The results showed that the discharge coefficient (Cd), efficiency (Ra) and, consequently, the increase in the passing flow discharge have a direct relationship with the inclination angle of the input and output keys, and for this reason, the weir of the piano key weir (PKW) with an angle of 90 degrees has the highest Cd and Ra. An increase in the HT/P ration in all angles leads to a decrease in Cd and Ra. However, the decrease rate of these parameters is less for 15-degree angle compared to other threshold angles of the input and output keys. In the PKWs with different input and output key angles and at low HT/P ratios (total head to weir height), the dropping flow nappe needs aeration due to its bonding. However, by increasing this ratio to the submergence condition and transforming into a linear weir, the dropping flow nappe is aerated. At low ratios of HT/P, the dropping flow nappes do not interfere and the highest discharge coefficient and the highest weir efficiency are obtained. While by increasing the HT/P ratio, the dropping flow nappes start to collide and interfere with each other in the output keys, causing the flow rise and the weir submergence, and consequently, the discharge coefficient and the weir efficiency is decreased.

  • Experimental investigation of the discharge coefficient and efficiency of the piano key weirs (PKWs).

  • Investigated the effect of the input and output key angle on the discharge coefficient.

  • Increasing the discharge coefficient in PKWs.

Weirs are the most important structures that control floods in dam reservoirs. In addition, spillways are used to measure discharge, divert or control flow in channels, rivers and dam reservoirs (Borghei et al. 2013; Azamathulla et al. 2016). Spillways are divided into linear and non-linear ones based on their plan shape. Non-linear spillways such as curved spillways in the plan and the labyrinth spillways increase the weir capacity by increasing the flow passage length. Labyrinth spillways are zigzag sequences of linear spillways, with a lower discharge coefficient than linear spillways and a longer length for which the product of the two parameters is greater than that of the linear spillways (Equation (1)). In other words, their hydraulic performance is three to four times greater than that of linear spillways (Tullis et al. 2007; Noroozi et al. 2021).
(1)
where Q is the weir flow discharge, Cd, Lt, and HT are the discharge coefficient, total threshold length, and total head (sum of the velocity and hydrostatic heads) over the weir, respectively.

The piano key weir (PKW) is a type of hydraulic infrastructure that aims to increase the passing flow discharge and discharge capacity, improve weir performance, and simultaneously decrease construction costs (Bhukya et al. 2022). PKWs are used with gravity dams and natural channels (Anderson & Tullis 2013; Erpicum et al. 2016; Crookston et al. 2019; Tullis et al. 2020) and can replace any affected gated weirs to increase operational performance and maintenance (Laugier 2007; Leite Ribeiro et al. 2012). The discharge coefficient of PKWs is very important, and it has been addressed in Machiels et al. (2009), Ghanbari & Heidarnejad (2020) and Roushangar et al. (2021). Kabiri-Samani & Javaheri (2012) studied the effect of weir geometry including weir length and height, upstream key and downstream key width, as well as upstream and downstream apex overhangs on the discharge coefficient of PKWs in free and submerged flow conditions. Machiels et al. (2014) performed a parametric study of the flow over PKWs and presented equations to determine the flow discharge passing over a cycle of PKWs. Safarzadeh & Noroozi (2017) used the Flow-3D numerical model to study the effects of the inlet key area and the angle of the side walls on the discharge coefficient of PKWs considering the 3D flow condition. Crookston et al. (2018) studied two approaches: (1) empirical prediction methods ranging from simple to sophisticated (including five experimental design methods) and (2) computational fluid dynamics (two different turbulence models) in PKWs. Kumar et al. (2019) considered the importance of the discharge coefficient in PKWs and compared the validity of four equations presented by different researchers under different experimental conditions and data to calculate the discharge coefficient. Akbari et al. (2019) studied the hydraulic performance of PKWs by adding a gate to the input keys. Abhash & Pandey (2021) experimentally and numerically studied the discharge capacity and sediment-carrying capacity of different geometries of PKWs. Behroozi & Vaghefi (2022) studied the discharge capacity and hydraulic behavior of type-A PKWs (with symmetrical consoles) considering different thresholds and geometries. In the laboratory, Mero et al. (2022) investigated the effect of the inlet-to-outlet width ratio (Wi/Wo) on the hydraulic performance of PKWs. Li et al. (2023) performed a numerical simulation to study the hydraulic characteristics of PKWs including the flow pattern. Kadia et al. (2023) used extensive experimental data to present a comprehensive equation with high efficiency and appropriate accuracy to calculate the flow coefficient of type-A PKWs.

In the present study, using experimental data and taking into account the importance of weirs in flood control, solutions were presented to increase the discharge coefficient (Cd) and efficiency (Ra) of rectangular PKWs. In other words, the effect of different input and output key inclination angles including 5°, 30°, 45°, 60°, 75°, and 90° on the discharge coefficient and efficiency of PKWs was studied.

Experimental setup

The experiments were carried out in the technical and engineering faculty of Bo-Ali-Sina University, Hamedan, Iran in a flume with a length of 14 m, height and width of 60 cm, and a width of 96 cm in one end meter of the flume (Figure 1(a)). At the end of the flume, different samples of rectangular PKWs with identical indentations were installed. In other words, the larger the size of the developed Labyrinth models, the less the effects of the scaling error and, accordingly, the more reliable results will be achieved in experimental conditions. For this reason, in the present study, by using a curved transition, the width of the flume end increased from 60 to 96 cm. It is worth mentioning that due to the limitations of the experimental flume, it was not possible to increase the width further.
Figure 1

Schematic view of the experimental devices. (a) Schematic view of the experimental flume. (b) Schematic view and location of the PKWs. (c) Experimental flume and the PKW.

Figure 1

Schematic view of the experimental devices. (a) Schematic view of the experimental flume. (b) Schematic view and location of the PKWs. (c) Experimental flume and the PKW.

Close modal

The reservoir was located under the main flume, and water was pumped to the flume using a pump and a pipe with an external diameter of 145 mm. Upon passing the flume and spillways, water was redirected to the reservoir. A hydraulic valve was used to control five different flow discharges.

