Introduction

The water retained in the upstream of various hydraulic control structures, such as road culverts, sluice gates, and spillways possesses a very high amount of potential energy. While flowing through the control structures, the potential energy of this retained water is mostly converted to kinetic energy at the base of the structures. Due to this, the flow of water downstream of these hydraulic structures is generally a supercritical flow. The highly energized supercritical flow enriched with powerful erosive actions moves fast at low depths downstream of the control structures. The energy of these supercritical flows, unless dissipated relatively in a shorter length of the channel, the surface of the channel and all other structures in the close vicinity of the control structures at their downstream remain under the threat of severe erosion. To prevent these structures from erosion, the energy of the powerful supercritical flow is reduced by practicing different methods like providing (i) Water Cushion (ii) Baffle Wall (iii) Biff Wall (iv) Deflector (v) Staggered Blocks (vi) Ribbed Pitching or Cellular Pitching (vii) Hydraulic Jump on Sloping Glacis. (i) Water cushion: In case of a drop spillway, a free-fall water jet impinges the downstream of the spillway with a high dynamic pressure which may cause significant scour in the downstream and in case of low air concentration, the jet may even cause significant scour in channel bed adjacent to the spillway or even downstream face of the spillway. The scour can cause significant stability problems in the dam body. A water cushion can be a practical tool to reduce the impact of the jet at the impingement area when the jet of water, instead of hitting the channel bed directly hits a pool of water (water cushion) created in the form of a cistern close to the downstream face of the spillway and thus dissipates its energy significantly. (ii) Baffle Wall: It is an obstruction in the form of a low-height wall constructed across the channel at a specified distance from the structure in its downstream due to which the water heads up just upstream of the baffle wall, which may work as a water cushion. Often, a baffle wall also facilitates the creation of a hydraulic jump in favorable flow conditions. (iii) Biff wall: The end wall of cistern created close to the downstream of face of the spillway consists of a vertical wall and a horizontal projection that extends into the cistern. The horizontal projection of the biff wall compels the water to flow back into the cistern instead of flowing out directly from the cistern. Thus, an additional obstruction is created to the fast-moving water down the fall over the hydraulic structures. As a result, the energy of flowing water is dissipated significantly. (iv) Deflectors: It is a short wall constructed at the end of a downstream apron to deflect the flowing water in the vertical direction, resulting in a reduced flow velocity. (v) Staggered Blocks: These are generally concrete rectangular blocks or cubes that are spaced out on the downward-facing horizontal apron. They function as an obstacle to the high velocity of flow and deflect it in a lateral direction, which causes a large loss of energy. These are rectangular blocks or cubes generally made of concrete arranged staggered on the downstream horizontal apron. They deflect the high velocity of flow in a lateral direction and act as an obstruction to the high velocity of flow, resulting in the dissipation of energy significantly. (vi) Ribbed pitching or cellular pitching is provided on the downstream side of the fall, which roughens the wetted perimeter of the channel, adding an increase in frictional resistance to the velocity of flow, resulting in loss of velocity of flow leading to significant energy dissipation of the flowing water. (vii) Hydraulic jump or standing wave is a hydraulic phenomenon in which a supercritical or hypercritical flow is converted to a subcritical flow resulting in significant amount of energy dissipation of the flowing water.

