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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°.

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  • 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.

branching angle, diversion flow, experimental investigation, numerical investigation, separation zone, validation
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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.

The construction of a branch channel to divert a percentage of the water from the main flow affects the flow of the main channel as well as the mechanics of the river bed, which results in a change in the bed shape, particularly in the transitional region, according to Allahyonesi 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).
Figure 1
Bifurcation of the open channel characteristics.
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Bifurcation of the open channel characteristics.

Figure 1
Bifurcation of the open channel characteristics.
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Bifurcation of the open channel characteristics.

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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 Qr for upstream Froude number (Fu) = 0.45 out of 45°, 60°, 75°, and 90°. According to Zhang 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.

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Experimental details

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The Fluid Laboratory at Babylon University in Iraq conducted experiments in the Civil Engineering Department. The laboratory channel was partitioned into two sections: the branch and the main channels. Figure 2 shows a schematic architecture of the experimental channel. The main canal flume was 10.0 m long, 0.3 m wide, and 0.45 m deep. The channel division corner was 5.5 m downstream of the main channel intake.
Figure 2
An illustration of the main channel (Flume) with the branch.
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An illustration of the main channel (Flume) with the branch.

Figure 2
An illustration of the main channel (Flume) with the branch.
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An illustration of the main channel (Flume) with the branch.

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The channel with the diversion is 3.2 m long, 0.3 m wide, and 0.45 m deep. The honeycomb at the Flume's start reduces turbulence and promotes smooth flow. To measure discharges, we built two 0.30 m wide, 0.08 cm deep weirs with sharp edges. Below is the calibrated weirs equation for discharge downstream main channel:
(1)
where is the total discharge measured by the flow meter (L/s) and is the water depth measured using an ultrasonic level up to a weir (cm).

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.

Table 1

Findings experimental for bifurcations angles

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 
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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

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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

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Alomari 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 as
(2)
where 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
(3)
Various terms have become accustomed to describe the Boussinesq supposition of eddy viscosity, including the Boussinesq approximation and the Boussinesq hypothesis, with flow incompressible situations being a particular emphasis (Prajapati et al. 2023).
(4)
The 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):
(5)
where ɛ is the turbulence of the dissipation rate and is a constant.

The numerical model assumptions

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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

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SketchUp 2023 developed the architectural parameters of the open channel design under investigation at several bifurcation angles. Tet/hybrid mesh with 0.5 cm mesh size is best. Figure 3 depicts the laboratory main channel in 3D and the mesh size used to study angle bifurcation (90°). The division and major channels are 3.2 and 10 m long. Connection is 5.5 m downstream of flume entrance. SketchUp offers an effective program that allows users to create and edit 3D models of various objects, including buildings, environments, and household items. It is frequently used in the domains of interior design and architecture.
Figure 3
An example of the system flow meshed at a 90° angle bifurcation.
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An example of the system flow meshed at a 90° angle bifurcation.

Figure 3
An example of the system flow meshed at a 90° angle bifurcation.
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An example of the system flow meshed at a 90° angle bifurcation.

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The conditions boundary

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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/s2. 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.

Applying the equations of mass and momentum conservation to free surface channels yields void fraction formulas. Ferziger & Perić (2002) indicate that air and water are one fluid with spatially varying characteristics near the free surface border at the highest point. Figure 4 shows the numerical work block diagram.
Figure 4
The numerical simulation outline for the system.
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The numerical simulation outline for the system.

Figure 4
The numerical simulation outline for the system.
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The numerical simulation outline for the system.

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For the rectangular portion, an open channel system was chosen. Both channels have a comparable size, measuring 0.30 m wide and 0.45 m deep. The mainstream has a flow rate of 17.3 and 12.3 L/s. The expected duct water depth at the inflow is approximately 20, 16, and 12 cm consequently. The outcome was the first average velocity of roughly 0.355 m/s at intake. The portion under investigation spans from x* = 0 to −1 and y* = 0 to −5 (Figure 5).
Figure 5
The dimensions of the numerical section.
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The dimensions of the numerical section.

Figure 5
The dimensions of the numerical section.
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The dimensions of the numerical section.

