Hydraulic Performance of Sluice Gates: A Review of Head Loss Estimation and Discharge Coefficients for Optimal Flow Control and Design Considerations

Abstract

Sluice gates are commonly used water management structures; they are able to deliver flow among both major and minor channels. When used for these operations, they are sometimes referred to as intake structures. These structures can be employed at the dam crest for controlling downstream flow. Irrespective of their simple structures, the hydraulics of sluice gates are complex. Sluice gates are used in various applications, from practical applications such as irrigation channels to research activities in universities. Accurate head loss calculations (ΔE) and discharge coefficients (Cd) are essential for the design of open canals. Calculations are needed for both free or submerged flow conditions. Although there have been some investigations on Cd for sluice gates, a comprehensive literature review shows that there are no studies on ΔE. Knowledge of ΔE is necessary for the design of intakes and irrigation canal inverts. The objective of this study is to investigate the head loss estimation and discharge coefficients for optimal flow control and design considerations. Both the vertical sluice gate and radial gate are investigated. This study experimentally explores ΔE and Cd using geometric scaling. It is found that ΔE for free flow exceeds that of submerged flow. In addition, free flow discharge coefficients exceed those for submerged flow. Relative energy losses (ΔE) range from 0.271 to 0.604. Energy losses of these magnitudes cannot be ignored, and their impact on minor canal inverts should be considered.

1. Introduction

1.1 Literature review

Hydraulic structures are used to manage the distribution of water for downstream use and to adjust water levels as they naturally fluctuate with precipitation, evaporation, and other effects. Gates are important water management structures. One common example, the sluice gate, has been used for many years [1]. The gates themselves have different types, each having advantages and disadvantages, and a gate type can be selected according to the specific application and its functional characteristics. Flow through sluice gates is divided into two types: (i) free surface flow and (ii) submerged flow, and the discharge coefficients for each of these two categories are different. Gates have been extensively used in irrigation canals, at the top of the spillway of dams, and at transitions between lakes and channels [2, 3], among others. Khaleel and Othman [4] studied clear water released from a sluice gate and its impact on a downstream alluvial laboratory channel.

Fuse gates are considered for increasing reservoir capacity. The result in increased water levels upstream of a spillway. Shahkarami [5] installed fuse gates along a river that provided a barrier for the flow without the need for extra flood-mitigating structures.

Sluice and radial gates are perhaps the most common types of gate-fluid structures [6]. When gates are used, it is important for hydraulic engineers to be able to make critical calculations of the flow behavior [7, 8]. Flow under gates has been studied for decades [6, 9, 10, 11, 12]. From the prior research, it has become known that the presence of a sill under a gate can improve the gate operation [1].

When a sill is deployed underneath a radial gate, the negative consequences of sedimentation are reduced. C_d subsequently increases the flow that is able to pass under the gate. A sill-under-gate arrangement reduces the required gate height and decreases the force of the water pressure. By reducing gate height, the weight is also decreased. Consequently, a sill can be expected to improve the gate performance, if the sill is properly positioned\ and shaped. On the other hand, when sills are installed in poor locations or without proper shape, they can have a negative consequence on the overall hydraulic performance. Recently, Salmasi et al. [13] studied the effect of sill location on C_d for radial gates. In that study, there were two test cases. For the first test case, a sill was located upstream of the gate, whereas in the second case, the sill was positioned underneath the gate. It was found that for case 1, the sill increased flow resistance and reduced the discharge coefficient – but the opposite was observed for case 2.

Belaud et al. [7] studied the contraction coefficient (C_c) for slide gates. In that study, coupled momentum and energy equations were solved. Contraction coefficient values of ∼0.6 were obtained – consistent with other researchers. Silva and Rijo [14] used an energy conservation approach method to estimate discharge below a sluice gate for both free and submerged flow conditions.

Khalili Shayan et al. [15] predicted discharge from radial gates using energy and momentum (E–M) conservation. Their study covered both free and submerged flows. Contemporaneously, Hoseini and Vatankhah [16] studied stage-discharge correlations for sluice gates located in circular open channels/pipes. Wu and Rajaratnam [17] provided direct solutions and iteration procedures for submerged flow problems for rectangular sluice gates. They also provided procedures to distinguish the flow state (free flow or submerged flow). Xu and Samuel Li [18] reported new experimental and computational results of underflow passing below a vertical sluice gate. The focus was on the flow curvature immediately downstream of the gate and the associated centripetal force on the gate lip. The computations successfully produced the two-phase (air-water) flow field by solving the Reynolds-averaged Navier–Stokes equations. Curvature-induced forces on sluice gates at hydroelectric power generating stations were determined. In addition, corrections to some existing formulations of the underflow problem were provided.

Daneshfaraz et al. [19] investigated how the gate opening, along with sill placement would affect the discharge. In that study, sills were placed in positions upstream, tangential to, and downstream of a gate at various locations. The discharge coefficient increases with increasing sill width and decreasing flow area. The discharge coefficient was also influenced by the installation of gates at various distance intervals. For similar openings, the discharge increases compared to the non-sill state (for sills underneath and tangent to the gate). It was also shown that the tangential sill leads to the largest discharge coefficient.

Aydin et al. [20] studied the importance of the geometry of the conduit cross-section. They connected this parameter with the air-demand ratio. Experiments covered multiple permutations of radial gates with high upstream heads. Their work showed that the cross-section geometry has an important effect on the air-demand ratio, particularly for small gate openings. Additionally, air-demand correlations were developed.

Investigation of free and submerged discharge coefficients for inclined sluice gates was the focus of the Salmasi et al. [21]‘s study. The experimental facility used an inclined gate with an angle (β) and gate opening (a). Inclination angles ranging up to 60 degrees were employed. The experiments showed greater flow convergence when the discharge coefficient increased – connecting the flow patterns with hydraulic performance. They also showed that when the submergence rate increases (y_t/a), the inclined gate discharge coefficient decreases. Various AI methods were used to develop performance metrics. Among the machine learning methods were regression, support vector machine (SVM), Gaussian process (GP), artificial neural networks (ANN), random forest (RF) regression, generalized regression neural network (GRNN), and random tree (RT)-based models. The models relied upon input of a relatively small number of dimensionless terms.

