# Experimental and Numerical Study of the Effects of Geometric Appendance Elements on Energy Dissipation over Stepped Spillway

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Flow Regimes on Stepped Spillway

_{c}/h) (y

_{c}is the critical flow depth and h is the step height), the skimming flow regime is observed for the higher discharges according to Chinnarasri and Wongwises [30], and with a large value of (y

_{c}/h). The factors that affect the generation of various types of flow on stepped spillways include the geometry of the steps (length and height of the steps, l and h) and the flow rate Q [1,2]. However, some researchers have proposed other criteria. The studies of many researchers such as Rajaratnam [3], Peyaras et al. [4], and Chanson [5] have shown that the type of flow regime depends on the normalized critical depth, y

_{c}/h, and a dimensionless variable of steps, h/l. According to Equations (1)–(3), criteria are presented for which a skimming flow regime occurs:

#### 2.2. Energy Dissipation Relations

_{c}) lies at the beginning and on the first step (y

_{c}= (Q/W)

^{2}/g

^{1/3}). In this case, Q (m

^{3}/s) is the discharge, W (m) the channel width, and g (m/s

^{2}) is the acceleration of gravity. On the other hand, the following equations can be used to obtain the total water energy upstream, downstream and energy dissipation of the spillway:

_{0}(m) is the total water energy upstream of the spillway; E

_{1}(m) is the total water energy downstream of the spillway; ΔE (m) is total energy dissipation; ∆Z (m) is spillway height; yc (m) is critical depth and g (m/s

^{2}) is the acceleration of gravity.

#### 2.3. Dimensional Analysis

- (i)
- Fluid properties: Water density (ρ); Dynamic viscosity (μ).
- (ii)
- Flow characteristics: Upstream water depth (y
_{0}); Water depth from upstream and downstream of hydraulic jump (y_{1}, y_{2}); the critical depth (yc); Water velocity just downstream the spillway (V_{1}); Water velocity downstream the hydraulic jump (V_{2}); length of hydraulic jump (L), and the acceleration of gravity (g). - (iii)
- Geometric properties: Spillway height (∆Z); Step length (l); Step height (h); slope of the chute (θ), The height of the elements (a) and Number of steps (N).

_{1}and Fr

_{2}are Froude number upstream and downstream of the hydraulic jump and Re represents the Reynolds number upstream of the hydraulic jump, i.e., Re = (ρ × V

_{1}× y

_{1})/μ [31].

#### 2.4. Experimental Work

_{BC}= 1 m, with the following equation by Felder and Chanson [12]:

_{1}the upstream total head and g the gravity acceleration constant.

#### 2.5. Numerical Modelling Analysis

^{®}[38] was used for the numerical simulations. Due to the limited supply of flow by pumps in experimental work, discharge from 60 to 75 l/s was generalized by FLOW3D

^{®}. The FLOW-3D uses the finite volume method in a Cartesian, staggered grid to solve the RANS equations (Reynolds Average Navier-Stokes) that describe continuity and momentum and are expressed as:

_{i}and u

_{i}are average and fluctuating velocities in the x

_{i}direction, respectively. The instantaneous velocity is defined as u

_{t}= U

_{t}+ u ’

_{t}for the three perpendicular directions (I = 1, 2, 3) in which X

_{i}= (x, y, z), U

_{t}= (U, V, W), u

_{t}= (u ’, v ’, w ’) and ρ, µ, P, g

_{i}are density, dynamic viscosity, kinematic pressure, and gravitational acceleration, respectively. FLOW-3D

^{®}uses an advanced algorithm for tracking free-surface flows, called Volume of Fluid (VOF), developed by Hirt and Nichols [39]. The VOF transport equation is expressed by the following Equation (12):

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

_{b}denotes the generation of turbulent kinetic energy caused by buoyancy, while S

_{k}and S

_{ε}are source terms. α

_{k}and α

_{ε}are inverse effective Prandtl numbers for k and ε, respectively. μ

_{eff}is the effective viscosity μ

_{eff}= μ+μ

_{t}, being that μ

_{t}is the eddy viscosity. The following equations provide detail on how the effective viscosity is determined.

_{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.

_{coarse}/G

_{fine}), which was a value of 1.3, as recommended by Celik et al. [48]. To this purpose, a refinement ratio of 1.37 was considered for reducing the grid sizes. Table 2 shows some characteristics of the computational grids.

_{1}, f

_{2}, f

_{3}are the parameters obtained from CFD simulations (f

_{1}corresponds to the fine mesh) and r is the refinement rate. The fine-grid convergence index is defined as [46]:

_{2}-f

_{1})/f

_{1}is relative error, f

_{2}and f

_{3}are medium- and fine-grid solutions, respectively. Dimensionless indices GCI

_{12}and GCI

_{23}can be calculated:

_{12}) are small as compared to the coarser grid (GCI

_{23}), it can be inferred that the grid-independent solution is almost obtained and no further mesh modification is required. Computed values of GCI

_{23}/r

^{p}GCI

_{12}close to 1 indicate that the numerical solutions are within the asymptotic range of convergence. As a result, the mesh consisting of a containing block with a cell size of 1.3 cm and a nested block of 0.95 cm was selected (see Figure 8).

