## ABSTRACT

Hydraulic jump is used to dissipate excessive flow energy in stilling basins to control erosion on the downstream side. The literature review revealed that the convergence of the side walls in a USBR type II stilling basin has enhanced energy dissipation by stabilizing the hydraulic jump. Taking this into account, a Computational Fluid Dynamics (CFD) model was created using CFD code to analyze the hydraulic efficiency of a USBR type III stilling basin with varying degrees of side wall convergence. Additionally, alterations were made to the standard Impact Blocks geometry to evaluate their effect on energy dissipation. The side walls of stilling basin were converged from 0.5° to 2.5° (with an increment of 0.5°). Study results indicated an increase in hydraulic jump efficiency from 1.6 to 14.5% due to increase in wall convergence. Modified Friction Blocks also enhanced the energy dissipation up to 2%. Post-jump Froude number values were found in acceptable range of 0.6 to 0.78. The optimal hydraulic performance of stilling basin was noted when wall convergence angle of 2.5° was used along with modified Friction Blocks. Hydraulic performance of modified stilling basin may be investigated during gated operation of the model.

## HIGHLIGHTS

Walls convergence in stilling basin.

Navier–Stokes equations.

Computational Fluid Dynamics (CFD) code.

## INTRODUCTION

Physical and numerical model studies play integral roles in hydraulic investigations, offering insights into the behavior of fluid systems and hydraulic structures. While physical models replicate real-world scenarios on a smaller scale, numerical models simulate hydraulic processes using mathematical equations. Both approaches have distinct advantages and are often used in tandem to complement each other. However, currently, numerical models are being used preferably because they are cost effective, allow researchers to simulate a wide range of hydraulic scenarios with ease, adjusting parameters and boundary conditions as needed, and simulate hydraulic processes with high precision and accuracy, particularly when calibrated and validated using field data or results from physical models. Recent research is developing a numerical model using physical model data for its calibration and validation, then carrying out scenario modeling on numerical models. Recently, Babaali *et al.* 2015; Ljubičić *et al.* 2018; Macián-Pérez *et al.* 2020a used a similar technique to assess the hydraulic performance of stilling. Murzyn & Chanson (2009); Lee & Wahab (2019); and Zaffar *et al.* (2023) employed a k-Ɛ model to study the turbulent flow characteristics of hydraulic jump numerically. Viti *et al.* (2018); Retsinis & Papanicolaou (2020); and Jayant & Jhamnani (2023) investigated the hydraulic jump through CFD modelling. Gomes *et al.* (2011) & Miao *et al.* (2023) evaluated numerical solutions for both turbulent and laminar flows and validated models by comparing experimental results and numerical solutions. The studies demonstrated the presence of downstream vortices near bottom that were hardly represented by physical models. In another study, Jian-gang *et al.* (2010) used two different turbulent models, i.e., VOF and mixture models and simulated the flows in stilling basins to assess their hydraulic efficiency. Literature survey indicated number of studies (Wu & Sugavanam 1978; Amorim *et al.* 2015; Tang & Lin 2015; Valero *et al.* 2016; Wang *et al.* 2016; Abbas *et al.* 2018; Karim & Mohammed 2020; Hunt & Kadavy 2021; Jiang *et al.* 2022) where numerical models were used to evaluate the hydraulic performance of stilling basins.

