# CFD Modelling of a Stepped Spillway with Various Step Layouts

Authors: Shicheng Li; Qiulin Li; James Yang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

A traditional stepped spillway is prone to cavitation risks. To improve its hydraulic behaviors, distorted step faces and pool weirs are devised. By numerical modelling, comparative studies are conducted to look into the flow features. The pressures on step surfaces of the unconventional layouts exhibit 3D distributions. Pool weirs are essential in increasing both the min. and max. pressure loads. Pressures on the downstream bed show a unique pattern for V- and inverted V-shaped models, with the extreme pressures at the sidewalls for the former and at the central plane for the latter. Symmetrical secondary flows are formed in V- and inverted V-shaped cases with different patterns. Distributions of turbulent kinetic energy suggest differences in flow motions in all cases. Furthermore, the relative energy loss of flat setups is ∼5.4% lower than that of the pooled ones with the same step face angle; inverting the face angle does not give rise to noticeable change. The results provide reference for relevant projects.


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## Body

## 1. Introduction

With the advances in roller compacted concrete (RCC) techniques, stepped spillways have gained popularity with a wide range of applications including hydropower [1–3]. As unique roughness elements, steps play an important role in improving energy dissipation and aeration efficiency [4–6]. As a result, construction costs are considerably low owing to the reduced dimensions of the downstream energy dissipation facility [7, 8]. Effective air entrainment contributes to reduction in cavitation damages [9–11]. Despite a large number of research projects, the complexity of the flows in stepped spillways makes the hydraulic design fairly difficult.Felder and Chanson [12] studied air concentration and energy dissipation of a stepped spillway with nonuniform step heights. Pfister and Hager [13] examined the self-aeration characteristics of flows in stepped spillways. Researchers [14–16] also investigated air-water mixture flows and turbulence properties. In addition, novel designs are also implemented to increase the hydraulic performances of traditional stepped spillways. For instance, Felder [6]; Zare and Doering [17]; and Kökpinar [18] looked into the aeration and flow resistance in stepped spillways with sills and baffles. Experiments by Zare and Doering [19] suggested higher energy dissipation by rounding the step edges. Inclining steps upwards is effective in lowering the outlet velocity and increasing energy loss [20]. 3D flow motions are produced by changing the step face angle, leading to higher head loss and aeration [21].Most of the studies of novel designs focused on the energy dissipation and air entrainment; examinations of step configurations are relatively limited. This paper illustrates, through 3D numerical simulations, the flow features of several step layouts. The aim is to reveal the effects of geometrical layouts of steps on pressure loads and flow structures, which would contribute to better understanding of their hydraulic characteristics and to effective engineering measures in design of stepped spillways.


## 2. Numerical Simulation

Felder and Chanson [22] carried out a series of experiments to study the hydraulic properties of the skimming flow over a stepped spillway, providing references for similar projects. Therefore, based on their model (shown in Figure 1), several modifications are made to investigate the effects of step layout on the flow characteristics. The spillway has 10 identical steps with step height h = 0.10 m and step length l = 0.20 m. The water is supplied from an upstream tank. A board-crested weir, with a length of Lcrest = 1.01 m and a rounded corner (r=0.08), conveys water the examined steps. The width of the spillway channel is w = 0.52 m. In this paper, spillway layouts are redesigned through changes in the step face angle (θ) and additions of pool weir, as displayed in Figure 2.Figure 1
Sketch of the physical model setup.Figure 2
Spillway configurations: (a) traditional stepped spillway (TS); (b) V-shaped stepped spillway (VS); (c) pooled V-shaped stepped spillway (PVS); (d) inverted V-shaped stepped spillway (IVS); and (e) pooled inverted V-shaped stepped spillway (PIVS).


