This study aims to investigate experimentally the variation of the coefficient of discharge Cd with the rectangular notch hydraulic and geometric parameters such as water head h, notch height p, notch width B, and notch thickness t. The results show that the coefficient of discharge Cd increases with an increase of h and B while it decreases with t. There are no changes in the variation of actual discharge Qact and consequently the discharge coefficient Cd with h for notch height p more than 6 cm. An empirical formula was developed based on the dimensional analysis principle that can be used to predict the coefficient of discharge Cd value for the rectangular notch with known hydraulic and geometric data (h, B, p, and t).

• There are no Qact changes and thus the discharge modulus Cd with h for crack height.

• The relationship between the theoretical discharge Qth and vertex h to be constant for the thickness of the rectangle Cd increases with increasing h/p because the actual discharge Qact increases with water head h.

• The discharge modulus Cd decreases with increasing slit thickness ratio t/p.

A notch is defined as an obstruction in the open channel over which water flows and is considered the simple, accurate, and classical device used both in the field and in the laboratory for flow measurement in open channels based on its geometry and head on its crest (Kumar et al. 2011). It consists of a plate set perpendicular to the flow in a rectangular channel. The horizontal crest of the notch crosses the full channel width. This feature means that the flow is essentially two-dimensional, without lateral contraction effects (Henderson 1966). Many relationships between the head and discharge for notches were developed. Generally, the discharge Qact over a sharp-crested notch under free-flow conditions in an open channel is expressed in terms of the following well-known equation (Henderson 1966).where B is the notch sill width, Cd is the discharge coefficient, h is the head over the notch sill, and g is the acceleration due to gravity. The coefficient of discharge Cd depends on flow characteristics and notch geometry. The earliest experimental studies on Cd were carried out by Rehbock (1929). Experiments were carried out on full-width weirs and proposed the following equation for h/p ≤ 5, which does not reflect the viscous and surface tension effects:where p is the notch height.
An intermediate zone is a continuous transition between two equations. Swamee (1988) developed a generalized weir equation for sharp-crested, narrow-crested, broad-crested, and long-crested weirs by combining the equations obtained from earlier works as follows:where L is the notch length in the direction of flow. Oshima et al. (2013) adopted the Rehbock equation for flow calculation of full-width weirs with the limitation on weir plate height set to 1 m, and the coefficient of discharge changed slightly as follows:

Kumar et al. (2011) conducted an experimental study on a sharp-crested weir under free-flow conditions and developed a discharge coefficient equation that is similar to the Kindsvater & Carter (1957) equation. Reviewing most of the proposed equations show that Cd primarily depends on the ratio h/p. Other flow characteristics may influence the discharge coefficient.

Zachoval & Rousar (2015) studied the flow characteristics over a broad-crested weir using numerical models. They found that Reynolds-averaged Navier–Stokes (RANS) equations and the two-layer shear stress transport (SST) turbulence model were suitable models. Results of their study show that numerical simulation using the Reynolds stress turbulence model gives better predictions for horizontal velocities than simulations with other turbulence models. Ghorban & Hadi (2018) experimentally examined the effect of h/y and Re on the Cd value of a rectangular sharp-crested weir and developed a discharge coefficient equation using the optimization method. They also conducted a numerical simulation to evaluate the ability of the numerical model and analyze the flow characteristics of the notch.

