## Abstract

Stilling basin of the Tuansa barrage is a modified form of the United States Bureau of Reclamation (USBR) Type-III basin, which includes impact baffle and friction blocks. Soon after the barrage's operation, baffle blocks were found to be uprooted. Additionally, previous studies also highlighted the issues of rectangular baffle blocks, i.e., less drag, smaller wake zone, and flow reattachment. On the contrary, the use of wedge-shaped splitter blocks is found limited downstream of diversion barrages. Therefore, this study focuses on the hydraulic effects of wedge-shaped splitter blocks on hydraulic jump (HJ) and energy dissipation using FLOW-3D and confirms its suitability downstream of the diversion barrage. The study mainly investigated the free surface profiles, roller lengths, relative energy losses, velocity profiles, and turbulent kinetic energies for three different stilling basins. The results of free surface profiles indicated that new basins stabilised the HJ and produced smaller lengths of HJ. The results also showed that as the flow increased, the roller lengths decreased in the new basins. Similarly, with an increase in the flow, the relative energy loss increased in new basins. Furthermore, within HJ, and at the basins' end, the results showed higher decay of velocity and turbulent kinetic energy in the new basins.

## HIGHLIGHTS

Investigation of flow behaviour in wedge-shaped stilling basins using FLOW-3D.

Comparison of hydraulic jump and flow characteristics between wedge-shaped and conventional stilling basins.

Investigation of energy dissipation indexes in wedge-shaped and conventional stilling basins.

## NOTATIONS

- 3D
three-dimensional

*A*_{x}, A_{y}, A_{z}flow areas in

*x*,*y*, and*z*directions (m^{2})*C*_{d}discharge coefficient

- CFD
computational fluid dynamics

*D*orifice opening (m)

*F*fraction of fluid

- FAVOR
Fractional area-volume obstacle representation

- Fr
_{1} Froude number

- FVM
finite volume method

*g*acceleration due to gravity (m/s

^{2})*G*,_{x}*G*,_{y}*G*_{z}body acceleration in

*x*,*y*, and*z*directions (m/s^{2})*H*_{1}specific energy heads upstream of hydraulic jump (m)

*h*_{1}water level in the supercritical region (m)

*H*_{2}specific energy heads downstream of hydraulic jump (m)

*h*_{2}water level in the subcritical region (m)

*H*_{c}centreline head (m)

*H*_{d}designed head (m)

*H*_{e}effective head (m)

*K*–*ɛ*turbulence model

*L*_{r}roller length of hydraulic jump (m)

*L*/_{r}*d*_{1}dimensionless roller length

*Q*volume flow rate (m

^{3}/s)- RANS
Reynolds averaged Navier–Stokes equations

- RNG
*K*–*ɛ* renormalization group

- RSM
Reynolds stress model

- Stl
stereolithography

*T*time (s)

- TKE
turbulent kinetic energy (m

^{2}/s^{2})*u*,*v*,*w*velocity components in

*x*,*y*, and*z*directions (m/s)*u*_{rms}root mean square velocity (m/s)

- USBR
United States Bureau of Reclamation

- VOF
volume of fluid models

*X*horizontal distance from hydraulic jump toe (m)

*X*_{min},*X*_{max}upstream and downstream boundaries (m)

*y*flow depth (m)

*Y*_{min},*Y*_{max}wall boundary in the lateral direction (m)

*Z*_{min},*Z*_{max}bottom and top boundaries (m)

turbulent dynamic viscosity (kg/m·s)

turbulent production (m

^{2}/s^{2})dynamic viscosity (kg/m·s)

dissipation of turbulent kinetic energy (m

^{2}/s^{3})fluid density (kg/m

^{3})fluctuation of specific weight (kg/m

^{2}·s^{2})- ,,,
model parameters

## INTRODUCTION

*et al.*2015; Bayon-Barrachina & Lopez-Jimenez 2015). Based on the Froude number (Fr), HJs are classified as weak (1.7 < Fr

_{1}≤ 2.5), oscillating (2.5 < Fr

_{1}≤ 4.5), steady (4.5 < Fr

_{1}≤ 9), and strong jumps (Fr

_{1}> 9) (Murzyn & Chanson 2009). Fr is a dimensionless number, which can be calculated by the following equation:where

*v*,

*h*, and

*g*are the stream-wise velocity, flow depth, and gravitational acceleration, respectively. To compute HJ efficiency , Hager & Sinniger (1985) proposed Equation (2), while for free surface profiles of the HJs, Bakhmeteff and Matzke (1936) developed Equation (3):where Fr

_{1}is the value of the Froude number before the HJ.where is the water depth at

*x*(

*hi*) in which the variable

*X*is the dimensionless longitudinal coordinates (

*x*) as shown in Equations (4) and (5),where

*h*

_{1}and

*h*

_{2}are the water depths upstream and downstream of the HJ, respectively.

*X*

_{1}and

*X*

_{2}are the function of variable

*X*and can be obtained at the toe of the HJ and at the roller's end, respectively. Figure 1 shows the components of Equations (4) and (5).

Many experimental investigations are carried out on HJs; however, few of the most relevant studies are highlighted here. Habibzadeh *et al.* (2011, 2012) carried out an experimental study on baffle blocks to investigate the hydraulic jump for low head hydraulic structures. Chachereau & Chanson (2011) and Wang & Chanson (2015) investigated the free surface profiles within the HJ for a wide range of initial Froude numbers (Fr_{1}). The result indicated that interfacial velocities in the HJs were identical to a wall jet-like profile, and turbulence intensities were high because of the longitudinal free surface fluctuations. Macián-Pérez *et al.* (2020b) carried out experiments on the USBR-II stilling basin to investigate HJ characteristics and studied velocity profiles. Kucukali & Chanson (2008) conducted experiments on the bubbly structures of the HJ. The results of the free surface profiles showed an unsteady turbulent structure of the HJ. Qasim *et al.* (2022) conducted experiments on bed discordance downstream of different weirs. The results indicated that as the bed discordance increased, the dimensionless flow depth decreased downstream of discordance, which increases the Froude number. The results further showed that as the configuration of bed discordance was changed, the free surface profiles were also changed, which affected the flow depths and velocity profiles. Jesudhas *et al.* (2018) investigated the turbulence characteristics of the classical hydraulic jump (CHJ) for Fr_{1} = 8 using detached-eddy simulation (DES). The results showed the upward movement of shear layers, which were found between the wall jet flow and roller region, which also created waves at the free surface.

