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Side weirs are widely used in hydraulic engineering applications. The studies on the subject have been generally focused on classical and labyrinth side weirs. However, the same is not true for piano key side weirs (PKSW) in a straight channel. The piano key weir (PKW) has high discharge capacity compared with classical weirs. In this study, the hydraulic characteristics of a trapezoidal piano key side weir (TPKSW) in straight channels were investigated experimentally. In all experiments, the hydraulic characteristics of nine TPKSW models were studied extensively using the De Marchi, Domínguez and Schmidt approaches in the subcritical flow regime, with Froude number range 0.12 < F1 < 0.87. The results show that a TPKSW provides better performance compared to traditional rectangular and triangular labyrinth side weirs. Specifically, for the 0.12 < F1 < 0.4 condition, the efficiencies of a TPKSW and trapezoidal labyrinth side weir are close to each other. A trapezoidal labyrinth side weir is more efficient than a TPKSW at larger Froude numbers. The discharge capacity of the TPKSW is 2.9 to 12 times higher than that of the rectangular side weir. Scatter diagrams were obtained for CPW and F1 numbers using various approaches available in the literature. The diagram generated by the De Marchi approach has much less scattering, compared to the diagrams generated by the Domínguez and Schmidt approaches. It has been determined that TPKSWs are an effective type of side weir in lateral flows. Lastly, an empirical equation was obtained for the discharge coefficient, which is in good agreement with the experimental data.

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  • TPKSWs provide better discharge performance compared to traditional rectangular and triangular labyrinth side weirs.

  • Most effective parameters discharge coefficient of TPKSW are Froude number and piezometric head.

  • CFD methods successfully model TPKSW structures and provide consistent results with experimental data.

  • The Schmidt and Domínguez methods produce very similar results. The De Marchi method provides less scatter data.

discharge coefficient, open-channel flow, piano key weir, side weir, water discharge
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b

width of main channel (L)

B

length of sidewall of PKW (L)

Bi

downstream overhang length of PKW (L)

Bo

upstream overhang length of PKW (L)

Cd

side weir discharge coefficient (–)

CPW

PKSW discharge coefficient (–)

E

specific energy (L)

F1

Froude number at upstream end of side weir (–)

F2

Froude number at downstream end of side weir (–)

acceleration due to gravity (LT−2)

h1

piezometric head on side weir at upstream end (L)

h2

piezometric head on side weir at downstream end (L)

y

the depth of flow at the section x (L)

y1

flow depth at upstream end of side weir at channel center (L)

y2

flow depth at downstream end of side weir at channel center (L)

W

opening length of side weir (L)

Wi

inlet key width of PKW (L)

Wo

outlet key width of PKW (L)

V1

flow velocity on the first edge of the weir (LT−1)

Ts

wall thickness of PKW (L)

L

total weir crest length of labyrinth and piano key side weir (L)

Nu

number of PKW-units (–)

P

height of weir crest (L)

σ

water surface tension (N/m)

Q

discharge in the main channel before m3/s beginning of side weir (L3T−1)

Q1

total discharge in main channel at upstream end of side weir (L3T−1)

Q2

total discharge in main channel at downstream end of side weir (L3T−1)

Qw

outflow weir discharge (L3T−1)

q

discharge per unit length over side weir (L2T−1)

α

sidewall angle of the trapezoidal piano key and labyrinth side weir (°)

α’

kinetic energy correction coefficient (–)

varied flow function of De Marchi (–)

δ

head angle (°)

η

outflow efficiency (-)

χ

dimensionless ratio dependent on downstream hydraulic conditions (–)

φ

angle of the flow deviation (°)

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Side weirs are often used to divert a certain amount of discharge from an open channel. Important hydraulic structures such as irrigation systems, wastewater facilities, sewerage networks, hydroelectric facilities and flood reduction or prevention networks often require components for efficient flow distribution and control. For this purpose, side weirs provide a spatially variable flow with decreasing discharge in the direction of the flow. In addition to directing water, another purpose of the side weirs is to measure the amount of discharge. The hydraulic characteristics of a side weir change depend on the type of weir, its cross-section, and the plan shapes of the main channel.

Since the hydraulic properties of a side weir depend on the type of weir, the subject still attracts the attention of researchers. Many important studies on classical sharp-crested weirs have been reported in the literature (Subramanya & Awasthy 1972; El-Khashab & Smith 1976; Ranga Raju et al. 1979; Kumar & Pathak 1987; Hager 1987; Borghei et al. 1999; Emiroglu et al. 2011; Ghaderi et al. 2020). The discharge coefficient equations for a conventional side weir, according to Subramanya and Awasthy (Subramanya & Awasthy 1972), Hager (Hager 1987) and Borghei et al. (1999) are given in Equations (1)–(3), respectively:
(1)
(2)
(3)
There have been several experimental (Emiroglu et al. 2007; Borghei et al. 2013; Nezami et al. 2015) and numerical (Aydin & Emiroglu 2013, 2016; Abbasi et al. 2020) studies on labyrinth side weirs which indicated high performance and discharge capacity. Borghei et al. (2013) studied a triangular labyrinth side weir with two cycles and proposed Equation (4):
(4)
where Cd is the discharge coefficient of the side weir (dimensionless), W is the opening length of the side weir (m), b is the channel width (m), δ is the head angle (degree), P is the crest height (m), y1 is the depth of flow at the upstream end of the side weir in the main channel center line (m) and F1 is the Froude number.
An extensive experimental study on a trapezoidal labyrinth side weir with two cycles was reported in Emiroglu et al. (2014). The coefficient of discharge for this structure is given by:
(5)
where L is the overflow length of the side weir (m) and α is the sidewall angle (degrees).

