1. Introduction
Sluice is a Dutch word for a channel controlled at its head by a movable gate which is called a sluice gate. A sluice gate can be considered as a bottom opening in a wall [
1] or an undershot gate that passes the flow through the bottom, similar to an orifice [
2] or a sort of nozzle. The simplicity of sluice gate design, construction, and operation, plus good safety and low maintenance costs [
2] result in them being among the most common hydraulic structures to control or measure flow [
3].
The volumetric flow rate passing through a sluice gate can be estimated if the opening of the sluice gate (
) and water depths upstream (
) and downstream (
) [
2,
3] are known. Low head loss and no need for new equipment make sluice gates preferable for measuring the flow rate where a device is installed [
3]. However, the accuracy of flow rates calculated based on sluice gate flows is typically less than weirs, and a complex calibration is needed to account for cases of free or submerged hydraulic jumps downstream of the gate [
2,
3,
4,
5]. The flow under sluice gates can be categorized as a free hydraulic jump (F), partially submerged hydraulic jump (PS), or submerged hydraulic jump (S) [
6,
7].
Figure 1 shows a sluice gate when a free hydraulic jump occurs. Here,
is the upstream depth,
is the opening of the gate,
is the minimum depth of flow after the sluice gate,
and
are the initial and secondary depths of the hydraulic jump, respectively, and
is the downstream depth.
Many studies focused on characterizing the free (classic) hydraulic jump [
8,
9,
10,
11,
12]. The flow after the sluice gate behaves like a stream coming out of an orifice (nozzle) [
1]. White (2011) proposed that a free discharge is expected for
(or
) [
1]. However, the literature indicates studies that relate the upstream flow depth to the maximum tailwater (downstream) depth to define flow regime distinguishing condition curves [
13]. To estimate the volume of flow, many studies focused on determining the coefficient of discharge (
). An early, significant study is Henry’s 1950 experiment to estimate the
for free and submerged flows [
14]. Henry developed a relationship between the
and
by neglecting energy losses and assuming a uniform velocity and a hydrostatic pressure distribution both upstream and at the vena contracta, providing a practical, widely used graph of this relationship represented in
Figure 2 [
14]. The highest value of
(0.611) occurs in the free flow regime [
15].
In 1967, Rajaratnam and Subramanya used the energy–momentum (EM) method to prove Henry’s (1950) results [
10]. In 1992, Swamee developed several relationships for free and submerged flows based on Henry’s (1950) graph [
15] to help prevent interpretation errors when interpolating discharge coefficient curves and provide an analytical and/or numerical method for determining the discharge coefficient for sluice gates. Lozano et al. Roth and Hager (in 1999) experimentally studied the effects of viscosity and surface tension on scaling sluice gate operations in free-flow conditions. Their research included studies on the contraction coefficient as well as other parameters such as the distribution of velocity and pressure on the gate and the channel bottom [
16].
Lozano et al., in 2009, performed field studies on four rectangular sluice gates by measuring the water depth and gate opening values [
12]. They reported that the EM model resulted in reasonable discharge estimations for three of the studied gates by calibrating the contraction coefficient. For the case where the EM method estimations were not accurate, the sluice gate had a unique nonsymmetric flow condition and was located at the channel’s head [
17].
Habibzadeh et al., in 2011, applied a theoretical method based on EM equations to find an equation for the discharge coefficient of sluice gates in rectangular channels based on orifice-flow conditions, applicable in both free and submerged flow conditions [
18]. In most sluice gate models, the energy losses are assumed negligible; however, Habibzadeh et al. reported that turbulence-related phenomena cause significant energy losses in the submerged-flow condition. Additionally, the recirculating region below the gate induces turbulence that results in energy loss in the upstream pool. They considered the magnitude of an additional energy-loss factor as a function of the sluice gate geometry and proposed that it could affect the discharge coefficient [
18].
Some studies divided submerged flow regime into two subcategories: 1. low and 2. high submerged regimes [
13]. In 2011, Habibzadeh et al. proposed an equation to calculate a parameter called “transitional value of tailwater depth”, based on several factors, including the contraction coefficient, upstream flow depth, gate’s opening, and an energy loss factor which was considered to be 0.062 [
18]. The free-flow regime is expected for downstream water depths less than this value, while a submerged flow regime is expected for higher downstream depths. They also proposed a measure to distinguish the submergence ratio of the flow as a function of the maximum tailwater depth for free flow and the downstream depth. According to this measure, the flow is considered low submerged for submergence ratios between 0 and 20 and considered high submerged for values higher than this range [
13,
18]. Castro-Orgaz et al., in 2010, proposed that for high submergence situations, the common EM method is not accurate. They proposed a new equation for submerged flow based on the energy–momentum method principles by applying correction factors on velocity and momentum [
19]. However, in 2012, Bijankhan et al. showed that this method has significant errors when the submergence is not significant [
13]. The analysis of the EM method indicates that the roller momentum flux and the energy loss could be significant. Therefore, in 2013, Castro-Orgaz et al. published a revised version of their 2010 research for estimating sluice gate discharge in submerged conditions. The new method introduces rationality in the EM equations for submerged gate flow. However, the results of this method were similar to the former one [
20].
