Mitigating the after-effect of climate-induced disaster through energy dissipation using stepped storm waterway in hilly roads

C_mean

cross-sectional mean void fraction

cross-sectional mean void fraction for nth flow depth

d_p

water depth in step cavity (m)

Fr

Froude's number at the inlet of the channel

g

gravity acceleration constant (m²/s)

h

step height (m)

h_i

equivalent clear water flow depth at different locations (m)

H_cha

height of the channel (m) above the step edge from where measurements are taken (Nh)

H_res(n)

residual energy at nth equivalent clear water depth

L_jet

distance from the start of a step cavity to the midpoint of the impinging jet (m)

L

length of the channel (m)

l

length of the step (m)

N

number of steps

Q

flow rate (m³/s)

q_w

specific flow rate (m²/s)

R_e

Reynolds number is defined in terms of the hydraulic diameter

U_w

equivalent clear water flow velocity (m/s)

w

channel width (m)

W_r

(q/h_o)²(h sin(θ))/ Weber number

X

coordinate perpendicular to horizontal step face at the end of Step-1 (m)

y_c

critical flow depth (m)

Y₉₀

bulked water depth where air concentration C = 0.90

Y₉₈

bulked water depth where air concentration C = 0.98

Y₉₉

bulked water depth where air concentration C = 0.99

surface tension (N/m)

θ

slope of the channel

ν

kinematic viscosity (m²/s)

The changing patterns of the atmosphere, ocean systems, and land as well as their relationships with one another at both global and local levels are being changed by the rise in worldwide temperatures (Wang et al. 2004; Soden & Held 2006; Vecchi & Soden 2007; Collins et al. 2010). Additionally, there has been a significant reduction in rainfall frequency (or wet days) during the Indian summer monsoon (ISM) over the past century. The prime differences are reflected as an alteration of the period of rainfall, increased intensity, or reduction in various rainfall factors. The agricultural system across the globe as well as the system of water resources including in India are being negatively impacted by changes in the frequency and intensity of precipitation, which are causing typical phases of disastrous floods in some regions and extreme drought in others (Arnell 1999; Lobell et al. 2011; Schiermeier 2011; Dai 2013; Trenberth et al. 2014; Roxy et al. 2017). As a result, a greater number of studies have been conducted over the last few decades to gain insights into the changing behaviour of the precipitation system over India. Rainfall patterns in India are altering with unusually catastrophic floods affecting areas of the Himalayan region specifically in Uttarakhand and Himachal Pradesh.

To address this lack in highway design, a stepped storm waterway downstream of the underpass rainwater drainage is suggested as an alternative solution. Its efficiency allows it to dissipate the huge potential energy of heavy rain throughout its entire length, thereby stabilizing the natural slope of the embankment. Stepped waterways have several applications such as sewers, water treatment plants, reoxygenation in canals, emergency spillways over embankment dams, road gutters, and city stormwater structures (Robison 1994; Chanson & Toombes 1997, 2002a; Chinnarasri & Wongwises 2004). Five artificial cascades were built near Chicago along a waterway system in order to help the process of reoxygenation of a dirty canal (Robison 1994). Some of the properties of stepped storm waterway systems, drains, and culverts for different slopes as well as geometries were experimentally tested (Chanson & Toombes 1997, 2002a). Stepped chutes are advantageous for various fields of civil engineering such as mountain drainage systems and emergency spillways over embankment dam downstream faces (Chinnarasri & Wongwises 2004). All these findings from the literature provide a clear image of its applicability in designing an excessive rainwater drainage system for highways in the Himalayan Mountain ranges. This study examines some of the design requirements for stepped channels as well as the hydraulic features associated to gain additional insights into them.

Numerous studies have been conducted to trace the correct range of C_mean in different regions. In experimental analysis, Toombes (2002) investigated that for low-gradient spillways the mean air concentrations vary from 40 to 50%. Similarly, observations made by Murillo Muñoz (2006) found C_mean as 35% for a 11° slope, and Takahashi et al. (2007) measured C_mean as 57% for a 19° slope for nappe flow. Detailed observation made by Felder et al. (2019) found C_mean to range from 0.05 to 0.65 at different locations of the spillway. However, Renna & Fratino (2010) took C_mean as 0.5 to calculate equivalent clear water depth. A detailed measurement of C_mean for nappe flow was well described by Toombes & Chanson (2008a), where C_mean for free-falling nappe varies from 0.05 to 0.3, and for spray region, from 0.25 to 0.5. However, the analysis by Toombes & Chanson (2008a) and Felder et al. (2019) took Y₉₀ as the upper limit of the free-surface level to evaluate C_mean. Taking the free-surface level of nappe flow as Y₉₀ will underestimate some of the aeration characteristics as bulking of water is more in the nappe flow compared with the skimming flow.

