Experimental and field verifications of radial gates as flow measurement structures Open Access

c₁, c₂, c₃, c₄

Constant parameters (−)

a

Relative gate opening (a = w/y₀) (–)

b

Gate width (m)

C_d

Discharge coefficient (−)

F

Approach Froude number (F = Q/(B.√(gy₀³))) (−)

f

f = Q/(B.√(g.w³))

F_f

Calculated value of approach Froude number from Equation (9) (−)

g

Acceleration due to gravity (m/s²)

k

Calibration coefficient (−)

N

Number of tests (−)

Q

Discharge (m³/s)

R

Gate arm radius (m)

S_r

Submergence ratio (–)

Y

Trunnion-pin height (m)

y_c

Critical depth (⁠⁠)(m)

y_j

Thickness of the vena contracta (m)

y₀

Upstream depth (m)

y₂

Flow depth just after the gate (m)

y_2L

Flow depth at the immediate section of the gate under transitional flow condition (m)

y_t

Tailwater depth (m)

y_tL

Maximum downstream depth for which the free flow condition occurs (m)

w

Gate opening (m)

α

Relative flow depth just after the gate (α = y₂/y₀) (−)

γ

Relative tailwater depth (γ = y_t/y₀) (−)

γ*

Relative tailwater depth under transitional flow conditions (γ* = y_tL/y₀) (−)

δ

Contraction coefficient (−)

Θ

Gate lip angle (radians)

Δ

Discriminant in Equation (12) (−)

Δ′

Discriminant in Equation (18) (−)

It seems that the key factor in saving irrigation water is to adjust the adequate water supply in proportion to the needs of the product. This is possible by accurately measuring the flow discharge and controlling the water level in irrigation channels. Therefore, measuring water in irrigation canals is essential for precision irrigation management.

Radial gates are common check structures and are frequently used in irrigation networks. Figure 1 shows some typical applications of these structures installed in the Dez irrigation network, Khuzestan province, Iran. They are used to regulate the water level and flow discharge in irrigation canals. Figure 2 depicts the characteristics of flow through a radial gate under free and submerged flow conditions. The discharge (Q) is a function of upstream and downstream geometries such as canal width (B), flow depths both at the upstream (y₀) and downstream of the gate (y_t), flow condition either free, submerged or transition, and geometric characteristics of the gate, such as gate radius (R), gate width (b), trunnion-pin height (Y) and gate opening (w).

An accurate estimate of flow discharge under radial gates is a classic problem in hydraulic engineering. Many studies are available to formulate the discharge of radial gates under free and submerged flow conditions (Buyalski 1983; Clemmens et al. 2003; Wahl 2005, 2011; Shahrokhnia & Javan 2006; Clemmens & Wahl 2012; Zahedani et al. 2012; Bijankhan et al. 2013; Abdelhaleem 2017). The proposed discharge curves by Toch (1955) are applicable to sharp-edged gates with only three relative heights of trunnion pin (Y/R). Clemmens et al. (2003) and Wahl (2005) presented an implicit (iterative) solution for estimating radial gate discharge under submerged flow conditions. In the studies of Toch (1955), Clemmens et al. (2003), Shahrokhnia & Javan (2006) and Zahedani et al. (2012), the type of gate seal was not considered in flow discharge modelling, and for each kind of gate (hard rubber bar, sharp-edged, and music note), a separate equation is proposed. Moreover, the proposed equations by Shahrokhnia & Javan (2006) and Zahedani et al. (2012) have been derived for calculating the submerged flow rate based on the flow depth immediate downstream of the gate (y₂), which may be subject to some complexities in measurement. In the studies of Buyalski (1983), Shahrokhnia & Javan (2006), Bijankhan et al. (2013), and Abdelhaleem (2017), the proposed discharge equations are regression-based, which could only be applicable for interpolation in the range of calibration and may result in considerable errors if extrapolation is needed.

A review of the reported works concerning flow measurement using radial gates shows that a general equation (with acceptable accuracy, simplicity in form, including all flow characteristics and gate dimensions) is lacking to determine the flow discharge of radial gates. Moreover, there are limited studies available regarding the distinguishing condition curve of radial gates. Bijankhan et al. (2011) reported three-fold errors in estimating discharge due to a lack of knowledge about the flow conditions. Since most of the previous approaches are based on the momentum equilibrium downstream of the gate and take benefit from the trial-and-error method, Bijankhan et al. (2011) suggested a new approach to identify the flow condition using the intersection of rating curves under free and submerged conditions. However, this implicit approach needs trial-and-error procedures. Also, under the transition zone, the flow depth immediately after the gate was assumed constant (y_2L = 1.035 × y_j) for all combinations of gate seal type and gate lip angle, requiring more clarifications.

