## Abstract

The term ‘branching flow’ describes water extraction from streams or main channels via secondary lateral channels. Through using 3D model simulation, the aim is to identify the ideal angle of diversion based on the maximum flow rate to the branching channel and the minimum zone of separation size attained at the entrance channel for the eight grades (90°, 75°, 60°, 45°, 30°, 25°, 20°, and 15°). An experimental study has previously been confirmed, and this paper provides a comprehensive implementation of the numerical solution (finite volume) using Flow 3D version 11.0.4 software. The validation study was conducted at the Babylon University/College of Engineering/Laboratory of the fluid. The study presented results for many different flow discharge ratios depending on two inflow discharges (12.5 and 18.5 L/s). The comparison between the numerical model and the experimental results revealed statistically a good agreement. The final results demonstrated that a diversion angle of 25° had the most significant optimum angle with the maximum discharge ratio, a minimum separation zone size, and minimum energy losses. Furthermore, the flow rate peaks in the bifurcating channel (5.76 and 8.11 L/s), which accounts for roughly 46.06 and 43.83%, respectively, from the main channel flow.

## HIGHLIGHTS

This paper identifies the angle of diversion using a 3D simulation model.

The best diversion angle that achieves the maximum flow was 25°.

This paper employs the separation zone in 3D.

## INTRODUCTION

In recent decades, an enormous amount of research has been undertaken to comprehensively understand the features of branching open channels via extensive theoretical and experimental investigations. The phenomenon of branching channel flow continues to captivate the attention of researchers in the field of water resource engineering, as it is commonly seen in many projects pertaining to water engineering. Because of the complexity of branching flow involving many interlinking factors, it is difficult to generalize the occurrence (Lama *et al.* 2002). The study of open channel flows at junctions is highly intricate and has garnered significant attention in the fields of environmental and hydraulic engineering. This phenomenon is commonly found in many hydraulic systems, encompassing wastewater treatment facilities, irrigation ditches, fish passage conveyance structures, and natural river channels. Water resources engineering designers must examine several elements that impact the design of bifurcating channels, such as the main flow channel, mechanics of the riverbed, and changes in the morphology of the bed, particularly in the junction zone (Allahyonesi *et al.* 2008). Numerous issues have arisen due to these alterations, encompassing alterations in the main channel gradient due to sedimentation and erosion occurring in both the branch and the main channels. Mathematical and numerical models use one or more fundamental equations, namely the continuity, momentum, and energy equations (Abu-Zaid 2023).

A recirculation zone immediately occurs upstream of the branching channel junction, a contracted flow region at both channels as a result of the recirculation zone, a stagnation point immediately upstream of the branch channel, a shear plane formed between the two diversion flows, and an increase in depth from the downstream to the upstream contributing channels. The momentum of the lateral branch flow causes the main flow to detach at the downstream corner of the junction, resulting in the zone of separation. A significant body of research has investigated the precise hydrodynamic characteristics of complicated junction flow due to the quantity of critical flow phenomena involved.

The first thorough experimental research in an open channel was carried out by Taylor (1944) who also provided a graphical approach using a trial-and-error process. Grace & Priest (1958) studied free overflowing branching flows with varying branch-to-main bed width ratios. The researchers partitioned the fluid dynamics into two distinct states: one characterized by the absence of standing waves, applicable to flows with relatively low Froude numbers, and another marked by the presence of localized standing waves along the branch channel. In their study, Weber *et al.* (2001) conducted a comprehensive experimental investigation on the merging of flows in a 90° open channel. The primary objective of their research was to generate a comprehensive dataset encompassing three components of the velocity, turbulence stresses, and mappings to the water surface. The flow diverted into branching channels will be significant features that vary depending on the diversion angle at which the junction of the channel is diverted. These characteristics include (a) the zone of the separation near the entrance of the channel bifurcating, (b) the constricted area of the flow, and (c) the stagnation point at the corner exiting to the junction flow. Due to flow expansion, there is a chance that there will be a sizable flow split just downstream of the junction (Sayed 2019).

