# Numerical Study on the Hydraulic Properties of Flow over Different Pooled Stepped Spillways

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Relations of Flow Regime Detection

_{c}/h (y

_{c}is the critical depth of the flow on the broad crested weir), and a dimensionless variable of steps, h/l. According to Equations (1)–(3), criteria are presented when a skimming flow regime occurs by Rajaratnam [7]; Peyras et al. [8] and Chanson [9], respectively:

## 3. Numerical Simulation and Simulation Setup

#### 3.1. Numerical Models

_{crest}, and width, W, of the broad-crested weir were 1 m and 0.52 m, respectively (see Figure 2a). Based on this physical model, 4 layouts were designed and studied numerically by modifying the step edges, using pooled weirs with same pool of height d

_{p}= 0.05 m and different configurations of the main steps: fully pooled steps (FPS) with the same channel width; zig-zag pooled steps (ZPS) with 2/3 channel width; central pooled steps (CPS) with 1/2 channel width; two-sided pooled steps with 1/3 channel width on each side (TPS) (see other details in Figure 2). Additionally, the pooled steps used were simple, with a notch (in short, “no” is a percentage of the notch) configuration. The notch heights of the pooled step sets were h/4 (no = 50%) and h/2 (no = 100%), with thickness t

_{p}= 1.5 cm (Figure 2 b-e). The modified cases are illustrated in Figure 2.

#### 3.2. FLOW-3D Model

^{®}is a CFD program allowing simulation of flow, turbulence, bed load, and suspended load under different boundary conditions [36]. FLOW-3D

^{®}uses the true volume of fluid (TruVOF) method based on the solution algorithm (SOLA) developed by Hirth and Nicolas [37]. The VOF method consists of three main steps: the definition of the volume of fluid function, the use of a method to solve the VOF transport equation, and the setting of boundary conditions at the free surface. Within the frame of VOF methods, the interface is determined from the volume fraction F. The F value varies between zero, when the grid cell contains no fluid, and unity, when the grid cell is fully occupied with fluid [30,36].

#### 3.3. Hydrodynamic Model

_{i}is the mean velocity; P is the hydrostatic pressure; A

_{i}is the fractional area open to flow in the subscript direction; V

_{F}is the volume fraction of fluid in each cell; G

_{i}is the gravitational acceleration in the subscript direction; f

_{i}is the represented Reynolds stress, which the turbulence model needs for closure; S

_{ij}is the strain rate tensor; τ

_{b,i}is the wall shear stress; ρ is the fluid density; μ

_{tot}is the total dynamic viscosity, which includes the effects of turbulence (μ

_{to}

_{t}= μ + μ

_{T}); μ is the dynamic viscosity; and μ

_{T}is the eddy viscosity. In the model setting, for computational cells full with fluid V

_{f}, A

_{j}is equal to one, leading the equations to reduce to single-fluid, incompressible, Reynolds-averaged Navier–Stokes (RANS) form.

#### 3.4. Turbulence Model

_{k}is the generation of turbulent kinetic energy caused by the average velocity gradient, G

_{b}is the generation of turbulent kinetic energy caused by buoyancy S

_{k}, and S

_{ε}represents source terms. Here, α

_{k}and α

_{ε}are inverse effective Prandtl numbers for k and ε, respectively. For the above equation:

_{µ}= 0.0845, C

_{1ε}= 1.42, C

_{2ε}= 1.68, C

_{3ε}= 1.0, σ

_{k}= 0.7194, σ

_{ε}= 0.7194, η

_{0}= 4.38, and β = 0.012.

#### 3.5. Numerical Domain

^{®}software was used to build the geometries of the models through an sterolithography (STL) file. Two mesh blocks were used: a containing mesh block was created for the entire spatial domain and a nested block was built, with refined cells for the area of interest where the stepped spillway is located. The areas where flow was not expected were cropped in the meshing process in order to increase the efficiency of the simulation without affecting the results, as shown in Figure 3. This technique (the nested mesh block technique) was adopted from previous studies by Choufu et al. [48] and Pourshahbaz et al. [49]. Best practice is to have fixed points aligning the mesh boxes and aspect ratios of no greater than 2.

_{ij}, of 1.3 recommended by Celik et al. [50].

_{32}= (u

_{s2}− u

_{s3})/u

_{s2}and is the approximate relative error between the medium and fine grids; u

_{s2}and u

_{s2}= medium and fine grid solutions for velocities, respectively; and p = local order of accuracy. For the three-grid solutions, p is obtained by solving the equation:

_{21}= u

_{s2}− u

_{s1}; e

_{32}= u

_{s3}− u

_{s2}; r

_{21}= G

_{2}/G

_{1}and is the grid refinement factor between a coarse and medium grid; and r

_{32}= G

_{3}/G

_{2}and is the grid refinement factor between a medium and fine grid. For the present three-grid comparisons, G

_{1}< G

_{2}< G

_{3}. Table 3 shows a summary of the GCI calculations for velocity distributions in step 7 in Q = 0.113 m

^{3}/s.