Radial gates may be designed in parallel across the downstream channels (Figure 1(a)). In this situation, some gates may be open, while others are closed (Figure 1(b)). Due to the difference in the gate openings, some gates may be free or submerged while others operate under transition flow conditions. Abdelhaleem (2016) used dimensional analysis to apply the incomplete self-similarity concept to estimate the discharge of submerged parallel radial gates. He introduced some constant parameters in his work, which were calibrated based on the field measurements in three control structures in the Delta irrigation district in Egypt. However, most of the previous methods are experimentally calibrated for an individual radial gate. Consequently, more studies are needed to evaluate the capability of prior discharge estimation methods for parallel radial gates.

In this study, firstly, the energy equation was used between the upstream and immediate contracted section of the gate and combined with the momentum equation between the immediate downstream and the tail section of the gate, with some simplifying assumptions. Then, the empirical values of the constant parameters are presented based on the experimental data. The accuracy of the proposed method was assessed in determining the discharge under free and submerged flow conditions. Under the flow limit conditions, the capability of the proposed method was carefully considered, and a novel solution is presented to decrease the discharge estimation errors. The paper also compares the accuracy of the different methods in estimating the discharge of radial gates and flow conditions. The effects of gate lip angle and gate seal type on the distinguishing condition curve and the discharge of the radial gates have also been evaluated. Finally, the field application of the proposed equations has been considered.

The experimental data by Buyalski (1983) were used to calibrate the proposed equations related to outflow from the radial gates. Buyalski (1983) performed 2,657 experiments on three types of radial gates, namely hard rubber bar, sharp-edged, and music note. In his experiments, the gate width was kept constant at 711 mm, while the downstream channel width was 762 mm. Each gate was installed at three different trunnion-pin heights. The experiments were conducted under three different flow conditions, which he called free, submerged, and jump (assumed to be the transition zone) flow. Table 1 shows the range of Buyalski's (1983) experimental observations. In each case, 80% of the data sets were randomly selected to determine the constant parameters and the remaining 20% were used to validate the proposed equations.

Since Buyalski (1983) performed his experiments for limited Y/R ratio values, additional tests have been conducted to extend the range of Buyalski's (1983) data domain. These experiments were performed in a long Plexiglas flume located at the hydraulic laboratory of the Irrigation and Reclamation Engineering Department, University of Tehran. The flume had a width of 0.973 m, and was 1.2 m in height and 18 m in length. All tests were conducted on a sharp-edged gate. The gate arm radius and width were 1.0 m and 0.972 m, respectively. An adjustable Lopac gate was used to regulate the tailwater depth. The upstream water depth (y₀) and downstream water depth (y_t) were measured using two gauges, with an accuracy of 0.1 mm, at 1.5 m upstream and 9.0 m downstream of the gate seal position, respectively. At this downstream distance, the flow was fully developed. During the experiments, the flow depth just after the gate (y₂) was measured. The discharge was measured by an electromagnetic flow-meter, which was installed in the feeding pipe. In total, 1,046 tests were performed on a wide range of flow conditions, gate lip angles, Y/R ratios, approach Froude numbers (F), and relative upstream and downstream depths (y₀/w and y_t/w).

Table 1 compares the ranges of Buyalski's (1983) data and those of this study. The data from this study covers the upper and lower ranges for the ratio of Y/R (0.388 ≤Y/R≤ 0.845). Some data were collected in the lower range of the gate lip angle (35.36° ≤θ < 49.01°). Figure 3 compares the range of discharge coefficients (C_d = Q/(B.w√(2 g.y₀)) determined from Buyalski (1983) and additional experimental data at equal values of θ and Y/R. As noted from the figure, the data of the present study are located in the lower range of f = Q/(B.√(g.w³)) and y₀/w. Consequently, the data of this study extend the range of Buyalski's (1983) experiments to the sharp-edged gates.