1.2 Theoretical background

In irrigation canals, gates are used to adjust the water level or measure the flow of water entering the intakes. In dams, gates are used to regulate the water flow in reservoirs and also on the crest of a dam to control the flood flow. The flow under a gate can be free or submerged. Figures 1 and 2 show free and submerged flow conditions, respectively, for a vertical slide/sluice gate. The important hydraulic factors in these figures are as follows: h₀ is the upstream water depth, h₁ is the water depth in the contraction section (vena contracta section) immediately downstream of the gate, h is the submergence depth, a is the gate opening, and h₂ is the downstream water depth or tailwater depth.

Figure 1.

Free flow under the vertical slide gate.

Figure 2.

Submerged flow under the vertical slide gate.

In Figure 1, if the energy equation between the upstream and downstream sections (immediately downstream of the gate) is used, it can be written as:

$$h_0+\frac{{v_0}^2}{2g}=h_1+\frac{{v_1}^2}{2g}\to h_0+\frac{q^2}{2g{h_0}^2}=h_1+\frac{q^2}{2g{h_1}^2}$$E1

where V₀ is the mean velocity of water upstream of the gate, V₁ is the mean velocity of water in the contraction section, g is the acceleration of the earth’s gravity and q is the flow rate or discharge passing through the gate per unit width of canal.

In Eq. (1), the energy loss between the two sections has not been considered. By arranging Eq. (1) and simplifying, the result is:

$$h_0-h_1=\frac{q^2}{2g}\times \frac{{h_0}^2-{h_1}^2}{{h_0}^2{h_1}^2}\to q=h_0{h}_1\sqrt{\frac{2g}{h_0+h_1}}$$E2

According to the definition, the contraction coefficient (C_c) is equal to the depth of water in Section 1 (h₁) divided by the gate opening, i.e., C_c = h₁/a. Therefore, by setting h₁ = a × C_c in Eq. (2), we have:

$$q=C_{c}a\sqrt{\frac{2g{h_0}^2}{h_0+C_{c}a}}\to q=C_{c}a\sqrt{\frac{2g{h}_0}{1+C_c\left(\frac{a}{h_0}\right)}}$$E3

Most researchers including Swamee [6], Rajaratnam [11] and Clements et al. [22], agree with the value of 0.61 for the contraction coefficient in practice. By defining the discharge coefficient (C_d) as follows, Eq. (3) can be written as Eq. (5) [23]:

$$C_d=\frac{C_c}{\sqrt{1+C_c\left(\frac{a}{h_0}\right)}}$$E4

$$q=C_{d}a\sqrt{2g{h}_0}$$E5

When the tailwater rises, there is a possibility that a submerged condition will occur. In this case, the tailwater level affects the discharge and the discharge decreases with a specific gate opening. Swamee [6] presented the condition for creating submerged flow as inequality (6):

$$h_2<h_0<0.81h_2{\left(\frac{h_2}{a}\right)}^{0.72}$$E6

Laboratory results are used to determine the discharge coefficient. The oldest and probably the most famous method is related to Henry [24]. He presented Figure 3 to determine the discharge coefficient.

Figure 3.

Henry’s diagram [24] to determine the discharge coefficient for vertical sliding gates in free and submerged flow conditions.

Although Eq. (5) was obtained for free flow conditions, it can be used to determine the discharge under submerged flow conditions. The difference is that in this case, based on Henry’s diagram [24], the discharge coefficient will depend on two dimensionless factors h₀/a and h₂/a. Calculation examples will now be presented.

Example (1): The depth of water upstream of a vertical slide gate is 1.2 m and the gate opening is 0.2 m. The depth of the tailwater is 0.8 m and the water velocity immediately after the gate is 3 m/s. According to Henry’s diagram, determine whether the flow is submerged or free and the discharge.

Solution: Using the following two dimensionless parameters on Henry’s diagram (Figure 4), it will be seen that the flow is submerged. Therefore, Eq. (4) cannot be used to determine the discharge coefficient. On Henry’s diagram, the discharge coefficient for submerged flow in the above example is equal to 0.4. Then, the discharge is calculated using Eq. (5):

Figure 4.

Henry’s diagram uses with an example.

$$\frac{h_0}{a}=\frac{1.2}{0.2}=6\mathit{\text{and}}\frac{h_2}{a}=\frac{0.8}{0.2}=4\to C_d=0.4$$

$$q=C_{d}a\sqrt{2g{h}_0}=0.4\times 0.2\times \sqrt{2\times 9.81\times 1.2}=0.388{\mathrm{m}}^2/\mathrm{s}$$

Example (2): If we want the discharge coefficient to reach its maximum with the same depth of tailwater and gate opening (free flow conditions), what should the upstream depth be?

Solution: On the h₂/a = 4 curve, move upwards to cut the free flow curve. In this case, it can be seen that h₀/a = 8.7, so that:

$$\frac{h_0}{a}=8.7\to h_0=8.7\times 0.2=1.74\mathrm{m}$$

In addition, the discharge coefficient is obtained from Henry’s diagram to be equal to 0.58, which is consistent with formula (4) (Figure 5). That is, we can write:

Figure 5.

Henry’s diagram uses with an example.

$$C_d=\frac{C_c}{\sqrt{1+C_c\left(\frac{a}{h_0}\right)}}=\frac{0.61}{\sqrt{1+0.61\left(\frac{0.2}{1.74}\right)}}=0.589\approx 0.59$$

If, in the case of submerged flow (Figure 2), the two equations of specific energy and specific force are written at appropriate points, we will have:

• Specific energy equation between two sections, upstream and immediately downstream of the gate (contraction section):

  $$h_0+\frac{Q^2}{2g{\left(b{h}_0\right)}^2}=h+\frac{Q^2}{2g{\left(b{h}_1\right)}^2}$$E7

• Specific force between the two sections immediately downstream of the gate (contraction section) and the downstream section (tailwater):

$$\frac{h}{2}\left(\mathit{bh}\right)+\frac{Q^2}{g\left(b{h}_1\right)}=\frac{h_2}{2}\left(b{h}_2\right)+\frac{Q^2}{g\left(b{h}_2\right)}$$E8

It should be noted that in the above equations, for the section immediately downstream of the gate, the depth of the water jet h₁ is used for the depth representing velocity head, and the depth h is used to represent the piezometric height of the water.