_{c}(here, V

_{c}is critical velocity V

_{c}= (g × y

_{c})

^{0.5}) over flat steps 8, 9, and 10, as shown in Figure 6. The maximum values for the relative error for steps 8, 9, and 10 were 6.25%, 5.25%, and 6.35% for Q = 0.113 m

^{3}/s and 6.42%, 5.33%, and 6.21% for Q = 0.090 m

^{3}/s, respectively. The maximum level of consistency between the numerical and experimental model was in the middle area of the depth up to the flow surface, indicating a high degree of accuracy of the numerical simulation.

_{crest}/L

_{crest}over a broad-crested weir for three different discharges. As shown in Figure 7, the simulated water surface profiles over a broad-crested weir length (L

_{crest}) were very similar to the experimental values. The mean relative error percentage of the flat stepped spillway was 4.85%. This figure indicates that the numerical results were fairly consistent with the experimental results. In addition, the numerical model was able to simulate the flow patterns with different discharges to an acceptable level.

## 4. Results and Discussions

_{c}) and The Froude number (F*)

#### 4.1. Flow Pattern

_{i}= h/4 (no = 50%) and h/2 (no = 100%), and the results were compared with the pooled step (no = 0%). It can be seen that the streamlines do not show significant changes in the axial plane between the pooled steps and the notch of the pooled steps. When we used the notch of the pooled steps, the aggregation of the streamlines, due to the passage of some vortexes of the notch, decreased.

#### 4.2. Velocity and Pressure Distribution

_{c}as a function of (y + d

_{p})/y

_{c}for Q = 0.105 m

^{3}/s. Herein, V

_{c}= ((Q × g)/W)

^{0.5}is the critical flow velocity. For all configurations, the measurements were performed at three transverse locations: Y/W = 0.25, 0.5 (centerline), and 0.75.

_{p})/y

_{c}> 0.6. However, the interfacial velocities on FPS steps are less than other pooled step configurations. In the bottom region, i.e., (y + w)/y

_{c}< 0.6, the velocity distributions for FPS with ZPS and CPS configurations were obviously different in the axial plane for Y/W = 0.25 and 0.75. However, at Y/W = 0.5, there was no major difference in the velocity distribution above different steps, except the free surface region. These cases have the opposite behavior between FPS and TPS at Y/W = 0.5. Figure 12 illustrates the velocity vectors in the axial plane Y/W = 0.5. In all configurations, the skimming flow transferring from the upstream to the downstream over the steps was partly trapped in the recirculating zones under the pseudo-bottom. However, it is evident that for pooled step configurations and in the recirculating zones, the vortex was significantly greater than for flat steps. Additionally, for each pooled step configuration, the velocity magnitude on steps was greater than on flat steps.

#### 4.3. Inception Point

_{i}/k

_{s}as a function of the Froude number F

^{*}in Figure 16. Herein, L

_{i}is the longitudinal distance from the first step edge to the inception point location and k

_{s}= (h + d

_{p}) × cosθ (in Flat step, d

_{p}= 0) is the roughness height. The Froude number F

^{*}is defined in terms of the step roughness:

_{p}= 0.03 cm) and a simple linear correlation proposed by Carosi and Chanson [53]:

#### 4.4. Residual Head and Energy Dissipation

_{max}) expresses the percentage of total energy loss along the stepped spillway relative to the upstream total head, H

_{max}. The total head can be evaluated as:

_{dam}is the height of the dam and y

_{c}is the critical flow depth. The total head loss ΔH can be estimated as ΔH = H

_{max}− H

_{res}, in which the residual head, H

_{res}, is:

_{p}, and g are the spillway slope, the water flow depth, the pool height (d

_{p}= 0 for a flat step), and the gravitational acceleration, respectively. Note that for notched pooled step configurations, the notch heights were 0.025 and 0.05 pool weir heights were 0.75w

_{p}and 0.5w

_{p}, respectively. The dimensionless residual head H

_{res}/y

_{c}for all discharges is illustrated in Figure 17 as a function of the dimensionless discharge (y

_{c}/h).

#### 4.5. Turbulent Kinetic Energy (TKE)

_{1}, u

_{2}, u

_{3}, …, u

_{n}), the value of the root mean square velocity, u

_{rms}, was obtained as:

^{3}/s are shown in Figure 19. It could be seen in this figure that the maximum TKE was created on flat steps rather than the different pooled step configurations. In contrast to the flat configurations, the maximum turbulent kinetic energy occurred in the middle of the horizontal step surface, while the maximum turbulent kinetic energy occurred near the step pool. The value of the TKE also increased along the spillway, which was similar to the results of previous studies, such as that by Bombardelli et al. [56]. The region of turbulence on the FPS was greater in intensity than for other types of pooled steps. With regard to Figure 20, it can be seen that with notched pools, the maximum TKE values increased near the step pool.

