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Weirs are used to control and regulate the flow in open channels. In gabion structures, the flow conditions are more complex due to the complexity of flow through the porous body of a gabion. The present study aims to investigate the water surface profile, the overflow velocity profile, and both the through-flow and overflow ratios. Six physical models of the three downstream slopes (V:H 1:1, 1:2, and 1:3) and two types of rockfill (crushed stone and rounded gravel) were investigated. Results show that for the same discharge, the milder slope model (1:3) shows higher water surface and higher velocity than the steeper slope (1:1) with about 9 and 8% on average respectively. The water surface was 60% higher on the lower steps than on the upper steps at the nappe flow regime. Moreover, the low porosity models show a slightly higher velocity and flow depth than higher porosity models for all sections. Furthermore, increasing the porosity from 0.38 to 0.42 led to about a 27% increment in the through-flow ratio. Finally, four relationships were developed to estimate the through-flow and overflow ratios at the upstream and inner sections of the gabion weir. The suggested relationships can be considered novel relationships.

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  • The water surface profile for different flow regimes was investigated.

  • The overflow velocity profile for different downstream slopes was investigated.

  • The through-flow and overflow to total flow ratios were investigated.

  • A relationship for the ratio of through-flow to total flow and overflow to total flow was developed.

Graphical Abstract

Graphical Abstract
Graphical Abstract
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downstream slop, flow regimes, gabion weir, overflow, porous media, through-flow
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Hporous

porous media thickness at the section of estimation

hs

the step height

Hw

weir height

L

weir length

ls

the step length

n

porosity of porous media

QOver

overflow

QThrough

through-flow

QTotal

total flow

Re

Reynolds number

X

the distance between the section and the upstream face of the weir

y

flow depth above the weir

YSection

water depth at the section of estimation

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A gabion weir is a type of hydraulic structure that is built by installing gabion baskets together to form the body of the weir. The characteristics of flow over gabion stepped weirs are more complex due to the complexity of the porous body of the weir. Investigating the characteristics of flow through and over the gabion weirs is problematic due to the complexity and variability of the interior structure of the porous media (Fathi-moghaddam et al. 2018). Salmasi et al. (2012) pointed out that the hydraulics of through-flow have complex flow patterns and have not received adequate attention in the literature. Michioku & Maeno (2005) used a one-dimensional analysis and laboratory experiment to investigate the discharge over a permeable rubble mound weir. Momentum and continuity equations were utilised to analyse the discharge of two layers: overflow above the weir and through-flow in the weir. A formula was obtained as a solution for the water depth, flow velocity, and discharge over the weir. Mohamed (2010) carried out an experimental study to calculate the discharge through and over the rectangular gabion weir. In this study, both the gabion and solid broad-crested weirs were tested to compare the results. Two equations were formulated, using regression analysis, for calculating the total discharge as a function of the depth of water, stone porosity, and weir dimensions. Mohammadpour et al. (2013) simulated a three-dimensional flow on gabion weirs. The model solves the RANS equation to estimate the water surface profile over the rectangular weir using the experimental data given by Leu et al. (2008) and Mohamed (2010). Results indicated that the standard k–ɛ has good accuracy for estimating the water surface profile and upstream streamwise velocity. Tavakol-Sadrabadi & Fathi-moghaddam (2016) studied numerically flow characteristics around triangular gabion weirs by using a three-dimensional CFD code. They studied three triangular models with upstream slopes of 30, 45, and 60°. They also analysed three models with downstream slopes of the same angles, all with the same height of 160 mm. The results of the simulation contained the water surface profile, distribution of velocity, and through the flow to total flow ratio. The main finding was for the same discharge, the weir of the upstream slope of 60° offers the minimum upstream head and minimum downstream velocity, as well as the maximum ratio of through to total flow. Zuhaira et al. (2017) used an advanced computational model to simulate the flow over the gabion stepped spillway. The two-dimensional model used the NEWFLUME code to solve the equations of Reynolds-Averaged Navier–Stokes. The model has been validated using the experimental data of Wuthrich & Chanson (2014). They studied the effects of the step configurations and slope on energy dissipation. Results indicate that the downstream slope is an important factor that affects energy dissipation. The low (mild) slope offers a longer contact surface than the high (deep) slope, which increases the rate of energy dissipation. Tavakol-Sadrabadi et al. (2018) simulated the flow characteristics of porous weirs with different upstream and downstream slopes numerically. They used the FLOW-3D software to simulate the flow characteristics through and over gabion weirs. The experimental data of Moradi (2015) were used to validate the numerical model. Results showed that increasing the size of the filling materials leads to an increase in the coefficient of discharge, discharge, and the rate of energy dissipation. Furthermore, non-linear multivariable regression was used to predict relationships to estimate the discharge through porous media. Fathi-moghaddam et al. (2018) studied the hydraulic performance of gabion weirs numerically. Further, laboratory experiments were conducted to validate the simulation results. Results showed that the size of the rockfill had a significant effect on the amount of through-flow. Furthermore, using a regression technique, two empirical relationships of the through-flow ratio were obtained for trapezoidal and triangular weirs. Reeve et al. (2019) studied numerically the hydraulic performance of gabion stepped spillways. The model solves the Reynolds-averaged Navier–Stokes equations. Four gabion models were used (normal, overlap, inclined, and pooled steps) to investigate the energy dissipation and inception point location. The results indicated that the normal steps have a higher rate of energy dissipation than other configurations. This result was inconsistent with the results obtained by Peyras et al. (1992). Rajaei et al. (2019) studied experimentally the energy dissipation on gabion and impervious spillways. The physical models of one, two, and three steps have been used. Results showed that the through-flow in the gabion spillways has a significant effect on increasing energy dissipations. Moreover, using multivariate regression, a formula to estimate the energy dissipation in terms of the drop number and steps number was obtained. Jalil et al. (2019) investigated the flow characteristics of rectangular gabion weirs experimentally. The effects of the weir height and rockfill material sizes on the upstream water depth were studied. Results indicated that the upstream water depth decreased as the rockfill material size increased for through-flow. Moreover, using non-linear regression analysis, three equations were predicted for calculating upstream water depth as a function of discharge, rockfill mean diameter and weir height. Al-Fawzy et al. (2020) investigated the energy dissipation on a stepped gabion weir of three steps experimentally. Results indicated that energy dissipation increases with increasing discharge. In addition, the energy dissipation decreases with increasing the ratio of the length of the third step to the total length of the weir. The porosity has a slight inverse proportion to the energy dissipation. Shariq et al. (2020) carried out an experimental study on rectangular gabion weirs to verify the accuracy of the relationships between the velocity and hydraulic gradient in porous media. The main finding is that Ergun's equation is more accurate in predicting the hydraulic gradient than other equations. Salmasi et al. (2021) studied experimentally the coefficient of discharge for rectangular gabion weirs. Results indicated that the coefficient of discharge on gabion weirs is about 10% more than that on impervious weirs. Multivariable non-linear regression and genetic programming have been used to predict the through-flow equations on gabion weirs for both submerged and free flow conditions. Shariq et al. (2022) used an experimental study to investigate the flow characteristics of rectangular gabion and impervious weirs. They proposed two equations to estimate the discharge coefficient for gabion and impervious weirs. Also, results indicated that the water depth above the weir-to-weir height is a significant factor affecting the discharge coefficient on gabion weirs.