### 2.1. Numerical Model

Performing CFD simulations is an effective tool to study complex flow problems [23–26]. Previous CFD studies [21, 27–29] reported the capability of the k−ε model in reproduction of the major hydraulic characteristics of the flows over stepped spillways. Therefore, in the current study, this turbulence model by Launder and Spalding [30] is used to look into the spillway flow features with different step layouts. The governing equations of the k−ε model are as follows:k equation:(1)∂ρk∂t+∂∂xiρuik=∂∂xiμ+μtσk∂k∂xi+Gk−ρε.ε equation:(2)∂ρε∂t+∂ρuiε∂xi=∂∂xiμ+μtσε∂ε∂xi+C1εεkG−C2ερε2k,where(3)μt=ρCμk2ε,Gk=μt∂ui∂xj+∂uj∂xi∂ui∂xj,in which ρ = flow density, ui and uj = velocity component, t = time and μ = dynamic viscosity. The empirical constants include Cμ = 0.09, σk = 1.0, σε = 1.3, C1ε = 1.44, and C2ε = 1.92.The volume of fluid (VOF) technique is employed to track the air-water interface. So,ρ and μ are obtained from(4)ρ=αwρw+1−αwρa,μ=αwμw+1−αwμa,where αw = volume fraction of water, ρw = water density, ρa = air density, μw = water viscosity, and μa = air viscosity.


### 2.2. Boundary Conditions

Figure3 shows the boundary conditions and local mesh pattern (for the VS layout). The upstream inlet includes two parts: the upper part as pressure inlet and the lower part as velocity inlet. The downstream boundary is set as pressure outlet; the top surface is given as pressure inlet. Moreover, the unmentioned boundaries, such as the channel bottom and sidewalls, are treated as solid wall. The inflow rate ranges from 0.063 to 0.113 m3/s.Figure 3
Computational domain with local enlargement and boundary conditions.


### 2.3. Model Calibration

The sensitivity of numerical solutions is checked using grid convergence index (GCI) [31–34]. The method, commonly used to estimate the uncertainty of CFD modelling, is expressed by indexes GCI and S:(5)GCI=1.25ϕ1−ϕ2/ϕ1h2−h1S−1,S=1lnh2/h1lnϕ3−ϕ2ϕ2−ϕ1+lnh2/h1S−sgnϕ3−ϕ2/ϕ2−ϕ1h3/h2S−sgnϕ3−ϕ2/ϕ2−ϕ1,where ϕii=1,2,3 = selected numerical results and hii=1,2,3 = grid size. Three grids with 41.5, 76.9, and 92.8 million cells are generated to evaluate grid independence.Figure4 shows the numerical solutions with error bars using GCI, where Vx = velocity in the x direction, Vc=g×dc = critical flow velocity, dc = critical flow depth, and P = flow pressure. It is obvious that the uncertainty of the results is small at most positions. The max. uncertainties of the normalized pressure and flow velocity are 8.4% and 6.6%, respectively. In view of this and affordable CPU time, the selection of the 76.9 million cells is reasonable.Figure 4
Numerical solutions with error bars by GCI (dc/h=1.45): (a) pressure and (b) velocity.
(a)
(b)With the grid independence checked, the numerical results are also compared with the experimental data from Felder and Chanson [22]. Figures 5(a)‒5(c) present comparisons of the free surface, velocity profile, and pressure distribution. As seen, the discrepancies between the computational data and laboratory work are small. The max. relative errors for the water surface, velocity, and pressure are 2.8%, 8.8%, and 4.6%, respectively. The good agreement between the results shows that the numerical model produces, with accuracy, the key hydraulic features of the stepped spillway.Figure 5
Comparisions between simulations and experiments: (a) free surface,H1/Lcrest=0.224; (b) velocity profile; (c) pressure distrbution.
(a)
(b)
(c)


## 2.1. Numerical Model

Performing CFD simulations is an effective tool to study complex flow problems [23–26]. Previous CFD studies [21, 27–29] reported the capability of the k−ε model in reproduction of the major hydraulic characteristics of the flows over stepped spillways. Therefore, in the current study, this turbulence model by Launder and Spalding [30] is used to look into the spillway flow features with different step layouts. The governing equations of the k−ε model are as follows:k equation:(1)∂ρk∂t+∂∂xiρuik=∂∂xiμ+μtσk∂k∂xi+Gk−ρε.ε equation:(2)∂ρε∂t+∂ρuiε∂xi=∂∂xiμ+μtσε∂ε∂xi+C1εεkG−C2ερε2k,where(3)μt=ρCμk2ε,Gk=μt∂ui∂xj+∂uj∂xi∂ui∂xj,in which ρ = flow density, ui and uj = velocity component, t = time and μ = dynamic viscosity. The empirical constants include Cμ = 0.09, σk = 1.0, σε = 1.3, C1ε = 1.44, and C2ε = 1.92.The volume of fluid (VOF) technique is employed to track the air-water interface. So,ρ and μ are obtained from(4)ρ=αwρw+1−αwρa,μ=αwμw+1−αwμa,where αw = volume fraction of water, ρw = water density, ρa = air density, μw = water viscosity, and μa = air viscosity.