The first theoretical study on the hydraulics of a rectangular side weir was reported by De Marchi (1934) and is widely used for different types of side weirs (Subramanya & Awasthy 1972; El-Khashab & Smith 1976; Hager 1987; Borghei et al. 1999; Borghei & Parvaneh 2011; Emiroglu et al. 2014). Other approaches that are of limited use were proposed by Schmidt (Borghei & Parvaneh 2011) and Domínguez (1935). Emiroglu & Ikinciogulları (2016) showed that the Schmidt approach provides valid results for discharge coefficient of classical rectangular side weirs. Bagheri et al. (2014a, 2014b) stated that the Domínguez approach is a reasonable method to determine the discharge coefficient of side weirs.

The piano key weir (PKW) is a particular shape of labyrinth weir involving up stream and downstream overhangs, to reduce the base length requirement (Figure 1). The PKW represents the evolution of the traditional labyrinth weir. Based on the slopes of inlet and outlet keys, the structure has a specific upstream and downstream overhang. This design reduces the footprint of the side weir compared to a rectangular labyrinth weir with vertical walls. As a result, the forces acting on the lateral walls are reduced, and the structural cost is reduced (Kabiri-Samani & Javaheri 2012). Therefore, in addition to the improved hydraulic performance, the PKW can be built even in very restricted foundation space, for example in the case of gravity dam crests. For a given upstream head, the PKW may improve the discharge capacity by up to four times, compared to an ogee-crested weir of the same width (Ouamane & Lempérière 2006), and by 10% compared to a labyrinth weir of the same crest footprint (Anderson & Tullis 2011).
Figure 1
Fundamental parameters of a PKW and unit PKW – plan view (left) and cross-section (right).
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Fundamental parameters of a PKW and unit PKW – plan view (left) and cross-section (right).

Figure 1
Fundamental parameters of a PKW and unit PKW – plan view (left) and cross-section (right).
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Fundamental parameters of a PKW and unit PKW – plan view (left) and cross-section (right).

Close modal

Various studies have been carried out in order to determine the best values for geometric characteristics that influence the efficiency of PKW as a frontal weir. The L/W ratio is the major parameter influencing discharge capacity, according to Ouamane & Lempérière (2006). Leite Ribeiro et al. (2012) stated that L/W value should be between 4 and 7, with L/W = 5 being the best. According to Lempérière et al. (2011), a L/W ratio of 5 is a good compromise between weir efficiency and structure complexity. Anderson & Tullis (2013) examined the performance of PKW with inlet–outlet key ratios ranging from 0.67 to 1.5 and found that increasing Wi/Wo (the inlet key width to the outlet key width) enhances flow coefficient. The best input and output key widths ratio, according to Lempérière et al. (2011), is between 1.2 and 1.5. Ouamane & Lempérière (2006) showed that the discharge efficiency depends on the P/Wu ratio and stated that when the weir height was increased by 20%, the discharge increase was between 5 and 10%. Gharibvand et al. (2016) stated that when the height of the PKW weir was increased by 50%, the flow efficiency increased by 26% at low nappe load values, thus supporting the relevant study.

The ‘PKW-unit’ represents the smallest extent of a complete structure of a PKW, comprising two transversal walls, two half outlets and an inlet. The basic geometric parameters of a PKW are the number of PKW-units (Nu), weir height (P), lateral crest length (B), up stream and downstream overhang lengths (Bo and Bi), inlet and outlet widths (Wi and Wo), and the wall thickness (Ts) (Pralong et al. 2011).

Depending on the existence of the upstream and downstream overhangs, PKW has been classified into four types, as presented in Figure 2. The basic geometry of a PKW, called type A, includes up stream and downstream overhangs. When the structure does not have a downstream or upstream overhang, the PKW is of type B or C, respectively. A PKW without overhangs is called type D (Lempérière et al. 2011). The structure is composed of two units placed along the side of a straight rectangular channel, in a fixed-bed (see Figure 3).
Figure 2
Types of PKW.
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Types of PKW.

Figure 3
Schematic layout of TPKSW: (a) plan, (b) front view.
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Schematic layout of TPKSW: (a) plan, (b) front view.

Figure 3
Schematic layout of TPKSW: (a) plan, (b) front view.
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Schematic layout of TPKSW: (a) plan, (b) front view.

Close modal

Studies on PKWs as side weirs are limited in the literature. Karimi et al. (2018) studied a C-type rectangular PKSW (RPKSW) in a straight channel. The study showed that C-type RPKSWs and rectangular labyrinth side weirs are more efficient than the equivalent linear side weirs, in terms of discharge capacity. Saghari et al. (2019) examined A-type trapezoidal piano key side weirs (TPKSWs) in a curved channel. As a result, an empirical equation is proposed to determine the discharge coefficient of A-type TPKSW. In an experimental study, Mehri et al. (2018) compared the discharge coefficient of C-type RPKSWs for channel curve angles of 30° and 120° in plan. The structure with 120° provides a higher weir discharge coefficient. In addition, Mehri et al. (2020) reported that P/h1 is the most effective parameter on the discharge coefficient of the side weir with the A, B, C and D type piano keys in the 120° curved channel and stated that the most efficient model is the B type.

Although TPKSW has been implemented in a curved channel, it has not yet been studied for a wide range of experiments as a side weir in a straight channel. The present study contributes to this by presenting experiments on the performance of a TPKSW. A wide range of design parameter values was studied under subcritical flow conditions. The performance of a TPKSW installed along a straight, rectangular channel was tested in a laboratory setting under fixed-bed and subcritical flow conditions. The main goals of this work were to (1) describe the hydraulic characteristics of a TPKSW flow (such as water surface profile, specific energy, and outflow efficiency); (2) offer a reliable evaluation of the discharge coefficient; (3) determine the suitability of the De Marchi, Domínguez and Schmidt methods, by comparing the results obtained for TPKSW; (4) find the outflow efficiency in comparison to the RPKSW, and (5) develop a useful and reliable equation for TPKSW which is a crucial requirement for applications.