In summary, the previous studies focused on the flow characteristics and the through-flow and overflow ratios on rectangular, triangular, and trapezoidal gabion weirs. Gabion stepped weirs have not been studied regarding the flow characteristics and the through-flow and overflow ratios. Therefore, the current study aims to investigate experimentally the characteristics of flow through and over gabion stepped weirs through three aspects. These are the flow surface profile, the velocity profile over gabion stepped weirs, and the through-flow to total flow ratio and overflow to total flow ratio. The studying of through-flow and overflow ratio on gabion stepped weirs can be considered a novel study as it has not been studied before. The experimental study was conducted in the civil laboratory at Deakin University – Australia.

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Experimental facilities and instrumentation

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The experimental work on gabion stepped weirs were conducted in a rectangular hydraulic flume of acrylic walls at Deakin University in Australia. The dimensions of the flume were 6,400 mm in length, 500 mm in width, and 600 mm in height (Figure 1). The water tank capacity is 2,200 L with a pumping system of two pumps, each with a maximum discharge of 35 l/s to provide a maximum flow rate of 70 L/s. The flow rate was regulated by a manual valve. The flume was equipped with a sluice gate at the downstream end to control tailwater depth and the hydraulic jump position. A flow meter with an accuracy of ±3% was installed to measure the flow rate. Point gauges were used with an accuracy of up to ±0.1 mm to measure water depth. The velocity profile was measured using a ME-2221 PASCO Pitot Tube, the tube diameter is Ø = 3 mm. The pitot tube works with the PS-2222 PASCO General Flow Sensor and PS-2002 PASCO Xplorer GLX data logger. The General Flow Sensor measures the pressure differences between the input tubes. It works as a versatile differential pressure measuring device. The range of measured velocity is 0–9.98 m/s, with a range of temperature 0–85 °C and an accuracy of ±0.03 m/s.
Figure 1
Photo of hydraulic flume/Deakin University – School of Engineering – Waurn Ponds Campus – Australia.
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Photo of hydraulic flume/Deakin University – School of Engineering – Waurn Ponds Campus – Australia.