- The free water surface was horizontal on flat and FSP steps, however on ZPS, CPS, and TPS configurations, certain instabilities were caused by the staggered nature of the configurations. With the notch of the pooled steps, the aggregation of the streamlines decreased due to the passage of some vortexes of the notch;
- The interfacial velocities of flat stepped spillways were smaller than for pooled step configurations. The velocities of the pooled stepped were almost identical distributions for (y + d
_{p})/y_{c}> 0.6. However, the interfacial velocities on FPS steps were lower than for other configurations; - The maximum velocity magnitude was observed in the overlaying flow over the pooled steps, while the minimum velocity value occurred in the bottom, but no changes were evident with different notched pooled steps;
- The pressure value at the beginning of the steps for the pooled configurations was larger than for the flat configuration and the maximum pressure was observed always near the step pool at the end of the step. Along the vertical step surface, negative pressures were observed for the flat stepped spillway. For all the proposed configurations (pooled and flat), pressure values along the horizontal step surfaces were never negative;
- The pool configuration (simple or notched) did not have a significant influence on the location of air entrainment;
- The lowest residual head was achieved with the flat step configuration. On the flat stepped spillway, the dimensionless residual head was the largest (~3.16), while for the same flow conditions the average dimensionless residual head on the other configurations of the pools was ~3.96. With the notched pooled steps, the mean residual head decreased to 3.49;
- The flat step configuration showed the best energy dissipation performance as compared with other configurations. Additionally, the FSP had a higher rate of energy dissipation than ZPS, CPS, or TPS. With the notched pooled steps, the performance of this structure improved;
- The maximum TKE was created on the flat steps as compared to different pooled step configurations. The region of turbulence on the FPS was greater in intensity than for other types of pooled steps. With the notched pool configuration, the maximum TKE values increased near the step pool.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Sketch of stepped spillway configurations with flat, pooled, and notch pooled steps: (

**A**) flat steps; (

**B**) FPS; (

**C**) ZPS; (

**D**) CPS; (

**E**) TPS.

**Figure 6.**Dimensionless interfacial velocity distributions of the flat stepped spillways; numerical simulation vs. experimental data (

**A**): Q = 0.113 m

^{3}/s; (

**B**): Q = 0.090 m

^{3}/s.

**Figure 11.**Velocity profile distributions on steps for different pool configurations, in terms of the dimensionless velocity V/Vc as a function of (y + dp)/yc for two steps (i.e., step 7 and 8). Vc = critical flow velocity.

**Figure 17.**Residual energy values at the downstream end of the spillway in different pooled step configurations.

**Figure 21.**The mean TKE (J/kg) values on pooled steps of different configurations (height and notched): (

**A**) Q = 0.090 m

^{3}/s; (

**B**) Q = 0.105 m

^{3}/s.

**Table 1.**Hydraulic characteristics and the type of flow regime on the stepped spillway in the present study.

Q (m^{3}/s) | y_{c} = (q^{2}/g)^{1/3} (m) | Step Height (m) | Step Length (m) | h/l | y_{c}/h | Fr | Re | Flow Regime |
---|---|---|---|---|---|---|---|---|

0.045 | 0.092 | 0.1 | 0.2 | 0.5 | 0.92 | 0.88 | 3.5 × 10^{5} | ✓Skimming flow |

0.063 | 0.115 | 0.1 | 0.2 | 0.5 | 1.15 | 1.23 | 4.9 × 10^{5} | ✓Skimming flow |

0.075 | 0.129 | 0.1 | 0.2 | 0.5 | 1.29 | 1.46 | 5.6 × 10^{5} | ✓Skimming flow |

0.09 | 0.145 | 0.1 | 0.2 | 0.5 | 1.45 | 1.74 | 6.9 × 10^{5} | ✓Skimming flow |

0.105 | 0.161 | 0.1 | 0.2 | 0.5 | 1.61 | 2.04 | 8 × 10^{5} | ✓Skimming flow |

0.113 | 0.17 | 0.1 | 0.2 | 0.5 | 1.7 | 2.21 | 8.7 × 10^{5} | ✓Skimming flow |

Mesh | Nested Block Cell Size | Containing Block Cell Size |
---|---|---|

G1 | 0.65 cm | 1.45 cm |

G2 | 0.85 cm | 1.80 cm |

G3 | 1.10 cm | 2.45 cm |

Mesh Size | R = G_{2}/G_{1} | Grid Convergence Index (GCI) |
---|---|---|

0.65 | - | - |

0.85 | 1.30 | 5.04% |

1.10 | 1.30 | 14.86% |

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**MDPI and ACS Style**

Ghaderi, A.; Abbasi, S.; Di Francesco, S.
Numerical Study on the Hydraulic Properties of Flow over Different Pooled Stepped Spillways. *Water* **2021**, *13*, 710.
https://doi.org/10.3390/w13050710

**AMA Style**

Ghaderi A, Abbasi S, Di Francesco S.
Numerical Study on the Hydraulic Properties of Flow over Different Pooled Stepped Spillways. *Water*. 2021; 13(5):710.
https://doi.org/10.3390/w13050710

**Chicago/Turabian Style**

Ghaderi, Amir, Saeed Abbasi, and Silvia Di Francesco.
2021. "Numerical Study on the Hydraulic Properties of Flow over Different Pooled Stepped Spillways" *Water* 13, no. 5: 710.
https://doi.org/10.3390/w13050710