https://orcid.org/0000-0001-7068-8291
ABSTRACT
The vertical U-shaped gate holds significant potential for widespread application in flow control within U-shaped channels, as it eliminates the necessity for constructing auxiliary hydraulic structures. The boundary conditions associated with the U-shaped gate are complex, offering distinctive hydraulic features. In this study, the hydraulic characteristics of a vertical U-shaped gate have been investigated by model test and numerical simulation on a U-shaped channel under different flow rates, and the hydraulic evolution process was analyzed. The results show that the minimum relative error of discharges is 0.4%, so the numerical simulation can accurately describe the hydraulic performance of the vertical U-shaped gate. The flow generates a contracted cross-section and presents rhomboid water waves with a ‘hump-like’ convex structure after passing the U-shaped gate, accompanied by large kinetic energy dissipation. The gate opening exerts notable influence on the free surface width. The width of the first contraction section increased by 53.88% as the gate opening ranged from 2.5 to 5.5 cm with a flow rate of 8.24 L/s. The power function relationship of upstream flow Froude number, the width of free surface and energy loss is established. The results are helpful for engineering designing and operation management of a U-shaped gate.
HIGHLIGHTS
Indoor experiments and numerical simulation are used to investigate the hydraulic performance of a U-shaped gate.
Water flow presented rhomboid-shaped water wave downstream for U-shaped gates.
Diamond-shaped water waves will result in kinetic energy dissipation.
The value of free surface width can estimate the kinetic energy loss.
The width of the free surface and the energy loss present a power function relationship.
INTRODUCTION
The U-shaped channel features a distinctive cross-sectional design, with a curved bottom and an upper portion consisting of straight segments at a specific inclination. Compared with trapezoidal and rectangular channels, the U-shaped channel exhibits a more uniform flow velocity distribution, enhanced water conveyance capacity, and commendable resistance to seepage and freezing. It is commonly employed as a cross-sectional form in channel water conveyance projects. However, when measuring the flow in such channels, it is typically necessary to construct specialized flow measurement facilities, such as twisted surface and water columns (Bijankhan & Ferro 2019; Kapoor et al. 2019), measuring water tank (Samani & Magallanez 1993, 2000; Xiaoyi et al. 2002; Sun et al. 2021), quantity horizontal plate. The construction of these facilities not only causes secondary head loss in the channel (Wahl 2005; Shayan & Farhoudi 2013; Mohamed & Abdelhaleem 2020), but also require additional construction and maintenance. Among all flow measurement devices, the flat-plate sluice gate stands out for its simplicity and compact structure, eliminating the need for additional structures. This not only reduces construction costs but also renders it widely applicable in small-scale channels. However, compared with other channels, the hydraulic performance of a flat-plate sluice gate in a U-shaped channel exhibits significant distinctions. Therefore, investigating the hydraulic characteristics and kinetic energy loss of a flat-plate sluice gate in a U-shaped channel is of crucial significance for the development of flow rate measurement devices tailored for U-shaped channels.
Presently, the hydraulic performance of open-channel flow measurement devices is commonly investigated through methods such as theoretical analysis, model experiments, and numerical simulations. For example, Ferro (2000) established the relationship between water level and discharge through dimensional analysis and incomplete similarity theory. Some researchers (Clemmens et al. 2003; Habibzadeh et al. 2011; Vaheddoost et al. 2021) derived the flow rate calculation formula of radial gate by simultaneously solving the energy and momentum equations, demonstrating the high accuracy of the modified formula; Akoz et al. (2009) proved that the k–ε turbulent closure model can simulate the velocity field and free surface profile of upstream and downstream flow of vertical sluice more accurately compared with the k–ω turbulence model; Kubrak et al. (2020) verified the feasibility of estimating the discharge flow using the gate calculation formula under submerged outflow conditions with experimental methods, and the accuracy of the flow coefficient was about 10%. These studies have provided numerous insights into selecting research methods for assessing the hydraulic performance of gates.