## 2.2. Boundary Conditions

Figure3 shows the boundary conditions and local mesh pattern (for the VS layout). The upstream inlet includes two parts: the upper part as pressure inlet and the lower part as velocity inlet. The downstream boundary is set as pressure outlet; the top surface is given as pressure inlet. Moreover, the unmentioned boundaries, such as the channel bottom and sidewalls, are treated as solid wall. The inflow rate ranges from 0.063 to 0.113 m3/s.Figure 3
Computational domain with local enlargement and boundary conditions.


## 2.3. Model Calibration

The sensitivity of numerical solutions is checked using grid convergence index (GCI) [31–34]. The method, commonly used to estimate the uncertainty of CFD modelling, is expressed by indexes GCI and S:(5)GCI=1.25ϕ1−ϕ2/ϕ1h2−h1S−1,S=1lnh2/h1lnϕ3−ϕ2ϕ2−ϕ1+lnh2/h1S−sgnϕ3−ϕ2/ϕ2−ϕ1h3/h2S−sgnϕ3−ϕ2/ϕ2−ϕ1,where ϕii=1,2,3 = selected numerical results and hii=1,2,3 = grid size. Three grids with 41.5, 76.9, and 92.8 million cells are generated to evaluate grid independence.Figure4 shows the numerical solutions with error bars using GCI, where Vx = velocity in the x direction, Vc=g×dc = critical flow velocity, dc = critical flow depth, and P = flow pressure. It is obvious that the uncertainty of the results is small at most positions. The max. uncertainties of the normalized pressure and flow velocity are 8.4% and 6.6%, respectively. In view of this and affordable CPU time, the selection of the 76.9 million cells is reasonable.Figure 4
Numerical solutions with error bars by GCI (dc/h=1.45): (a) pressure and (b) velocity.
(a)
(b)With the grid independence checked, the numerical results are also compared with the experimental data from Felder and Chanson [22]. Figures 5(a)‒5(c) present comparisons of the free surface, velocity profile, and pressure distribution. As seen, the discrepancies between the computational data and laboratory work are small. The max. relative errors for the water surface, velocity, and pressure are 2.8%, 8.8%, and 4.6%, respectively. The good agreement between the results shows that the numerical model produces, with accuracy, the key hydraulic features of the stepped spillway.Figure 5
Comparisions between simulations and experiments: (a) free surface,H1/Lcrest=0.224; (b) velocity profile; (c) pressure distrbution.
(a)
(b)
(c)


## 3. Results and Analysis

Due to the transverse motion of the water flow, the pressures on the spillway bottom vary greatly and exhibit a typical 3D pattern. Letz be transverse axis with z=0 at the central plane of the spillway channel. To show the pressure distribution pattern, three planes are selected (Figure 6): z/w=0 (axial plane), z/w=0.25, and z/w=0.5 (sidewall), where ρ = density of water and g = gravitational acceleration.Figure 6
Pressures acting on step surfaces (step 10,dc/h=1.45): (a) horizontal surface; (b) vertical surface.
(a)
(b)


### 3.1. Pressures on Horizontal Step Surface

For the TS layout, the pressures exhibit S-shaped variations, whose patterns remain almost unchanged in the transverse direction. The VS and IVS pressure fluctuations show an opposite trend in the spanwise direction. For the former, a large change is found atz/w=0.5 and a little variation at z/w=0. As for PVS, the pressure fluctuations increase from z/w = 0 to 0.5, showing an U-shaped distribution. Conversely, the PIVS pressures change most at z/w=0 and little at z/w=0.5. In addition, at z/w=0.25, the VS and IVS pressures are almost the same, with an S-shaped distribution. PVS and PIVS have similar pressure distribution and magnitude. Therefore, for a given θ (for either V or inverted V), pool weirs do not impose any significant effect on the pressures at z/w=0.25.