Gumus et al. (2016) studied the velocity field and surface profile of the submerged hydraulic jump of a vertical sluice gate using 2D CFD modeling. The results of this numerical study were compared with experimental data, and they concluded that the accuracy of the Reynolds Stress Model (RSM) is more than other studied turbulence models in predicting horizontal velocities and computing the free-surface profile of the hydraulic jump characteristics [
21].
Rady (2016) applied developed multilayer perceptron (MLP) artificial neural networks (ANNs) to predict
of vertical and inclined sluice gates. This study applies the MLP using the steepest descent back-propagation training algorithm and one hidden layer. For free flow, ANNs used
, the sluice gate’s inclination angle, Froude number and Reynolds number. For submerged flow,
was used in addition to the former inputs. Rady reported that using this method led to reasonable accuracy [
22].
Silva and Rijo (2017) studied several methods to determine
including EM-bases models, orifice flow rate relationships, and dimensional analysis using Buckinghum’s Π-theorem. They concluded that the EM-based method led to better results for all free, submerged, and partially submerged flows. Additionally, they reported that there were no improvements in discharge estimation results of the methods that divide the partially and fully submerged flows for the studied sluice gate openings [
3].
Kubrak et al. (2020) investigated measuring the volumetric flow rate using sluice gates under submerged flow conditions. A laboratory experiment on a model made on a 1:2 scale of an irrigation sluice gate was conducted to collect the experimental data. This study utilized Swamee’s (1992) model as a basis for the determination of
. Using this experimental data and empirical methods to adopt corrections, they achieved a reasonable accuracy for estimated discharge coefficients of Swamee’s (1992) model for this case study. Based on this and as stated by Boiten (1992) [
23], they concluded that this method is useful in estimating flow through the sluice gate [
24].
Nasrabadi et al. (2021) compared the Group Method Data Handling (GMDH) and Developed Group Method of Data Handling (DGMDH) machine learning methods to predict the characteristics of a submerged hydraulic jump of a sluice gate. Their study indicated a reasonable accuracy for both models in estimating relative submergence depth, jump length, and relative energy loss [
25].
Based on the literature, the flow under the sluice gate is a classic problem that has been extensively studied experimentally and numerically. Hence, the literature indicates that in recent years the advancements in new measurement methods, such as using a laser Doppler anemometer to measure velocity fields [
21], etc., improved the accuracy and domain of experimental measurements on this topic. Moreover, beside the classic studies, modern methods such as machine learning techniques, AI, and CFD opened new avenues for investigations on this topic. However, in practice, modern methods are not easy to implement or cost-effective: the new measurement methods are technology-based and require new instruments to be installed, and the application of modern techniques is still highly dependent on the operator’s skills and experience.
It is worth mentioning that in practice, some sluice gate models are available in software packages such as SIC and HEC-RAS. The Simulation of Irrigation Canals (SIC) is a one-dimensional hydrodynamic model for river and irrigation canal modeling and regulation [
6]. This commercial software has been released by Cemagref since 1989. SIC utilizes empirical relationships [
6] based on orifice flow equations [
3] to estimate the flow rate under sluice gates. The Hydrologic Engineering Center’s River Analysis System (HEC-RAS) has been released since 1995 and supports 1-D steady flow, 1-D and 2-D unsteady flow calculations [
7]. However, these software packages cannot estimate
. For SIC, some literature suggests considering
as 0.6 [
26], and the HEC-RAS manual suggests that typically
[
7]. Such a drawback makes the accuracy of results depend on choosing an appropriate value for
and the experience of the operators and makes this software not desirable for the design of sluice gates.
In practice, for existing sluice gates, calibration and determination of the optimum is performed using field or experimental measurements. However, models and tools used to characterize flow under sluice gates are essential, especially where the experimental data is not available for calibration, such as for design purposes. On top of that, in system optimization, non-iterative and fast analytical calculations for a component are important for maintaining the overall efficiency of optimizing the whole system of many components. Hence, many available models in the literature still require further investigations, require adequate knowledge to apply, and are not simplified enough or represented in a standard method for such purposes. The essence of developing standard and simple analytical methods is even more significant when sluice gates need to be modeled as a component in a complex hydraulic system, especially when there are many scenarios to be studied, managed, or optimized.