Figure 6 shows the use of gates in irrigation canals. Figure 6a shows the use of a gate for delivering water to the secondary canal (minor canal), and Figure 6b shows the use of a gate for adjusting the water level (keeping the water level at a normal level).

Figure 6.

Application of gates in irrigation canals. (a) a canal intake equipped with a vertical slide gate, and (b) use of a gate for adjusting the water level.

Example (3): The required discharge for an irrigation canal, which is controlled by a slide gate, is equal to 11 m³/s. The width of the canal is 4 m and the height of the water in the dam reservoir is 2 m with respect to the bottom of the canal entrance. If the depth of the tailwater (h₂) in the canal is equal to 1.8 m, find the required opening of the gate. Is the flow from under the gate free or submerged?

Solution: By writing the two equations of energy and specific force at the appropriate sections and inserting the assumed information given in the problem, we will have:

$$2+\frac{{11}^2}{2\times 9.81{\left(4\times 2\right)}^2}=y+\frac{{11}^2}{2\times 9.81{\left(4\times 0.61\times a\right)}^2}\to 2.096=y+\frac{1.036}{a^2}$$

$$\frac{y}{2}\left(4y\right)+\frac{{11}^2}{9.81\left(4\times 0.61\times a\right)}=\frac{1.8}{2}\left(4\times 1.8\right)+\frac{{11}^2}{9.81\left(4\times 1.8\right)}\to 2y^2+\frac{5.055}{a}=8.193$$

By solving the above two non-linear equations, the following four pairs of solutions are finally obtained, the last solution is acceptable; a negative root is meaningless from the point of view of hydraulics:

$$y=2.086\mathit{\text{and}}a=-9.938$$

$$y=-3.126\mathit{\text{and}}a=-0.445$$

$$y=-0.408\mathit{\text{and}}a=0.643$$

$$y=1.448\mathit{\text{and}}a=1.264$$

Note: To solve the above two non-linear equations, the following website was used, and the diagram of the two equations and their roots are shown in Figure 7.

Figure 7.

Roots for non-linear equations.

https://www.wolframalpha.com/calculators/system-equation-calculator.

To check whether the flow is free or submerged, the criterion proposed by Swamee [6] is used:

$$h_2<h_0<0.81h_2{\left(\frac{h_2}{a}\right)}^{0.72}$$

$$1.8<2<0.81\times 1.8{\left(\frac{1.8}{1.264}\right)}^{0.72}\to 1.8<2<1.881$$

Because the condition 2<1.881 is not true, the flow under the gate is free.

Example (4): The flow rate of 4 m³/s enters the secondary canal through an intake installed in the main canal through a vertical slide gate. The width of the secondary canal with a rectangular section is 1.8 m, and the normal water depth (h₂) is 0.8 m. The normal depth of water in the main canal is 2.2 m. Calculate the required opening of the gate. Is the flow from under the gate free or submerged? What is the discharge coefficient of the gate?

Solution: By applying the two equations of energy and specific force (Eqs. (7) and (8)) at appropriate sections and placing the assumed data given in the problem, it can be written (Figure 8).

$$h_0+\frac{Q^2}{2g{\left(b{h}_0\right)}^2}=h+\frac{Q^2}{2g{\left(b{h}_1\right)}^2}\to$$

$$2.2+\frac{4^2}{2\times 9.81{\left(1.8\times 2.2\right)}^2}=h+\frac{4^2}{2\times 9.81{\left(1.8\times 0.61\times a\right)}^2}\to 2.259=h+\frac{0.676}{a^2}$$

$$\frac{h}{2}\left(\mathit{bh}\right)+\frac{Q^2}{g\left(b{h}_1\right)}=\frac{h_2}{2}\left(b{h}_2\right)+\frac{Q^2}{g\left(b{h}_2\right)}\to$$

$$\frac{1}{2}\times 1.8{\left(h\right)}^2+\frac{4^2}{9.81\left(1.8\times 0.61\times a\right)}=\frac{1}{2}\times 1.8{\left(0.8\right)}^2+\frac{4^2}{9.81\left(1.8\times 0.8\right)}\to 1.791\times a+0.9h^2=2.925$$

Figure 8.

Illustration of flow for example (4).

By solving the above two non-linear equations, the following four pairs of solutions are finally obtained, and the last solution is accepted. As before, negative roots have no physical meaning:

$$h=2.275+0.409i\mathit{\text{and}}a=-0.884-0.934i$$

$$h=2.275-0.408i\mathit{\text{and}}a=-0.844+0.934i$$

$$h=-1.553\mathit{\text{and}}a=0.422$$

$$h=1.266\mathit{\text{and}}a=0.828$$

The graph of the two equations and their roots are shown below (Figure 9).

Figure 9.

Roots for non-linear equations.

To check whether the flow is free or submerged, the criterion proposed by Swamee [6] is used.

$$h_2<h_0<0.81h_2{\left(\frac{h_2}{a}\right)}^{0.72}$$

$$0.8<2.2<0.81\times 0.8{\left(\frac{0.8}{0.828}\right)}^{0.72}\to 0.8<2.2<0.632$$

Because the condition 2.2 < 0.632 is not true, the flow is free.

The following equation is used to calculate the flow coefficient:

$$q=C_{d}a\sqrt{2g{h}_0}\to =C_d=\frac{q}{a\times \sqrt{2g{h}_0}}=\frac{4/1.8}{0.828\times \sqrt{2\times 9.81\times 2.2}}=0.409$$

Swamee [6] presented the Eqs. (9) and (10) using the experimental results of Henry [24]. Eqs. (9) and (10) are for free and submerged flow conditions under a vertical slide gate, respectively:

$$C_d=0.611\times {\left(\frac{h_0-a}{h_0+15a}\right)}^{0.072}$$E9

$$C_d=0.611\times {\left(\frac{h_0-a}{h_0+15a}\right)}^{0.072}{\left\{0.32\times {\left(\frac{0.81\times h_2{\left(h_2/a\right)}^{0.72}-h_0}{h_0-h_2}\right)}^{0.7}+1\right\}}^{-1}$$E10

Nasehi Oskuyi and Salmasi [25] produced 5200 data points by simultaneously solving two Eqs. (7) and (8) for different hydraulic parameters of vertical slide gates. By combining two Eqs. (7) and (8), Eq. (11) is obtained:

$$f\left(h_0,h_1,h_2,q\right)={\left\{h_0+\frac{q^2}{2g}\left(\frac{1}{{h_0}^2}-\frac{1}{{h_1}^2}\right)\right\}}^2+\frac{2q^2}{g}\left(\frac{1}{h_1}-\frac{1}{h_2}\right)-{h_2}^2$$E11

In the design of gates, the upstream water depth (h₀), the downstream water depth (h₂) and the discharge per channel width unit (q) are usually known. Therefore, by solving Eq. (11), it is possible to calculate the water depth immediately downstream of the gate (h₁) and then obtain the opening value of the gate with the relation a = h₂/0.61. It is clear that instead of solving Eq. (11), two Eqs. (7) and (8) can be solved simultaneously.

