Abstract

Piano-key weirs can be used instead of classic rectangular side-weirs (CRSWs) to increase the discharge capacity of side weirs. So far, no research has been done on trapezoidal piano-key side weirs (TPKSWs) in a curved channel. This study examines the effect of using one or two cycles in TPKSWs on discharge capacity, having the same total width and upstream–downstream length. Dimensional analysis has been performed to determine the dimensionless parameters affecting the discharge coefficient related to the developed length (CdL) of TPKSWs in a curved channel. An empirical equation for CdL has been proposed based on the experimental results. There is a good agreement between estimated and measured data. Results showed that the discharge coefficient related to the total width of a TPKSW is 1.7 to 5.6 times higher than that of a CRSW. Also, the CdL of a one-cycle TPKSW is 1.4 to 2 times higher than that of a two-cycle TPKSW.

INTRODUCTION

Side weirs are hydraulic structures used for flow diversion in irrigation and drainage systems. The best way to enhance the discharge coefficient () of side weirs is to increase the crest length of the weir without changing the channel width. Labyrinth and piano-key weirs are folded in plan-view and they have this advantage. Some studies have been conducted on labyrinth side-weirs (LSWs). Triangular LSWs were examined by Emiroglu et al. (2010), who reported that of the triangular LSW was 1.5 to 4.5 times greater than that of the classic rectangular side-weir (CRSW). Emiroglu & Kaya (2011) investigated LSWs and reported that of the trapezoidal LSW was 1.5 to 5.0 times greater than that of the CRSW. For a specified channel width, the number of LSW cycles has considerable effect on developed crest length of the weir that influences construction cost and efficiency of the LSW. Zahedi-Khameneh et al. (2014) found that increasing the number of cycles in triangular LSWs reduced the eddy flow, effective length and weir input. They deduced that one- and two-cycle weirs had higher than four-cycle weirs.

The piano-key weir (PKW) is a special type of labyrinth weir that can be used as a side weir. The height of the vertical sidewalls of PKWs gradually decreases towards the weir apex; thus, the required reinforcement steel decreases (Erpicum et al. 2017). Moreover, since the footprint of piano-key side weirs (PKSWs) is smaller than that of LSWs, it could be a more appropriate choice to confront space or other construction-site restrictions (Karimi et al. 2018). PKWs are as efficient or more efficient than a labyrinth weir under specific conditions (Blancher et al. 2011). Trapezoidal PKWs are more efficient than rectangular PKWs (Mehboudi et al. 2016).

Despite extensive studies on traditional PKWs (flow direction perpendicular to the weir), few studies have been conducted on PKSWs. Karimi et al. (2017, 2018) studied type-C rectangular PKSWs (RPKSWs) in a straight channel. They investigated the effects of Froude number, upstream head and crest length on the efficiency of the RPKSW and reported that it was more efficient than the CRSW. There is no more information about other types and other plan-views of the PKSW. Also, the effect of the number of cycles on of the PKSW has not been examined.

The abovementioned studies were performed in straight channels. The outer bank of a curved channel is often the best location for water intake; therefore, understanding the flow characteristics in this region is of great importance. Lateral flows and spiral motions are generated by the side weir and curved channel, respectively. Secondary currents are created by both lateral and spiral flows. These secondary flows grow as the upstream Froude number () increases or the ratio of water depth to channel width increases (Choudhary & Narasimhan 1977). Coşar & Agaccioglu (2004) studied classic triangular side weirs (CTSWs) in a curved channel. They concluded that spiral motions are created at and the of side weirs in the curved channel is greater than that in the straight channel. Agaccioglu et al. (2012) examined CRSWs in a curved channel and proposed an equation for the . Based on our knowledge, no study has been done on of LSWs and PKSWs in curved channels.

The present study aims at investigating the effect of geometrical and flow parameters on of type-A trapezoidal piano-key side weirs (TPKSWs) in a curved channel and focuses on the effect of using one or two cycles in TPKSWs on the .