At the end of the flume and before the spillways, a point depth gauge with an accuracy of 0.1 mm was used to read the flow depth on the weir congresses (Figure 2). Upon the installation of each weir, the zero point of the depth gauge reading corresponded to the weir threshold and, accordingly, water depth on the weir threshold was accurately observed and read at 1.5, 4.5, 9.5, 14.5, 19.5, 24.5 and 29 cm from the beginning of each weir location and at different discharges. A calibrated triangular weir was used to measure the discharge (Figure 2).
Figure 2

Experimental equipment. (a) Depth gauge. (b) Triangular weir for flow measurement.

Figure 2

Experimental equipment. (a) Depth gauge. (b) Triangular weir for flow measurement.

Close modal

Experimental models

In the present study, in order to study the discharge coefficient, six experimental models of rectangular PKWs with the same length of indentations were developed. The six models were designed in such a way that all of them had four cycles with the same effective length of 333 cm to yield the same effect of the two parameters on the weir discharge coefficient in all samples. In the present study, the threshold angles of 15°, 30°, 45°, 60°, 75°, and 90° (Figure 3) were used for the input and output keys of the weirs.
Figure 3

Characteristics of the PKWs with different angles.

Figure 3

Characteristics of the PKWs with different angles.

Close modal
In all tests, the weir height (P) and the total threshold length (Lt) were considered as 20 and 333 cm, respectively. In Tables 16, the values of total head (HT), passing discharge over the PKW (QL), passing discharge over the Ogee linear weir (QN) (calculated using Equation (2)), weir discharge coefficient (Cd), HT/P ratio and weir efficiency (Ra) (equal to the QL/QN ratio) are listed for different angles.
(2)
Table 1

The results obtained for the PKW with a 15° threshold angle

x (cm)HT (cm)QL (L/s)QN (L/s)CdHT /PRa
1.5 1.25 5.76 2.956 0.419 0.0625 1.949 
4.5 1.3 3.135 0.395 0.065 1.837 
9.5 1.32 3.208 0.386 0.066 1.796 
14.5 1.32 3.208 0.386 0.066 1.796 
19.5 1.35 3.317 0.373 0.0675 1.736 
24.5 1.35 3.317 0.373 0.0675 1.736 
29 1.36 3.354 0.369 0.068 1.717 
1.5 1.78 9.4 5.023 0.403 0.089 1.872 
4.5 1.85 5.322 0.380 0.0925 1.766 
9.5 1.9 5.539 0.365 0.095 1.697 
14.5 1.92 5.627 0.359 0.096 1.671 
19.5 1.95 5.759 0.351 0.0975 1.632 
24.5 1.97 5.848 0.346 0.0985 1.607 
29 1.98 5.893 0.343 0.099 1.595 
1.5 3.1 20.86 11.544 0.389 0.155 1.807 
4.5 3.22 12.221 0.367 0.161 1.707 
9.5 3.3 12.679 0.354 0.165 1.645 
14.5 3.32 12.794 0.351 0.166 1.630 
19.5 3.38 13.143 0.341 0.169 1.587 
24.5 3.4 13.259 0.338 0.17 1.573 
29 3.46 13.612 0.330 0.173 1.532 
1.5 4.45 35.15 19.854 0.381 0.2225 1.770 
4.5 4.55 20.527 0.368 0.2275 1.712 
9.5 4.7 21.550 0.351 0.235 1.631 
14.5 4.75 21.895 0.345 0.2375 1.605 
19.5 4.76 21.964 0.344 0.238 1.600 
24.5 4.78 22.103 0.342 0.239 1.590 
29 4.78 22.103 0.342 0.239 1.590 
1.5 5.3 44.63 25.806 0.372 0.265 1.729 
4.5 5.48 27.132 0.354 0.274 1.645 
9.5 5.55 27.653 0.347 0.2775 1.614 
14.5 5.66 28.480 0.337 0.283 1.567 
19.5 5.68 28.631 0.335 0.284 1.559 
24.5 5.69 28.706 0.334 0.2845 1.555 
29 5.7 28.782 0.334 0.285 1.551 
x (cm)HT (cm)QL (L/s)QN (L/s)CdHT /PRa
1.5 1.25 5.76 2.956 0.419 0.0625 1.949 
4.5 1.3 3.135 0.395 0.065 1.837 
9.5 1.32 3.208 0.386 0.066 1.796 
14.5 1.32 3.208 0.386 0.066 1.796 
19.5 1.35 3.317 0.373 0.0675 1.736 
24.5 1.35 3.317 0.373 0.0675 1.736 
29 1.36 3.354 0.369 0.068 1.717 
1.5 1.78 9.4 5.023 0.403 0.089 1.872 
4.5 1.85 5.322 0.380 0.0925 1.766 
9.5 1.9 5.539 0.365 0.095 1.697 
14.5 1.92 5.627 0.359 0.096 1.671 
19.5 1.95 5.759 0.351 0.0975 1.632 
24.5 1.97 5.848 0.346 0.0985 1.607 
29 1.98 5.893 0.343 0.099 1.595 
1.5 3.1 20.86 11.544 0.389 0.155 1.807 
4.5 3.22 12.221 0.367 0.161 1.707 
9.5 3.3 12.679 0.354 0.165 1.645 
14.5 3.32 12.794 0.351 0.166 1.630 
19.5 3.38 13.143 0.341 0.169 1.587 
24.5 3.4 13.259 0.338 0.17 1.573 
29 3.46 13.612 0.330 0.173 1.532 
1.5 4.45 35.15 19.854 0.381 0.2225 1.770 
4.5 4.55 20.527 0.368 0.2275 1.712 
9.5 4.7 21.550 0.351 0.235 1.631 
14.5 4.75 21.895 0.345 0.2375 1.605 
19.5 4.76 21.964 0.344 0.238 1.600 
24.5 4.78 22.103 0.342 0.239 1.590 
29 4.78 22.103 0.342 0.239 1.590 
1.5 5.3 44.63 25.806 0.372 0.265 1.729 
4.5 5.48 27.132 0.354 0.274 1.645 
9.5 5.55 27.653 0.347 0.2775 1.614 
14.5 5.66 28.480 0.337 0.283 1.567 
19.5 5.68 28.631 0.335 0.284 1.559 
24.5 5.69 28.706 0.334 0.2845 1.555 
29 5.7 28.782 0.334 0.285 1.551 
Table 2