It should be noted that when solving Eq. (11), four roots are obtained for h₁. Two negative roots which are hydraulically meaningless. A complex root that is unacceptable and a positive root that will be acceptable.

Inequality (6) can be used to distinguish free or submerged flow conditions. If the flow is submerged, the piezometric water depth (h) downstream of the gate can be calculated from the energy Eq. (7).

The equations fitted to the data set are presented below, where Fr is the Froude number of the gate opening and is defined as follows:

$${F_r}^2=\frac{q^2}{g{a}^3}$$E12

For free flow conditions, the discharge coefficient was obtained from Nasehi Oskuyi and Salmasi [25]. In the following equations, R² is the coefficient of determination (the square of the regression coefficient):

$$C_d=0.4445{\left(\frac{h_0}{a}\right)}^{0.1289}{\left(\frac{1}{{F_r}^2}\right)}^{0.0107},R^2=0.806$$E13

$$C_d=0.44457{\left(\frac{h_0}{a}\right)}^{0.1219},R^2=0.7894$$E14

For submerged flow, the discharge coefficient was obtained from Nasehi Oskuyi and Salmasi [25]:

$$C_d=0.8275{\left(\frac{h_0}{a}\right)}^{0.077}{\left(\frac{h_2}{a}\right)}^{-0.9898}{\left(\frac{h}{a}\right)}^{0.1637}{\left(\frac{1}{{F_r}^2}\right)}^{-0.4132},R^2=0.9883$$E15

$$C_d=0.7482{\left(\frac{h_2}{a}\right)}^{-0.6825}{\left(\frac{1}{{F_r}^2}\right)}^{-0.3929},R^2=0.9831$$E16

$$C_d=0.3865{\left(\frac{h_0}{a}\right)}^{1.0676}{\left(\frac{h_2}{a}\right)}^{-1.4486},R^2=0.82$$E17

One of the significant points in the research of Nasehi Oskuyi and Salmasi [25] for free flow under the gate is hysteresis where two discharge coefficients are obtained for one Froude number. This phenomenon needs more investigation in theory and experiment (Figure 10).

Figure 10.

Variation of discharge coefficients versus Froude number for a gate and the phenomenon of hysteresis in the research of Nasehi Oskuyi and Salmasi [25].

2. Materials and methods

2.1 Hydraulics of radial gates

From the hydraulic point of view, the flow under a radial gate can be analyzed like the flow under a vertical slide gate, and Eq. (5) can be used to calculate the discharge. It should be noted that C_c and C_d will depend on more factors. The angle of the radial gate edge with horizontal (Ɵ) is a function of the height of the gate opening (G). In addition, the radius of the radial gate (R) and the height of the horizontal axis of the gate (a) are other important factors. The following empirical equation provided by Toch [26] can be used to determine C_c in free flow conditions:

$$C_c=1-0.75\left(\frac{\mathrm{Ɵ}}{90}\right)+0.36{\left(\frac{\mathrm{Ɵ}}{90}\right)}^2$$E18

where Ɵ is in degrees. For submerged flow conditions, C_d should be obtained from charts like Figure 11.

Figure 11.

Discharge coefficient for radial gates in free flow conditions [27].

The superiority of radial gates over sliding gates is that a lot of force is not required to open and close the gate during operation. In addition, the edge of the radial gates has an inclined surface compared to the horizon compared to the vertical gate edge, and this causes less compression in the flow downstream of the gate. As a result, C_c for radial gates is greater than 0.61.

In the overflow spillways with gates (gated spillways), gates are installed on the spillway crest. Each gate is a movable buffer surface that is used to regulate the flow through the spillway. Usually, sector-type (hinged) gates are used for this purpose. Sector gates rotate around an axis, and in this way, by adjusting the opening of the gate, the discharge from the spillway can be controlled.

The gated spillways discharge formula for the case where part of the gate is open is:

$$Q=C\times G_0\times B\times \sqrt{2\mathit{gH}}$$E19

where Q is the discharge (m³/s), C is the discharge coefficient, G₀ is the shortest distance from the lower edge of the spillway to the curve of the crest (m), B is the opening length of the gate (m) or the effective length of the spillway crest, g is the acceleration of gravity (m/s²), and H is the head on the center of the gate opening (m). H includes static head and approach velocity head.

The discharge coefficient (C) can be determined from Figure 11. There, β is the angle between the tangent line on the lower edge of the gate and the tangent line on the crest curve at the closest point of the crest curve, x is the horizontal distance between the gate seat and the crest apex. Figure 11 was prepared based on information from the US Army Corps of Engineers [27].

Using laboratory data, Shahrokhnia and Javan [28] provided relations to estimate the discharge coefficient in radial gates under free and submerged conditions. Among the relations presented in their study, which was the result of fitting the non-linear multivariable equation, Eqs. (20) and (21) had the best results. In these equations, H is the water depth upstream of the gate, y_t is the water depth downstream of the gate, φ is the angle of the gate edge with the horizon, and G is the gate opening (Figure 12).

Figure 12.