MATERIALS AND METHODS

The tests were conducted in the hydraulic laboratory of the Soil Conservation and Watershed Management Research Institute in Tehran, Iran. The laboratory flume consists of a curved main channel (17 m long, 0.35 m wide and 0.5 m deep), three intake channels and a movable channel (Figure 1). The intake channels were perpendicular to the main channel at 60°, 105° and 150° of bend curvature. The flow conditions were free and subcritical. A calibrated 90° triangular weir was used to measure the outflow discharge () of the main channel. A calibrated rectangular weir at the end of the movable channel was used to estimate the side weir discharge (). The total discharge () was calculated as the sum of and . A slide gate installed near the end of the main channel controlled the surface water level. The water level was measured using a Profile Indicator PV-09 (Delft Hydraulics) with ±0.1 mm accuracy.

Figure 1

Laboratory flume.

Figure 1

Laboratory flume.

Figure 2 shows the main geometric parameters of a type-A TPKSW. The one- or two-cycle TPKSWs were made of 12-mm thick polyvinyl chloride plate. The ratio of inlet to outlet crest length was 1 (). The total width () of all TPKSWs was 0.25 m. The TPKSWs were installed at the beginning of the intake channel.

Figure 2

Geometric parameters of a type-A TPKSW: (left) half-element unit in plan view; (right) elevated view (Cicero et al. 2013).

Figure 2

Geometric parameters of a type-A TPKSW: (left) half-element unit in plan view; (right) elevated view (Cicero et al. 2013).

Table 1 shows the range of experimental variables for the 324 test runs conducted to determine the discharge coefficient of TPKSW. Three CRSWs (0.25 m long, 0.012 m thick) and 54 test runs were conducted to estimate the discharge coefficient of the CRSWs.

Table 1

Range of experimental variables

Parameter symbolDefinitionUnitValueNon-dimensional parameterValue
 Number of cycles – 1, 2 Nu 1, 2 
δ Side-wall angle of the TPKSWs degree 3, 5, 7 sin δ 0.052, 0.087, 0.122 
β Angle of bend curvature (angle in which the intake is positioned) degree 60, 105, 150 sin (β/4) 0.259, 0.442, 0.609 
B Upstream–downstream length of the TPKSW 0.34 B/P 2.125, 2.429, 2.833 
P Weir height 0.12, 0.14, 0.16 
y1 Flow depth at the upstream end of the side weir and the centre line of the main channel 0.15–0.22 P/y1 0.73, 0.763, 0.8 
Fr1 Froude number at the measuring point of  – 0.35, 0.5 Fr1 0.35, 0.5 
Number of runs 324 
Parameter symbolDefinitionUnitValueNon-dimensional parameterValue
 Number of cycles – 1, 2 Nu 1, 2 
δ Side-wall angle of the TPKSWs degree 3, 5, 7 sin δ 0.052, 0.087, 0.122 
β Angle of bend curvature (angle in which the intake is positioned) degree 60, 105, 150 sin (β/4) 0.259, 0.442, 0.609 
B Upstream–downstream length of the TPKSW 0.34 B/P 2.125, 2.429, 2.833 
P Weir height 0.12, 0.14, 0.16 
y1 Flow depth at the upstream end of the side weir and the centre line of the main channel 0.15–0.22 P/y1 0.73, 0.763, 0.8 
Fr1 Froude number at the measuring point of  – 0.35, 0.5 Fr1 0.35, 0.5 
Number of runs 324 