Of all the above approaches and many others, the dissipation of energy of supercritical flows through hydraulic jumps is widely used as most of the approaches have their limitations, especially in small hydraulic structures. Through a hydraulic jump, highly energized pre-jump supercritical flow quickly changed into a substantially less energized flow (mostly subcritical) relatively in a shorter channel length. In the process of transition from a higher energy level to a lower energy level during a hydraulic jump, the entropy rises as kinetic energy and is transformed into potential energy, as stated by the laws of thermodynamics. As a result, the transition process involves an energy dissipation mechanism. Since there are no boundary features that can help break up streamlines, the depth must change on its own from low to high, causing the flow to slow down and create turbulent eddies to get rid of its kinetic energy. Stilling basins are typically employed in the downstream of the hydraulic structures to establish and maintain a hydraulic jump in order to reduce the exit velocity and dissipate the surplus energy, associated with the supercritical flow (Chow 1959). In addition to this hydraulic jump-type energy dissipaters, drops (Chanson 1999; Carvalho and Leandro 2011), baffled outlets, vertical stilling wells, and sky jumps (Burgi 1975) are some impact-type energy dissipators. In the recent past, screens have been examined as an alternative in dissipating the energy of the supercritical flows for small hydraulic structures. Even though numerous designs of hydraulic structures for dissipating energy match the practical requirements, the researchers recommended using screen walls to replace these hydraulic structures because of their performance and ease of use. An early investigation into a head loss for using screens with various shapes of openings and their placement in the open channel predicted the head loss coefficient (Karrh 1950). Numerical solutions for the mass and momentum balance equations through the screen model have been checked with the experimental model by Koo and James (1973). The test shows an agreement between the two results. Rajaratnam and Hurting (2000) investigated a screen having a square-shaped opening with a porosity of 40% as an energy dissipator in single, double wall mode as well as a triangular screen. In their experiment, they placed the screen at a distance of 1.25 m from the gate and tested it with a fixed gate opening of 24.5 mm. The Froude number (F1) ranged from 4 to 13 in this investigation. ÇCakir (2003) used screens with 20% and 60% porosities and circular shapes of openings to dissipate the excess energy of the supercritical flow having Froude numbers (F1) 5–18. Energy dissipation for flows with Froude numbers (F1) ranging from 5 to 22.5 was investigated by Bozkus and Aslankara (2008) using screens with the circular shape of openings and porosity of 40%. Sadeghfam et al. (2014) investigated dual vertical screens for energy dissipation using two different porosity ratios and reported that dual screens are more effective in reducing energy for free or submerged hydraulic jumps. He used screens with the shape of circular openings of 40% and 50% porosity to dissipate energy from supercritical flows with Froude numbers (F1) ranging from 2.5 to 8.5. Shaker et al. (2013) used circular, square, and hexagonal shapes of openings in screens with 40% porosity to dissipate extra energy in supercritical flows having Froude numbers (F1) ranging from 8 to 15. Daneshfaraz et al. (2017) used Flow-3D software to study how supercritical flows loose energy for a range of Froude numbers (F1) from 2 to 10 using the circular shape of the opening in the screen and porosity of between 40 and 50%. Abbaspour et al. (2019) investigated energy reduction for supercritical flows with Froude numbers (F1) ranging from 4.5 to 10.6 using a screen with square shape openings and 50% porosity. In a series of laboratory tests, Daneshfaraz et al. (2019a, b) used a screen with the circular shape of openings with porosities of 40 and 50% to dissipate the energy of supercritical flows with Froude numbers (F1) ranging from 5 to 18. According to the studies, use of screens increased energy dissipation efficiency by 400–900% compared to a plain inclined drop (Norouzi et al. 2011). Singh and Roy (2022) used screens both single and double wall mode with triangular shape of opening having porosity of 45% per unit depth to reduce the energy of supercritical flow with Froude number (F1) varied from 3 to 19 and reported that screens with triangular shape of openings are also equally efficient with screens having circular and square shape of openings. Figure 1 presents the details of the above investigations and their findings in terms of relative energy dissipation by using perforated screens in the case of small hydraulic structures. The relative energy dissipation for supercritical flows corresponding to the Froude numbers (F1) in a conventional hydraulic jump is also shown in Fig. 1.

Fig. 1
figure 1

Comparison of relative energy loss with respect to the Froude number (F1) as reported by the previous investigators

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While flowing through the screen openings of any shape, the jet of water leaves the screen having its shape similar to the shape of the screen openings. Thus, the shape of the jet of water through the screen openings remains the same while flowing through the single or multiple layers of screens with similar shapes. The present study investigated energy dissipation performance by using a triple layer of screens, with each layer having a different shape of openings and spaced at a sufficient gap between the screen layers. The gap between the screens allows the water jet to change its shape while flowing through the first screen to the second screen and finally through the third screen. The shapes of openings considered for the screens are circular, square and triangular, with the same area for each opening. It may be noted that for the circular, square and triangular shapes of openings, the perimeter is minimum in the case of circular followed by square and triangular shapes. The screen combination for the triple layers the placing of the screens has been considered as (i) most upstream screen with circular shape openings, followed by square and triangular shape openings, (ii) screen with square shape openings followed by a screen with triangular shapes and circular shape openings, (iii) screen with triangular shape openings followed by a screen with circular and finally screen with square shape openings and vice versa of the combinations. Different investigators used screens with the same opening shape having porosities between 40 and 50%. The current investigation employs the screen with different opening shapes in each layer and a porosity of 45% per unit depth of the screens for each of the layers.