Flow under the radial gate and factors affecting the discharge coefficient.

$$\mathit{\text{Free flow}}\to C_d=0.46{\left(\frac{\phi }{90}\right)}^{-0.36}{\left(\frac{H}{G}\right)}^{0.12}$$E20

$$\mathit{\text{Submerged flow}}\to C_d=0.53{\left(\frac{\phi }{90}\right)}^{0.87}{\left(\frac{H-y_t}{G}\right)}^{0.33}$$E21

In Eqs. (20) and (21), the angle of the gate edge with the horizon is:

$$\mathit{\text{cosφ}}=\frac{a-G}{R}$$E22

$$\phi =\mathit{Arc}\mathit{cos}\left(\frac{a-G}{R}\right)$$E23

Figures 13–16 shows examples of radial gates.

Figure 13.

Downstream view of a radial gate.

Figure 14.

Movement of a radial gate installed on a dam spillway.

Figure 15.

Placement of the radial gate on the spillway of a dam and its components [29].

Figure 16.

Downstream view of a radial gate.

2.2 Dam gates

The Howell Bunger valve was introduced by Howell and Bunger in 1935. This valve was first used in El Vado Dam in Charna, New Mexico. This valve includes a conical part and discharges the outgoing water to the atmosphere by reducing its energy. The Howell Bunger valve is also known as a hollow jet. The flow of water toward the outlet of the valve is not convergent, and the discharge is in the form of a hollow jet. Strong interaction of air with water jet reduces the kinetic energy of water. The Howell Bunger valve is designed in such a way that it releases a large amount of flow without causing erosion on the river bed and vibration, so it is suitable for draining water from high dams.

A Howell Bunger valve consists of a shell, a cylindrical body, and a conical jet nozzle at the outlet of the valve. Disconnecting, connecting, and controlling the flow is done by the movement of the outer shell (shut-off sleeve) that surrounds the output cone. The primary sealing between the outer shell and the output conical surface by the metal-to-metal sealing system and the secondary rubber sealing system creates a reliable sealing system for this type of valve. In Howell Bunger valves, the jet has a circular shape symmetric to the center of the valve in all open positions of the valve. This water jet exits at a high velocity and becomes an open umbrella by the output cone. Due to the friction between air and water on a large surface, a mixture of air–water (white water) in the dam outlet will be observed. This phenomenon effectively consumes the energy of the water jet.

A hood can be installed at the end of the valve to reduce the risk of erosion caused by the water jet on the river bottom. A Howell Bunger valve without a hood is used in situations where the water jet coming out of the valve does not have an obstacle such as a concrete wall (Figure 17).

Figure 17.

Howell Bunger valve without a hood [30].

2.2.1 Howell bunger hooded valve

Where the water jet must be discharged into a channel to prevent the destruction of the concrete wall and flow of water back into the valve chamber, Howell Bunger hooded valves are used. The diameter of the hood pipe must be at least twice the nominal diameter of the valve, and the hood must be equipped with an aeration system (Figure 18). Figure 19 shows the geometric characteristics of a Howell Bunger valve.

Figure 18.

Howell Bunger valve with a hood.

Figure 19.

Geometric characteristics of Howell Bunger valve [30].

The chart below can be used to choose the diameter of the pipe or the flow rate of a Howell Bunger valve (Figure 20).

Figure 20.

Choosing the diameter of the pipe for the Howell Bunger valve [31].

Three Howell Bunger valves have been installed in the Dez Dam in Khuzestan province of Iran. In Figure 21, the exit of the water jet from these three valves can be seen.

Figure 21.

Jet flow through Howell Bunger valves in Dez Dam, Iran. (a) a close-up view of the three Howell Bunger valves installed in Dez Dam in Khuzestan province, Iran, and (b) creation of air-water mixture in jet flow through Dez Dam outlets.

2.2.2 Sleeve valve

Sleeve valves are often used as discharge valves at the ends of transmission lines, outlet of dams and hydropower plants for bypass discharge. Along with needle valves and Howell Bunger valves, sleeve valves also play a big role as energy dissipators, and for connecting and disconnecting and adjusting at the end of storage lines. With this valve, it is possible to keep the flow rate constant for a long time, or to increase the amount of fluid passing through it in case of an emergency, such as when a flood occurs, so that the walls of the dam are under less stress. Therefore, it is necessary to have stable conditions to prevent cavitation and damage to the buildings.

These valves are used as outlet control valves, and they control the downstream irrigation canals easily and at a relatively low cost. These valves are installed inside the concrete pond and the energy dissipation takes place inside the pond and the water is transferred to the irrigation canals after entering the pond. The amount of water entering the canals is controlled by a valve. In Iran, these taps are produced with diameters of 400–1200 mm.

In Figures 17 and 18, the sleeve valve installed inside the concrete pond and its components are presented (Figures 22 and 23).

Figure 22.

Sleeve valve for energy dissipation. (a) sleeve valve installed inside the concrete pond, and (b) a real sleeve valve.

Figure 23.

Components of a sleeve valve.

3. Results and discussion

3.1 Energy dissipation in vertical slide gates

Sluice gates are capable of measuring water discharge flow rates and also are able to adjust water levels in canals. Accurate calculations of energy changes and discharge are important to hydraulic engineers. As already discussed but worthy of revisiting, the flow can be either free or submerged – depending on the downstream water levels. Unfortunately, there is little research on energy changes for flow passing through sluice gates. Knowledge of ΔE is necessary for the design of intakes and irrigation canal inverts. Salmasi and Abraham [32] performed a series of experiments to quantify both energy changes and discharge. They showed that energy changes for free flow exceed that for submerged flow. Furthermore, discharge in free flow is greater than for submerged flow conditions. Relative energy losses (ΔE) range from 0.271 to 0.604; these energy losses cannot be ignored, and their impact on minor canal inverts should be considered. Non-linear regression (MNR) was used to predict both ΔE and C_d. The MNR method yielded suitably accurate predictions.

The correlation between relative energy losses with h₀ and a is shown in Eq. (24) with a determination coefficient of R² = 0.97 [32]:

$$∆E=0.604{\left(\frac{h_0-a}{h_0+0.174a}\right)}^{0.303}$$E24

Figures 23 and 24 show the contours of the discharge coefficient (C_d) and h₂/a, respectively. Figure 24 is similar to Henry’s [24] diagram; however, Figure 23 is easier to use. With values of h₀/a and h₂/a, available, C_d is directly available from Figure 23. On the other hand, Figure 24 requires interpolation between h₂/a contours. To improve readability, the values corresponding to the vertical axis in Figure 25 have been multiplied by 10.

