eq. 3
For a full pipe, hydraulic radius is calculated as (Eq. 4 and Fig. 2):
1
𝑅ℎ = 4 𝐷ℎ
Eq. 4
Figure 2. A graphic presentation of a wetted perimeter
V-notch weir in open channel flow:
Weirs are overflow structures built across open flow channels to measure the discharge (or
flowrate) (Fig. 3). Some general terms pertaining to weirs are:
•
•
•
•
Notch: the opening which water flows through
Crest: the edge which water flows over
Nappe: the overflowing sheet of water
Length: the “width” of the weir notch
Figure 1. A sketch of V-notch weir in open channel
The basic principle is that discharge is directly related to the water depth above the bottom of
the V; this distance is called head (h). The V-notch design causes small changes in discharge
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to have a large change in depth allowing more accurate head measurement than with a
rectangular weir.
To calculate the flow rate of an open channel flow, the Bernoulli equation (Eq. 4) for a 0
surface pressure is used (Eq. 5). Q is fluid discharge at any given point, which is the product
of fluid velocity (V) and cross-sectional area (A). The cross-sectional area can be calculated
as the product of the width of the free surface and the constant head (Eq. 5).
𝑃
𝑣2
Bernoulli’s equation: 𝜌𝜌1 + 2𝑔 + 𝑧1 = ℎ1
Eq. 4
𝑣2
𝑃
Where 𝜌𝜌1 is the pressure head, 2𝑔 is the velocity head, and z1 is the velocity or potential head.
𝑉2
ℎ1 = 2𝑔
Eq. 5
𝐴 = 𝑏ℎ
Eq. 6
where h1 is the head associated with velocity at a point in the weir; and g is gravitational
constant.
where b is the width of free surface.
𝛼
𝑏 = 2(ℎ2 − ℎ1 ) tan 2
Eq. 7
where h2 is head associated with the total height of the weir; and α is the angle of the V-notch
weir.
The values of b and α are used to calculate the cross-sectional area.
However, the flowrate changes when hydraulic head changes (Eqs. 8 and 9);
𝑑𝑑 = 𝑉𝑉 𝑑ℎ
𝛼
and ∫ 𝑑𝑑 = ∫��2𝑔ℎ1 �[2(ℎ2 − ℎ1 ) tan 2 ] 𝑑ℎ1
Eq. 8
h1
b
α
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h2
Figure 4. A sketch of discharge calculation
The result of integration is represented by Eq. 8:
𝛼
5
𝑄 = 𝐶𝐷 �𝑔 tan � 2 � ℎ2 2 𝑎𝑎𝑎 𝐶𝐷 =
8√2
Eq. 9
15
where CD is discharge coefficient. The theoretical value represented by Equation 9 precludes
losses, upstream kinetic energy, and jet contraction. Neat, handwritten calculations are
appropriate, with inclusion of a control volume sketch. State all assumptions.
Hydraulic jump:
In an open channel flow, a supercritical flow can change rapidly to a subcritical flow by
passing through a hydraulic jump (Figure 5). The dimensionless number which indicates the
relative effects of gravity and inertial forces on the state of flow is the Froude number. It is
also derived through dimensional analysis and is defined as (Eq. 10):
Fr =
𝑉
Eq. 10
√𝑔𝑔
Where v is the average flow velocity, g is the gravitational constant, and y is the channel
depth.
Figure 2. A sketch of hydraulic jump in open channel flow
The Froude number defines three regimes of flow in open channels: critical, subcritical, and
supercritical flow (Table 1).
Fr
1
<1
>1
Table 1. The three regimes in an open channel
Type of flow
Criteria
Critical flow
Y = yc; V= Vc
Sub-critical flow
Y>yc, V<Vc
Super-critical flow (fast flow-high energy state) Y<yc, V>Vc
The critical depth (yc in m) and critical velocity (Vc in m/s) are calculated as follows (Eqs.
10-13):
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𝐹𝐹𝑐 =
𝑉𝑐
=1
�𝑔𝑦𝑐
Eq. 10
Where Frc is the critical Froude number.
where 𝑉𝑐 =
and 𝐴𝑐 = (
𝑄
𝐴𝑐
𝑏𝑄 2 1
3
𝑔
𝐴𝑐 𝑔
=�
)
Eq. 11
𝑏
Eq. 12
𝑄2 1
and 𝑦𝑐 = (𝑏2𝑔)3
Eq. 13
Figure 6. A picture and schematic of a sluice gate for a hydraulic jump
For this experiment, the discharge per unit width of a rectangular channel is calculated as (Eq.