Theoretical Background

Figure 2 shows the definition sketch of the phenomenon and the water depth upstream and downstream of the screen for the supercritical flow under sluice gates, which could be forced to transform its kinetic energy into other forms of energy.

Fig. 2
figure 2

Definition sketch

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The specific energy at any section of the flowing fluid having depth of flow y, velocity of flow v can be expressed as Eq. (1).

$$E=y+\frac{{V}^{2}}{2g}$$
(1)

Equation (1) in terms of Froude number (Fr) and flow rate per unit width (q) can be written as

$$E=\sqrt[3]{{\frac{q}{g{F}^{2}}}^{2}}\left(1+\frac{{F}^{2}}{2}\right)$$
(2)

The relative energy loss between Sects. 1 and  2, as shown in Fig. 2, can be written as Eq. (3)

$$\frac{E1-E2}{{E}_{1}}=\frac{\Delta E}{{E}_{1}}=\frac{\left[{y}_{1}+\left(\frac{{V}_{1}^{2}}{2g}\right)\right]-\left[{y}_{d}+\left(\frac{{V}_{2}^{2}}{2g}\right)\right]}{\left[{y}_{1}+\left(\frac{{V}_{1}^{2}}{2g}\right)\right]}$$
(3)

Experimental setup

The experiments have been performed in a horizontal laboratory channel of uniform rectangular cross section, 0.30 m wide, 0.45 m deep and 6.0 m long. The sidewalls walls of the channel are made of glass and the bed made of steel coated with paint. The channel is equipped with a gear lifting mechanism to adjust the slope. It is connected to a head tank fitted with a sharp-edged sluice gate of adjustable opening at its upstream end and a tail water gate at the downstream end. A reservoir tank installed at the downstream end of the channel collects the water leaving the channel. A pipeline fitted with an orifice meter supplies water from the reservoir tank to the head tank. A regulating valve is provided in the pipe line between the orifice meter and the reservoir tank for feeding water into the head tank at a controlled rate. During experiments, water flows at constant rate into the head tank without allowing any overflow and subsequently flows through the sluice gate into the channel. The reservoir tank at the downstream end of the channel collects the water from channel which is re-circulated into the head tank through the connecting pipeline. Thus, the channel is allowed to flow continuously at regulated flow during the experiments. The perforated screen is made of 6 mm thick acrylic sheet, 300 mm wide and 200 mm deep into which a series of openings were drilled, maintaining 45% porosity per unit depth of the screen as shown in Fig. 3. The openings in screens are drilled in the shape of equilateral triangles of sides 7.6 mm, circular shape of radius 2.8 mm and square shape of side 5 mm, respectively. The openings in the screens are arranged in a rectangular grid spaced at 5.5 mm horizontal distance and 10 mm vertical distance between the consecutive openings. Three such vertical layers of screen are used in the experiment maintaining a gap of 25 mm between the consecutive layers with the layer being placed at a distance of 1.5 m downstream of the sluice gate. Placement of screen from sluice gate was estimated such that it is with in the transition zone of classical jump length. It is ensured that toe of the jumps forced due to screens are at sufficient distance from sluice gate so that flow through the sluice gate is not affected. It is observed that the flow leaves the screen as a supercritical flow with reduced Froude Number. The gap between the screen was chosen such that no jump is created in the gap.

Fig. 3
figure 3

Perforated screen with a circular, b square and c triangular shape openings

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Figure 4 shows the experimental setup used in the present study. During the experimental runs supercritical flows of different Froude numbers, F1 were generated in the channel in the downstream of the sluice gate by adjusting the sluice gate opening. When the screens were approached and hit by these supercritical flows forced jumps were created by the screens result in insignificant loss of energy. During experiments, the tailgate was left entirely open, resulting in primarily supercritical flows with significantly reduced Froude number, F2 (< < F1) in the downstream of the screen. There was significant mixing of turbulence in the pool of water created between the perforated screen layers. In all of the cases of experimentations the flow downstream of the screens were uniform flows since they were created by large number of turbulent jets through the successive layer of the screen openings. Figure 5 shows the forced hydraulic jump by screens.