Literature has reported various research studies around the globe to optimize the energy dissipation phenomenon in stilling basins. In a few studies side walls of stilling basins were converged to enhance their hydraulic performance (Babaali *et al.* 2015; Ljubičić *et al.* 2018; Macián-Pérez *et al.* 2020a). The convergence of walls stabilized the jump within the basin which increased the energy dissipation in the stilling basin (Pirestani *et al.* 2011; Khadka *et al.* 2020; Macián-Pérez *et al.* 2020b). Soori *et al.* (2017) attempted to enhance energy dissipation by installing the obstacles (a sort of auxiliary device) in USBR stilling basin Type-II of Nazloochay spillway model in Iran. Authors used numerical models for the optimal design of obstacles. In an experimental study, Frizell *et al.* (2016) examined the efficiency of stilling basins with stepped spillways by testing USBR Type-III stilling basins on 3 different slopes. Akib *et al.* (2015) conducted experimental research to investigate how different shapes of artificial uneven/grooved beds influence the characteristics of hydraulic jumps. The study revealed that a circular-shaped tire waste grooved bed had a substantial impact on hydraulic jump by reducing its length and depth. Eshkou *et al.* (2018) investigated the impact of sloping baffle blocks on the hydraulic jump in a gradually diverging stilling basin with a negative bed slope. Al-Mansori *et al.* (2020) evaluated hydraulic performance of stilling basins by replacing the standard baffle blocks with the new blocks on the physical model. In 2021, Nasralla (2022) performed an experimental investigation on stepped spillways using baffled stilling basins to observe energy dissipation. The results showed that installation of baffles downstream of the stepped spillway increases energy dissipation. Daneshfaraz *et al.* (2020) investigated energy dissipation in supercritical fluid that showed a positive correlation between numerical and experimental study. Daneshfaraz *et al.* (2021a, 2021b, 2021c) studied the effect of the rough elements on energy dissipation. Babaali *et al.* (2019) investigated the hydraulic parameters of flow, i.e., Froude number, pressure, turbulent dissipation, and air entrainment in USBR Type II stilling basin utilizing FLOW-3D. An adverse slope (3: l) was installed at the end of the USBR Type II stilling basin. Adverse slopes were provided to increase the energy loss, but this did nothing to stabilize the hydraulic jump, especially for higher discharges. On the other hand, wall convergence yielded a favorable impact on energy dissipation and 5 degree of wall convergence showed the best performance. It is evident from the literature review that the hydraulic performance of stilling basin USBR Type-III by modifying the baffle block geometry along with wall convergence has not been studied yet. Hence, there is a need for systematic study to see the impact of wall convergence and modified baffle blocks on hydraulic performance of stilling basin USBR Type-III. For this purpose, the lower stilling basin of Mohmand Dam spillway, Pakistan was selected as a case study.

## STUDY AREA

## MATERIAL AND METHODS

A comprehensive methodology was worked out and formulated after a detailed technical literature review and the gray area was assessed for current study. A numerical model of Mohmand Dam Spillway was developed in FLOW-3D by employing the following sequential steps.

### Numerical simulations in CFD Code FLOW-3D

*ρ*i, the isotropic stress resulting from mean pressure field

*δ*ij, the viscous stresses , and apparent stress as a result of the unstable velocity field, usually represented as the Reynolds stress.

### Geometry and mesh

The geometry of Mohmand Dam Spillway consists of an ogee type control structure, upper chute with 1:1.6 slope, upper stilling basin with impact blocks, and a weir of 10 m height at the end of the upper chute. The weir is followed by lower chute and lower stilling basin. The lower chute has a longitudinal slope of 1:1.6. Auxiliary devices, namely chute blocks, impact blocks and end sill, are installed in the lower stilling basin. Two bays of the spillway were selected for current study. Length and width of selected geometry was 592 and 30 m respectively. To enchase the computational efficiency of the model and given the specific area of focus, the computational domain was delimited from the weir of the upper stilling basin to the end sill of the lower stilling basin. Tail water levels of the upper stilling basin were used as a head water for computational domain after ensuring that the model is accurately representing the flow conditions and hydraulic behavior within the lower stilling basin. To ensure the accuracy and reliability of the model, results were validated against experimental data.

_{1}, f

_{2}and f

_{3}represent the solutions generated from fine, medium, and coarse mesh as an expression of discharge. ɸ

_{21}is the difference between volume flow rates of finer and medium mesh whereas ɸ

_{32}is the difference between volume flow rates of medium and coarser mesh, respectively. , are the approximate relative errors with finer and coarser meshes, respectively. , are Grid Convergence Index for fine and coarse meshes respectively, where r

^{p}is the refinement factor with order of accuracy ‘p’ and it is defined as the ratio of the sizes of two consecutive grids such as r = h

_{course}/h

_{fine}. The order of accuracy p is calculated by using Equation (3).

Sr. No. . | Reservoir Elevation . | Volume Flow Rate (m^{3}/sec). | ɸ_{21}
. | ɸ_{32}
. | e_{21}
. | e_{32}
. | GCI_{21}
. | GCI_{32}
. | ||
---|---|---|---|---|---|---|---|---|---|---|

m asl . | f_{1} (0.3 m)
. | f_{2} (0.6 m)
. | f_{3} (1.2 m)
. | (m^{3}/s)
. | (m^{3}/s)
. | – . | – . | % . | % . | |

1 | 551 | 2,600 | 2,605 | 2,613 | 5 | 8 | 1.9 × 10^{−3} | 3.07 × 10^{−3} | 4 × 10^{−3} | 6.4 × 10^{−3} |

2 | 555 | 4,871.4 | 4,873 | 4,882 | 1.60 | 9 | 3.3 × 10^{−4} | 1.8 × 10^{−3} | 8.8 × 10^{−5} | 4.8 × 10^{−4} |