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Validating results for the system

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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.

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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.

Table 2

Numerical analysis of the system's data and outflow uses the Flow3D Program at various branching angles

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 
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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.

As a statistically significant test that is regularly used in engineering, the chi-square test is an acceptable choice for the current investigation because it is a test that is frequently utilized. According to Williams et al. (2007), the equation that represents the chi-square statistic is generally represented as follows:
(6)
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.

Table 3

The statistical results

FlowNSEFactor of correlationχ2 estimatedχ2 α = 5%χ2 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 
FlowNSEFactor of correlationχ2 estimatedχ2 α = 5%χ2 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 
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The assessment of the anticipated accuracy of hydrological simulations is performed utilizing the NSE index, which was proposed by McCuen et al. (2006). As explicitly stated, it is
(7)
where 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.
Figure 6
NSE experimental and theoretical diversion channel comparison of discharge ratio Qr.
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NSE experimental and theoretical diversion channel comparison of discharge ratio Qr.

Figure 6
NSE experimental and theoretical diversion channel comparison of discharge ratio Qr.
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NSE experimental and theoretical diversion channel comparison of discharge ratio Qr.

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Figures 7–38 provide information on various parameters in the diversion channel, including separation (recirculation) zones, contracted flow, total hydraulic head, turbulent intensity (%), velocity streamlines, and velocity magnitudes. These parameters are given for all angles (90°, 75°, 60°, 45°, 30°, 25°, 20°, and 15°) and two total inflow rates from the main channel (17 and 12.3 L/s).
Figure 7
Contours of intensity turbulent (%) for 90° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 90° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 7
Contours of intensity turbulent (%) for 90° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 90° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 8
Contours of intensity turbulent (%) for 75° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 75° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 8
Contours of intensity turbulent (%) for 75° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 75° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 9
Contours of intensity turbulent (%) for 60° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 60° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 9
Contours of intensity turbulent (%) for 60° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 60° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 10
Contours of intensity turbulent (%) for 45° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 45° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 10
Contours of intensity turbulent (%) for 45° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 45° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 11
Contours of intensity turbulent (%) for 30° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 30° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 11
Contours of intensity turbulent (%) for 30° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 30° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 12
Contours of intensity turbulent (%) for 25° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 25° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 12
Contours of intensity turbulent (%) for 25° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 25° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 13
Contours of intensity turbulent (%) for 20° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 20° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 13
Contours of intensity turbulent (%) for 20° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 20° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 14
Contours of intensity turbulent (%) for 15° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 15° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 14
Contours of intensity turbulent (%) for 15° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of intensity turbulent (%) for 15° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 15
Contours of X/vorticity for the 90° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 90° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 15
Contours of X/vorticity for the 90° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 90° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 16
Contours of X/vorticity for the 75° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 75° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 16
Contours of X/vorticity for the 75° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 75° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 17
Contours of X/vorticity for the 60° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 60° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 17
Contours of X/vorticity for the 60° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 60° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 18
Contours of X/vorticity for the 45° angle branch: (a) Qr = 12.3 L/s; (b) Qr = 17 L/s.
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Contours of X/vorticity for the 45° angle branch: (a) Qr = 12.3 L/s; (b) Qr = 17 L/s.

Figure 18
Contours of X/vorticity for the 45° angle branch: (a) Qr = 12.3 L/s; (b) Qr = 17 L/s.
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Contours of X/vorticity for the 45° angle branch: (a) Qr = 12.3 L/s; (b) Qr = 17 L/s.

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Figure 19
Contours of X/vorticity for the 30° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 30° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 19
Contours of X/vorticity for the 30° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 30° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 20
Contours of X/vorticity for the 25° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 25° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 20
Contours of X/vorticity for the 25° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 25° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 21
Contours of X/vorticity for the 20° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 20° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 21
Contours of X/vorticity for the 20° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 20° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