### 3.2. Pressures on Vertical Step Surface

It is noticed that, for all configurations, the pressures acting on the vertical step surfaces decline generally with an increase in water depth. In addition, the pressures of pooled layouts are, regardless ofθ, larger than the flat ones. At the selected locations, no negative pressure is found for PVS and PIVS. However, TS shows subpressures in the entire cross section. At z/w=0, TS and IVS display a similar level of pressures; the PVS and PIVS pressures are close. At z/w=0.25, the VS pressures are almost the same as IVS, which is also true of PVS and PIVS. This implies that, at z/w=0.25, inverting the step face angle does not seem to exert any influence on the pressures. At z/w=0.5, the TS and VS pressures exhibit a similar distribution pattern; the PIVS pressures are slightly greater than the PVS.


### 3.3. Extreme Pressures on Step Faces

LetPmin and Pmax denote the min. and max. pressures acting upon the step surfaces. Figure 7 compares their values for all the layouts. On the horizontal surfaces, the pressures are positive and increase as flow rate increases. In comparison with the TS layout, VS and IVS with changed step face angles lead to lower Pmin; PVS and PIVS give roughly three times as large Pmin values as TS. For Pmax, alternation of step face angle and addition of pool weirs give rise to considerably lower Pmax. VS and IVS exhibit a similar level of Pmax; PVS and PIVS show an almost same pattern of distribution. In the pooled layouts, the large Pmin and Pmax values are associated with the larger water depth below the pseudobottom.Figure 7
Min. and max. pressures on step surfaces: (a) horizontal surface and (b) vertical surface.
(a)
(b)On the vertical step surfaces, negative pressures occur only in the configurations without pool weirs; VS and IVS result in slightly higherPmin than TS. For PVS and PIVS, their Pmin values are all positive and much higher than other layouts. Moreover, for PVS and PIVS, the difference in Pmax is insignificant and their values are almost twice those of TS. VS and IVS lead to almost identical Pmax and are moderately larger than TS. A pool weir contains more water that is trapped in the cavities, leading to lager Pmax values. Flow separations move from the step edges to the weirs, which contributes to higher Pmin values on the vertical step face.


### 3.4. Pressures on Downstream Bed

The downstream area is of importance to the adjacent hydraulic structures and environment. Therefore, flow pressures on the downstream bed are examined to provide basis for protection measures. Figures8(a)‒8(e) show the contours of pressure on the downstream bottom. Figure 8(f) illustrates their max. pressures. It is evident that the TS layout does not lead to any pronounced change in the transverse direction, with extreme pressures occurring along the entire width. The other layouts exhibit a typical 3D pattern. For VS and PVS, their max. pressures occur at the sidewalls, while for IVS and PIVS, they appear on the axial plane. It seems that the locations of extreme pressures are associated with the step face angle, rather than the pool weir. Engineering measures should be undertaken to prevent undesirable scouring at these positions.Figure 8
Flow pressures acting on the downstream bed (dc/h=1.45): (a) TS; (b) VS; (c) PVS; (d) IVS; (e) PIVS; (f) max. pressure.
(a)
(b)
(c)
(d)
(e)
(f)As shown in Figure8(f), for all the layouts, Pmax exhibits an almost linear growth as the flow discharge increases, with TS showing significantly lower values than others. For VS and IVS, Pmax is slightly larger than that of PVS and PIVS. For the same face angle, the pool weirs reduce Pmax on the bed. However, inverting the step face angle leads to a higher Pmax.