Theoretical considerations

The flow over a side weir falls within the class of spatially varied flow. Henderson (1966) presented the differential equation of spatially varied flow with decreasing discharge as: 
formula
(1)
where, y is flow depth at the centre line of the main channel, x is longitudinal axis, is longitudinal slope of the main channel, is energy grade line, Q is main channel discharge, is diverted discharge per unit length of the side weir, A is cross-sectional area of the flow in the main channel, g is gravitational acceleration and b is main channel width. The discharge equations based on the total width () or total developed length () of the side weir are defined as: 
formula
(2)
 
formula
(3)
where, is diverted discharge, is discharge coefficient related to the total width of the side weir, is flow depth at the upstream end of the side weir on the centre line of the main channel, P is height of the side weir and is discharge coefficient for the total developed length of the side weir.
In this research, has been used to compare the discharge coefficients of the TPKSWs because L influences their construction cost. has been used to compare the discharge coefficient of the TPKSWs and CRSWs because L and W are equal in CRSWs. The discharge coefficient of the TPKSWs in the curved channel can be described as: 
formula
(4)
where, is number of cycles, is side-wall angle of the TPKSWs, B is upstream–downstream length of the TPKSWs, is angle of bend curvature, is average flow velocity at the measuring point , is deviation angle of flow, is dynamic viscosity of water, is surface tension of water and is mass density of water.
If is formulated using non-dimensional parameters, the number of variables and number of test runs decreases and the test runs are easier to carry out. Equation (4) can be simplified using the Buckingham theorem as: 
formula
(5)
where, is upstream Froude number, is Reynolds number (ratio of inertia force to viscous force) and is Weber number (ratio of inertia force to surface tension force). Because the flow is turbulent and the inertia force is more effective than the viscous force, Re can be removed (Henderson 1966). If the upstream total head is greater than 30 mm, the scale effects due to surface tension can be neglected; thus, can be dropped from the formula (Erpicum et al. 2016). The ψ value, which depends on , changes along the side weir (Agaccioglu & Yüksel 1998). By considering , ψ can be eliminated from Equation (5) and a non-dimensional equation for can be obtained as: 
formula
(6)
The independent variables in Equation (6) were considered to be experimental variables and the test runs were planned based on them. After running the tests, a regression function was fitted to the experimental data using a non-linear regression module in Statistica software. The independent variables of Equation (6) and algebraic products of all their pairs produced 27 independent variables for the regression function. The t-values and p-values were calculated for the independent variables of the regression function. Independent variables with |t-value| < 2 and p-value > 0.05 were eliminated from the regression function. The elimination of these variables did not considerably affect the accuracy of the regression function. The independent variables with higher t-values had a greater effect on . The following error indices were calculated to estimate the precision of the regression function: 
formula
(7)
 
formula
(8)
 
formula
(9)
 
formula
(10)
 
formula
(11)
where, C is estimated using the regression function, O is observed in laboratory tests, is mean absolute percentage error, is mean overall error, is maximum relative error, is mean value of the observed to estimated and is coefficient of determination.

RESULTS AND DISCUSSION

The experimental data were analyzed to understand the flow characteristics of the TPKSWs situated on a curved channel under free and subcritical flow conditions.

Variation of with

Figure 3 shows variation of versus . The one-cycle TPKSW had a wider inlet key than the two-cycle TPKSW; thus, the flow velocity decreased in the inlet key and was better distributed over the crests of the inlet key and sidewalls. The one-cycle TPKSW had a wider outlet key than the two-cycle TPKSW. Thus, overcrossing of the spilling jets of the side-walls occurred later in the outlet key. For these promising reasons, the of the one-cycle TPKSW was greater than that of the two-cycle TPKSW. It should be noted that reflects the effect of developed crest length on the discharge coefficient and all TPKSWs had the same total width and upstream–downstream length.

Figure 3

versus .

Figure 3

versus .

Comparison of of TPKSWs and CRSWs

The values for the TPKSWs and CRSWs are plotted versus the angle of curvature in Figure 4. The discharge coefficient of the TPKSW is 1.7 to 5.6 times greater than that of the CRSW. Because the of the TPKSWs is affected by more parameters than that of the CRSWs, the of the TPKSWs has a far greater variation range. The values are similar at β = 60° and β = 150°. However, the value decreases at β = 105°. Coşar & Agaccioglu (2004) similarly tested CTSWs located in a curved channel and found that the values of CTSWs for β = 60° were greater than β = 90°.