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The discharge over the side weir per unit length, q, can be estimated using:
(6)
where Qw is the weir overflow discharge, y1 is the flow depth at the upstream end of the side weir at channel center, and x is the distance from the beginning of the side weir. (y1-P) is the piezometric head over the side weir.

De Marchi approach

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Along the side weir, the equation for the closed form solution of the water surface differential was first presented by De Marchi, assuming constant energy in the main channel:
(7)
where is given by:
(8)
Therefore, the relation between weir opening length (W) and other hydraulic variables of the flow is:
(9)
where E is the specific energy and y is the flow depth in the main channel. is the flow function of De Marchi. Equation (8) was first derived by De Marchi (1934) and can be used beneficially as a discharge equation for the side weir. and indicate upstream and downstream of the weir, respectively, and Cd is determined experimentally.

Schmidt approach

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The Schmidt approach is valid where the flow arrives and continues in the river regime. The Poleni equation is used in the Schmidt approach (Schmidt, 1954):
(10)
(11)
where ha is the average piezometric head over the weir (m), h1 is the piezometric head over the weir at the upstream end of the side weir (m), and h2 is the piezometric head over the weir at the downstream end of the side weir (m).

Domínguez approach

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The method developed by Domínguez provides a simple yet accurate way to determine the discharge of the side weir flow (Bagheri et al. 2014b). Assuming that specific energy along the side weir is constant and the water surface profile in the side weir changes linearly along the channel, the overflow discharge, Qw (m3/s), is given by:
(12)
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The experiments were conducted in a rectangular main channel (12 m long, 0.50 m wide and 0.50 deep) and a discharge evacuation channel (12 m long, 0.50 m wide and 0.70 deep). The bed slope of the main channel was 0.1%. The channels were made of a smooth, well painted steel bed with vertical glass sidewalls. The bed had an average Manning's roughness coefficient of 0.0099. A sluice gate was placed at the end of the main channel to control the depth of the flow. The width of the evacuation channel across the side weir was 1.3 m and constructed in a circular to provide free overflow conditions. Figure 4 shows a scheme of the experimental setup.
Figure 4
Plan view of the experimental setup.
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Plan view of the experimental setup.

Figure 4
Plan view of the experimental setup.
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Plan view of the experimental setup.

Close modal

A sharp-crested rectangular weir was placed at the end of the evacuation channel in order to measure the discharge of the side weir. A digital point gauge with ±0.01 mm accuracy was placed upstream at a distance of 0.4 m from the sharp-crested weir. TPKSWs were fabricated from fully aerated steel plates with sharp edges. These were fixed flush to the main channel wall.

The water was fed to the main channel from a tank via a supply pipe under the influence of gravitational forces. The discharge was controlled by a valve and measured by an electromagnetic flow meter with an accuracy of ± 0.01 L/s. The results were compared to a calibrated 90° V-notched weir. The overflow discharge was measured via a calibrated standard rectangular weir, located at the downstream end of the evacuation channel. The water depth in the channel was measured using a digital point meter for steady-state flow conditions. For the water surface measurements, a measuring car was used on a rail that could move both along the side weir and across the main channel.

The experiments were carried out in the Hydraulic Laboratory at Firat University, Elazig, Turkey. The experiments were repeated under conditions of subcritical flow, steady-state flow, and free overflow, which frequently occur in most weir applications (such as rivers, irrigation, and land drainage systems). In all of the experiments, the piezometric head over the weir was greater than the required value for minimizing the surface tension effect (30 mm above the side weir height, as proposed by Novák & Cabelka 1981). TPKSW structures with different widths (W = 0.25, 0.50 and 0.75 m) and heights (P = 0.12, 0.16 and 0.20) were tested. The variation of discharge coefficient with respect to the Froude number, P/y1, W/b and W/L ratios was analyzed (Table 1). In total, 211 tests were carried out to study the behavior of the discharge coefficient.

Table 1

The range of variables used in the channel and TPKSW models

VariablesLimits of the value
The main channel width, b (m) 0.5 
The main channel deep, z (m) 0.5 
The main channel slope,  0.001 
Weir opening length, W (m) 0.25–0.50–0.75 
Weir high, P (m) 0.12–0.16–0.20 
Flow into the system, (m3/s) 0.0119–0.1420 
Froude number 0.12–0.87 
Head, h1 (m) 0.03–0.09 
The inlet crest width Wi (m) 0.03–0.11 
The outlet crest width Wo (m) 0.02–0.09 
Length of sidewall B (m) 0.03–0.81 
Downstream overhang length Bi (m) 0.13–0.40 
Upstream overhang length Bo (m) 0 
Wall thickness Ts (m) 0.002 
Sidewall angle (°) 6 
VariablesLimits of the value
The main channel width, b (m) 0.5 
The main channel deep, z (m) 0.5 
The main channel slope,  0.001 
Weir opening length, W (m) 0.25–0.50–0.75 
Weir high, P (m) 0.12–0.16–0.20 
Flow into the system, (m3/s) 0.0119–0.1420 
Froude number 0.12–0.87 
Head, h1 (m) 0.03–0.09 
The inlet crest width Wi (m) 0.03–0.11 
The outlet crest width Wo (m) 0.02–0.09 
Length of sidewall B (m) 0.03–0.81 
Downstream overhang length Bi (m) 0.13–0.40 
Upstream overhang length Bo (m) 0 
Wall thickness Ts (m) 0.002 
Sidewall angle (°) 6 
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In this section, experimental results are presented and analyzed in order to describe the main features of the TPKSW flow and find the dimensionless parameters that primarily influence discharge capacity. The discharge coefficient of the TPKSW is named CPW for ease of comparison, and the equations given for Cd in Section 1 are used. Some of the results along with the hydraulic and physical conditions of the experiments are presented in Table 2.