Sr. No. . | Reservoir Elevation . | Volume Flow Rate (m^{3}/sec). | ɸ_{21}
. | ɸ_{32}
. | e_{21}
. | e_{32}
. | GCI_{21}
. | GCI_{32}
. | ||
---|---|---|---|---|---|---|---|---|---|---|

m asl . | f_{1} (0.3 m)
. | f_{2} (0.6 m)
. | f_{3} (1.2 m)
. | (m^{3}/s)
. | (m^{3}/s)
. | – . | – . | % . | % . | |

1 | 551 | 2,600 | 2,605 | 2,613 | 5 | 8 | 1.9 × 10^{−3} | 3.07 × 10^{−3} | 4 × 10^{−3} | 6.4 × 10^{−3} |

2 | 555 | 4,871.4 | 4,873 | 4,882 | 1.60 | 9 | 3.3 × 10^{−4} | 1.8 × 10^{−3} | 8.8 × 10^{−5} | 4.8 × 10^{−4} |

The results, in Table 1, indicated that numerical solutions achieved with 0.3 m cell size have a smaller grid convergence index value (GCI_{21}) as compared to coarse-grid convergence index (GCI_{32}). Moreover, the approximate relative error also decreased with a decrease in cell size of grid. However, mesh was further refined (from 0.3 to 0.25 m) which showed a negligible difference in flow rate as indicated in Table 2. As a result, a grid size of 0.3 m was selected for scenario modelling.

Mesh Configuration . | Mesh Size (m) (cumecs) . | Computed Discharge . |
---|---|---|

Coarse | 1.2 | 4,882.14 |

Medium | 0.6 | 4,873.22 |

Optimal | 0.3 | 4,871.40 |

Fine | 0.25 | 4,871.36 |

Mesh Configuration . | Mesh Size (m) (cumecs) . | Computed Discharge . |
---|---|---|

Coarse | 1.2 | 4,882.14 |

Medium | 0.6 | 4,873.22 |

Optimal | 0.3 | 4,871.40 |

Fine | 0.25 | 4,871.36 |

### Turbulence modelling

Simulations were carried out using the Renormalization Group Turbulence (RNG) k-Ɛ model. The RNG k-Ɛ model is a refined form of standard k-Ɛ model. It is better than the standard k-Ɛ model because it is based on statistical methods and equation constants are derived explicitly which are found empirically in the standard k-Ɛ model. Moreover, the RNG model has wider applicability. Hence, the RNG model, being robust and most accurate, was used in simulations.

### Free surface modelling

^{th}phase, this equation has the following form:where

*α*

_{q}is the volume fraction of q

^{th}phase. The following three conditions are possible for each cell of the computational domain:

*α*_{q}= 0: the cell is empty (of the q^{th}phase)*α*_{q}= 1: the cell is full (of the q^{th}phase)0<

*α*_{q}< 1: the cell contains the interface between q^{th}phase and one or more other phases.

It can be assumed that free surface is on the volume fraction of 0.5.

### Initial and boundary conditions

_{min}, X

_{max}, Y

_{min}, Y

_{max}, Z

_{min}, and Z

_{max}. The boundary conditions were employed for these extents of the computational domain as shown in Figure 2.

_{min}and X

_{max}represent the extent of computational domain in the upstream and downstream direction, respectively. Y

_{min}and Y

_{max}indicate the right and left sides of the computational domain. Whereas, Z

_{min}represents the bottom of the computational domain and Z

_{max}represents the top of the domain. ‘Wall Boundary Condition’ was used for Zmin. ‘Specified Pressure’ was opted for X

_{min}, X

_{max}, and Z

_{max}. ‘Symmetry’ was applied at Y

_{min}and Y

_{max}. instead of ‘wall’ boundary condition. ‘Symmetry’ boundary condition was selected considering the balance between computational efficiency and accuracy in representing the physical phenomena of interest. In FLOW-3D, ‘wall’ is a boundary condition that describes a surface in the computational domain where the fluid velocity normal to the surface is zero. This is important for accurately simulating fluid behavior near solid surfaces. ‘Symmetry’ boundary condition is used to specify a plane of symmetry in the computational domain. This condition is used when there is a repeating pattern in the domain, and only one part of the domain needs to be simulated. The symmetry boundary condition specifies that the flow variables (pressure and velocity) on one side of the plane are equal to those on the other side of the plane. This reduces the size of the computational domain and makes the simulation more efficient. ‘Specific pressure’ boundary condition applies the pressure value at a specific location in the computational domain. This boundary condition is used when the pressure value at a certain point in the domain is known, such as at an inlet or outlet boundary. Fluid region is added in the domain by using the initial tab in Flow -3D. For fluid initialization, fluid elevation is used. In this study, fluid region, namely ‘Fluid Region I’ was added upstream of the structure. Limits of fluid region were then specified. Limits of fluid region I at Reservoir Level 555 m were 585 m and Z 492.6 m as shown in Figure 3.

