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Discharge formula for flows over open-check dams
Journal Pre-proof Discharge formula for flows over open-check dams Hsun-Chuan Chan, Hsin-Kai Yang, Po-Wei Lin, Jung-Tai Lee PII:
S0955-5986(19)30230-4
DOI:
https://doi.org/10.1016/j.flowmeasinst.2019.101690
Reference:
JFMI 101690
To appear in:
Flow Measurement and Instrumentation
Received Date: 21 June 2019 Revised Date:
30 December 2019
Accepted Date: 30 December 2019
Please cite this article as: H.-C. Chan, H.-K. Yang, P.-W. Lin, J.-T. Lee, Discharge formula for flows over open-check dams, Flow Measurement and Instrumentation (2020), doi: https://doi.org/10.1016/ j.flowmeasinst.2019.101690. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
1
Discharge formula for Flows over Open-Check Dams
2
Hsun-Chuan Chan 1, Hsin-Kai Yang 2, Po-Wei Lin 3, and Jung-Tai Lee 4,*
3 4 5 6 7 8 9 10 11 12
1
Professor, Department of Soil and Water Conservation, National Chung Hsing University, 145 Xingda Rd., Taichung 402, Taiwan (ROC). (Email: [email protected])
2
Ph.D. Student, Department of Soil and Water Conservation, National Chung Hsing University, 145 Xingda Rd., Taichung 402, Taiwan (ROC). (Email: [email protected])
3
Ph.D. Student, Department of Soil and Water Conservation, National Chung Hsing University, 145 Xingda Rd., Taichung 402, Taiwan (ROC). (Email: [email protected])
4
Assistant Professor, Department of Forestry and Natural Resources, National Chiayi University, 300 Syuefu Rd., Chiayi City 60004, Taiwan (ROC). (Email: [email protected])
* Correspondence: [email protected]; Tel: 886-5-2717482; Fax: 886-5-2717467.
Abstract
13
The present study proposed a discharge prediction model for open-check dams.
14
The open-check dam is composed of a trapezoidal spillway and a rectangular opening
15
located at the bottom of the spillway. The flow conditions, determined using the ratio
16
of water head and dam crest length, for flow over open-check dams were divided into
17
broad-crested and sharp-crested weir flow conditions. For the broad-crested weir flow
18
condition, the discharge equation for rectangular broad-crested weir was proposed to
19
estimate the discharge. For the sharp-crested weir flow condition, a linear
20
combination of discharge equations for rectangular and trapezoidal sharp-crested
21
weirs was proposed to describe the discharge. A series of laboratory experiments were
22
performed in order to calibrate the proposed discharge model. The results showed the
23
discharge coefficient and cross-sectional area ratio ( Ar ) were highly correlated under
24
two flow conditions. The averaged error between measured and predicted discharge
25
was less than 2%. The previous models displayed averaged errors between 4% and
26
25% under the present experimental conditions, whereas that of the present model
27
was considerably low. The present model demonstrated favorable accurate and
28
convenient estimation of discharge for flows over open-check dams.
29
Keywords: open-check dam; discharge; discharge coefficient; open channel flow 1
1
1. Introduction
2
Check dams are commonly constructed in upstream catchment areas to minimize
3
sediment disasters. However, because check dam design involves accumulating a
4
substantial amount of sediment, dam heights are often increased to expand sediment
5
storage space. Thus, check dam construction projects often have adverse effects on
6
river habitats and the continuity of sediment transport. In recent years, the increased
7
awareness of ecological conservation involved adding a rectangular opening
8
underneath the conventional spillway in check dams. Figs. 1(a) and (b) show a
9
conventional check dam with a trapezoidal spillway and that of an open-check dam
10
with a trapezoidal spillway and a rectangular opening respectively. The rectangular
11
opening in a check dam reduces the gap in elevation between the upstream and
12
downstream of the riverbed which maintains the continuity of sediment and aquatic
13
organism, thereby ameliorating the negative effects of the check dam on the natural
14
environments. In the design process of a conventional check dam (Piton & Recking,
15
2016), the discharge formula for sharp-crested weirs proposed by Zollinger (1983) is
16
commonly employed to decide the size of the spillway. However, open-check dams
17
differ considerably from conventional check dams in cross-sectional shape (the
18
addition of a rectangular opening). The use of the aforementioned discharge formula
19
to predict discharge may yield erroneous results. To effectively predict the discharge
20
of open-check dams, the present study superposed the discharge of the spillway and
21
opening of open-check dams. Considering the variations of water depth resulted in the
22
change in flow conditions, relevant data were incorporated into the discharge
23
coefficients to build a discharge prediction model of open-check dams under different
24
flow conditions. Five open-check dam models were tested in a laboratory flume for
25
large range of discharges. The predicted results are in agreement with the
26
experimental measurements. The model developed in the present study is expected as 2
1
the reference of the future researches in disaster prevention, sediment transport, and
2
habitat stability of open-check dams.
(a) Conventional check dam
(b) Open-check dam
Fig. 1. Different types of check dams. 3
2. Background
4
Check dams are transverse structures of rivers, which block water flow and cause
5
water levels to rise. Roushangar et al. (2014) asserted that water passing through
6
check dam spillways exhibits flow characteristics, including nappe flow, curved
7
streamlines, and air vent, similar to those of weir flows. Fig. 2 shows the flow patterns
8
of water flowing through a check dam. Regarding the flow conditions and the
9
geometry of the spillways, numerous studies have predicted the discharge of check
10
dams by using weir flow equations (Zollinger, 1983; Fang & Fang, 2012; Piton &
11
Recking, 2016).
Fig. 2. Flow patterns of water flowing through a check dam. 12 13
2.1. Flows over Weir According to Govinda Rao and Muralidhar (1963), weir flow conditions can be 3
1
divided into broad-crested and sharp-crested on the basis of weir water head and weir
2
crest length ratios ( h L ; where h and L are the water depth above crest weir and
3
the weir crest length, respectively).
4
2.1.1. Broad-crested Weir Flow Condition
5
When h L ≤ 1.5 , water flow makes contact with the weir crest, and the
6
friction created will cause energy loss in the water flow. This flow condition is
7
referred to as a broad-crested weir (Fig. 3(a)). The water touching the weir crest
8
upstream undergoes water flow separation, which, because of the effect of gravity,
9
causes the water to again come into contact with the weir crest. The critical water
10
depth occurs at the highest point of the water flow separation. Thus, discharge
11
predictions are often made using critical flow conditions. However, because
12
discharge prediction formulas ignore energy losses in the water flow due to
13
friction with the weir crest, the discharge predicted is generally underestimated
14
(Govinda Rao & Muralidhar, 1963; Azimi & Rajaratnam, 2009; Azimi et al.,
15
2014; Bijankhan et al., 2014).
16
2.1.2. Sharp-crested Weir Flow Condition
17
When h L > 1.5 , flows are considered to those of sharp-crested weir flows
18
(Fig. 3(b)). Because the water depth of a sharp-crested weir is greater than that of
19
a broad-crested weir, water flow in a sharp-crested weir possesses greater
20
potential energy. In a sharp-crested weir, when the water flow makes contact with
21
the weir crest upstream, the energy lost in the water flow is less than that lost in a
22
broad-crested weir. Because most of the potential energy converts into kinetic
23
energy, the water flows directly over the weir crest and lose contact with the crest.
24
In this situation, the streamlines are highly curved. Because discharge prediction
25
formulas for sharp-crested weirs ignore energy losses in the water flow due to 4
1
highly curved nappe flow, the discharge coefficients predicted are generally
2
overestimated (French, 1986; Swamee, 1988; Bos, 1989; Rehbock, 1929; Aydin et
3
al., 2011).
(a) Broad-crested weir flow condition
(b) Sharp-crested weir flow condition
Fig. 3. Weir flow condition classification. 4
2.2. Discharge Estimation of Weir Flows
5
Weirs can have a variety of cross sections, depending on the requirements. The
6
simplest type is the weir with a rectangular cross-section. The shapes of the single
7
cross-section opening are mostly rectangular and trapezoidal. By contrast, when the
8
opening shapes are a combination of different shapes, the weirs are referred to as the
9
compound cross-section weirs. Many stage-discharge relationships were used to
10 11
predict discharge of weir flows. 2.2.1. Weirs with a rectangular cross-section
12
For rectangular sharp-crested weirs, Rehbock (1929) conducted experimental
13
research and found that the discharge coefficients are determined by their water
14
head–height ratios
15
passing through a weir, French (1986) proposed using constriction factors to
16
identify the effect of cross section constriction on discharge coefficients, where
17
the discharge coefficients range from 0.59 to 0.64. In the rectangular
18
broad-crested weirs, Govinda Rao and Muralidhar (1963) considered the
(h P ) .
Because flow exhibits a constriction effect while
5
1
discharge coefficient to be a function of h L in order to take into account the
2
friction loss caused by weir crest. In addition, they revealed that the discharge
3
coefficient of a rectangular broad-crested weir ranged from 1.02 to 1.28. A
4
comprehensive review of the experimental investigations on finite crest length
5
weirs can be found in Azimi and Rajaratnam (2009). They also developed
6
empirical correlations for the discharge coefficient.
