Introduction

The confluence of streams in different situations, such as river networks, irrigation and drainage channels, leads to the changes on erodible beds. The bed morphology at confluences induced by the turbulence and augmentation of shear stress, including the formation of scour holes and sediment deposition in the tranquil flow zone. Flow pattern changes due to the collision of the tributary and main channel flows are very complex and have been the subject of research by many hydraulic scholars. Best (1987) was the first researcher who presented the hydraulic of confluences and corresponding zones and distinguished six regions as presented in Fig. 1.

Fig. 1
figure 1

Flow pattern in channel junction (After Best 1987)

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Taylor (1994) studied the flow characteristics of a rectangular confluence flume theoretically using momentum equations. He investigated the theoretical accuracies by comparing with experimental results for angles of 45 and 135 degrees and showed that theoretical approaches were more correlated for small angles than for large ones.

Mosley (1976) found from laboratory experiments that the maximum scour depth in a confluence decreases by increasing sediment discharge.

Lin and Song (1979) addressed the issue of erosion at the intersection of open channels. They showed that the decrease on discharge ratios of two intersected canals, conducts to reduction on wall erosion. They categorized energy losses into two terms: frictional energy loss and turbulent energy loss.

Bryan and Kuhn (2002) studied experimentally the effect of geometry on flow characteristics and erosion patterns in small waterways. They used symmetric and asymmetric junctions with angles varying from 19 to 90 degrees. They showed that the scour hole dimensions raise by increasing the angle of two joined channels. Also, they observed that symmetry or asymmetry of the junction shape has a significant effect on the erosion pattern. Symmetric junctions conducted to symmetric scour hole and the asymmetric junctions, produced more complicated scour hole.

Ramamurthy et al. (2006) investigated the accuracy of a three-dimensional turbulence model for two perpendicular channels and found a good correlation between numerical and experimental results.

Balouchi (2014) investigated experimentally the effect of sediment discharge carried by flow on erosion pattern. He reported that for clear water condition, the erosion process attained to an equilibrium state; however, under live-bed conditions and by the presence of sediment discharge, the maximum scour hole fluctuated by time.

Nargess et al. (2017) conducted an experimental study to investigate the local scour at two intersected channels. They discovered that the depth of scouring is dependent on the flow ratio, Froude number, and confluence angle.

Ramos et al. (2019) investigate influence of bed elevation discordance between the main channel and tributary on flow pattern and head losses in a right-angled confluence of a rectangular open channel. They found that increasing the bed elevation discordance leads to introduce tributary flow into the recirculating zone and conducts more head losses in confluence zone.

Yu et al. (2019) performed experiments on morphodynamic changes of open channel confluences aimed to the implications for accumulating of contaminated sediments. They found out that the bed morphology and sediment transport pattern based on the contact angle and discharge.

Zhu et al. (2021) studied the impact of connecting angle and discharge ratio on the velocity distribution within the contraction zone. They used a 1D theoretical model and a 3D numerical model to find the relationships between the energy correction coefficient (α) and momentum correction coefficient (β) with the discharge ratio and junction angle. They found that α and β increase by increasing the connection angle and decreasing the discharge ratio.

Xia Shen et al. (2022) investigated the confluence angles and discharge ratios impact on creating and developing of secondary flows, as well as the separation region, using a numerical model. Their results revealed that twin surface-convergent secondary flows occur on both sides of the shear plane, with a low-velocity separation region forming immediately downstream of the confluence. Varying the confluence angle and discharge ratio leads to a complex reversal in the process of secondary flow formation within the separation zone and conducts the significant changes on bed morphology.

Materials and methods

Experimental setup

In this study, experimental results of Weber et al. (2001) obtained from two perpendicular rigid canals are used for verification test. Main canal and its tributary had 21.95 m and 3.57 m long, respectively. The widths of both canals were 0.914 m with horizontal beds (Fig. 2). The positive directions for X, Y and Z are toward the upstream of main channel, flow direction in tributary, and upwards, respectively. The origin coordinates is located at the junction in the corner of the upstream wall. All distances are normalized by dividing with the channel width (W). The normalized coordinates Z*, Y*, and X* are equivalent to z/w, y/w, and x/w, respectively. The measured velocities were also normalized by dividing with the mean velocity at downstream boundary.

Fig. 2
figure 2

Schematic diagram of laboratory Flume (After Weber et al. 2001)

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An Acoustic Doppler Velocimeter (ADV) was used to measure velocities and turbulence intensities. The velocity measurements were taken at each sampling location for 60 s at a sampling rate of 10 Hz. To validate the numerical model, experimental results for discharge of 0.127 and 0.043 m3/s were used.