Advanced numerical and experimental studies were used to investigate hydraulic phenomena. Liu et al. (2002) studied numerically the water surface profile on semi-circular notches using the k-ε turbulence model. Aydin et al. (2011) after their experimental studies proposed that the discharge in rectangular weirs can better be formulated in terms of average weir velocity, which has a universal distribution easy to fit empirically, rather than the discharge coefficient which exaggerates the experimental error by changing the curvatures. The study also proposed that for precise measurement of h, the maximum velocity in the channel should be limited to 0.55 m/s. Ferro (2012) examined the geometrical shapes of sharp-crested weirs. A stage-discharge relationship was developed for triangular sharp-crested weirs using dimensional analysis and the self-similarity theory. He concluded that a power equation can be used for establishing the stage-discharge equation with a coefficient and an exponent depending upon the weir geometry. Aydin et al. (2014) introduced a physical quantity known as weir velocity, i.e. the average velocity over the weir section, which is directly formulated as a function of weir geometry and head over the weir. The weir velocity plotted against the weir head has a universal behavior for constant weir width to channel width ratio which is independent of weir size. This unique behavior is described in terms of weir parameters to calculate the discharge without involving the discharge coefficient. Akoz et al. (2014) carried out experiments to measure the flow characteristics over a semi-cylindrical notch and compared them with those obtained numerically. Bin Shaharin (2013) showed in a numerical study that the important variable governing discharge over sharp-crested weir was the water head over weir per weir divided by weir height, h/P. He also highlighted the advantages of an ANSYS CFX-14 as a tool for examining velocity vectors and pressure patterns over rectangular sharp-crested weirs. In an experimental study, Zbyněk et al. (2014) determined a relationship for the calculation of the discharge coefficient at free overflow over a rectangular sharp-edged broad-crested weir without lateral contraction. The developed formula, expressed using the relative height of the weir, was subjected to verification made by an independent laboratory confirming its accuracy. Alwan and Al- Mohammed (2018) used a dimensional analysis technique to estimate the values of the coefficient of discharge for various rectangular notch dimensions and developed an empirical formula to estimate the discharge coefficient using a regression procedure.

Eltoukhy & Alsaydalani (2021) carried out experimental runs to study the notch thickness on the discharge coefficient for V-notch. Formulas for predicting the V-notch discharge coefficient, Cd, were developed for different vertex angles, Ɵ, and then the predicted values of the discharge coefficient, Cd, using the developed formulas were plotted against the calculated values with a coefficient of determination (R2 = 0.9372), showing a good agreement between the predicted and measured values.

There are no studies that deal with the rectangular notch thickness effect on the discharge coefficient. In this study, experimental runs examined the effect of the rectangular notch hydraulic and geometric data as water head h, notch width B, notch height p, and notch thickness, t on its discharge coefficient Cd value. Based on the analysis of the experimental results with the use of the dimensional analysis principle a new empirical equation is developed for predicting the coefficient of discharge Cd for given rectangular notch data (h, B, p, and t).

The experimental runs were carried out using a rectangular flume 4.0 m long, 0.30 m wide, and 0.50 m height. The Nontilting type was used in the experimental runs. The experimental setup is a self-contained one having a closed cycle for the water. A three-horse power pump makes the water circulating mechanism work efficiently in the flume. The flume is installed with glassy sheets as side walls which make the viewing of the experimental run easy. At the entrance into the channel, baffle vertical plates were installed to prevent vortex motion and regulate the flow to control the damp fluctuations at the entry of the flume. The water after the notch was then collected into a hydraulic bench (Figure 1). Actual discharge was calculated using the hydraulic bench by dividing the collected water volume in the hydraulic bench by the corresponding time. For each experimental run, the actual discharge was calculated as the average of three recording discharge values. A vernier-type gauge with accuracy ± 1 mm was used for measuring the bed elevation and water surface elevation. Calibration was done before every experimental run to avoid instrumental errors. The depth rod was adjusted accurately to the surface of the water to get the value of ‘h’. While measuring h it was ensured that the flow in the channel was stable and constant. Discharge is maintained for an individual experimental run. Rectangular notch plates made up of fiberglass with different thicknesses were used for the experimental study. Different notch sections were selected having different sill widths B, notch thickness t, and the notch crest height P. The used notches have sill widths B of 3, 4, 6, and 8 cm, notch thicknesses t of 1, 3, 4, and 6 mm and notch crest heights P was varied as 4, 6, 8, and 10 cm only for t = 1 mm and B = 3 cm for estimating P that has a negligible effect on the discharge coefficient Cd. According to Bos (1989), the minimum notch height P was suggested as 10 cm.

Once the calibration process was completed the accuracy of discharge measurement depends on the measurement of water level h on the upstream side of the notch. The point gauge with a vernier scale having accuracy ± 1 mm was used for measuring the water level. The point gauge was fixed at an upstream distance of four times the maximum head over the notch (Bos 1989). Because the bottom boundary affects the nature of flow crossing through the weir section should be a free-flow condition, therefore notch section is used for discharge measurement and discharge can be determined by measuring the head over the notch.