7
2.2.2 Weirs with Compound Cross-Section
8
Discharge over weirs with the compound cross-section have investigated by
9
Jan et al. (2005), Salmasi (2011), and Gogus (2016). Jan et al. (2005) discussed
10
four different compound weirs: weirs with rectangular compound cross-section
11
opening at both the top and bottom, weirs with a trapezoidal compound
12
cross-section opening at the top and rectangular compound cross-section opening
13
at the bottom, weirs with rectangular compound cross-section opening at the top
14
and triangular cross-section opening at the bottom, and weirs with trapezoidal
15
compound cross-section opening at the top and triangular cross-section opening at
16
the bottom. They calculated the discharges of the various cross sections and
17
linearly superposed them. However, Jan et al. (2005) only examined the high
18
discharge (i.e., when the flow flowed through both openings), and ignored
19
instances when the discharge was low (i.e., when the flow flowed through only the
20
bottom cross-section opening), consequently limiting the applicability of the study
21
findings. Salmasi (2011) and Gogus (2016) stated that the discharge coefficients
22
of compound weirs should change when water depths change. They investigated
23
the flow discharge for water flowing through only the bottom rectangular
24
compound cross-section opening and water flowing through both the top and
25
bottom openings. Salmasi (2011) showed that when 0.1 < h L < 0.6 , the 6
1
discharge coefficients of compound weirs change with h L ; in addition, the
2
trends for water flowing through only the bottom rectangular cross-section
3
opening differ from those for water flowing through both the top and bottom
4
rectangular cross-section openings, with discharge coefficients for these two
5
opening types ranging from 0.7 to 1.0. Gogus (2016) utilized flow coefficient Cv
6
to correct the negative effects created when the influence of velocity heads was
7
ignored during the theoretical derivation process, thereby elevating discharge
8
prediction accuracy. Gogus (2016) conducted experiments and revealed that when
9
h L < 1.7 , the discharge coefficients of compound weirs range from 0.74 to 1.10.
10
3. Theoretical Analysis
11
When water flows through check dams, the water depth ( hd ) and crest length
12
( Ld ) ratio ( hd Ld ) changes constantly due to changes in water level. Fig. 4 shows the
13
water flowing through a check dam for different discharges. Roushangar et al. (2014)
14
suggested that flow conditions through check dams are similar to that through weirs,
15
and that weir flow condition changes with the ratio of weir water depth and weir crest
16
length. Therefore, to ensure the prediction model effectively estimates the discharge
17
in different flow conditions, the present study used the weir flow classification
18
proposed by Govinda Rao and Muralidhar (1963) and divided the flows of check
19
dams into broad-crested weir and sharp-crested weir flow conditions according to the
20
ratios of dam water depth and dam crest length. Furthermore, the discharge prediction
21
model for open-check dams was developed under different flow conditions.
22 23 7
1 2
(a) Low discharge
(b) High discharge
Fig. 4. Water flows over open-check dam with different discharges. 3
3.1. Broad-Crested Weir Flow
4
When flowing through open-check dam with hd Ld ≤ 1.5 , the amount of
5
water passing through the dams is considered to have a low discharge. At this
6
time, most of the water flows through the rectangular opening below the
7
trapezoidal spillway. Fig. 5(a) shows the dimensions of an open-check dam, where
8
b0 and hd are the width of the rectangular opening and the water depth at the
9
head measurement section, respectively. Here, because the water in the upstream
10
of the dam contains scant potential energy, it converts into little kinetic energy
11
when the water flows through the weir crest, resulting in relatively short water
12
flow separation distances and causing the water to come in contact with the weir
13
crest. Regarding hydraulics, the water flow is classified as a broad-crested weir
14
flow. In this flow condition, the discharge of open-check dam was predicted using
15
the broad-crested weir water flow method.
8
(a) Front view
(b) Side view
Fig. 5. Definition sketch for broad-crested weir flow over open-check dam. 1
Fig. 5(b) presents the height, width energy head of a dam. Cross Section 1 shows
2
a stable water flow in the upstream of the dam, whereas Cross Section 2 reveals the
3
separation of water when the water comes in contact with the weir crest. Gogus
4
(2016) hypothesized that there are no energy losses from Cross Section 1 to Cross
5
Section 2 and that critical flow conditions occur at Cross Section 2. The energy
6
relationships between the two cross sections can be obtained using the following
7
energy equation:
8
H 1 = h2 +
9
where H 1 is the total energy head of Cross Section 1, h2 is the water depth of
10
Cross Section 2, V2 is the flow velocity of Cross Section 2, g is the gravitational
11
acceleration, yc is the critical water depth , and Vc is the critical velocity. By
12
incorporating the relationship between critical velocity and total energy head obtained
13
from Equation (3-1) into the flow rate formula (Q = A × V ) , the equation of discharge
14
can be obtained as:
15
Qb = Ac 2g(H1 − yc )
V22 V2 = yc + c 2g 2g
(1)
(2)
9
1
where Ac is the cross-sectional area where critical flow occurred and Qb is the
2
discharge of the check dam under broad-crested weir water flow condition. Because
3
hd Ld ≤ 1.5 , the cross-sectional shape of the water passing through the open-check
4
dam is rectangular. Assuming that the velocity head of Cross Section 1 can be
5
ignored, total energy head is therefore equal to the water depth at the head
6
measurement section (H1 = hd ) . Because the flow through the control section is
7
critical, the discharge equation of the flow passing through the rectangular opening at
8
the bottom of the open-check dam can be written as:
9
Qb =
10
2 2 1. 5 b0 g hd 3 3
(3)
3.2. Sharp-Crested Weir Flow
11
When hd Ld > 1.5 , water flows through both the bottom rectangular opening
12
and the top trapezoidal spillway of open-check dams. Fig. 6(a) shows the dimensions
13
of a check dam, where b0 is the width of the rectangular opening, b1 is the bottom
14
width of the trapezoidal spillway, b2 is the top width of the trapezoidal spillway, hrd
15
is the water depth of the rectangular opening, and htd is the water depth of the
16
trapezoidal spillway. Considering the kinetic energy of sharp-crested weir flow is
17
high, the water no longer coming into contact with the weir crest and creates a
18
sharp-crested weir water flow pattern. Therefore, the present study used sharp-crested
19
weir theories to predict discharge. The water passes both the rectangular opening and
20
trapezoidal spillway. The linear superposition method (Martínez et al., 2005;
21
Piratheepan et al., 2006; Jan et al., 2005) was adopted to superpose the discharges of
22
the rectangular opening and trapezoidal spillway.
10
1
In Fig. 6(b), Cross Section 1 presents stable water flow in the upstream of the
2
dam, whereas Cross Section 2 shows the water flowing over the weir crest. Assuming
3
that no energy losses occurred between Cross Section 1 to Cross Section 2, the energy
4
relationships between the two cross sections can be obtained as:
5
H1 = (hd − ∆h) +
V22 2g
(4)
6
Assuming that the approaching velocity of progressive flow in Cross Section 1 is
7
zero, then the velocity head in Cross Section 1 (V12 2 g ) can be ignored. The upstream
8
total energy head ( H1 ) is therefore equal to check dam water depth hd ( H1 = hd ),
9
and the flow velocity of Cross Section 2 ( V2 ) can be simplified as:
10
V2 = 2g∆h
(3–5)
11
Using the flow velocity ( V2 ) and cross-sectional areas of Cross Section 2, one
12
can obtain the discharge passing though the weir. Bos (1989) expressed the discharge
13
for flow over a rectangular sharp-crested weir in the following form:
14
2 1.5 Qr = b0 2g hrd 3
15
where Qr is the discharge of the rectangular opening of the open-check dam. For a
16
trapezoidal sharp-crested weir, Henderson (1966) expressed its discharge equation in
17
the form as:
18
Qt =
19
where Qt is the discharge of the trapezoidal spillway. Theoretical discharge equation
20
for an open-check dam with the sharp-crested weir Flow is developed by linearly
(6)
2 4 2 g htd1.5 b1 + mhtd 3 5
(7)
11
1
superposing the discharge of the rectangular opening and the trapezoidal spillway. The
2
equation can be rearranged as follows:
3
Qc =
1. 5 2 4 2 g b0 hrd + htd1.5 b1 + mhtd 3 5
(8)
(a) Front view
(b) Side view
Fig. 6. Definition sketch for sharp-crested weir flow over open-check dam. 4
3.3. Discharge Coefficient ࢊ
5
In the previous developments of the discharge equations, a number of
6
assumptions are made for the sake of simplicity and therefore a discharge coefficient
7
was usually introduced in the real fluid conditions. (Swame, 1988; Bos, 1989;
8
Rehbock, 1929; Govinda Rao & Muralidhar, 1963; Azimi & Rajaratnam, 2009; Aydin
9
et al., 2011; Salmasi, 2011; Bijankhan et al., 2014; Gogus, 2016). Eqs. (9) and (10)
10
are the theoretical discharge prediction models containing the discharge coefficient
11
for open-check dams under sharp-crested and broad-crested weir flow conditions,
12
respectively.
13
Broad-crested weir flow conditions ( hd Ld ≤ 1.5 )
Q p1 = Cd 1 × Qb 14
2 2 1.5 = Cd 1 × b0 g hd 3 3
(9)
12
1
Sharp-crested weir flow conditions ( hd Ld > 1.5 ) Q p 2 = C d 2 × Qc
2
= Cd 2 ×
2 4 2 g b0 hrd1.5 + htd1.5 b1 + mhtd 3 5
(10)
3
where Qp1 is the discharge through the rectangular opening, Qp 2 is the discharge
4
through both the rectangular opening and trapezoidal spillway, Cd1 is discharge
5
coefficient of broad-crested weir flow condition, and Cd 2 is the discharge coefficient
6
of sharp-crested weir flow condition. Aforementioned analysis revealed the discharge
7
coefficients of weir flows, applications of these discharge coefficients have
8
limitations and must be met for the applicable conditions. To effectively and
9
accurately predict the discharge of open-check dams, it is necessary to explore the
10
discharge coefficients under desired flow conditions.
