Numerical Simulation of Turbulent Flow in River Bends and Confluences Using the k-ω SST Turbulence Model and Comparison with Standard and Realizable k-ε Models

Abstract

River bends and confluences are critical features in fluvial environments where complex flow patterns, including secondary currents, turbulence, and surface changes, strongly influence sediment transport, river morphology, and water quality. The accurate prediction of these flow characteristics is essential for hydraulic engineering applications. In this study, we present a numerical simulation of turbulent flow in river bends and confluences, with special consideration given to the dynamic interaction between free-surface variations and closed-surface constraints. The simulations were performed using OpenFOAM, an open-source computational fluid dynamics (CFDs) platform, with the k-ω SST (Shear Stress Transport) turbulence model, which is well-suited for capturing boundary layer behavior and complex turbulence structures. The finite volume method (FVM) is used to simulate and examine the behavior of the secondary current in channel bends and confluences. Two sets of experimental data, one with a sharply curved channel and the other with a confluent channel, were used to compare the numerical results and to evaluate the validity of the model. This study focuses on investigating to what extent the k-ω SST turbulence model can capture the effects of secondary flow and surface changes on flow hydrodynamics, analyzing velocity profiles and turbulence effects. The results are validated against experimental data, demonstrating the model’s ability to reasonably replicate flow features under both free- and closed-surface conditions. This study provides insights into the performance of the k-ω SST model in simulating the impact of geometrical constraints on flow regimes, offering a computationally robust and reasonable tool for river engineering and water resources management, particularly in the context of hydraulic structure design and erosion control in curved and confluence regions.

1. Introduction

The river’s bends and confluences are among the most critical hydraulic structures in natural water courses. Their complex flow dynamics, characterized by the interaction between secondary currents, turbulence, and sediment transport, make them essential areas of study for river engineering and geomorphology. Understanding these complex flow structures is vital for designing flood control measures, hydraulic structures, and effective riverbank protection systems, as well as for predicting morphological changes in river systems.

The flow in river bends is influenced by secondary currents, which are induced by the imbalance of centrifugal forces and pressure gradients along the bend curvature. These secondary flows significantly affect sediment transport, erosion, and deposition patterns, leading to dynamic channel changes over time [1]. Additionally, at river confluences, flow dynamics are further complicated by the merging of two streams, resulting in mixing layers, flow separation, and the development of shear zones. This convergence of flows often leads to highly turbulent regions and complex surface interactions that influence sediment dynamics and water quality [2,3].

Various studies have been performed on physical models and experimental studies to analyze flow structures in river bends and confluences (e.g., [4,5,6,7]). However, physical modeling can be time-consuming and cost-prohibitive, particularly for large-scale river systems. Advances in computational fluid dynamics (CFDs) have provided researchers with robust numerical tools to simulate these complex hydrodynamic systems with high accuracy. Among these tools, OpenFOAM has emerged as a powerful open-source CFD platform capable of solving a wide range of turbulent flow problems in natural and engineered systems [8].

The k-ω Shear Stress Transport (SST) turbulence model, developed by Menter [9], has proven particularly effective for simulating boundary layer flows and resolving complex turbulent structures in both internal and external flows. The k-ω SST model combines the advantages of the k-ε model in free stream regions and the k-ω model in near-wall regions, making it well-suited for accurately capturing the secondary currents and turbulent stresses in river bends and confluences [9,10].

Romanova et al. [11] explored the calibration of the k–ω SST turbulence model to improve the accuracy of 3D simulations of dense turbulent flows on slopes.

Du et al. [12] tackled the overprediction of flow separation in 3D shock-separated flows using conventional eddy-viscosity turbulence models due to poor Reynolds stress predictions. This study focuses on numerically simulating turbulent flow in river bends and confluences using the k-ω SST turbulence model within the OpenFOAM framework. Two solvers were employed: pisoFoam, which is primarily used for incompressible and turbulent flows in closed-channel scenarios (also referred to as the rigid-lid model, RLM), and interFoam, which is mainly utilized for incompressible and turbulent flows in open-channel cases (also known as the Free-Surface Model, FSM). The present study examines the accuracy of the k-ω SST model compared to the standard and realizable k-ε models of Shaheed et al. [13]. The objective is to develop a comprehensive understanding of the performance of the k-ω SST model in simulating the effect of geometric and surface constraints on the hydrodynamics of critical flow features, such as secondary flow caused by centrifugal forces and momentum redistribution, with practical applications in simulating sediment transport problems, hydraulic design, and riverbank erosion control.