Fig. 4
figure 4

Experimental setup

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Fig. 5
figure 5

Forced hydraulic jumps by the screens

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A series of tests were carried out for different values of F1 varying from 3.2 to 19.3 and vertical triple screen layers with every layer having the opening shapes either circular (C), square (S) or triangular (T). Column 1 of Table 1 shows the arrangement of screen layers starting from most upstream position (1.5 m downstream of the sluice gate) to downstream position. A point gauge and pre-marked levels on channel walls were used to measure the flow depth y1 of the pre-jump supercritical flow between the sluice gate and jump toe, the flow depth, y2 downstream of the screen, maximum height of the jump, ym at the screen.

Table 1 Tailwater depth deficit parameter with respective upstream Froude Number (F1)
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Dimensional analysis

For the present study the relationship between the energy dissipation and other depending parameters is expressed as Eq. (4) for dimensional analysis to evaluate the functional relationship among the parameters.

$$ \Delta E = \, f_{1} \left( {Q, \, a, \, b, \, V_{1} , \, y_{1} , \, y_{2} , \, y_{m} , \, X, \, S_{s} , \, S_{g} , \, P, \, t, \, g, \, \rho , \, \mu } \right) $$
(4)

where ∆E is total energy dissipation between Sects. 1 and 2 due to the screen, whose dimension is [L]. Q is the discharge [L3 T−1], a is the gate opening [L], b is the width of the channel [L], y1 is the upstream depth at Sect. 1 [L], y2 is the downstream depth [L], X is the distance between the screen and the gate [L], Ss is the shape of opening of the screen [L], Sg is the gap between the screen [L], P is the porosity of the screen[M0L0T0], t is the thickness of the screen [L], g is the acceleration due to gravity [LT−2], ρ is the water density [ML−3], µ is dynamic viscosity of the water [ML−1 T−1].

Similar to Eq. (4), Eqs. (5–7) are also expressed.

$$ E_{1} = \, f_{2} \left( {Q, \, a, \, b, \, y_{1} , \, g} \right) $$
(5)
$$ L_{j} = \, f_{3} \left( {Q, \, a, \, y_{1} , \, g} \right) $$
(6)
$$ F_{2} = \, f_{4} \left( {Q, \, b, \, y_{2} , \, g} \right) $$
(7)

where E1 is the specific energy at Sect. 1, length of free hydraulic jump (Lj), and Froude number at Sect. 2 (F2) can be described as the following functions:

Thus, substituting b, y1 and yd into Eq. (4) for E1, Lj, F2, respectively, leads to:

$$ \Delta E = \, f_{5} (Q, \, a, \, E_{1} , \, L_{j} , \, F_{2} , \, X, \, S_{s} , \, S_{g} , \, P, \, t, \, g,\rho , \, \mu ) $$
(8)

Using Buckingham’s theorem and with ρ, g, and y1 as repeated variables, non-dimensional parameters can be extracted as:

$$ \frac{\Delta E}{{y_{1} }} = f_{6} \left( {\frac{{E_{1} }}{{y_{1} }},F1,\frac{{L_{j} }}{{y_{1} }}, \frac{X}{{y_{1} }}, P, \frac{{S_{s} }}{{y_{1} }} \frac{{S_{g} }}{{y_{1} }},\frac{t}{{y_{1} }}\frac{{y_{1} }}{a},\frac{{E_{1} }}{{y_{1} }},{\text{Re}} } \right) $$
(9)

Here, Re is the upstream Reynolds number, F1 is the upstream Froude number. Equation (8) can be rewritten as follows:

$$ \frac{\Delta E}{{E_{1} }} = \, f_{7} \left( {F_{1} , \, P,\frac{{S_{s} }}{t},\frac{{S_{g} }}{t}\left| {F_{2} ,\frac{{E_{1} }}{a},\frac{{L_{j} }}{{y_{1} }},\frac{X}{a}, \frac{t}{a}, C_{c} Re} \right|} \right) $$
(10)

