## ABSTRACT

This study aims to establish the optimal diversion angle for bifurcating channels to minimize separation zone size in the intake channel while maximizing discharge in the bifurcating channel through experimental and numerical investigations. The study successfully accomplished its goals by employing the Flow-3D 11.0.4 software. The software was utilized to examine the flow diversion into bifurcating channels with various diversion angles, including 900°, 750°, 600°, 450°, 300°, 250°, 200° and 150°. The experimental investigation has confirmed the theoretical predictions regarding the expected flow characteristics. The conclusive findings demonstrated that the diverted flow is most effectively impacted by a diversion angle of 25°. The study provided findings for various discharges flowing (12.3 and 17 L/s); a total of 95 runs were performed, and investigations revealed that the branching discharge depends on several interconnected parameters. It rises with an increase in the depth ratio. In subcritical flow, the main channel always has a lower water depth than the branch channel. The flowing diversion to the branch channel causes a reduction in water depth downstream of the main channel. The study found that the optimal angle for branching was 25°.

## HIGHLIGHTS

The zone of separation dimensions was estimated.

In this study, we ensured that the angle of diversion is properly graded for optimal diversion of the flow system.

Experimental data were statistically validated.

Flow-3D program was used to calculate the flow ratio.

Multiobjective optimization techniques were utilized to determine the most favorable branch angle.

## INTRODUCTION

The existence of secondary flow in channel branches greatly increases the complexity of flow characteristics compared to straight channels. The phenomenon of branching channel flow remains fascinating for water resources engineering researchers. It is common in various water engineering projects, but its complexity and multiple factors of interlinking make it difficult to generalize the phenomena (Lama *et al*. 2002; Alfatlawi & Hussein 2024a, 2024b). A principal flow channel, riverbed mechanics, and shifting bed form are some variables that water resources engineering designers must consider while designing bifurcating channels, especially in the zone of the junction (Allahyonesi *et al*. 2008). Various research studies arose for these generations, such as shifts in the main channel slope for erosion and sedimentation for the main and branching channels. A combination of both the main and branch flows creates the so-called helical flow at the branch.

*et al*. (2008), Abdalhafedh & Alomari (2021), and Alomari

*et al*. (2020). The slope changes due to erosion and sedimentation in the main and branch channels. The flow study of the branching channel focuses on discharge and regimes. Taylor (1944) first studied branch channel flow discharge measurement. He suggested a graphical trial and error method for determining free flow branching flow discharge based on experimental findings. In their study, Grace & Priest (1958) studied branching flow by altering the branch-to-main bed width ratio. They observed that the flow may be categorized into two regimes: one without standing waves that occurs when the Froude number is small, and another with local standing waves near the branch channel. Some characteristics that set a flow junction with the open channel apart are shown in Figure 1 (Best & Reid 1984).

The study of branching channel flow progressed further by investigating theoretical equations. Ramamurthy & Satish (1988), Hsu *et al.* (2002), and Ramamurthy *et al.* (1990) developed a theoretical model to explain the phenomenon of branching flow into a channel with a short and right-angle branch. Kesserwani *et al*. (2010) and Ghostine *et al*. (2013) derived their theoretical equations by modeling branching flow as lateral flow over side weirs without height. Most branching channel flow experiments have used a fixed boundary and 90° branching angle. However, only a few researchers have examined different branching angles (Keshavarzi & Habibi 2005; Al Omari & Khaleel 2012; Alomari *et al*. 2015) or considered a condition where the bed can move, such as Kerssens & van Urk (1986). Herrero *et al*. (2015) conducted a study on a flow that diverts at a right angle where the bed of sand can be moved. They witnessed the creation of a scour hole at the downstream end of the entrance to the branch channel.

