Investigations of velocity and pressure fluctuations over a stepped spillway with new step configuration Open Access

Symbols used are as follows:

h_s

step height (m).

l_s

step length (m).

L_c

length of the crest (m).

N_s

Number of steps (unitless).

P

pressure (pa).

Q

discharge (m³/sec).

q

discharge per unit width ((m³/sec)/m).

V_c

velocity at critical depth (m/sec).

V_o

mean flow velocity at upstream (m/sec).

V₁

the downstream velocity (m/sec).

W

the width of flume (m).

X

the distance (m).

y_c

critical depth of the flow (m).

γ

water density (KN/m³).

θ

the angle of the slope (degree).

Stepped spillways are significant hydraulic structures. Flow characteristic, pressure, velocity distribution and step geometry are factors affecting stepped spillway design and safety (Boes & Hager 2003). Chinnarasri & Wongwises (2006) presented flow characteristics including flow regimes for the flow on a stepped spillway. Moreover, these factors were affecting the energy dissipation for different step shapes: a horizontal step, inclined step, and step with end sill. The outcomes showed that energy dissipation was lower at skimming flow than for a nappe flow regime. The step with end sills was the most efficient step geometry in terms of energy dissipation. The effect of the number of steps was evident since the energy dissipation increased with the number of steps. Also, it looks like stepped spillways with a modified step configuration might be a successful solution that can be applied (Jahad et al. 2018). Three different kinds of flow regimes can be seen in the flow over a stepped spillway depending on the effecting factors: nappe flow for low discharges, skimming flow for high discharges, and transition flow (Wan et al. 2019; Kaouachi et al. 2021). Furthermore, it is important to investigate other influences such as pressure distribution and velocity distribution across the step faces. Boes & Hager (2003) and Ghaderi et al. (2020) explained that hydrodynamic pressure and flow velocity are two of the significant factors that affect the design and safety of a stepped spillway. Takahashi & Ohtsu (2012) stated that as the impact region increases the relative critical depth (y_c/h_s) decreases whereas the location with the highest pressure on a sloping chute moves toward the inner step edge (x/l_s increases). According to Frizell et al. (2013) two zones (vertical step face and horizontal step face) have drawn attention in all the previous investigations of pressure distribution over a stepped spillway.

At present, one of the most popular methods used to successfully apply hydrodynamics to hydraulic structures is the computational fluid dynamic (CFD) method. Chow (1979) noted that CFD is the best choice to solve the problem because of the number of assumptions that may be utilized numerically to obtain CFD outcomes. Using CFD modelling incorporated with free surface effects has been reported in the last decade. The models typically employ Reynolds-averaged Navier–Stokes (RANS) equations coupled with turbulence closure solved numerically on a computational grid. Biringen & Chow (2011), Dastgheib et al. (2012), Castillo et al. (2014), Shahheydari et al. (2015), Peng (2015), Hamedi & Ketabdar (2016), Daneshfaraz et al. (2020), Kaouachi et al. (2021) and Lebdiri et al. (2022) modelled flow characteristics over stepped spillways. Predictions of the velocity field over the spillways revealed the transition between different flow regimes (nappe, skimming and transition) for changing flow conditions, pressure distributions, and energy dissipation. Tabbara et al. (2005) predicted the water surface profile and the energy dissipation rate for the flow on a stepped spillway by using the CFD method. The turbulent flow model was k-ε with a skimming flow regime. The results agreed with and were close to the experimental work. Aydin & Ozturk (2009) verified and validated that the outcome of a CFD model predicted by FLUENT software is compared with the experimental data for the physical model. The verification was based on the American Society of Mechanical Engineers editorial policy statement (Roache et al. 1986) which provides a framework for CFD uncertainty analysis. Vosoughifar et al. (2013) developed a MATLAB code called V-Flow for CFD with a 2D flow model on a stepped spillway. The assumed flow model was laminar without turbulence. GAMBIT software coupled with V-Flow was used to solve the problem with different configurations of the spillway. FLUENT software and V-Flow were used in validation according to Gonzalez's experimental work. The results showed good agreement between the experimental and CFD data for velocity vectors, flow streamlines, static pressure, dynamic pressure, and total pressure on the spillway. Rajaa & Kamela (2020) used CFD ANSYS Fluent to simulate 2D flow over a spillway. Volume of fluid (VOF) was applied with renormalisation group (RNG) k-ε of RANS equations for turbulent flow. The results showed good agreement and that supports use of the program in future work. Lebdiri et al. (2020) investigated the impacts of step number on the energy dissipation rate, the actual shear stresses on the walls for a flow stepped spillway. Ansys-Fluent was used in the modelling to solve Navier-Stokes equations with both VOF and turbulent k-ε. The results encouraged the researchers to use the CFD approach. A CFD model can show the inside hydraulic variations which can sometimes not be shown physically.