To stabilise the HJ and to dissipate the left-over energy of the HJ, energy dissipaters are provided in the stilling basin and at the channels' bed (Bradley & Peterka 1958; Peterka 1984). Many researchers have developed energy dissipators downstream of open channel flow and pipe outlets, i.e., rectangular (Habibzadeh *et al.* 2011, 2012), curved (Eloubaidy *et al.* 1999), T-shaped (Tiwari *et al.* 2010), triangular (Tiwari & Goel 2016), porous structures (Widyastuti *et al.* 2022), and wedge-shaped (Pillai *et al.* 1989; Goel 2007, 2008). However, after the extensive laboratory tests for a low head dam and diversion structures, the USBR developed a Type-III basin for an inflow Froude number of 5–9, for which they recommended different energy breaking appurtenances, i.e., chute blocks, rectangular baffle blocks, baffle pier with rectangular front face, and sloping end sill (Bradley & Peterka 1958). However, with the use of such stilling basins, low pressure on the sides of the baffle pier was noticed, which caused cavitation problems. A similar type of issue occurred on McNary, Bonneville, and Kanopolis (Pillai 2022), and Folsom dams (Frizell & Svoboda 2012). Despite the fact, still, hydraulic and civil engineers follow UBSR guidelines for hydraulic structures. Many experimental and numerical studies were carried out for basins' appurtenances; however, few of the relevant studies are highlighted here. Al-Mansori *et al.* (2020) investigated different shapes of baffle blocks for energy dissipation and compared the results with the USBR baffle blocks. As compared to the USBR baffle blocks, the results indicated that a new baffle reduced the HJ length and increased the energy dissipation of up to 9.31%. The results also showed an increase in the drag force up to 98.6% as compared to the USBR baffle block. Chaudary & Sarwar (2014) and Ali & Kaleem (2015) studied the chute blocks and baffle blocks' stilling basins for different tailwater levels. The results indicated that as compared to the impact baffle blocks’ basin, the chute blocks and end sill basin were dissipating less energy, which caused the launching of stone apron and river drift. Pillai *et al.* (1989) compared three different stilling basins for the Fr_{1} values less than 4.5. The results indicated that a stilling basin with wedge-shaped baffle blocks at a vertex angle of 150° and a cutback angle of 90° reduced the basin length and scour depth. Goel (2007, 2008) and Tiwari *et al.* (2010) studied the flow characteristic using wedge-shaped splitter blocks downstream of circular and square pipe outlets. The results showed that as compared to the impact-type USBR-VI stilling basin, the splitter blocks’ basins reduced the length of the stilling basin up to 50%, and the splitter blocks spread the fluid in the lateral direction, which increased the energy dissipation.

Conventionally, reduced scale modelling is usually applied to investigate the flow over the hydraulic structures, but due to scaling effects, errors in measuring devices and difficulties in displaying the terrain and concrete roughness (Ikhsan *et al.* 2022) compromised the accuracy of results. Additionally, such a modelling technique is becoming very expensive and time-consuming. On the other hand, recent developments in the computer technology have facilitated researchers to address the hydraulic issues, mainly using turbulence models (Siuta 2018). Additionally, these computer-based models also allow to develop the full-scale computer models such as FLOW-3D. These models not only facilitate in flow investigation but also help in examining the internal structures of HJs and air entrainment (Chanson & Gualtieri 2008). Consequently, the use of numerical modelling tools such as Open Foam (Bayon-Barrachina & Lopez-Jimenez 2015), (Bayon *et al.* 2016), ANSYS Fluent (Aydogdu *et al.* 2022), Complete two-fluid models (CTFM) (Moghadam *et al.* 2020), REEF3D (Kamath *et al.* 2019), and FLOW-3D (Leuker *et al.* 2008) is becoming prevalent. Few of most recent numerical studies on the HJ and flow behaviour in different types of stilling basins are highlighted here. Yamini *et al.* (2022) used FLOW-3D to investigate the hydraulic performance of the sea water intake system using different energy breaking devices, i.e., curtain walls, baffles, fillets, and splitters. The results showed no vertex in the pumping zone of scenarios I, II, and III. However, the results indicated a small number of vortexes on the free surface between the suction pump and intake tank. With an increase in the flow depth, the results showed a decrease in the vortex. The study concluded that dividing the wall between the pump decreased the velocity. The study recommended to further investigate the intake structure for energy dissipation in the basin of the sea water intake system. Zaffar & Hassan (2023) investigated different characteristics of the HJ in different stilling basins using FLOW-3D. The results showed higher values of velocity, turbulent kinetic energy (TKE), and roller lengths in the remodelled stilling basin. Nikmehr & Aminpour (2020) investigated HJ on a rough bed using FLOW-3D models and validated the outputs with the experimental data. Mirzaei & Tootoonchi (2020) investigated the effects of sluice gate and bump on the characteristics of the HJ for various hydraulic conditions, and compared the results with FLOW-3D models. Macián-Pérez *et al.* (2020a) analysed the structural properties of the HJ, i.e., HJ shape, free surface profile, sequent depth ratio, and HJ efficiency, using FLOW-3D models. Tohamy *et al.* (2022) implemented vertical screens downstream of the hydraulic structure for energy dissipation using FLOW-3D. The results showed that with the use of screen, the water surface profiles decreased and an inverse relation of flow depths with Fr_{1} was observed. Ebrahimiyan *et al.* (2021) investigated the characteristics of HJ using different sediment concentrations. The results showed a direct correlation of sediment concentration with bed shear stress and energy dissipation. However, as the sediment concentration increased, the dimensionless length of the HJ ratio decreased. For Fr_{1} = 7, the results indicated an 18.1 and 17.2% decrease in the jump length and sequent depths, respectively, while as compared to clear water, a 3.2% increase was noticed in energy dissipation.

