Stepped spillways have gained a huge attention recently for the release of excess water from the reservoir since energy loss in a stepped spillway is higher than the smooth spillways. Due to an increase in energy loss, the stilling basin length at the bottom part of the spillway would be shorter than the smooth spillway. Because the length of a hydraulic jump is shorter in the bottom of the spillway, called stilling basin, in the case of a stepped spillway than the smooth spillways. Engineers prefer the stepped spillways where a large amount of energy dissipation is required due to the short stilling basin length. Use of roller-compacted concrete (RCC) reduces cost and construction of steps become faster which increases dissipation energy as compared to a straight type of smooth spillway which was used in the past. Stepped spillways subsidize the elimination of downstream energy loss The steps’ roughness increases the dissipation energy in the stepped spillways. This means the steps water depth increases, and flow velocity reduces, which causes a higher dissipation of energy in stepped than smooth spillways. Another advantage of stepped spillways is the increase of size of the boundary layer turbulence, which enables free surface air to move along the spillway. This free surface air reduces the cavitation damage on the spillway surface. The boundary layer thickness depends mostly upon the roughness height and the stream wise location [1
]. After a specific location, this boundary layer turbulence reach the free surface; this is termed as surface inception point [3
]. If this turbulence incapacitates the surface tension, the air starts entering into the water and this is called the start of the aerated zone. After that point, the aerated zone thickness is increased in the way scattering to the steps until the total flow is aerated [4
]. The pseudo-bottom inception point is found at distance Li, and is in fact the space among the spillway crest and location where the pseudo-bottom air concentration extends to Cb = 0.01 [5
]. In addition, this point is very relevant to measure the cavitation potential in stepped spillways. The pseudo-bottom is formed due to joining the corners of the steps by a line or a plane tangent to the corners of the steps. Both points, the inception surface and pseudo-bottom inception point, are both situated very close to each other [6
The concept of air entrainment in the stepped spillway is determined between entrained and entangled air in the stepped spillway. Entrained air is conveyed through the air bubbles while the trapped air is conveyed overhead in the coherent water body. The addition of entrained air and trapped air is entitled total air absorption [7
]. The inception point location in the stepped spillway is of particular interest because of the upstream cavitation risk if the discharge exceeding a certain limit requires aeration upstream of the inception point [8
]. The locality of the inception point for the smooth spillway is closer to the bottom than the stepped spillway. The maximum black water discharge is reduced in the stepped spillway than the uniform spillway [9
]. Aerated flows in the stepped spillways are more disposed to inhibit cavitation loss than the non-aerated flows [10
]. Energy dissipation in the white water flow is less than the black water flow because of drag reduction in the aerated flow region. Drag reduction is caused by the boundary of air bubbles and mixing layers [11
Frequently stepped spillways are planned to operate in skimming flow regions because most spillways work on moderate discharge [12
]. The water flows as a clear stream in the skimming flow region over the pseudo-bottom; beneath the pseudo-bottom, the three-dimensional vortices are formed [13
]. Energy loss during the skimming flow is due to these three-dimensional vortices. Skimming flow occurs for higher discharge, and smaller step size in which water flows over the step edges and the recirculation zone is developed in the triangular region, which is formed by the step faces and pseudo-bottom [14
]. For small discharge and large step size, there is flow occurring in the stepped spillway called nappe flow. In the flow region which is called nappe, the stream forms a nappe which is felled from one step to another. The free-falling nappes dissipate more energy than the skimming flow region due to more impact of water with a step face [16
]. After nappe to skimming flow region, there is also a flow arise, which is called transition flow. In the transition flow region, mutually, nappe and skimming flow arise in different portions of the stepped spillway [17
At the inception point, the air enters into the flow due to the large amount of turbulence, when the boundary layer turbulence extents the free surface the air starts entering into the water. Cavitation is a very serious problem in the stepped spillways. It will occur on the step edges, or step surfaces when pressure falls below the vapor pressure [18
]. A spillway could be severely damaged with the cavitation, one way to reduce cavitation damage to the stepped spillway is entering significant air amount along the spillway. In order to decide the location of inception point (Li) on the stepped spillway, the un-aerated zone, which is disposed to cavitation damage during the flow, should be determined [19
]. As the length of the non-aerated region over the stepped spillway increases, the cavitation risk heightens, and more area of the stepped spillway could be affected by the cavitation. An average of 5–8% air concentration is essential to prevent cavitation loss in the stepped spillway. For specific stepped spillway geometries, point of inception or fright of air entrainment transfers downstream with flow rate rises and the magnitude of the non-aerated flow region rises with the rise in the discharge so chances of cavitation in the stepped spillway increases when the discharge value is increased [20
]. The geometry of a stepped spillway through skimming flow regime is presented in Figure 1
. It also shows the length of non-aerated flow zone and aerated flow and the point where the air starts entering into the water.