The results obtained for the PKW with a 30° threshold angle

x (cm)HT (cm)QL (L/s)QN (L/s)CdHT/PRa
1.5 0.9 4.29 1.806 0.511 0.045 2.376 
4.5 0.95 1.958 0.471 0.0475 2.191 
9.5 0.95 1.958 0.471 0.0475 2.191 
14.5 0.97 2.021 0.457 0.0485 2.123 
19.5 0.98 2.052 0.450 0.049 2.091 
24.5 2.115 0.436 0.05 2.028 
29 2.115 0.436 0.05 2.028 
1.5 0.95 4.99 1.958 0.548 0.0475 2.548 
4.5 2.115 0.507 0.05 2.359 
9.5 1.06 2.308 0.465 0.053 2.162 
14.5 1.11 2.473 0.434 0.0555 2.017 
19.5 1.15 2.608 0.411 0.0575 1.913 
24.5 1.17 2.677 0.401 0.0585 1.864 
29 1.18 2.711 0.396 0.059 1.841 
1.5 2.18 15.17 6.808 0.479 0.109 2.228 
4.5 2.28 7.281 0.448 0.114 2.083 
9.5 2.36 7.668 0.426 0.118 1.978 
14.5 2.4 7.864 0.415 0.12 1.929 
19.5 2.4 7.864 0.415 0.12 1.929 
24.5 2.42 7.962 0.410 0.121 1.905 
29 2.45 8.111 0.402 0.1225 1.870 
1.5 2.8 19.93 9.909 0.433 0.14 2.011 
4.5 2.9 10.445 0.410 0.145 1.908 
9.5 2.95 10.716 0.400 0.1475 1.860 
14.5 2.97 10.825 0.396 0.1485 1.841 
19.5 2.97 10.825 0.396 0.1485 1.841 
24.5 2.99 10.935 0.392 0.1495 1.823 
29 10.990 0.390 0.15 1.813 
1.5 3.62 28.69 14.567 0.424 0.181 1.970 
4.5 3.75 15.359 0.402 0.1875 1.868 
9.5 3.82 15.791 0.391 0.191 1.817 
14.5 3.85 15.977 0.386 0.1925 1.796 
19.5 3.9 16.289 0.379 0.195 1.761 
24.5 3.92 16.415 0.376 0.196 1.748 
29 3.92 16.415 0.376 0.196 1.748 
x (cm)HT (cm)QL (L/s)QN (L/s)CdHT/PRa
1.5 0.9 4.29 1.806 0.511 0.045 2.376 
4.5 0.95 1.958 0.471 0.0475 2.191 
9.5 0.95 1.958 0.471 0.0475 2.191 
14.5 0.97 2.021 0.457 0.0485 2.123 
19.5 0.98 2.052 0.450 0.049 2.091 
24.5 2.115 0.436 0.05 2.028 
29 2.115 0.436 0.05 2.028 
1.5 0.95 4.99 1.958 0.548 0.0475 2.548 
4.5 2.115 0.507 0.05 2.359 
9.5 1.06 2.308 0.465 0.053 2.162 
14.5 1.11 2.473 0.434 0.0555 2.017 
19.5 1.15 2.608 0.411 0.0575 1.913 
24.5 1.17 2.677 0.401 0.0585 1.864 
29 1.18 2.711 0.396 0.059 1.841 
1.5 2.18 15.17 6.808 0.479 0.109 2.228 
4.5 2.28 7.281 0.448 0.114 2.083 
9.5 2.36 7.668 0.426 0.118 1.978 
14.5 2.4 7.864 0.415 0.12 1.929 
19.5 2.4 7.864 0.415 0.12 1.929 
24.5 2.42 7.962 0.410 0.121 1.905 
29 2.45 8.111 0.402 0.1225 1.870 
1.5 2.8 19.93 9.909 0.433 0.14 2.011 
4.5 2.9 10.445 0.410 0.145 1.908 
9.5 2.95 10.716 0.400 0.1475 1.860 
14.5 2.97 10.825 0.396 0.1485 1.841 
19.5 2.97 10.825 0.396 0.1485 1.841 
24.5 2.99 10.935 0.392 0.1495 1.823 
29 10.990 0.390 0.15 1.813 
1.5 3.62 28.69 14.567 0.424 0.181 1.970 
4.5 3.75 15.359 0.402 0.1875 1.868 
9.5 3.82 15.791 0.391 0.191 1.817 
14.5 3.85 15.977 0.386 0.1925 1.796 
19.5 3.9 16.289 0.379 0.195 1.761 
24.5 3.92 16.415 0.376 0.196 1.748 
29 3.92 16.415 0.376 0.196 1.748 
Table 3