### 3.5. Downstream Flow Patterns

The flow patterns in the downstream channel are investigated, shown in Figure9. As seen from Figures 9(a)‒9(e), secondary flows are formed in all the layouts exclusive of TS. In TS, the velocity vectors are approximately parallel; the free surface does not show much change in the transverse direction. For VS and PVS, symmetrical secondary flows occur near the axial planes; the water-surface elevations increase from the sidewalls to the central plane. For IVS and PIVS, secondary flows exist at the sidewalls, where the water depth reaches a maximum. Furthermore, for the same step face angle, larger fluctuations of the water surfaces are seen in the configurations without pool weirs. Therefore, the step face angle affects the locations of secondary flows, while the pool weirs influence the water-surface variations.Figure 9
Flow patterns (x=3m, dc/h=1.45): (a) TS; (b) VS; (c) PVS; (d) IVS; (e) PIVS; (f) free surface fluctuation.
(a)
(b)
(c)
(d)
(e)
(f)Figure9(f) shows the free surface fluctuation, expressed in terms of ΔEl=Elmax−Elmin, where Elmax and Elmin are the max. and min. water-surface elevation, respectively. As the flow rate increases, ΔEl becomes larger for all the layouts excluding TS. The ΔEl values of TS remained unchanged regardless of flow discharge. In addition, for the unpooled layouts, the average ΔEl value is ∼70% higher for the pooled ones. For the inverted V-shaped layouts, it is ∼60% larger than the V-shaped ones.


### 3.6. Energy Dissipation

On the step surfaces, contours of the turbulent kinetic energy (TKE) are shown in Figure10(a), providing information about the velocity gradient. The overlaying flow hits the horizontal step faces when flowing downstream and the TKE exhibits extreme values at the downstream end (see the TS plot). For VS, the collision between the flow and step faces occurs near the sidewalls [21], resulting in high velocity gradient in these regions. For PVS, large TKE values are at the pool weirs, which is due to the larger bottom roughness. For the inverted V-shaped layouts, the TKE distributions are similar to the V-shaped ones, however with different locations.Figure 10
Energy loss: (a) turbulent kinetic energy (dc/h=1.45); (b) energy dissipation rate.
(a)
(b)The residual head at the chute end is a key parameter for evaluation of the hydraulic performance of a spillway. The relative energy loss (η) is commonly used to quantify its energy dissipation [11, 19, 27, 35]:(6)η=E2−E1E2,where E1 and E2  =  are the energy heads at the up- and downstream cross sections, respectively, calculated by(7)E1=h1+qw22gh12,E2=HT+hus+qw22gHT+hus2,where h1 = upstream water depth, qw=Q/w = unit discharge, HT = dam height, and hus = water head above the spillway crest. Figure 10(b) illustrates the results of the energy dissipation. It is obvious that η for all configurations exhibits a downward trend with an increase in flow discharge, which is consistent with the findings in Li et al. [36], Zare and Doering [17], and Chinnarasri and Wongwises [20]. The TS layout leads to the lowest efficiency. In addition, η is ranked in the order of PVS≈PIVS > VS≈IVS > TS. For the pooled configurations, their η values are, for the same θ, ∼5.4% larger than those for the flat ones. So all proposed modifications from TS enhance the energy dissipation. The V- and inverted V-shaped ones do not show any significant differences in η, which is similar to the result of Bai et al. [27].


## 3.1. Pressures on Horizontal Step Surface

For the TS layout, the pressures exhibit S-shaped variations, whose patterns remain almost unchanged in the transverse direction. The VS and IVS pressure fluctuations show an opposite trend in the spanwise direction. For the former, a large change is found atz/w=0.5 and a little variation at z/w=0. As for PVS, the pressure fluctuations increase from z/w = 0 to 0.5, showing an U-shaped distribution. Conversely, the PIVS pressures change most at z/w=0 and little at z/w=0.5. In addition, at z/w=0.25, the VS and IVS pressures are almost the same, with an S-shaped distribution. PVS and PIVS have similar pressure distribution and magnitude. Therefore, for a given θ (for either V or inverted V), pool weirs do not impose any significant effect on the pressures at z/w=0.25.