11
4. Experimental Materials and Procedure
12
4.1. Experimental Setup
13
Experiments of this study were performed in a rectangular flume. The flume was
14
supplied by pumping water from the reservoir to the constant-head water tank. An
15
inflow control valve was used to adjust the discharge during the experiments, and the
16
water head of a V-shaped weir was used to determine the discharge. The check dam
17
model was built using 6-mm thick acrylic sheets. The inside of the model was filled
18
with quartz sand to eliminate the effect of buoyancy during the experiments. Silicone
19
was used to seal the contact areas between the flume sidewalls and the bed to prevent
20
experimental errors due to water leakage and dam movement. Fig. 7 shows definition
21
sketch of model dam and flume used in the experiment. Experimental analyses were
22
performed for the five check dams with different opening sizes. The geometric
23
dimensions of all the models are summarized in Table 1. 13
Fig. 7. Definition sketch of model dam and flume used in the experiment 1 Table 1. Parameters of the check dam model Distance between the Experimental
rectangular
group
opening and the riverbed
Rectangular opening width b0
Rectangular
Trapezoidal
Trapezoidal
opening
spillway
spillway
height hr
side slope
height ht
(m)
m
(m)
(m)
Pd (m)
2
Check dam crest length
Ld (m)
E1
0.15
0.05
0.10
0.5
0.15
0.07
E2
0.15
0.075
0.10
0.5
0.15
0.07
E3
0.15
0.10
0.10
0.5
0.15
0.07
E4
0.15
0.05
0.10
0.3
0.15
0.07
E5
0.15
0.05
0.10
1.0
0.15
0.07
4.2. Experimental Procedure
3
During the experiments, it was ensured that the water flow passed through the
4
check dam and became free flow with nappe flow characteristics. A needle gauge with
5
a precision of ±0.1 mm was used to measure the water depth of the flume.
6
Measurements were made from the nappe drop region to the upstream region until the
7
change in water depth was less than 1 mm (i.e., the change in water depth was 14
1
minimal), indicating a stabilized water depth. This water depth of the measurement
2
point was considered as the upstream water head ( h0 ). The check dam water depth
3
hd (= h0 − Pd ) was obtained by subtracting the weir height ( Pd ) from the upstream
4
water head ( h0 ). A total of 110 discharge experiments were performed, in which 22
5
discharge experiments were performed for five check dams. The experimental cases
6
and ranges of variables are summarized in Table 2, where Qm is the discharge, h0
7
is the upstream water head, and Fr is the Froude number based on the upstream
8
water head.
9 Table 2. Flow characteristics of the experimental cases. Experimental Number of
Discharge Qm
Check dam water depth h0
Froude number Fr
group
experiments
(m3/s)
E1
22
0.00023~0.03074
0.169~0.387
0.005~0.131
E2
22
0.00020~0.03416
0.168~0.383
0.005~0.149
E3
22
0.00018~0.02818
0.166~0.384
0.004~0.122
E4
22
0.00052~0.03340
0.171~0.383
0.011~0.145
E5
22
0.00038~0.03375
0.171~0.390
0.008~0.142
(m)
10 11
4.3. Deducing Reduction Factor for Discharge Coefficient
12
The discharge coefficient is generally described as a function of the relevant
13
dimensionless parameters based on experimental data. For a weir, the relevant
14
parameters of discharge ( Qm ) were fluid mass density ( ρ ), gravitational acceleration
15
( g ), dam water depth ( hd ), dynamic viscosity coefficient ( µ ), weir height ( Pd ), and
16
flume slope ( S ) (Aydin et al., 2011). For an open-check dam, the water flowing
17
through the dam may create a contraction effect. Therefore, the present study
15
1
considered the flow area of dam ( A ) as one of the factors that influenced discharge.
2
The discharge can be expressed by the following functional relationship:
3
Qm = f1 (ρ、g、h、 d µ、P、 d S、A)
(11)
4
Because the functional relationship in Eq. (11) represents a physical phenomenon
5
that does not depend on the choice of measurement units, using the Buckingham π
6
theorem, Eq. (11) can be expressed in a dimensionless form as follows:
Qm
ρ g hd1.5 hd A = f2 、 、 、S µ Pd A0
7
Cd =
8
where B is the width of the channel; A0 ( = B× h0 ) is the cross-sectional area
9
ρ g hd1.5 h upstream of the dam; is the Reynolds number of the flow, d is the ratio µ Pd
B g hd1.5
(12)
A is the ratio between the A0
10
of the check dam water depth and the weir height, and
11
flow area of the dam and that of the upstream. In the experiments flume slope is
12
settled as constant and the research only focuses on the effect of the ratio between the
13
flow area of the dam and that of the upstream on discharge coefficient of open-check
14
dam. The dimensionless parameter for an open-check dam then can be simplified as
15
follows:
16
Cd =
17
where Ar is the ratio between the flow area of the dam and that of the upstream. The
18
use of Ar enables the discharge coefficient to account for the contraction effect and
19
energy changes created by the differences between the cross-sectional areas of the
20
check dam opening and that of the channel.
A Qm = f 3 1.5 B g hd A0
= f 3 ( Ar )
(13)
16
1
5. Results and Discussion
2
Figs. 8(a) and (b) show the variations of experimental discharge Qm and the
3
corresponding discharge coefficients Cd1 and Cd 2 at the broad-crested weir flow
4
and sharp-crested weir flow conditions, respectively. For the broad-crested weir flow
5
condition, discharge coefficient increases with increasing discharge or rectangular
6
opening width ( b0 ). For sharp-crested weir flow condition, the behavior of discharge
7
coefficient with discharge is reversed; i.e. discharge coefficient decreases with
8
increasing discharge. As the side slope of spillway m increases, moreover, there is a
9
decrease in discharge coefficient. It is clear from Figs. 8(a) and (b) that the discharge
10
coefficients of open-check dams change with discharge, the geometric shapes of the
11
check dams, and flow conditions change. The discharge coefficient is difficult to
12
represent as a simple trend. An attempt to directly fit the discharge to experimental
13
data with a simple mathematical expression involve using the results of the
14
dimensional analysis.
15
Figs. 9(a) and (b) plot the discharge coefficients
(i.e., Cd1 and C d 2 ) with
16
corresponding dimensionless flow area ratio Ar at different flow conditions,
17
respectively. The importance of flow area ratio Ar for contraction weir flow is
18
obvious. It is possible to develop best fit expressions of discharge coefficients. For the
19
broad-crested weir flow condition, C d 1 and Ar ranged from 1.0 to 1.19 and from
20
0.01 to 0.08, respectively, where Cd1 increases with Ar . The best fit coefficients are
21
obtained by a simple linear regression analysis and the relationship between Cd1 and
22
Ar is given as:
23
Cd1 = 3.278× Ar + 0.958
24
where the coefficient of determination ( R2 ) is 0.87, and discharge coefficient Cd1
25
and dimensionless parameter Ar are highly correlated. When water flow is
(14)
17
1
concentrated in the rectangular opening of the open-check dam, the flow experiences
2
a significant contraction effect and backwater occurs. It increases the potential energy
3
of the upstream water. The increased potential energy is subsequently converted into
4
kinetic energy while passing through the dam. As the potential energy is increased,
5
there is increasing upstream water level which increases the streamline curvature on
6
horizontal plane around the dam. This reduces the flow passing capacity of the weir
7
and corresponding an increase of discharge coefficient. For the sharp-crested weir
8
flow condition, Ar and Cd 2 ranged from 0.06 to 0.26 and from 0.71 to 0.79,
9
respectively, where Cd 2 decreases linearly when Ar increases. A simple linear
10
regression analysis also revealed a high correlation ( R2 = 0.848) between Ar and
11
Cd 2 . The best fit function for Cd 2 and Ar is as follows:
12
Cd 2 = −0.408× Ar + 0.817
(15)
13
When water flooded the spillways, because the geometry of trapezoid is narrow
14
at the bottom and wide at the top, the contraction effect partly vanished with
15
increasing flow depth and side slope (m). Thus, the discharge coefficient decreased
16
with increasing Ar . Figs. 10(a) and (b) show the comparisons of the experimental
17
discharge ( Qm ) and the calculated discharge ( Qp1 and Q p 2 ) obtained by using Eqs.
18
(14) and (15) for discharge coefficient under two flow conditions, respectively. For
19
the broad-crested weir flow condition, the error percentages between the experimental
20
data and predicted value ( (Qm − Q p 1 ) / Qm ) ranged between 0.14% to 2.88% with an
21
averaged value of 1.39%. For the sharp-crested weir flow condition, the error
22
percentages ( (Qm − Q p 2 ) / Qm ) ranged between 0.02% to 1.95% with an averaged
23
value of 0.99%. The prediction models successfully estimate the discharge passing
24
through open-check dams with error less than 3%. 18
1
To evaluate the performance of the discharge prediction model developed in the
2
present study, the results obtained using present model were compared with those
3
obtained using the previous models. For the broad-crested weir flow condition, the
4
previous models of Govinda Rao and Muralidhar (1963), Swamee (1988), Azimi and
5
Rajaratnam (2009), Salmasi (2011), and Gogus (2016) were used. Fig. 11 shows the
6
comparisons of the experimental discharge and the predicted discharge. It is seen that
7
previous models generally underestimated the discharge and the prediction errors
8
increased as discharge increased. The averaged error of different models is
9
summarized in Table 3. The error of the previous models ranges between 4.03% and
10
24.96% and three models have averaged error greater than 10%. From hydraulics
11
point of view, the approaching velocity must be low enough and unsteady phenomena
12
must be eliminated for the purpose of weir flow measurements. Therefore, the
13
contraction rate Ar of the broad-crested weir is sufficient to prevent possible
14
vorticity and turbulence produced at the upstream. In the present cases of weir flow,
15
the contraction effect was pronounced. It resulted in relatively large discharge errors
16
of the previous models than that of the present model. The model proposed by
17
Salmasi (2011) has a largest averaged error of approximately 25%. The discharge
18
prediction models presented by Gogus (2016) and Swamee (1988) displayes relatively
19
favorable prediction results among the previous models. The Gogus’s model shows
20
prediction errors ranging between 0.25% and 11.87% with an average of about 4%.
21
Nevertheless, Gogus’s model showed large deviations at high discharge values which
22
suggests further examinations. The errors of Swamee’s model range from 0.16% to
23
10.45% with an average of about 4%. However, this model involves a complex
24
calculation of parameters that gives a limitation for practically application. Among all
25
the models, the present model provides a lowest averaged error of 1.39%. It is seen
26
that the present model can successfully represent the discharge data with similar 19
1
accuracy at the broad-crested weir flow condition.