A numerical model for diverting flow into a right angle with a short branch channel was developed by Ramamurthy & Satish (1988), Ramamurthy *et al.* (1990), and Hsu *et al.* (2002). It assumes that energy losses along the main channel are neglected, and it incorporates the concepts of energy, momentum, and mass conservation. The analysis makes use of the similarity in flow configuration between the division of flow in a branch channel and in a two-dimensional lateral conduit outlet fitted with a barrier. This similarity of flow is used to estimate the contraction coefficient of the converging jet entering the branch channel. The ratio of the branch channel flow to the main channel flow is related to the Froude number in the main channel section downstream of the junction. In their study, Wu & Mao (2003) conducted 90° for the open channel flow numerical simulation to analyze a combining junction. They employed the Hanjalic Launder modification model (HLMM) to assess the relation significance to the different parameters and compared their findings with laboratory measurements. Chong (2006) proposed a theoretical model employing finite element methods (FEMs) to analyze the diversion to the open channel junction. Additionally, a straightforward comparison with experimental data was conducted to assess the accuracy of the velocity profiles. In addition, Table 1 presents a comprehensive overview of the numerical research investigations and their principal conclusions. The goal of this study is to make some educated guesses about the properties of the diverted flow that passes to the branching channel and determine the ideal angle of the diversion to the channels bifurcating, so it may achieve the least zone size of the separation in the intake of the branch channel, the minimum amount of energy loss that occurs at the branch channel, and the greatest flow amount ratio that goes into the branching channel.

Authors . | Cross-section . | Branching angle . | Numerical model . | Key findings . |
---|---|---|---|---|

Abdalhafedh & Alomari (2021) | Rectangular | 90° | k–ɛ (RNG) model | The numerical model and experimental data were compared, and it was found that there was a sufficient level of agreement |

Hameed Al-Mussawi et al. (2009) | Rectangular | 90° | k–ɛ model | In comparison to preceding turbulence models, the RSM demonstrated superior accuracy in characterizing the features of deviated flow |

Hassan & Shabat (2023) | Rectangular | 30°, 45°, 60°, 90° | The use of the Fluent program, in conjunction with the k–ɛ turbulence model, is employed to address the (RANS) equations | The combined numerical model validates with the past experimental data to get a velocity profile |

Mirzaei et al. (2014) | Rectangular | 90° | The Reynolds Stress Model (RSM) | The findings obtained from the RSM turbulence model exhibit a high degree of similarity to the experimental data |

Mignot et al. (2013) | Rectangular | 90°, 60° | Using the Fluent program and the k–ω models are employed for solving the RANS equations | The Fluent has been demonstrated as a valuable tool to predict patterns of flow in branch intakes |

Pandey & Mohapatra (2021) | Rectangular | 90° | k–ɛ (RNG) model | Both the bed roughness and the discharge ratio have an impact on the equivalent of Manning's roughness in a flow at a channel junction |

Nikpour & Khosravinia (2021) | Trapezoidal | 90° | The k–ω turbulence model | An investigation was conducted to compare and contrast the hydraulic parameters of flow in the main channel when subjected to a 45° side slope versus a 90° side slope |

Li & Zeng (2009) | Rectangular | 90° | 3D RANS model | A 3D RANS model simulated flowing at a T-junction open channel accurately |

Vasquez (2005) | Rectangular | 30° | The hydrodynamic model River 2D, which is a two-dimensional (2D) model, utilizes the Boussinesq model | The performance of River2D was found to be satisfactory, as indicated by the results. These findings suggest that River2D holds potential as a valuable tool for designing the preliminary branch intakes and the examination of bifurcation rivers |

Authors . | Cross-section . | Branching angle . | Numerical model . | Key findings . |
---|---|---|---|---|

Abdalhafedh & Alomari (2021) | Rectangular | 90° | k–ɛ (RNG) model | The numerical model and experimental data were compared, and it was found that there was a sufficient level of agreement |

Hameed Al-Mussawi et al. (2009) | Rectangular | 90° | k–ɛ model | In comparison to preceding turbulence models, the RSM demonstrated superior accuracy in characterizing the features of deviated flow |

Hassan & Shabat (2023) | Rectangular | 30°, 45°, 60°, 90° | The use of the Fluent program, in conjunction with the k–ɛ turbulence model, is employed to address the (RANS) equations | The combined numerical model validates with the past experimental data to get a velocity profile |

Mirzaei et al. (2014) | Rectangular | 90° | The Reynolds Stress Model (RSM) | The findings obtained from the RSM turbulence model exhibit a high degree of similarity to the experimental data |

Mignot et al. (2013) | Rectangular | 90°, 60° | Using the Fluent program and the k–ω models are employed for solving the RANS equations | The Fluent has been demonstrated as a valuable tool to predict patterns of flow in branch intakes |