In Pakistan, barrages were built about 50–100 years ago, and stilling basins of these hydraulic structures are jump-type. The design of these basins is the modified form of the USBR Type-III basin, which consists of two rows of the baffle and friction blocks. Such basins not only enhance turbulence but also stabilise HJ in the case of less tailwater depth. The stilling basin of the Taunsa barrage was also a modified Type-III basin, which was made operational during the year 1958. Due to the vertical face of USBR-III impact baffle blocks, soon after the barrage operation, the uprooting of baffle blocks in front of many bays was reported (Zaidi *et al.* 2004, 2011). Additionally, these traditional shapes of baffle blocks also encounter flow reattachment on the side, which reduces the drag force (Frizell & Svoboda 2012; Pillai & Kansal 2022).

On the contrary, in the literature, many experimental and numerical investigations have been carried out on different shapes of energy dissipators downstream of open channel flow, but the use of wedge-shaped baffle blocks (Pillai *et al.* 1989; Verma & Goel 2003; Verma *et al.* 2004; Goel 2007, 2008; Tiwari *et al.* 2010) downstream of the hydraulic structures, i.e., river diversion barrage, is found to be limited. Previously, these dissipators only investigated downstream of the pipe outlets’ stilling basin for a Froude number of 4.5 (Pillai *et al.* 1989). Therefore, due to the splitting ability of wedge-shaped blocks, the focus of this study is to investigate the hydraulic performance of wedge-shaped splitter blocks for different gated flows (44 and 88 m^{3}/s) on the investigated barrage. Based on the recommendation of previous studies, in the present study, the wedge-shaped baffle block with a vertex angle of 150° and cutback angle 90° was employed. The main objective of this numerical study is to investigate HJ and flow characteristics with different arrangements of wedge-shaped splitter blocks on the studied stilling basin using FLOW-3D. The study applies a higher value of incoming Froude number up to 5.75, and investigates free surface profiles, roller lengths, hydraulic jump efficiencies, velocity profiles, and turbulent kinetic energies. Furthermore, to ensure the hydraulic performance of the proposed wedge-shaped baffle block basins, the results of the present study are also compared with USBR baffle blocks’ basin and with the relevant literature.

### Study area

^{3}/s. The width of the barrage between the abatements is 1,324.60 m, while 1,176.5 m is a clear waterway (Zaidi

*et al.*2004, 2011). The stilling basin of the barrage was a modified form of the USBR Type-III basin, which consists of two rows of the baffle and friction blocks. These devices stabilise the HJ even in the case of less tailwater depth and help in dissipating excessive kinetic energy. Figure 2 shows the typical cross section of the barrage.

## MATERIALS AND METHODS

### Numerical model implementation

In Equations (6)–(8), *u, v*, and *w* are the velocity components in *x*, *y*, and *z* directions, respectively. and *P* are the total pressure and fluid density. The terms in Equation (7) are known as the Reynolds stresses. *A _{x}*,

*A*, and

_{y}*A*are the flow areas, while

_{z}*R*, , and

*R*

_{SOR}are the model coefficient, flow generic property, and mass source term, respectively.

One of the critical aspects of computation fluid dynamic (CFD) models is turbulence modelling. CFD models use the Reynolds averaged Navier Stokes equations (RANS) to solve the turbulence and high Reynolds numbers, which create instabilities within the flow. Turbulence models find closure by averaging the Reynolds stress terms in Navier–Stokes' equation and append additional variables for turbulent viscosity and transport equations. Six turbulence models are available in FLOW-3D, which employ numerous equations to solve the closure problems. Among various models, the two equation turbulence models such as standard *K*–*ɛ* (Jones & Launder 1972), RNG *K*–*ɛ* (Yakhot *et al.* 1991), and k-*ω* (Wilcox 2008) are most widely used in hydraulic investigations.

*et al.*(2020a, 2020b) for flow behaviour in the USBR-II stilling basin, and the study indicated that the RNG

*K*–

*ɛ*model showed better accuracy for free surface, roller lengths, and hydraulic jump efficiency. Also, the RNG

*K*–

*ɛ*model was used to investigate free surface profiles (Shirkavand & Badiei 2014, 2015) and velocity profiles (Moghadam

*et al.*2019, 2020). Based on the bibliographical results, this numerical investigation has implemented the RNG

*K*–

*ɛ*model and transport Equations (9) and (10) are used to model turbulent kinetic energy (

*k*) and its dissipation (ɛ), respectively:where is the coordinate in the x direction, is the dynamic viscosity, is the turbulent dynamic viscosity,

*k*is the turbulent kinetic energy (TKE), is the turbulent dissipation, is the fluid density, and is the production of TKE. Finally, the terms , , , and are model parameters whose values are given in Yakhot

*et al.*(1991).

*F*), in which

*F*represents the proportion of fluid. The following equation is used to compute the evolution of F throughout the domain:

In FLOW-3D, the fluid fraction (*F*) in a cell is usually presented by three possibilities as:

- (A)
*F*= 0, the cell is empty. - (B)
*F*= 1, the cell is fully occupied by fluid. - (C)
0 <

*F*< 1, the cell represents the surface between the two fluids.

In the present study, only one fluid (water) with free surface is considered, and FLOW-3D models have automatically selected the free surface method from the available advection scheme. For free surface tracking, a value of 0.5 is assigned in each computation cell.

### Geometry and meshing

*.*file) were imported to FLOW-3D as shown in Figure 4. Three structural hexahedral mesh blocks were used to resolve the geometries and flow domain. stl files of different stilling basins are shown in Figure 4. Figure 4(a) shows that the two rows of the baffle consist of USBR baffle blocks, while Figure 4(b) shows that the USBR baffle blocks are replaced with wedge-shaped blocks. In Figure 4(c), the baffle region consists of both wedge and USBR baffle blocks, in which the first row contains the wedge, while the second row has the USBR blocks.