2
For the sharp-crested weir flow condition, the present model was compared with
3
the discharge equation proposed by Jan et al. (2005) for the compound sharp-crested
4
weir with similar geometry as the open-check dam. A comparison between the
5
experimental discharge and the corresponding predicted discharge using the present
6
model and the model proposed by Jan et al. (2005) is presented in Fig. 12. The errors
7
of the model proposed by Jan et al. (2005) increases as discharge increases with the
8
averaged and largest errors of approximately 10% and 19%, respectively. Jan et al.
9
(2005) formulated the discharge by dividing into five parts as three rectangular and
10
two triangular sharp-crested weirs and linearly superposed discharge of each part.
11
However, the discharge coefficient of Jan et al. (2005) for the triangular sharp-crested
12
weir considered exclusively as a function of the geometric parameter. The most
13
important characteristic of the weir flow is the independence of the discharge
14
coefficient from the channel width, B, because of the completely contracted nature of
15
the flow over the weir. Errors of the present model fall far below the aforementioned
16
model due to incorporate contracted aspects of flow behavior into the formulation of
17
discharge coefficient. Because the compound cross sections were manufactured using
18
cross sections of varying shapes, the model proposed by Jan et al. (2005) first
19
predicted the discharge and discharge coefficients for each cross sections and linearly
20
superposed discharge of each cross section to obtain the total discharge of the
21
compound cross sections. The discharge coefficient used by Jan et al. (2005) could
22
not account for the changes in the cross sections of where water discharged as check
23
dam water level increased or decreased. Additionally, slit-check dams were exposed
24
to substantial contraction effects because of the changes in the cross sections where
25
water discharged. As a result, the discharge prediction errors created in the model
26
proposed by Jan et al. (2005) were greater than those developed in the present study 20
1
were.
2 3
(a)
(b) 21
Fig. 8. Discharge coefficient versus measured discharge: (a) broad-crested weir flow condition; (b) sharp-crested weir flow condition. 1
(a)
(b) 22
Fig. 9. Discharge coefficient versus flow area ratio: (a) broad-crested weir flow condition; (b) sharp-crested weir flow condition. 1
23
(a)
(b) Fig. 10. Comparison between calculated and measured discharge: (a) broad-crested weir flow condition; (b) sharp-crested weir flow condition. 1
24
Fig. 11. Comparison of predicted discharges by using the various models under broad-crested weir flow condition. 0.04 45 degrees exact line Present study Jan et al. (2005)
0.03
0.02
0.01
0 0
0.01
0.02
0.03
0.04
Measured Discharge (m3/sec) Fig. 12. Comparison of predicted discharges by using the various models under sharp-crested weir flow condition.
25
Table 3. Summary of the errors in the predicted discharges for different models. Error (%) Flow condition
Model Minimum
Maximum
Average
Standard deviation
0.14
2.88
1.39
0.77
4.76
18.15
11.13
3.01
0.16
10.45
4.05
2.49
5.77
19.98
12.48
3.12
17.65
31.20
24.96
3.02
Gogus (2016)
0.25
11.87
4.03
2.69
Sharp-crested Present study weir flow Jan et al. ( hd Ld > 1.5 ) (2005)
0.02
1.95
0.99
0.52
1.43
18.84
9.69
3.84
Present study Govinda Rao & Muralidhar (1963) Swamee Broad-crested (1988) weir flow Azimi & ( hd Ld ≤ 1.5 ) Rajaratnam (2009) Salmasi (2011)
1
6. Conclusions
2
In this study, the discharge equations for open-check dams with a trapezoidal
3
spillway at the top and a rectangular opening at the bottom were developed. The ratio
4
of check dam water depth ( hd ) and crest length ( Ld ) was used to determine the flow
5
conditions of the open-check dams. Theoretical discharge equations were developed
6
for different flow conditions. Considering the contraction effect created by dams, the
7
ratio between the cross-sectional areas of upstream and dam ( Ar ) was incorporated
8
into the discharge coefficient. Experiments were performed to investigate the effect of
9
Ar on discharge coefficients . The predicted discharges of the present study were
10
compared with those of other studies. From the analysis of these results, the following
11
conclusions can be stated: 26
1
1. Formulation of open-check dams depends on the ratio between the check dam
2
water depth and crest length. When hd Ld ≤ 1.5 , the water flow touched the weir
3
crest, creating flow condition that resembled broad-crested weir flow condition.
4
Thus, the broad-crested weir discharge formula can use to predict discharge; when
5
hd Ld > 1.5 , the flow condition resembled sharp-crested weir flow condition, and
6
the sharp-crested weir discharge formula can use to predict discharge.
7
2. Experimental observations indicate the discharge coefficients are significantly
8
affected by Ar and empirical equations of discharge coefficients for different
9
flow conditions were developed, respectively. Under broad-crested weir flow
10
conditions, the averaged error between the predicted and measured discharge was
11
within 2%; under sharp-crested weir flow conditions, the averaged error was
12
within 1%, indicating that the discharge prediction model of the present study
13
produced reasonable results.
14
3. In contrast with other relevant studies, the present model formulates discharge in
15
open-check dams in terms of the water depth and geometric dimensions of
16
open-check dams, which is easy to use practically. For the range of the
17
experiments, it is also demonstrated the present model provides favorable
18
prediction results.
19
7. References
20
1.
Finite Crest Length.” Journal of Hydraulic Engineering, 135(12), 120-125.
21 22
Azimi, A.H. and Rajaratnam, N. (2009). “Discharge Characteristics of Weirs of
2.
Aydin, I., Altan-Sakarya, A.B. and Sismanb, C. (2011). “Discharge formula for
23
rectangular
sharp-crested
weirs.”
24
Instrumentation, 22, 144-151.
Journal
27
of
Flow
Measurement
and
1
3.
Azimi, A.H., Rajaratnam, N. and Zhu, D. (2014). “Submerged Flows over
2
Rectangular Weirs of Finite Crest Length.” Journal of Irrigation and Drainage
3
Engineering, 140(5), 10.1061/(ASCE)IR.1943-4774.0000728, 06014001.
4
4.
Institute for Land Reclamation and Improvement, Wageningen. The Netherlands.
5 6
Bos, M.G. (1989). “Discharge measurement structures”, 2nd Ed., International
5.
Bijankhan, M., Stefano, C.D., Ferro, V. and Kouchakzadeh, S. (2014). “New
7
Stage-Discharge Relationship for Weirs of Finite Crest Length.” Journal of
8
Irrigation
9
10.1061/(ASCE)IR.1943-4774.0000670, 06013006.
and
Drainage
Engineering,
140(3),
10
6.
French, R.H. (1986). “Open-channel hydraulics.”, McGraw-Hill, New York.
11
7.
Fang, C. and Fang, K. (2012). “Derivation and Verification of New Common
12
Formulas for the Calculation of Instantaneous Dam-break Maximum Discharge.”
13
Journal of Procedia Engineering, 28, 648-656.
14
8.
Weirs of Finite-Crest Width.” Houille Blanche, 18(5), 537-545.
15 16
Govinda Rao, N.S. and Muralidhar, D. (1963). “Discharge Characteristics of
9.
Gogus, M., Al-Khatib, I.A., Atalay, A.E. and Khatib, J.I. (2016). “Discharge
17
prediction in flow measurement flumes with different downstream transition
18
slopes.” Journal of Flow Measurement and Instrumentation, 47, 28-34.
19 20
10. Henderson, F.M. (1966). “Open channel flow.” Prentice-Hall, Englewood Cliffs, N.J.
21
11. Jan, C.D., Chang, C.J. and Lee, M.C. (2005). “Discussion of Design and
22
Calibration of a Compound Sharp-Crested Weir.” Journal of Hydraulic
23
Engineering, 131(2), 112-116.
24
12. Martínez, J., Reca, K., Morillas, M.T. and López, J.G. (2005). “Design and
25
Calibration of a Compound Sharp-Crested Weir.” Journal of Hydraulic
26
Engineering, 13(2), 112-116. 28
1
13. Piratheepan, M., Winston, N.E.F. and Pathirana, K.P.P. (2006). “Discharge
2
Measurements in Open Channels Using Compound Sharp-Crested Weirs.”
3
Journal of the Institution of Engineers, 3, 31-38.
4
14. Piton, R. and Recking, A. (2016). “Design of Sediment Traps with Open Check
5
Dams. I: Hydraulic and Deposition Processes.” Journal of Hydraulic
6
Engineering, 142(2), 10.1061/(ASCE)HY.1943-7900.0001048, 04015045.
7 8
15. Rehbock, T. (1929). “Discussion of Precise Weir Measurements.” Journal of Transportation Engineering, 93, 1143-1162.
9
16. Roushangar, K., Akhgar, A., Salmasi, F. (2014). “Estimating Discharge
10
Coefficient of Stepped Spillways under Nappe and Skimming Flow Regime
11
Using Data Driven Approaches.” Journal of Flow Measurement and
12
Instrumentation, 59, 78-87.
13 14
17. Swamee, P.K. (1988). “Generalized Rectangular Weir Equations.” Journal of Hydraulic Engineering, 114(8), 945-949.
15
18. Salmasi, F., Sanaz, P., Dalir, A.H. and Zadeh, D.F. (2011). “Discharge Relations
16
for Rectangular Broad-Crested Weirs.” Journal of Agricultural Science, 17(4),
17
324-336.
18
19. Zollinger, F. (1983). “Die Vorgänge in einem Geschiebeabl Agerungsplatz ihre
19
Morphologie und die Möglichkeiten einer Steuerung.” Ph.D. thesis, ETH Zürich,
20
Zurich.
29
Highlights: •
Discharge prediction model for open-check dams was developed.
•
Flow conditions are divided as broad-crested weir flow and sharp-crested weir flow.
•
The results confirm the model was generalized with high accuracy.
Hsun-Chuan Chan: Conceptualization, Methodology. Hsin-Kai Yang: Data curation, Writing- Original draft preparation. Po-Wei Lin: Data curation. Jung-Tai Lee: Supervision, Writing- Reviewing and Editing.