Table 2

Conditions and results of tests for TPKSW

No.W/b (-)P (m)L/W (-)y1 (m)y2 (m)F1 (-)Q1 (m3/s)η (-)
1 1.00 0.12 4.82 0.1675 0.1960 0.7694 82.60 0.42 
2 1.00 0.12 4.82 0.1696 0.1969 0.7681 84.00 0.35 
3 1.00 0.12 4.82 0.1519 0.1555 0.2264 21.00 0.93 
4 1.00 0.12 4.82 0.1728 0.1773 0.2862 32.20 0.94 
5 1.00 0.12 4.82 0.1628 0.1701 0.3132 32.20 0.82 
6 1.00 0.12 4.82 0.1574 0.1637 0.3293 32.20 0.82 
7 1.00 0.12 4.82 0.1800 0.1884 0.3689 44.10 0.86 
8 1.00 0.12 4.82 0.1644 0.1710 0.4225 44.10 0.69 
9 1.00 0.16 4.82 0.1947 0.1997 0.2400 32.30 0.79 
10 1.00 0.16 4.82 0.2057 0.2111 0.2636 38.50 0.82 
11 1.00 0.16 4.82 0.2175 0.2237 0.2902 46.10 0.82 
12 1.00 0.16 4.82 0.2038 0.2116 0.3201 46.10 0.72 
13 1.00 0.16 4.82 0.2021 0.2098 0.3387 48.20 0.71 
14 1.00 0.16 4.82 0.2016 0.2067 0.4084 57.90 0.61 
15 1.00 0.16 4.82 0.2032 0.2052 0.4197 60.20 0.60 
16 1.00 0.20 4.82 0.2350 0.2388 0.1884 33.60 0.75 
17 1.00 0.20 4.82 0.2419 0.2602 0.4760 88.70 0.39 
18 1.00 0.20 4.82 0.2453 0.2619 0.5184 98.60 0.35 
19 1.00 0.20 4.82 0.2350 0.2559 0.6002 107.10 0.28 
20 1.00 0.20 4.82 0.2311 0.2542 0.6485 112.80 0.32 
21 1.00 0.20 4.82 0.2343 0.2606 0.6809 120.90 0.26 
22 1.00 0.20 4.82 0.2331 0.2374 0.2015 35.50 0.84 
23 0.50 0.12 4.82 0.1882 0.1903 0.1729 22.10 0.73 
24 0.50 0.12 4.82 0.1734 0.1734 0.1530 17.30 0.68 
25 0.50 0.12 4.82 0.1512 0.1529 0.1293 11.90 0.54 
26 0.50 0.12 4.82 0.1796 0.1828 0.2878 34.30 0.41 
27 0.50 0.12 4.82 0.1684 0.1716 0.3170 34.30 0.32 
28 0.50 0.16 4.82 0.1994 0.2014 0.1413 19.70 0.67 
29 0.50 0.16 4.82 0.1901 0.1913 0.1518 19.70 0.48 
30 0.50 0.16 4.82 0.1962 0.1984 0.1322 18.00 0.65 
31 0.50 0.16 4.82 0.2136 0.2151 0.1792 27.70 0.56 
32 0.50 0.16 4.82 0.1973 0.1995 0.2018 27.70 0.42 
33 0.50 0.20 4.82 0.2582 0.2598 0.1246 25.60 0.69 
34 0.50 0.20 4.82 0.2384 0.2409 0.1405 25.60 0.46 
35 0.50 0.20 4.82 0.2475 0.2491 0.1826 35.20 0.42 
36 0.50 0.20 4.82 0.2311 0.2335 0.2023 35.20 0.28 
37 0.50 0.20 4.82 0.2585 0.2623 0.2333 48.00 0.39 
38 1.50 0.12 4.82 0.1501 0.1589 0.4006 36.50 0.85 
39 1.50 0.12 4.82 0.1500 0.1572 0.3451 31.40 0.97 
40 1.50 0.12 4.82 0.1552 0.1656 0.4480 42.90 0.85 
41 1.50 0.12 4.82 0.1527 0.1627 0.4593 42.90 0.78 
42 1.50 0.16 4.82 0.1900 0.1957 0.2907 37.70 0.93 
43 1.50 0.16 4.82 0.1939 0.2023 0.3485 46.60 0.85 
44 1.50 0.16 4.82 0.1939 0.2063 0.4121 55.10 0.74 
45 1.50 0.16 4.82 0.1949 0.2085 0.4503 60.70 0.69 
46 1.50 0.16 4.82 0.1937 0.2099 0.4946 66.00 0.63 
47 1.50 0.20 4.82 0.2300 0.2343 0.2217 38.30 0.92 
48 1.50 0.20 4.82 0.2401 0.2470 0.2795 51.50 0.92 
49 1.50 0.20 4.82 0.2339 0.2419 0.2907 51.50 0.80 
50 1.50 0.20 4.82 0.2426 0.2549 0.3734 69.90 0.73 
51 1.50 0.20 4.82 0.2339 0.2472 0.3945 69.90 0.63 