However, it is observed that many methods of estimating mean air concentration are given in the literature (Toombes & Chanson 2008a; Renna & Fratino 2010; Felder et al. 2019). Therefore, it is necessary to identify the mean air concentration and other properties of the jet with respect to its location at various steps. Since the steps are the key factor of energy dissipation, further designing the steps for better efficiency will help. Equations (1) and (2) are a few fundamental expressions proposed on the stepped channel to predict energy dissipation where y_c/h and height of the channel are the most significant parameters to predict energy dissipation. Further studies on multiple parameters are needed to predict energy dissipation for various channel geometries.

As nappe flow is the combination of a series of jets, the first objective of this study is to find out the different jet properties, such as jet length, pool water depth, and mean air concentration, at different locations. Hence, the study determines the behaviour of jets (L_jet, d_p) at various locations of a single step as well as at different steps. The second objective is to identify the appropriate free surface of the aerated flow by taking three different bulked water depths such as Y₉₉ (water depth where air concentration C = 0.99), Y₉₈, and Y₉₀. It also includes the effect of different aerated flow depths on air–water flow properties and energy dissipation. Further, to provide more clarity, the air–water flow characteristics at various phases of a jet and various locations are also presented.

As seen from the literature (Table 1), several experiments were conducted in nappe flow by considering different geometries of a channel such as height of the step (⁠ length of the step (l), critical depth of flow (⁠⁠), number of steps (N), width of the channel (w), and maximum height of the channel. Therefore, the third objective of this research is to identify the set of input parameters and their contribution in estimating energy dissipation at various steps and to develop a functional relationship that can predict energy dissipation more precisely.

Three different locations are selected in each step for measuring flow properties of the channel as shown in Figure 3(c). Section-1 is selected at the inward side of the jet, Section-2 at the outward side of the jet, and Section-3 at the step edge. The corresponding equivalent clear water depth is denoted as h₁, h₂, and h₃. The ranges of the different variables involved in the experimental program are given in Tables 2 and 3, which include the direct measurements of air–water flow properties and jet properties. According to the literature, stepped channels are used for two distinctive purposes, i.e., aeration and energy dissipation. Similarly, the parametric study of these two criteria is also important. That's why the current study includes all the jet properties and air concentration (C) on the rigid stepped channel. All the experiments are conducted on each step by taking three different sections such as free-falling region (Section-1/inward side of the jet), nappe spray region (Section-2/outward side of a jet), and step edge (Section-3).

The probe was vertically supported by a trolley to measure flow depth normal to the flow direction. An error of less than 0.028 mm is measured in the vertical position of the probe. An interval of 0.5 cm is taken as the longitudinal movement of the probe. Similarly, in the crosswise direction, the interval of the probe is taken as 0.15 cm. A maximum error of 4.65% is measured for all values of void fraction varying from a minimum air concentration of 0.06 to a maximum of 0.99, for a flow range of unit discharges 0.008925 < q_w < 0.0657 m²/s (0.201 < y_c/h < 0.761), with Reynold number Re varying from 0.80 × 10⁴ to 5.088 × 10⁴ and Weber number W_r ranging from 42 to 770. Visual observation of the propagation of jet impact on the horizontal steps provides clarity on the nappe flow pattern. The nappe flow appeared to be very regular along the flow and agreed well with a common description that the flow pattern may be observed as a succession of the free-falling jet.

As the experiment was conducted on a fully aerated flow, the measurement of bulked water depth, as well as air–water flow properties are essential to study the flow behaviour. Among the three flow regimes of the stepped flow, the nappe flow regime is considered for further study. To identify the behaviour of the nappe at each particular step, the jet length (L_jet) and pool water depth (d_p) were studied at different flow conditions. Air–water flow measurements were conducted at three different locations of each step to identify the behaviour of the jet as shown in Figure 3(c).

