elocation-id: e3317

The sustainability of water use in agriculture is very important, and to achieve this,
proper management of surface water resources is required, which, in small basins,
depends on good measurement of flow rate data. This work analyzes the hydraulic performance
of three proposed modified long-throated flumes with a 10% slope supercritical channel
by using the WinFlume^{®} v.1.06 software for a maximum flow of 50 L s^{-1}, for clean water and in conditions in which the cross section of the inlet was modified
by sediments. Hydraulic simulations were performed for triangular, rectangular, and
trapezoidal cross sections in the throat and the supercritical channel by using the
Iber^{®} v.2.5.2 software to compare the pattern of the WinFlume^{®} gauging curve with the results of Iber^{®}; the difference between both gauging curves was a maximum error of 3% in the three
cross-sectional areas. For hydraulic simulation in the supercritical modified cross-sectional
area for clean flow water and sediment flow water, data located at half the total
length of the supercritical channel were taken. Variations in flow depth assuming
sedimentation were negligible, and it was found that the water profile in the supercritical
channel was not affected, so it can be used as an indicator. In addition, it was observed
that the best cross-sectional area for the channel was the rectangular one with a
maximum of 0.9997 R^{2} for the flow rate curve (Q-h).

hydraulic simulation, natural runoff, sediment flow, water measurement.

Mexico is a country of great territorial area (1 959 million km^{2}), which lacks the capacity to monitor its water resources since it only has 815 hydrometric
stations operating (CONAGUA, 2019), located in the main rivers and strategic sites for the development of the country.
Therefore, it is urgent to find alternatives to improve knowledge about the quantity,
quality, and temporality of water available in the country.

Considering that, in developing countries such as Mexico, approximately 76% of the
available water is used for irrigation (CONAGUA, 2019), its monitoring becomes more relevant, especially for sustainable management. Since,
as pointed out by De Fraiture *et al*. (2010) in relation to the agricultural sector, some poorly conceived or implemented interventions
have caused high social and environmental costs, such as inequitable allocation of
water and adverse impacts on ecosystems.

Given the need to quantify water, the use of gauging structures that allow reliable
knowledge of the available volumes becomes relevant. At present, there is a great
variety of flumes, where the most used are the weirs, which can be broad-crested or
sharp-crested (WMO, 2010); in addition, their measurement section can be rectangular,
triangular, or semicircular (Ibrahim, 2015; Haghshenas and Imanian *et al*., 2021; Vatankhah, 2021). Other widely used structures are long-throated flumes, which are constructed for
a wide variety of shapes and are generally very accurate when operated under non-submerged
flow conditions (Clemmens *et al*., 2001).

In all cases, the structures work well with clean water flow; however, in natural
surface runoff, there is sediment entrainment (Prado *et al*., 2017) and a frequent problem is their accumulation in the place where the water level
is measured, to the point that the measurements lose validity (Castro-Orgaz and Mateos, 2014).

Given this situation, Carrillo (1999) suggested, as a possible solution to prevent errors caused by sedimentation, the
addition of a rapid to a long-throated flume and finding a site with an adequate correlation
between the depth of flow and the circulating flow rate; this alternative could be
viable since it has been found that velocities greater than 0.25 m s^{-1} are sufficient to promote sediment entrainment and thus prevent sedimentation (Ackers *et al*., 2001; Blokker *et al*., 2011).

Therefore, this work presented an initial advance that considered the hydraulic analysis through numerical simulations of three different geometries for long-throated flumes, to which a rapid was added after the throat with a 10% slope and the elevation of the bottom of the structure was modified.

Three long-throated flumes were designed with the WinFlume^{®} v.1.06 software, selecting different geometries for the control section (throat):
rectangular, trapezoidal, and triangular. In all cases, an operating range from 10
to 50 L s^{-1} and material with absolute roughness ks of 0.00006 m were proposed; in addition,
the flow rate gauging table of each structure was obtained, measured with flow depth
increments of 1 cm (17 flow rates). A rapid was added to each designed structure and
two scenarios were generated to study the effects of sediment accumulation on the
gauging structures.

Table 1 shows the dimensions of the structures designed, including the quick proposal for their study. In the convergence section, only the length is indicated since their width, depth and side slopes coincide upstream with the inlet channel and downstream with the control section (Figure 1).

To test sedimentation effects on the development of the flow in the rapid, the elevation of the bottom of the inlet channel and convergence section of each flume was modified until reaching the elevation of the control section, generating structures with new dimensions of width and depth (Table 2 and Figure 2).

From the data of the gauging table, a value of the Manning roughness coefficient was calculated for the results of each expense obtained, considering the hydraulic characteristics of the inlet of the structure (hydraulic area, wet perimeter, hydraulic diameter), relative roughness and Reynolds number, using an equation derived from the Manning and Darcy-Weisbach equations (equation 1) and the Colebrook equation (equation 2);

Where:n is the Manning roughness coefficient (dimensionless); g is the acceleration
of gravity (m∙s^{-2}); R_{h} is the hydraulic radius (m); f is the Darcy friction factor (dimensionless); k is
the absolute roughness (m);D is the hydraulic diameter (m); and Re is the Reynolds
number (dimensionless). An average value for the Manning coefficients was then calculated,
obtaining a value of 0.0121 for the rectangular flume, 0.0123 for the trapezoidal
flume and 0.0133 for the triangular flume.