*X*

_{min}= 15 m to

*X*

_{max}= 51 m, while the second mesh block was implemented between

*X*

_{min}= 51 m and

*X*

_{max}= 56 m. However, the third mesh block was ranged between

*X*

_{min}= 56 m and the end of the stilling basin (rigid floor,

*X*

_{max}= 71 m). It is worth mentioning here that the use of three mesh blocks was made to fully resolve the mesh geometry, especially the baffle blocks’ region whose effects were checked on different hydraulic parameters. Overall, 56 m long, 9 m wide, and 11 m high models were generated. The details of each mesh block, cell sizes, and their quality indicators are provided in Figure 5 and Table 1. For gated flow analysis, a 7 m wide, 0.53 m long, and 6.10 m high vertical gate was mounted upstream of the weir crest.

Mesh block . | Number of cells . | Maximum adjacent ratio . | Maximum aspect ratio . | ||||
---|---|---|---|---|---|---|---|

X
. | Y
. | Z
. | X–Y
. | Y–Z
. | Z–X
. | ||

Block-1 | X = 144 | 1.00 | 1.00 | 1.00 | 1.016 | 1.016 | 1.061 |

Y = 36 | |||||||

Z = 60 | |||||||

Block-2 | X = 40 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

Y = 72 | |||||||

Z = 120 | |||||||

Block-3 | X = 60 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

Y = 36 | |||||||

Z = 60 |

Mesh block . | Number of cells . | Maximum adjacent ratio . | Maximum aspect ratio . | ||||
---|---|---|---|---|---|---|---|

X
. | Y
. | Z
. | X–Y
. | Y–Z
. | Z–X
. | ||

Block-1 | X = 144 | 1.00 | 1.00 | 1.00 | 1.016 | 1.016 | 1.061 |

Y = 36 | |||||||

Z = 60 | |||||||

Block-2 | X = 40 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

Y = 72 | |||||||

Z = 120 | |||||||

Block-3 | X = 60 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

Y = 36 | |||||||

Z = 60 |

### Boundary conditions

*P*), while the sides and floor were set to wall (

*W*) boundaries, which stated no-slip, zero tangential, and normal velocities (

*u*=

*v*=

*w*= 0) to the walls, where

*u*,

*v*, and

*w*, are the velocities in

*x*,

*y*, and

*z*directions, respectively. Except pressure, which was assigned to zero, for all the other variables, the upper boundaries were set to atmospheric pressure, which permitted the fluid as null von Neumann. For the second mesh block, except upstream boundary (

*X*

_{min}) and downstream boundary (

*X*

_{max}), which were set to symmetry (

*S*), all other boundaries were like block 1. On the contrary, in the third mesh blocks, the upstream and downstream boundaries were set to symmetry (

*S*) and pressure (

*P*), respectively, while the rest of the four boundaries were like the other blocks. The governing boundary conditions for the models are shown in Figure 6.

For the investigation of HJs and flow parameters on the rigid bed, the solution convergence and steady state of the models are usually monitored by volume flow rates and mass-averaged mean kinetic energy at the inlet and outlet boundaries. At each time step, the stability and convergence were controlled by the Courant number (Ghaderi *et al.* 2020). In each iteration, stability and convergence checked the time step, which varied from 0.0023 to 0.0025 and 0.015 to 0.0025 for free and gated flows, respectively. The hydraulic analysis for gated flow analysis indicated that for 44 and 88 m^{3}/s discharge, *T* = 170 s was appropriate to reach the steady state. However, the time for the initial steady state of various models was found different and varied from *T* = 100 to 165 s. In the case of free flow analysis, at *T* = 60 s, models achieved steady state. However, for both free and gated flows, to accommodate the small fluctuations in volume flow rates, the finish time of *T* = 170 s was assigned to the models.

### Operation of numerical models

^{3}/s) of the Taunsa barrage, and the models were run on

*h*/

_{e}*H*= 0.998 (Savage & Johnson 2001; Johnson & Savage 2006; Gadge

_{d}*et al.*2018), whereas

*h*and

_{e}*H*are the effective and designed heads, respectively, as can be seen from Figure 7.

_{d}^{3}/s discharges, the gate opening of

*D*= 0.28 and 0.60 m was set, respectively, while a constant designed head (i.e., upstream pond level) of

*H*= 136.24 m was maintained for all the discharges. The schematic diagram of model operations is provided in Figure 7. All the simulations were run on boundary conditions as described in Section 2.3, while initial conditions for free and gated flows are provided in Table 2:whereas

_{d}*Q*(m

^{3}/s),

*A*(m

^{2}), and

*g*(m/s

^{2}) are the volume flow rate, the orifice area, and the gravitational acceleration, respectively, while

*h*is the centreline head (

_{c}*h*=

_{c}*H*/2).

_{d}–DFlow conditions . | Discharge (m^{3}/s)
. | Pond levels (m) . | TWLs attained during 2010 flood . |
---|---|---|---|

Gate flow | 44 | 136.24 | 130.40 |

88 | 136.24 | 131.23 | |

Free flow | 440 | 135.93 | 134.00 |

Flow conditions . | Discharge (m^{3}/s)
. | Pond levels (m) . | TWLs attained during 2010 flood . |
---|---|---|---|

Gate flow | 44 | 136.24 | 130.40 |

88 | 136.24 | 131.23 | |

Free flow | 440 | 135.93 | 134.00 |

## RESULTS AND DISCUSSION

### Results of free and gated discharge rating curves

To check the accuracy of numerical models, each of the three stilling basins was investigated for the free flow and gated discharge rating curves of the prototype (Savage & Johnson 2001; Johnson & Savage 2006; Chanel & Doering 2009). After steady state, the discharge values of numerical models were compared with prototype's discharge. In gated flow for 44 m^{3}/s discharge, the models produced 51, 47, and 46 m^{3}/s discharge in Type-A, -B, and -C stilling basins, respectively, which indicated that all models overestimated the discharge. Similarly, at 88 m^{3}/s, the discharge values reached 94, 96, and 97 m^{3}/s in Type-A, -B, and -C stilling basins, respectively. The trend of the volume flow rate at the investigated discharge was found to be similar, which indicated that the present model overestimated the discharge value. However, as compared to the lower discharge, as the models were run for a higher value, the errors were found to be reduced.