Likewise, the air–water flow properties at each step were conducted. One of the major studies was undertaken to find out the upper limit of the free surface as the flow is highly aerated. To identify that, three different bulked water depths are thoroughly studied such as Y₉₀, Y₉₈, and Y₉₉. Because the flow has different behaviours at different flow depths, the effect of the relative increase or decrease in bulked water depth at different flow depth conditions (Y₉₀, Y₉₈, and Y₉₉) was studied. Likewise, its effect on mean air concentration (C_mean) was also verified at these flow depths.

A few parametric studies have been carried out in order to get further insights into the jet flow over the stepped channels. The investigation was initially carried out by measuring the pool water depth and the jet length at various steps. It helps in understanding the jet's movement and fluctuations at various locations. It will most likely assist in designing various step geometries and their spacings. Furthermore, there have not been many studies on determining the aerated flow's free-surface level and how it affects energy dissipation. Therefore, a study on the effect of flow depth variations on aeration as well as energy dissipation will help in optimizing the design stepped channels. The results also identify the effect of channel width on energy dissipation and aeration.

Here in this study, the variation of jet length at each step was found. For that, two main parameters, jet length L_jet and pool depth d_p, were measured in each step at different flow rates.

The jet is highly stable and no aeration is observed throughout the flow for the first step, as shown in Figure 4(a). A small amount of air entrainment is observed at the impingement point, where the depth of flow is much lower to measure flow properties. Aeration started from the edge of the second step and full aeration was achieved at the free-falling region of the third step. The present study shows the variations of jet length (L_jet, the distance from the step corner to the centre of the impinging jet), jet impact point, and pool water depth (d_p) for each step at different flow rates.

A minimum value of L_jet was observed at the first step and the flow curvature was ogee-shaped. There was no similarity in the shape of the falling jet between the first step and the other adjacent steps. That is why during the prediction of flow properties, the first step is not taken into consideration. Figure 4(a) and 4(b) clearly displays the difference in flow patterns from Step 3 to Step 6 and agrees with the statement given by Toombes & Chanson (2008a) that some deviation has been observed in Steps 3 and 5. However, regular flow patterns are observed from Step 7 onwards. Observations were made on each location where the jet strikes on the horizontal face of each step. It is found that with an increase in discharge, the deflection of the jet increases, and at a certain points it crosses the step and jumps over the subsequent step. The flow regime at that point is called transition flow.

The above analysis confirmed that the flow becomes stable at X/L ≥ 0.6, where X is the horizontal distance from Step 1, and L is the length of the channel. Based on the present observation, it is clear that after crossing the horizontal face of Step 7 (60–70% of the total length of the channel) for all discharges, the flow seems to be uniform, whereas Felder et al. (2019) observed a uniform flow at the edge of Step 4 out of 6 (66.67% of the total length of the channel) the number of steps for a slope of 15° and a width of 0.2 m. This may be due to variations in the slope and width as well. It is also observed that air entrainment starts from the edge of Step 3 for almost all flow rates. Accordingly, Steps 1 and 2 come under the non-aerated zone, while the partially aerated flow region starts from the Step 2 edge to Step 3. Similarly, a fully aerated zone has been achieved from the Step 3 edge to Step 7.

The aeration increases with the flow rate, which leads to the enlargement of the jets at each successive step. To further develop different designs for the steps (non-uniform steps or any modification to the step's shape), it is necessary to understand the variations in both jet length and pool depth in a uniform step. By studying jet behaviour, it was also determined that each jet must stick to the horizontal face of the step to dissipate kinetic energy.

The study on the free-surface level of nappe flow is essential to estimate mean air concentration (C_mean). Therefore, one of the most important parts of this study is identifying the free-surface level as the bulking of flow depth is high for nappe flow. In this regard, few trials have been made by taking three free-surface levels (Y₉₀, Y₉₈, and Y₉₉) to estimate C_mean. As described in the literature, the calculation of C_mean is based on an accurate measurement of bulked water depth. In some cases, the upper limit of the bulked water depth was taken as Y₉₀ (Toombes & Chanson 2008a; Felder et al. 2019). In some cases, C_mean is considered as 0.5 (Renna & Fratino 2010). Therefore, to identify the variation of bulked water depth on the estimation of C_mean, three depths were selected.