The results obtained for the PKW with a 45° threshold angle

x (cm)HT (cm)QL (L/s)QN (L/s)CdHT /PRa
1.5 1.28 7.14 3.063 0.501 0.064 2.331 
4.5 1.3 3.135 0.490 0.065 2.278 
9.5 1.3 3.135 0.490 0.065 2.278 
14.5 1.3 3.135 0.490 0.065 2.278 
19.5 1.31 3.171 0.484 0.0655 2.252 
24.5 1.35 3.317 0.463 0.0675 2.152 
29 1.4 3.503 0.438 0.07 2.038 
1.5 1.45 8.07 3.693 0.470 0.0725 2.185 
4.5 1.45 3.693 0.470 0.0725 2.185 
9.5 1.45 3.693 0.470 0.0725 2.185 
14.5 1.46 3.731 0.465 0.073 2.163 
19.5 1.46 3.731 0.465 0.073 2.163 
24.5 1.48 3.808 0.456 0.074 2.119 
29 1.5 3.885 0.447 0.075 2.077 
1.5 13.68 5.982 0.492 0.1 2.287 
4.5 2.1 6.436 0.457 0.105 2.125 
9.5 2.15 6.668 0.441 0.1075 2.052 
14.5 2.2 6.901 0.426 0.11 1.982 
19.5 2.23 7.043 0.418 0.1115 1.942 
24.5 2.25 7.138 0.412 0.1125 1.916 
29 2.3 7.377 0.399 0.115 1.854 
1.5 3.18 25.86 11.994 0.464 0.159 2.156 
4.5 3.2 12.107 0.459 0.16 2.136 
9.5 3.3 12.679 0.439 0.165 2.040 
14.5 3.32 12.794 0.435 0.166 2.021 
19.5 3.35 12.968 0.429 0.1675 1.994 
24.5 3.35 12.968 0.429 0.1675 1.994 
29 3.37 13.084 0.425 0.1685 1.976 
1.5 37.28 16.920 0.474 0.2 2.203 
4.5 4.1 17.558 0.457 0.205 2.123 
9.5 4.25 18.531 0.433 0.2125 2.012 
14.5 4.3 18.859 0.425 0.215 1.977 
19.5 4.3 18.859 0.425 0.215 1.977 
24.5 4.3 18.859 0.425 0.215 1.977 
29 4.32 18.990 0.422 0.216 1.963 
x (cm)HT (cm)QL (L/s)QN (L/s)CdHT /PRa
1.5 1.28 7.14 3.063 0.501 0.064 2.331 
4.5 1.3 3.135 0.490 0.065 2.278 
9.5 1.3 3.135 0.490 0.065 2.278 
14.5 1.3 3.135 0.490 0.065 2.278 
19.5 1.31 3.171 0.484 0.0655 2.252 
24.5 1.35 3.317 0.463 0.0675 2.152 
29 1.4 3.503 0.438 0.07 2.038 
1.5 1.45 8.07 3.693 0.470 0.0725 2.185 
4.5 1.45 3.693 0.470 0.0725 2.185 
9.5 1.45 3.693 0.470 0.0725 2.185 
14.5 1.46 3.731 0.465 0.073 2.163 
19.5 1.46 3.731 0.465 0.073 2.163 
24.5 1.48 3.808 0.456 0.074 2.119 
29 1.5 3.885 0.447 0.075 2.077 
1.5 13.68 5.982 0.492 0.1 2.287 
4.5 2.1 6.436 0.457 0.105 2.125 
9.5 2.15 6.668 0.441 0.1075 2.052 
14.5 2.2 6.901 0.426 0.11 1.982 
19.5 2.23 7.043 0.418 0.1115 1.942 
24.5 2.25 7.138 0.412 0.1125 1.916 
29 2.3 7.377 0.399 0.115 1.854 
1.5 3.18 25.86 11.994 0.464 0.159 2.156 
4.5 3.2 12.107 0.459 0.16 2.136 
9.5 3.3 12.679 0.439 0.165 2.040 
14.5 3.32 12.794 0.435 0.166 2.021 
19.5 3.35 12.968 0.429 0.1675 1.994 
24.5 3.35 12.968 0.429 0.1675 1.994 
29 3.37 13.084 0.425 0.1685 1.976 
1.5 37.28 16.920 0.474 0.2 2.203 
4.5 4.1 17.558 0.457 0.205 2.123 
9.5 4.25 18.531 0.433 0.2125 2.012 
14.5 4.3 18.859 0.425 0.215 1.977 
19.5 4.3 18.859 0.425 0.215 1.977 
24.5 4.3 18.859 0.425 0.215 1.977 
29 4.32 18.990 0.422 0.216 1.963 
Table 4

The results obtained for the PKW with a 60° threshold angle

x (cm)HT (cm)QL (L/s)QN (L/s)CdHT /PRa
1.5 0.75 4.23 1.374 0.662 0.0375 3.079 
4.5 0.8 1.513 0.601 0.04 2.795 
9.5 0.85 1.657 0.549 0.0425 2.552 
14.5 0.9 1.806 0.504 0.045 2.342 
19.5 0.9 1.806 0.504 0.045 2.342 
24.5 0.9 1.806 0.504 0.045 2.342 
29 0.9 1.806 0.504 0.045 2.342 
1.5 1.28 8.46 3.063 0.594 0.064 2.762 
4.5 1.3 3.135 0.580 0.065 2.699 
9.5 1.4 3.503 0.519 0.07 2.415 
14.5 1.45 3.693 0.493 0.0725 2.291 
19.5 1.5 3.885 0.468 0.075 2.177 
24.5 1.5 3.885 0.468 0.075 2.177 
29 1.5 3.885 0.468 0.075 2.177 
1.5 2.1 16.21 6.436 0.542 0.105 2.519 
4.5 2.15 6.668 0.523 0.1075 2.431 
9.5 2.2 6.901 0.505 0.11 2.349 
14.5 2.25 7.138 0.488 0.1125 2.271 
19.5 2.27 7.233 0.482 0.1135 2.241 
24.5 2.27 7.233 0.482 0.1135 2.241 
29 2.28 7.281 0.479 0.114 2.226 
1.5 3.15 28.44 11.824 0.517 0.1575 2.405 
4.5 3.25 12.392 0.494 0.1625 2.295 
9.5 3.37 13.084 0.468 0.1685 2.174 
14.5 3.4 13.259 0.461 0.17 2.145 
19.5 3.4 13.259 0.461 0.17 2.145 
24.5 3.4 13.259 0.461 0.17 2.145 
29 3.42 13.377 0.457 0.171 2.126 
1.5 3.7 35.1 15.053 0.502 0.185 2.332 
4.5 3.88 16.164 0.467 0.194 2.171 
9.5 3.94 16.604 0.456 0.197 2.122 
14.5 3.98 16.793 0.450 0.199 2.090 
19.5 16.920 0.446 0.2 2.074 
24.5 16.920 0.446 0.2 2.074 
29 16.920 0.446 0.2 2.074 
x (cm)HT (cm)QL (L/s)QN (L/s)CdHT /PRa
1.5 0.75 4.23 1.374 0.662 0.0375 3.079 
4.5 0.8 1.513 0.601 0.04 2.795 
9.5 0.85 1.657 0.549 0.0425 2.552 
14.5 0.9 1.806 0.504 0.045 2.342 
19.5 0.9 1.806 0.504 0.045 2.342 
24.5 0.9 1.806 0.504 0.045 2.342 
29 0.9 1.806 0.504 0.045 2.342 
1.5 1.28 8.46 3.063 0.594 0.064 2.762 
4.5 1.3 3.135 0.580 0.065 2.699 
9.5 1.4 3.503 0.519 0.07 2.415 
14.5 1.45 3.693 0.493 0.0725 2.291 
19.5 1.5 3.885 0.468 0.075 2.177 
24.5 1.5 3.885 0.468 0.075 2.177 
29 1.5 3.885 0.468 0.075 2.177 
1.5 2.1 16.21 6.436 0.542 0.105 2.519 
4.5 2.15 6.668 0.523 0.1075 2.431 
9.5 2.2 6.901 0.505 0.11 2.349 
14.5 2.25 7.138 0.488 0.1125 2.271 
19.5 2.27 7.233 0.482 0.1135 2.241 
24.5 2.27 7.233 0.482 0.1135 2.241 
29 2.28 7.281 0.479 0.114 2.226 
1.5 3.15 28.44 11.824 0.517 0.1575 2.405 
4.5 3.25 12.392 0.494 0.1625 2.295 
9.5 3.37 13.084 0.468 0.1685 2.174 
14.5 3.4 13.259 0.461 0.17 2.145 
19.5 3.4 13.259 0.461 0.17 2.145 
24.5 3.4 13.259 0.461 0.17 2.145 
29 3.42 13.377 0.457 0.171 2.126 
1.5 3.7 35.1 15.053 0.502 0.185 2.332 
4.5 3.88 16.164 0.467 0.194 2.171 
9.5 3.94 16.604 0.456 0.197 2.122 
14.5 3.98 16.793 0.450 0.199 2.090 
19.5 16.920 0.446 0.2 2.074 
24.5 16.920 0.446 0.2 2.074 
29 16.920 0.446 0.2 2.074 
Table 5