Studying the velocity and pressure distributions on steeped spillways by using new designs for the end sill (curved end sill) is significant. The characteristics of the velocity and pressure distributions have not been comprehensively studied in transition and skimming flow regimes over stepped spillways. In this study, the pressure measurement studied on both the horizontal and vertical faces and the velocity distributions for the steps on a stepped spillway with various step configurations were considered by using physical and numerical models to decrease the negative pressure and the cavitation impacts. The study aims are: (i) simulating flow regimes for different flow conditions with different step shapes (Step models (flat step), Sill models (normal end sill), and Curve models (quarter circle shape end sill which is the new step configuration); (ii) simulating the velocity distribution at the first step, mid-step, and last step for different step shapes; (iii) simulating the pressure distribution for the flow regimes at the first step, mid-step, and the last step for different step shapes; and (iv) simulating the water surface profile with streamlines of the flow along a stepped spillway.

The current work was conducted at Deakin University in Australia. Laboratory experiments were carried out in a 7.0 m long, 0.5 m wide and 0.6 m deep flume (Figure 1). Discharge Q (L/s) was measured by a flowmeter with an accuracy of ±1 L/s. Flow was supplied from a 3.0 m³ storage tank via a 0.2 m diameter pipe to the inlet tank and was straightened before entrance into the flume before exiting via a free overfall. Three different step shapes, each with 2 different geometries, resulting in a total of 6 experimental setups, were tested. All six setups had a total length L = 0.7 m and spillway height H_d = 0.3 m, giving a chute angle θ = 26.6°. The three shapes comprised: (i) rectangular shape or Step Model (Figures 2(a) and 2(b)), where Step_1 has ten flat steps with step height (h_s = 0.03 m) and Step_2 has six flat steps with step height (h_s = 0.05 m), (ii) rectangular shape with an end sill or Sill Model (Figures 2(c) and 2(d)), where Sill_1 has ten steps (height of the end sill equals step height h_s = 0.03 m) and Sill_2 has six steps (height of the end sill equals step height h_s = 0.05 m, and (iii) a quarter circle curve step or Curve Model (Figures 2(e) and 2(f)), where Curve_1 has ten steps (height of the quarter circle equals h_s = 0.03 m) and Curve_2 has six steps (height of the quarter circle equals h_s = 0.05 m). All the test setups were made from plywood and finished with a coat of marine paint to a smooth finish. The Step Model was chosen as the basic shape, for comparison with previous studies conducted with the same chute angle, and ultimately with the Sill and Curve Models. The ranges of experimental parameters tested are provided in Table 1, where h_s and l_s are the step height and length, respectively, N_s is the steps number, and y_c is the critical flow depth. All tests were carried out twice, to test for repeatability. A total of 144 experimental runs were therefore conducted for unit discharges, q ranging from 9.0 L/s.m. to a maximum of 111 L/s.m. Water depths were measured by point gauges, with an accuracy of ±0.2%.

Velocity was determined from head measurements using a Prandtl-Pitot tube, with 0.01 mm accuracy. The tube has an L-shape with diameter Ø = 3 mm. Tapping of the static pressure occurred 2.5 cm after the tip of the tube at the total head measurement opening. The Prandtl‒Pitot tube was attached at an inclination angle of 30° to reduce the tolerance of the tube reading to ±0.2 mm. In addition, the tube was installed on a digital gauge system, enabling its vertical placement with a 0.01 mm error. The Prandtl‒Pitot tube was supported by a horizontal beam to prevent vibration and movement during high flow. The Prandtl‒Pitot tube data were used to obtain the velocity distributions of non-aerated flows only.