Figure 4

in the TPKSWs and CRSWs versus β.

Figure 4

in the TPKSWs and CRSWs versus β.

Empirical equation of discharge coefficient

An empirical equation was written to determine the discharge coefficient () of TPKSWs in a curved channel based on the experimental data using the non-linear regression module. The correlation of 27 independent variables and a dependent variable () was calculated and the independent variables having |t-value| < 2 and p-value > 0.05 were excluded from the estimation steps of the regression function. The number of independent variables was decreased to eight and the empirical equation was obtained as follows. Details of the estimated coefficients () for Equation (12) are shown in Table 2. 
formula
(12)
Table 2

Details of estimated coefficients () and other statistical parameters for Equation (12)

VariableCoefficient
t-valuep-value95% confidence interval
NameValueLowerUpper
constant  +0.44860 +19.21 0.00E + 0 +0.40265 +0.49455 
  −1.65210 −19.13 0.00E + 0 −1.82202 −1.48217 
  +1.98853 +21.10 0.00E + 0 +1.80310 +2.17396 
  −0.06684 −3.71 2.49E − 4 −0.10233 −0.03135 
  −0.08730 −20.23 0.00E + 0 −0.09580 −0.07881 
  +0.09286 +3.09 2.20E − 3 +0.03368 +0.15203 
  +0.12468 +5.62 4.27E − 8 +0.08101 +0.16836 
  +0.34133 +23.88 0.00E + 0 +0.31321 +0.36946 
  −0.13594 −9.47 0.00E + 0 −0.16419 −0.10769 
VariableCoefficient
t-valuep-value95% confidence interval
NameValueLowerUpper
constant  +0.44860 +19.21 0.00E + 0 +0.40265 +0.49455 
  −1.65210 −19.13 0.00E + 0 −1.82202 −1.48217 
  +1.98853 +21.10 0.00E + 0 +1.80310 +2.17396 
  −0.06684 −3.71 2.49E − 4 −0.10233 −0.03135 
  −0.08730 −20.23 0.00E + 0 −0.09580 −0.07881 
  +0.09286 +3.09 2.20E − 3 +0.03368 +0.15203 
  +0.12468 +5.62 4.27E − 8 +0.08101 +0.16836 
  +0.34133 +23.88 0.00E + 0 +0.31321 +0.36946 
  −0.13594 −9.47 0.00E + 0 −0.16419 −0.10769 

As was stated before, the independent variables having higher t-values have the greatest effect on . Hence, has the greatest effect on . Then, , and affect the . Other parameters have smaller effects on . The parameter is repeated in four of the independent variables. Thus, the number of cycles () also has a significant effect on . The error indices for Equation (12) are computed using Equations (7)–(11). The , , values are 5.07%, 1.86% and 16.91%, respectively. MR and are 1.0006 and 0.9478, respectively.

CONCLUSION

In this study, experiments have been performed to determine the discharge coefficient of one- and two-cycle type-A TPKSWs in a curved channel. The TPKSWs had the same total width and upstream–downstream length. The following conclusions have been extracted based on the results of this study:

  • A dimensionless equation has been developed for of the TPKSW in a curved channel. The predicted results by this equation are shown to be very satisfactory. The equation is valid for . To increase the applicability of the equation, it is suggested to perform additional experiments for a wider range of parameters.

  • In a curved rectangular channel, the of TPKSWs is 1.7 to 5.6 times greater than that of CRSWs.

  • Under the same conditions, the of a one-cycle TPKSW is 1.4 to 2 times greater than that of a two-cycle TPKSW.

Future studies should be performed on other types and other plan-views of PKSW. In design of channel intakes and diversion structures, the use of TPKSWs in the outer wall of the curved channel provides the maximum possible discharge capacity. Results of this study in curved channels could be applied to meandering rivers in flat plains.

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