In present study the parameters F2, \(\frac{{E}_{1}}{a}\), \(\frac{X}{a}\), \(\frac{{L}_{j}}{{y}_{1}}\) are out of the current study, these parameters can be considered as a case in future work scope. Since in our investigation the type of flow is free surface flow, in which gravitational effects are more dominant, the importance of the Reynolds number is secondary and it resultantly is neglected. The thickness parameter of screens was introduced as an insignificant one in the literature (Cakir 2003; Balkis 2004; Bozkus et al. 2005), so \(\frac{t}{a}\) can be neglected.

The final dimensionless equation is constructed after taking into account all of the previously supplied explanations and deleting the previously described parameters as presented in Eq. (11). It is evident that Froude number of supercritical flow at upstream (F1), screen porosity P, triple wall mode arrangement \(\frac{{S}_{g}}{t}\) and shape of the screen openings \(\frac{{S}_{s}}{t}\) play an important role in the present study.

$$ \frac{\Delta E}{{E_{1} }} = f_{7} \left( {F1, P,\frac{{S_{s} }}{t}, \frac{{S_{g} }}{t}} \right) $$
(11)

Experiment result

Equation (3) is used to calculate the total relative energy dissipation ∆E/E1 in all cases between Sects. 1 and 2, as shown in Fig. 2. Equation (1) or Eq. (2) which subscript 1 and 2 represents the energy at Sect. 1 upstream of the screen and at Sect. 2 downstream of the screen, respectively; ∆E is the relative energy loss between E1 and E2; y1 represents the flow depth in Sect. 1; similarly, y2 represents the flow depths in Sect. 2, while V1 and V2 represent the mean flow velocity for the corresponding locations. Figure 6 depicts the relationship between the supercritical Froude number F1 and the relative energy loss ∆E/E1 for mixed shape openings in triple wall mode. Figure 6 also depicts the variation of ∆E/E1 with respect to F1 corresponding to the classical hydraulic jump.

Fig. 6
figure 6

Relative energy loss, ∆E/E1 with respect to F1

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The relative energy loss presented in Fig. 6 depicts that the energy loss due to screen in triple wall is more than that of classical hydraulic jump. Furthermore, the energy loss found in the present study is comparable to that of Rajaratnam and Hurting (2000). In all cases, the flow was supercritical just downstream of the screens. The value of the Froude number, F2, downstream of the screen are in the range of was 1.1 to 1.81, with an average value of 1.35. It was found that the maximum flow depth, ym occurred at the screens and was about 1.35y2*. Figure 7 compares the downstream Froude number (F2) corresponding to the pre-jump supercritical upstream Froude number (F1) for the perforated screen with mixed shape openings. It is noticed from Fig. 7 that, like ∆E/E1, F2 is also increases linearly with an increase in F1 for the mixed shape of openings in triple wall mode.

Fig. 7
figure 7

Variation of downstream Froude number (F2) vs upstream Froude number (F1)

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Table 1 shows the tailwater deficit parameter D, calculated by using Eq. (12) (Rajaratnam and Hurting 2000) for all the experiments. In Eq. (12), y2* represents the subcritical sequent depth for classical hydraulic jump corresponding to u/s supercritical flows with Froude number F1 is obtained from the Ballenger equation. Parameter D indicates the deficit in tailwater depth in comparison to y2* and its value could vary from 0 to almost 1. D equals to 0 indicates the tailwater depth becomes equal to y2*, a free hydraulic jump. In the present study the value of D varied from 0.66 to 0.90, with an average of 0.82. Table 1 presents the values of parameter D while Fig. 8 presents a graphical representation of D with respect to F1.

Fig. 8
figure 8

Parameter D vs Froude number (F1)

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$$D=\frac{\left({y}_{2}*-{y}_{2}\right)}{{y}_{2}*}$$
(12)

Further, the experiments are extended by increasing the tailwater depth with help of tailwater gate provided downstream end of the channel. During the experiments, peripheral free jumps were observed downstream of the perforated screen due to supercritical flow downstream of the Froude number F2 and depth y2 for a tailwater depth yt equal to a sequent depth corresponding to F2 and y2. For the series of experiments D = 0.66, F2 = 1.11 and yt/y2* approximately equal to 0.52. In this case, the additional energy loss in the second jump is approximately 5.2%. Table 2 provides a summary of the experimental flow condition under investigation. Figure 9 presents the comparison in energy dissipation for present and earlier studies.