Figure 24.

C_d variations with h₀/a and h₂/a.

Figure 25.

Contours of h₂/a vs. discharge coefficient (C_d) and h₀/a.

Figure 26 shows relative energy loss (ΔE, %) vs. h₀/a and h₂/a. For constant values of h₂/a, it is seen that as the upstream head (h₀) increases, the change in energy also increases (ΔE) – see the lower-right portion of Figure 26.

Figure 26.

Contours of relative energy loss (ΔE) vs. h₀/a and h₂/a.

A sluice gate results in relatively large energy losses – even when h₀/a is small. Increases in ho or decreases in an increase in the outlet jet intensity. This change in jet intensity causes downstream energy dissipation.

Energy loss depends on h₀/a and h₂/a (Figure 26). Taking measurements of h₀, h_2, and a are easier than calculations of V_2, as denoted by Habibzadeh et al. [33]. Habibzadeh et al. [33] presented energy loss coefficients (k) for determining C_d. They defined this loss as kV₂²/2 g, where V₂ is the mean velocity in the vena contracta. Constant values of 0.062 and 0.088 for k were obtained for free and submerged flow conditions, respectively. Insertion of V₂ into the analysis produces an implicit equation, and this increases the difficulty for using the proposed equation. However, the simple approach of treating k as constant for free or submerged flow conditions needs more investigation.

Figure 27 provides contours of relative energy loss (ΔE) vs. h₀/a and h₂/a [32]. The contours of (ΔE) make it possible to distinguish free flow from submerged flow. Rearrangement of Eq. (6) yields Eq. (25).

Figure 27.

Data used to identify free flow (below curve) and submerged flow (above curve) based on two variables of h₀/a and h₂/a [32].

$$\frac{h_2}{a}>1.13{\left(\frac{h_0}{a}\right)}^{0.581}$$E25

By selecting points where contours of ΔE change direction (Figure 27) and by fitting a curve through these points, Eq. (26) is obtained and is comparable with Swamee’s [6] relation (Eq. (25)). Eq. (26) is the preferred relation with determination coefficient R² = 0.9246:

$$\frac{h_2}{a}>1.725{\left(\frac{h_0}{a}\right)}^{0.437}$$E26

The curve in Figure 27 separates areas above (submerged) from areas below (free flow). There are some areas in Figure 27 with limited data, and those areas require further exploration. Such a paucity of data exists between h₀/a = 4.3 and h₂/a = 3 to h₀/a = 5.9 and h₂/a = 4.

The MNR model was used to estimate discharge coefficients in submerged flow conditions. During the regression mapping, the objective of minimizing the sum of the squares of errors (SSE) was applied, as shown here:

$$\mathit{SSE}=\sum _{i=1}^n{\left(O_i-P_i\right)}^2$$E27

where O_i is the observed value obtained from tests, P_i is the predicted value from MNR, and n is the number of observations:

$$C_d=\frac{0.312{\left(\frac{h_0}{a}\right)}^{1.220}}{1.569{\left(\frac{h_2}{a}\right)}^{2.387}}+\frac{0.729{\left(\frac{h_0}{a}-\frac{h_2}{a}\right)}^{0.570}}{{\left(\frac{h_0}{a}\right)}^{0.635}}$$E28

The accuracy metrics of Eq. (28) are R² = 0.992 and SSE = 0.048; excellent improvement in terms of R² and SSE is observed using this approach.

The accuracy metrics of Eq. (29) are R² = 0.999 and SSE = 10.184, respectively. Note that ΔE is calculated as a percentage (%):

$$∆E=\frac{92.158{\left(\frac{h_0}{a}-\frac{h_2}{a}\right)}^{0.985}}{{\left(\frac{h_0}{a}\right)}^{0.958}}$$E29

3.2 Design example

In order to elucidate this design methodology, a calculation example is provided. In this example, a main canal conveys water to an irrigation area. The normal water depth (h₀) is 1.5 m; a sluice gate is present with an opening of 20 cm. An intake delivers water for a secondary canal with 1.2 m normal depth (h₂). A calculation of the secondary bed invert if the main canal bed invert is 100 from mean sea level (MSL).

Solution: The head loss of the sluice gate is calculated using Eq. (19):

$$∆E=\frac{92.158{\left(\frac{h_0}{a}-\frac{h_2}{a}\right)}^{0.985}}{{\left(\frac{h_0}{a}\right)}^{0.958}}=\frac{92.158{\left(\frac{1.5}{0.2}-\frac{1.2}{0.2}\right)}^{0.985}}{{\left(\frac{1.5}{0.2}\right)}^{0.958}}=19.94\%$$

The water surface level in the main canal (M-WS_El.) equal to that in the main canal bed invert plus h₀, so that,

$$M-{\mathit{WS}}_{\mathit{El}.}=100+1.5=101.5\mathrm{m}\text{from}\mathrm{MSL}$$

The secondary canal bed invert (S-Bed_El.) is equal to the water surface level in the main canal (M-WS_El.) minus the energy loss through the sluice gate and minus the secondary canal normal depth (h₂). Thus,

$$S-{\mathit{Bed}}_{\mathit{El}.}=101.5-\frac{19.94}{100}-1.2=100.1\mathrm{m}\text{from}\mathrm{MSL}$$

The sluice gate discharge coefficient (C_d) can be written as:

$$\mathit{Eq}.\left(18\right)\to C_d=\frac{0.312{\left(\frac{h_0}{a}\right)}^{1.220}}{1.569{\left(\frac{h_2}{a}\right)}^{2.387}}+\frac{0.729{\left(\frac{h_0}{a}-\frac{h_2}{a}\right)}^{0.570}}{{\left(\frac{h_0}{a}\right)}^{0.635}}\to$$

$$C_d=\frac{0.312{\left(\frac{1.5}{0.2}\right)}^{1.220}}{1.569{\left(\frac{1.2}{0.2}\right)}^{2.387}}+\frac{0.729{\left(\frac{1.5}{0.2}-\frac{1.2}{0.2}\right)}^{0.570}}{{\left(\frac{1.5}{0.2}\right)}^{0.635}}=0.288$$