As the experimental Weber et al.’s tests were performed on rigid boundary, the numerical model should be validated by datasets obtained on loose boundary. So, for validating the erosion pattern in numerical model, datasets of the experimental study reported by Yuan et al. (2017) were used. The flow model had a main channel and a tributary channel with 10 and 3 m in lengths, respectively. The width of main channel and its tributary at upstream of confluence was 30 cm and increased to 40 cm right after side channel (Fig. 3).

Fig. 3
figure 3

Schematic view of experimental model (After Yuan et al. 2017)

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Sediment with poor sorting was selected for the bed material with a d50 equals to 0.9 mm and d90 equals to 2.5 mm. Since d50 is much larger than 0.3 mm, there was very little sediment entrainment in the downstream flow recovery zone. Similarly, negligible sediment entrainment occurred in the two upstream channels, as their velocities were lower than the post-confluence channel. The bed was covered with a 5.5 cm-thick layer of poorly sorted sediment mixture, as shown in Fig. 3. Flow conditions and sediment characteristics are detailed in Table 1.

Table 1 Hydraulic and sediment data of experiment (Yuan et al. 2017)
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Bed elevation in the dredged zones where sediment erosion and deposition mainly occurred was measured using a laser bed profiler with an accuracy of 0.1 mm. Three-dimensional flow velocities were measured using an Acoustic Doppler Velocimeter with a sampling frequency of 100 Hz. Seeding materials consisted of neutrally buoyant hollow glass spheres (Potter Industries Spherical, Valley Forge, USA) were added to provide acoustic scatting signals for ADV during the tests.

Numerical model

The morphological pattern in a confluence is studied numerically using SSIIM 2.0. This software utilizes a finite-volume approach to discrete the differential equations of current by solving Reynolds-averaged Navier–Stokes equations using standard k–ε and k–ε turbulence model with RNG extensions. SSIIM 2.0 also utilizes the SIMPLE method to compute pressure and a power-law scheme to discrete the convective terms (Olsen 2006). The default formula for suspended sediments in SSIIM 2.0 is Van Rijn formula. Three well known bed load formulas defined in software were examined (Olsen 2006). Also, the current equations, the software computes both bed load and suspended load and profits the transient convection diffusion equation (Eq. 1) for sediment concentration c to calculate suspended load.

$$ \frac{\partial c}{{\partial t}} + u_{i} + \frac{\partial c}{{\partial x_{i} }} + w\frac{\partial c}{{\partial z}} = \frac{\partial }{{\partial x_{i} }}\left( {\Gamma_{T} \frac{\partial c}{{\partial x_{i} }}} \right) $$
(1)

in which u, w and Γ are the Reynolds-averaged velocity, sediment fall velocity and the diffusion coefficient, respectively.

Van Rijn (1987) formula (Eq. 2) is used for the cells very close to the bed to specify the concentration

$$ C_{bed} = 0.015\frac{{d^{0.3} \left[ {\frac{{\tau - \tau_{c} }}{\tau }} \right]}}{{a\left[ {\frac{{\left( {\rho_{s} - \rho } \right)}}{{\rho_{w} \nu^{2} }}} \right]^{0.1} }}^{1.5} $$
(2)

in which d, τ, τc, ρw, ρs, ν and g are sediment diameter, shear stress, critical shear stress (obtained from Shields’ diagram), density of water, density of sediment and gravitational acceleration, respectively.

The bed load, qb, is computed from formulae proposed by Van Rijn (Eq. 3).

$$ \frac{{q_{b} }}{{d_{50}^{0.3} \sqrt {\frac{{\left( {\rho s - \rho } \right)}}{{\rho_{w} }}} g}} = 0.053\frac{{\left[ {\frac{{\tau - \tau_{c} }}{\tau }} \right]}}{{d_{50}^{0.3} \left[ {\frac{{\left( {\rho s - \rho } \right)g}}{{\rho_{w} \nu^{2} }}} \right]^{0.1} }}^{2.1} $$
(3)

The standard wall function is defined by Eq. 4.

$$ \frac{U}{{u^{*} }} = \frac{1}{k}\ln \left( {\frac{30y}{{k_{s} }}} \right) $$
(4)

where ks is the bed roughness, k is the Prandtl constant equals to 0.4, and y is the distance from the wall.