In the free flow analysis of 444 m^{3}/s, Type-A stilling produced 441 m^{3}/s discharge, while 438 and 440 m^{3}/s discharge values were noticed in Type-B and -C stilling basins. The comparison of gated and free flows with the rating curve of the prototype is given in Table 3.

Stilling basins . | Free flow (m^{3}/s). | Error % . | Gated flow (m^{3}/s). | Error % . | Gated flow (m^{3}/s). | Error % . | |||
---|---|---|---|---|---|---|---|---|---|

Prototype . | FLOW 3D . | Prototype . | FLOW 3D . | Prototype . | FLOW 3D . | ||||

Type-A | 444 | 441 | −0.70 | 44 | 51 | 15.90 | 88 | 94 | 6.80 |

Type-B | 444 | 438 | −1.35 | 44 | 47 | 6.80 | 88 | 97 | 10 |

Type-C | 444 | 440 | −0.90 | 44 | 46 | 4.50 | 88 | 96 | 9 |

Stilling basins . | Free flow (m^{3}/s). | Error % . | Gated flow (m^{3}/s). | Error % . | Gated flow (m^{3}/s). | Error % . | |||
---|---|---|---|---|---|---|---|---|---|

Prototype . | FLOW 3D . | Prototype . | FLOW 3D . | Prototype . | FLOW 3D . | ||||

Type-A | 444 | 441 | −0.70 | 44 | 51 | 15.90 | 88 | 94 | 6.80 |

Type-B | 444 | 438 | −1.35 | 44 | 47 | 6.80 | 88 | 97 | 10 |

Type-C | 444 | 440 | −0.90 | 44 | 46 | 4.50 | 88 | 96 | 9 |

Hence, it can be believed that FLOW-3D has produced gated and free flows with acceptable accuracy, which allowed for the further analysis of HJ and other flow parameters in Type-A, -B, and -C stilling basins.

### Free surface profiles

Volume of fluid (VOF) obtained the free surface profiles in different stilling basins at various discharges.

^{3}/s discharge.

All the stilling basins showed a similar pattern of free surface profiles up to the jump initiating points, while the locations of the HJ were found different. The results indicated 4.90, 4.66, and 4.50 m distances of HJs initiating location from the toe of the glacis in Type-A, -B, and -C stilling basins. In the transition region of HJ, free surface profiles in the Type-A stilling basin showed dissimilar behaviour in which a free surface drop was noticed before HJ termination while a smooth transition of HJs was noticed in Type-B and -C basins. After the HJs, a similar pattern of free surface profiles was noticed in all the stilling basins. However, in the subcritical region, the free surface profiles were found different due to the shape of HJ formed by different dissipating arrangements. The results showed 9.7, 9.5, and 9.3 m lengths of HJs in Type-A, -B, and -C stilling basins, respectively. Figure 8(b) compares the results of the free surface profiles of HJs with Bayon-Barrachina & Lopez-Jimenez (2015) (RNG *K*–*ɛ*) and Wang & Chanson (2015) at 44 m^{3}/s. The present models showed a good agreement with the free surface profile of Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015). At the HJ initiating location, the free surface profile of the present models deviated from the compared studies; however, the free surface profiles showed agreement as the distance from the HJ was increased up to the HJ termination point. After comparing the results with Wang & Chanson (2015), the free surface profiles in all the studied stilling basins showed a close agreement.

^{3}/s discharge. A similar trend of free surface profiles was noticed in the supercritical region, while variations were noticed in the HJ regions. As compared to the lower discharge, at 88 m

^{3}/s discharge, the length of the location of HJs was found different in all the stilling basins. At 88 m

^{3}/s discharge, the locations of the HJs reached 6.10, 5.19, and 4.86 m from the downstream toe of glacis in Type-A, -B, and -C basins, respectively.

As compared to Type-B and -C basins, the location of the HJ was found in upper regions of downstream glacis. However, it is important to mention that in all the basins, the locations of HJs were found on the glacis. From Figure 9(a), in the Type-A basin, free surface profiles were found to be different from Type-A and -B basins. However, in Type-A and -B basins, the free surface profiles were identical due to similar type dissipators in the baffle blocks’ region, which characterised the HJ. In the subcritical region, Type-B and Type-C showed the identical free surface profiles, which differed from that noticed in the Type-A basin. As compared to the lower discharge (i.e., 44 m^{3}/s), the length of HJs in Type-A, -B, and -C stilling basins increased and reached 11.7, 10.7, and 10.5 m, respectively. However, it is worth mentioning that as compared to the Type-A basin, Type-B, and Type-C basins produced a lesser length of HJs. Figure 9(b) compares the free surface profiles of HJs with Bayon-Barrachina & Lopez-Jimenez (2015) and Wang & Chanson (2015). In the Type-A stilling basin, near the HJ initiating location, the present model overestimated the free surface profile, which deviated from the previous studies, while in the transition region of the HJ, the free surface showed a good agreement with previous studies. On the other hand, in Type-B and -C stilling basins, the free surface profiles are in close agreement with the compared studies. Compared to Bayon-Barrachina & Lopez-Jimenez (2015), the results of the present study showed more promising results with Wang & Chanson (2015).

### Roller lengths

Torkamanzad *et al.* 2019 showed that as compared to the length of the HJ, the roller length is a better characteristic parameter of the HJ which is measured between the toe of the HJ and the end of rollers. In addition, as compared to the HJ length, rollers can easily be computed and observed to specify the flow conditions.

The results showed that as the discharge increased, the dimensionless roller length was found to be reduced in all the stilling basins. The results further indicated that at the lower discharge, as compared to Type-A and -B basins, the Type-C basin produced a larger dimensionless roller length, which reached 35.80. However, with an increase in the discharge, the roller length in Type-C was found to be smaller, which reached 19.81, while at the lower discharge, the maximum dimensionless roller length was found in the Type-A basin, which reached 21.30.