## 3. Results and Discussions

#### 3.1. Validity of the FLOW-3D^{®} Model Results

#### 3.2. Flow Pattern

#### 3.3. Energy Dissipation and Residual Head

#### 3.3.1. Flat Step Configuration

_{c}/h and ∆Z/y

_{c}, respectively. The present data for the flat steps are compared to a previous study with the same chute slope (θ = 26.60°), but where the step dimensions are not generally the same conducted by Felder and Chanson [14], in which ∆Z is the height of the dam, y

_{c}is the critical flow depth, and h is step height. The residual head, H

_{res}, is:

_{1}represents the mean flow velocity at the downstream end of the spillway. Also, θ, y, and g are the slope of the chute, the water flow depth, and the gravitational acceleration, respectively. From the results in Figure 11, the energy dissipation rate increases with increasing ∆Z/y

_{c}, which means that the energy dissipation rate decreases with increasing discharge, which is consistent with results from earlier studies of stepped spillways [14]. In addition, from the comparison of the results, as expected, a higher rate of energy dissipation was observed for the flat stepped spillway at low flow rates. According to Bung [10], energy dissipation is not related to step height. At low flow rates, the residual head is decreased due to the increasing discharge for all cases and it remained almost constant for the largest flow rates. However, for the largest flow rates, the discharge was not fully developed at the downstream end of the spillway and the residual energy might be overestimated [16,55]. The present findings are in good agreement with previous studies on stepped spillways with slopes of θ = 26.60°.

#### 3.3.2. Appendance Elements on Steps

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

_{1}, u

_{2}, u

_{3}, ..., u

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

_{rms}, is obtained as:

^{3}/s is provided in Figure 17. This figure highlights that adding the appendance elements increases the deflected jets on the steps and maximum TKE is created on the appendance elements step compared to the flat step. Along the spillway, the value of the TKE also increases, which was in agreement with previous research by Bombardelli et al., [57]. The region of turbulence on hybrid appendance elements was a greater intensity than other types of arrangement of appendance elements step.

#### 3.5. Flow Resistance

_{e}). The Darcy–Weisbach friction factor is determined by measuring the air–water flow properties and is calculated from Equation (23):

_{1}is the velocity at the downstream end of the spillway and g is the gravitational acceleration. Figure 19 shows the friction factor results of all models as a function of the dimensionless (k

_{s}/y

_{c}). k

_{s}is step roughness height and it is defined by k

_{s}= h$\times $cos(θ). It could be seen that the minimum friction factor (f

_{e}) was obtained for the flat stepped spillway compared to when appendance elements were used on a step of the stepped spillway. It could also be seen that as given Reynolds numbers increase the k

_{s}/y

_{c}, the value of friction factor increases for all models. The hybrid appendance elements step is a greater friction factor than other types of arrangement of appendance elements step.

#### 3.6. Inception Points with the Appendance Elements on Step

_{i}/k

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

^{*}in Figure 21. Herein, L

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

_{s}is the roughness height. The Froude number F

^{*}is defined:

#### 3.7. Scale Effects

^{3})

^{0.5}, the Weber number We = ū/(σ/(ρL

_{S}))

^{0.5}, and the Reynolds number Re

**=**q/υ are relevant. Here, y is the water depth, ū = q/y

_{i}the average velocity with y

_{i}as the mixture flow depth, σ the surface tension between air and water, ρ the water density, L

_{s}= (h/cosθ) the distance between two step edges, and υ the kinematic water viscosity. Boes and Hager [59] investigated scale effects. For a standard prototype step height of 0.60 m, minimum Reynolds and Weber numbers and a maximum scaling factor to minimize scale effects were suggested. For a standard step height of 0.60 m, 1 m downstream of the spillway crest, the ranges of Fr, We, Re, and yc/h for the herein-studied physical models using λ = 10 are 2.70–5.81, 48–93, 1.80–3.50 × 10

^{5}, and 0.79–1.23, respectively, thus partially satisfying the above conditions.

## 4. Conclusions

^{®}model. To simulate the free surface, the Volume of Fluid (VOF) method was adopted and the turbulence RNG k-ε model was used. To validate the present model, comparisons between numerical and experimental results were performed for the flat stepped spillway. The following conclusions are summarized from the present study:

- With appendance elements on a step, the vortex was divided into two areas in the recirculating zones on the step and appendance elements and presented some instabilities caused by the staggered configurations. However, the flow surface over the flat step was approximately parallel to the pseudo-bottom.
- Along the horizontal step surface, the pressure value on the step in flat steps is larger compared to other configurations with appendance elements. The maximum pressure moves from a point in the step center towards the end of the step.
- Along the vertical step surface, negative pressures can be observed in the flat stepped spillway. With appendance elements on the step, negative pressures are divided into smaller, and are, therefore, less subject to cavitation.
- The rate of energy dissipation with appendance elements on step is greater. Adding appendance elements increases the streamlines interference and deflected jets on the steps. With an increase in flow discharge at all step configurations, the energy dissipation becomes less effective (lower energy loss).
- On the flat stepped spillway, the dimensionless residual head is the largest (~3.47) at a given flow condition; however, the average dimensionless residual head on the hybrid appendance elements step is ~3.02.
- A decrease in the height of the appendance elements leads to increase in the rate of energy dissipation and decreases the residual head to 3.20% and 7.40%, respectively.
- Adding appendance elements increases the deflected jets on the steps and maximum TKE is created on the appendance elements step compared to the flat step. The area of turbulence on hybrid appendance elements was of greater intensity than other types of arrangement of appendance elements step.
- As the height of the appendance elements decreases, the value of TKE increases. With an increase in flow discharge, the value of TKE increases for all models.
- It is observed that the friction factor (fe) in the flat stepped spillway is lower compared to when appendance elements are used on a step of the stepped spillway. The maximum friction factor demonstrates an increasing trend with an increase in the k
_{s}/y_{c}values. - By decreasing the heights of appendance elements, the friction factor increases, and the performance of the appendance elements improves.
- The air entrainment inception location can be changed through the appendance of elements on the step, leading to a positioning further upstream with respect to a flat step stepped spillway. However, the heights of appendance elements did not have an impact on the location of the inception point of air entrainment.
- To exclude scale effects on stepped spillway models, the ranges of hydraulic parameters (Froude, Weber, and Reynolds numbers) were established on the basis of literature review, considering a scale ratio less than 1:15.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Flow regimes on stepped spillways: (

**a**) Nappe flow, (

**b**) Transition flow, (

**c**) Skimming flow.

**Figure 5.**Stepped spillway configurations with flat and appendance elements (AE) on steps (θ = 26.60°).

**Figure 9.**Comparison between the experimental and numerical results of the free surface (Q = 0.052 m

^{3}/s).

**Figure 13.**The rate of energy dissipation on flat with appendance elements (AE) on steps versus (∆Z/y

_{c}).

**Figure 14.**Flow over the stepped spillway and its interference on transverse sides of the appendance elements on step (

**A)**: Flat step (

**B)**: Hybrid appendance elements step.

**Figure 16.**The effect of the appendance elements height on the rate of energy dissipation and the residual head.

**Figure 18.**The mean TKE (J/kg) value on appendance elements step with different heights and discharge.

**Figure 19.**Darcy friction factors of stepped spillways with appendance elements on step versus (k

_{s}/y

_{c}).

**Figure 21.**Inception points of free-surface aeration for flat step and appendance elements on step of stepped spillway.

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

✓Skimming flow | 0.79 | 0.5 | 0.12 | 0.06 | 0.048 | 0.040 |

✓Skimming flow | 0.87 | 0.5 | 0.12 | 0.06 | 0.052 | 0.045 |

✓Skimming flow | 0.93 | 0.5 | 0.12 | 0.06 | 0.056 | 0.050 |

✓Skimming flow | 1 | 0.5 | 0.12 | 0.06 | 0.06 | 0.055 |

✓Skimming flow | 1.05 | 0.5 | 0.12 | 0.06 | 0.063 | 0.06 |

✓Skimming flow | 1.11 | 0.5 | 0.12 | 0.06 | 0.067 | 0.065 |

✓Skimming flow | 1.17 | 0.5 | 0.12 | 0.06 | 0.07 | 0.07 |

✓Skimming flow | 1.23 | 0.5 | 0.12 | 0.06 | 0.074 | 0.075 |

Type | Nested Block Cell Size | Containing Block Cell Size | Number of Cells | Mesh |
---|---|---|---|---|

1 | 1.65cm | 2.2cm | 1,425,876 | Coarse |

2 | 1.25cm | 1.65cm | 2,518,228 | Medium |

3 | 0.95cm | 1.30cm | 3,972,156 | Fine |

Quantity | f_{3} | f_{2} | f_{1} | p | GCI_{12} | GCI_{23} | Asymptotic Range |
---|---|---|---|---|---|---|---|

Discharge flow rate (m^{3}/s) | 0.0505 | 0.0514 | 0.0518 | 2.64 | 0.007 | 0.017 | 1.007 |

Water depth (m) | 0.68 | 0.698 | 0.705 | 3.08 | 0.008 | 0.02 | 1.01 |

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Ghaderi, A.; Abbasi, S.
Experimental and Numerical Study of the Effects of Geometric Appendance Elements on Energy Dissipation over Stepped Spillway. *Water* **2021**, *13*, 957.
https://doi.org/10.3390/w13070957

**AMA Style**

Ghaderi A, Abbasi S.
Experimental and Numerical Study of the Effects of Geometric Appendance Elements on Energy Dissipation over Stepped Spillway. *Water*. 2021; 13(7):957.
https://doi.org/10.3390/w13070957

**Chicago/Turabian Style**

Ghaderi, Amir, and Saeed Abbasi.
2021. "Experimental and Numerical Study of the Effects of Geometric Appendance Elements on Energy Dissipation over Stepped Spillway" *Water* 13, no. 7: 957.
https://doi.org/10.3390/w13070957