Figure 1
Photo of hydraulic flume/Deakin University – School of Engineering – Waurn Ponds Campus – Australia.
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Photo of hydraulic flume/Deakin University – School of Engineering – Waurn Ponds Campus – Australia.

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Physical models

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Six physical models of gabion stepped weirs were used to study the hydraulic of flow through and over gabion stepped weirs. The models have three downstream slopes 1:1, 1:2, and 1:3 (V:H) filled with two types of rockfill materials (Table 1). All models were designed to a scale of 1:10 with four steps and had the same height, width, step height, and broad crest (height 400 mm, width 500 mm, step height 100 mm, and broad crest 200 mm) (Figure 2). The rockfill materials were crushed stone of nominal size (37.5–13.2 mm) D50 = 23 mm and rounded gravel of nominal size (26.5–13.2 mm) D50 = 16 mm. The average porosity was 0.42 and 0.38, respectively, which was measured three times by direct method. The gabion baskets were made of 1.5 mm galvanised wire mesh with square openings of 12.7 × 12.7 mm.
Table 1

Details of physical models

Weir slope V:HModel number
Crushed stone (n = 0.42)Rounded gravel (n = 0.38)
1:1 N-CS-1 N-RG-1 
1:2 N-CS-2 N-RG-2 
1:3 N-CS-3 N-RG-3 
Weir slope V:HModel number
Crushed stone (n = 0.42)Rounded gravel (n = 0.38)
1:1 N-CS-1 N-RG-1 
1:2 N-CS-2 N-RG-2 
1:3 N-CS-3 N-RG-3 
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Figure 2
Photos of physical models.
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Photos of physical models.

Figure 2
Photos of physical models.
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Photos of physical models.

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Measurements

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Measurements differed according to the test purpose. In the present study, three different measurements were considered according to the aim of the study: discharge, velocity, and flow depth. The discharge was measured for each test using a flow meter that was installed on the inlet pipe. For the water surface profile, in addition to discharge, the flow depth should be measured at the selected sections along the centre line of the flume. The pointer gauges were utilised to measure the depth of flow. The flume bed was adopted as a datum of zero levels (i.e., the measured flow depth was the height of water above the bed of the flume). When the water surface is submerged inside the weir, the mesh grid of gabion baskets was used to estimate the flow depth and extract the water surface profile. In this case, the water surface profile was for flow on the side of the flume. It is assumed to represent the water surface profile along the centre line. The readings of flow depth started at 700 mm upstream of the model and ended at 1,000 mm downstream of the model at intervals of 100 mm with additional readings on the crest and steps depending on the flow conditions. The overflow velocity profile has been measured experimentally on gabion stepped weirs at five sections. Namely, they are the upstream end of the weir crest, the downstream end of the weir crest, and the edge of each step. In each section, the velocity has been measured using the pitot tube at different levels, which depends on the flow depth above the weir at the section. The velocity profile has been measured for nappe, transition, and skimming flow regimes on models N-CS-1, N-RG-1, N-CS-2, N-RG-2, N-CS-3, and N-RG-3. Estimating the amount of through-flow in gabion weirs is more complicated because of the nature of these structures and the complexity of flow patterns. Many parameters affect the through-flow characteristics. Some of these parameters are related to rockfill conditions, while others relate to flow conditions. Rockfill porosity, stone shape, and stone size are the main parameters that affect the amount of through-flow. This is in addition to the geometrical parameters, such as weir height, weir width, weir downstream slope, and the number of steps. Furthermore, the overall flow conditions, such as the discharge, velocity, and upstream water depth, also affect the amount of through-flow. As a result, it is difficult to measure the amount of through-flow practically. Then, the amount of through-flow will be calculated by subtracting the amount of overflow from the total flow. The overflow can be calculated through the velocity profile, which is obtained using the pitot tube in selected sections on the weir crest and on steps. The total flow can be found from the reading of the flowmeter:
where QThrough is the through-flow, QTotal is the total flow, and QOver is the overflow.