The results obtained for the PKW with a 75° threshold angle

x (cm)HT (cm)QL (L/s)QN (L/s)CdHT/PRa
1.5 0.95 5.84 1.958 0.641 0.0475 2.982 
4.5 2.115 0.594 0.05 2.761 
9.5 1.05 2.276 0.552 0.0525 2.566 
14.5 1.05 2.276 0.552 0.0525 2.566 
19.5 1.05 2.276 0.552 0.0525 2.566 
24.5 1.05 2.276 0.552 0.0525 2.566 
29 1.07 2.341 0.537 0.0535 2.495 
1.5 1.4 9.73 3.503 0.597 0.07 2.777 
4.5 1.45 3.693 0.567 0.0725 2.635 
9.5 1.48 3.808 0.550 0.074 2.555 
14.5 1.5 3.885 0.539 0.075 2.504 
19.5 1.5 3.885 0.539 0.075 2.504 
24.5 1.52 3.963 0.528 0.076 2.455 
29 1.52 3.963 0.528 0.076 2.455 
1.5 2.2 18.75 6.901 0.584 0.11 2.717 
4.5 2.35 7.619 0.529 0.1175 2.461 
9.5 2.45 8.111 0.497 0.1225 2.312 
14.5 2.5 8.360 0.482 0.125 2.243 
19.5 2.55 8.612 0.468 0.1275 2.177 
24.5 2.55 8.612 0.468 0.1275 2.177 
29 2.57 8.714 0.463 0.1285 2.152 
1.5 3.4 33.4 13.259 0.542 0.17 2.519 
4.5 3.6 14.446 0.497 0.18 2.312 
9.5 3.65 14.748 0.487 0.1825 2.265 
14.5 3.72 15.175 0.473 0.186 2.201 
19.5 3.75 15.359 0.468 0.1875 2.175 
24.5 3.75 15.359 0.468 0.1875 2.175 
29 3.78 15.543 0.462 0.189 2.149 
1.5 3.85 40.5 15.977 0.545 0.1925 2.535 
4.5 4.1 17.558 0.496 0.205 2.307 
9.5 4.3 18.859 0.462 0.215 2.148 
14.5 4.35 19.189 0.454 0.2175 2.111 
19.5 4.35 19.189 0.454 0.2175 2.111 
24.5 4.35 19.189 0.454 0.2175 2.111 
29 4.38 19.387 0.449 0.219 2.089 
x (cm)HT (cm)QL (L/s)QN (L/s)CdHT/PRa
1.5 0.95 5.84 1.958 0.641 0.0475 2.982 
4.5 2.115 0.594 0.05 2.761 
9.5 1.05 2.276 0.552 0.0525 2.566 
14.5 1.05 2.276 0.552 0.0525 2.566 
19.5 1.05 2.276 0.552 0.0525 2.566 
24.5 1.05 2.276 0.552 0.0525 2.566 
29 1.07 2.341 0.537 0.0535 2.495 
1.5 1.4 9.73 3.503 0.597 0.07 2.777 
4.5 1.45 3.693 0.567 0.0725 2.635 
9.5 1.48 3.808 0.550 0.074 2.555 
14.5 1.5 3.885 0.539 0.075 2.504 
19.5 1.5 3.885 0.539 0.075 2.504 
24.5 1.52 3.963 0.528 0.076 2.455 
29 1.52 3.963 0.528 0.076 2.455 
1.5 2.2 18.75 6.901 0.584 0.11 2.717 
4.5 2.35 7.619 0.529 0.1175 2.461 
9.5 2.45 8.111 0.497 0.1225 2.312 
14.5 2.5 8.360 0.482 0.125 2.243 
19.5 2.55 8.612 0.468 0.1275 2.177 
24.5 2.55 8.612 0.468 0.1275 2.177 
29 2.57 8.714 0.463 0.1285 2.152 
1.5 3.4 33.4 13.259 0.542 0.17 2.519 
4.5 3.6 14.446 0.497 0.18 2.312 
9.5 3.65 14.748 0.487 0.1825 2.265 
14.5 3.72 15.175 0.473 0.186 2.201 
19.5 3.75 15.359 0.468 0.1875 2.175 
24.5 3.75 15.359 0.468 0.1875 2.175 
29 3.78 15.543 0.462 0.189 2.149 
1.5 3.85 40.5 15.977 0.545 0.1925 2.535 
4.5 4.1 17.558 0.496 0.205 2.307 
9.5 4.3 18.859 0.462 0.215 2.148 
14.5 4.35 19.189 0.454 0.2175 2.111 
19.5 4.35 19.189 0.454 0.2175 2.111 
24.5 4.35 19.189 0.454 0.2175 2.111 
29 4.38 19.387 0.449 0.219 2.089 
Table 6