14):
𝑞=
𝑄
𝑏
= 𝑉𝑉
Eq. 14
where q is the discharge per unit width of a rectangular channel.
To get the sluice gate force, 𝐹𝑒𝑒𝑒 is calculated as (Eq. 15):
1
1
𝑏
𝐹𝑒𝑒𝑒 = 𝜌𝜌𝑞 2 �𝑦 − 𝑦 � + 𝜌𝜌 2 (𝑦22 − 𝑦12 )
2
1
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Eq. 15
Examples of Froude number is given in Table 1.
Table 1. Description of Froude number
Fr
Jump
1
Dissipation
Impossible
1-1.7
Standing wave (undular jump) about 4 y2 long
<5 %
1.7-2.5
Weak jump
5-15%
2.5-4.5
Unstable, oscillating
15-45%
4.5-9
Stable, well-balanced, steady jump
45-70%
>9
Rough jump-intermittant and strong
70-85%
For Today’s Experiment:
Apparatus:
The apparatus consists of the following equipment, manufactured by G. Cussons, Limited, of
Great Britain, and maintained in Marston 10:
• Hydraulic bench (P. 6100)
• Hook gauge and scale (P. 6107)
• Rectangular open channel (P. 6245)
• 90 degree (α) V-notch weir
• Stop watch and scale
Procedure:
Part 1: V-Notch Weir
1. Note the water temperature in the hydraulic bench.
2. The cross sectional area of the drainage canal approaching the weir should be noted.
3. Measure the height of the bottom of the “V” to the bottom of the weir, as well as the
cross sectional area that will be used to measure the volume of flow. (Referring to the
low and high flow regions in the stepped measuring tank of the hydraulic bench).
4. Establish a low recirculating flow in the bench by turning the pump on and
maintaining a low setting on the rheostat, with the measuring tank drain open. The
discharge should be sufficient to create a free jet through the V-notch weir. Observe
the progression of flow through the drainage canal, V-notch weir, and into the
measuring tank and sump.
5. Record the water surface elevation in the drainage canal using the hook gauge
6. Be ready with a stopwatch. When measuring the flow rate, select a volume in the
lower step of the tank (as well as the high step if chosen to do so). Turn the knob at
the sump horizontal to stop drainage and allow for flow to accumulate to the selected
volume. The timer should start at the time the drainage of the sump is stopped. There
should be a total of 3 replicates for this flow.
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7. Increase the discharge for two more values. Make replicate measurements for each
new flow, so that you'll have a total of 3 replicated flows for 3 different flows (giving
a total of 9 data points) for the V-notch weir experiment.
8. Turn off the pump and open the measuring tank drain.
Part 2: Hydraulic Jump
1. Measure the width of the open channel and the opening under the sluice gate.
2. Carefully turn on the pump and establish a subcritical depth y1 upstream of the sluice
gate (without flooding the lab), a supercritical jet of depth y2 downstream of the
sluice gate, a hydraulic jump, and a downstream subcritical depth y3.
3. Note the progression of flow from the inlet section of the flume, through the sluice
gate, through the exit gate, and into the hydraulic bench measuring tank. Measure the
discharge into the tank with a stopwatch and sight glass. Use a ruler to measure y1, y2,
and y3.
4. Measure the downstream locations of the sluice gate, start of the hydraulic jump and
end of the hydraulic jump.
5. Turn off the pump and open the measuring tank drain when you're through.
Data Analysis:
Part 1: V-notch weir
Estimate observed values for CD and the open channel Reynolds number (defined in class,
based on upstream section) for the 3 discharges of the V-notch weir.
Part 2: Hydraulic Jump
Compute the approach, jet, and exit Froude numbers Fr1, Fr2, and Fr3 for the jump.
Compute the power dissipated across the hydraulic jump and the magnitude of the sluice gate
force FEXT. Compute the discharge per unit width q.