Validation of the numerical model

Optimal mesh size, convenient bed roughness, turbulence model, and sediment transport formula were investigated to validate the numerical model. Sensitivity analysis on bed roughness revealed that 6d50 was the most appropriate value for simulating scour depth in a junction. Olsen proposed a bed roughness ranging from d50 to 100d50, while d50 and 3d50 have been suggested by Wu et al. (2000) and Khosronejad et al. (2007).

Coefficients of determination such as R2, MAE and RMSE Eqs. (5, 6, and 7) are used to evaluate the numerical model.

$$ R^{2} = \frac{{{\text{cov}} \left( {E_{i} - N_{i} } \right)}}{{\sigma_{Ei} \times \sigma_{Ni} }} $$
(5)
$$ {\text{MAE}} = \frac{{\sum {\left| {E_{i} - N_{i} } \right|} }}{n} $$
(6)
$$ {\text{RMSE}} = \sqrt {\frac{{\sum {\left( {E_{i} - N_{i} } \right)^{2} } }}{n}} $$
(7)

where Ei, Ni and represent experimental and numerical data, respectively. In Eq. (5), cov(Ei-Ni), σEi, and σNi denote the covariance of numerical and experimental data, standard deviation of laboratory and numerical data, respectively. The closer R2 to unity and MAE and RMSE to zero, the more accuracy achieved for numerical model.

Biron et al. (2002) suggested by comparing two grid sizes, if the difference between the hydraulic parameters obtained from a certain grid and a finer grid remains inferior than 10% in comparison with experimental model, the coarser grid is chosen for its minimum simulation costs.

In Fig. 4, the domain of flow and the corresponding meshes are shown. Boundary conditions applied for numerical model are as follows: Dirichlet boundary condition at upstream, specified by velocity, zero gradient at downstream for all variables, wall law for rough boundaries and symmetry the water surface boundary.

Fig. 4
figure 4

Optimal mesh generated for simulating channel confluences

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To find optimal mesh size for numerical model, 3 sizes of cells were examined. In Table 2, the determination coefficients are compared for three sizes of mesh. As shown in this table, for the cell sizes of 5 cm, values of R2, MAE and RMSE are higher than for the cell sizes 2.5 and 2 Centimeters. As the same determination coefficients for 2 and 2.5 of mesh sizes attained, while the simulation costs are different, the cell sizes of 2.5 cm are chosen for flow field.

Table 2 Comparison the determination parameters for three size meshes
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To determine an appropriate turbulence model, the k-ɛ turbulence model and the k-ɛ with RNG extension turbulence model were compared, and the determination coefficients, R2, MAE and RMSE are presented in Table 3. According to the results, the standard k-ɛ turbulence model has better coefficients of determination on evaluating the vertical velocity profiles compared to k-ɛ with RNG extension turbulence model.

Table 3 Determination coefficients for k-ɛ standard and k-ɛ with RNG extension turbulence model
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Figure 5 shows the vertical velocity profiles for the standard k-ɛ model. In this figure, velocity profile at different points of flow field for both experimental and numerical model are presented. As shown in this figure, a good correlation between numerical and experimental results is achieved.

Fig. 5
figure 5

Vertical velocity profiles in the main channel for Standard k- ɛ turbulence model at X* = 0 and X* = -1

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Based on a verification test conducted to determine the appropriate sediment transport formula, Van Rijn formula (Van Rijn 1987) was agreed with the measured topography. The determination coefficients, presented in Table 4. In Fig. 6, bed change on cross sections downstream of confluence extracted from experimental and mathematical models are compared. Regarding the experimental and numerical results, we can conclude that numerical model is capable to model accurately the bed changes.

Table 4 Comparison of results from the examined sediment transport formulas in the simulation
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Fig. 6
figure 6

Comparison the cross-section profiles of experimental and numerical model downstream of confluence using Van Rijn’s sediment transport formula. A X =  + 6 cm, B X =  + 15 cm, C X =  + 24 cm D X =  + 36

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Results and discussions

After verification of numerical model in the previous section, SSIIM2 software is applied to simulate the current and the changes of fluvial bed. In this section, different parameters influencing the bed changes in a confluence are investigated. These Parameters are presented in the form non-dimensional.

According to the present study, to investigate the effect of hydraulic and geometric parameters on the maximum scour depth and erosion pattern for five connecting angles: 45, 60, 75, 90, and 105 degrees, three width ratios (Br = Bt/Bc = 0.5, 0.74, and 1), and three Froude number ratios (Frr = Frt/Frc = 0.68, 2, and 3) were investigated. The subscripts r, t, and c refer to relative, tributary, and main channel, respectively.