The recirculation zone contracts flow for all canals immediately upstream of the junction to the branch channel, the stagnation point appears immediately at the beginning of the branching channel, and a planning shear appears between the two-flow bifurcation, and the depth increases downstream to upstream contribution channel. A zone of separation is created when the flow of the main channel separates at the corner downstream to the bifurcation due to a momentum of lateral branch flow. Because numerous important flow phenomena are involved, much research has been done to determine the precise hydrodynamics typical of intricate junction flow. Herrero Casas (2013) conducted a laboratory study and compared separation zone sizes at various diversion angles (45°, 56°, 67°, 79°, and 90°) to determine the optimal diversion angle 55° based on intake channel separation zone size depending only on the discharge ratio with total head energy, the minimum energy losses found in angle 55°, and the maximum ratio of the discharge. According to the highest branch channel discharge, the best angle for the diversion channel is 60° out of 30°, 60°, and 90° (Alomari *et al*. 2018).

Weber *et al*. (2001) exhaustively analyzed the stream flowing in a 90° angle channel. The three velocity components, turbulence stresses, and surface water map were their key objectives for a huge dataset. According to the angle diversion, a diverted flow into branching channels will have different major properties. The properties encompass (a) a separation zone in close proximity to the entry of the bifurcating channel, (b) a confined area of flow, and (c) the location of stagnation positioned at the corner where the flowing substance exits and merges with the junction flow. A significant flow split is possible immediately downstream of the junction from the expansion flow (Sayed 2019). The study by Masjedi & Taeedi (2011) found that a 45° branching channel angle resulted in the highest *Q _{r}* for upstream Froude number (

*F*) = 0.45 out of 45°, 60°, 75°, and 90°. According to Zhang

_{u}*et al*. (2009), employing a 115° branching channel from the branch flow reduces the upstream scour length compared to 150°. Finally, the branching channel system's geometry, velocity, Froude number, moment, controlling gates at the main and branch channels, and so on, make the flow complex. After this analysis, branching channel geometries, including branching angle, must be examined to determine how they affect main and branch channel flows.

## METHODS AND MATERIALS

### Experimental details

Every investigation was conducted under the conditions of flow steady. A channel's principal flow remained constant throughout studies at 12.3 and 17 L/s. Table 1 shows the experimental data of the main flume's intake and outflow and the rate flowing in the channel diversion.

Branch angle (°) . | Main channel inflow (L/s) . | Main channel outflow (L/s) . | Branch channel outflow (L/s) . | Flow ratio % . |
---|---|---|---|---|

90 | 17 | 15.96 | 1.04 | 6.12 |

12.3 | 11.74 | 0.56 | 4.55 | |

75 | 17 | 15.63 | 1.37 | 8.06 |

12.3 | 11.67 | 0.63 | 5.12 | |

60 | 17 | 15.02 | 1.98 | 11.65 |

12.3 | 10.24 | 1.26 | 10.23 | |

45 | 17 | 14.51 | 2.49 | 14.65 |

12.3 | 10.69 | 1.61 | 13.12 | |

30 | 17 | 13.25 | 3.75 | 22.06 |

12.3 | 9.78 | 2.52 | 20.49 | |

15 | 17 | 9.10 | 7.9 | 46.47 |

12.3 | 6.9 | 5.4 | 43.88 |

Branch angle (°) . | Main channel inflow (L/s) . | Main channel outflow (L/s) . | Branch channel outflow (L/s) . | Flow ratio % . |
---|---|---|---|---|

90 | 17 | 15.96 | 1.04 | 6.12 |

12.3 | 11.74 | 0.56 | 4.55 | |

75 | 17 | 15.63 | 1.37 | 8.06 |

12.3 | 11.67 | 0.63 | 5.12 | |

60 | 17 | 15.02 | 1.98 | 11.65 |

12.3 | 10.24 | 1.26 | 10.23 | |

45 | 17 | 14.51 | 2.49 | 14.65 |

12.3 | 10.69 | 1.61 | 13.12 | |

30 | 17 | 13.25 | 3.75 | 22.06 |

12.3 | 9.78 | 2.52 | 20.49 | |

15 | 17 | 9.10 | 7.9 | 46.47 |

12.3 | 6.9 | 5.4 | 43.88 |

Surface turbulence and divided zones make measuring water depths challenging, especially near diversion subchannel starting points. Practically measuring the recirculation zone is impossible since water depths vary across the channel. Therefore, the zone dimensions must be measured theoretically.