Channel slope in the stepped spillway is an important parameter for the construction of any type of stepped spillway. Channel slope is actually the quotient of step height to the step length or whole elevation of the channel to the channel length. On the basis of the slope of channel, the stepped spillway is categorized into three types, mild slope stepped spillway; moderate slope stepped spillway; steep slope stepped spillway. The stepped spillway having a slope range of 10 to 14 is called a mild slope stepped spillway. A stepped spillway with a slope range of 14 ≤ ѳ ≤ 22 is called a moderate slope stepped spillway. A stepped spillway with a slope range of ѳ ≥ 22 is called a steep slope stepped spillway [22
Numerical modeling is a technique that is used frequently nowadays in many engineering fields. Computational fluid dynamics (CFD) is a type of numerical method which is used to solve fluid problems. The use of CFD has received significant attention for hydraulic engineers for constructing the difficult spillway design and other hydraulic structures. In CFD there is no need for construction of a physical model or prototype model, so it proves to be very cost-effective throughout the design process. Physical modeling is very time consuming, and the creation of physical models, as well as contracting the facility and hiring the soft researchers to accomplish experiments can be very expensive. Hydraulic engineers are consequently attracted to CFD and use various software packages for numerical modeling. CFD is subsequently attractive in relations of cost and time. More prominently, the complete flow explanation is attained with sufficient accuracy [24
Numerical simulation of over stepped spillways is accomplished by the Reynolds Averaged Navier–Strokes coupled with different turbulence simulation Reynolds stresses model, sst k-ὠ model, v2
-f, and LES models [27
]. These Navier–Strokes equations are problematic to solve, these equations are resolved by using different viable software like Fluent and Flow-3D and other software. This software uses finite volume methods to solve the Navier–Strokes equations [29
]. Qian et al. [30
] analysed and compare the impact of four turbulence models and the extent of the velocity of the mean water flow. The obtained results are compared with those obtained with PVC. The V2
-f turbulence model underestimates the mean velocity while the results found by realizable k-ϵ, k-ὠ SST models are reliable and superior to those values obtained by PIV. Wei et al. [31
] developed a model to measure the amount of air in nature-aerated open channel flows. The air-water flow contains the two regions’ low flow areas, where the amount of air is less than 0.5, and the higher flow region where the amount of air is greater than 0.5. Zhong Dong et al. [32
] used the Fluent software to simulate the flow above the flat stepped spillway. They determined that the k-ϵ model is the most efficient model to simulate the flow over the stepped spillways. Chen et al. [33
] used the Flow-3D software and compared the different turbulence models. They measured the velocity outline and water surface length through experimental and numerical modeling. Morovati et al. [34
] studied the five different types of pooled stepped spillways and offered the effect of alignment of the different pools and considered the vortex flow, velocity contours, and standing side wall waves. Morovati and Eghbalzadeh [35
] considered the inception point, pressure, and a void fraction over the pooled stepped spillway using the Flow-3D model. They used the volume of fluid (VOF) technique and k-ϵ turbulence model to simulate the free surface. They studied the different pressure values on the crest of the spillway. There is no negative pressure arising on the crest and horizontal face of the step. Negative pressure only happens on the vertical face of the step. Chen et al. [36
] studied and examined the dam slope and ogee of the spillway bottom on the energy loss proportion. They decided that loss in energy increase as the spillway rise increases. In addition, they studied that energy loss in the stepped spillway without ogee is much more than the stepped spillway with the ogee at the toe of the spillway. They studied the characteristics of the turbulent flow on the stepped spillways. They utilized the different models to study the turbulence over the stepped spillway. Wan et al. [37
] calculated the location of the inception point in diverse types of stepped spillways. They used the volume of fluid and realizable k-ϵ models to study the inception point in different types of stepped spillways. They studied the effect of step height and geometry of the step on the inception point location. They concluded that the increase of step height inception point moves upward toward the crest of the spillway. Wan et al. [38
] calculated the cavitation loss in high-speed smooth spillways by using Fluent software. They use the volume of fluid (VOF) and standard k-ϵ models to measure the cavitation in the high-speed smooth spillway. They concluded that cavitation mostly occurs at the end of the chute where pressure is minimum below the vapor pressure. Craft et al. [39
] used nonlinear models for different step heights. Cheng et al. [40