The results obtained for the PKW with a 90° threshold angle

x (cm)HT (cm)QL (L/s)QN (L/s)CdHT/PRa
1.5 0.78 5.47 1.457 0.808 0.039 3.754 
4.5 0.92 1.866 0.630 0.046 2.931 
9.5 0.95 1.958 0.601 0.0475 2.793 
14.5 0.95 1.958 0.601 0.0475 2.793 
19.5 0.97 2.021 0.582 0.0485 2.707 
24.5 0.98 2.052 0.573 0.049 2.666 
29 2.115 0.556 0.05 2.586 
1.5 1.15 7.41 2.608 0.611 0.0575 2.841 
4.5 1.2 2.780 0.573 0.06 2.665 
9.5 1.2 2.780 0.573 0.06 2.665 
14.5 1.33 3.244 0.491 0.0665 2.284 
19.5 1.35 3.317 0.480 0.0675 2.234 
24.5 1.35 3.317 0.480 0.0675 2.234 
29 1.36 3.354 0.475 0.068 2.209 
1.5 2.35 19.88 7.619 0.561 0.1175 2.609 
4.5 2.5 8.360 0.511 0.125 2.378 
9.5 2.52 8.461 0.505 0.126 2.350 
14.5 2.56 8.663 0.494 0.128 2.295 
19.5 2.6 8.867 0.482 0.13 2.242 
24.5 2.6 8.867 0.482 0.13 2.242 
29 2.65 9.124 0.469 0.1325 2.179 
1.5 2.7 23.58 9.383 0.540 0.135 2.513 
4.5 2.8 9.909 0.512 0.14 2.380 
9.5 2.85 10.176 0.498 0.1425 2.317 
14.5 2.9 10.445 0.486 0.145 2.258 
19.5 2.92 10.553 0.481 0.146 2.234 
24.5 2.95 10.716 0.473 0.1475 2.200 
29 10.990 0.461 0.15 2.146 
1.5 4.35 46.07 19.189 0.516 0.2175 2.401 
4.5 4.47 19.988 0.496 0.2235 2.305 
9.5 4.52 20.324 0.488 0.226 2.267 
14.5 4.55 20.527 0.483 0.2275 2.244 
19.5 4.61 20.934 0.473 0.2305 2.201 
24.5 4.61 20.934 0.473 0.2305 2.201 
29 4.62 21.002 0.472 0.231 2.194 
x (cm)HT (cm)QL (L/s)QN (L/s)CdHT/PRa
1.5 0.78 5.47 1.457 0.808 0.039 3.754 
4.5 0.92 1.866 0.630 0.046 2.931 
9.5 0.95 1.958 0.601 0.0475 2.793 
14.5 0.95 1.958 0.601 0.0475 2.793 
19.5 0.97 2.021 0.582 0.0485 2.707 
24.5 0.98 2.052 0.573 0.049 2.666 
29 2.115 0.556 0.05 2.586 
1.5 1.15 7.41 2.608 0.611 0.0575 2.841 
4.5 1.2 2.780 0.573 0.06 2.665 
9.5 1.2 2.780 0.573 0.06 2.665 
14.5 1.33 3.244 0.491 0.0665 2.284 
19.5 1.35 3.317 0.480 0.0675 2.234 
24.5 1.35 3.317 0.480 0.0675 2.234 
29 1.36 3.354 0.475 0.068 2.209 
1.5 2.35 19.88 7.619 0.561 0.1175 2.609 
4.5 2.5 8.360 0.511 0.125 2.378 
9.5 2.52 8.461 0.505 0.126 2.350 
14.5 2.56 8.663 0.494 0.128 2.295 
19.5 2.6 8.867 0.482 0.13 2.242 
24.5 2.6 8.867 0.482 0.13 2.242 
29 2.65 9.124 0.469 0.1325 2.179 
1.5 2.7 23.58 9.383 0.540 0.135 2.513 
4.5 2.8 9.909 0.512 0.14 2.380 
9.5 2.85 10.176 0.498 0.1425 2.317 
14.5 2.9 10.445 0.486 0.145 2.258 
19.5 2.92 10.553 0.481 0.146 2.234 
24.5 2.95 10.716 0.473 0.1475 2.200 
29 10.990 0.461 0.15 2.146 
1.5 4.35 46.07 19.189 0.516 0.2175 2.401 
4.5 4.47 19.988 0.496 0.2235 2.305 
9.5 4.52 20.324 0.488 0.226 2.267 
14.5 4.55 20.527 0.483 0.2275 2.244 
19.5 4.61 20.934 0.473 0.2305 2.201 
24.5 4.61 20.934 0.473 0.2305 2.201 
29 4.62 21.002 0.472 0.231 2.194 

Equation (2) is in the metric system in which QN is the passing discharge over the Ogee linear weir, L is the length of the weir threshold, HT is the height of the water nappe over the weir threshold, and Cd is the weir discharge coefficient, which was considered as 2.1804.

It is worth noting that since the P/HT ratio was greater than 3 for all tests, the value of Cd was considered the highest value for the USBR threshold.