Under the orifice flow regime, three hydraulic flow conditions (free, transition, and submerged) can be observed at a gate located in an open channel. However, the discharge error estimation may be increased under three flow conditions, so-called flow limit conditions, as the following (Figure 4): (a) the transition between the orifice and non-orifice flow regimes, which is named the gate lip flow (1 ≤y₀/w≤ 2) in which the downstream water depth firstly hits the gate lip, (b) the transition between free and submerged flows ((−2.5% ≤S_r≤ +2.5%), where S_r is the submergence ratio = 100 × (y_t − y_tL)/y_tL, and y_tL is the transitional value of the tailwater depth) in which the downstream water fluctuations due to the hydraulic jump will be transmitted to the gate lip, and (c) high submerged flow (y_t/y₀ ≥ 0.95).

Equations (1) and (2) were based on the following assumptions:

• (1)

  energy conservation between sections 1 and 2;

• (2)

  momentum conservation between sections 2 and 3;

• (3)

  presence of rectangular sections both upstream and downstream of the gate;

• (4)

  equal width for upstream canal and gate;

• (5)

  hydrostatic pressure distribution at sections 1, 2, and 3;

• (6)

  uniform flow velocity distribution at the vena contracta section;

• (7)

  zero roller velocity at the vena contracta section and uniform flow velocity distribution at the downstream section allowing the use of a power-law velocity distribution at this section (Castro-Orgaz et al. 2013); and

• (8)

  the drag force between sections (2) and (3) is neglected.

One fundamental assumption in the development of explicit solutions is an equal width of the gate and the downstream channel. This assumption limits the application of equations for the field cases in which the downstream channel may be wider than the gate. However, it is possible to provide explicit solutions when the gate and downstream channel widths are different.

In this equation, the angle of θ is in radians. Table 2 shows the empirical values of constants as a function of the gate seal types and flow conditions, which were determined using Buyalski's (1983) data.

Buyalski's (1983) data were combined with observations of this study to extend the application of the proposed equations for estimating the contraction coefficient of the radial gates, where the values of c₁, c₂, c₃, and c₄ have been recalibrated for the sharp-edged gates.

Table 4 shows the values of the parameters as a function of flow conditions using Buyalski's (1983) data and the present study. Table 5 presents the accuracy of the proposed method with new calibrated constant parameters for the δ function under the different flow conditions.

Based on the data from Buyalski (1983) and the present study, the proposed method for sharp-edged gates showed MARE values of 1.4% and 2.8%, respectively, under free and submerged flow conditions (Table 5). However, the proposed equation with calibrated parameters only based on the data from Buyalski (1983) estimates discharge with MARE values of 1.1% and 1.2%, respectively, for the free and submerged flow conditions (Table 3). Consequently, it is recommended to use the constant parameters in Table 2 for the discharge estimation of sharp-edged gates with more accuracy under the limited range of Buyalski's (1983) data.

Tables 3 and 5 show an increase in the MARE values for the discharge estimation of radial gates under jump flow conditions. Figure 5 presents the variation of MARE in the discharge estimation of radial gates using the proposed method with y₀/w and y_t/y₀. By increasing the ratio of y_t/y₀, the discharge of the radial gate decreases, and the values of MARE computed by the proposed method considerably increase. As shown in Figure 5(a), the MARE for discharge estimation of the hard rubber bar gate is about 2.5% for the ratio of y_t/y₀< 0.95 (and C_d > 0.26). However, the MARE increases by approximately 3.5% for 0.95 <y_t/y₀< 0.99 (mean value of the discharge coefficient is about 0.16). Under high submergence i.e., 0.99 ≤y_t/y₀< 1.0), the mean value of the discharge coefficient tends to be about 0.07, which results in a significant increase in MARE up to 11.7%. It was also observed that the discharge estimation of radial gates is associated with more error when the upstream depth drops relative to the gate opening (Figure 5(b)). For the range of y₀/w > 2, the value of MARE is less than 4.25%. However, for gate lip flow (i.e., 1 <y₀/w≤ 2), the MARE increases to 6.3%, which corresponds to the transition zone between the orifice and non-orifice flow conditions.

Figure 6 compares the measured and calculated discharge of sharp-edged gates from the proposed method under three different flow limit conditions. It can be seen that the amounts of MARE for high submerged flow, transition flow, and gate lip flow are 15.5%, 6.2%, and 30.1%, respectively (Figure 6(a), 6(c) and 6(e)).