### Description of the numerical model

In recent years, there has been notable progress in numerical modeling, primarily driven by enhancements in computer power. It has facilitated the computation of numerical solutions for a wide range of problems in many domains. Numerical modeling is an exceptionally adaptable tool with a broad range of practical applications. Nevertheless, the foundation of numerical models in different research areas relies on similar models and is constructed by formulating partial differential equations. These equations provide the mathematical structure for the specific situation. The process of representing partial differential equations through mathematical equations requires the application of several numerical approaches, as explained by Hassan & Shabat (2023), such as a method of finite volume and analysis of finite elements. ‘Flow-3D 11.0.4’ is a three-dimensional (3D) computer application utilized for studying and simulating the dynamic principles of computational fluid dynamics (CFD). The CFD tool, Flow-3D version 11.0.4, is widely recognized and offers several applications for diverse scenarios. Internal properties of such models include porous media, multiple phase flows, turbulent conditions, free-standing surfaces, and other factors. Effectively representing Flow-3D with restricted computer resources is the objective of the FAVOR software. Subsequent sections of this software contain exhaustive conditions, retinal requirements, and extremely precise equations.

#### CFD governing equations

*et al*. (2018) state that energy equations, Navier–Stokes equations, and continuity equations regulate CFD. In mathematics, the Navier–Stokes basic equations are among the most difficult. Yan

*et al*. (2020) demonstrate that to obtain analytical solutions to a coherent, time-dependent nonlinear system of equations studied, several approximations must be used. The Navier–Stokes (RANS) equations with the Reynolds mean are one of the easiest to solve since they span the most regions. RANS formulas distinguish steady-state fixes from system changes. These apparent variations restore system-distinct traits. The Boussinesq assumptions supposed a tensor of the stress of Reynolds represented as , shows a correlation linearly with the less mean trace strain of tensor rate, indicated as . This formula can be conveyed aswhere is named the eddy viscosity (scalar property),

*k*is the turbulent kinetic energy, and is the density of water. The previous relation may be clearer as

*et al*. 2023).

*k*-

*ω*and models are the most widely used numerical turbulence models (Abu-Zaid 2023). In this investigation, the model was utilized. During the calculating process, a temporary state is first used until a steady state is reached, which usually happens 120 s after the water enters the intake. The eddy viscosity that may be calculated using a certain mathematical equation is known as the kinetic eddy viscosity (Alfatlawi & Hussein 2024a, 2024b):where

*ɛ*is the turbulence of the dissipation rate and is a constant.

#### The numerical model assumptions

Multiple hypotheses represent the flow at the specified junction to simplify the issue. Assume an incompressible, constant flow with average velocity components over the *u*, *v*, and *w* axes. At crossroads, they observed equal water depths in both the branch and main canals. Experimental studies and prior analytical frameworks have provided support for the notion. Through numerical configuration the side walls and beds were observed to have flat surfaces. Their properties were investigated utilizing a flume featuring a sharp-edged mixed flow configuration and a horizontal slope.

#### The structure of geometry

#### The conditions boundary

The fluid parameters for all simulations were changed to match 20 °C water. To effectively represent this search request, only the two important physical factors were considered. Gravity was enabled when the *z*-direction vertical acceleration reached −9.81 m/s^{2}. It was necessary to use turbulence and viscosity choices when choosing the right Newtonian viscosity and turbulence model for the flow. The volume of fluid approach is commonly used to explain a specific free surface impact.

*x** = 0 to −1 and

*y** = 0 to −5 (Figure 5).

### Validating results for the system

Based on the 3D theoretical principles, the results for the CFD simulation acquired with Flow-3D 11.0.4 program must be evaluated to determine the accuracy of the theoretical modeling. The validation process should include comparing the results of the simulations with the experimental data for all six scenarios in reality. Experimental and computational data show a negative velocity at the upstream branch channel's inner wall. There is backflow toward the upstream area if the velocity in the separation zone is negative.

