# Stepped Spillway Slope Effect on Air Entrainment and Inception Point Location

^{*}

## Abstract

**:**

## 1. Introduction

^{2}-f, and LES models [27,28]. 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 V

^{2}-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.

## 2. Materials and Methods

#### 2.1. Mesh Gridance

#### 2.2. Boundary Conditions

#### 2.2.1. Inlet Boundary

#### 2.2.2. Outlet Boundary

#### 2.3. Walls

#### 2.4. Volume of Fluid

_{a}denotes the volume part of air in Equation (1).

#### 2.5. Turbulence Model

## 3. Results and Discussions

#### 3.1. The Commencement of the Skimming Flow Regime

#### 3.2. Effect of Channel Slope on Inception Point Location

#### 3.3. Variation of Inception Length with Critical Depth

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${A}_{0}$ | model constant (4.04) |

${A}_{s}$ | model constant ($\sqrt{6}\mathrm{cos}\mathsf{\varphi}$) |

B | width of the step (m) |

${C}_{2}$ | 1.9 (model constant) |

${C}_{1\mathsf{\epsilon}}$ | 1.44 (model constant) |

${C}_{u}$ | coefficient of uniformity ${C}_{\mathsf{\mu}}=\frac{1}{{A}_{0}+{A}_{s}\frac{kU*}{\mathsf{\epsilon}}}$ |

F | Froude surface roughness |

${G}_{b}$ | generation of k due to buoyancy (kg/(ms3)) |

${G}_{k}$ | generation of k due to fluid Shear ${G}_{k}={\mathsf{\mu}}_{t}{S}^{2}$(kg/ms^{3}) |

g | gravitational acceleration (m/sec^{2}) |

h | step height (m) |

hp | height of Pool (m) |

k | turbulent kinetic energy (m^{2}/s^{2}) |

ks | roughness height, ks = hcosθ (m) |

l | step length (m) |

lp | length of Pool (m) |

Li | distance of inception point location from spillway crest (m) |

$\frac{Li}{{k}_{s}}$ | normalized distance from spillway crest to Inception Point |

M1-M4 | number of stepped spillway model |

N | number of steps |

S | modulus of mean rate of tensor $S=\sqrt{2{S}_{ij}{S}_{ij}}$ |

${S}_{k}$ | source term of kinetic energy ($kg/(m{s}^{3})$) |

${S}_{\mathsf{\epsilon}}$ | source term of dissipation rate ($kg/(m{s}^{4})$) |

${S}_{ij}$ | mean rate of deformation |

P | pressure in (N/m^{2}) |

Q | unit discharge (m^{2}/s) |

R | radius of the edge (m) |

W | width of the step |

${Y}_{m}$ | effect of compressibility on turbulence (kg/ms^{3}) |

${\mathsf{\sigma}}_{k}$ | turbulent prandtl number |

${\mathsf{\sigma}}_{\mathsf{\epsilon}}$ | turbulent prandtl number |

${u}_{i}$ | velocity in ${x}_{i}$ direction (m/s) |

${u}_{j}$ | velocity in ${x}_{j}$ direction (m/s) |

${\alpha}_{a}$ | volume fraction of air (%) |

${\alpha}_{w}$ | volume fraction of water (%) |

$\mathsf{\epsilon}$ | turbulent dissipation rate (m^{2}/s^{3}) |

θ | spillway slope (°) |

$\mathsf{\mu}$ | molecular dynamic viscosity (kg/ms) |

${\mathsf{\mu}}_{t}$ | turbulent dynamic viscosity (kg/ms) |

$\mathsf{\rho}$ | cell density (kg/m^{3}) |

${\mathsf{\rho}}_{a}$ | density of air (kg/m^{3}) |

${\mathsf{\rho}}_{w}$ | density of water (kg/m^{3}) |

t | time in (sec) |

$\mathsf{\varphi}$ | channel slope (°) |

$\upsilon $ | kinematic viscosity (m^{2}/s) |

dc | critical depth |

$\frac{dc}{h}$ | Relative critical depth |

${\mathsf{\sigma}}_{k}$ | 1.0 (model constant) |

${\mathsf{\sigma}}_{\u03f5}$ | 1.2(model constant) |

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**Figure 2.**Three-dimensional stepped spillway models used for numerical experiments; (

**a**) M

_{1}stepped spillway model with slope of 12.5°; (

**b**) M

_{2}stepped spillway model with slope of 20°; (

**c**) M

_{3}stepped spillway model with slope of 29°; (

**d**) M

_{4}stepped spillway model through a slope of 35°.

**Figure 4.**Variation of relative critical depth (dc/h) of different slope stepped spillway with (h/l) values with critical values (derived from the equation).

**Figure 5.**Water volume fraction contours of stepped spillway models with slopes of 12.5° and 20° at different discharge rates; (

**a**) 12.5° model with discharge of 0.625 m

^{2}/s; (

**b**) 12.5° model with discharge of 0.75 m

^{2}/s; (

**c**) 20° model with discharge of 0.625 m

^{2}/s; (

**d**) 20° model with discharge of 0.75 m

^{2}/s.