Effect of intersecting angle on the evolution and the maximum of scour hole for Br = 0.5, Br = 0.74, and Br = 1 are presented in Fig. 7. The results showed that for all Br ratios, the maximum depth of erosion increases by increasing the intersecting angel. Furthermore, the range of variation in the maximum scour depth decreases by increasing the width ratio for different angles of connection.

Fig. 7
figure 7

Trend of scour hole for different angles and Br = 0.5, Br = 0.74, Br = 1

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Figure 8 illustrates the changes on bed morphology, erosion and deposition patterns for different angles. According to this figure, by reducing the intersecting angle, the amount and the pattern of deposition and erosion is more limited and also, the maximum scour hole moves to the intercept point of two currents. Decreasing rate in maximum scour depth is higher for the intersecting angle reduced from 105 to 60 degrees, while the trend of decreasing scour depth has a lower speed for the junction angle varied from 60 to 45.

Fig. 8
figure 8

Bed changes for different angles Qr = 0.5, Br = 0.5, a 105°, b 90°, c 75°, d 60°, e 45°

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In Table 5, the position and amount of scour hole and deposition mound are shown for Qr = 0.5 and Br = 0.5. As presented in this table, maximum scour hole and deposition mound increase about 20 times, by increasing intersection angle. The same trend could be observed for deposition mound.

Table 5 Coordinates of the maximum scour hole and deposition mound for different angles Qr = 0.5, Br = 0.5
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The effect of the intersecting angle on bed shear stress for Qr = 0.5 and Br = 0.74 was investigated and presented in Figure 9, which indicating a direct relationship between the intersecting angle and the magnitude of bed shear stress. The maximum bed shear stress for angles of 45, 60, 75, 90, and 105 degrees are 39.0, 44.0, 58.0, 74.0, and 89.0 N/m2, respectively. The bed shear stress decreases up to 56% by decreasing the angle of junction from 105 to 45 degrees.

Fig. 9
figure 9

variations in bed shear stress for different angles Qr = 0.5, Br = 0.74, a 45°, b 105°

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Influence of different ratios of Froude number (Frr=Frt/Frc), which illustrates relative flow intensity in tributary, on morphological pattern of a confluence are investigated. Figure 10 shows erosion and deposition pattern for two longitudinal profiles; one situated in the middle of the main channel, and the other near the inner wall of confluence. As shown in this figure, by increasing Frr, the depth and volume of the scour hole and the height and volume of sediment deposition are increased. Generally, increasing tributary discharge and its Froude number, leads to flow disturbance at the confluence and localized erosion.

Fig. 10
figure 10

Longitudinal profiles of bed for different Frr,Br = 0.5, θ = 90°, a in the middle, b in vicinity of inner wall

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Conclusions

In the current study, SSIIM2.0 software was used to simulate erosion and sediment deposition patterns in open channel confluences. Validation results showed that the SSIIM2.0 model could accurately simulate velocity profiles and bed changes for a 90-degree confluence. After validating the numerical model, effects of intersecting angle, width ratio, and Froude number on the erosion pattern and scour depth were investigated. Five tributary angles (i.e. 45, 60, 75, 90, and 105 degrees) were considered, along with three width ratios of 0.5, 0.74, and 1, and three discharge ratios including 0.4, 0.5, and 0.6. The following results are achieved:

  1. (1)

    Numerical results on rigid bed showed a good correlation between the location of maximum bed shear stress and the location of the scour hole. Also, by reducing the angle of junction from 105 to 45 degrees, a significant decrease in the magnitude of bed shear stress attained.

  2. (2)

    Comparing the results of different intersecting angle, the maximum scour depth decreases by 92%, 95%, and 96% for width ratios of 0.5, 0.74, and 1, respectively, by decreasing the intersecting angle from 105 to 45 degrees.

  3. (3)

    Effect of width ratio for a 105 degree and a discharge ratio equals to 0.6, showed the increase in the width ratio from 0.5 to 1, resulted 25% reduction in the maximum scour depth.

  4. (4)

    Impact of ratios of Froude number between tributary and main channel (i.e. 0.68, 2 and 3) for a 90 degree confluence and a width ratio equals to 0.5 revealed that the scour depth decreases by 30% when the Froude number ratio decreased from 3 to 0.68.

  5. (5)

    By investigating three parameters, including the angle of junction, Br and Frr, the angle of junction has the greater influences on the maximum scour depth and dimensions of the scour hole and deposition mound compared with other factors.