2. Numerical Simulations

In this paper, a 3D approach is employed for flow simulation, which involves solving the full three-dimensional Navier–Stokes equations under the assumption of incompressible and turbulent flow. This method is essential for capturing complex flow features like secondary currents, vertical shear, and flow separation. The Shear Stress Transport (SST) k-ω model was developed by Menter [9] as a combination of the standard k-ω model and a modified k-ε model. Its primary distinction lies in calculating turbulent viscosity, specifically to account for the transport of the primary turbulent shear stress. This model includes a cross-diffusion term in the ω equation and a blending function that enables accurate calculations across both near-wall and far-field regions. The blending function activates the standard k-ω model near walls and switches to the k-ε model in areas farther from surfaces. The core idea behind the Shear Stress Transport (SST) turbulence model is to combine the strengths of the k-ϵ and k-ω models, applying them in different regions of the flow domain. As previously mentioned, the k-ϵ model has difficulty accurately capturing energy dissipation in areas where viscous forces dominate, such as viscous sublayers or flow fields with adverse pressure gradients. On the other hand, the k-ω model performs better in these situations, but it is highly sensitive to freestream turbulent kinetic energy conditions at the inlet. By blending these two models, the SST model leverages its complementary strengths. The reader is referred to Menter [9] and Mohammadian et al. [14] for more details.

3. Verification of the Numerical Model

3.1. Curved Channel (Open-Channel (Free-Surface Model) and Closed-Channel (Rigid-Lid Model) Cases)

One of the most influential experimental studies on curved channels was conducted by Rozovskii [15], which provided comprehensive measurements of velocity fields and water surface profiles in a 180-degree sharply curved channel. This research has served as a fundamental benchmark for validating numerical models that simulate flow behavior in meandering and curved open-channel systems. To assess the reliability of computational approaches, Rozovskii’s experiment was replicated in a numerical model, with the channel configuration illustrated in Figure 1. The experimental setup included the following:

• A 6 m long straight approach section allows the flow to stabilize before entering the curved reach.
• A 180-degree bend with an average radius of curvature of 0.8 m is where strong secondary currents and transverse velocity gradients are developed due to centrifugal forces and pressure redistribution.
• A 3 m long straight outlet section facilitates the transition of flow back to a more uniform velocity distribution after exiting the bend.

The channel had a rectangular cross-section, maintaining a constant width of 0.8 m, and was positioned on a horizontal bed (Figure 1), eliminating any slope effects on the flow. The study provided detailed velocity measurements at multiple cross-sections along the channel, capturing the evolution of the flow structure throughout the bend. These measurements included longitudinal velocity profiles, secondary flow patterns, and surface elevation variations, offering valuable insights into the complex hydrodynamics of curved channels. Rozovskii’s findings are particularly important for understanding momentum redistribution, secondary current formation, turbulence characteristics, and flow separation in curved open-channel systems. These insights have significant implications for river engineering, sediment transport modeling, and hydraulic structure design. The key hydraulic parameters and geometric characteristics from Rozovskii’s experiment are summarized in

Table 1. Hydraulic conditions and geometric dimensions of the curved channel [15].

Variable	Symbol	Value
Upstream discharge	$Q$	0.0123 m3/s
Downstream water depth	$h$	0.058 m
Channel width	$B$	0.8 m
Bend angle	$\theta$	180°
Internal radius of curvature	$r_i$	0.4 m
Mean radius of curvature	$r_c$	0.8 m
Mean radius to width ratio	$r_c/B$	1.0
Chézy factor	$C$	60
Mean velocity at downstream	${ U}_1$	0.265 m/s
Tailwater Froude number	${Fr}_d$	0.35

, providing a crucial dataset for numerical model calibration and validation.

${Fr}_d=$ ${ U}_1/\sqrt{gh}$ where ${ U}_1$ is the mean velocity downstream; $g$ is the acceleration due to gravity (approximately 9.81 m/s²); and $h$ is the downstream water depth.

3.1.1. Boundary Conditions and Mesh Generation

Four distinct boundary conditions were applied to ensure realistic flow behavior and turbulence representation:

• Inlet Boundary Conditions:

  o

  At the channel inlet, a uniform velocity profile was imposed.

  o

  For the open-channel case, the inflow depth was assigned based on upstream conditions, while for the closed-channel case, a no-slip condition was maintained at the rigid-lid surface.

• Outlet Boundary Conditions:

  o

  At the channel outlet, a zero-gradient condition was applied for velocity and turbulence parameters, allowing the flow to exit freely without artificial disturbances.

  o

  To prevent numerical reflections, an outflow boundary was used, ensuring a smooth transition of velocity and pressure fields.

• Wall Boundary Conditions (Sidewalls and Bottom):

  o

  The channel walls (sidewalls and bottom) were modeled as no-slip boundaries, meaning that velocity components at the solid surfaces were set to zero.

  o

  A wall function approach was applied to resolve the near-wall turbulence effects.