Pandey & Mohapatra (2021) | Rectangular | 90° | k–ɛ (RNG) model | Both the bed roughness and the discharge ratio have an impact on the equivalent of Manning's roughness in a flow at a channel junction |

Nikpour & Khosravinia (2021) | Trapezoidal | 90° | The k–ω turbulence model | An investigation was conducted to compare and contrast the hydraulic parameters of flow in the main channel when subjected to a 45° side slope versus a 90° side slope |

Li & Zeng (2009) | Rectangular | 90° | 3D RANS model | A 3D RANS model simulated flowing at a T-junction open channel accurately |

Vasquez (2005) | Rectangular | 30° | The hydrodynamic model River 2D, which is a two-dimensional (2D) model, utilizes the Boussinesq model | The performance of River2D was found to be satisfactory, as indicated by the results. These findings suggest that River2D holds potential as a valuable tool for designing the preliminary branch intakes and the examination of bifurcation rivers |

## MATERIAL AND METHODS

### Numerical model description

In recent years, there has been rapid progress in numerical modeling, mostly driven by advancements in computational capability. This has enabled the numerical solution of a wide range of problems across many applications. Numerical modeling is a versatile tool with many practical uses. However, at its core, numerical models in various fields of study are dependent on analogous models and are developed through the formulation of partial differential equations, which serve as the mathematical framework for the given scenario. The procedure for constructing a set of algebraic equations to represent partial differential equations includes the application of various numerical methodologies, such as finite element analysis and the finite volume method (Hassan & Shabat 2023). The software utilized in this study is Flow-3D V 11.0.4, a numerical three-dimensional (3D) software that employs computational fluid dynamics (CFD) principles. FLOW-3D version 11.0.4 is a CFD software that offers a wide range of models to suit each operational scenario, including free surfaces, turbulence, multi-phase flows, porous media, and more. The software's unique Fractional Area-Volume Obstacle Representation (FAVOR) approach accurately models 3D flow while using compact computational resources. The software incorporates certain governing equations, mesh type and size, and initial and boundary conditions, as outlined in the subsequent sections.

#### CFD governing equations

The fundamental equations that govern the physics of fluid mechanics and thermal sciences, which are crucial in CFD, include the continuity equations, Navier–Stokes equations, and energy equations (AlOmari *et al.* 2018). The Navier–Stokes equations are often regarded as being among the most intricate and challenging equations to employ and answer in mathematical physics. The set of coupled time-dependent nonlinear equations under consideration necessitates the utilization of several approximations in order to get analytical solutions (Yan *et al.* 2020).

*k*is the turbulence kinetic energy, and is the water density. The equation above can be expressed more explicitly as follows:

*k*–

*ω*and

*k*– models (Abu-Zaid 2023). In the present study, the two

*k*– models were adopted. A temporary state is first employed during the computation process until a steady state is achieved, typically after around 140 s from the moment of water entry and intake. The kinetic eddy viscosity refers to the eddy viscosity that may be determined using a specific mathematical equation:where ϵ is the rate of dissipation turbulence, is constant, and

*k*is the thermal conductivity.

#### Numerical model assumptions

To simplify the problem, some assumptions were made in order to represent the flow at the junction as described. The flow was hypothesized to possess incompressibility and stability, exhibiting mean velocity components along the *u*, *v*, and *w* axes. Upon reaching an intersection, it was determined that the water depth in both the branch and the main channels was equivalent. The theory has been substantiated by experimental investigation as well as previous analytical models. The bed and the side walls of the numerical setup were observed to possess a smooth surface. To investigate their characteristics, a flume with a horizontal slope and a sharp-edged, combined flow configuration was employed.

#### Structure of the geometry

#### Boundary conditions

In all simulations, the fluid properties were calibrated to match those of water at a temperature of 20 °C. While several other physical variables might have been included, only the two relevant ones were incorporated to ensure accurate modeling of the data necessary for this inquiry. The gravity option became active when the acceleration of gravity in the *Z*-direction or vertical reached −9.81 m/s^{2}. The turbulence and the viscosity options were also involved when a suitable turbulence and Newtonian viscosity model were applied to the flow. The volume of fluid scheme is extensively employed for modeling a particular free surface phenomenon.

*x** = 0 to

*x** = −1 and

*y** = 0 to

*y** = −5, as shown in Figure 4.