As a result, this study focused on analyzing, evaluating, and simplifying models to improve the application of the models with higher accuracy in the design of sluice gates. Five models were reviewed, analyzed, represented in a standard and easy to use form, and their performance was evaluated in distinguishing conditions of flow regimes, estimating the
and flow rate. For this purpose, a series of lab experiments were conducted at the University of Guelph to study the flow on different sluice gate openings, flow rates, upstream and downstream conditions. The effects of physical scale on models were investigated, and recommendations were provided. Moreover, new analytical equations are proposed to improve the accuracy or applicability of some models. The unscalable initial studies costs are a large burden on small projects [
27,
28]. The presented new equations and simplified models could facilitate their applicability in the design of sluice gates and initial studies of small projects such as the irrigation channels or pico- and micro-hydropower plants. In addition, this study contributes to simplifying these models in a standard form to facilitate the application of models, particularly for modeling sluice gates as a component in complex hydraulic systems such as hydropower plants. The contributions of the analytical equations and models proposed in this study facilitate the development of models for such complex hydraulic systems for optimization purposes, development of management or operation plans, etc.
5. Conclusions
Sluice gates are important components in designing, operating, and maintaining water delivery systems and hydropower plants, which are essential elements in the sustainable development of water resources and power generation. This study investigated five models’ performances in distinguishing conditions and estimating the coefficient of discharge () and flow rate in sluice gates for free flow and submerged flows. Experiments in a laboratory flume at the University of Guelph were used to characterize steady-state flow through a sluice gate for different flow rates, gate openings, upstream and downstream conditions. New equations were proposed to make them easier to analyze and compare and facilitate the application of the studied models.
For this experiment, analysis of the EM, EML, and Swamee methods indicated relatively similar curves and performance in distinguishing the sluice gate flow regimes. However, the HEC-RAS model’s estimations of the flow regime were not accurate.
Estimating the flow rate results of this experiment indicated a reasonable performance of EM, EML, and Henry’s models in free flow. For submerged flow, the performances of EM and EML models were reasonable. The performance of Henry’s model was significantly lower than EM-based models for submerged flow. Swamee’s models’ performance was not acceptable for free and submerged flows. Several other studies report the same issue for this model. The results of this experiment, the fact that the nonlinear regression of Swamee’s (1992) method is based on Henry’s (1950) nomogram, and similar reports in several studies could raise concerns about the generality of this model and the scaling effects on this method’s accuracy. Therefore, it seems that Swamee’s (1992) method should be used with extra caution.
This study results also indicated that the scaling effects on loss factor in the EML model need to be considered for different cases. Investigations showed that calibration k using this experiment data increases the EML accuracy. In practice, most of the losses will be captured through the calibration of using real field or experiment measurements. For design purposes, the EML model could be considered as a model with reasonable accuracy. Further investigations on this model are recommended.
An analytical equation and method were proposed to determine the loss factor due to the importance of in the EML model, which makes it easier to use and improved its performance. Additionally, a new equation form was proposed for the EML model to determine the discharge coefficient in submerged flow.
In addition, a solution was proposed for the HEC-RAS model drawback in not providing any analytical methods to estimate the coefficient of discharge . Technically, could be calibrated when the experimental data is available. However, for design purposes, or when the experimental measurements are not available, such a drawback makes the application and accuracy of HEC-RAS highly dependent on the operator’s experience. To address this issue, a method is offered and evaluated to dynamically estimate for HEC-RAS. It led to a reasonable accuracy and enables utilizing this model for design purposes where the experimental data is not available for calibration. Further studies could reveal more facts about this method.
In system optimization, non-iterative and fast analytical calculations for a component are important to maintain the overall efficiency of optimizing the whole system of many components. The new analytical equations, methods, and simplified models presented in this study could assist with modeling sluice gates as a component of complex hydraulic systems, especially when there are many scenarios to be studied, managed, or optimized. Moreover, they could assist where the experimental data is not available for calibration, such as for design purposes or where the flow in open channels needs to be controlled or measured using the sluice gates. This study could benefit the design, optimization, development of management or operation plans, etc., of sluice gates in complex hydraulic systems such as water delivery systems and hydropower plants, which are essential components in the sustainable development of water resources and power generation. Further studies are recommended.