## RESULTS AND DISCUSSION

The CFD investigation produced information regarding the rate of flow emanating for both the flume and branch channels. The results were achieved by altering the diverting angle of the dividing conduit to 90°, 75°, 60°, 45°, 30°, and 15°. Table 2 displays the precise values.

Diversion angle (°) . | Main channel inflow (L/s) . | Main channel outflow (L/s) . | Branch channel outflow (L/s) . | Flow ratio % . |
---|---|---|---|---|

90 | 17 | 15.97 | 1.03 | 8.22 |

12.3 | 11.61 | 0.69 | 5.65 | |

75 | 17 | 15.22 | 1.78 | 10.45 |

12.3 | 11.4 | 0.9 | 7.32 | |

60 | 17 | 15.02 | 1.98 | 13.75 |

12.3 | 10.76 | 1.54 | 12.53 | |

45 | 17 | 14.15 | 2.85 | 16.76 |

12.3 | 10.44 | 1.86 | 15.12 | |

30 | 17 | 12.74 | 4.26 | 25.06 |

12.3 | 9.53 | 2.77 | 22.5 | |

15 | 17 | 7.86 | 9.14 | 53.77 |

12.3 | 6.61 | 5.69 | 46.21 |

Diversion angle (°) . | Main channel inflow (L/s) . | Main channel outflow (L/s) . | Branch channel outflow (L/s) . | Flow ratio % . |
---|---|---|---|---|

90 | 17 | 15.97 | 1.03 | 8.22 |

12.3 | 11.61 | 0.69 | 5.65 | |

75 | 17 | 15.22 | 1.78 | 10.45 |

12.3 | 11.4 | 0.9 | 7.32 | |

60 | 17 | 15.02 | 1.98 | 13.75 |

12.3 | 10.76 | 1.54 | 12.53 | |

45 | 17 | 14.15 | 2.85 | 16.76 |

12.3 | 10.44 | 1.86 | 15.12 | |

30 | 17 | 12.74 | 4.26 | 25.06 |

12.3 | 9.53 | 2.77 | 22.5 | |

15 | 17 | 7.86 | 9.14 | 53.77 |

12.3 | 6.61 | 5.69 | 46.21 |

These two results show that the diversion angles and the branch channel flow rate are directly related, which decreases as direction flow shifts. With the assistance of the Excel program, the gathered data were subjected to statistical analysis. The analysis used the chi-square test for goodness-of-fit and the Nash–Sutcliffe efficiency (NSE) coefficient. The goal was to verify the consistency of the theoretical and experimental results.

*et al*. (2007), the equation that represents the chi-square statistic is generally represented as follows:where is the observed value and is the expected value.

The discharge data were analyzed using the chi-square equation, with significant levels of *α* = 5 and 1%. Furthermore, the link between the factors influencing each outcome was assessed to determine the level of convergence among theoretical and experimental data. Table 3 presents a statistical summary of the results.

Flow . | NSE . | Factor of correlation . | χ^{2} estimated
. | χ^{2} α = 5%
. | χ^{2} α = 1%
. |
---|---|---|---|---|---|

Main channel DS of outflow | 95.9% | 98.76% | 0.09543 | 12.18 | 16.78 |

Branch channel DS of outflow | 95.9% | 99.10% | 0.2251 | 12.18 | 16.78 |

Ratio of flow rate | 95.9% | 99.91% | 1.4712 | 12.18 | 16.78 |

Flow . | NSE . | Factor of correlation . | χ^{2} estimated
. | χ^{2} α = 5%
. | χ^{2} α = 1%
. |
---|---|---|---|---|---|

Main channel DS of outflow | 95.9% | 98.76% | 0.09543 | 12.18 | 16.78 |

Branch channel DS of outflow | 95.9% | 99.10% | 0.2251 | 12.18 | 16.78 |

Ratio of flow rate | 95.9% | 99.91% | 1.4712 | 12.18 | 16.78 |

*et al*. (2006). As explicitly stated, it iswhere is the discharge mean observed, is the discharge modeled, and is the discharge observed for any time

*t*. The NSE resultant should be equal (1) when the variance error prediction is zero, and it is assumed as a perfected modeling. Conversely, according to Nash & Sutcliffe (1970), the NSE value is set to 0 when modeling produces a calculated variance of error equal to the observed time series variances. A negative result for the NSE, shown as NSE less than 0, indicated that the mean observed outperforms the tested model as a predictor. There is a significant agreement between the two sets of findings when the experimental and theoretical findings are compared. A goodness of fit (NSE) is displayed in Figure 6.