The value of MARE in discharge estimation at the three above-mentioned zones can be further reduced by using the energy principle between the upstream and downstream of the gate (using Equation (5)) and measured y₂ values. This method significantly decreases the MARE amount for discharge estimation of radial gates under high submerged flow, transition flow, and gate lip flow with about 10.6%, 2.2%, and 8.2%, respectively (Figure 6(b), 6(d) and 6(f)). As a result, it is expected that a systematic error in reading the tailwater depth during the experimental observations is of significant importance in increasing the value of MARE for discharge estimation from Equation (8), especially under high submergence. Consequently, it is recommended that a gauge is installed after the gate for direct measurement of y₂. Under the high submerged and gate lip flows, the water level after the gate has no turbulence, making possible the direct measurement of y₂ (Figure 4(a) and 4(c)). However, there are some difficulties for direct measurement of y₂ due to the gate's turbulence under the transition flow condition (Figure 4(b)). As a result, it is recommended to avoid adjusting the gates under the transition zone for field applications to reduce the discharge estimation error.

Figure 7 shows the effects of gate seal type, gate lip angle (θ), and Y/R ratio on the discharge coefficient of radial gates. The discharge coefficient (C_d = Q/(b.w√(2g.y₀)) was obtained from Equations (5) and (8) using the functional relationship between F and C_d as C_d = F/(a.√2). Figure 7 shows that the discharge coefficient decreases as θ increases or the ratio of Y/R decreases. This result is in agreement with Salmasi et al.’s (2020) observations, which showed that the gate angle causes the discharge coefficient to increase. The sluice gates are a particular type of radial gates, where R → ∞ and θ →π/2. Consequently, under certain conditions, the outflow from a radial gate is more significant than that passing through a sluice gate. It is also notable that the hard rubber bar gate conveys a higher discharge than the sharp-edged and the music note gates. Additionally, the music note gates transport more discharge than the sluice gates.

Most of the previous methods have been calibrated based on Buyalski's (1983) data. In this study, the graphical process of Toch (1955), because of multiple interpolations, is avoided. Also, Buyalski's (1983) method was eliminated because of the complex set of equations.

Tables 6 and 7 compare the capability of different methods for discharge estimation of radial gates for free and submerged flow conditions, respectively. The results were obtained based on Buyalski's (1983) experimental data on the three types of radial gates. It should be mentioned that the proposed method of the present study leads to the least error for all flow conditions with the three types of radial gates. Generally, energy–momentum methods (Clemmens et al. 2003; Wahl 2005, and WinGate by Clemmens & Wahl 2012 and the present study) give a more accurate estimation of discharge than those of the regression-based methods (Shahrokhnia & Javan 2006; Zahedani et al. 2012; Bijankhan et al. 2013). Zahedani et al.’s (2012) approach provides the least accuracy in discharge estimation. On the other hand, Wahl's (2005) equation is more accurate than previous methods due to considering the effect of gate seal type.

Under submerged flow conditions, particularly in the low submergence case, the methods which are based on the energy–momentum equations may not provide a convergent solution. Table 7 shows that the proposed method in this research and WinGate software has the maximum convergence level over those offered by Clemmens et al. (2003) and Wahl (2005).

Among the previous methods, WinGate software provides a more reliable discharge estimation for the three types of radial gates. Accordingly, WinGate software is recommended as a superior tool compared with the other previous methods.

Table 8 shows that the MARE values for WinGate software, as the selected method in the previous section, are 5.6% and 4.6%, respectively, for free and submerged flow conditions. The MARE for WinGate software at high submerged, transition, and gate lip flows increases respectively to about 16.8%, 5.2%, and 13.5%. Consequently, the WinGate software needs to be calibrated for a wide range of Y/R and θ. Hence, the proposed method in this research is recommended for the discharge estimation of sharp-edged gates in a wider range.

Table 3 shows the percentage of precise flow estimation using Equation (15) and its comparison with the observations of Buyalski (1983). Table 3 demonstrates that Equation (15) correctly predicts the flow conditions for 98.7% of the measurements in the hard rubber bar gates, 100% of the measurements in the sharp-edged gates, and 100% of the measurements in the music note gates. According to the experimental observations in the present study, Equation (15) correctly estimates the flow conditions in most cases (97%) for the sharp-edged gates (Table 5). Therefore, it can be concluded that Equation (15) is efficient in predicting the flow conditions for all three types of radial gates.