No.W/b (-)P (m)L/W (-)y1 (m)y2 (m)F1 (-)Q1 (m3/s)η (-)
1 1.00 0.12 4.82 0.1675 0.1960 0.7694 82.60 0.42 
2 1.00 0.12 4.82 0.1696 0.1969 0.7681 84.00 0.35 
3 1.00 0.12 4.82 0.1519 0.1555 0.2264 21.00 0.93 
4 1.00 0.12 4.82 0.1728 0.1773 0.2862 32.20 0.94 
5 1.00 0.12 4.82 0.1628 0.1701 0.3132 32.20 0.82 
6 1.00 0.12 4.82 0.1574 0.1637 0.3293 32.20 0.82 
7 1.00 0.12 4.82 0.1800 0.1884 0.3689 44.10 0.86 
8 1.00 0.12 4.82 0.1644 0.1710 0.4225 44.10 0.69 
9 1.00 0.16 4.82 0.1947 0.1997 0.2400 32.30 0.79 
10 1.00 0.16 4.82 0.2057 0.2111 0.2636 38.50 0.82 
11 1.00 0.16 4.82 0.2175 0.2237 0.2902 46.10 0.82 
12 1.00 0.16 4.82 0.2038 0.2116 0.3201 46.10 0.72 
13 1.00 0.16 4.82 0.2021 0.2098 0.3387 48.20 0.71 
14 1.00 0.16 4.82 0.2016 0.2067 0.4084 57.90 0.61 
15 1.00 0.16 4.82 0.2032 0.2052 0.4197 60.20 0.60 
16 1.00 0.20 4.82 0.2350 0.2388 0.1884 33.60 0.75 
17 1.00 0.20 4.82 0.2419 0.2602 0.4760 88.70 0.39 
18 1.00 0.20 4.82 0.2453 0.2619 0.5184 98.60 0.35 
19 1.00 0.20 4.82 0.2350 0.2559 0.6002 107.10 0.28 
20 1.00 0.20 4.82 0.2311 0.2542 0.6485 112.80 0.32 
21 1.00 0.20 4.82 0.2343 0.2606 0.6809 120.90 0.26 
22 1.00 0.20 4.82 0.2331 0.2374 0.2015 35.50 0.84 
23 0.50 0.12 4.82 0.1882 0.1903 0.1729 22.10 0.73 
24 0.50 0.12 4.82 0.1734 0.1734 0.1530 17.30 0.68 
25 0.50 0.12 4.82 0.1512 0.1529 0.1293 11.90 0.54 
26 0.50 0.12 4.82 0.1796 0.1828 0.2878 34.30 0.41 
27 0.50 0.12 4.82 0.1684 0.1716 0.3170 34.30 0.32 
28 0.50 0.16 4.82 0.1994 0.2014 0.1413 19.70 0.67 
29 0.50 0.16 4.82 0.1901 0.1913 0.1518 19.70 0.48 
30 0.50 0.16 4.82 0.1962 0.1984 0.1322 18.00 0.65 
31 0.50 0.16 4.82 0.2136 0.2151 0.1792 27.70 0.56 
32 0.50 0.16 4.82 0.1973 0.1995 0.2018 27.70 0.42 
33 0.50 0.20 4.82 0.2582 0.2598 0.1246 25.60 0.69 
34 0.50 0.20 4.82 0.2384 0.2409 0.1405 25.60 0.46 
35 0.50 0.20 4.82 0.2475 0.2491 0.1826 35.20 0.42 
36 0.50 0.20 4.82 0.2311 0.2335 0.2023 35.20 0.28 
37 0.50 0.20 4.82 0.2585 0.2623 0.2333 48.00 0.39 
38 1.50 0.12 4.82 0.1501 0.1589 0.4006 36.50 0.85 
39 1.50 0.12 4.82 0.1500 0.1572 0.3451 31.40 0.97 
40 1.50 0.12 4.82 0.1552 0.1656 0.4480 42.90 0.85 
41 1.50 0.12 4.82 0.1527 0.1627 0.4593 42.90 0.78 
42 1.50 0.16 4.82 0.1900 0.1957 0.2907 37.70 0.93 
43 1.50 0.16 4.82 0.1939 0.2023 0.3485 46.60 0.85 
44 1.50 0.16 4.82 0.1939 0.2063 0.4121 55.10 0.74 
45 1.50 0.16 4.82 0.1949 0.2085 0.4503 60.70 0.69 
46 1.50 0.16 4.82 0.1937 0.2099 0.4946 66.00 0.63 
47 1.50 0.20 4.82 0.2300 0.2343 0.2217 38.30 0.92 
48 1.50 0.20 4.82 0.2401 0.2470 0.2795 51.50 0.92 
49 1.50 0.20 4.82 0.2339 0.2419 0.2907 51.50 0.80 
50 1.50 0.20 4.82 0.2426 0.2549 0.3734 69.90 0.73 
51 1.50 0.20 4.82 0.2339 0.2472 0.3945 69.90 0.63 
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Assumption of constant specific energy