S0955-5986(19)30230-4
DOI:
https://doi.org/10.1016/j.flowmeasinst.2019.101690
Reference:
JFMI 101690
To appear in:
Flow Measurement and Instrumentation
Received Date: 21 June 2019 Revised Date:
30 December 2019
Accepted Date: 30 December 2019
Please cite this article as: H.-C. Chan, H.-K. Yang, P.-W. Lin, J.-T. Lee, Discharge formula for flows over open-check dams, Flow Measurement and Instrumentation (2020), doi: https://doi.org/10.1016/ j.flowmeasinst.2019.101690. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
1
Discharge formula for Flows over Open-Check Dams
2
Hsun-Chuan Chan 1, Hsin-Kai Yang 2, Po-Wei Lin 3, and Jung-Tai Lee 4,*
3 4 5 6 7 8 9 10 11 12
1
Professor, Department of Soil and Water Conservation, National Chung Hsing University, 145 Xingda Rd., Taichung 402, Taiwan (ROC). (Email: [email protected])
2
Ph.D. Student, Department of Soil and Water Conservation, National Chung Hsing University, 145 Xingda Rd., Taichung 402, Taiwan (ROC). (Email: [email protected])
3
Ph.D. Student, Department of Soil and Water Conservation, National Chung Hsing University, 145 Xingda Rd., Taichung 402, Taiwan (ROC). (Email: [email protected])
4
Assistant Professor, Department of Forestry and Natural Resources, National Chiayi University, 300 Syuefu Rd., Chiayi City 60004, Taiwan (ROC). (Email: [email protected])
* Correspondence: [email protected]; Tel: 886-5-2717482; Fax: 886-5-2717467.
Abstract
13
The present study proposed a discharge prediction model for open-check dams.
14
The open-check dam is composed of a trapezoidal spillway and a rectangular opening
15
located at the bottom of the spillway. The flow conditions, determined using the ratio
16
of water head and dam crest length, for flow over open-check dams were divided into
17
broad-crested and sharp-crested weir flow conditions. For the broad-crested weir flow
18
condition, the discharge equation for rectangular broad-crested weir was proposed to
19
estimate the discharge. For the sharp-crested weir flow condition, a linear
20
combination of discharge equations for rectangular and trapezoidal sharp-crested
21
weirs was proposed to describe the discharge. A series of laboratory experiments were
22
performed in order to calibrate the proposed discharge model. The results showed the
23
discharge coefficient and cross-sectional area ratio ( Ar ) were highly correlated under
24
two flow conditions. The averaged error between measured and predicted discharge
25
was less than 2%. The previous models displayed averaged errors between 4% and
26
25% under the present experimental conditions, whereas that of the present model
27
was considerably low. The present model demonstrated favorable accurate and
28
convenient estimation of discharge for flows over open-check dams.
29
Keywords: open-check dam; discharge; discharge coefficient; open channel flow 1
1
1. Introduction
2
Check dams are commonly constructed in upstream catchment areas to minimize
3
sediment disasters. However, because check dam design involves accumulating a
4
substantial amount of sediment, dam heights are often increased to expand sediment
5
storage space. Thus, check dam construction projects often have adverse effects on
6
river habitats and the continuity of sediment transport. In recent years, the increased
7
awareness of ecological conservation involved adding a rectangular opening
8
underneath the conventional spillway in check dams. Figs. 1(a) and (b) show a
9
conventional check dam with a trapezoidal spillway and that of an open-check dam
10
with a trapezoidal spillway and a rectangular opening respectively. The rectangular
11
opening in a check dam reduces the gap in elevation between the upstream and
12
downstream of the riverbed which maintains the continuity of sediment and aquatic
13
organism, thereby ameliorating the negative effects of the check dam on the natural
14
environments. In the design process of a conventional check dam (Piton & Recking,
15
2016), the discharge formula for sharp-crested weirs proposed by Zollinger (1983) is
16
commonly employed to decide the size of the spillway. However, open-check dams
17
differ considerably from conventional check dams in cross-sectional shape (the
18
addition of a rectangular opening). The use of the aforementioned discharge formula
19
to predict discharge may yield erroneous results. To effectively predict the discharge
20
of open-check dams, the present study superposed the discharge of the spillway and
21
opening of open-check dams. Considering the variations of water depth resulted in the
22
change in flow conditions, relevant data were incorporated into the discharge
23
coefficients to build a discharge prediction model of open-check dams under different
24
flow conditions. Five open-check dam models were tested in a laboratory flume for
25
large range of discharges. The predicted results are in agreement with the
26
experimental measurements. The model developed in the present study is expected as 2
1
the reference of the future researches in disaster prevention, sediment transport, and
2
habitat stability of open-check dams.
(a) Conventional check dam
(b) Open-check dam
Fig. 1. Different types of check dams. 3
2. Background
4
Check dams are transverse structures of rivers, which block water flow and cause
5
water levels to rise. Roushangar et al. (2014) asserted that water passing through
6
check dam spillways exhibits flow characteristics, including nappe flow, curved
7
streamlines, and air vent, similar to those of weir flows. Fig. 2 shows the flow patterns
8
of water flowing through a check dam. Regarding the flow conditions and the
9
geometry of the spillways, numerous studies have predicted the discharge of check
10
dams by using weir flow equations (Zollinger, 1983; Fang & Fang, 2012; Piton &
11
Recking, 2016).
Fig. 2. Flow patterns of water flowing through a check dam. 12 13
2.1. Flows over Weir According to Govinda Rao and Muralidhar (1963), weir flow conditions can be 3
1
divided into broad-crested and sharp-crested on the basis of weir water head and weir
2
crest length ratios ( h L ; where h and L are the water depth above crest weir and
3
the weir crest length, respectively).
4
2.1.1. Broad-crested Weir Flow Condition
5
When h L ≤ 1.5 , water flow makes contact with the weir crest, and the
6
friction created will cause energy loss in the water flow. This flow condition is
7
referred to as a broad-crested weir (Fig. 3(a)). The water touching the weir crest
8
upstream undergoes water flow separation, which, because of the effect of gravity,
9
causes the water to again come into contact with the weir crest. The critical water
10
depth occurs at the highest point of the water flow separation. Thus, discharge
11
predictions are often made using critical flow conditions. However, because
12
discharge prediction formulas ignore energy losses in the water flow due to
13
friction with the weir crest, the discharge predicted is generally underestimated
14
(Govinda Rao & Muralidhar, 1963; Azimi & Rajaratnam, 2009; Azimi et al.,
15
2014; Bijankhan et al., 2014).
16
2.1.2. Sharp-crested Weir Flow Condition
17
When h L > 1.5 , flows are considered to those of sharp-crested weir flows
18
(Fig. 3(b)). Because the water depth of a sharp-crested weir is greater than that of
19
a broad-crested weir, water flow in a sharp-crested weir possesses greater
20
potential energy. In a sharp-crested weir, when the water flow makes contact with
21
the weir crest upstream, the energy lost in the water flow is less than that lost in a
22
broad-crested weir. Because most of the potential energy converts into kinetic
23
energy, the water flows directly over the weir crest and lose contact with the crest.
24
In this situation, the streamlines are highly curved. Because discharge prediction
25
formulas for sharp-crested weirs ignore energy losses in the water flow due to 4
1
highly curved nappe flow, the discharge coefficients predicted are generally
2
overestimated (French, 1986; Swamee, 1988; Bos, 1989; Rehbock, 1929; Aydin et
3
al., 2011).
(a) Broad-crested weir flow condition
(b) Sharp-crested weir flow condition
Fig. 3. Weir flow condition classification. 4
2.2. Discharge Estimation of Weir Flows
5
Weirs can have a variety of cross sections, depending on the requirements. The
6
simplest type is the weir with a rectangular cross-section. The shapes of the single
7
cross-section opening are mostly rectangular and trapezoidal. By contrast, when the
8
opening shapes are a combination of different shapes, the weirs are referred to as the
9
compound cross-section weirs. Many stage-discharge relationships were used to
10 11
predict discharge of weir flows. 2.2.1. Weirs with a rectangular cross-section
12
For rectangular sharp-crested weirs, Rehbock (1929) conducted experimental
13
research and found that the discharge coefficients are determined by their water
14
head–height ratios
15
passing through a weir, French (1986) proposed using constriction factors to
16
identify the effect of cross section constriction on discharge coefficients, where
17
the discharge coefficients range from 0.59 to 0.64. In the rectangular
18
broad-crested weirs, Govinda Rao and Muralidhar (1963) considered the
(h P ) .
Because flow exhibits a constriction effect while
5
1
discharge coefficient to be a function of h L in order to take into account the
2
friction loss caused by weir crest. In addition, they revealed that the discharge
3
coefficient of a rectangular broad-crested weir ranged from 1.02 to 1.28. A
4
comprehensive review of the experimental investigations on finite crest length
5
weirs can be found in Azimi and Rajaratnam (2009). They also developed
6
empirical correlations for the discharge coefficient.