Pressure measurements were recorded by piezoresistive transducers (Measure X, MRB20) with a measurement range of –2 m to +2 m and error of ±0.5%. Four pressure transducers were mounted on the horizontal face to measure the pressure (P1 at 2 cm, and P2 at 1 cm from the corner between the horizontal and vertical faces, P_mid at the mid point of the horizontal face, and P_last at 1 cm before the step edge). After that, for the same conditions, two pressure transducers were mounted on the vertical face (P3 at 1 cm and P4 at 2 cm from the corner between the horizontal and vertical faces). The pressure data was acquired using a data logger (HIOKI, MR 8875) set at a rate of 100 Hz with data capture over 60 secs period. Locations where the transducers were placed are shown in Figure 3. The transducers were used for measuring the pressure on the horizontal face and vertical face for step number one. The transducers were shifted to the mid-step (step number three for models with h_s = 0.03 m, and step number 5 for models with h_s = 0.05 m) and the last step (step number six for models with h_s = 0.03 m, and step number ten for models with h_s = 0.05 m) with the same conditions. Figure 3 shows step_2 model as an example. The process was repeated with changes to the discharge value.

In the CFD model, it is important to treat the turbulence of the flow. Solving RANS equations is the method that has been widely utilized in engineering applications. The models of this type depend on averaging flow leading to the RANS equations. The transport equations were added to the Navier–Stokes equations to achieve a closer mathematical flow model. In particular, the transport equations are used to represent the turbulence behavior of the flow to relate that with the turbulent viscosity, which is related to Reynolds stresses (Pope & Pope 2000). Among the available turbulent models, a widely utilized model is the standard k-ε model (Lopez et al. 2015).

A numerical model (2D flow) was run to steady state for all experimental conditions listed in Table 1. For the inlet, the boundary conditions were separated into two parts to consider the air and water fluid components in all tested models. Velocity equals to average velocity. Velocity was specified for the water flow component, and atmospheric pressure was specified for the air layer at the inlet boundary. Then, the pressure outlet was specified as the outlet boundary condition for the water layer, and atmospheric pressure was specified for the air layer at the inlet, outlet, and top boundary. The flume bed and the surfaces of the steps were defined as the wall boundary conditions.

To represent the domain of the geometry taken by the flow of the air‒water, two meshing approaches are used and both were considered: unstructured mesh and structured mesh. The most appropriate meshing approach depends on the specific case. According to Kim & Boysan (1999), unstructured mesh is suitable for selective refinement to prevent over-refinement in the zones that have expected small gradients. Also, unstructured mesh is best in multi-component geometry. This mesh has fewer closure problems, and its arbitrary topology makes automating the meshing process easier (Biswas & Strawn 1998). In some models, the mesh was slightly refined in the proximity of solid boundaries to accurately resolve the flow features in boundary layers, where larger gradients occur. This may cause the generation of very deformed elements, and it is not an actual problem if the mesh axes remain perpendicular to the solid boundaries (Hirsch 2007). For all these reasons, a static structured rectangular hexahedral mesh is considered the best choice for the cases in the current study. Moreover, inflation layers were used in the mesh because they have the critical components of a good CFD.

In the current study, two geometry types were considered: low complex geometry (the rectangular channel) and very complex geometry (the flow over the stepped spillway). Based on the geometry type, the mesh grid was created of quadrilaterals with two different mesh sizes. The mesh shape is very significant in the analysis, and it affects the total number of elements, run time, and accuracy. For the low complex geometry, a mesh size of 2 mm and element size of 3 mm without inflation layers were used. The very complex geometry has a mesh size of 1 mm, element size of 2 mm, and 10 inflation layers for more accuracy. Figure 4 presents the section of the geometry which has the same dimensions as the physical model and the mesh. The number of elements was 368,490, and the number of nodes was 369,709. The geometry details are listed in Table 1.

The boundary conditions do not change from one case to another except for the inlet velocity. The velocity inlet was calculated using continuity equations for five discharges (Q = 4.35 L/s, 10.10 L/s, 25 L/s, 38.46 L/s, and 55.55 L/s) from experimental work. The velocity was defined as velocity magnitude and the x-direction was used as a velocity direction in the ANSYS-Fluent software. The velocity used was v_o = 0.0275, 0.0605, 0.1326, 0.1895, and 0.2566 (m/s) as the data input in ANSYS-Fluent software for all the models, where v_o is the inlet velocity. A VOF model was used to simulate the two-phase flow (air and water) with a k-ε model to identify the turbulent flow turbulence in ANSYS-Fluent software. A numerical model was run to steady state for all experimental conditions listed in Table 1. The inlet boundary conditions were separated into two parts to account for air and water fluid components. The average velocity was set for the water flow component. The atmospheric pressure was specified for the air layer. Pressure outlet was taken as the outlet boundary condition for the water layer. In addition, atmospheric pressure was specified for the outlet air layer and top boundary. The flume bed and the surfaces of the steps were defined by the wall boundary (nonslip) condition.