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Since De Marchi equations were developed under constant specific energy assumptions, verification of the assumptions is required. Cross-sectional average flow velocities were calculated by using the flow depths measured at the upstream and downstream ends of the weir. The average specific energy was found using these velocities. Figure 5 illustrates that the average specific energy difference in the main channel between the two ends of the weir (ΔE = E1 − E2) is 0.2%. Borghei et al. (1999) and Emiroglu et al. (2014) obtained a mean value of 3.7% and 1.0%, respectively. El-Khashab & Smith (1976) estimated a 5% value for subcritical flow. In this study, the energy difference is lower than 1.0% for all experiments involving TPKSW. Therefore, the specific energy is assumed to be constant.
Figure 5
Specific energy at the ends of the TPKSW.
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Specific energy at the ends of the TPKSW.

Figure 5
Specific energy at the ends of the TPKSW.
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Specific energy at the ends of the TPKSW.

Close modal

Water surface profile

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The water surface profiles were measured at 18 points along the center of the main channel and along the side weir. In addition to the existing data, water surface profiles were generated by calculating intermediate values with the extrapolation method (Figures 6 and 7). The water level over the TPKSW generally dropped sharply at the upstream end of the side weir, due to acceleration induced by the lateral flow, and then ascended downstream. This rising profile is compatible with the principle of spatially varied flow over side weirs. Similar actions have been seen for subcritical flow over trapezoidal sharp-crested side weirs and triangular labyrinth side weirs (e.g., Emiroglu et al. 2010; Ghaderi et al. 2020). Increasing the Froude number increases the flow velocity and makes it difficult to divert the flow to the side weir. Therefore, increasing the Froude number increases the difference between upstream and downstream water levels. The effect of the Froude number is also in agreement with the theory of spatially varied flow over side weirs. It was observed that the water level fell in two regions above the TPKSW, especially in conditions with large Froude numbers. The regions are above the inlet keys. The flow rate decreases at the entrance of the inlet key and thus helps flow uniformity. Karimi et al. (2018) experimentally demonstrated that a similar situation occurs for RPKSW.
Figure 6
Water surface profile for P = 0.12 m TPKSW.
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Water surface profile for P = 0.12 m TPKSW.

Figure 6
Water surface profile for P = 0.12 m TPKSW.
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Water surface profile for P = 0.12 m TPKSW.

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Figure 7
Water surface profile for P = 0.16 m TPKSW.
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Water surface profile for P = 0.16 m TPKSW.

Figure 7
Water surface profile for P = 0.16 m TPKSW.
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Water surface profile for P = 0.16 m TPKSW.

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Discharge coefficient of trapezoidal piano key side weir using different approaches

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Figure 8 shows that the flow coefficient values obtained using the Schmidt and Domínguez methods are quite compatible with each other. Similarly, Emiroglu et al. (2010) stated that the data obtained by the Schmidt and Domínguez methods for the triangular labyrinth side weir almost overlap with each other. In the current study, compared to the De Marchi method, the other two methods have more scattered results, as presented in Figure 8. Between the De Marchi method and other methods, there is 71 and 89% agreement for F1 = 0.2 and F1 = 0.8, respectively. Contrary to the current study, in Emiroglu et al. (2016), the data obtained with the De Marchi method were more scattered and closer to the data of the Schmidt and Domínguez methods.
Figure 8
Discharge coefficient of TPKSW according to different approaches.
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Discharge coefficient of TPKSW according to different approaches.

Figure 8
Discharge coefficient of TPKSW according to different approaches.
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Discharge coefficient of TPKSW according to different approaches.

Close modal

It was observed that the discharge coefficient decreases as the Froude number increases for all approaches (see Figure 8). Mehri et al. (2018) used the De Marchi approach in their study for low Froude numbers (0.05–0.30) and obtained a decreasing trend in some experiments and an increasing trend in other experiments, between F1 and CPW. Moreover, the data obtained by the De Marchi method were very scattered. However, a decreasing trend was obtained between F1- CPW in the studies that used the De Marchi method (Borghei et al. 1999, 2013; Emiroglu et al. 2014). Different from other studies, this study provides a better understanding of the relationship between F1 and CPW by performing experiments over a wide range of Froude numbers. Bagheri et al. (2014a, 2014b) stated that the Domínguez method provides reliable results in a sharp-edged weir. In addition, Emiroglu & Ikinciogullari (2016) showed that the Schmidt method is reliable for conventional side weirs. Similarly, all approaches provide reasonable results for TPKSW in the present study.

Effects of upstream head

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As the head (h1) approaches a certain value, the effective weir length gradually decreases and, eventually, the behavior of the TPKSW converges with the behavior of a linear weir. When the piezometric head over the weir is small, the flow falls freely over the full length of the weir. Crookston & Tullis (2011) stated that there is no significant gain for h1/P > 1.0 at the frontal labyrinth weirs. According to studies conducted for PKW, small head values maximize the contribution of increased crest length (L) (Lempérière et al. 2011; Anderson & Tullis 2013). Similarly, Figure 9 shows that the discharge coefficient is a decreasing function of the h1/P value. The rate of change of CPW decreases as h1/P increases. The trend of CPW as a function of h1/P is in agreement with previous studies conducted for different types of side weirs (Singh et al. 1994; Jalili & Borghei 1996; Borghei et al. 1999; Emiroglu et al. 2011; Abbasi et al. 2020).
Figure 9
Values of CPW versus h1/P for TPKSW (a) W/b = 0.5, (b) W/b = 1.0, (c) W/b = 1.5.
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Values of CPW versus h1/P for TPKSW (a) W/b = 0.5, (b) W/b = 1.0, (c) W/b = 1.5.

Figure 9
Values of CPW versus h1/P for TPKSW (a) W/b = 0.5, (b) W/b = 1.0, (c) W/b = 1.5.
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Values of CPW versus h1/P for TPKSW (a) W/b = 0.5, (b) W/b = 1.0, (c) W/b = 1.5.

Close modal

In this study, head value is between 3 < h1 < 9 cm, which corresponds with the range of 0.15 < h1/P < 0.57. For W/b = 0.5, P/W = 0.64 gives higher CPW values of TPKSW, compared to P/W values of 0.48 and 0.80. In the range of 0.2 ≤ h1/P ≤ 0.5, the most stable behavior of CPW is observed at P/W = 0.80, with an average slope value of −0.15. A mean slope of −0.44 for P/W = 0.48 and −0.80 for P/W = 0.64 is observed. For a given value of h1, CPW increases as crest height increases. The reason for this behavior is the fact that it has a wider inlet key cross-sectional area. As a result, the velocity decreases as the flow approaches the side weir and the flow becomes uniform.