Table 2 Details of experimental flow conditions for hydraulic jump studies
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Fig. 9
figure 9

Screen performance comparison between current and former studies

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As a practical matter, concrete walls should be used to dissipate energy beneath small hydraulic structures such as drops, chutes, small dams, and other similar structures. Concrete walls should be replaced with an evenly distributed grid of circular, triangular and square holes with an altitude equal to about 0.3–0.7 of the depth of the supercritical stream and a porosity of approximately 45%. The scaling considerations of the Froude law would serve as the similarity criteria. There may be significant differences in the flow patterns through the screens used in the laboratory study and through concrete walls with holes used as a prototype in the field with the same porosity because the Reynolds numbers of the jets leaving the first screen and entering the pool between the screens in the prototype are estimated to be in excess of 105. In contrast, the jets running through screens in the laboratory were turbulent for small-scale investigation, with Reynolds values ranging from 41,000 to 81,000. On the other hand, the global behavior of turbulent jets is not dependent on the precise value of the Reynolds number by Rajaratnam (1976). Second, when using thin screens in a laboratory investigation, the jets have little contact with the sidewalls of the holes in the screens. Despite this, the jets make greater contact with the hole sides when the concrete walls are quite thick, resulting in some energy loss due to friction. However, this is little as compared to the energy lost in the jumps upstream of the displays, which are many. The height of the wall should be at least equal to the subcritical sequent depth y2*, which corresponds to the design values of F1 and y1 to prevent flow over the wall. For twin screens, the distance between the two baffle walls may be equal to y1. The length of the basin with the porous baffle is roughly 2.7y2*, which is the same as the length of the Type III basin by Peterka (1958). Furthermore, suppose the flow contains a large number of debris. In that case, it will limit the porosity and cause it to behave like a weir, which, although it may not have an impact on energy dissipation, the flow over screens may cause erosion below the screens themselves. The installation of a debris boom on the structure's downstream side and its regular maintenance is thus required.

Conclusion

Laboratory experimental results show that multilayer screens with circular, square and triangular shape openings and a porosity of approximately 45% per unit depth of screen for each layer can be used as effective energy dissipators under hydraulic structures in mixed multi-wall mode. Mixed multi-wall screen mode's performance in energy dissipation is greater than the energy dissipation for the classical hydraulic jump. The current experimental work investigated the use of mixed type multi-wall mode for energy dissipation for Froude numbers ranging from 3.2 to 19.3 for the supercritical flow approaching the screen. For the range of Froude number 3.2–13, the performance of the multi-wall mode mixed type screen is almost similar to that of other screens that have been documented in the literature. However, for higher values of Froude number of approaching flow, the performance is marginally better than that between 3.16 and 13.

The flow downstream of the screen remains supercritical, with Froude numbers ranging from 1.01 to 1.81 and tailwater depths 0.12 to 0.37 times y2*, where y2* is subcritical sequent depth in the case of classical hydraulic jump. The energy dissipation for the screen arrangement CST is marginally more than that of any other screen arrangement like STC, TSC, etc. In the most of the previous studies including both computational and experimental the performance of the screen toward energy dissipation is investigated for the range of the Froude number F1 narrower than present study for similar experimentations. The range of F1 used for investigation by Rajaratnam and Hurting (2000) was 4–13 while it is 2–10 by Daneshfaraz et al. (2017). For the present investigation F1 varies from F1 3.16 to 19.3. Percentage energy loss corresponds to F1 = 4 in case Rajaratnam, Daneshfaraz and present study is 55%, 64% and 74%, respectively. Corresponding to F1 = 10 the respective values are 88%, 88% and 90% while for F1 = 13 Rajaratnam reported 90% energy dissipation and present study finds it approximately 94%. Further, tailwater deficit parameter D found to be varying the range 0.66–0.90.