### Rectangular notch models

Sixteen rectangular notch models were fabricated from acrylic glass sheets with different sill widths B and thicknesses t, Table 1. Figure 2 shows the form for each model, where h is the water head, H is the height of the notch plate above the sill level, and P is the notch height. Four rectangular notches with sill widths B of 3, 4, 6, and 8 cm each with thicknesses of 1, 3, 4, and 6 mm were used in this study on different models. The experimental programs are indicated in Table 1.
Table 1

The experimental programs

Sill width, B cmNotch plate thickness, t mmNotch height, p cmWater heads h cm
4, 6, 8, and 10 Six head values
3, 4, and 6
1, 3, 4, and 6
1, 3, 4, and 6
1, 3, 4, and 6
Sill width, B cmNotch plate thickness, t mmNotch height, p cmWater heads h cm
4, 6, 8, and 10 Six head values
3, 4, and 6
1, 3, 4, and 6
1, 3, 4, and 6
1, 3, 4, and 6

### Tests procedure

The Experimental runs procedures for each rectangular notch model were carried out as follows:

• 1.

Installing the notch model in its allocated position at the channel of a hydraulic bench.

• 2.

Adjust the control valve to establish the flow rate pumped to the bench.

• 3.
After developing a stable flow, head over notch, h was measured about 4hmax away from the upstream of the weir where max is the maximum head over the weir (Franzini & Finnemore 1997), and theoretical discharge was calculated,
• 4.

Recording the volume of water, V accumulated in the bench tank over time, T for each run and actual discharge was calculated, Qact=V/T.

• 5.

Determining the coefficient of discharge, Cd = Qact/Qth

• 6.

Repeat the procedures from points 2 to 5 for further runs.

• 7.

Repeating steps 2–6 for other notch models as indicated in Table 1.

The discharge coefficient, Cd of the rectangular notch is a function of several parameters which is mathematically expressed by Equation (2):where Cd is the discharge coefficient, ρ is the water density, σ is the surface tension, μ is water viscosity, h is the head over the notch sill, B is the notch width, p is the notch height, t is the notch plate thickness, and g is the gravitational acceleration. A dimensional analysis is performed to find a relation between the discharge coefficient and other parameters stated above:where Re is the Reynolds number and We is the Weber number. In most practical cases, however, the Reynolds and Weber numbers effects are negligible for water at normal temperatures and notch geometry is the main element.

Before the commencement of the experimental runs, the volume of water, V accumulated in the bench tank over time, T which was used to calculate the actual discharge as Qact=V/T, was calibrated with the use of a graduated jar. The calibration process was carried out three times for the same head h and the average water volume and corresponding time values were compared with V and T. The results were consistent as shown in Table 2.

Table 2

Calibration of the bench volume tank, t = 1 mm, p = 8 cm, and B = 3 cm

h (cm)V (l)T (s)Jar vol. (l)Jar time (s)Qact (l/s)Jar Qact (l/s)Error
2.74 15.85 16.17 0.189274 0.1855 2%
3.75 9.63 9.84 0.311526 0.3047 2.2%
4.88 6.28 6.39 0.477707 0.4691 1.8%
h (cm)V (l)T (s)Jar vol. (l)Jar time (s)Qact (l/s)Jar Qact (l/s)Error
2.74 15.85 16.17 0.189274 0.1855 2%
3.75 9.63 9.84 0.311526 0.3047 2.2%
4.88 6.28 6.39 0.477707 0.4691 1.8%

### Notch height p

First of all, for the prediction of discharge coefficient, Cd for the rectangular notch with different thicknesses, 24 experimental runs were carried out to estimate the notch height, p which is used for achieving the purpose of this study. Notch models with thickness, t of 1 mm, notch height, p values of 4, 6, 8, and 10 cm, and six head, h values for each p were used for p estimating. The actual discharge, Qact – head relationships were presented in Figure 3, which shows that the actual discharge, Qact increases with the head, h. Also, it was shown that there are no changes in the variation of Qact and h for p more than 6 cm i.e. p = 8 and 10 cm. The notch height, p = 8 cm was selected to diminish the effect of the bottom boundary. So the discharge coefficient, Cd will become independent of the value of h/p. This is consistent with Sisman's (2009) and Bos's (1989) observations. Thus, any p-value greater than the recommended value will hydraulically imply that the flow over the notch is no longer relying on the height of the notch. In addition, it is realizable that the chosen p may remain valid for the experimental range of water heads only. Above this range, it can be expected that larger notch plate heights might be required to suppress boundary layer development.