Figure 8 shows the distribution of experimental pairs (y_t/w, y₀/w) relative to the distinguishing curve defined by Equation (15) using the combined data of Buyalski (1983) and the present study. It can be seen that, in most cases, the pairs of (y_t/w, y₀/w) corresponding to free and submerged flow conditions are located in the lower and upper zone of the distinguishing curve, respectively. Among the experiments on the sharp-edged gates at the present study, several observations are related to the transition zone, which is shown with blue points in Figure 8. These points are located approximately on the distinguishing curve.

Figure 9 shows the effects of θ, Y/R, and gate seal type on the distinguishing curve. It can be seen that the gate with a more significant angle is more likely to become submerged. Hence, under certain conditions of the gate opening and upstream depth, radial gates are less likely to become submerged than sluice gates. Also, the hard rubber bar gate operates under free flow in wider ranges than the sharp-edged and music note gates.

Table 9 compares the capability of different methods to predict the flow condition based on Buyalski's (1983) data and the present study (all data set). Based on Buyalski's (1983) data, the WinGate software provides a more accurate prediction of flow conditions than the other methods. Based on the data from the present study, Equation (15) correctly predicts the flow conditions in 97% of all cases. However, in this series of data, the WinGate software identifies the flow conditions with lower precision than the others. This is due to the limited range of Y/R, and θ was taken from Buyalski's (1983) data to calibrate the WinGate software. It can be seen that the method of Bijankhan et al. (2011) provides an accurate prediction of flow conditions. However, the method of Bijankhan et al. (2011) is implicit and uses a trial-and-error procedure to determine the tailwater depth under the transition zone.

Figure 11(b) and 11(d) show that Equations (28) and (30) give the negative answers to identify the limit of the radial gates, which physically cannot be correct (⁠ and < 0). Also, it can be seen from Figure 11(a) and 11(c) that < γ_c and > γ_c. Therefore, Equation (27) is the only solution which is valid for the subcritical flow condition at the tailwater section.

Figure 8 compares the distinguishing limits from Equations (15) and (27) on the base of a classical hydraulic jump. As shown in the figure, both limits give similar results. However, the use of Equation (15) is preferable due to its simplified form.

By replacing the relative tailwater depth from Equation (27) in Equation (6), the relative flow depth just after the gate (y_2L/(δ.w)) can be estimated for transitional flow conditions. Figure 12 shows a typical variation of y_2L/(δ.w) with θ and w/y₀ for the hard rubber bar gates. As can be noted, the flow depth just after the gate increases with decreasing the relative gate opening and increasing gate lip angle. This result is an alternative to the finding of Bijankhan et al. (2011), which assumed y_2L = 1.5 (δ.w) at a transition zone.

Generally, for the transition zone, y_2L > (δ.w), as is shown in Figure 12. This result agrees with the assumptions made in concluding the classical hydraulic jump. When the tailwater depth is slightly greater than the threshold tailwater depth determined from Equation (15), the hydraulic jump moves towards the gate, while the gate still operates under the free flow condition with no changes in upstream depth. This continues to intersect the supercritical flow jet with the gate lip, where the flow becomes submerged. Therefore, it is believed that the transition occurs when the tailwater depth is slightly greater than the values revealed in Equation (15).

The proposed methods in this study are specifically developed to estimate the discharge of a radial gate with a width equal to the upstream and downstream channel. For field applications, a set of parallel radial gates may be installed and operated (Figure 1(a)). The opening of the gates may differ from each other, such as in the passage of low discharge. In this case, some gates may be open while others are closed (Figure 1(a)). Also, the difference in the gate opening in parallel applications may cause different flow conditions in each gate as free, submerged, or transition flows (Figure 1(b)). The gates may also be designed at higher or lower levels than the upstream or downstream channels or may be affected by the curvature of the approaching flow lines in a channel intake. These issues can affect the accuracy of the newly proposed methods to calculate the discharge of a radial gate.

The WinGate software has been developed to compute the flow through a complete check structure made up of multiple gates. However, a systematic study covering the field evaluation of WinGate is lacking in the literature. To investigate the effects of parameters as mentioned above on the efficiency of the proposed equations and WinGate software, some available field data from the operation of 12 check structures in the United States' irrigation canals (468 data) and 16 check structures in the Doroodzan irrigation network of Iran (65 data) were utilized. Moreover, a series of 2,240 data was collected related to some measurements since 1977 on the check structure located at the entrance of the main canal in the west of the Dez irrigation network, Khuzestan province, Iran.