⁠, ⁠, and are the mean air concentrations at flow depths of Y₉₉, Y₉₈, and Y₉₀. Equations (5)–(7) are applicable for flow depth of Y₉₉, Y₉₈, and Y₉₀. Similarly, the variation of C_mean is also different for all three zones as shown in Figure 8(d). As per Toombes & Chanson (2008a), C_mean for free-falling nappe varies from 0.05 to 0.35, and for the spray region, from 0.25 to 0.55 under the slope of 2.6° by using Y₉₀ as the free surface. While in the present case, variations of C_mean in all three regions are shown in Figure 8(d) with three different depths (Y₉₀, Y₉₈, and Y₉₉). In the free-falling section, the variation in C_mean varies from 0.259 to 0.518 for Y₉₀, 0.303 to 0.619 for Y₉₈, and 0.37 to 0.75 for Y₉₉. Similarly, for the nappe-spray region, it varies from 0.45 to 0.608 for Y₉₀, 0.57 to 0.731 for Y₉₈, and 0.58 to 0.738 for Y₉₉. While for the step edge, it varies from 0.38 to 0.53 for Y₉₀, 0.63 to 0.70 for Y₉₈, and 0.66 to 0.765 for Y₉₉. As the selection of bulked water depth is very sensitive to the calculation of C_mean, taking Y₉₈ or Y₉₉ as the free-surface level will produce better results compared with Y₉₀.

However, the difference between Y₉₈ and Y₉₉ in terms of C_mean is much less. So, Y₉₈ can be taken as the free-surface level for the present study. It is also critical to examine the impact of these increased water depths on energy dissipation. Therefore, all these three flow depths are further considered to identify their effect on energy dissipation.

Estimation of energy dissipation is one of the most vital parts of designing a stepped channel. Energy dissipation is the basic parameter to evaluate the efficiency of stepped channels. It identifies the functional ability of stepped channels at a particular step geometry. To obtain variation of flow depth on energy dissipation, all three of the aerated flow depths (Y₉₀, Y₉₈, and Y₉₉) are considered. By considering equivalent clear water depth, energy dissipation at different locations (Section-1, Section-2, and Section-3) is also evaluated.

For the step edge, the variation of for depth of flow Y₉₉ to Y₉₀, Y₉₈ to Y₉₀, and Y₉₉ to Y₉₈ varied from −2.49 to −6.72%, −2.67 to −6.17%, and −0.515 to 0.502%, respectively. It clearly shows that Y₉₀ overpredicts energy loss. In another way, the parameter is also used to show energy dissipation efficiency by using three flow depth conditions as shown in Figure 9(b). The relative energy loss varies from 0.228 to 2.16% for depth of flow Y₉₉ to Y₉₀, 0.22 to 2.16% for Y₉₈ to Y₉₀, and −0.03 to 0.11% for Y₉₉ to Y₉₈. It again shows that by using Y₉₀, the energy dissipation rate is mostly underpredicted. Figure 9(c) shows the variation of for three different depths, which shows that the variation of for Y₉₈ and Y₉₉ is much less compared with Y₉₀ for different channel heights. The study mentioned above leads to the finding that using Y₉₈ flow depth is more advantageous than using Y₉₀ and Y₉₉. Therefore, the aerated flow depth Y₉₈ is chosen as the free-surface level for further study.

Here in this section, the study of air concentration and its distribution across the flow depth have been discussed. For measurement of aeration parameters, three different locations (Section-1, placed at the inward side of jet; Section-2, placed at the outward side of the jet; and Section-3, placed at step edge) are selected to identify the behaviour of the jet. One of the most important parameters of aerated flow is the estimation of mean air concentration (C_mean). Ultimately, it facilitates the construction of the stepped channel for better aeration efficiency.

However, in the case of the free-falling region, the pattern of the curve has been changed, although some variation has been observed in the range of from 0.22 to 0.6 owing to the presence of a pool. The observed pattern of air concentration distribution against matches well with the air concentration contour at the free-falling region. One of the important considerations in computing air concentration is defining the depth over which air concentration needs to be averaged out. The existing literature indicates the consideration of this region as 90% of flow water depth. Therefore, use of Y₉₀ is very common in aerated flows. Since the region between Y₉₀ to free-surface can have larger air concentration because of exchange processes happening near the surface, therefore it is also obvious to consider a certain proportion of air concentration between Y₉₀ to free-surface.