Figure 5 illustrates the initial boundary conditions applied in the CFD model. The upstream boundary was separated into water and air inlets to distinguish between the two-fluid media. The velocity inlet was set as the boundary condition for the part occupied by water flow (BC). The atmospheric pressure was defined for AB and AD boundaries. The nonslip wall boundary condition was used for the steps and flume bed. Pressure outlet was specified in the outlet section (DE). All the dimensions in the model are similar to the physical model (spillway height = 0.3 m, section length (AD line) = 3.0 m, and depth (DE line) = 0.6 m).

Table 2 presents a comparison between the experimental and CFD results. The statistical values such as R, MAPE and RMSE were computed for verification. The benefit of MAPE and RMSE were used as an indicator for the error which should be zero or close to it. In the present study, the model has an index value nearest to zero which is considered more precise. As shown in Table 2, MAPE and RMSE are 4.32 and 0.21 respectively with R = 0.996 for five values of discharges. RMSE was close to zero, signifying the best agreement. In additional, Figure 6 presents critical flow depth (y_c) on the x-axis and downstream velocity on the y-axis, where V₁ was the flow velocity downstream of the stepped spillway with h_s being the step height and v_c the flow velocity at the critical depth. When the discharge changed, V₁ and the flow depth downstream had significant effects on the hydraulic characteristics because they were the main variables in this process. The results present the minimum and maximum errors for validation data, being 4.32% and 8.21% respectively. Figure 6 shows the downstream velocity V₁ that was obtained by the developed 2D flow model. It shows a close correlation within a (±6.27%) tolerance when compare with the experimental work. There was a good agreement between the experimental data and the simulated data.

Figure 7 compares the experimental data of the pressure for the models (Step_2, Sill_2, Curve_2) as a function for x/l_s with a previous study where x is the horizontal distance from point O and l_s is the step length in Figure 2. The present research obtained low pressure at low discharges in all the models and the relationship between the pressure and the discharge was a positive relationship. In general, pressure increases when discharge increases, consistent with the study by Sánchez-Juny et al. (2007). The differences between the results can be attributed to the influence of the flow conditions and geometry on the flow characteristics.

The CFD model results are presented by model Step_2 as an example. Figure 8 shows the experimental observations and the predicted flow regimes under different flow conditions. A nappe flow regime (NA) occurred in the experiments for q≈20.0 L/s.m and was reproduced by the numerical model (Figure 8(a)). The modelled free surface was noticed to be wavy and disturbed because of the plunging flow. In the experiments, plunging flows were accompanied by substantial amounts of air entrainment, especially for the upper steps, in the flat-step mode. This manifests as regions of white water in the experiments and low (<1) water to air (w/a) ratios. The white water was observed on every step, intensifying as the flow progresses down the steps. The NA CFD results show good similarity with the experimental observations.

The progress to a transition flow regime (TR) was also well predicted by the numerical model (Figure 8(b)). The longitudinal water surface profile was predicted to change from a skimming flow regime (SK) (first three steps) to NA (lower steps) when 20.0<q <39.0 L/s.m. It was consistent with the experimental observations. In the experiments (Figures 8(c)), the results show that the flow started out as SK at the uppermost steps and showed that as the available energy was unable to support SK further downstream, the flow changed to NA, accompanied by increased air entrainment. These observations were also well supported by the numerical model results. SK was predicted by the model for larger flow rates (q≥39.0 L/s.m) where the flow surface appears to be relatively undisturbed, especially for the flat-step model. For the normal sill and quarter- circle sill configurations, some waviness was seen because of the obstruction to the flow by the forward end sills. The observations showed flow regimes delayed in the flow over the new step shape (step with quarter-circle end sill) because of the effects of the quarter circle sill on the recirculated flow. Also, in the pooled stepped models, NA and TR regimes were noted at higher discharges than in the flat-step models.

Moreover, the differences in flow regimes on the pooled step models for the corresponding unit discharge and y_c/h_s in the flat step came from the impacts of the end sill, especially for the steps with quarter-circle end sill models because of circulation. In all models, the flow of water can be divided into two parts: (1) the first part goes straight to the downstream section, similar to the method in flat-step models; (2) the second part circulates back to the step surface or to the end sills that were oriented perpendicular to the main discharge. Moreover, in the second part of the flow, a small hydraulic jump is produced on the surface of the same step, while the main flow generates a hydraulic jump on the subsequent step. The two parts of the flow then collide in the next step. The hydraulic jump generated by the main part of the discharge disappears, the discharge is disturbed, and the hydraulic jumps become drowned, with more air (bubbles) becoming entrained. Because of these effects, the step height, step length, and end sills affect the flow pattern and its characteristics (hydraulic jump, air entrainment, inception point, and turbulence). The quarter circle end sill shape affected the flow circulation positively, producing a high flow circulation.