7
2.2.2 Weirs with Compound Cross-Section
8
Discharge over weirs with the compound cross-section have investigated by
9
Jan et al. (2005), Salmasi (2011), and Gogus (2016). Jan et al. (2005) discussed
10
four different compound weirs: weirs with rectangular compound cross-section
11
opening at both the top and bottom, weirs with a trapezoidal compound
12
cross-section opening at the top and rectangular compound cross-section opening
13
at the bottom, weirs with rectangular compound cross-section opening at the top
14
and triangular cross-section opening at the bottom, and weirs with trapezoidal
15
compound cross-section opening at the top and triangular cross-section opening at
16
the bottom. They calculated the discharges of the various cross sections and
17
linearly superposed them. However, Jan et al. (2005) only examined the high
18
discharge (i.e., when the flow flowed through both openings), and ignored
19
instances when the discharge was low (i.e., when the flow flowed through only the
20
bottom cross-section opening), consequently limiting the applicability of the study
21
findings. Salmasi (2011) and Gogus (2016) stated that the discharge coefficients
22
of compound weirs should change when water depths change. They investigated
23
the flow discharge for water flowing through only the bottom rectangular
24
compound cross-section opening and water flowing through both the top and
25
bottom openings. Salmasi (2011) showed that when 0.1 < h L < 0.6 , the 6
1
discharge coefficients of compound weirs change with h L ; in addition, the
2
trends for water flowing through only the bottom rectangular cross-section
3
opening differ from those for water flowing through both the top and bottom
4
rectangular cross-section openings, with discharge coefficients for these two
5
opening types ranging from 0.7 to 1.0. Gogus (2016) utilized flow coefficient Cv
6
to correct the negative effects created when the influence of velocity heads was
7
ignored during the theoretical derivation process, thereby elevating discharge
8
prediction accuracy. Gogus (2016) conducted experiments and revealed that when
9
h L < 1.7 , the discharge coefficients of compound weirs range from 0.74 to 1.10.
10
3. Theoretical Analysis
11
When water flows through check dams, the water depth ( hd ) and crest length
12
( Ld ) ratio ( hd Ld ) changes constantly due to changes in water level. Fig. 4 shows the
13
water flowing through a check dam for different discharges. Roushangar et al. (2014)
14
suggested that flow conditions through check dams are similar to that through weirs,
15
and that weir flow condition changes with the ratio of weir water depth and weir crest
16
length. Therefore, to ensure the prediction model effectively estimates the discharge
17
in different flow conditions, the present study used the weir flow classification
18
proposed by Govinda Rao and Muralidhar (1963) and divided the flows of check
19
dams into broad-crested weir and sharp-crested weir flow conditions according to the
20
ratios of dam water depth and dam crest length. Furthermore, the discharge prediction
21
model for open-check dams was developed under different flow conditions.
22 23 7
1 2
(a) Low discharge
(b) High discharge
Fig. 4. Water flows over open-check dam with different discharges. 3
3.1. Broad-Crested Weir Flow
4
When flowing through open-check dam with hd Ld ≤ 1.5 , the amount of
5
water passing through the dams is considered to have a low discharge. At this
6
time, most of the water flows through the rectangular opening below the
7
trapezoidal spillway. Fig. 5(a) shows the dimensions of an open-check dam, where
8
b0 and hd are the width of the rectangular opening and the water depth at the
9
head measurement section, respectively. Here, because the water in the upstream
10
of the dam contains scant potential energy, it converts into little kinetic energy
11
when the water flows through the weir crest, resulting in relatively short water
12
flow separation distances and causing the water to come in contact with the weir
13
crest. Regarding hydraulics, the water flow is classified as a broad-crested weir
14
flow. In this flow condition, the discharge of open-check dam was predicted using
15
the broad-crested weir water flow method.
8
(a) Front view
(b) Side view
Fig. 5. Definition sketch for broad-crested weir flow over open-check dam. 1
Fig. 5(b) presents the height, width energy head of a dam. Cross Section 1 shows
2
a stable water flow in the upstream of the dam, whereas Cross Section 2 reveals the
3
separation of water when the water comes in contact with the weir crest. Gogus
4
(2016) hypothesized that there are no energy losses from Cross Section 1 to Cross
5
Section 2 and that critical flow conditions occur at Cross Section 2. The energy
6
relationships between the two cross sections can be obtained using the following
7
energy equation:
8
H 1 = h2 +
9
where H 1 is the total energy head of Cross Section 1, h2 is the water depth of
10
Cross Section 2, V2 is the flow velocity of Cross Section 2, g is the gravitational
11
acceleration, yc is the critical water depth , and Vc is the critical velocity. By
12
incorporating the relationship between critical velocity and total energy head obtained
13
from Equation (3-1) into the flow rate formula (Q = A × V ) , the equation of discharge
14
can be obtained as:
15
Qb = Ac 2g(H1 − yc )
V22 V2 = yc + c 2g 2g
(1)
(2)
9
1
where Ac is the cross-sectional area where critical flow occurred and Qb is the
2
discharge of the check dam under broad-crested weir water flow condition. Because
3
hd Ld ≤ 1.5 , the cross-sectional shape of the water passing through the open-check
4
dam is rectangular. Assuming that the velocity head of Cross Section 1 can be
5
ignored, total energy head is therefore equal to the water depth at the head
6
measurement section (H1 = hd ) . Because the flow through the control section is
7
critical, the discharge equation of the flow passing through the rectangular opening at
8
the bottom of the open-check dam can be written as:
9
Qb =
10
2 2 1. 5 b0 g hd 3 3
(3)
3.2. Sharp-Crested Weir Flow
11
When hd Ld > 1.5 , water flows through both the bottom rectangular opening
12
and the top trapezoidal spillway of open-check dams. Fig. 6(a) shows the dimensions
13
of a check dam, where b0 is the width of the rectangular opening, b1 is the bottom
14
width of the trapezoidal spillway, b2 is the top width of the trapezoidal spillway, hrd
15
is the water depth of the rectangular opening, and htd is the water depth of the
16
trapezoidal spillway. Considering the kinetic energy of sharp-crested weir flow is
17
high, the water no longer coming into contact with the weir crest and creates a
18
sharp-crested weir water flow pattern. Therefore, the present study used sharp-crested
19
weir theories to predict discharge. The water passes both the rectangular opening and
20
trapezoidal spillway. The linear superposition method (Martínez et al., 2005;
21
Piratheepan et al., 2006; Jan et al., 2005) was adopted to superpose the discharges of
22
the rectangular opening and trapezoidal spillway.
10
1
In Fig. 6(b), Cross Section 1 presents stable water flow in the upstream of the
2
dam, whereas Cross Section 2 shows the water flowing over the weir crest. Assuming
3
that no energy losses occurred between Cross Section 1 to Cross Section 2, the energy
4
relationships between the two cross sections can be obtained as:
5
H1 = (hd − ∆h) +
V22 2g
(4)
6
Assuming that the approaching velocity of progressive flow in Cross Section 1 is
7
zero, then the velocity head in Cross Section 1 (V12 2 g ) can be ignored. The upstream
8
total energy head ( H1 ) is therefore equal to check dam water depth hd ( H1 = hd ),
9
and the flow velocity of Cross Section 2 ( V2 ) can be simplified as:
10
V2 = 2g∆h
(3–5)
11
Using the flow velocity ( V2 ) and cross-sectional areas of Cross Section 2, one
12
can obtain the discharge passing though the weir. Bos (1989) expressed the discharge
13
for flow over a rectangular sharp-crested weir in the following form:
14
2 1.5 Qr = b0 2g hrd 3
15
where Qr is the discharge of the rectangular opening of the open-check dam. For a
16
trapezoidal sharp-crested weir, Henderson (1966) expressed its discharge equation in
17
the form as:
18
Qt =
19
where Qt is the discharge of the trapezoidal spillway. Theoretical discharge equation
20
for an open-check dam with the sharp-crested weir Flow is developed by linearly
(6)
2 4 2 g htd1.5 b1 + mhtd 3 5
(7)
11
1
superposing the discharge of the rectangular opening and the trapezoidal spillway. The
2
equation can be rearranged as follows:
3
Qc =
1. 5 2 4 2 g b0 hrd + htd1.5 b1 + mhtd 3 5
(8)
(a) Front view
(b) Side view
Fig. 6. Definition sketch for sharp-crested weir flow over open-check dam. 4
3.3. Discharge Coefficient ࢊ
5
In the previous developments of the discharge equations, a number of
6
assumptions are made for the sake of simplicity and therefore a discharge coefficient
7
was usually introduced in the real fluid conditions. (Swame, 1988; Bos, 1989;
8
Rehbock, 1929; Govinda Rao & Muralidhar, 1963; Azimi & Rajaratnam, 2009; Aydin
9
et al., 2011; Salmasi, 2011; Bijankhan et al., 2014; Gogus, 2016). Eqs. (9) and (10)
10
are the theoretical discharge prediction models containing the discharge coefficient
11
for open-check dams under sharp-crested and broad-crested weir flow conditions,
12
respectively.
13
Broad-crested weir flow conditions ( hd Ld ≤ 1.5 )
Q p1 = Cd 1 × Qb 14
2 2 1.5 = Cd 1 × b0 g hd 3 3
(9)
12
1
Sharp-crested weir flow conditions ( hd Ld > 1.5 ) Q p 2 = C d 2 × Qc
2
= Cd 2 ×
2 4 2 g b0 hrd1.5 + htd1.5 b1 + mhtd 3 5
(10)
3
where Qp1 is the discharge through the rectangular opening, Qp 2 is the discharge
4
through both the rectangular opening and trapezoidal spillway, Cd1 is discharge
5
coefficient of broad-crested weir flow condition, and Cd 2 is the discharge coefficient
6
of sharp-crested weir flow condition. Aforementioned analysis revealed the discharge
7
coefficients of weir flows, applications of these discharge coefficients have
8
limitations and must be met for the applicable conditions. To effectively and
9
accurately predict the discharge of open-check dams, it is necessary to explore the
10
discharge coefficients under desired flow conditions.