In addition to the air–water flow distribution, various typical air–water flow parameters were also considered for all data and measurements. The C_mean is one of the distinguishing factors that have been represented for different y_c/h in Figure 11. It has been observed that for the wider channel, the difference between Y₉₀ and Y₉₈ varies up to 41%, whereas for the 0.28 m width channel, it varies up to 51.68%. Further, it is observed that there is not much variation in C_mean computed between Y₉₈ and Y₉₉ for both the models. Therefore, in the subsequent analysis, Y₉₈ has been considered to compute mean air concentration for all further calculations. As per the analysis, C_mean at Y₉₈ varies from 0.47 to 0.82 for the wider channel at different flow rates as shown in Figure 11(a) and 11(b). Whereas for Model-2, C_mean at Y₉₈ ranges up to 0.78.

The difference between C_mean for two specific widths varies up to 16% for the initial few steps whereas the difference becomes insignificant as the number of steps increases. However, this analysis identifies the variation of C_mean as per different widths of the channel. So, during the design, a channel of wider width must be recommended to enhance the aeration properties in it.

In this phase of the study, the energy dissipation of a channel is examined for three separate locations (inward side of the jet, outward side of the jet, and step edge of a step) in a single step. Consecutively, the energy dissipation at the step edge of two different models (Model-1 and Model-2) is also included. However, it helps in identifying the effect of step geometry and location on energy dissipation.

Designing a stepped channel for drainage in a hilly area is one of the climate-induced disaster mitigation strategies that have been suggested here. To achieve a wide view of this investigation, data from the current study were obtained. This modelling approach encourages more research to make stepped channel design simpler. This is probably a means of reducing the after-effects of a natural catastrophe and enhancing its use in the actual world.

Here, the few main functions of the flow over stepped channel have been discussed: primarily, its capacity of aeration, and subsequently, its capacity for dissipating high potential energy. Different properties of jets such as jet length and pool water depth are studied to get an idea of the steady state of the jet region where the fluctuations of L_jet/h and d_p/h in successive steps are less. From this study, it is observed that at nearly 60–70% of the entire length or Step 7 onward, a steady state of the jet has been achieved. This finding will also be beneficial during the design of steps with varying geometric shapes rather than uniform steps, to ensure flow stability. To obtain even more clarity on the free-surface of the aerated flow depth, three flow depths (h₁, h₂, h₃) were taken into consideration (Figures 8–11). It was discovered how flow depth affects the mean air concentration (aeration). However, the free-surface level was taken to be Y₉₈. This finding aids in accurately calculating energy dissipation and reduces the possibility of over- or underestimating energy loss. Additionally, finding the set of input parameters and their role in predicting energy dissipation at different stages, and creating a functional relationship that can accurately predict energy dissipation, was the final objective of this research. Finally, the effect of aerated flow depths on energy dissipation (Figure 9) as well as the expression (Figures 12–14) to predict energy dissipation is proposed based on the geometry of the channel. With the proposed expression (Equation (15)), design of stepped channels becomes more feasible.

Further, some statistical analysis has been conducted to analyse the efficiency of the predicted energy dissipation datasets. In this regard, the following goodness-of-fit parameters are selected:

• Correlation coefficient: 0 ≤ CC ≤ 1

  

  (16)
• Squared correlation coefficient: 0 ≤ R² ≤ 1

  

  (17)
• Root mean squared error: 0 ≤ RMSE ≤ 1

  

  (18)

  where N is the number of data, y_exp is the experimental data, and y_sim is the simulated data.

To further compare the present results, experimental data with varying slopes as well as widths of channels from the literature (Essery & Horner 1978; Pinheiro & Fael 2000; Chanson & Toombes 2002a; Renna & Fratino 2010; Felder et al. 2019) are well compared with the present experimental data in Figure 15(a)–15(c). Eventually, the current results showed a decrease in energy dissipation performance with increasing flow rate, which is also consistent with the earlier findings (Essery & Horner 1978; Pinheiro & Fael 2000; Chanson & Toombes 2002a; Renna & Fratino 2010; Felder et al. 2019). These results of existing experimental studies are based on different instrumentation, different methods of estimation, and different ranges of input parameters as mentioned in Table 1. After having all these differences in experimental conditions, the energy dissipation data of the present study as well as the existing study can be explained by the fact that the energy dissipation decreases with the increase in flow rate.