### Model validation

The numerical model was validated to verify the accuracy of numerically obtained results. For this purpose, discharge, flow depth and depth average velocity values were observed on the physical model which were then compared with the numerical model results. Physical modelling of Mohmand Dam spillway was carried out by the Water and Power Development Authority (WAPDA) in October 2017 at a scale of 1:60. Flow parameters (flow depth & velocity) computed by numerical model were noted at two locations, i.e., at start of basin and end of stilling basin. Table 3 shows the comparison of observed flow parameters, i.e. discharge, flow depth and flow velocity with that of computed values from the numerical model at reservoir levels of 555 m asl and 563 m asl. Comparison shows a difference of less than 3% which indicated that the numerical model is a good representative of prototype.

Sr. No. . | Flow Characteristic . | Scale Model Results . | CFD Model Results . |
---|---|---|---|

(a). (Operating Condition: Reservoir level = 555 m asl) | |||

1. | Discharge (cumecs) | 3,842.9 | 3,900 |

2. | Flow Depth (m) | 26 | 25.7 |

3. | Depth average velocity (m/s) | 11 | 11.3 |

(b). (Operating Condition: Reservoir level = 563 m asl) | |||

1. | Discharge (cumecs) | 7,314.3 | 7,450 |

2. | Flow Depth (m) | 32 | 32.4 |

3. | Depth average velocity (m/s) | 13 | 12.7 |

Sr. No. . | Flow Characteristic . | Scale Model Results . | CFD Model Results . |
---|---|---|---|

(a). (Operating Condition: Reservoir level = 555 m asl) | |||

1. | Discharge (cumecs) | 3,842.9 | 3,900 |

2. | Flow Depth (m) | 26 | 25.7 |

3. | Depth average velocity (m/s) | 11 | 11.3 |

(b). (Operating Condition: Reservoir level = 563 m asl) | |||

1. | Discharge (cumecs) | 7,314.3 | 7,450 |

2. | Flow Depth (m) | 32 | 32.4 |

3. | Depth average velocity (m/s) | 13 | 12.7 |

### 3.7. Model operation

After validation of the numerical model, scenario modelling was performed to evaluate the hydraulic performance of the stilling basin. Flow parameters (flow depth, Froude number, and depth average velocity) were computed from the numerical model for each geometric scenario of wall convergence (0°, 0.5°, 1°, 1.5°, 2°, 2.5°) and impact blocks (standard and modified) by varying reservoir levels between 555 m asl (above sea level) and 563 m asl (555, 558, 561.3, 563) under free flow condition.

## RESULTS AND DISCUSSION

### Flow velocity profiles

The high flow velocity causes erosion downstream of the spillway. Assessment of flow velocity is essential to find out the potential for erosion.

Location Points with Coordinates (x,y,z) . | Flow Velocity at 0° Wall convergence . | Flow Velocity at 0.5° Wall convergence . | Flow Velocity at 1.5° Wall convergence . | Flow Velocity at 2.5° Wall convergence . |
---|---|---|---|---|

(m) | (m/s) | (m/s) | (m/s) | (m/s) |

Pre-Jump (505.6, 90.5, 348) | 57 | 57.1 | 57.09 | 57.11 |

Post Jump (550, 90.5, 353.2) | 13.5 | 13 | 12.88 | 11.3 |

At the End Sill (591.6, 90.5, 353.2) | 9 | 7.5 | 7 | 6.48 |

Location Points with Coordinates (x,y,z) . | Flow Velocity at 0° Wall convergence . | Flow Velocity at 0.5° Wall convergence . | Flow Velocity at 1.5° Wall convergence . | Flow Velocity at 2.5° Wall convergence . |
---|---|---|---|---|

(m) | (m/s) | (m/s) | (m/s) | (m/s) |

Pre-Jump (505.6, 90.5, 348) | 57 | 57.1 | 57.09 | 57.11 |

Post Jump (550, 90.5, 353.2) | 13.5 | 13 | 12.88 | 11.3 |

At the End Sill (591.6, 90.5, 353.2) | 9 | 7.5 | 7 | 6.48 |

### Computation of flow depths

Sr. No. . | Operating Condition . | Free Flow (563 m asl) . | ||||
---|---|---|---|---|---|---|