The changes in the Cd, Ra, and QL versus HT/P ratio and changes in the Cd and Ra for different angles of the input and output keys for all the tested states are shown in Figure 4.
Figure 4

Changes of effective parameters in PKWs in different models. (a) Changes in Cd versus HT/P ratio of the PKW for all the threshold angles. (b) Changes in Ra versus HT/P ratio of the PKW for all the threshold angles. (c) Changes in QL versus HT/P ratio of the PKW for all the threshold angles. (d) Changes in Cd versus the angle of the input and output keys of the PKW for all threshold angles. (e) Changes in Ra versus the angle of the input and output keys of the PKW for all threshold angles.

Figure 4

Changes of effective parameters in PKWs in different models. (a) Changes in Cd versus HT/P ratio of the PKW for all the threshold angles. (b) Changes in Ra versus HT/P ratio of the PKW for all the threshold angles. (c) Changes in QL versus HT/P ratio of the PKW for all the threshold angles. (d) Changes in Cd versus the angle of the input and output keys of the PKW for all threshold angles. (e) Changes in Ra versus the angle of the input and output keys of the PKW for all threshold angles.

Close modal

In order to study the changes in the weir discharge coefficient, efficiency, passing flow discharge and flow nappe height over the weir, a distance of 9.5 cm from the weir was considered to be studied. In other words, five different flow discharges were passed over each weir at different weir installation angles. Then, for each flow discharge, the mentioned parameters were measured and calculated at different distances such as 9.5 cm from the weir. Different parts of Figure 4 were then plotted. Similar figures can be plotted and presented at all distances.

According to Figure 4(a), Cd and HT/P are inversely related. Therefore, the highest discharge coefficient occurred at the lowest HT/P, resulting in the lowest flow discharge. This can be attributed to the low interference of the water nappes on the weir. The reduction in Cd occurred in the 15° PKW with a lower intensity and, in other weirs, with a higher intensity. In addition, at constant HT/P values, weirs with lower angles had lower Cd. In other words, a 90° PKW had the highest Cd among all cases.

According to Figure 4(b), Ra is inversely related to the HT/P ratio. Due to the low interference of the water nappes on the weir, the highest efficiency occurred at the lowest discharge, and consequently, at the lowest HT/P ratio. By increasing the flow nappe height on the weir, the PKW was out of its desired efficiency and acted like a linear weir. This reduction in PKW efficiency occurred with a lower intensity for a 15° threshold angle and a higher intensity for other threshold angles. The PKW with a 90° threshold angle also had the highest efficiency.

According to Figure 4(c), there was a direct relationship between QL and HT/P. At constant HT/P values, weirs with a higher angle had a higher passing flow discharge. Among all the PKWs, the highest discharge occurred for the 90° threshold angle.

Figure 4(d) shows the changes in Cd versus the angle of the input and output keys of the weir for five different values of HT/P (from 0.05 to 0.25). For all constant values of HT/P, Cd was directly related to the angle of the input and output keys. The increase in Cd occurred with more intensity at HT/P = 0.05 and less intensity in other cases. In addition, at a constant angle, weirs with higher HT/P had lower Cd. Among all cases, the highest Cd was related to a 90° PKW and HT/P = 0.05.

Figure 4(e) shows the changes in Ra versus the angle of the input and output keys of the weir for five different values of HT/P (from 0.05 to 0.25). For all constant values of HT/P, Ra was directly related to the angle of the input and output keys. The increase in efficiency occurred with more intensity at HT/P = 0.05 and less intensity in other cases. In addition, at a constant angle, weirs with higher HT/P had lower Ra. Among all weirs, the highest efficiency was related to a 90° PKW and HT/P = 0.05.

It is worth noting that the fitted binomial relations between the discharge coefficient (Cd) and the efficiency (Ra) considering the angle of the input and output keys for different HT/P ratios are also shown in Figure 4.

According to Figure 5, the tests conducted in the present study on all six PKW models with input and output key angles of 15–90° indicated that at low ratios of HT/P, the dropping flow nappe is bonding and requires aeration. By increasing the HT/P ratio, the flow nappes drop in the aerated form and do not need aeration. The process continues until the submergence of the weir and, as a result, a performance similar to a linear weir occurs, in which the PKW does not function as expected. At low ratios of HT/P, the dropping flow nappes do not interfere with each other, and the highest discharge coefficient and the highest weir efficiency occur. However, by increasing the HT/P ratio, the dropping flow nappes at the output keys start to interfere with each other, causing the flow to rise and the weir to become submerged. As a result, the discharge coefficient and the weir efficiency also decrease.
Figure 5

Flow passing over the PKWs at different experimental conditions.

Figure 5

Flow passing over the PKWs at different experimental conditions.

Close modal

According to Tables 16 and Figures 4 and 5, the discharge coefficient, efficiency, and passing flow discharge are directly related to the inclination angle of the input and output keys. In low HT/P ratios, these types of weirs do not need aeration. In other words, according to the equation of the flow discharge passing over the PKWs (Equation (1)), the hydraulic characteristics such as the discharge coefficient and the efficiency of the PKWs generally improve with the inclination angle of the input and output keys. Consequently, the efficiency of this type of weir in passing flow discharge increases.

Weirs are one of the most important structures in the design of dams. They are responsible for transferring water over the dam's capacity during floods. The PKWs have high efficiency, and as a result, have a higher passing discharge capacity in flood conditions in comparison to the linear weirs. In the present study, the effect of the angle of the input and output keys on the discharge coefficient (Cd) and efficiency (Ra) (the ratio of the passing flow over a PKW to that of a linear weir) of the rectangular PKWs with identical indentations was studied experimentally for six different angles 15°, 30°, 45°, 60°, 75°, and 90°.

In general, the results of the present study include the following:

  • - The discharge coefficient, efficiency and passing discharge for constant values of the HT/P ratio are directly related to the angle of the input and output keys. A decrease in these mentioned parameters occurs at an angle of 15° with less intensity than other angles.

  • - In the PKWs with different input and output key angles, the dropping nappe bonds at low HT/P ratios and needs aeration. By increasing the HT/P ratio, the flow nappes drop in the aerated condition and do not need aeration. The increase in the ratio continues until the submergence of the weir with its performance converting to that of a linear weir. After submergence, the performance of the PKW decreases.