And finally, the discharge per unit width of canal is computed as:

$$q=C_{d}a\sqrt{2g{h}_0}=0.288\times 0.2\times \sqrt{2\times 9.81\times 1.5}=0.952\frac{m^2}{s}$$

Some of the other examples are presented in Table 1.

h0 (m)	a (m)	h2 (m)	h0/a	h2/a	ΔE (%), Eq. (19)	Main bed invert El. (m)	M-WSEl (m)	S-BedEl (m)	Cd, Eq. (18)	q (m^3/s/m)
1.5	0.2	1.2	7.5	6.0	19.94	100	101.5	100.10	0.288	0.952
1.5	0.3	1.2	5.0	4.0	19.72	100	101.5	100.10	0.314	1.040
1.2	0.2	1	6.0	5.0	16.56	100	101.2	100.03	0.272	0.804
1.2	0.3	1	4.0	3.3	16.38	100	101.2	100.03	0.301	0.890
1	0.2	0.8	5.0	4.0	19.72	100	101.0	100.00	0.314	0.849
1	0.3	0.8	3.3	2.7	19.51	100	101.0	100.00	0.352	0.952
0.8	0.2	0.6	4.0	3.0	24.42	100	100.8	99.955	0.381	0.920
0.8	0.3	0.6	2.7	2.0	24.15	100	100.8	99.958	0.436	1.054

Table 1.

Design examples results.

3.3 New research on gates

As mentioned earlier, because the force required to open the vertical slide gate is larger, hydraulic structure designers pay attention to radial gates as water level adjusting and flow control gates. Radial gates have a cylindrical shell. Therefore, the result of the water pressure on the gate passes through its axis and does not create a torque around it. As a result, the force required to pull up the gate should only be against the force of the weight of the gate. The superiority of the radial gate over the vertical sliding gate is the ease of operation and maintenance, no need for grooves in the supports, no need for wheels and other facilities such as rails and pulleys, and a significant reduction in force to open and close the gate. The height of radial gates varies from 2 to 14 m, and their opening width from 3 to 40 m. The maximum product of width and height is 300 m².

According to various studies, the presence of a sill on the bottom of the canal can improve the performance of the gate. Salmasi et al. [13] used different sill shapes (a circle, a semicircle, a triangle, a rectangle, and a trapezoid). Length, upstream slope, downstream slope, and sill height were investigated. The dependency of discharge on sill location was explored using multiple cases. In case 1, there is an open gate and the sill is located upstream of the gate. In the second case, the sill is located under the gate. There are a total of 43 physical models that were tested.

The results indicate that when the radial gate is open and when upstream sills are used (case 1), the sill presents a resistance to flow and reduces the discharge. In case 2, however, the sill location increases the discharge by ∼30%. The experiments also reveal that the rectangular and trapezoidal sills always increase the discharge. The magnitude of the effect depends on the sill length and height – small values of L/Z yield increase in discharge of ∼13%.

The radial gate used by Salmasi et al. [13] is shown in Figure 28. Figures 29 and 30 show different sill locations for both open and closed gates. In case 1, with an open gate, the sill was located upstream of the gate. The specific position of the sill is 5 cm upstream from the center of the gate. The second case, the sill is positioned under the gate.

Figure 28.

An example of the studied gate with a rectangular sill [13].

Figure 29.

Sill location under the gate (state 1). (a) sill location upstream of the open gate (case 1), and (b) sill location upstream of the closed gate (case 1).

Figure 30.

Sill location under the gate (state 2). (a) sill location under the open gate (case 2), and (b) sill location under the closed gate (case 2).

To determine the discharge coefficient, the Buckingham π theorem will be relied upon. Effective dimensional parameters relation can be rewritten as Eq. (30):

$$C_D=F_4\left(\frac{H-Z}{G},\frac{G}{R},\frac{G}{Z},\frac{L}{Z}\right)$$E30

Circular sills and semicircular sills, yield larger changes to the discharge coefficients (30% increase) than do other shapes.

Correlating equations can be developed to allow designers to calculate discharge coefficients during design selection. Some representative correlations are shown in Figure 31.

Figure 31.

Discharge coefficients for circular and semicircular sills [13].

These formulae are limited to ranges of 5.1 < (H-Z)/G < 26.4 for circular sills and 6 < (H-Z)/G < 26 for semicircular sills. Standard deviations for the circular-sill and semicircular-sill equations are, respectively, 0.047 and 0.30.

Salmasi and Abraham [34] conducted a series of experiments for the prediction of discharge coefficients for sluice gates equipped with different geometric sills under the gate using multiple non-linear regression (MNLR). Figure 32 indicates the longitudinal cross-section of a vertical sluice gate with a circular sill in a free flow condition.

Figure 32.

Free flow condition in the sluice gate [34].

Considering Figure 32 and its combination with energy conservation, Eqs. (31) and (32) are obtained [9]:

$$q=C_d.\mathrm{G}\sqrt{2g\left(H-Z\right)}$$E31

$$C_d=\frac{C_c}{\sqrt{1+\frac{C_c.G}{H-Z}}}$$E32

Here, the symbol q represents the discharge per unit width of the canal; other terms include:

• C_d is the gate discharge coefficient

• G is the gate opening

• g is the acceleration due to gravity

• H is the upstream water depth

• Z is the sill height

• C_c is the gate contraction coefficient

The ratio of water depth downstream of the sluice gate (y₁) to the gate opening (G) is defined as C_c. The minimum depth of y₁ occurs downstream of a sluice gate and is known as the vena contracta – a localized acceleration in a converging flow. Rajaratnam and Subramanya [10] noted that the vena contracta occurs at a distance of about 1.15 times the gate opening, G. Twelve different sills were studied. The cross-sections have five different shapes: triangular, trapezoidal, circular, semicircular, and rounded-faced sills. These sill shapes are illustrated in Figure 33.

Figure 33.

Different configurations of sills (all units are in cm); the flow direction is from left to right [34].