Block Type . | Standard Blocks . | |||||

Convergence of Wall (Degrees) . | 0° . | 0.5° . | 1.5° . | 2.5° . | ||

1 | Flow Depth (m) | Pre-Jump | 7.98 | 7.99 | 8.00 | 8.05 |

2 | Post-Jump | 32.70 | 33.30 | 33.70 | 33.90 | |

3 | Near end sill | 30 | 31 | 32.5 | 33 |

Sr. No. . | Operating Condition . | Free Flow (563 m asl) . | ||||
---|---|---|---|---|---|---|

Block Type . | Standard Blocks . | |||||

Convergence of Wall (Degrees) . | 0° . | 0.5° . | 1.5° . | 2.5° . | ||

1 | Flow Depth (m) | Pre-Jump | 7.98 | 7.99 | 8.00 | 8.05 |

2 | Post-Jump | 32.70 | 33.30 | 33.70 | 33.90 | |

3 | Near end sill | 30 | 31 | 32.5 | 33 |

Sr. No. . | Operating Condition . | Free Flow (563 m asl) . | ||||
---|---|---|---|---|---|---|

Block Type . | Modified Blocks . | |||||

Convergence of Wall (Degrees) . | 0° . | 0.5° . | 1.5° . | 2.5° . | ||

1 | Flow Depth (m) | Pre-Jump | 8.00 | 8.00 | 8.00 | 8.05 |

2 | Post-Jump | 33.00 | 33.70 | 33.80 | 34.00 | |

3 | Near end sill | 30.5 | 31.8 | 32.70 | 33.5 |

Sr. No. . | Operating Condition . | Free Flow (563 m asl) . | ||||
---|---|---|---|---|---|---|

Block Type . | Modified Blocks . | |||||

Convergence of Wall (Degrees) . | 0° . | 0.5° . | 1.5° . | 2.5° . | ||

1 | Flow Depth (m) | Pre-Jump | 8.00 | 8.00 | 8.00 | 8.05 |

2 | Post-Jump | 33.00 | 33.70 | 33.80 | 34.00 | |

3 | Near end sill | 30.5 | 31.8 | 32.70 | 33.5 |

### Computation of Froude number

### Energy loss

_{1}is flow depth just before the hydraulic jump, y

_{2}is flow depth after the formation of hydraulic jump and is head loss.

### Efficiency of the hydraulic jump

_{1}is the specific energy of the supercritical flow, i.e., prior to hydraulic jump formation, E

_{2}is specific energy of the subcritical flow, i.e. after the formation of the hydraulic jump, and

*η*is the relative energy loss that shows the efficiency of the hydraulic jump.

### Comparison of results with previous studies

*et al.*2015; Raza

*et al.*2023). Analyzing the data from both the current and past studies allowed for a comprehensive understanding of how changes in converging angles impact the hydraulic performance of stilling basins. Figure 11 showing an increasing trend in hydraulic efficiency with the increase in covering angle. In all three cases hydraulic efficiency of the stilling basin has increased after wall convergence. Maximum hydraulic efficiency is noted at 2.5 degree wall convergence. By examining the trends in efficiency, researchers can identify optimal design parameters for stilling basins, contributing to the safety and economy of hydraulic structures.

### Improved design

## CONCLUSIONS AND RECOMMENDATIONS

A CFD model study was conducted on the lower stilling basin of Mohmand Dam spillway by varying the reservoir levels at free flow conditions. The study was conducted to investigate the effect of wall convergence and modified impact block on the hydraulic efficiency of the stilling basin. The side walls of the stilling basin were converged from 0.5° to 2.5° (with an increment of 0.5°). At all operating conditions, converged walls improved the hydraulic performance of stilling and contained the hydraulic jump in the stilling basin. The optimum efficiency of the hydraulic jump was observed at 2.5̊ wall convergence with modified impact blocks. The replacement of modified impact blocks with standard blocks increased the flow depth which in turn reduced the post-jump velocity. As a result, modified impact block increased the hydraulic efficiency up to 1%. Even with these modification, pre-jump and post-jump Froude numbers were found in an acceptable range of 4.83 to 6.75 and 0.66 to 0.78, respectively. Consequently, the current study provided improved design of USBR Type-III stilling basin. Improved design is an effort to achieve economy in the length of the basin with optimum hydraulic efficiency. The present study recommends checking the hydraulic performance of the modified stilling basin for gated flow condition. Moreover, for implementation of modified design, economic feasibility is required.

## COMPETING INTERESTS

The authors have no competing interests to declare that are relevant to the content of this article.

## FUNDING

No funding was received for conducting this study.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Symmetry*

**13**(9), 1643.

*Performance of Type III Stilling Basins for Stepped Spillways*

*Proc. 2nd Int. Seminar on Dam Protection against Overtopping*, Fort Collins, 7–9 September 2016