### Experimental data

#### Experimental approach

The present work employed the data from Abu-Zaid (2023) to validate the numerical model. The length of the main channel was 10 m, with 30 cm width and 30 cm depth. A bifurcating channel corner division was positioned at 5.0 m downstream of the intake of the main channel. The diversion channel measures 3.0 m in length, 30 cm in width, and 30 cm in depth. A honeycomb was installed in place at the entrance of the main channel to ensure adequate flow expansion and minimal turbulence. To measure both discharges, two 30 cm wide by 10 cm deep sharp-edged weirs were built at the ends of the channels.

*H*is the headwater above the weir (cm).

All experiments were carried out under steady-flow conditions. The main discharge of the channel held constant at 16.40 L/s during all experiments. Table 2 presents the experimental measurements of the inflow and the outflow inside the main flume and the flow rate within the diversion channel.

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

90 | 16.4 | 15.96 | 0.44 | 2.68 |

75 | 16.4 | 15.63 | 0.77 | 4.69 |

60 | 16.4 | 15.02 | 1.38 | 8.41 |

45 | 16.4 | 14.51 | 1.89 | 11.52 |

30 | 16.4 | 13.25 | 3.15 | 19.21 |

15 | 16.4 | 9.10 | 7.3 | 44.51 |

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

90 | 16.4 | 15.96 | 0.44 | 2.68 |

75 | 16.4 | 15.63 | 0.77 | 4.69 |

60 | 16.4 | 15.02 | 1.38 | 8.41 |

45 | 16.4 | 14.51 | 1.89 | 11.52 |

30 | 16.4 | 13.25 | 3.15 | 19.21 |

15 | 16.4 | 9.10 | 7.3 | 44.51 |

Due to the surface of the water turbulency and the presence of a separation zone, it is challenging to measure water depths, particularly at the beginning of the branching channel. The variance in water depths across the entire channel length makes it difficult to empirically measure the separation (recirculation) zone dimensions, so it is essential to measure the zone dimensions theoretically.

#### Validating the results of the simulation

In order to assess the precision of the theoretical modeling, it is necessary to evaluate the CFD simulation results obtained using the Flow-3D V 11.0.4 program, which is based on 3D theoretical principles. This validation process should involve comparing the simulation results with experimental data for the six empirical instances. The numerical and experimental charts depicting the velocity at the inner wall of the upstream branch channel both indicate a negative value. Within the separation zone, a negative velocity denotes the occurrence of backflow directed toward the upstream region.

## RESULTS AND DISCUSSION

The CFD analysis yielded data on the flow rate departing from the main and branching channels. These findings were obtained by varying the diversion angles of the bifurcating channel, namely at 90°, 75°, 60°, 45°, 30°, and 15°. The specific values can be found in Table 3.

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

90 | 16.4 | 15.65 | 0.75 | 4.75 |

75 | 16.4 | 15.13 | 1.27 | 7.74 |

60 | 16.4 | 14.85 | 1.55 | 9.45 |

45 | 16.4 | 14.20 | 2.2 | 13.41 |

30 | 16.4 | 13.61 | 2.79 | 17.01 |

15 | 16.4 | 7.8 | 8.6 | 52.43 |

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

90 | 16.4 | 15.65 | 0.75 | 4.75 |

75 | 16.4 | 15.13 | 1.27 | 7.74 |

60 | 16.4 | 14.85 | 1.55 | 9.45 |

45 | 16.4 | 14.20 | 2.2 | 13.41 |

30 | 16.4 | 13.61 | 2.79 | 17.01 |

15 | 16.4 | 7.8 | 8.6 | 52.43 |

However, both results indicate a positive correlation between the flow rate of the branch channel and the diversion angle decreasing with the direction of flow. The collected results were subjected to statistical analysis using the *χ*^{2} test for goodness of fit and Nash–Sutcliffe model efficiency coefficient (NSE) with Excel Program. It was done to validate the consistency between the experimental and theoretical results.

*χ*

^{2}test is a statistically significant test commonly employed in the field of engineering, making it a suitable choice for the present investigation. The equation for the

*χ*

^{2}statistic is frequently referred to as follows (Williams

*et al.*2007):where is the observed values and is the expected values.

The *χ*^{2} equation was applied to the discharge data, with a significance level of *α* = 0.05 and *α* = 0.01. In addition, the correlation factors for all the results were estimated to measure the agreement between the experimental and theoretical data. Table 4 shows the statistical summary of the results.