• Free-Surface or Symmetric Boundary Condition:

  o

  In the closed-channel case, the top boundary was treated as a symmetric boundary condition, which assumes that there is no velocity gradient across the surface (Figure 2). This mimics a rigid-lid approximation, where surface deformations are neglected.

  o

  In the open-channel case, the free surface was treated as an atmospheric boundary, meaning that the pressure was set to zero pressure, allowing surface fluctuations to be captured (Figure 3).

Mesh sensitivity analysis was conducted in this study. Specifically, simulations with at least three different mesh resolutions (coarse, medium, and fine) were performed. The key flow parameters were compared across these meshes to evaluate the impact of grid refinement on the numerical solution. The results showed that differences between the medium and fine meshes were negligible, confirming mesh independence.

To improve the accuracy of the numerical model, a refined computational mesh was employed, as shown in Figure 4. The mesh was carefully structured to enhance resolution in key flow regions:

• Near the banks of the channel, where velocity gradients and secondary currents are most pronounced.
• Within the bend, where flow separation and turbulence intensification occur due to curvature-induced pressure redistribution.

This mesh refinement strategy significantly improved the resolution of flow structures, leading to more accurate velocity profiles, turbulence characteristics, and secondary current predictions.

3.1.2. Analysis and Discussion of Results for the Curved Channel

Figure 5 presents a comparison of transverse velocity profiles between experimental data and numerical simulations, along with the experimental results from Rozovskii, for a 90-degree section, where $\zeta$ is the dimensionless elevation ($z/h$) and $h$ is the water depth, $u_{\mathrm{t}}$ is the transverse velocity, and $U_1$ is the mean velocity. Based on the results shown in Figure 5, there is an agreement between Rozovskii’s data and the observed numerical values throughout the depth. The k-ω SST in both the RLM and FSM demonstrate good alignment with experimental data, particularly in the lower half of the channel’s cross-section. To evaluate the level of agreement, the root mean square error (RMSE) and the coefficient of determination (R²) were calculated, as presented in

Table 2. Sectional RMSE and R² values of transverse velocity from Figure 5.

	RMSE	R2
The models	Rozovskii experiment	Rozovskii experiment
Standard k-ε model [13]	0.0571	0.9517
Realizable k-ε model [13]	0.0642	0.9776
k-ω SST (RLM) model	0.1262	0.9264
k-ω SST (FSM) model	0.0891	0.9095

. While the results for both models are similar, slight differences were observed in their RMSE and R² values. Both models exhibit relatively low RMSE and high R² values, highlighting their performance in predicting flow dynamics with an overall good degree of accuracy. The low RMSE values indicate low deviations between the numerical and experimental results, while the high R² values confirm a good correlation between the simulated and observed data. These findings suggest that both models effectively capture the essential characteristics, velocity distributions, and overall flow behavior within the simulated domain. However, slight variations in the RMSE and R² values between the models may indicate differences in their ability to resolve specific flow features, such as secondary currents and shear layers. Overall, while the k-ω SST model’s accuracy is reasonable, the realizable and standard k-ε models perform better than the k-ω SST model. The realizable k-ε model of Shaheed et al. [13] presents the highest R² and the RSME values of the standard and realizable k-ε models of Shaheed et al. [13], which are lower than the k-ω SST model. Allowing free-surface variations in the k-ω SST model leads to a significant reduction in the RMSE value and a moderate reduction in the R² value.

3.1.3. Velocity Patterns and Flow Velocity Properties

Figure 6 presents the velocity magnitude pattern at the top of the curved, closed channel, while Figure 7 shows the velocity magnitude distribution at the surface in the curved, open channel after the flow reaches a steady state. As expected, in both cases, the flow behavior is influenced by curvature-induced secondary currents, leading to noticeable velocity redistribution along the bend. The k-ω SST model can reasonably simulate the expected flow features as explained in the following. At the entrance of the bend, the maximum velocity initially shifts toward the inner bank due to the development of centrifugal forces and pressure gradients. In the closed-channel scenario (Figure 6), the rigid-lid assumption constrains surface movement, leading to different flow structures compared to the free-surface case. These differences are particularly evident in the velocity magnitudes. In the open-channel case (Figure 7), free surface deformation plays a significant role in modifying the velocity pattern, as surface gradients contribute to the redistribution of momentum.

