## Abstract

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 < F_{1} < 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 < F_{1} < 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 *C _{PW}* and F

_{1}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.

## HIGHLIGHTS

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.

## NOTATIONS

*b*width of main channel (L)

*B*length of sidewall of PKW (L)

*B*_{i}downstream overhang length of PKW (L)

*B*_{o}upstream overhang length of PKW (L)

*C*_{d}side weir discharge coefficient (–)

*C*_{PW}PKSW discharge coefficient (–)

*E*specific energy (L)

- F
_{1} Froude number at upstream end of side weir (–)

- F
_{2} Froude number at downstream end of side weir (–)

acceleration due to gravity (LT

^{−2})*h*_{1}piezometric head on side weir at upstream end (L)

*h*_{2}piezometric head on side weir at downstream end (L)

*y*the depth of flow at the section

*x*(L)*y*_{1}flow depth at upstream end of side weir at channel center (L)

*y*_{2}flow depth at downstream end of side weir at channel center (L)

*W*opening length of side weir (L)

*W*_{i}inlet key width of PKW (L)

*W*_{o}outlet key width of PKW (L)

*V*_{1}flow velocity on the first edge of the weir (LT

^{−1})*T*_{s}wall thickness of PKW (L)

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

*N*_{u}number of PKW-units (–)

*P*height of weir crest (L)

*σ*water surface tension (N/m)

*Q*discharge in the main channel before m

^{3}/s beginning of side weir (L^{3}T^{−1})*Q*_{1}total discharge in main channel at upstream end of side weir (L

^{3}T^{−1})*Q*_{2}total discharge in main channel at downstream end of side weir (L

^{3}T^{−1})*Q*_{w}outflow weir discharge (L

^{3}T^{−1})*q*discharge per unit length over side weir (L

^{2}T^{−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 (°)

## INTRODUCTION

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.

*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:

*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):where

*C*is the discharge coefficient of the side weir (dimensionless),

_{d}*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),

*y*

_{1}is the depth of flow at the upstream end of the side weir in the main channel center line (m) and F

_{1}is the Froude number.

*et al. (*2014). The coefficient of discharge for this structure is given by: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.

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 *W _{i}*/

*W*(the inlet key width to the outlet key width) enhances flow coefficient. The best input and output key widths ratio, according to Lempérière

_{o}*et al*. (2011), is between 1.2 and 1.5. Ouamane & Lempérière (2006) showed that the discharge efficiency depends on the

*P*/

*W*ratio and stated that when the weir height was increased by 20%, the discharge increase was between 5 and 10%. Gharibvand

_{u}*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 (*N _{u}*), weir height (

*P*), lateral crest length (

*B*), up stream and downstream overhang lengths (

*B*and

_{o}*B*), inlet and outlet widths (

_{i}*W*and

_{i}*W*), and the wall thickness (

_{o}*T*) (Pralong

_{s}*et al.*2011).

*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).

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/h*_{1} 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.

## BASIC THEORY

*q*, can be estimated using:where

*Q*is the weir overflow discharge,

_{w}*y*

_{1}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. (

*y*

_{1}-

*P*) is the piezometric head over the side weir.

### De Marchi approach

*W*) and other hydraulic variables of the flow is: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

*C*is determined experimentally.

_{d}### Schmidt approach

*h*is the average piezometric head over the weir (m),

_{a}*h*

_{1}is the piezometric head over the weir at the upstream end of the side weir (m), and

*h*

_{2}is the piezometric head over the weir at the downstream end of the side weir (m).

### Domínguez approach

*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,

*Q*(m

_{w}^{3}/s), is given by:

## EXPERIMENTAL SETUP AND PROCEDURE

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*/*y*_{1}, *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.

Variables . | Limits 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, (m^{3}/s) | 0.0119–0.1420 |

Froude number | 0.12–0.87 |

Head, h_{1} (m) | 0.03–0.09 |

The inlet crest width W (m) _{i} | 0.03–0.11 |

The outlet crest width W (m) _{o} | 0.02–0.09 |

Length of sidewall B (m) | 0.03–0.81 |

Downstream overhang length B (m) _{i} | 0.13–0.40 |

Upstream overhang length B (m) _{o} | 0 |

Wall thickness T (m) _{s} | 0.002 |

Sidewall angle (°) | 6 |

Variables . | Limits 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, (m^{3}/s) | 0.0119–0.1420 |

Froude number | 0.12–0.87 |

Head, h_{1} (m) | 0.03–0.09 |

The inlet crest width W (m) _{i} | 0.03–0.11 |

The outlet crest width W (m) _{o} | 0.02–0.09 |

Length of sidewall B (m) | 0.03–0.81 |

Downstream overhang length B (m) _{i} | 0.13–0.40 |

Upstream overhang length B (m) _{o} | 0 |

Wall thickness T (m) _{s} | 0.002 |

Sidewall angle (°) | 6 |

## RESULTS AND DISCUSSION

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 *C _{PW}* for ease of comparison, and the equations given for

*C*in Section 1 are used. Some of the results along with the hydraulic and physical conditions of the experiments are presented in Table 2.