  • - Due to the lack of interference of the dropping nappes, the highest discharge coefficient and efficiency occur at low HT/P ratios. By increasing the ratio, the dropping nappes in the output keys start to interfere with each other, causing the flow to rise and the weir to become submerged. As a result, the discharge coefficient and efficiency of the weir decrease.

In general, the highest discharge coefficient of the rectangular PKWs occurs at a 90° angle of the input and output keys with the best efficiency.

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

The authors declare there is no conflict.

Akbari
M.
,
Salmasi
F.
,
Arvanaghi
H.
,
Karbasi
M.
&
Farsadizadeh
D.
2019
Application of Gaussian process regression model to predict discharge coefficient of gated piano key weir
.
Water Resources Management
33
,
3929
3947
.
Anderson
R. M.
&
Tullis
B. P.
2013
Piano key weir hydraulics and labyrinth weir comparison
.
Journal of Irrigation and Drainage Engineering
139
(
3
),
246
253
.
Azamathulla
H. M.
,
Haghiabi
A. H.
&
Parsaie
A.
2016
Prediction of side weir discharge coefficient by support vector machine technique
.
Water Science and Technology: Water Supply
16
(
4
),
1002
1016
.
Bhukya
R. K.
,
Pandey
M.
,
Valyrakis
M.
&
Michalis
P.
2022
Discharge estimation over piano key weirs: A review of recent developments
.
Water
14
(
19
),
3029
.
Borghei, S. M., Nekooie, M. A., Sadeghian, H. & Ghazizadeh, M. R. J. 2013 Triangular labyrinth side weirs with one and two cycles. In Proceedings of the Institution of Civil Engineers-Water Management 166 (1), 27–42. https://doi.org/10.1680/wama.11.00032.
Crookston
B. M.
,
Anderson
R. M.
&
Tullis
B. P.
2018
Free-flow discharge estimation method for piano key weir geometries
.
Journal of Hydroenvironment Research
19
(
Mar
),
160
167
.
https://doi.org/10.1016/j.jher.2017.10.003
.
Crookston
B. M.
,
Erpicum
S.
,
Tullis
B. P.
&
Laugier
F.
2019
Hydraulics of labyrinth and piano key weirs: 100 years of prototype structures, advancements, and future research needs
.
Journal of Hydraulic Engineering
145
(
12
),
02519004
.
Erpicum
S.
,
Tullis
B. P.
,
Lodomez
M.
,
Archambeau
P.
,
Dewals
B. J.
&
Pirotton
M.
2016
Scale effects in physical piano key weirs models
.
Journal of Hydraulic Research
54
(
6
),
692
698
.
Kabiri-Samani
A.
&
Javaheri
A.
2012
Discharge coefficients for free and submerged flow over piano key weirs
.
Journal of Hydraulic Research
50
(
1
),
114
120
.
Kadia
S.
,
Pummer
E.
,
Kumar
B.
,
Ruther
N.
&
Ahmad
Z.
2023
A reformed empirical equation for the discharge coefficient of free-flowing type – A piano key weirs
.
Journal of Irrigation and Drainage Engineering
149
(
4
).
http://dx.doi.org/10.1061/JIDEDH.IRENG-9886
.
Kumar
B.
,
Kadia
S.
&
Ahmad
Z.
2019
Evaluation of discharge equations of the piano key weirs
.
Flow Measurement and Instrumentation
68
(
Aug
),
101577
.
Laugier
F.
2007
Design and construction of the first piano key weir spillway at Goulours dam
.
International Journal on Hydropower & Dams
14
(
5
),
94
100
.
Leite Ribeiro
M.
,
Bieri
M.
,
Boillat
J. L.
,
Schleiss
A. J.
,
Singhal
G.
&
Sharma
N.
2012
Discharge capacity of piano key weirs
.
Journal of Hydraulic Engineering
138
(
2
),
199
203
.
Li
Z.
,
Xu
J.
,
Li
Y.
&
Han
C.
2023
Analysis of piano key weir drainage characteristics
. In:
Proceedings of PIANC Smart Rivers 2022. PIANC 2022. Lecture Notes in Civil Engineering
(
Li
Y.
,
Hu
Y.
,
Rigo
P.
,
Lefler
F. E.
&
Zhao
G.
, eds.), Vol.
264
.
Springer
,
Singapore
.
https://doi.org/10.1007/978-981-19-6138-0_25
.
Machiels
O.
,
Erpicum
S.
,
Archambeau
P.
,
Dewals
B.
&
Pirotton
M.
2009
Large scale experimental study of piano key weirs
. In
33rd IAHR Congress
,
IAHR
,
Vancouver, Canada
.
Machiels
O.
,
Pirotton
M.
,
Pierre
A.
,
Dewals
B.
&
Erpicum
S.
2014
Experimental parametric study and design of piano key weirs
.
Journal of Hydraulic Research
52
(
3
),
326
335
.
Mero
S. K.
,
Haleem
D. A. J.
&
Yousif
A. A.
2022
The influence of inlet to outlet width ratio on the hydraulic performance of piano key weir (PKW-type A)
.
Water Practice & Technology
17
(
6
),
1273
1283
.
Noroozi
B.
,
Bazargan
J.
&
Safarzadeh
A.
2021
Introducing the T-shaped weir: A new nonlinear weir
.
Water Supply
21
(
7
),
3772
3789
.
Roushangar
K.
,
Majedi Asl
M.
&
Shahnazi
S.
2021
Hydraulic performance of PK weirs based on experimental study and kernel-based modeling
.
Water Resources Management
35
,
3571
3592
.
Safarzadeh
A.
&
Noroozi
B.
2017
3D hydrodynamics of trapezoidal piano key spillways
.
International Journal of Civil Engineering
15
,
89
101
.
Tullis
B. P.
,
Young
J. C.
&
Chandler
M. A.
2007
Head-discharge relationships for submerged labyrinth weirs
.
Journal of Hydraulic Engineering
133
(
3
),
248
254
.
Tullis
B. P.
,
Crookston
B. M.
&
Young
N.
2020
Scale effects in free-flow nonlinear weir head-discharge relationships
.
Journal of Hydraulic Engineering
146
(
2
),
04019056
.
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