*L*/

_{r}*d*

_{1}) with the previous studies at 44 and 88 m

^{3}/s discharges, whereas Lr and d

_{1}are the roller length of HJs and the flow depth before the HJ, respectively. In comparison with the previous study, at 44 m

^{3}/s discharge, the present models overestimated the roller lengths, which deviated from the previous studies, and the maximum deviation was noticed in the Type-C basin. However, the results of roller lengths in types-A and B are in close agreement with Murzyn & Chanson (2009) as shown in Figure 10(a). With an increase in the discharge (88 m

^{3}/s), as compared to the Type-A basin, the results of roller lengths in types-B and C are in good agreement with Kucukali & Chanson (2008) and Wang & Chanson (2015) as can be seen from Figure 10(b).

### Relative energy loss (*R*_{L})

_{L}

*R*) within the HJs in different stilling basins. The (

_{L}*R*) is the ratio of energy loss between upstream and downstream of the HJ to the upstream hydraulic head. Flow depth (

_{L}*h*), velocity (

_{i}*v*), and acceleration due to gravity (

_{i}*g*) are the variables used in the computation of elative energy loss:whereas

*H*

_{1}and

*H*

_{2}are the specific energy heads upstream and downstream of HJs, respectively.

At 44 and 88 m^{3}/s discharges, the results of *R _{L}* in Type-A, -B, and -C stilling basins are presented in Table 4.

Discharge (m^{3}/s)
. | Type of stilling basin . | Before HJ (E_{1})
. | After HJ (E_{2})
. | Relative energy loss (%) . |
---|---|---|---|---|

44 | Type-A | 4.39 | 3.54 | 19 |

Type-B | 4.50 | 3.70 | 18 | |

Type-C | 4.54 | 3.68 | 19 | |

88 | Type-A | 4.79 | 4.38 | 8 |

Type-B | 5.64 | 4.56 | 19 | |

Type-C | 5.10 | 3.78 | 26 |

Discharge (m^{3}/s)
. | Type of stilling basin . | Before HJ (E_{1})
. | After HJ (E_{2})
. | Relative energy loss (%) . |
---|---|---|---|---|

44 | Type-A | 4.39 | 3.54 | 19 |

Type-B | 4.50 | 3.70 | 18 | |

Type-C | 4.54 | 3.68 | 19 | |

88 | Type-A | 4.79 | 4.38 | 8 |

Type-B | 5.64 | 4.56 | 19 | |

Type-C | 5.10 | 3.78 | 26 |

From Table 4, it can be observed that in Type-A stilling basins, as the discharge increased, *R _{L}* decreased, an increase in

*R*was observed in Type-B and -C stilling basins.

_{L}At the studied flows, the maximum relative energy loss of the hydraulic jump was seen at 88 m^{3}/s discharge in the Type-C stilling basin, which reached 26%.

### Velocity profiles

*X*). The velocity profiles were measured before, within, and after the HJs at different flow depths. At 44 m

^{3}/s, the maximum velocity was noticed after at

*X*= 2 m (from the toe of the HJ), and their values reached 5.31, 5.17, and 5.46 m/s, in Type-A, -B, and -C stilling basins, as shown in Figures 11(a)–11(c), respectively. From

*X*= 2 to 8 m, forward velocity profiles were noticed in all the stilling basins, while as the distances increased, the velocity declined, which levelled off at the end of the stilling basins. The results showed that in all the stilling basins, as the distance from the jump initiating location increased, the velocity near the bed decreased; hence, boundary growth layers also reduced. From HJ initiating to terminating locations, the trends of forward velocity profiles in Type-A, -B, and -C stilling basins were found to be identical, which is in agreement with Ead & Rajaratnam (2002) and Nikmehr & Aminpour (2020). Up to

*X*= 8 m (from the HJ toe) in all the stilling basins, the results showed the backward velocity profiles in the upper fluid regions. However, as compared to Type-B and C basins, the backward velocity profiles in the Type-A basin were initiated from the lower fluid depths (

*Z*= 1.92 m). At

*X*= 12 m, as compared to the Type-A stilling basin, Type-B and -C stilling basins showed different velocity profiles, which displayed two forward regions, i.e., near the bed and at

*Z*= 1.5 m as shown in Figures 11(b) and 11(c). Overall, the velocity distribution in Type-A, -B, and -C basins showed a wall jet-like structure, which is in agreement with Ead & Rajaratnam (2002) and Nikmehr & Aminpour (2020).

^{3}/s discharge, Figure 12 shows the two-dimensional (2D) illustration of velocity profiles (

*Y*–

*Z*) at two different locations, i.e., after the HJ and at the end of the stilling basin. From Figure 12(a), after the HJ, the results showed a higher velocity magnitude near the floor between the baffle blocks. However, behind the baffle blocks, wake zones were noticed, which showed a considerable decrease in the velocity magnitude. After the HJ, at the free surface, the maximum magnitude of the velocity reached 1 m/s at the sides, while in the central region, its value reached 0.7 m/s. In the Type-A stilling basin, at the end of the stilling basin, the maximum velocity magnitude was seen at the free surface, which was found to be reduced in the lower fluid depths as shown in Figure 12(b).

In the Type-B stilling basin, after the HJ, the results showed different flow patterns due to the change in the geometry of baffle blocks. The maximum velocity was observed in the gully flow between the wedge-shaped baffle blocks, while large wake zones were noticed on the sides of the wedge blocks. After the HJ in the Type-B stilling basin, the maximum and minimum velocity magnitudes reached 1.5 and 0.6 m/s at the floor and free surfaces, respectively, as shown in Figure 12(c). At the end of the Type-B stilling basin, the maximum velocity was noticed at the free surface, which reached 1 m/s as noticed in the Type-A stilling basin, as shown in Figure 12(d). In the Type-C stilling basin, the pattern of velocity contours after the HJ was found identical to the Type-B stilling basin as shown in Figure 12(e). However, in the Type-C stilling basin lower velocity magnitude was observed at the basin's end at both floor and free surfaces, as can be seen from Figure 12(f).

^{3}/s discharge, at 88 m

^{3}/s, up to

*X*= 10 m (from the HJ toes), the results showed the identical patterns of forward velocity profiles in Type-A, -B, and -C stilling basins as shown in Figures 13(a)–13(c). However, at 88 m

^{3}/s discharge, higher velocity values were witnessed at

*X*= 4, 6, 8, and 10 m in all the investigated stilling basins.