*Q*grows as input drops, requiring two flume inflows (12.3 and 17 L/s). Connection between branch and main channels, surface roughness, energy losses, and velocity can alter the diverted quantity, angles, and flow rate. As shown in the figures above, it is clear that the lowest discharge ratio, highest energy losses, and largest size of zone of recirculation happen at an angle of 90°. With reducing branch angles, there is a decrease in the loss of energy, recirculating zone region, and contraction duration. The experimental study representation of the separation and contraction zones is shown in Figure 39, which contrasts their depths.

_{r}Equation No. . | Diversion angle (θ). | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

90° . | 75° . | 60° . | 45° . | 30° . | ||||||

a_{1}
. | a_{2}
. | a_{1}
. | a_{2}
. | a_{1}
. | a_{2}
. | a_{1}
. | a_{2}
. | a_{1}
. | a_{2}
. | |

Equation (8) | −18.34 | 38.34 | −19.76 | 42.12 | −33.93 | 52.73 | −99.53 | 101.47 | −110.57 | 112.20 |

Equation (9) | −24.22 | 42.15 | −36.33 | 47.51 | −71.11 | 67.34 | −141.90 | 157.23 | −161.59 | 131.31 |

Equation No. . | Diversion angle (θ). | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

90° . | 75° . | 60° . | 45° . | 30° . | ||||||

a_{1}
. | a_{2}
. | a_{1}
. | a_{2}
. | a_{1}
. | a_{2}
. | a_{1}
. | a_{2}
. | a_{1}
. | a_{2}
. | |

Equation (8) | −18.34 | 38.34 | −19.76 | 42.12 | −33.93 | 52.73 | −99.53 | 101.47 | −110.57 | 112.20 |

Equation (9) | −24.22 | 42.15 | −36.33 | 47.51 | −71.11 | 67.34 | −141.90 | 157.23 | −161.59 | 131.31 |

Given that the turbulence intensity percentage values show a positive relationship between the zone of recirculation size and angle diversion, the zone of separation almost finishes at 25° and then stops affecting the ratio of the discharge for grades beyond that. The depth of the contracting region upstream of the lateral channel grows as it moves away from the zone, and this increase is opposite to the depth of the separation zone. The best diverting angle to the entire system is 25°, which results in a higher discharging percentage, a system with the smallest energy losses, the shortest separation zone area, and a massive contraction zone.

## CONCLUSION

A computational study and an experimental examination were conducted to validate the open channel flow system. The study used 12.3 and 17 L/s inputs for six diversion angles (90°, 75°, 60°, 45°, 30°, and 15°). Eight distinct branch angles are used in the 3D theoretical analysis: 90°, 75°, 60°, 45°, 30°, 25°, 20°, and 15°.

1. The results showed statistically significant agreement using the chi-square test and the NSE, with respective parameters 99 and 95.9%.

2. A CFD simulation model was used in the numerical research to obtain the results.

3. The model was designed with the program SketchUp 2023 and used the two equations of

*k-ɛ*. The building model used the Flow-3D 11.0.4 software.4. The diverted flow into the branching channel increases as the diversion angle decreases.

5. The bifurcation angle correlates positively with the quantity of the recirculation zone.

6. Optimal outcomes are achieved by mitigating risks associated with construction projects, environmental degradation, erosion, and flooding. The effects would be intensified if the diverted angle were reduced with 5°.

7. The highest

*Q*is discovered between 25° and 15° (52.17 and 58.23, respectively) with the most negligible influence of ratio discharge variations, and the maximization zone of contraction is 25° with the most minor losses of energy, indicating that 25° is the ideal angle for the system._{r}

## 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.