For sill 12 (polyhedral sill), the shape of the crest and downstream slope were the main shape parameters. A form factor based on the wetted perimeter (p) and hydraulic radius R_s = A/p was utilized (A is the sill cross-section area). Non-linear regression yields the following correlation:

$${\mathrm{C}}_{\mathrm{d}}=0.63{\left(\frac{{\mathrm{H}}_1-\mathrm{G}}{{\mathrm{H}}_1+15\mathrm{G}}\right)}^{0.067}\frac{{\left(1+\frac{\mathrm{Z}}{\mathrm{G}}\right)}^{0.366}{\left(1+\frac{{\mathrm{H}}_1}{\mathrm{P}}\right)}^{0.045}}{{\left(1+\frac{{\mathrm{R}}_{\mathrm{s}}}{\mathrm{G}}\right)}^{0.523}{\left(1-\frac{{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{H}}_1}\right)}^{0.413}}$$E33

Eq. (33) can be used for gates with or without a sill and for free flow conditions. The statistical measures used to characterize Eq. (33) are R² = 0.867 and RMSE = 0.042.

The range of validity for Eq. (33) is $0.0\le H_1/p\le 16.3$, $0.0\le R_s/G\le 0.13$, $0.0\le Z/G\le 0.4$ and $0.0\le R_s/H_1\le 0.04$.

With Eq. (33), C_d can be predicted with a maximum error of less than 6%. For gates without a sill, Z is zero, p is infinite, and H₁ is equal to H. Under these circumstances, Eq. (33) can be transformed into Eq. (34):

$${\mathrm{C}}_{\mathrm{d}}=0.63{\left(\frac{\mathrm{H}-\mathrm{G}}{\mathrm{H}+15\mathrm{G}}\right)}^{0.067}$$E34

Previous studies showed that the inclination of slide gates has a progressive effect on C_d and increases capacity under the gate. Salmasi and Abraham [35] experimentally determined C_d for inclined slide gates. They evaluated both free and submerged flows with inclination angles of 0, 15, 30, and 45 degrees and with different gate openings. The C_d is used to introduce equations either by classical multiple regression or genetic programming (GP) for predicting C_d. The increase in C_d relates to the convergence of the flow through the gate opening. Salmasi et al. [36] previously used genetic programming (GP) and artificial neural network (ANN) techniques for predicting discharge coefficients of compound broad-crested weirs. Akbari et al. [37] used GP to calculate C_d in Piano Key (PK) weir. A Gated Piano Key (GPK) weir was constructed and tested for discharge ranges of between 10 and 130 liter per second. Figure 29 provides a longitudinal cross-section of the inclined slide gate – the discharge coefficient for the slide gate dependency is provided in Eqs. (35) and (36) for free and submerged flows, respectively:

$$C_d=f\left(a,y,\beta \right)\to C_d=f\left(\frac{y}{a},\beta \right)$$E35

$$C_d=f\left(a,y,y_t,\beta \right)\to C_d=f\left(\frac{y}{a},\frac{y_t}{a},\beta \right)$$E36

In these equations, $\beta$ is the angle of the slide gate with respect to the vertical direction (Figure 34).

Figure 34.

Longitudinal cross-section of the inclined slide gate for free and submerged flow conditions [35]. (a) free flow condition, and (b) submerged flow condition.

Figure 35 is for inclination angle β =15 degrees. The discharge coefficients converge to 0.57 for larger values of y/a up to 20. This indicates an increase of 7.5% in C_d for submerged flow through the inclined gate relative to the vertical gate.

Figure 35.

Discharge coefficient variations against y/a for an inclined gate (β = 15°) for different values of y_t/a in a submerged flow.

The GP model was used to estimate the discharge coefficient. For GP applications, 70% of the data was used for model training, and 30% of the data was reserved for testing. In Eqs. (37) and (38), the units of $\beta$ are degrees. Table 2 shows results from the testing phase using the GP method [35]:

Equation No.	Testing phase		Training phase
	R^2	RE	R^2	RE
37	0.9431	0.0014	0.9488	0.0019
38	0.6546	−0.0861	0.6603	−0.1043

Table 2.

Comparison of evaluation parameters for the GP models.

$$C_d=0.33+\frac{\frac{y_t}{a}+\beta +8.86}{266.48\left(\frac{y_t}{a}+1\right)}+\frac{11.11\left(\frac{y_t}{a}-\frac{y}{a}+1\right)}{\left(\frac{y}{a}\right)\left(\frac{2y_t}{a}-44.57\right)}$$E37

$$C_d=1.19\left(3-\frac{y}{a}\right)-1-\frac{0.066}{\beta }+2\left(\frac{y}{a}\right)+\frac{\frac{y}{a}-10.16}{5.52}+\frac{0.61-\frac{y_t}{a}}{2.61\left(\frac{y}{a}\right)}-\frac{y}{a}$$E38

Figure 36 shows the comparison of C_d in the present study with the other studies. This figure includes C_d for the sluice gate in free flow conditions.

Figure 36.

Variation of C_d against y/a for free flow conditions and comparison with other studies [35].

Radi [38] collected four groups of data sets from previously published experimental work. These data were for free and submerged flow conditions under vertical sluice gates as well as for inclined sluice gates. The collected information on inclined sluice gates was extracted from Montes [39] and Nago [40]. The comparison is for 45 and 60 degrees of inclination. Radi [38] found different patterns, especially for β = 60^o. These patterns show five different curve trends for β = 60^o, each for a separate gate opening. On the other hand, gate opening (a) appears in the x-axis (y/a), and Radi [38] did not provide the reason for these separated trends. Some part of Radi [38]‘s collected data show good agreement with the present study for 1.5 < y/a < 15, but the remaining data do not show proper agreement (y/a > 15).

For submerged flow conditions, both y/a and y_t/a must be included in the determination of C_d, but Radi [38] only has provided y/a effects. Thus, a comparison with the present study was not available.

Notations

Q = sluice gate discharge (m³/s);

a = sluice gate opening (m);

b = width of gate (m);

q = discharge per unit width of sluice gate (m²/s);

h₀ = flow depth upstream of the sluice gate (m);

h₂ = tailwater or downstream water depth (m);

g = gravitational acceleration (m/s²);

C_d = discharge coefficient of the slide gate (dimensionless);

C_c = contraction coefficient (dimensionless);

E₀ = specific energy at the upstream of sluice gate (m);

E₁ = specific energy at the downstream of sluice gate (m);

ΔE = relative energy losses (dimensionless);

V₀ = mean velocity at the upstream of sluice gate (m/s);

V₁ = mean velocity at the downstream of sluice gate (m/s).