Results and Discussion:
• Why would you expect CD observed to be less than the theoretical Eq. (1b)? Does CD
vary with Reynolds number?
• Verify the sub or supercriticality of the three open channel flow regions. Where are
the controls located for y1, y2, and y3?
• What fraction of the approach energy is dissipated across the jump? What is the ratio
y2 to the sluice gate opening?
• Compare the observed y2 and y3 values with predicted values and comment.
Basic contents of lab report:
• Introduction
• Materials and methods
• Results and discussion
• Conclusions
• References
• Appendix
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Open Channel Flow Laboratory – Part 1
Lab Group: _________
Name: _______________________
Date: _________
V-Notch Weir
Temp (C): ___________
Approach width, cm: _________
Height of the bottom of the “V” to the bottom of the weir, cm: ___________
Area high Q section cm2: ________
Replicate
∆𝜉 (cm)
Replicate
∆𝜉 (cm)
Replicate
∆𝜉 (cm)
∆𝑡 (𝑠)
∆𝑡 (𝑠)
∆𝑡 (𝑠)
Area low Q, cm2: ____________
Low Flow
Q (cm3/s)
h1 (cm)
h2 (cm)
CD
Mid Flow
Q (cm3/s)
h1 (cm)
h2 (cm)
CD
High Flow
Q (cm3/s)
h1 (cm)
h2 (cm)
CD
Open Channel Flow Laboratory – Part 2
Lab Group: ______
Name: _______________________
Date: ___________
Hydraulic Jump and Sluice Gate
Temp (C): ___________
Channel width, cm: ___________
Sluice gate opening, cm: ___________
Channel Depth, Y1, cm: ___________ Distance from flow source, X1, cm: ___________
Channel Depth, Y2, cm: ___________ Distance from flow source, X2, cm: ___________
Channel Depth, Y3, cm: ___________ Distance from flow source, X3, cm: ___________
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Jump start, XSTART, cm: ___________
Jump end XEND, cm: ___________
XGATE, cm: ___________
∆𝜉 (cm): ___________
∆𝑡 (𝑠): ___________
Q, cm3/s: ___________
Example:
Water flows in a wide channel at q = 10 m3/(s-m) and y1 = 8 m. If the flow undergoes a
hydraulic jump, compute
a) Y2
b) V2
c) Fr2
d) Y3
e) V3
f) Fr3
g) hf
h) % dissipation of energy
1. Find the upstream velocity, V1
q 10m 3 / s ⋅ m
=
= 1.25m / s
V1 =
y1
8m
2. The upstream Froude number, Fr1 is
1.25m / s
V
Fr1 = 1 =
= 0.14 (This is a subcritical flow)
gy1
(9.8m / s 2 )(8m)
3. Is the hydraulic jump possible?
a. Find the depth y2
𝑦23
𝑉12
𝑉12 𝑦12
2
− � + 𝑦1 � 𝑦2 +
= 0;
2𝑔
2𝑔
𝑦2 = 0.84 𝑚
b. Find the velocity after the gate
V2 =
V1 y1 1.25m / s × 8m
=
= 11.9m / s
y2
0.84m
c. The supercritical Froude number is
Fr2 =
11.9m / s
V2
=
= 4.15
gy2
(0.84m)(9.8m / s 2 )
d. The depth, y3, is
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2 y3
2 y3
= −1 + [1 + 8 Fr22 ]1/ 2 =
= −1 + [1 + 8(4.15) 2 ]1/ 2
y2
0.84m
y3 = 4.53m
e. The downstream velocity, V3, is
V y
(11.9m / s )(0.84m)
= 2.2m / s
V3 = 2 2 =
y3
4.53m
f. The downstream Froude number
V
2.2m / s
Fr3 = 3 =
= 0.33
gy3
(9.8m / s 2 )(4.53m)
g. Dissipation head loss across the jump is
( y − y2 ) 3 (4.53m − 0.84m) 3
=
= 3.3m
hf = 3
4 y 2 y3
4(4.53m)(0.84m)
h. % Dissipation
%loss =
hf
E2
(100)
V22
(11.9m / s) 2
E2 = y 2 +
= 0.84m +
= 8.07m
2g
2(9.8m / s 2 )
%loss =
(3.3m)
(100) = 41%
8.07m
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