Error analysis and uncertainty

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In the current study, three main measurements have been taken in place, flow depth, velocity, and discharge. Pointer gauges were utilised to measure the depth of the flow at different locations with an accuracy of ±0.1 mm. The velocity profile was measured using a Pitot Tube with an accuracy of ±0.03 m/s, and the flow meter was installed to measure the flow rate, with an accuracy of ±0.001 m3/s. All measurements were repeated three times to obtain the true value and avoid errors. However, according to the accuracy of each instrument used in the measurement, there was an error in each measured value. The common technique to show the closest range of values to the true value is: [measurement = best estimate ± uncertainty]. The relative or fractional uncertainty has been used to report the precision quantitively:
(1)

For the flow depth measurements, the measured depth ranged between 4–558 mm, the average depth is 281 mm, and the uncertainty is ±0.1 mm. Then the relative uncertainty equals 0.04%. The measured velocity ranged between 0.25–2.5 m/s, the average velocity is 1.5 m/s, and the uncertainty is ±0.03 m/s. Therefore, the relative uncertainty is equal to 2%. Finally, the measured discharge ranged between 0.003–0.0525 m3/s, the average discharge is 0.0278 m3/s, and the uncertainty is ±0.001 m3/s. The relative uncertainty is equal to 3.6%.

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Water surface profile

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Studying the water surface profile of flow above the gabion stepped weir is an essential matter to provide more details about the nature of flow and the type of flow regime. In the current study, the water surface profile has been investigated along the centre line of the flow by using the pointer gauge. Figure 3 shows the water surface profile for the models N-CS-1, N-CS-2, and N-CS-3 each for through, nappe, transition, and skimming flow regimes. At the lowest discharge, the shape of the water surface inside the porous media was convex, and it is similar for all models when the flow is through the porous body of the weir. However, the water surface was higher at upstream and inside the weir for the milder slope model (N-CS-3). This is due to more internal resistance being provided by a longer flow path inside the milder slope models than in stepper slope models. At the nappe flow regime, in general, the flow depth above the steps for all models was higher on lower steps than on upper steps due to the flow seeping through the vertical face of the steps. The shape of the water surface on the steeper slope model (N-CS-1) was started as a nappe at the upper steps. It looks like it is skimming on the lower steps due to an increase in the overflow part on the lower steps. For the other models, the increment of the overflow part did not affect the flow regime because the steps were longer. The nappe surface on all steps has been observed. For all models, the water surface on the lower steps was about 60% higher than that on the upper steps at the nappe flow regime. The reason is the higher overflow ratio on the lower steps due to water seeping out from the vertical face of each step. At the transition flow regime, the water surface was undulated at the upper steps and smooth at the lower steps due to the increment in overflow at the lower steps. At the skimming flow regime, the increment in overflow at lower steps did not affect the water surface. Indeed, the total discharge was much higher than this increment. Therefore, the water surface profile was similar for all models. For all flow regimes, the water surface was higher on the milder slope model (1:3) than steeper slope model (1:1). At the upstream weir crest, the water surface was higher on the model N-CS-3 than on model N-CS-1 with about 9% in average. This is perhaps due to more resistance to flow provided by milder slopes through longer flow paths for both the through-flow part and the overflow part.
Figure 3
Water surface profile: (a) model N-CS-1, (b) model N-CS-2, and (c) model N-CS-3.
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Water surface profile: (a) model N-CS-1, (b) model N-CS-2, and (c) model N-CS-3.

Figure 3
Water surface profile: (a) model N-CS-1, (b) model N-CS-2, and (c) model N-CS-3.
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Water surface profile: (a) model N-CS-1, (b) model N-CS-2, and (c) model N-CS-3.

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Velocity profile

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Figure 4 illustrates the velocity profile in section 1 on models N-CS-1, N-CS-2, and N-CS-3 for skimming, transition, and nappe flow regimes as an example. The models of the milder slope (N-CS-3) show a higher velocity than that of the steeper models (N-CS-1 and N-CS-2) for all flow regimes. This is due to the higher ratio of overflow on model N-CS-3. The path of the through-flow inside the model of the milder downstream slope is longer than that on the model of the steeper downstream slope. This means that there is more internal resistance to flow through the weir body on the models of the milder slope. This reduced the through-flow and increased the overflow. The average velocity has been higher on model N-CS-3 than model N-CS-1, with about 11, 7, and 5% for nappe, transition, and skimming flow regimes, respectively. The higher percentage at the nappe flow regime is due to lower velocity than transition and skimming flow regimes. Therefore, the percentage of the same increment is higher.
Figure 4
The velocity profile in section 1 on models N-CS-1, N-CS-2, and N-CS-3: (a) nappe flow, (b) transition flow, and (c) skimming flow.
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The velocity profile in section 1 on models N-CS-1, N-CS-2, and N-CS-3: (a) nappe flow, (b) transition flow, and (c) skimming flow.

Figure 4
The velocity profile in section 1 on models N-CS-1, N-CS-2, and N-CS-3: (a) nappe flow, (b) transition flow, and (c) skimming flow.
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The velocity profile in section 1 on models N-CS-1, N-CS-2, and N-CS-3: (a) nappe flow, (b) transition flow, and (c) skimming flow.