_{d}No. . | W/b (-)
. | P (m)
. | L/W (-)
. | y_{1} (m)
. | y (m)
. _{2} | F_{1} (-)
. | Q_{1} (m^{3}/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 (-)
. | y_{1} (m)
. | y (m)
. _{2} | F_{1} (-)
. | Q_{1} (m^{3}/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 |

### Assumption of constant specific energy

*ΔE*=

*E*

_{1}−

*E*

_{2}) 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.

### Water surface profile

*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.

### Discharge coefficient of trapezoidal piano key side weir using different approaches

*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 F

_{1}= 0.2 and F

_{1}= 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.

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 F_{1} and *C _{PW}*. Moreover, the data obtained by the De Marchi method were very scattered. However, a decreasing trend was obtained between F

_{1}-

*C*in the studies that used the De Marchi method (Borghei

_{PW}*et al.*1999, 2013; Emiroglu

*et al.*2014). Different from other studies, this study provides a better understanding of the relationship between F

_{1}and

*C*by performing experiments over a wide range of Froude numbers. Bagheri

_{PW}*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

*h*

_{1}) 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

*h*

_{1}/

*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

*h*

_{1}/

*P*value. The rate of change of

*C*decreases as

_{PW}*h*

_{1}/

*P*increases. The trend of

*C*as a function of

_{PW}*h*

_{1}/

*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).

In this study, head value is between 3 < *h*_{1} < 9 cm, which corresponds with the range of 0.15 < *h*_{1}/*P* < 0.57. For *W*/*b* = 0.5, *P*/*W* = 0.64 gives higher *C _{PW}* values of TPKSW, compared to

*P*/

*W*values of 0.48 and 0.80. In the range of 0.2 ≤

*h*

_{1}/

*P*≤ 0.5, the most stable behavior of

*C*is observed at

_{PW}*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

*h*

_{1},

*C*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.

_{PW}### Effects of upstream Froude number

*C*and Froude number is in the form of . In all cases, the

_{PW}*b*takes negative values, therefore

*C*is a decreasing function of the Froude number. Similar behaviors are observed in labyrinth side weirs (Aydın & Emiroglu 2016; Abbasi

_{PW}*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 F

_{1}= 0.2, and −0.10 for F

_{1}= 0.6. For

*P*/

*W*= 0.24, the slope is equal to −0.46 at F

_{1}= 0.2 and −0.14 at F

_{1}= 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

*C*. It is seen that the Froude number is more effective on

_{PW}*C*at

_{PW}*P*/

*W*= 0.16. At high Froude numbers, the best performance is obtained at

*P*/

*W*= 0.27 and the effect of the F

_{1}number decreases. The ineffective region at the corners of the keys expands and lowers the outflow discharge when F

_{1}value is raised (see Figure 11). In the case of large F

_{1}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.

### Outflow efficiency

*η*is correlated with the dimensionless parameter :where F

_{2}is the downstream Froude number and

*y*

_{2}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*=

*aχ*with a correlation coefficient above 0.95. For a fixed

^{b}*P*/

*y*

_{2}value,

*η*decreases with increasing F

_{2}. Similarly, if the F

_{2}value is constant, the

*η*values decrease as the

*P*/

*y*

_{2}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 F

_{2}or

*P*/

*y*

_{2}. 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.

### Comparison with literature

*et al.*(1999) developed empirical formulas for the flow coefficient

*C*in the case of rectangular weir lateral flow in a linear channel. The discharge coefficient

_{d}*C*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

_{d}*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).

*h*

_{1}/

*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

*C*. As head increases, PKSW and trapezoidal labyrinth side weir discharge coefficients get very close to each other.

_{PW}*C*over

_{PW}*C*(triangular labyrinth side weir) was found to be 2.1.

_{d}*C*over

_{PW}*C*(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

_{d}*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.

### Proposed equation for calculating the weir discharge coefficient

^{2}) criteria were used. For Equation (15), RMSE and R

^{2}are 0.16968 and 0.88, respectively. Figure 15 compares the measured

*C*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:

_{PW}Equation (15) is subject to the limitations of present tests: 0.13 ≤ F_{1} ≤ 0.88, 0.17 ≤ *H*_{1}/*P* ≤ 0.85, 0.5 ≤ *W*/*b* ≤ 1.5, 1.4 ≤ *B*/*P* ≤ 6.73, 0.15 ≤ *h*_{1}/*P* ≤ 0.57, *W _{i}*/

*W*= 1.23,

_{o}*n*=

*L*/

*W*= 4.83, and

*α*= 6°.

## CONCLUSIONS

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 < F

_{1}< 0.76.

## DATA AVAILABILITY STATEMENT

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

## CONFLICT OF INTEREST

The authors declare there is no conflict.

L' Energia Electtrica11, 849–860.