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Figure 22
Contours of X/vorticity for the 15° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 15° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 22
Contours of X/vorticity for the 15° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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Contours of X/vorticity for the 15° angle branch: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 23
The contours of the velocity magnitude for 90° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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The contours of the velocity magnitude for 90° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 23
The contours of the velocity magnitude for 90° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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The contours of the velocity magnitude for 90° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 24
The contours of the velocity magnitude for 75° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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The contours of the velocity magnitude for 75° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 24
The contours of the velocity magnitude for 75° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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The contours of the velocity magnitude for 75° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 25
The contours of the velocity magnitude for 60° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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The contours of the velocity magnitude for 60° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 25
The contours of the velocity magnitude for 60° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contours of the velocity magnitude for 60° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 26
The contours of the velocity magnitude for 45° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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The contours of the velocity magnitude for 45° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 26
The contours of the velocity magnitude for 45° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contours of the velocity magnitude for 45° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 27
The contours of the velocity magnitude for 30° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contours of the velocity magnitude for 30° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 27
The contours of the velocity magnitude for 30° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contours of the velocity magnitude for 30° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 28
The contours of the velocity magnitude for 25° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contours of the velocity magnitude for 25° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 28
The contours of the velocity magnitude for 25° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contours of the velocity magnitude for 25° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 29
The contours of the velocity magnitude for 20° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contours of the velocity magnitude for 20° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 29
The contours of the velocity magnitude for 20° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contours of the velocity magnitude for 20° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 30
The contours of the velocity magnitude for 15° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s
View largeDownload slide

The contours of the velocity magnitude for 15° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s

Figure 30
The contours of the velocity magnitude for 15° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s
View largeDownload slide

The contours of the velocity magnitude for 15° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s

Close modal
Figure 31
The contour of the total hydraulic head 3D for 90° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
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The contour of the total hydraulic head 3D for 90° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 31
The contour of the total hydraulic head 3D for 90° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 90° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 32
The contour of the total hydraulic head 3D for 75° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 75° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 32
The contour of the total hydraulic head 3D for 75° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 75° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 33
The contour of the total hydraulic head 3D for 60° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 60° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 33
The contour of the total hydraulic head 3D for 60° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 60° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 34
The contour of the total hydraulic head 3D for 45° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 45° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 34
The contour of the total hydraulic head 3D for 45° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 45° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 35
The contour of the total hydraulic head 3D for 30° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 30° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 35
The contour of the total hydraulic head 3D for 30° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 30° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 36
The contour of the total hydraulic head 3D for 25° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 25° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 36
The contour of the total hydraulic head 3D for 25° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 25° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 37
The contour of the total hydraulic head 3D for 20° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 20° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 37
The contour of the total hydraulic head 3D for 20° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 20° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
Figure 38
The contour of the total hydraulic head 3D for 15° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 15° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Figure 38
The contour of the total hydraulic head 3D for 15° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.
View largeDownload slide

The contour of the total hydraulic head 3D for 15° branch angle: (a) Qr = 12.3 L/s and (b) Qr = 17 L/s.

Close modal
The flow entering the branching channel increases as the diversion angle decreases. For the same system, the discharge ratio Qr 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.
Figure 39
Depths of contraction and zone of separation (example of 90°).
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Depths of contraction and zone of separation (example of 90°).

Figure 39
Depths of contraction and zone of separation (example of 90°).
View largeDownload slide

Depths of contraction and zone of separation (example of 90°).

Close modal
Based on the contour of the magnitude of the velocity, it can be seen that the largest value is found at an angle of 25° degrees, which also has the smallest recirculation zone. Afterward, the zone of separation begins to develop at an angle of 20°–15°, depending on the type of connection between the principal channel and the diversion canals. Beyond this point, the separating zone begins to expand. For instance, it can be expressed in the following manner based on the experimental data and simple and repeated linear regression analyses as shown in Table 4:
(8)
(9)
Table 4

The values of empirical coefficients according to Equations (8) and (9)

Equation No.Diversion angle (θ)
90°
75°
60°
45°
30°
a1a2a1a2a1a2a1a2a1a2
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°
a1a2a1a2a1a2a1a2a1a2
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 
View Large

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.

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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 Qr 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.

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

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The authors declare there is no conflict.

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© 2024 The Authors
This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY 4.0), which permits copying, adaptation and redistribution, provided the original work is properly cited (http://creativecommons.org/licenses/by/4.0/).