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A comparison of the velocity profile between the models of N-CS-2 and N-RG-2 for the skimming flow regime is presented in Figure 5. The figure shows a slightly higher velocity and flow depth on model N-RG-2 for all sections. The reason for this is that the lower porosity of model N-RG-2 causes more internal resistance to flow through the weir body. As a result, the through-flow ratio will decrease, and the overflow ratio increases. In general, model N-RG-2 has a higher average velocity than model N-CS-2 with about 6, 4, 3.5, 3, and 2.4% for sections 1, 2, 3, 4, and 5, respectively. The upstream sections have higher percentages of increases in the average velocity than the downstream sections due to lower average velocity on the upstream sections than the downstream sections. The average velocity in section 1 was 0.62 and 0.58 m/s, while in section 5 it was 2.05 and 2.0 for models N-RG-2 and N-CS-2, respectively. Moreover, the flow depth was 0.518, 0.456, 0.343, 0.24, and 0.142 m for model N-RG-2 and 0.505, 0.445, 0.338, 0.2378, and 0.140 for model N-CS-2 at sections 1, 2, 3, 4, and 5, respectively.
Figure 5
Comparison of the velocity profile between models N-CS-2 and N-RG-2 for the skimming flow regime: (a) section 1, (b) section 2, (c) section 3, (d) section 4, and (e) section 5.
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Comparison of the velocity profile between models N-CS-2 and N-RG-2 for the skimming flow regime: (a) section 1, (b) section 2, (c) section 3, (d) section 4, and (e) section 5.

Figure 5
Comparison of the velocity profile between models N-CS-2 and N-RG-2 for the skimming flow regime: (a) section 1, (b) section 2, (c) section 3, (d) section 4, and (e) section 5.
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Comparison of the velocity profile between models N-CS-2 and N-RG-2 for the skimming flow regime: (a) section 1, (b) section 2, (c) section 3, (d) section 4, and (e) section 5.

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Flow through and over gabion stepped weir

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Experimentally, the overflow velocity profiles at sections 1–5 have been measured for models N-RG-1, N-CS-1, N-RG-2, N-CS-2, N-RG-3, and N-CS-3. Then, the amount of the overflow has been calculated using the velocity profiles. The amount of through-flow can be calculated by subtracting the amount of the overflow from the total flow at each section of each model. The results of the overflow ratio and the through-flow ratio for nappe, transition, and the skimming flow regimes of models N-RG-1 and N-CS-1 are presented in Figure 6, models N-RG-2 and N-CS-2 in Figure 7, and models N-RG-3 and N-CS-3 in Figure 8. The discussion of results will be for the through-flow ratio only, given that the overflow ratio has an exactly opposite behaviour. The ratio of through-flow was higher at the downstream edge (section 2) than at the upstream edge of the crest (section 1) for all flow regimes and all models. This is due to water entering the porous body near the brink. The seepage next to the downstream edge of the weir crest appeared to accelerate inside the gabion because of direct interaction with the weir overflow. On the steps of the weir (sections 3–5), the through-flow ratio decreased due to water seeping out from the vertical face of each step. Moreover, the through-flow ratio increased by increasing the porosity of porous media for all flow regimes. In the skimming flow regime, increasing the porosity from 0.38 to 0.42 increased the through-flow ratio by about 21, 25, and 35% for downstream models 1:1, 1:2, and 1:3, respectively. The average increment in the through-flow ratio was about 27%. The through-flow ratio was higher at the nappe flow and decreased with an increase in the discharge. The reason is the amount of through-flow depends on the properties of the porous media. Therefore, for a certain porous body, there is a capacity to pass flow through the body. By increasing the discharge more than its capacity, the increment will, then, pass over the body. Furthermore, the through-flow ratio was higher on models N-RG-1 and N-CS-1 (Figure 6) than on models N-RG-3 and N-CS-3 (Figure 8). On the milder slopes (models N-RG-3 and N-CS-3) the weir was longer. The flow path inside the weir was longer, as well, which means it had more internal resistance to the flow and a lower through-flow ratio. As a result, it can be stated that the through-flow ratio increased as the downstream slope steepened.
Figure 6
Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-1 and N-CS-1: (a) nappe flow, (b) transition flow, and (c) skimming flow.
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Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-1 and N-CS-1: (a) nappe flow, (b) transition flow, and (c) skimming flow.

Figure 6
Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-1 and N-CS-1: (a) nappe flow, (b) transition flow, and (c) skimming flow.
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Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-1 and N-CS-1: (a) nappe flow, (b) transition flow, and (c) skimming flow.