Discharge . | Nash coefficient . | Correlation factor . | calculated . | tabulated (α = 0.05)
. | tabulated (α = 0.01)
. |
---|---|---|---|---|---|

Outflow DS main channel | 0.929 | 0.9889 | 0.090132 | 11.07 | 15.09 |

Outflow DS branch channel | 0.929 | 0.9889 | 0.210823 | 11.07 | 15.09 |

Flow rate ratio | 0.929 | 0.9887 | 1.347945 | 11.07 | 15.09 |

Discharge . | Nash coefficient . | Correlation factor . | calculated . | tabulated (α = 0.05)
. | tabulated (α = 0.01)
. |
---|---|---|---|---|---|

Outflow DS main channel | 0.929 | 0.9889 | 0.090132 | 11.07 | 15.09 |

Outflow DS branch channel | 0.929 | 0.9889 | 0.210823 | 11.07 | 15.09 |

Flow rate ratio | 0.929 | 0.9887 | 1.347945 | 11.07 | 15.09 |

DS = downstream.

*et al.*2006). As defined, it is:where is the mean of observed discharges, is the modeled discharge, and is the observed discharge at time

*t*. When a perfect model is assumed, with an estimated error variance of zero. The resultant NSE is equal to 1. On the other hand, if a model generates an estimated error variance that is equivalent to the variance of the observed time series, it leads to an NSE value of 0.0 (Nash & Sutcliffe 1970). When the NSE takes on a negative value, denoted as NSE < 0, it indicates that the observed mean is a more effective predictor than the evaluated model. Figure 5 shows the goodness of fit (NSE).

The comparison of the experimental and theoretical findings reveals that there is a high degree of concordance between the two sets of findings.

*Q** increases with a decreasing inflow rate for the same system, so it takes two inflows for the flume (12.5 and 18.5 L/s). The relationship between the volume of the diverted flow rate and the angle may be attributed to many factors, such as the way main and branching channels are connected, the surface roughness, the amount of energy losses, and velocity. As shown in Figures 6–21, it is evident that the minimum discharge ratio, maximum energy losses, and the maximum size of the recirculation zone occur at the diversion angle of 90°. Consequently, as the branch angle decreases, there is a decrease in energy losses, a reduction in the scale of the recirculation zone, and a raise in the length of contraction. Figure 38 shows the 3D depth comparison to the separation and contraction zones.

The velocity magnitude contour revealed that the maximum quantity occurs at an angle of 25° with the smallest separation zone size, after which the separation zone begins to expand at 20 and 15° depending on the method of the main channel connected to the branch channel.

It has been demonstrated by the turbulent intensity % figures that there is a positive correlation between the angle diversion and the size of the separation zone. Hence, the separation zone ends at 25° and no longer affects the discharge ratio for the subsequent grades.

The contraction zone depth upstream of the branch channel is inversely related to the separation zone depth and then rises after the zone.

The optimum diversion angle for the system is 25°, which ensures maximum discharge ratio, minimum energy losses, minimum separation zone size, and maximum contraction zone.

## CONCLUSION

The open channel flow system was numerically investigated and validated with the previous experimental data for inflow (16.4 L/s) for six diversion angles (90°, 75°, 60°, 45°, 30°, and 15°), and it showed statistically good agreement by two methods (Χ^{2} test and Nash–Sutcliffe Efficiency (NSE)), with the factors of 0.9889 and 0.92, respectively. The theoretical investigation in 3D takes eight branch angles (90°, 75°, 60°, 45°, 30°, 25°, 20°, and 15°) for two inflow discharge rates (12.5 and 18.5 L/s).

The numerical research employed a CFD simulation model to get the results, which was created using the Flow 3D V 11.0.4 software; after designing the model with the SketchUp 2023 program, the two *k*–*ɛ* equations model was adopted.

As the diversion angle decreases, the diverted flow into the branching channel is observed to increase. A positive correlation exists between the bifurcation angle and the magnitude of the zone of separation. The optimum results benefit construction works, environmental degradation, erosion, and flooding to reduce associated risks. The outcomes would be enhanced if the diversion angle were reduced by 5°.

The maximum discharge ratio *Q** was found at 25 to 15° (46.06 and 52.81, respectively) with the least impact of discharge ratio changes and the maximum contraction zone was observed at 25° with minimal energy losses, so the optimal angle for the system is 25°.

## 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.