11
4. Experimental Materials and Procedure
12
4.1. Experimental Setup
13
Experiments of this study were performed in a rectangular flume. The flume was
14
supplied by pumping water from the reservoir to the constant-head water tank. An
15
inflow control valve was used to adjust the discharge during the experiments, and the
16
water head of a V-shaped weir was used to determine the discharge. The check dam
17
model was built using 6-mm thick acrylic sheets. The inside of the model was filled
18
with quartz sand to eliminate the effect of buoyancy during the experiments. Silicone
19
was used to seal the contact areas between the flume sidewalls and the bed to prevent
20
experimental errors due to water leakage and dam movement. Fig. 7 shows definition
21
sketch of model dam and flume used in the experiment. Experimental analyses were
22
performed for the five check dams with different opening sizes. The geometric
23
dimensions of all the models are summarized in Table 1. 13
Fig. 7. Definition sketch of model dam and flume used in the experiment 1 Table 1. Parameters of the check dam model Distance between the Experimental
rectangular
group
opening and the riverbed
Rectangular opening width b0
Rectangular
Trapezoidal
Trapezoidal
opening
spillway
spillway
height hr
side slope
height ht
(m)
m
(m)
(m)
Pd (m)
2
Check dam crest length
Ld (m)
E1
0.15
0.05
0.10
0.5
0.15
0.07
E2
0.15
0.075
0.10
0.5
0.15
0.07
E3
0.15
0.10
0.10
0.5
0.15
0.07
E4
0.15
0.05
0.10
0.3
0.15
0.07
E5
0.15
0.05
0.10
1.0
0.15
0.07
4.2. Experimental Procedure
3
During the experiments, it was ensured that the water flow passed through the
4
check dam and became free flow with nappe flow characteristics. A needle gauge with
5
a precision of ±0.1 mm was used to measure the water depth of the flume.
6
Measurements were made from the nappe drop region to the upstream region until the
7
change in water depth was less than 1 mm (i.e., the change in water depth was 14
1
minimal), indicating a stabilized water depth. This water depth of the measurement
2
point was considered as the upstream water head ( h0 ). The check dam water depth
3
hd (= h0 − Pd ) was obtained by subtracting the weir height ( Pd ) from the upstream
4
water head ( h0 ). A total of 110 discharge experiments were performed, in which 22
5
discharge experiments were performed for five check dams. The experimental cases
6
and ranges of variables are summarized in Table 2, where Qm is the discharge, h0
7
is the upstream water head, and Fr is the Froude number based on the upstream
8
water head.
9 Table 2. Flow characteristics of the experimental cases. Experimental Number of
Discharge Qm
Check dam water depth h0
Froude number Fr
group
experiments
(m3/s)
E1
22
0.00023~0.03074
0.169~0.387
0.005~0.131
E2
22
0.00020~0.03416
0.168~0.383
0.005~0.149
E3
22
0.00018~0.02818
0.166~0.384
0.004~0.122
E4
22
0.00052~0.03340
0.171~0.383
0.011~0.145
E5
22
0.00038~0.03375
0.171~0.390
0.008~0.142
(m)
10 11
4.3. Deducing Reduction Factor for Discharge Coefficient
12
The discharge coefficient is generally described as a function of the relevant
13
dimensionless parameters based on experimental data. For a weir, the relevant
14
parameters of discharge ( Qm ) were fluid mass density ( ρ ), gravitational acceleration
15
( g ), dam water depth ( hd ), dynamic viscosity coefficient ( µ ), weir height ( Pd ), and
16
flume slope ( S ) (Aydin et al., 2011). For an open-check dam, the water flowing
17
through the dam may create a contraction effect. Therefore, the present study
15
1
considered the flow area of dam ( A ) as one of the factors that influenced discharge.
2
The discharge can be expressed by the following functional relationship:
3
Qm = f1 (ρ、g、h、 d µ、P、 d S、A)
(11)
4
Because the functional relationship in Eq. (11) represents a physical phenomenon
5
that does not depend on the choice of measurement units, using the Buckingham π
6
theorem, Eq. (11) can be expressed in a dimensionless form as follows:
Qm
ρ g hd1.5 hd A = f2 、 、 、S µ Pd A0
7
Cd =
8
where B is the width of the channel; A0 ( = B× h0 ) is the cross-sectional area
9
ρ g hd1.5 h upstream of the dam; is the Reynolds number of the flow, d is the ratio µ Pd
B g hd1.5
(12)
A is the ratio between the A0
10
of the check dam water depth and the weir height, and
11
flow area of the dam and that of the upstream. In the experiments flume slope is
12
settled as constant and the research only focuses on the effect of the ratio between the
13
flow area of the dam and that of the upstream on discharge coefficient of open-check
14
dam. The dimensionless parameter for an open-check dam then can be simplified as
15
follows:
16
Cd =
17
where Ar is the ratio between the flow area of the dam and that of the upstream. The
18
use of Ar enables the discharge coefficient to account for the contraction effect and
19
energy changes created by the differences between the cross-sectional areas of the
20
check dam opening and that of the channel.
A Qm = f 3 1.5 B g hd A0
= f 3 ( Ar )
(13)
16
1
5. Results and Discussion
2
Figs. 8(a) and (b) show the variations of experimental discharge Qm and the
3
corresponding discharge coefficients Cd1 and Cd 2 at the broad-crested weir flow
4
and sharp-crested weir flow conditions, respectively. For the broad-crested weir flow
5
condition, discharge coefficient increases with increasing discharge or rectangular
6
opening width ( b0 ). For sharp-crested weir flow condition, the behavior of discharge
7
coefficient with discharge is reversed; i.e. discharge coefficient decreases with
8
increasing discharge. As the side slope of spillway m increases, moreover, there is a
9
decrease in discharge coefficient. It is clear from Figs. 8(a) and (b) that the discharge
10
coefficients of open-check dams change with discharge, the geometric shapes of the
11
check dams, and flow conditions change. The discharge coefficient is difficult to
12
represent as a simple trend. An attempt to directly fit the discharge to experimental
13
data with a simple mathematical expression involve using the results of the
14
dimensional analysis.
15
Figs. 9(a) and (b) plot the discharge coefficients
(i.e., Cd1 and C d 2 ) with
16
corresponding dimensionless flow area ratio Ar at different flow conditions,
17
respectively. The importance of flow area ratio Ar for contraction weir flow is
18
obvious. It is possible to develop best fit expressions of discharge coefficients. For the
19
broad-crested weir flow condition, C d 1 and Ar ranged from 1.0 to 1.19 and from
20
0.01 to 0.08, respectively, where Cd1 increases with Ar . The best fit coefficients are
21
obtained by a simple linear regression analysis and the relationship between Cd1 and
22
Ar is given as:
23
Cd1 = 3.278× Ar + 0.958
24
where the coefficient of determination ( R2 ) is 0.87, and discharge coefficient Cd1
25
and dimensionless parameter Ar are highly correlated. When water flow is
(14)
17
1
concentrated in the rectangular opening of the open-check dam, the flow experiences
2
a significant contraction effect and backwater occurs. It increases the potential energy
3
of the upstream water. The increased potential energy is subsequently converted into
4
kinetic energy while passing through the dam. As the potential energy is increased,
5
there is increasing upstream water level which increases the streamline curvature on
6
horizontal plane around the dam. This reduces the flow passing capacity of the weir
7
and corresponding an increase of discharge coefficient. For the sharp-crested weir
8
flow condition, Ar and Cd 2 ranged from 0.06 to 0.26 and from 0.71 to 0.79,
9
respectively, where Cd 2 decreases linearly when Ar increases. A simple linear
10
regression analysis also revealed a high correlation ( R2 = 0.848) between Ar and
11
Cd 2 . The best fit function for Cd 2 and Ar is as follows:
12
Cd 2 = −0.408× Ar + 0.817
(15)
13
When water flooded the spillways, because the geometry of trapezoid is narrow
14
at the bottom and wide at the top, the contraction effect partly vanished with
15
increasing flow depth and side slope (m). Thus, the discharge coefficient decreased
16
with increasing Ar . Figs. 10(a) and (b) show the comparisons of the experimental
17
discharge ( Qm ) and the calculated discharge ( Qp1 and Q p 2 ) obtained by using Eqs.
18
(14) and (15) for discharge coefficient under two flow conditions, respectively. For
19
the broad-crested weir flow condition, the error percentages between the experimental
20
data and predicted value ( (Qm − Q p 1 ) / Qm ) ranged between 0.14% to 2.88% with an
21
averaged value of 1.39%. For the sharp-crested weir flow condition, the error
22
percentages ( (Qm − Q p 2 ) / Qm ) ranged between 0.02% to 1.95% with an averaged
23
value of 0.99%. The prediction models successfully estimate the discharge passing
24
through open-check dams with error less than 3%. 18
1
To evaluate the performance of the discharge prediction model developed in the
2
present study, the results obtained using present model were compared with those
3
obtained using the previous models. For the broad-crested weir flow condition, the
4
previous models of Govinda Rao and Muralidhar (1963), Swamee (1988), Azimi and
5
Rajaratnam (2009), Salmasi (2011), and Gogus (2016) were used. Fig. 11 shows the
6
comparisons of the experimental discharge and the predicted discharge. It is seen that
7
previous models generally underestimated the discharge and the prediction errors
8
increased as discharge increased. The averaged error of different models is
9
summarized in Table 3. The error of the previous models ranges between 4.03% and
10
24.96% and three models have averaged error greater than 10%. From hydraulics
11
point of view, the approaching velocity must be low enough and unsteady phenomena
12
must be eliminated for the purpose of weir flow measurements. Therefore, the
13
contraction rate Ar of the broad-crested weir is sufficient to prevent possible
14
vorticity and turbulence produced at the upstream. In the present cases of weir flow,
15
the contraction effect was pronounced. It resulted in relatively large discharge errors
16
of the previous models than that of the present model. The model proposed by
17
Salmasi (2011) has a largest averaged error of approximately 25%. The discharge
18
prediction models presented by Gogus (2016) and Swamee (1988) displayes relatively
19
favorable prediction results among the previous models. The Gogus’s model shows
20
prediction errors ranging between 0.25% and 11.87% with an average of about 4%.
21
Nevertheless, Gogus’s model showed large deviations at high discharge values which
22
suggests further examinations. The errors of Swamee’s model range from 0.16% to
23
10.45% with an average of about 4%. However, this model involves a complex
24
calculation of parameters that gives a limitation for practically application. Among all
25
the models, the present model provides a lowest averaged error of 1.39%. It is seen
26
that the present model can successfully represent the discharge data with similar 19
1
accuracy at the broad-crested weir flow condition.