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Figure 7
Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-2 and N-CS-2: (a) nappe flow, (b) transition flow, and (c) skimming flow.
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Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-2 and N-CS-2: (a) nappe flow, (b) transition flow, and (c) skimming flow.

Figure 7
Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-2 and N-CS-2: (a) nappe flow, (b) transition flow, and (c) skimming flow.
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Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-2 and N-CS-2: (a) nappe flow, (b) transition flow, and (c) skimming flow.

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Figure 8
Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-3 and N-CS-3: (a) nappe flow, (b) transition flow, and (c) skimming flow.
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Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-3 and N-CS-3: (a) nappe flow, (b) transition flow, and (c) skimming flow.

Figure 8
Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-3 and N-CS-3: (a) nappe flow, (b) transition flow, and (c) skimming flow.
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Through-flow to total flow ratio and overflow to total flow ratio for models N-RG-3 and N-CS-3: (a) nappe flow, (b) transition flow, and (c) skimming flow.

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Through-flow ratio and overflow ratio

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Estimating the through-flow ratio on gabion stepped weirs is a significant matter for operating the structure. The amount of through-flow on a gabion stepped weir depends on the flow conditions, porous media properties, and weir geometry. The flow conditions can be represented by the Reynolds number. The porosity can be used to represent the porous media properties. The weir geometry can be represented by the ratio of weir length to weir height. Therefore, at the upstream of the weir (section 1), the empirical equation for calculating the through-flow ratio in gabion stepped weirs was developed using multilinear regression analysis. The suggested equations estimate the through-flow ratio in terms of the Reynolds number (Re), the porosity of porous media (n), and the ratio of weir length to weir height ⁠. The experimental data obtained in the present study for the models N-RG-1, N-CS-1, N-RG-2, N-CS-2, N-RG-3, and N-CS-3 at section 1 have been used in the regression analysis to develop the following empirical equation for estimating the through-flow ratio at the upstream section of the weir:
(2)

The correlation coefficient is R= 0.98, the root mean square error RMSE = 0.026, and the mean absolute percentage error (MAPE) = 4.20. The performance of the estimated through-flow ratio obtained using Equation (2) against the experimentally calculated ratio is presented in Figure 10(a). As shown in the figure, the estimated values of the through-flow ratio using Equation (2) describe the through-flow ratio. Accordingly, the suggested equation can be recommended to estimate the through-flow ratio with high accuracy.

Furthermore, the experimental data of models N-RG-1, N-CS-1, N-RG-2, N-CS-2, N-RG-3, and N-CS-3 in sections 2, 3, 4, and 5 have been used to develop a general empirical equation. The equation estimates the through-flow ratio at the edge of the weir crest and each step. The multilinear regression technique has been utilised to develop the equation. The suggested equations estimate the through-flow ratio concerning the following parameters: the Reynolds number (Re), the porosity of porous media (n), the ratio of porous media thickness at the section of estimation to the distance between the section and upstream face of the weir ⁠, and the ratio of water depth at the section of estimation to the distance between the section and upstream face of the weir ⁠, as shown in Figure 9. The obtained general empirical equation:
(3)
Figure 9
Details of through-flow and overflow ratio parameters.
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Details of through-flow and overflow ratio parameters.

Figure 9
Details of through-flow and overflow ratio parameters.
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Details of through-flow and overflow ratio parameters.

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Figure 10
Performance of developed equations to estimate the flow ratio against experimentally calculated flow ratio: (a) Equation (2), (b) Equation (3), (c) Equation (4), and (d) Equation (5).
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Performance of developed equations to estimate the flow ratio against experimentally calculated flow ratio: (a) Equation (2), (b) Equation (3), (c) Equation (4), and (d) Equation (5).

Figure 10
Performance of developed equations to estimate the flow ratio against experimentally calculated flow ratio: (a) Equation (2), (b) Equation (3), (c) Equation (4), and (d) Equation (5).
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Performance of developed equations to estimate the flow ratio against experimentally calculated flow ratio: (a) Equation (2), (b) Equation (3), (c) Equation (4), and (d) Equation (5).

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The correlation coefficient is R= 0.93, the root means square error (RMSE) = 0.063, and the MAPE = 11.20. Figure 10(b) presents the performance of the estimated through-flow ratio obtained using Equation (3) against the experimentally calculated ratio. It can be stated that Equation (3) estimates the values of the through-flow ratio at the weir crest edge and each step of the edge with high reliability. The negative sign of the Reynolds number in Equations (2) and (3 indicated that the through-flow ratio reduces when the flow rate increases.