At present, the research on the hydraulic performance of channel flow measurement devices is predominantly focused on rectangular and trapezoidal channels. Omid et al. (2007) investigated the hydraulic jump phenomenon formed in the stilling pool with a gradually expanding cross-section in trapezoidal channels, and an implicit equation was proposed for the sequent depth and the energy loss. Mitchell (2008) proposed a series of algorithms to predict the ratio of conjugate depths for trapezoidal and circular channels, it can be applied to predict energy loss in civil engineering hydraulics design. Nedim (2021) presents an analysis of water flow and flow velocity in the rectangular channel for free flow conditions, and a quadratic function of water velocity and flow depending on the water depth in the channel was proposed; Lamri et al. (2021) proposed two direct solutions for head loss and normal water depth in rectangular and triangular open channels, the explicit equations for the normal depths are developed, the results presented a high accuracy. Daneshfaraz et al. (2020, 2022) studied the energy dissipation of supercritical water flowing through a sudden contraction, which found that the rate of relative depreciation of energy is increased and hysteresis occurs when the supercritical flow is dealing with contraction. The formation of hysteresis increases the relative residual energy of different sections by 49.47–56.18%. Also, the crescent-shaped contraction has a significant effect on energy dissipating for supercritical flow, the energy dissipation is increased with increasing the upstream Froude number of crescent-shaped contraction (Daneshfaraz et al. 2021).
For U-shaped channels, Bushra & Afzal (2006) simplified the three-dimensional flow in the U-shaped channel to two-dimensional flow and analyzed the turbulent structure of hydraulic jump in the channel employing the Reynolds equation. Liu et al. (2014) proposed a water-measuring column with a round head to measure flow based on the cylindrical flow around theory, and a calculation formula for water column flow with a circular head was obtained by regression analysis. Azimi et al. (2017) conducted numerical studies on the three-dimensional morphology of hydraulic jump in the U-shaped channel, the comparison between the computational results and experimental findings indicates that the numerical model exhibits good accuracy. Mingyu et al. (2023) studied the flow capacity of the gate under different working conditions, and a formula to estimating flow rates was derived based on the hydraulic characteristics of the gate and channel, it is helpful for designing U-shaped channels. These researches focus on the hydraulic characteristics of various channels. Nevertheless, the flow pattern behind the vertical U-shaped sluice gate under the condition of free discharge has not been systematically examined, meanwhile, energy dissipation is a critical parameter for the evaluation of a sluice gate in the channel (Kalateh et al. 2024), but little research have been conducted for U-shaped sluice gates.
The U-shaped channel has been widely used in the irrigated area, but the review of previous studies indicates that a thorough investigation into the hydraulic characteristics of U-shaped gates under free discharge conditions is necessary, especially for the changes of flow pattern and energy dissipation. In order to solve these problems, this paper investigates the hydraulic characteristics of a vertical U-shaped gate under varying flow rates and gate openings through model testing cooperating with numerical simulation. The flow pattern behind the gate is observed, and flow parameters such as water depth, free surface width and velocity distribution at different measuring points are recorded, and the energy losses under different working conditions are calculated.
MATERIALS AND METHODS
Dimensional analysis
is the channel width, C is the arc length, e is the gate opening, l is the contraction width, y is the flow depth after the jump, g represents the gravity acceleration;
is the density; 
is the dynamic viscosity; 
is the specific energy upstream; 
is the specific energy downstream. On the basis of 
theorem, the non-dimensional parameters are produced as follows:
Experimental setup
Experimental setup.
In this research, five gate openings with five different flow rates were designed to investigate the influence of gate opening and upstream depth on hydraulic characteristics and energy dissipation under free flow. The experimental factors and levels are shown in Table 1, and the total number of experiments and simulations are both 29. The flow rate is controlled by a valve installed in the water supply pipe, the gate opening is measured by the ruler, upstream and downstream flow depths are measured by needle water level gauge, and the width of the water surface is measured by steel tape. The velocity is measured by a velocity meter (HD-LS300-A, propeller-type), and the duration of velocity data collection is about 30 s.