## 3.2. Pressures on Vertical Step Surface

It is noticed that, for all configurations, the pressures acting on the vertical step surfaces decline generally with an increase in water depth. In addition, the pressures of pooled layouts are, regardless ofθ, larger than the flat ones. At the selected locations, no negative pressure is found for PVS and PIVS. However, TS shows subpressures in the entire cross section. At z/w=0, TS and IVS display a similar level of pressures; the PVS and PIVS pressures are close. At z/w=0.25, the VS pressures are almost the same as IVS, which is also true of PVS and PIVS. This implies that, at z/w=0.25, inverting the step face angle does not seem to exert any influence on the pressures. At z/w=0.5, the TS and VS pressures exhibit a similar distribution pattern; the PIVS pressures are slightly greater than the PVS.


## 3.3. Extreme Pressures on Step Faces

LetPmin and Pmax denote the min. and max. pressures acting upon the step surfaces. Figure 7 compares their values for all the layouts. On the horizontal surfaces, the pressures are positive and increase as flow rate increases. In comparison with the TS layout, VS and IVS with changed step face angles lead to lower Pmin; PVS and PIVS give roughly three times as large Pmin values as TS. For Pmax, alternation of step face angle and addition of pool weirs give rise to considerably lower Pmax. VS and IVS exhibit a similar level of Pmax; PVS and PIVS show an almost same pattern of distribution. In the pooled layouts, the large Pmin and Pmax values are associated with the larger water depth below the pseudobottom.Figure 7
Min. and max. pressures on step surfaces: (a) horizontal surface and (b) vertical surface.
(a)
(b)On the vertical step surfaces, negative pressures occur only in the configurations without pool weirs; VS and IVS result in slightly higherPmin than TS. For PVS and PIVS, their Pmin values are all positive and much higher than other layouts. Moreover, for PVS and PIVS, the difference in Pmax is insignificant and their values are almost twice those of TS. VS and IVS lead to almost identical Pmax and are moderately larger than TS. A pool weir contains more water that is trapped in the cavities, leading to lager Pmax values. Flow separations move from the step edges to the weirs, which contributes to higher Pmin values on the vertical step face.


## 3.4. Pressures on Downstream Bed

The downstream area is of importance to the adjacent hydraulic structures and environment. Therefore, flow pressures on the downstream bed are examined to provide basis for protection measures. Figures8(a)‒8(e) show the contours of pressure on the downstream bottom. Figure 8(f) illustrates their max. pressures. It is evident that the TS layout does not lead to any pronounced change in the transverse direction, with extreme pressures occurring along the entire width. The other layouts exhibit a typical 3D pattern. For VS and PVS, their max. pressures occur at the sidewalls, while for IVS and PIVS, they appear on the axial plane. It seems that the locations of extreme pressures are associated with the step face angle, rather than the pool weir. Engineering measures should be undertaken to prevent undesirable scouring at these positions.Figure 8
Flow pressures acting on the downstream bed (dc/h=1.45): (a) TS; (b) VS; (c) PVS; (d) IVS; (e) PIVS; (f) max. pressure.
(a)
(b)
(c)
(d)
(e)
(f)As shown in Figure8(f), for all the layouts, Pmax exhibits an almost linear growth as the flow discharge increases, with TS showing significantly lower values than others. For VS and IVS, Pmax is slightly larger than that of PVS and PIVS. For the same face angle, the pool weirs reduce Pmax on the bed. However, inverting the step face angle leads to a higher Pmax.


## 3.5. Downstream Flow Patterns

The flow patterns in the downstream channel are investigated, shown in Figure9. As seen from Figures 9(a)‒9(e), secondary flows are formed in all the layouts exclusive of TS. In TS, the velocity vectors are approximately parallel; the free surface does not show much change in the transverse direction. For VS and PVS, symmetrical secondary flows occur near the axial planes; the water-surface elevations increase from the sidewalls to the central plane. For IVS and PIVS, secondary flows exist at the sidewalls, where the water depth reaches a maximum. Furthermore, for the same step face angle, larger fluctuations of the water surfaces are seen in the configurations without pool weirs. Therefore, the step face angle affects the locations of secondary flows, while the pool weirs influence the water-surface variations.Figure 9
Flow patterns (x=3m, dc/h=1.45): (a) TS; (b) VS; (c) PVS; (d) IVS; (e) PIVS; (f) free surface fluctuation.
(a)
(b)
(c)
(d)
(e)
(f)Figure9(f) shows the free surface fluctuation, expressed in terms of ΔEl=Elmax−Elmin, where Elmax and Elmin are the max. and min. water-surface elevation, respectively. As the flow rate increases, ΔEl becomes larger for all the layouts excluding TS. The ΔEl values of TS remained unchanged regardless of flow discharge. In addition, for the unpooled layouts, the average ΔEl value is ∼70% higher for the pooled ones. For the inverted V-shaped layouts, it is ∼60% larger than the V-shaped ones.