Up to *X* = 10 m (from the HJ toe), the distribution of forward velocity profiles showed a wall jet-like profiles, which is in agreement with Ead & Rajaratnam (2002) and Nikmehr & Aminpour (2020). After *X* = 10 m, the backward velocity profiles in the upper fluid region recovered, and the results showed only forward velocity profiles in all the stilling basins. As compared to the lower discharge, at 88 m^{3}/s, after the HJ (at *X* = 14 m), the velocity in the lower fluid regions was less. Also, at *X* = 14 m, as compared to the Type-A stilling basin, velocities were found less in Type-B and -C stilling basins as shown in Figures 13(b) and 13(c). However, as compared to the Type-A stilling basin, in the central region (*Z* = 1.5 to 2.5 m), the velocity in Type-B and -C stilling basins was higher. The maximum velocity magnitude near the bed reached 2.55, 0.42, and 0.15 m/s in Type-A, -B, and -C stilling basins, respectively. The difference in the velocity near the bed was due to the different shapes of baffle blocks in different stilling basins. Conclusively, the results showed that as compared to Type-A and -B basins, as the discharge increased, the rate of velocity decay was higher in the Type-C basin.

*Y*–

*Z*plane at two different horizontal sections in the Type-A stilling basin. After the baffle blocks, the velocity near the bed decreased, which reached 1.2 m/s, while a velocity value of 1.9 m/s was observed between the baffle blocks as shown in Figure 14(a). In the HJ region, higher values of velocity were noticed near the bed, while recirculation was noticed in the upper fluid region, for which the velocity value near the free surface reduced, which reached 1.2 m/s after the baffle blocks. Compared to the lower discharge (44 m

^{3}/s), at 88 m

^{3}/s discharge, velocities at the end of the Type-A basin were higher at both floor and free surfaces. In the Type-A stilling basin, the maximum value of the velocity at the end of the stilling basin reached 1.8 m/s in lower fluid depths as shown in Figure 14(b).

^{3}/s/m discharge. Like the Type-A basin, near the floor, the velocity values in the Type-B stilling basin showed identical trends. As compared to the Type-A stilling basin, the velocity values after the baffle blocks were less because the wedge-shaped baffle blocks produced more wake areas in the lateral directions. In the Type-B stilling basin, the velocity after the baffle blocks remained between 1.2 and 1.5 m/s, while a velocity of 2.8 m/s was found between the baffle blocks region as shown in Figure 15(c). At the end of the Type-B basin, the maximum value of the velocity reached 1.5 m/s in the central fluid region, while 1.2 m/s was noticed at floor and free surfaces as shown in Figure 15(d). Figures 14(e) and 14(f) show the velocity profiles after the baffle blocks and at the end of the Type-C basin. From Figure 14(e), the results showed that the velocity from the basin's floor to the central fluid region declined, while a slight increase in the velocity magnitude was noticed at the free surface, which reached 1.1 m/s. Additionally, at this section, different flow patterns were noticed, i.e., near the floor, central fluid region, and upper fluid region. In these regions, different velocity values were observed; however, the maximum velocity magnitude was seen at the central part, which reached 1.9 m/s. On the other hand, velocity magnitudes of 0.7 and 1.3 m/s were noticed at the floor and upper fluid regions, respectively, as shown in Figure 14(e). In the Type-C stilling basin, the maximum velocity at the basin's end was noticed at the free surface, which reached 1.5 m/s, while near the floor, a velocity of 0.6 m/s was observed, as can be seen from Figure 14(f).

### Turbulent kinetic energy and turbulent intensity

*x*,

*y*, and

*z*directions and is measured by the root mean square of velocity fluctuations, which can be computed by the following expression:where

*u*

_{1},

*u*

_{2}, and

*u*

_{3}are the successive velocities, and using these values, TKE can be computed by the below expression:where

*u*

_{rms},

*v*

_{rms}, and

*w*

_{rms}are the root mean square velocities in

*x*,

*y*, and

*z*directions, respectively.

^{3}/s flows, Figure 16 shows the TKEs from the centerline of the Type-A, -B, and -C stilling basins.

Figure 15(a) shows the TKE in the Type-A stilling basin at 44 m^{3}/s discharge. Due to the contracted jet of the upcoming supercritical velocity, the maximum TKE magnitude was found within and foreside of the HJ, which reached 3.1 m^{2}/s^{2}. From the HJ initiating location, the amount of TKE was gradually reduced. Furthermore, the maximum magnitude of TKEs was observed from the lower flow depth (*Z* = 1.22 m) up to the free surface. In the Type-A stilling basin, the maximum TKE reached below 0.21 m^{2}/s^{2} at the end of the basin. Figure 15(b) indicates the turbulent intensity (TI) in the Type-A stilling basin at 44 m^{3}/s. The results indicated that maximum TI was found within the HJ region at the locations of maximum TKEs, and ranged from 115 to 171%. In the Type-A basin, after the baffle blocks, the TI was found to be reduced, which reached 26% at the basin's end as shown in Figure 15(b). Figures 15(c) and 15(d) show the TKE and TI in the type-B stilling basin at 44 m^{3}/s. The pattern of TKE and TI is Type-B stilling was found to be identical as noticed in the type-A stilling basin. Similarly, the maximum TKE was noticed at the foreside and within the HJ as shown in Figure 16(c), which reached 3 m^{2}/s^{2}. After the wedge baffle blocks in the Type-B basin, the TKE was below 0.23 m^{2}/s^{2}. From Figure 15(c), as compared to the Type-A stilling basin, in the Type-B basin, less TKE was observed at the basin's end. Figure 15(d) represents the TI in the Type-B stilling basin, and the maximum TI was 180.7% at the impingement point of the supercritical jet and after the HJ, the TI reached 100%. The maximum TI at the end of the Type-C basin reached 26%.