The flow over the stepped spillway was characterised by high velocity with a big aeration process after the inception point in TR and SK flow regimes. It is important to investigate the velocity distribution for the flow to show its effect on the aeration mechanism. The vertical velocity distribution on the stepped spillway was measured at the midpoint of the horizontal face of the first, middle, and last steps in the experiments (Figure 3). The measurements were taken across the range of the discharge (4.35–55.55 l/s) which included TR (Q = 10.10 l/s) and SK (Q = 55.55 l/s) flow regimes. The CFD model results were validated from comparisons of the modelled velocity distribution over the spillways with laboratory measurements.

Figure 9(a) compares the laboratory measured and modelled velocity profiles for transition and skimming flow conditions. The water depth (y) was normalized with respect to the tread of the step. The water depth y₉₀ is equal to 0.9 of y, whilst v is the velocity at y and v90 is the velocity at 0.9 the depth. In general, laboratory measured profiles agree with power law distributions observed in open channel flow for step number one. In subsequent steps, the measured velocity profiles are relatively large for approximately 90% of the flow depth. The influence of the recirculating pool on the step had a small influence on the velocity profile at the first step. However, the profiles of the velocities at step 5 and the last step of the stepped spillway showed the interaction effects of the recirculating pool. The minimum velocities were measured close to the surface of the steps because of the high impact roughness and shear boundary. The greatest velocities were measured at the top because the roughness effect was very low there.

The velocity profiles over the stepped spillway pointed out some differences which are related to inflow condition and step configurations. Figure 9(b) presents the CFD and experiment results at the three locations (first step, mid step, and last step) for all the step configurations. The measurements were taken in the SK regime. The CFD profiles of the velocities had slight differences with the flow of ideal fluid theory. However, the velocity distributions presented a similar behaviour to the velocity profile of free overfalls at SK especially at the last step (Henderson 1966). The minimum velocities were greater at SK than TR because the impact of the roughness and shear boundary became less with high discharge. Furthermore, the results showed the impact of gravitational force when the air boundary has its maximum effect on the flow. The velocity profile with the pooled step models (step with quarter circle end sill) was steady being low velocity at the beginning and then rapidly changing when the step pool fill with water. This is a good impact on the velocity because it leads to increase the residence time which was the contribution of the new model (step with quarter circle end sill).

The total pressure distribution included static and dynamic pressures. Several pressure measurements were taken over the stepped spillway for all step configurations. Six pressure ports (pressure transducers) were installed on the first, mid and last steps. The measurements were taken for aerated and non-aerated flows on horizontal and vertical step surfaces. For the horizontal step face, four points were used to carry out the pressure measurements on point P1, P2, P3 and P4 as shown in Figure 3. The dimensional less value of y_c/h_s verses p/γh_s were used to see the effects of the flow regimes, step configurations, and chute angle.

The data for the first step is described and discussed below as an example.

Several pressure measurements were carried out on the first, mid, and last steps of all models to see the pressure changes regarding the flow, step shape and the port location. In addition to the four pressure ports installed along the horizontal step face of each step, pressure ports were also installed on the vertical step face for points P5 and P6 at y_s = 1 and 2 cm from the corner of the step.

The total pressure measurements on the horizontal step face (P1, P2, P3, and P4) for all the discharge range are shown in Figures 10 and 11. Figure 10 presents the non-dimensional total pressure (P/γy_c) at y-axis and non-dimensional value (y_c/h_s) at x-axis to see the pressure variation at each point. The results show that there was no similarity for the pressure distribution between the step configurations especially for low discharges in nappe flow. The pressure values showed slight convergence with increasing discharge and divergence with decreasing discharge. In Step_1 and Step_2 models, the differences between the measurements were clear because of the impacts of the step height. This happened earlier for the skimming regime compared to the other models. However, a slight difference between the pressure results for models with an end sill (Sill_1, Sill_2, Curve_1, and Curve _2) because of the pooled water effect over the step. The results showed negative pressure at point P1 in both models at low discharge (nappe flow regime) because of the air pocket effect. On the other hand, the total pressure distributions had slight similarity at intermediate discharge and had good similarity at large discharges (transition and skimming flow) because the flow became partial aerated flow or non-aerated flow. Moreover, the results of total pressure distributions in the pooled models (Sill_1, Sill_2, Curve_1, and Curve_2) were all positive values and they had good similarity because the flow was non-aerated flow (there was no chance for air-pocket impact (no air-pocket)).