The simulation of the water flow in the gauging structures was carried out using the
Iber® v.2.5.2 program, which solves the Saint Venant two-dimensional equations, incorporating
the effects of turbulence and surface friction by wind through equations 3, 4 and
5 (Bladé *et al*., 2014):

Where: h is the depth (m);U_{x} and U_{y} are the horizontal velocities averaged in depth in the main and transverse direction
to the flow (m s^{-1});g is the acceleration of gravity (m s^{-2}); g is the density of water;Z_{b} is the height of the bottom; τ s is the friction on the free surface due to friction
produced by the wind; Ƭ_{s} is the friction due to bottom friction; and v_{t} is turbulent viscosity.

The geometric models for scenarios 1 and 2 were configured, defining the Manning roughness coefficients for the walls as obtained: 0.0121 for the rectangular flume, 0.0123 for the trapezoidal and 0.0133 for the triangular flume, except the lower part of the inlet and convergence sections of the proposed flumes of scenario 2, in which case a coefficient of 0.023 was used, considering the characteristics of the deposited material (Chow, 1988).

In all cases, a numerical discretization was performed based on a structured mesh consisting of elements of 0.5 cm × 1 cm, placing the largest side along the structure (predominance of flow). The imposed boundary conditions were constant flow rate in subcritical regime at the inlet of the flumes and flow in critical/supercritical regime at the outlet.

The k-ε turbulence model of Rastogi and Rodi was used since it is one of the most recurrent for shallow turbulent flows and with good results obtained through experimentation (Cea et al., 2007; Bladé et al., 2014). The simulation for the case of each flume was carried out in permanent regime for each flow rate, with the aim of obtaining the best precision of the gauging tables of the designed structures (scenarios 1 and 2); a total of 132 simulations were performed.

The depth of flow in the inlet section was determined according to the requirements specified for each design (Clemmens et al., 2001), the determination sites were located 0.3 m, 0.5 m, and 0.32 m upstream of the beginning of the convergence in the rectangular, trapezoidal, and triangular flumes, respectively. The site selected in the rapids was located in all cases at half its length. In addition, at each site, five readings were made in a direction transversal to the flow and an average value was obtained.

To evaluate the performance of the numerical simulation carried out in Iber^{®} with respect to the flow at the inlet of the flumes, the values of flow depth were
measured to be compared with those obtained with WinFlume^{®}, three statistical indicators were used: scatter index (SI), mean absolute percentage
error (MAPE) and the root mean square error (RMSE), expressed mathematically in equations
6, 7 and 8 (Yousif et al., 2019).

Where: φ_{i-calc} is the flow depth value (cm) obtained in the design with WinFlume; φ_{i-sim} is the i-th value obtained for the flow depth (cm) in the hydraulic simulation with
Iber; 𝑁 is the number of data analyzed; and φ_{i-calc} is the average of the flow depth values (cm) obtained with WinFlume.

With the data from the gauging table obtained with WinFlume and the measurements made
in the rapid, the Q-h curves of each flume (both scenarios) were plotted and the data
were fitted to an equation characteristic of weirs, which is usually expressed in
the form (Saul, 1997):Q= k·h^{n} 9). Where Q is the flow rate (L s^{-1}); ℎ is the depth of flow (cm); and k and n are constants (dimensionless). In addition,
the coefficient of determination (R^{2}) was calculated, which represents the percentage of variability in the dependent
variable; that is, the variance with respect to the mean (Chicco et al., 2021).

To verify the reliability of the measurements made in the rapid, despite the modifications made in the inlet of the structures, using the same indicators shown in equations 6, 7 and 8, the variation between the data measured in the rapids of the structures proposed in scenario 1 and the data measured in the rapids of the structures included in scenario 2 was evaluated.

Based on the original design of the gauging structures, with the data of the gauging tables obtained from each structure Figure 3 shows the behavior of the depth of the water, in the inlet section of the flumes, corresponding to the circulating flow rates.

Regarding the hydraulic simulations performed with Iber^{®}Figure 4 shows the results obtained, particularly the results for flow depth in the rectangular
flume, although in this case the total depth is shown, while the value reported in
WinFlume^{®} is referenced to the bottom of the throat; that is, the result must be subtracted
5 cm.

When processing the information, it was obtained that the values of the depth of flow
at the inlet of the structures were slightly higher compared to those obtained with
WinFlume^{®}; however, the difference was constant for the entire range of flow rates, the results
can be observed in Table 3.

The results obtained with the simulations were used to obtain the statistical indicators
shown in Table 4, where it can be seen that, in all cases, the error obtained was less than 3%, even
in the case of the triangular flume the calculated error did not exceed 2%. These
results support the reliability of the results obtained through numerical simulation
since, if it is considered that the authors of WinFlume^{®} indicate that the error supported analytically and experimentally in the designed
flumes is approximately 2%, it can be inferred that in the worst case scenario, there
would be an error of ±5% with respect to the true value.