The same procedure for estimating the through-flow ratio will be used to estimate the overflow ratio in section 1. The multilinear regression analysis has been used to develop an empirical equation to estimate the overflow ratio on the gabion stepped weirs concerning the Reynolds number (Re), the porosity of porous media (n), and the ratio of flow depth above the weir to weir length ⁠. The experimental data of the models N-RG-1, N-CS-1, N-RG-2, N-CS-2, N-RG-3, and N-CS-3 at section 1 have been used in the regression analysis to develop the following empirical equation for estimating the overflow ratio at the upstream section of the weir:
(4)

The correlation coefficient is R= 0.98, the RMSE = 0.027, and the MAPE = 4.81. Figure 10(c) shows the performance of the overflow ratio calculated by using Equation (4) against the experimentally calculated ratio. As shown in the figure, the estimated value of the overflow ratio is calculated by using Equation (4), which well-represents the overflow ratio. Thus, the suggested equation can be used to estimate the overflow ratio with high accuracy.

Moreover, an empirical equation for estimating the overflow ratio at the edge of the weir crest and each step has been developed using multilinear regression analysis. The experimental data of models N-RG-1, N-CS-1, N-RG-2, N-CS-2, N-RG-3, and N-CS-3 in sections 2, 3, 4, and 5 were used to develop the equation. The suggested equations estimate the overflow ratio concerning the following parameters: the Reynolds number (Re), the porosity of porous media (n), the ratio of porous media thickness ⁠, the ratio of water depth to distance ⁠, and the weir downstream slope ⁠. Where hs is the step height and ls is the step length. The developed equation is as follows:
(5)

The correlation coefficient is R= 0.95, the RMSE = 0.056, and the MAPE = 10.47. Figure 10(d) presents the performance of the estimated overflow ratio obtained by using Equation (5) against the experimentally calculated ratio. According to the figure, Equation (5) can estimate the values of the overflow ratio at the weir crest edge and each step edge with high accuracy. The positive sign of the Reynolds number in Equations (4) and (5 indicated that the overflow ratio increases as the flow rate increases.

Table 2 shows the results of the estimated regression line including the estimated coefficients, the standard error of the coefficients, the calculated t-statistic, the corresponding p-value, and the bounds of the 95% confidence intervals.

Table 2

Regression coefficients table for Equations (2)–(5)

VariableCoefficientsStandard Errort StatP-valueLower 95%Upper 95%
Equation (2) for estimating the through–flow ratio in section 1 
Intercept 2.203265 0.268499831 8.20583343 1.01761 × 10–6 1.627390028 2.779139755 
 –0.71844 0.051171953 –14.03965409 1.21738 × 10–9 –0.828189438 –0.608683592 
n 3.63647 0.343960924 10.57233606 4.66833 × 10–8 2.898747667 4.374193288 
 –0.0091 0.00938583 –0.96978954 0.348608548 –0.029232882 0.011028323 
Equation (3) for estimating the through–flow ratio at the edge of the weir crest and each step 
Intercept 1.978494766 0.359399821 5.504996526 6.28928 × 10–7 1.261129721 2.695859811 
 –0.547659485 0.081396574 –6.72828672 4.62728 × 10–9 –0.710127726 –0.385191244 
n 2.011168407 0.38753714 5.189614615 2.12966 × 10–6 1.237641037 2.784695776 
 1.093646271 0.223567104 4.891803183 6.57458 × 10–6 0.647404464 1.539888079 
 –0.906794736 0.210234452 –4.313254681 5.41151 × 10–5 –1.326424459 –0.487165012 
Equation (4) for estimating the overflow ratio in section 1 
Intercept –1.350067951 0.30493922 –4.427334577 0.000573841 –2.00409753 –0.696038372 
 0.7775316 0.07075 10.98984594 2.86706 × 10–8 0.625787942 0.929275259 
n –3.811979807 0.340880674 –11.18273959 2.30036 × 10–8 –4.543096139 –3.080863475 
 –0.229292729 0.184662603 –1.241684703 0.234753651 –0.625354621 0.166769163 
Equation (5) for estimating the overflow ratio at the edge of the weir crest and each step 
Intercept –0.681457476 0.307359032 –2.217138279 0.030063029 –1.295119584 –0.067795368 
 0.484827239 0.069465634 6.979382573 1.76248 × 10–9 0.346134625 0.623519853 
n –1.905798548 0.326576253 –5.835692367 1.7781 × 10–7 –2.557829075 –1.253768021 
 –1.329685449 0.193151081 –6.884172975 2.60149 × 10–9 –1.715324024 –0.944046874 
 1.133872385 0.181852553 6.235119415 3.61663 × 10–8 0.770792048 1.496952722 
 –0.122931895 0.022950286 –5.356442806 1.1557 × 10–6 –0.168753619 –0.07711017 
VariableCoefficientsStandard Errort StatP-valueLower 95%Upper 95%
Equation (2) for estimating the through–flow ratio in section 1 
Intercept 2.203265 0.268499831 8.20583343 1.01761 × 10–6 1.627390028 2.779139755 
 –0.71844 0.051171953 –14.03965409 1.21738 × 10–9 –0.828189438 –0.608683592 
n 3.63647 0.343960924 10.57233606 4.66833 × 10–8 2.898747667 4.374193288 
 –0.0091 0.00938583 –0.96978954 0.348608548 –0.029232882 0.011028323 
Equation (3) for estimating the through–flow ratio at the edge of the weir crest and each step 
Intercept 1.978494766 0.359399821 5.504996526 6.28928 × 10–7 1.261129721 2.695859811 
 –0.547659485 0.081396574 –6.72828672 4.62728 × 10–9 –0.710127726 –0.385191244 
n 2.011168407 0.38753714 5.189614615 2.12966 × 10–6 1.237641037 2.784695776 
 1.093646271 0.223567104 4.891803183 6.57458 × 10–6 0.647404464 1.539888079 
 –0.906794736 0.210234452 –4.313254681 5.41151 × 10–5 –1.326424459 –0.487165012 
Equation (4) for estimating the overflow ratio in section 1 
Intercept –1.350067951 0.30493922 –4.427334577 0.000573841 –2.00409753 –0.696038372 
 0.7775316 0.07075 10.98984594 2.86706 × 10–8 0.625787942 0.929275259 
n –3.811979807 0.340880674 –11.18273959 2.30036 × 10–8 –4.543096139 –3.080863475 
 –0.229292729 0.184662603 –1.241684703 0.234753651 –0.625354621 0.166769163 
Equation (5) for estimating the overflow ratio at the edge of the weir crest and each step 
Intercept –0.681457476 0.307359032 –2.217138279 0.030063029 –1.295119584 –0.067795368 
 0.484827239 0.069465634 6.979382573 1.76248 × 10–9 0.346134625 0.623519853 
n –1.905798548 0.326576253 –5.835692367 1.7781 × 10–7 –2.557829075 –1.253768021 
 –1.329685449 0.193151081 –6.884172975 2.60149 × 10–9 –1.715324024 –0.944046874 
 1.133872385 0.181852553 6.235119415 3.61663 × 10–8 0.770792048 1.496952722 
 –0.122931895 0.022950286 –5.356442806 1.1557 × 10–6 –0.168753619 –0.07711017 
View Large