## 3.6. Energy Dissipation

On the step surfaces, contours of the turbulent kinetic energy (TKE) are shown in Figure10(a), providing information about the velocity gradient. The overlaying flow hits the horizontal step faces when flowing downstream and the TKE exhibits extreme values at the downstream end (see the TS plot). For VS, the collision between the flow and step faces occurs near the sidewalls [21], resulting in high velocity gradient in these regions. For PVS, large TKE values are at the pool weirs, which is due to the larger bottom roughness. For the inverted V-shaped layouts, the TKE distributions are similar to the V-shaped ones, however with different locations.Figure 10
Energy loss: (a) turbulent kinetic energy (dc/h=1.45); (b) energy dissipation rate.
(a)
(b)The residual head at the chute end is a key parameter for evaluation of the hydraulic performance of a spillway. The relative energy loss (η) is commonly used to quantify its energy dissipation [11, 19, 27, 35]:(6)η=E2−E1E2,where E1 and E2  =  are the energy heads at the up- and downstream cross sections, respectively, calculated by(7)E1=h1+qw22gh12,E2=HT+hus+qw22gHT+hus2,where h1 = upstream water depth, qw=Q/w = unit discharge, HT = dam height, and hus = water head above the spillway crest. Figure 10(b) illustrates the results of the energy dissipation. It is obvious that η for all configurations exhibits a downward trend with an increase in flow discharge, which is consistent with the findings in Li et al. [36], Zare and Doering [17], and Chinnarasri and Wongwises [20]. The TS layout leads to the lowest efficiency. In addition, η is ranked in the order of PVS≈PIVS > VS≈IVS > TS. For the pooled configurations, their η values are, for the same θ, ∼5.4% larger than those for the flat ones. So all proposed modifications from TS enhance the energy dissipation. The V- and inverted V-shaped ones do not show any significant differences in η, which is similar to the result of Bai et al. [27].


## 4. Conclusions

With the increasing use of stepped spillways, step configurations become an issue of concern for an optimized design adapted to varying flow conditions. The study aims to provide insight into the hydraulic features of a stepped spillway with modified step layouts. The modifications refer to distorted step faces and addition of pool weirs. In combination with thek−ε turbulence model, 3D numerical studies are performed using the two-phase VOF model. The major conclusions include the following:(i)
The pressures on step faces of the V- and inverted V-shaped layouts exhibit a 3D pattern. For the vertical step surface, cavitation risk is reduced if a pool weir is installed on the horizontal step surface. In addition, the pressures in all the layouts show an upward trend with an increase in flow discharge.(ii)
With the same step face angle, the patterns of pressure distribution on the downstream bottom are similar. However, compared with the pooled ones, the flat stepped layouts result in higher pressures.(iii)
No secondary flow is found in the conventional configuration. If the step face angle is not equal to 180°, it would appear and also lead to differences in water-surface fluctuations.(iv)
As for the energy dissipation rate, the V- and inverted V-shaped layouts do not lead to any pronounced difference. However, with same step face angle, the flat layouts give ∼5.4% lower values than the pooled ones.The paper proposes several modified layouts of a stepped spillway, illustrating their effects on the pressure and flow characteristics. Numerical results reveal both their advantages and disadvantages. For instance, negative pressures acting on the vertical step surfaces are reduced with the help of the pool weirs, which is beneficial to the reduction in cavitation risk. However, they also increase the flow pressure loads. Compared with the modified ones, a conventional layout shows favorable flow patterns in the downstream channel; its energy dissipation is however less effective. In addition, owing to the 3D flow patterns, the modified versions are suitable alternatives for landscape and aeration purposes.



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*Source: 6215739-2019-11-28.xml*