### Actual and theoretical discharges variation with head

Once the notch height was decided to be kept at 8 cm, experimental runs continued with different notch widths. There were four different notch widths were tested in this study (B = 3, 4, 6, and 8 cm) and results were recorded. Figure 4 shows the obtained results points for actual Qact and theoretical Qth discharges at different water heads, different notch widths B, and notch plate thickness t. All discharge–head relationships have the same trend and the theoretical discharge Qth is independent of t, as in Equation (6). On the other hand, the Qth and Qact increase as the notch width B increase. The theoretical discharge Qth increases with B with the same increasing ratio. For example at the same head h and notch thickness t, increasing B from 4 to 6 cm (50%) the Qth at B = 6 cm equals 6/4 × Qth at B = 4 cm (50%). But the actual discharge Qact increases with a different ratio, for example at h = 3.8 cm and t = 1 mm, increasing B from 4 to 6 cm (50%), the Qact increases from 0.810811 to 0.898356 l/s (10.79%).

### Variation of the discharge Cd coefficient with h/p

Figure 5 shows the variation of the discharge coefficient Cd with h/p for different notch widths B and thicknesses t. It can be seen that Cd which equals Qact/Qth changes abruptly with changes in h/p. At the same notch width B and notch thickness t, Cd increases as h/p increases this is because the actual discharge Qact increases with water head h with a value more than that of the theoretical discharge Qth. For example for B = 4 cm and t = 1 mm, for increasing h/p from 0.2588 to 0.4788 (0.85%), Qact increases from 0.293 to 0.81081 l/s (177%), and Qth increases from 0.5276 to 1.328 l/s (152%).

### Effect of the notch thickness ratio t/p on the discharge coefficient Cd

Throughout the analysis of the obtained results of the experimental runs using the different notch models with thicknesses of 1, 3, 4, and 6 mm, the effect of the notch thickness ratio t/p on the discharge coefficient Cd is presented in Figure 6. It can be seen that the discharge coefficient and the notch thickness ratios have the same trend for different notch widths, where the discharge coefficient decreases as the notch thickness ratio increases. For example, at h/p = 0.512, increasing the notch thickness t by 100% i.e. from 3 to 6 mm results in decreasing in the actual discharge Qact by 17.22, 13.08, 21.13, and 10.9% for notch width B of 3, 4, 6, and 8 cm, respectively, and decreasing the discharge coefficient Cd by the same ratios of the actual discharge decreasing. This is because the theoretical discharge has a constant value at a given head for different notch thicknesses.

### Discharge coefficient based on dimensional analysis technique

Now the effect of the notch thickness t was checked on the discharge coefficient Cd. Then, the results from the four notch thicknesses were analyzed. Using nonlinear regression analysis and all the observed data, the relation for Equation (8) can be found and the suggested relation is shown in Equation (9) (Figures 7 and 8).

A series of experiments were carried out to investigate the effect of water head, h, and rectangular notch geometry (width B, height p, and thickness t) on the discharge coefficient Cd values. Based on the analysis of the experimental run results of this study the following conclusions were obtained:

• There are no changes in the variation of actual discharge Qact and consequently the discharge coefficient Cd with h for notch height p more than 6 cm, i.e. p = 8 and 10 cm.

• Theoretical discharge Qth – head h relationship was found to be constant for different rectangular notch thicknesses t but the actual discharge Qact – head h relationship varies with the notch thickness.

• Theoretical discharge Qth increases with the same percent as the rectangular notch width B increases, but the actual discharge increases with different percent.

• At the same notch width B and notch thickness t, Cd increases as h/p increases; this is because the actual discharge Qact increases with water head h with a value more than that of the theoretical discharge Qth.

• The discharge coefficient Cd decreases as the notch thickness ratio t/p increases.

• The developed empirical formula Equation (9) can be used for predicting the discharge coefficient Cd value for the rectangular notch with known hydraulic and geometric data (h, B, p, and t).

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

The authors declare there is no conflict.

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