Equations (2)–(5) have been obtained from the normal steps gabion weirs of normal downstream slopes, which ranged between 1:1 and 1:3. Further, the discharge per unit width (q) ranged from 0.05 to 0.105 ⁠, and the porosity ranged from 0.38 to 0.42. Therefore, it is recommended to estimate the overflow ratio within the above limitations.

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The current study investigated the flow characteristics on gabion stepped weir for different downstream slopes and rockfill materials experimentally. The water surface profile, the velocity profile over the gabion stepped weir, and the through-flow and overflow ratio have been investigated. Results indicated that the milder slope model (slope 1:3) shows higher water surface and overflow velocity than the steeper slope (slope 1:1) for all flow regimes. This is because the milder slope models have a longer flow path inside and over the weir than the stepper slope models, which creates more internal and surface flow resistance. Moreover, the flow depth and overflow velocity were slightly higher on models of lower porosity (0.38) than on models of higher porosity (0.42). At the nappe flow regime, the water depth was higher on the lower steps than on the upper steps by about 60%. This was due to flow seeping through the vertical face of the steps. Consequently, an accumulative increment of overflow on the lower steps. For all flow regimes, increasing the porosity of porous media resulted in an increase in the through-flow ratio. The through-flow ratio was higher at low discharge (nappe flow regime) and decreased as the discharge increased (skimming flow regime). Furthermore, on the weir crest, the through-flow ratio was higher at the downstream edge than the upstream one due to water entering inside the porous body near the brink. While on the step edges, the through-flow ratio decreased towards the weir toe due to water seeping from the vertical face of each step. The multilinear regression analysis has been used to develop two empirical equations to estimate the through-flow ratio and the overflow ratio at the upstream section of the weir. Furthermore, two equations have been developed to estimate the through-flow ratio and the overflow ratio at the inner sections of the weir. The suggested equations can estimate the through-flow and overflow ratios with high accuracy. The significance of Equations (3) and (5) is the ability to estimate the flow ratio at any section of the weir.

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The authors acknowledge the support provided by the technical staff in the School of Engineering at Deakin University, Australia.

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

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The authors declare there is no conflict.

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© 2023 The Authors
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