] used the standard k-ϵ model to simulate the flow over the stepped spillway model and after that they used the RNG (renormalized group) k-ϵ model to simulate the same stepped spillway model. They found that the results found by the RNG turbulence model are more precise than the results gained by the standard k-ϵ model. So, they resolved that the turbulence model plays an important part in the precision of simulation of flow by using computational fluid dynamics to simulate the stepped spillway models. Tebbara et al. [41
] used the ADINA software to simulate the flow above the stepped spillway. They used the different step configurations to determine the skimming flow development region and surface water profile, and the resolve of the energy dissipation ratio. They compared the numerical results with the laboratory experiments and they concluded that the numerical results have a close agreement with the laboratory experiments. Bai and Zhang [42
] used the k-ϵ model to study the pressure values in three types of the stepped spillways (V-formed, inverted v-formed, and flat stepped spillway). The value of negative pressure in case of V-shaped stepped spillway is followed near both sidewalls of each step, while in the case of the inverted v-shaped stepped negative spillway pressure occurs on the axial plane of the step, while in the case of the traditional stepped spillway the negative pressure occurs along the entire cross-section. The value of negative pressure is decreased with the increase in Froude number in all types of stepped spillways. They decided that from all three kinds of stepped spillway models, the inverted v-formed stepped spillway model is most expected to lead to cavitation damage. Daneshfaraz et al. [43
] used four types of stepped spillways with different step sizes; they use three changed turbulence models by using Fluent software to simulate different stepped spillway models. They found that the RNG k-ϵ model gives more appropriate results than the other two turbulence models by comparing the results with laboratory experiments. By choosing the appropriate turbulence model, they measured the pressure distribution on the steps and concluded that pressure distribution on the steps is the same for all spillway models. Abbasi and Kamanbedast [44
] numerically solved the three groups of stepped spillways using the Flow-3D model; they studied the energy loss and critical depth over the stepped spillways and compared the values with the experiments. Roushanger et al. [45
] used various types of modeling by using artificial neural networks and genetic programming techniques (GEP) by using an empirical data set to determine the energy loss in rotating and slipping flows along the stepped spillways.
This study is concerned with determining the air entrainment and location of the inception point by using computation fluid dynamics (CFD). For this purpose, different stepped spillway models are simulated with different slopes to determine the relation of how the size of non-aerated flow area varies with the change of slope. Different stepped spillways models with various slopes, i.e., 12.5°, 19°, 29°, and 35° are modeled to study how the inception point and size of non-aerated flow area vary with the change of slope of stepped spillways. For this, the Fluent software is used to simulate the stepped spillway models with different slopes. The volume of fluid model is utilized to track the boundary between water and air, and also, a realizable k-ϵ model is utilized to measure the turbulence inflow of stepped spillways. All the four stepped spillway models are simulated at different discharge rates. Overall, sixteen different spillways are modeled to analyze the relation between slope, non-aerated flow zone length, inception point, and discharge. For this purpose, four different discharge values are used to simulate the stepped spillway models with different slopes to study how the inception points location and non-aerated flow zone length vary with discharge and slope.
Four different stepped spillway models with different slopes varying from mild slopes to steep slopes were simulated using VOF and k-ϵ realizable models to calculate the inception point location and span of the non-aerated flow zone.
An increase of flow rate of a slope of a stepped spillway might prompt the cause of the skimming flow regime. The skimming flow regime in the stepped spillway depends upon the discharge, step height, and step length. For all discharges to determine the skimming flow regime, the discharge value should be larger than critical value.
Location of air entrainment (inception point) moves downward as the discharge rate increases, and the span of non-aerated flow zone Li rises with the increase in discharge in all four models of stepped spillways.
With the increase of slope from mild to steep slope the start of air entrainment is close to the crest, and the inception point moves toward the crest as the slope of the stepped spillway increases.
The normalized Li decreases with the increase in surface roughness ks. So normalized Li has a direct relationship with the Froude number, Li/ks increases with the increase in the Froude number. The minimum value of Li/ks is shown at the minimum Froude number.
Critical depth (dc) is the depth of the flow where the increase in discharge causes depth to increase. The length of the non-aerated flow zone Li has a direct relationship with the critical depth, with the increase in critical depth the Li increases in all the slope channels of stepped spillways.
Relative flow depth is the quotient of critical depth to step height. Relative flow depth indicates that at lower channel slope values the relative flow depth increases and the length of non-aerated flow zone also increases.