Figure 15(e) shows the TKE in the type-C basin, and the maximum TKE reached 3.6 m^{2}/s^{2} at the HJ initiating location, and was observed to be higher than Type-A and -B stilling basins. However, as compared to Type-A and -B basins, the TKE in Type-C travelled less distance, which reached 0.2 m^{2}/s^{2} (at *X* = 18 m, from the HJ toe). Like the Type-B basin, the TKE in the Type-C basin decayed before the basin's end. Figure 15(f) shows the TI at 44 m^{3}/s in the Type-C basin, and as compared to Type-A and -B basins, the TI was found more in the HJ region, which reached 195%. However, the values of the TI reduced within the baffle block region, which ranged between 68 and 90%. From the baffle to the friction region, the TI ranged between 20 and 68% and even lesser values were observed at the end of the stilling basin.

Figure 16 shows the variation in the TKE at 88 m^{3}/s discharge in Type-A, -B, and -C basins. The section is drawn in the *X*–Z plane at *Y*/2. From Figure 16(a), in the Type-A basin, the results showed that the maximum TKE reached 3.34 m^{2}/s^{2} at the foreside of the HJ. The results also showed that as the discharge increased, the TKE also increased in the Type-A basin.

Figure 16(b) shows the TKE at 88 m^{3}/s in the Type-B basin. The pattern of TKE was found to be similar to the lower discharge; however, at the higher flow rate, the maximum TKE reached 4.22 m^{2}/s^{2}, which was considerably higher than the lower discharge (44 m^{3}/s). In the Type-B stilling basin, TKEs contained between the HJ and baffle blocks’ region, which decreased as the distance from the HJ increased. Compared to the Type-A basin, in the Type-B basin, TKEs remained above the top level of wedge-shaped baffle blocks. At the end of the Type-B basin, values of the TKE in the lower fluid layers were found to be dissipated. At 88 m^{3}/s, from Figure 16(c), in the Type-C basin, the maximum amount of TKEs was found in the foreside of the HJ and reduced as the distance from the HJ initiating location increased. The vertical profiles and patterns of TKEs in the Type-C basin were similar to Type-A and -B basins. However, due to the change in the baffle blocks’ geometry, the trends of TKEs between the HJ and baffle block region were found to be different in the Type-C basin. After the HJ, the maximum TKE in the Type-C basin was observed at *Z* = 3.4 m flow depth, which indicated that the TKE was shifted towards the free surface in the Type-C stilling basin.

*X*= 70 m) reached 30%. Figure 17(b) shows the TI intensity after the HJ and at the end of the Type-B basin. Soon after the HJ, the maximum amount of TI was found at

*Z*= 3.25 m, which reached 94% and gradually reduced to 91% at the free surface. At the end of the Type-B stilling basin, the TI reduced to 30% at the free surface. From Figure 17(c), in the Type-C stilling basin, the trend of the TI after the HJ was similar to the Type-B stilling basin. However, at the end of the Type-C stilling basin, the TI reached 29% (

*Z*= 4.5 m) at the free surface as shown in Figure 17(c). It is worth mentioning here that the stilling basins with wedge-shaped baffle (types-B and C) showed the identical trend of the TI, which differed from the Type-A stilling basin.

## CONCLUSIONS

This numerical study investigated the HJ and flow characteristics for modified USBR-III and wedge-shaped baffle block stilling basins using FLOW-3D. The wedge-shaped baffle block with a vertex angle of 150° and cutback angle of 90° was employed in the baffle region of the Taunsa barrage stilling basin. The study investigated the hydraulic parameters, i.e., free surface profiles, roller lengths, relative energy losses, velocity profiles, and turbulent kinetic energies. It also compared the results of the proposed stilling basins with the modified USBR-III basin and with the literature. However, the present study is limited to 44 and 88 m^{3}/s discharges and implemented only RNG *K*–*ɛ* for turbulence modelling. Additionally, the present study has only focused hydraulic parameters on the rigid basin. The following main findings are drawn from the present study:

At the studied discharges, locations of the HJ toe were noticed at the glacis, and as compared to the USBR basin, the proposed wedge baffle block basins produced smaller hydraulic jump lengths. However, the location of the HJ in wedge-shaped basins was noticed below that observed in the USBR basin. Upon comparison with the literature, the results of free surface profiles were found to be in good agreement.

The results showed that as the discharge was increased the dimensionless roller lengths reduced in the wedge-shaped basins, which indicated that the proposed stilling basins contained the rollers and flow recirculation between the HJ toe and baffle region. In comparison with the Type-A basin, the dimensionless roller lengths in Type-B and -C stilling basins agreed well with the previous studies.

In all the investigated basins, the results of forward velocity profiles showed a wall jet-like structure, which agreed well with the previous studies. However, at the higher discharge, two forward regions were noticed in Type-B and -C basins, which indicated more velocity decay in the HJ region.

At the lower flow, the relative energy in the new stilling basin was found to be equal to that noticed in the USBR basin. However, with an increase in the discharge, as compared to the USBR basin, Type-B and -C basins increased the relative energy loss, which reached 58 and 70% in Type-A and -B basins, respectively.

RNG

*K*–*ɛ*model captured the turbulent kinetic energy and turbulent intensity. In all the basins, the flow was strongly turbulent on the foreside of the HJs, which showed less turbulent flow as the distance from the jump termination point increased. As compared to Type-A and -B basins, with an increase in the discharge, in the Type-C basin, the turbulent kinetic energy and turbulent intensity were directed towards free surface. In the wedge-shaped baffle block basins, the results showed large wake zones on sides of the wedge blocks, and no flow reattachment was noticed downstream of the wedge blocks.

Conclusively, it can be said that the wedge-shaped block in combination with the USBR block produced promising results of the HJ and flow behaviour within the stilling basin of the Taunsa barrage. However, the study recommends studying these energy dissipators for higher discharges using other turbulence schemes (i.e., large eddy simulation (LES) and standard *K*–*ɛ* model) and a finer mesh. Moreover, as the present study focuses on the constant geometry of the wedge block, it is suggested to employ other vertex and cutback angles to check its suitability for the stilling basins of diversion barrages. The study further suggests developing scour models to investigate the performance of wedge-shaped baffle block stilling basins.

## FUNDING

The authors received no research grants or funding.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its supplementary information.

## CONFLICT OF INTEREST

The authors declare that there is no conflict.