Experiment factors and levels
| Gate opening (cm) | Upstream water level (m) | Downstream water level (m) | Flow rate (L/s) | Experimental group number | Test number | Flow condition |
|---|---|---|---|---|---|---|
| 2.5 | 0.082–0.143 | 0.052–0.078 | 5.01–8.05 | 4 | A11 A12 B11 B12 | Free discharge |
| 4.0 | 0.055–0.222 | 0.050–0.122 | 5.01–16.88 | 9 | A21 A22 A23 B21 B22 C11 C12 D11 D12 | Free discharge |
| 5.5 | 0.075–0.195 | 0.068–0.125 | 8.05–19.44 | 8 | B31 B32 C21 C22 D21 D22 E11 E12 | Free discharge |
| 7.5 | 0.105–0.148 | 0.081–0.123 | 14.09–19.44 | 6 | C31 C32 D31 D32 E21 E22 | Free discharge |
| 9.5 | 0.122–0.125 | 0.099–0.122 | 19.24–20.60 | 2 | E31 E32 | Free discharge |
| Gate opening (cm) | Upstream water level (m) | Downstream water level (m) | Flow rate (L/s) | Experimental group number | Test number | Flow condition |
|---|---|---|---|---|---|---|
| 2.5 | 0.082–0.143 | 0.052–0.078 | 5.01–8.05 | 4 | A11 A12 B11 B12 | Free discharge |
| 4.0 | 0.055–0.222 | 0.050–0.122 | 5.01–16.88 | 9 | A21 A22 A23 B21 B22 C11 C12 D11 D12 | Free discharge |
| 5.5 | 0.075–0.195 | 0.068–0.125 | 8.05–19.44 | 8 | B31 B32 C21 C22 D21 D22 E11 E12 | Free discharge |
| 7.5 | 0.105–0.148 | 0.081–0.123 | 14.09–19.44 | 6 | C31 C32 D31 D32 E21 E22 | Free discharge |
| 9.5 | 0.122–0.125 | 0.099–0.122 | 19.24–20.60 | 2 | E31 E32 | Free discharge |
Numerical simulation
Fluent software (ANSYS FLUENT 21.0.) is used to investigate the hydraulic characteristics of channel with U-shaped plate gate. The continuity and Navier–Stokes equations for fluid flow are adopted to describe the flow state (Temam 2001; Liu et al. 2014), standard k–epsilon model is chosen to solve the unsteady calculation. The continuity and momentum equation was written as follows:
,
,
) are the body accelerations in (
,
,
) directions; (
,
,
) are the viscous accelerations in (
,
,
) directions; (
,
,
) are the fraction of flowable area in (
, y, 
) directions. (
,
,
) are velocity components in (
, y, 
) directions, respectively; p is the pressure; t is the time; 
is the density of the fluid, and the object of study is water; 
is the fractional volume open to flow.
The three-dimensional model is created using SolidWorks software, with the U-shaped gate positioned 1.5 m from the upstream entrance. A tetrahedral mesh is generated for the model using ANSYS ICEM software, with a cell grid size set to 2.5 cm and a total of 681,114 grid cells. The boundary conditions are configured as follows: The channel's water inlet is designated as a pressure inlet, while the water outlet employs a pressure outlet boundary (Figure 8). The inlet and outlet boundary conditions are assigned with values derived from experimental measurements, encompassing water depth and velocities. A pressure inlet is applied at the air inlet boundary atop the channel. The sidewall and the U-shaped gate are defined as no-slip wall boundaries. The SIMPLE algorithm is employed for solving the mass conservation and Navier–Stokes equations. A time step size (t) of 0.01 s is chosen, with a convergence precision of 0.0001 applied to all equations.
Mesh Independence check
Three different grid sizes were selected for meshing the numerical model, resulting in a total of 681,114; 711,358; and 880,506 grid cells, respectively. Under the same boundary conditions, the comparison of the width in contraction and diffusion sections at the same location in the numerical models with three different grid sizes was conducted to test grid independence (Table 2). It was observed that the width at the same location remained essentially unchanged for different numbers of grid cells. Therefore, the mesh size with 681,114 grid cells was selected to save computational resources and reduce simulation time while achieving the desired results.
Validation
The comparison of velocities in test C11 and E11 for simulation and experiment method.
The comparison of velocities in test C11 and E11 for simulation and experiment method.
RESULTS AND DISCUSSION
Flow pattern
The comparison of flow regime. (a) Top view of test number B11 (b) Top view of test number D32 (c) Side view of test number B11 (d) Side view of test number D32
The comparison of flow regime. (a) Top view of test number B11 (b) Top view of test number D32 (c) Side view of test number B11 (d) Side view of test number D32
The numerical simulation results show that the water flow presented a rhomboid flow pattern along the channel, and the central water level of the contraction section was increased, these findings are consistent with the experimental observation results.
- (1)
Water surface profile
- (2)
The characteristic of rhomboid-shaped water wave
The comparison of simulated and experimental water surface profile distribution in the central section.
The comparison of simulated and experimental water surface profile distribution in the central section.
The comparison of hydraulic characteristic of rhomboid-shaped water wave with simulation and experimental method.
The comparison of hydraulic characteristic of rhomboid-shaped water wave with simulation and experimental method.
Discharge and velocity
- (1)
Discharges
A triangle weir is set in the return canal section to measure the flow rate, and the flow rate can be calculated by Equation (4) as soon as the water head on the weir is obtained. The comparison of flow rates are presented in Table 3. It can be seen that the relative error between the experiments and the simulations ranges from 0.40 to 7.25%, which indicates that the numerical simulation method is accurate in describing the flow rate of the U-shaped sluice gate.