The data from the gauging tables obtained with WinFlume^{®} for the structures included in scenario 1, in addition to the results obtained with
the numerical simulations, were used to obtain fitted equations between the flow rate
and flow depth at the inlet of the section (Table 5). A variation in the fitted equations is observed; however, in all cases, values
very close to 1 were obtained for the coefficients of determination (R^{2}).

It is possible to notice, according to the value of R^{2}, that the best fit for WinFlume^{®} data is achieved with the triangular and rectangular flumes. However, the data obtained
with Iber^{®} indicate a better fit with the rectangular flume, followed by the triangular one
and finally the trapezoidal one.

Figure 5 shows the Q-h relationships obtained in the rapids of the flumes when analyzing the results of the numerical simulations, where the cross of the curves of the trapezoidal and rectangular channels is explained by the greater rate of increase of the hydraulic area of a trapezoidal section with respect to a rectangular one when increasing its depth, in addition to the behavior of the supercritical regime that generates transverse waves resulting in the formation of the ‘oblique jump’.

The equation models fitted to the data measured in the rapids of the structures belonging
to scenario 1, together with their coefficient of determination (R^{2}), are shown in Table 6. Where, despite having a good fit in the three flumes, it can be noted again that
the best fit is obtained with the rectangular section flume with an R^{2} closer to 1.

Regarding the variation between the results of the numerical simulations in the rapids of the flumes, for the analyzed scenarios, it was found that a reduction of the hydraulic area at the inlet of the flume does not have a relevant effect on the behavior of the flow developed in the rapid because, as can be seen in Table 7, in none of the cases was an error greater than 1% reached.

Rectangular flume | Trapezoidal flume | Triangular flume | |
---|---|---|---|

SI (%) | 0.2 | 0.5 | 0.13 |

MAPE (%) | 0.16 | 0.39 | 0.12 |

RMSE (cm) | 0.016 | 0.04 | 0.019 |

This is because, although the structure is modified at the bottom by the accumulation of sediments, it still maintains the design of a flume with lateral constrictions, which, according to Bos and Wijbenga (1997), have already been used previously for the measurement of flow rates in channels with sediment transport; also recently, the work by Aali and Vatankhah (2023) reported an average error close to 2.2% in flow rate measurement using simple channels with a trapezoidal contraction.

In this work, it was also found that in the two scenarios, the Froude number in the inlet channel always remained below 0.4, which is a requirement to achieve uniform surfaces of the water surface. By maintaining the conditions upstream of the throat, the conditions of flow developed in the throat and the rapid are also maintained.

Based on Table 7, the results indicate that the performance of the rectangular and triangular structure is similar; however, it is necessary to consider other aspects, such as those observed by the following researchers, in order to select the type of geometry suitable for use in natural flow conditions.

The research by Agazadeh and Mohammadi (2013) on the conditions of the incipient movement speed of sediment particles in channels of triangular section showed that the section of the channel has a significant impact because the walls of the channel are very close at the bottom, the flow is very weak, so a higher speed is necessary to reach the incipient movement of the sediment.

On the other hand, Aksoy et al. (2017) conducted an experimental work analyzing the flow of water with sediments in channels of trapezoidal, rectangular, circular, U-shaped, and V-bottom sections, where they found that, for the same slope, the channels of rectangular sections provide the lowest incipient sediment deposition rate, while the V-bottom channel has the highest incipient sediment deposition rate.

In this same sense, Unal (2018) reported that rectangular channels have a higher efficiency in sediment transport since sediment particles begin to be deposited under a lower shear stress in the rectangular channel compared to non-rectangular channels. Therefore, considering the results of these studies, together with the results found in this work, the convenience of using rectangular sections was observed.

Since the problem in all flumes is sedimentation, the basis of this research was the
proposal to measure the flow rates in the rapid added, where the results obtained
show the development of velocities in a range from 0.5 to 2 m s^{-1}, being greater than those indicated by Ackers et al. (2001); Blokker et al. (2011) to promote sediment entrainment, the problem of sedimentation accumulation is prevented
and consequently a constant reading of circulating flow rates is ensured.

Although variations were observed in the flow depths determined in Iber with respect
to those originally obtained with WinFlume^{®}, the results showed a maximum error of ±5%. Derived from the analysis between scenarios
1 and 2, it was found that the accumulation of sediments at the inlet of the structure
has a minimal effect on the development of flow conditions because in none of the
simulated flow rates was a Froude number higher than 0.4 found, this condition allows
the structure to continue working correctly and consequently that the variations in
the depth of the flow in the rapid are less to 1%.

Based on the results of simulations and reviews of other experimental studies that have shown that the presence of bottom sediments in the flow is a reason to consider in more detail the selection of the type of flume to install, it was concluded that a flume with rectangular geometry is more suitable for its construction and flow rate conditions in natural channels.

It is recommended to carry out more detailed research to achieve the best performance of the selected structure, proposing other slopes for the rapids and also other measurement sites in them; all this aimed at determining the characteristics of a structure that could offer the best alternative for its implementation in the field, which includes its experimental study.