**Figure 6.**Water volume fraction contours of stepped spillway models with slopes of 29° and 35° at different discharge rates; (

**a**) 29° model with discharge of 0.625 m

^{2}/s; (

**b**) 29° model with discharge of 0.75 m

^{2}/s; (

**c**) 35° model with discharge of 0.625 m

^{2}/s; (

**d**) 35° model with discharge of 0.75 m

^{2}/s.

**Figure 7.**Relationship between discharge (q) and inception point length of different sloped stepped spillways.

**Figure 8.**Variation of normalized Li with the Froude number of different sloped stepped spillway models.

**Figure 9.**Variation of inception point length with relative critical depth (dc/h) of different sloped stepped spillways.

**Figure 10.**Normalized inception point (Li/ks) relationship with the relative critical depth of different sloped stepped spillways.

Model | Step Height (h) | Step Length (m) | Slope |
---|---|---|---|

M1 | 0.1 | 0.45 | 12.5° |

M2 | 0.163 | 0.45 | 19° |

M3 | 0.25 | 0.45 | 29° |

M4 | 0.315 | 0.45 | 35° |

**Table 2.**Table of unit discharge (q), step height, inception point (Li), and normalized Li for 12.5°, 20°, 29°, and 35° stepped spillways.

Unit Discharge q (m^{2}/s) | Channel Slope ° | Step Height (m) | Froude Number F | Inception Point Length Li | Surface Roughness ks | Li/ks |
---|---|---|---|---|---|---|

0.625 | 12.5° | 0.1 | 13.58 | 1.35 | 0.097 | 13.91 |

0.75 | 12.5° | 0.1 | 16.3 | 1.8 | 0.097 | 18.55 |

0.875 | 12.5° | 0.1 | 19.02 | 2.25 | 0.097 | 23.19 |

1 | 12.5° | 0.1 | 22.69 | 2.7 | 0.097 | 27.83 |

0.625 | 20° | 0.163 | 5.73 | 0.9 | 0.153 | 5.88 |

0.75 | 20° | 0.163 | 6.88 | 1.35 | 0.153 | 8.82 |

0.875 | 20° | 0.163 | 8.02 | 1.8 | 0.153 | 11.76 |

1 | 20° | 0.163 | 9.12 | 2.25 | 0.153 | 14.7 |

0.625 | 29° | 0.25 | 2.82 | 0.45 | 0.218 | 2.06 |

0.75 | 29° | 0.25 | 3.39 | 0.9 | 0.218 | 4.12 |

0.875 | 29° | 0.25 | 3.95 | 1.35 | 0.218 | 6.19 |

1 | 29° | 0.25 | 4.5 | 1.8 | 0.218 | 8.25 |

0.625 | 35° | 0.315 | 1.49 | 0.3 | 0.258 | 1.16 |

0.75 | 35° | 0.315 | 1.78 | 0.6 | 0.258 | 2.32 |

0.875 | 35° | 0.315 | 2.08 | 0.9 | 0.258 | 3.48 |

1 | 35° | 0.25 | 3.21 | 1.35 | 0.258 | 5.23 |

**Table 3.**Summary of critical depth (dc), relative critical depth (dc/h), and ks/h for 12.5°, 20°, 29°.

Critical Depth dc | Channel Slope ° | Step Height | Surface Roughness ks | dc/h | ks/h |
---|---|---|---|---|---|

0.345 | 12.5° | 0.1 | 0.097 | 3.45 | 0.97 |

0.4 | 12.5° | 0.1 | 0.097 | 4 | 0.97 |

0.43 | 12.5° | 0.1 | 0.097 | 4.3 | 0.97 |

0.47 | 12.5 | 0.1 | 0.097 | 4.7 | 0.97 |

0.345 | 20° | 0.163 | 0.153 | 2.11 | 0.938 |

0.4 | 20° | 0.163 | 0.153 | 2.45 | 0.938 |

0.43 | 20° | 0.163 | 0.153 | 2.63 | 0.938 |

0.47 | 20° | 0.163 | 0.153 | 2.88 | 0.938 |

0.345 | 29° | 0.25 | 0.218 | 1.38 | 0.872 |

0.4 | 29° | 0.25 | 0.218 | 1.6 | 0.872 |

0.43 | 29° | 0.25 | 0.218 | 1.72 | 0.872 |

0.47 | 29° | 0.25 | 0.218 | 1.88 | 0.872 |

0.345 | 35° | 0.315 | 0.258 | 1.09 | 0.819 |

0.4 | 35° | 0.315 | 0.258 | 1.26 | 0.819 |

0.43 | 35° | 0.315 | 0.258 | 1.36 | 0.819 |

0.47 | 35° | 0.315 | 0.258 | 1.49 | 0.819 |

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Raza, A.; Wan, W.; Mehmood, K. Stepped Spillway Slope Effect on Air Entrainment and Inception Point Location. *Water* **2021**, *13*, 1428.
https://doi.org/10.3390/w13101428

**AMA Style**

Raza A, Wan W, Mehmood K. Stepped Spillway Slope Effect on Air Entrainment and Inception Point Location. *Water*. 2021; 13(10):1428.
https://doi.org/10.3390/w13101428

**Chicago/Turabian Style**

Raza, Awais, Wuyi Wan, and Kashif Mehmood. 2021. "Stepped Spillway Slope Effect on Air Entrainment and Inception Point Location" *Water* 13, no. 10: 1428.
https://doi.org/10.3390/w13101428