- (2)
Velocities
Mesh independence study of the width in contraction and diffusion section
| Numbers of cells | The width in contraction and diffusion section (cm) | ||
|---|---|---|---|
| First contraction section | First diffusion section | Second contraction section | |
| 681,114 | 19.62 | 32.30 | 24.48 |
| 761,358 | 19.54 | 31.20 | 24.40 |
| 880,506 | 19.55 | 31.24 | 24.47 |
| Numbers of cells | The width in contraction and diffusion section (cm) | ||
|---|---|---|---|
| First contraction section | First diffusion section | Second contraction section | |
| 681,114 | 19.62 | 32.30 | 24.48 |
| 761,358 | 19.54 | 31.20 | 24.40 |
| 880,506 | 19.55 | 31.24 | 24.47 |
The comparison of flow rate with simulation and experiment method
| Flow regime | Test number | Discharge (L/s) | Relative deviation | |
|---|---|---|---|---|
| Experimental value | Simulation data | |||
| Free discharge | A | 5.01 | 5.03 | 0.40% |
| B | 8.05 | 8.24 | 2.36% | |
| C | 14.09 | 14.75 | 4.68% | |
| D | 16.88 | 18.25 | 8.12% | |
| E | 19.44 | 20.85 | 7.25% | |
| Flow regime | Test number | Discharge (L/s) | Relative deviation | |
|---|---|---|---|---|
| Experimental value | Simulation data | |||
| Free discharge | A | 5.01 | 5.03 | 0.40% |
| B | 8.05 | 8.24 | 2.36% | |
| C | 14.09 | 14.75 | 4.68% | |
| D | 16.88 | 18.25 | 8.12% | |
| E | 19.44 | 20.85 | 7.25% | |
The comparison of velocity distribution at the center section for simulation and experiment method.
The comparison of velocity distribution at the center section for simulation and experiment method.
Dissipation of kinetic energy
- (1)
Evolution of rhomboid water waves
The contour of streamline with a flow rate of 8.24 L/s. (a) e = 2.5 cm (b) e = 4.0 cm (c) ) e = 5.5 cm.
The contour of streamline with a flow rate of 8.24 L/s. (a) e = 2.5 cm (b) e = 4.0 cm (c) ) e = 5.5 cm.
Figure 7 also shows that the greater the velocity of the impacting flow, the smaller the contracted cross-sectional width in the first contraction section, and gate opening has a remarkable influence on the width, the width is 16.11 and 24.79 cm as the gate opening increased from 2.5 to 5.5 cm, it indicates that the width of free surface in first contraction section can estimate the kinetic energy loss.
- (2)
Kinetic energy loss
Boundary conditions.
The power function fit curve of energy loss with Froude number and e/l. (a) Kinetic energy loss for different Froude number (b) Kinetic energy loss in different e/l
The power function fit curve of energy loss with Froude number and e/l. (a) Kinetic energy loss for different Froude number (b) Kinetic energy loss in different e/l
CONCLUSION
In this research, the hydraulic characteristics of a vertical flat gate were investigated under varying flow rates and gate openings through model testing cooperating with numerical simulation, the main results can be summarized as follows:
- (1)
The flow generates a contracted cross-section after passing the U-shaped gate, and a raised structure resembling a camel hump is formed. Meanwhile, the water flow presented rhomboid-shaped water wave downstream; the intensity of the diamond-shaped waves diminishes with the increasing flow distance, leading to a stabilized water surface eventually.
- (2)
The flow field distribution obtained by model testing is consistent with the results obtained through numerical simulation; in addition, the disparities between experimental and simulated values, including water depth, velocity, and discharge, are small. The minimum relative error of discharges is 0.4% while the maximum relative error of the width in the first diffusion section is −8.59%, which indicates that the numerical simulation results can be used to describe the hydraulic performance of the vertical U-shaped sluice gate under free flow conditions.
- (3)
Gate opening and flow rate have a remarkable influence on free surface width in the first contraction section. The width is 16.11 and 24.79 cm as the gate opening increased from 2.5 to 5.5 cm with a flow rate of 8.24 L/s. The greater the velocity of the impacting flow, the smaller the width, i.e. diamond-shaped water waves lead to kinetic energy dissipation.
- (4)
The power function relationship of upstream flow Froude number, the width of free surface and the energy loss is established, which can estimate the kinetic energy loss for a constant gate opening.
ACKNOWLEDGEMENTS
I would like to express my deepest gratitude to the editors and reviewers.
FUNDING
This research was supported by The Scientific and Technological Research Program of Henan Province (No. 242102321001), for which the authors are grateful.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.
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