Effects of upstream Froude number

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For all TPKSW models, the relation between CPW and Froude number is in the form of ⁠. In all cases, the b takes negative values, therefore CPW is a decreasing function of the Froude number. Similar behaviors are observed in labyrinth side weirs (Aydın & Emiroglu 2016; Abbasi et al. 2020) and in the curved channel RPKSW (Mehri et al. 2018). Figure 10 presents the flow coefficient as a function of Froude number. The flow coefficient is a decreasing function of the Froude number. The lowest discharge coefficients were observed at P/W = 0.64 and P/W = 0.48, respectively (Figure 10(a)). For W/b = 1.0, P/W = 0.32 provides the most stable results at low and high Froude numbers, with a slope of −0.32 at F1 = 0.2, and −0.10 for F1 = 0.6. For P/W = 0.24, the slope is equal to −0.46 at F1 = 0.2 and −0.14 at F1 = 0.6. In the experiments carried out for W/b = 1.5, the amount of discharged flow is too much for the capacity of the collection channel. When the flow rate entering the system increases, in order to obtain the desired flow depths, the tail water rises and exceeds the weir crest. Few experiments have been possible under these conditions. Missing data are predicted by the extrapolation method. Figure 10(c) shows that, for low Froude number values, P/W = 0.21 provides the highest and P/W = 0.16 gives the lowest CPW. It is seen that the Froude number is more effective on CPW at P/W = 0.16. At high Froude numbers, the best performance is obtained at P/W = 0.27 and the effect of the F1 number decreases. The ineffective region at the corners of the keys expands and lowers the outflow discharge when F1 value is raised (see Figure 11). In the case of large F1 values, (starting from the upstream) in all of the 1st half outlet key crest and a certain part of the 1st side wall of the TPKSW do not provide discharge (see Figure 11). This is due to the fact that the velocity of the flow increases as the flow rate in the main channel is increased. With larger velocity vectors, changing direction of the flow to the side weir becomes harder.
Figure 10
(a–c) CPW versus F1 for different dimensionless weir lengths of TPKSW (a) W/b = 0.5, (b) W/b = 1.0, (c) W/b = 1.5.
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(a–c) CPW versus F1 for different dimensionless weir lengths of TPKSW (a) W/b = 0.5, (b) W/b = 1.0, (c) W/b = 1.5.

Figure 10
(a–c) CPW versus F1 for different dimensionless weir lengths of TPKSW (a) W/b = 0.5, (b) W/b = 1.0, (c) W/b = 1.5.
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(a–c) CPW versus F1 for different dimensionless weir lengths of TPKSW (a) W/b = 0.5, (b) W/b = 1.0, (c) W/b = 1.5.

Close modal
Figure 11
Change of flow on TPKSW depending on F1 (a) F1 = 0.22, and (b) F1 = 0.76.
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Change of flow on TPKSW depending on F1 (a) F1 = 0.22, and (b) F1 = 0.76.

Figure 11
Change of flow on TPKSW depending on F1 (a) F1 = 0.22, and (b) F1 = 0.76.
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Change of flow on TPKSW depending on F1 (a) F1 = 0.22, and (b) F1 = 0.76.

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Outflow efficiency

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Outflow efficiency is defined as:
(13)
η is correlated with the dimensionless parameter  :
(14)
where F2 is the downstream Froude number and y2 is the depth in the main channel at the downstream end of a linear side weir. Figure 12 shows η versus χ for rectangular and TPKSWs with different W/b ratios. The discharge efficiency η is an increasing function of χ and can be expressed as η = aχb with a correlation coefficient above 0.95. For a fixed P/y2 value, η decreases with increasing F2. Similarly, if the F2 value is constant, the η values decrease as the P/y2 value increases. Maranzoni et al. (2017) obtained similar findings for the short-crested side weir in a converging channel. It was observed that the η increases with an increase in the W/b ratio, and there is no significant change depending on the crest height (see Figure 12). The results are in agreement with the findings obtained for type C rectangular PKSW (Karimi et al. 2018). The design introduced in Karimi et al. (2018) was found to be slightly more efficient, compared to the current study in the range of 0 < χ < 12, where χ is very small. This occurs from the relationship between increasing values of either F2 or P/y2. In this range, the PKSW model starts with the inlet key upstream of the side weir opening. This provides an advantage by increasing the efficiency of the upstream part of the weir, where lateral flow effects with high velocity are dominant.
Figure 12
Outflow efficiency of TPKSW (a) = 0.12 m, (b) = 0.16 m, (c) = 0.20 m.
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Outflow efficiency of TPKSW (a) = 0.12 m, (b) = 0.16 m, (c) = 0.20 m.

Figure 12
Outflow efficiency of TPKSW (a) = 0.12 m, (b) = 0.16 m, (c) = 0.20 m.
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Outflow efficiency of TPKSW (a) = 0.12 m, (b) = 0.16 m, (c) = 0.20 m.