Table 10 shows the physical properties of the radial gates used in the field verification. It should be emphasized that the field data were collected from a wide number of parallel gates (N) with different gate openings, seal types, sill levels relative to downstream channel bed, and flow conditions. The simplest method for estimating outflow from parallel gates is based on using the proposed equation by neglecting the effects of interaction between the parallel jets emerging from multiple gates and adjusting the water depths at upstream and downstream channels relative to the sill elevation. Table 10 shows the values of MARE and Percentage of Correct Estimation (PCE) of flow conditions in the data series. As previously discussed, the percentage of errors in discharge estimation increases when the ratio of y_t/y₀ rises to 0.9 (Figure 5(a)). In the data collection from California Aqueduct check No. 21, the ratio of y_t/y₀ changes in the range of 0.939 < y_t/y₀ < 0.98. Also, the discharge coefficient varies in the range of 0.096 < C_d < 0.199. Accordingly, significant errors are expected in the estimation of discharge with this data series, where the value of MARE for this data set was found as 18.3% (see Table 10).

In the proposed approach for deducing Equations (8) and (9), the gate was installed at the same level of upstream and downstream channels' levels. The gate sill causes an additional force on the control volume at the downstream side of the gate. It is noteworthy that overlooking the effects of the gate sill height can result in significant errors in estimating the discharge, such as for Tehama-Colusa Canal check No. 2, in which the gates are designed about 1.4 m above the downstream channel level.

When multiple gates are under operation, adjusting all gates to an equal opening would be more desirable in flow estimation, although it is difficult (Clemmens 2004). The complexity of discharge estimation from multiple gates has been mentioned by some researchers such as Clemmens et al. (2003), Clemmens (2004), Sauida (2014), and Bijankhan & Kouchakzadeh (2015). The complexity is due to the interaction between the jets emerging from the multiple gates, which cause a momentum exchange and additional shear force.

Based on all field data sets, the MARE values in estimating the discharge through the proposed equations are 5.1% (Table 10). The proposed methods in this study also predict the flow conditions at a level of confidence of 99.2%. Also, by ignoring the high submerged data from California Aqueduct check No. 21 and using other field data series from 11 check structures in the United States' irrigation canals, the MARE for discharge estimation using WinGate software is about 5.9%. Hence, the new proposed equations in this study and WinGate's method provide acceptable accuracy. Interestingly, the new suggested method, with its simplifying assumptions, presents an equal accuracy in comparison with WinGate, which has been developed for a complete check structure made up of multiple gates. Consequently, further studies are needed to clarify the effects of the parameters mentioned on the discharge passing under radial gates; particularly, some efforts are required to calibrate multiple gates with different openings.

This paper deals with flow passage through radial gates under free and submerged flow conditions, taking recourse from E-M equations. The theoretical equations have been verified based on the experimental data of Buyalski (1983) and collected data in the present study for the three types of radial gates.

Based on all experimental data, the MARE value to estimate the discharge is 1.4% and 2.8% for free and submerged flow conditions, respectively. The error considerably decreases for flow limit conditions by replacing the tailwater depth with the flow depth just after the gate (y₂) and using the energy equation for the sections before and after the gate. Although the direct measurement of y₂ depth is possible in the case of gate lip and high submerged flows, there are some difficulties due to the gate's turbulence under the transition between free and submerged flow conditions. As a result, it is recommended that a gauge is installed just after the gate (in the sidewalls) when the field measurement of y₂ is required.

The results showed that the proposed method in this study and WinGate software provide an acceptable uncertainty of about ±5% for discharge estimating of radial gates under field applications.

It was observed that the error may increase by ignoring the effects of difference in the gate seat and downstream canal levels and by operating multiple parallel radial gates with different openings. Consequently, further studies are needed to clarify the effects of the parameters mentioned above, particularly when more accuracy is desired.

Data cannot be made publicly available; readers should contact the corresponding author for details.

Some data used during the study are available from the corresponding author by request [experimental new data sets from the present study related to sharp-edged gates]. Other data used during this study appeared in Buyalski (1983).

There is no funding for this article.

The authors declare that they have no conflict of interest.

None.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

This article does not contain any studies with human participants or animals performed by any of the authors.