2
For the sharp-crested weir flow condition, the present model was compared with
3
the discharge equation proposed by Jan et al. (2005) for the compound sharp-crested
4
weir with similar geometry as the open-check dam. A comparison between the
5
experimental discharge and the corresponding predicted discharge using the present
6
model and the model proposed by Jan et al. (2005) is presented in Fig. 12. The errors
7
of the model proposed by Jan et al. (2005) increases as discharge increases with the
8
averaged and largest errors of approximately 10% and 19%, respectively. Jan et al.
9
(2005) formulated the discharge by dividing into five parts as three rectangular and
10
two triangular sharp-crested weirs and linearly superposed discharge of each part.
11
However, the discharge coefficient of Jan et al. (2005) for the triangular sharp-crested
12
weir considered exclusively as a function of the geometric parameter. The most
13
important characteristic of the weir flow is the independence of the discharge
14
coefficient from the channel width, B, because of the completely contracted nature of
15
the flow over the weir. Errors of the present model fall far below the aforementioned
16
model due to incorporate contracted aspects of flow behavior into the formulation of
17
discharge coefficient. Because the compound cross sections were manufactured using
18
cross sections of varying shapes, the model proposed by Jan et al. (2005) first
19
predicted the discharge and discharge coefficients for each cross sections and linearly
20
superposed discharge of each cross section to obtain the total discharge of the
21
compound cross sections. The discharge coefficient used by Jan et al. (2005) could
22
not account for the changes in the cross sections of where water discharged as check
23
dam water level increased or decreased. Additionally, slit-check dams were exposed
24
to substantial contraction effects because of the changes in the cross sections where
25
water discharged. As a result, the discharge prediction errors created in the model
26
proposed by Jan et al. (2005) were greater than those developed in the present study 20
1
were.
2 3
(a)
(b) 21
Fig. 8. Discharge coefficient versus measured discharge: (a) broad-crested weir flow condition; (b) sharp-crested weir flow condition. 1
(a)
(b) 22
Fig. 9. Discharge coefficient versus flow area ratio: (a) broad-crested weir flow condition; (b) sharp-crested weir flow condition. 1
23
(a)
(b) Fig. 10. Comparison between calculated and measured discharge: (a) broad-crested weir flow condition; (b) sharp-crested weir flow condition. 1
24
Fig. 11. Comparison of predicted discharges by using the various models under broad-crested weir flow condition. 0.04 45 degrees exact line Present study Jan et al. (2005)
0.03
0.02
0.01
0 0
0.01
0.02
0.03
0.04
Measured Discharge (m3/sec) Fig. 12. Comparison of predicted discharges by using the various models under sharp-crested weir flow condition.
25
Table 3. Summary of the errors in the predicted discharges for different models. Error (%) Flow condition
Model Minimum
Maximum
Average
Standard deviation
0.14
2.88
1.39
0.77
4.76
18.15
11.13
3.01
0.16
10.45
4.05
2.49
5.77
19.98
12.48
3.12
17.65
31.20
24.96
3.02
Gogus (2016)
0.25
11.87
4.03
2.69
Sharp-crested Present study weir flow Jan et al. ( hd Ld > 1.5 ) (2005)
0.02
1.95
0.99
0.52
1.43
18.84
9.69
3.84
Present study Govinda Rao & Muralidhar (1963) Swamee Broad-crested (1988) weir flow Azimi & ( hd Ld ≤ 1.5 ) Rajaratnam (2009) Salmasi (2011)
1
6. Conclusions
2
In this study, the discharge equations for open-check dams with a trapezoidal
3
spillway at the top and a rectangular opening at the bottom were developed. The ratio
4
of check dam water depth ( hd ) and crest length ( Ld ) was used to determine the flow
5
conditions of the open-check dams. Theoretical discharge equations were developed
6
for different flow conditions. Considering the contraction effect created by dams, the
7
ratio between the cross-sectional areas of upstream and dam ( Ar ) was incorporated
8
into the discharge coefficient. Experiments were performed to investigate the effect of
9
Ar on discharge coefficients . The predicted discharges of the present study were
10
compared with those of other studies. From the analysis of these results, the following
11
conclusions can be stated: 26
1
1. Formulation of open-check dams depends on the ratio between the check dam
2
water depth and crest length. When hd Ld ≤ 1.5 , the water flow touched the weir
3
crest, creating flow condition that resembled broad-crested weir flow condition.
4
Thus, the broad-crested weir discharge formula can use to predict discharge; when
5
hd Ld > 1.5 , the flow condition resembled sharp-crested weir flow condition, and
6
the sharp-crested weir discharge formula can use to predict discharge.
7
2. Experimental observations indicate the discharge coefficients are significantly
8
affected by Ar and empirical equations of discharge coefficients for different
9
flow conditions were developed, respectively. Under broad-crested weir flow
10
conditions, the averaged error between the predicted and measured discharge was
11
within 2%; under sharp-crested weir flow conditions, the averaged error was
12
within 1%, indicating that the discharge prediction model of the present study
13
produced reasonable results.
14
3. In contrast with other relevant studies, the present model formulates discharge in
15
open-check dams in terms of the water depth and geometric dimensions of
16
open-check dams, which is easy to use practically. For the range of the
17
experiments, it is also demonstrated the present model provides favorable
18
prediction results.
19
7. References
20
1.
Finite Crest Length.” Journal of Hydraulic Engineering, 135(12), 120-125.
21 22
Azimi, A.H. and Rajaratnam, N. (2009). “Discharge Characteristics of Weirs of
2.
Aydin, I., Altan-Sakarya, A.B. and Sismanb, C. (2011). “Discharge formula for
23
rectangular
sharp-crested
weirs.”
24
Instrumentation, 22, 144-151.
Journal
27
of
Flow
Measurement
and
1
3.
Azimi, A.H., Rajaratnam, N. and Zhu, D. (2014). “Submerged Flows over
2
Rectangular Weirs of Finite Crest Length.” Journal of Irrigation and Drainage
3
Engineering, 140(5), 10.1061/(ASCE)IR.1943-4774.0000728, 06014001.
4
4.
Institute for Land Reclamation and Improvement, Wageningen. The Netherlands.
5 6
Bos, M.G. (1989). “Discharge measurement structures”, 2nd Ed., International
5.
Bijankhan, M., Stefano, C.D., Ferro, V. and Kouchakzadeh, S. (2014). “New
7
Stage-Discharge Relationship for Weirs of Finite Crest Length.” Journal of
8
Irrigation
9
10.1061/(ASCE)IR.1943-4774.0000670, 06013006.
and
Drainage
Engineering,
140(3),
10
6.
French, R.H. (1986). “Open-channel hydraulics.”, McGraw-Hill, New York.
11
7.
Fang, C. and Fang, K. (2012). “Derivation and Verification of New Common
12
Formulas for the Calculation of Instantaneous Dam-break Maximum Discharge.”
13
Journal of Procedia Engineering, 28, 648-656.
14
8.
Weirs of Finite-Crest Width.” Houille Blanche, 18(5), 537-545.
15 16
Govinda Rao, N.S. and Muralidhar, D. (1963). “Discharge Characteristics of
9.
Gogus, M., Al-Khatib, I.A., Atalay, A.E. and Khatib, J.I. (2016). “Discharge
17
prediction in flow measurement flumes with different downstream transition
18
slopes.” Journal of Flow Measurement and Instrumentation, 47, 28-34.
19 20
10. Henderson, F.M. (1966). “Open channel flow.” Prentice-Hall, Englewood Cliffs, N.J.
21
11. Jan, C.D., Chang, C.J. and Lee, M.C. (2005). “Discussion of Design and
22
Calibration of a Compound Sharp-Crested Weir.” Journal of Hydraulic
23
Engineering, 131(2), 112-116.
24
12. Martínez, J., Reca, K., Morillas, M.T. and López, J.G. (2005). “Design and
25
Calibration of a Compound Sharp-Crested Weir.” Journal of Hydraulic
26
Engineering, 13(2), 112-116. 28
1
13. Piratheepan, M., Winston, N.E.F. and Pathirana, K.P.P. (2006). “Discharge
2
Measurements in Open Channels Using Compound Sharp-Crested Weirs.”
3
Journal of the Institution of Engineers, 3, 31-38.
4
14. Piton, R. and Recking, A. (2016). “Design of Sediment Traps with Open Check
5
Dams. I: Hydraulic and Deposition Processes.” Journal of Hydraulic
6
Engineering, 142(2), 10.1061/(ASCE)HY.1943-7900.0001048, 04015045.
7 8
15. Rehbock, T. (1929). “Discussion of Precise Weir Measurements.” Journal of Transportation Engineering, 93, 1143-1162.
9
16. Roushangar, K., Akhgar, A., Salmasi, F. (2014). “Estimating Discharge
10
Coefficient of Stepped Spillways under Nappe and Skimming Flow Regime
11
Using Data Driven Approaches.” Journal of Flow Measurement and
12
Instrumentation, 59, 78-87.
13 14
17. Swamee, P.K. (1988). “Generalized Rectangular Weir Equations.” Journal of Hydraulic Engineering, 114(8), 945-949.
15
18. Salmasi, F., Sanaz, P., Dalir, A.H. and Zadeh, D.F. (2011). “Discharge Relations
16
for Rectangular Broad-Crested Weirs.” Journal of Agricultural Science, 17(4),
17
324-336.
18
19. Zollinger, F. (1983). “Die Vorgänge in einem Geschiebeabl Agerungsplatz ihre
19
Morphologie und die Möglichkeiten einer Steuerung.” Ph.D. thesis, ETH Zürich,
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Zurich.
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Highlights: •
Discharge prediction model for open-check dams was developed.
•
Flow conditions are divided as broad-crested weir flow and sharp-crested weir flow.
•
The results confirm the model was generalized with high accuracy.
Hsun-Chuan Chan: Conceptualization, Methodology. Hsin-Kai Yang: Data curation, Writing- Original draft preparation. Po-Wei Lin: Data curation. Jung-Tai Lee: Supervision, Writing- Reviewing and Editing.