However, few proposed expressions are again compared with the present datasets. The observed and predicted datasets by using the expression given by Chanson (1994a) (Equation (2)) and Chamani & Rajaratnam (1994) (Equation (1)) are also compared in Figure 15(e) and 15(f). These expressions show good agreement with the present dataset with an error of ±29.30% for the expression given in Equation (2) and ±37.44% for the expression given in Equation (1). However, the proposed expression (Equation (15)) from the present study shows a better result with an error of ±10%.

Finally, based on the flow parameters (Table 4), the regression model was developed to estimate the energy dissipation ⁠. Equation (15) can be used effectively for the initial design of a stepped channel. Even though it has been derived based on certain broad assumptions and laboratory studies (w, l, h, N, α, discharge), it can still predict the energy dissipation efficiently. This study is limited to nappe flow. Therefore, the expressions suggested in this study are less appropriate for transitional or skimming flows, because the energy dissipation processes in the skimming flow and transition flow are different.

However, this study describes a climate change–mitigation approach that uses nappe flow to construct stepped storm waterways on mountainous roads to address climate change challenges. It regulates the flow throughout every step with a high impact that would likely reduce the energy downstream and reduce the risk of landslides and highway erosion. In summary, a thorough study on stepped channels was conducted by taking both air and water flow properties and the energy dissipation potential into consideration. In this regard, two different models with varying widths were taken. To predict the energy dissipation of a stepped channel, different parametric studies were conducted at different flow rates.

One of the major goals of the present work is to design a stepped storm waterway that can regulate surplus water during periods of heavy rainfall, distribute the potential energy of flowing water, and protect the hilly road against damages caused by heavy rain in the Himalayan range. To accomplish these objectives, two types of models with different datasets of air concentrations with various jet parameters were studied. A stepped channel with two different widths was experimentally modelled to verify its utility and efficiency in successfully handling the underpass rainwater drainage system as of a hilly road or highway. In this regard, both air and water flow properties and energy dissipation parameters were systematically studied in the current research. The following results are obtained from the present study.

Fluctuations of pool water depth and jet length become stable from Step 7 onwards. The present analysis confirmed that the flow gets stable at X/L ≥ 0.6. One of the important considerations in computing air concentration is to define the depth over which air concentration needs to be averaged out. As per the air–water flow properties, three different bulked water depths were studied to decide the most appropriate depth for the free-surface level. From the computation of the C_mean, the results are found to be sensitive to the selection of flow depths. The use of Y₉₀ predicts a lesser value of C_mean, which means it is underpredicting the mean air concentration. In the present study, mean air concentration was computed based on Y₉₀, Y₉₈, and Y₉₉. However, the difference in the results of Y₉₈ and Y₉₉ appears to be insignificant. Likewise, the effect of these flow depths on energy dissipation was also found. The variation of the residual head at the step edge in terms of was found to be minimum between Y₉₉ and Y_98, as −0.515%, whereas between Y₉₈ and Y₉₀, it was found to range from −2.67 to −6.17%. This means the residual head is overpredicted by using Y₉₀ flow depth. Similarly, the variation in for the flow depths, Y₉₈ and Y₉₀, was found to range from 0.22 to 2.16%. Therefore, it is suggested to use Y₉₈ for mean air concentration as well as energy dissipation.

The mean air concentration and energy dissipation rate of the stepped channel for two specific widths have been studied. The maximum energy dissipation has been observed at the step-edge region. Likewise, the wider channel has a better potential ability to dissipate high potential head. For a few initial steps, the dissipation efficiency is lower for a narrower channel than the broader channel, with a relative difference of 26%. The present study also provides multi-nonlinear regression expression for predicting energy dissipation by taking channel property and flow conditions as the known input parameters (Nh/y_c, y_c/h, and w/y_c). The current study may be useful in predicting energy dissipation at the step edge by taking channel properties as input parameters.

To obtain the optimum design for stepped channel at any specific location, an integrated study of the hydrological incidence of severe rainfall events and hydraulics of the stepped storm waterway is necessary for effective modelling. Further, in designing a stepped storm waterway, variation of slopes as well as step geometry (pooled steps, non-uniform steps, flip-bucket shaped steps) can be studied taking the stability of flow, variation of jet, and energy dissipation efficiency into consideration.

The authors are thankful to the entire staff of Hydraulics Laboratory, Department of Civil Engineering, IIT Roorkee, India, for the efficient experimental set-up of the channel.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.