For the vertical step face, the comparison between the step configurations for points P5 and P6 are presented in Figure 12(a) and 12(b) the same behaviours were indicated for pressure distribution similarity between the step configurations. In Step_1 and Step_2 models, the changes in the pressure values were higher than the other models (Sill_1, Sill_2, Curve_1, and Curve _2) because of the impacts of the step height, flow depth, and the pooled water over the step.

This section presents the results for five discharges (q = 20.0 L/s.m with y_c/h_s = 0.69 to q = 39.0 l/m.s with y_c/h_s = 1.08) by using the simulation. The free surface shape upstream and above the crest of the stepped spillway was smooth. Figure 13 shows the free-surface profiles obtained by the CFD for Step_1 model as an example. The data highlighted that the water surface profile had a curvature profile, as indicated by (Woodburn 1932), especially at low discharges. At high discharges, the water surface profile has a wavy shape due to the effect of the interactions of the developing boundary layer with the main flow, as explained by Isaacs (1981). The results agreed with the experimental data including the water surface profiles (CFD data) above the crest.

Based on the findings of this study, the following conclusions can be drawn regarding stepped spillways with modified step configurations.

Nappe flow, transition flow, and skimming flow regimes were observed for all step configurations and geometries investigated. These observations were consistent with those made in past studies on stepped spillways with various step heights, step lengths, step numbers, and discharges. However, the onset of the TR and SK regimes was dependent on step shape and their modification (the new step shape is referred to as a ‘step with a quarter-circle end sill’), and the flow regime change from NA to TR and to SK was delayed as the discharge increased.

The results showed that the velocity profiles were agreed with a 1/10 power law distribution in all the steps over the stepped spillway. Also, the recirculating pool effect was small on step 1 (first step for 0<x≤L_crest) but high at the mid-step and last step (middle of the stepped spillway for x = L/2). The maximum velocities were recorded at 90% of flow depth on step 1 and close to the water surface at mid- and last steps.

The total pressure and the flow measurements illustrated the total pressure fluctuations in all the step shapes including the modified models. The total pressure fluctuations were induced by a combination of step-generated turbulence and density fluctuations. Two different regions on the step have been identified: the outer edge of the step (points P1 and P5), governed by impact with the upper jet, and the inner region (points P2, P3, P4 and P6) controlled by recirculating internal eddies. At the flat step shape with nappe flow regime, the pressure distribution was a non-uniform distribution due to the effects of the air pocket. The air pocket was affected by the step shape and the flow characteristics. In the air pocket zone at both the horizontal and vertical faces, with a nappe flow regime some of the pressure measurements were negative at points P2 and P6. However, the pressure distribution on the horizontal step face was a uniform distribution for the TR and SK regimes. The flow was partial aerated flow and fully aerated flow respectively. In the pooled step shape, there was not a negative pressure because the stored water prevents the generation of the air pocket. Also, when the discharge increases, the effect of friction trapped the circulation which leads to reducing the vortices and the circulations. This process affects pressure distribution. In general, the steps with an end sill (normal and curve end sill) have better aeration with no cavitation risk regarding negative pressure.

Comparisons with numerical predictions show that the CFD model can capture the main flow features based on the laboratory results. The water profiles predicted by the CFD model showed good similarity throughout the range of experimental conditions tested. The onset of nappe, transition, and skimming flow regimes predicted by the model for all step shapes corresponded well with laboratory observations. In addition, strong recirculation with a single vortex of regular shape was predicted for steps with quarter-circle end sill models. For steps with normal end sill models, the vortex shape was irregular and occasionally accompanied by a secondary, shear-induced counter-rotating vortex at the upstream corner of the sill. The traditional step model is accompanied by a single shear-induced vortex that is similar to the flow over a backward-facing step, albeit with a step length much smaller than the reattachment length. From the above, the curve models (quarter circle shape end sill which is the new step configuration) showed better performance than other models regarding the chosen flow characteristics.

The authors would like to thank all the staff of the Digital Manufacturing and Civil Laboratories at Deakin University for their assistance and support. The first author would like to acknowledge the full scholarship support provided by the Ministry of Higher Education and Scientific Research of Iraq.

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

The authors declare there is no conflict.