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Comparison with literature

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Subramanya & Awasthy (1972), Hager (1987) and Borghei et al. (1999) developed empirical formulas for the flow coefficient Cd in the case of rectangular weir lateral flow in a linear channel. The discharge coefficient Cd was calculated by using Equations (1)–(3) and comparing them with the C type TPKSW data generated in the current study. With the increase of Froude number value, the discharge coefficients of both TPKSW and rectangular side weirs decrease in a similar trend. The ratio of the discharge coefficient of TPKSW to the discharge coefficient of the rectangular side weir was found to be between 2.9 and 12 (Figure 13). The TPKSW effective crest length is equal to 4.82 times the weir opening. Therefore, a positive effect of the increased weir length on the discharge efficiency is observed. At large values of the Froude number, The TPKSW achieves a very significant increase in flow coefficient compared to rectangular weirs. This difference is too large to be explained by the contribution of effective weir length only, reflecting that the hydraulic behavior of piano key weirs in lateral flow is quite different from the behavior of rectangular side weirs. The secondary flow in TPKSWs which is the cause by lateral flow is more severe as in trapezoidal labyrinth side weirs compared to rectangular side weirs (Emiroglu et al. 2014). Intensity of secondary movement increases with increasing overflow length. As the relative side weir length grows, an increase in the secondary flow causes an increase in the deflection angle and change in momentum towards the side weir. The intensity of the secondary flow occurring inside the side weir decreases as the flow velocity inside the side weir reduces (Abbasi et al. 2020).
Figure 13
Comparison of a TPKSW with a rectangular side weir.
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Comparison of a TPKSW with a rectangular side weir.

Figure 13
Comparison of a TPKSW with a rectangular side weir.
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Comparison of a TPKSW with a rectangular side weir.

Close modal
The effects of h1/P on discharge coefficients of the trapezoidal labyrinth side weir, triangular labyrinth side weir and TPKSW are similar (Figure 14(a)). For constant crest height and small head values, the discharge coefficient of trapezoidal labyrinth side weir is greater than CPW. As head increases, PKSW and trapezoidal labyrinth side weir discharge coefficients get very close to each other. CPW over Cd (triangular labyrinth side weir) was found to be 2.1. CPW over Cd (trapezoidal labyrinth side weir) was found to be between 0.92 and 1. The triangular labyrinth side weir data are calculated using Equation (4). The weirs used in all three studies consist of two units and W/b = 1.0. Figure 14(b) presents the relationship between Froude number and the discharge coefficient. Increasing the Froude number decreases the discharge coefficient of the PKSW. Conversely discharge coefficients of the trapezoidal labyrinth and triangular labyrinth side weirs increase. The highest discharge coefficients are observed in the trapezoidal labyrinth weir. The lowest values of the discharge coefficients were observed in the triangular labyrinth weir. Considering that the ratio of effective weir length (L) to weir opening (W) of TPKSW is L/W = 4.82, the discharge coefficients of the trapezoidal labyrinth weir are quite high compared to TPKSW. The most significant difference between the labyrinth weir and PKSW geometries is the inlet and outlet key ramps in the PKSW. This suggests that the performance difference between the two weir models is related to the slopes of the ramps. Higher velocities are required to fill the inlet key and to reach crest of outlet key due to the presence of ramps. It is expected that the decrease in the slope of the inlet key contributes to the discharge capacity. However, increasing the slope of the outlet key improves the discharge as the velocity increases.
Figure 14
(a, b) Comparison of TPKSW with trapezoidal and triangular labyrinth side weirs.
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(a, b) Comparison of TPKSW with trapezoidal and triangular labyrinth side weirs.

Figure 14
(a, b) Comparison of TPKSW with trapezoidal and triangular labyrinth side weirs.
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(a, b) Comparison of TPKSW with trapezoidal and triangular labyrinth side weirs.

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Proposed equation for calculating the weir discharge coefficient

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In this study, by using the results of the Domínguez approach and nonlinear regression analysis, Equation (15) is developed for the discharge coefficient of TPKSW. As presented in Figure 15, the weir discharge coefficient is successfully predicted for the experimental conditions. To evaluate the accuracy of the estimation using the nonlinear model, the root-mean square error (RMSE) and coefficient of determination (R2) criteria were used. For Equation (15), RMSE and R2 are 0.16968 and 0.88, respectively. Figure 15 compares the measured CPW values with those predicted by Equation (15). Good correlation was observed between the measured values and the values computed from the equation. Thus, a reliable equation is introduced for the discharge coefficient of a two unit TPKSW located on a straight channel in subcritical flow conditions:
(15)
Figure 15
Comparison of measured CPW values with those calculated from Equation (15).
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Comparison of measured CPW values with those calculated from Equation (15).

Figure 15
Comparison of measured CPW values with those calculated from Equation (15).
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Comparison of measured CPW values with those calculated from Equation (15).

Close modal

Equation (15) is subject to the limitations of present tests: 0.13 ≤ F1 ≤ 0.88, 0.17 ≤ H1/P ≤ 0.85, 0.5 ≤ W/b ≤ 1.5, 1.4 ≤ B/P ≤ 6.73, 0.15 ≤ h1/P ≤ 0.57, Wi/Wo = 1.23, n = L/W = 4.83, and α = 6°.

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In the current study, the hydraulic characteristics of a TPKSW were experimentally studied. The values of discharge coefficient for TPKSW were obtained using De Marchi, Schmidt and Domínguez approaches. The study leads to the primary results listed below:

  • (a)

    The fundamental constant energy assumption commonly used in side weir flow modeling is valid for TPKSW flow.

  • (b)

    The location of the low levels observed in the water surface profile shows that the contribution of the inlet key to the discharge capacity is higher compared to the other TPKSW parts as in frontal flow.

  • (c)

    Contrary to the frontal weir, in the case of lateral flow, the positions of the inlet and outlet keys along the weir are more effective for the behavior of the flow.

  • (d)

    As the P/W value increases, the flow coefficient increases up to a certain value (0.64) and then decreases. Key slopes should be optimized for new TPKSW designs since the change in key slopes is effective.

  • (e)

    In comparison to the traditional rectangular side weir and triangular labyrinth side weir discharge coefficients, the TPKSW discharge coefficient has greater values.

  • (f)

    A higher discharge capacity is observed in the trapezoidal labyrinth side weir compared to TPKSW especially for 0.38 < F1 < 0.76.

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All relevant data are included in the paper or its Supplementary Information.

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The authors declare there is no conflict.

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