The depth-averaged velocity across the width of the channel is illustrated in Figure 8, where $U$ is the depth-averaged velocity magnitude:

$$U={\displaystyle \frac{1}{h}} {\int }_0^h\sqrt{u_x^2+u_y^2+u_z^2 }dz$$

and $u_x, u_y, \mathrm{a}\mathrm{n}\mathrm{d} u_z$ are the longitudinal, vertical, and lateral velocities, respectively. Note that the depth-averaged velocity is non-dimensionalized by $U_1,$ which is the downstream mean velocity. The horizontal axis represents the radial distance $(r-r_i)/B$, non-dimensionalized by the channel width, where $B$ is the channel width, $r$ is the radial distance and $r_i$ is the radius of curvature for the inner bank. The velocity distribution within the bend was analyzed by examining multiple cross-sections along the channel, specifically at 0°, 35°, 65°, 100°, and 143° (see Figure 1). The results indicate that both numerical models produced comparable velocity patterns, demonstrating a good level of agreement. The model performance in simulating the expected general flow features is reasonable, as explained in the following. Due to centrifugal forces and the formation of secondary flows, at the initial stages of the bend, the flow exhibits an increase in velocity near the inner bank, while a decrease in velocity is observed near the outer bank due to adverse pressure gradients, as expected. This velocity pattern persists throughout most of the bend, highlighting the characteristic redistribution of momentum in curved channels. However, as the flow approaches the final section of the bend (cross-section 186°), the velocity pattern undergoes a reversal. The flow near the inner bank begins to decelerate while the velocity near the outer bank increases, suggesting a shift in momentum redistribution as the flow transitions toward the downstream straight section. Both numerical models showed reasonable agreement with previous computational studies and experimental data, capturing the essential flow dynamics with minor variations. These discrepancies were more pronounced near the inner and outer banks. Despite these small variations, the overall accuracy of both models confirmed their effectiveness in simulating essential features of flow in curved channel hydrodynamics. To further assess the performance of the numerical models, the index root-mean-squared error (RMSE) was used to quantify the agreement between the two models and the experiment.

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\sum _{i=1}^n{\displaystyle \frac{{(y_i^{^}-y_i)}^2}{n}}}$$

Here, $n$ is the number of observations, $y_i^{^}$ represents the predicted values, and $y_i$ represents the observed values.

Table 3. RMSE estimation for resultant velocities in Figure 8.

θ (°)	Models
	Standard k-ε Model (RLM) [13]	Realizable k-ε Model (RLM) [13]	Lien et al. [21]	Yeh and Kennedy [22]	Song et al. [23]	Johannesson [24]	Standard k-ε Model (FSM) [25]	Realizable k-ε Model (FSM) [25]	k-ω SST Model (RLM)	k-ω SST Model (FSM)
0	0.0972	0.1706	0.1462	0.1515	0.1555	0.1663	0.0848	0.0869	0.0973	0.0878
35	0.0757	0.1741	0.1163	0.1790	0.1026	0.0782	0.0718	0.0948	0.0894	0.0813
65	0.0733	0.1668	0.1063	0.1771	0.0963	0.1070	0.0846	0.1174	0.0857	0.0587
100	0.0855	0.1635	0.1020	0.1287	0.0731	0.1339	0.1063	0.1388	0.0998	0.1092
143	0.0975	0.1574	0.1255	0.0476	0.0887	0.0866	0.1618	0.1903	0.2216	0.1761
186	0.1183	0.1262	0.0978	0.2838	0.1514	0.1682	0.1917	0.2297	0.2463	0.3385
MeanRMSE	0.0912	0.1597	0.1157	0.1613	0.1112	0.1234	0.1168	0.1429	0.1400	0.1419

shows the sectional RMSE of the resultant velocity for each model. The results of the K-omega SST model exhibited a noticeable convergence between the RLM and FSM cases, particularly in the first three sections, where their performance was highly comparable. However, a slight divergence was observed in the last two sections. Furthermore, the RMSE values for both cases were relatively close, indicating similar accuracy. Nevertheless, the RMSE value for the RLM case was marginally lower, suggesting slightly better predictive performance compared to FSM. The R² values, as presented in

Table 4. R² values of resultant velocities shown in Figure 8.

θ (°)	Models
	Standard k-ε Model (RLM) [13]	Realizable k-ε Model (RLM) [13]	Standard k-ε Model (FSM) [25]	Realizable k-ε Model (FSM) [25]	k-ω SST Model (RLM)	k-ω SST Model (FSM)
0	0.9668	0.9272	0.8939	0.8994	0.9536	0.8947
35	0.9895	0.8383	0.9897	0.99	0.9895	0.9874
65	0.9892	0.8575	0.994	0.9922	0.9847	0.9821
100	0.9948	0.8677	0.993	0.9916	0.975	0.9868
143	0.9637	0.884	0.8859	0.8956	0.4052	0.6142
186	0.7494	0.7406	0.5116	0.5062	0.867	0.7602
Mean R2	0.9422	0.8525	0.878	0.8791	0.8625	0.8709

, indicate a convergence between the RLM and FSM cases, with only a slight variation observed in the last two sections. Notably, in all the cross-sections analyzed, both RLM and FSM models demonstrated superior predictive accuracy for the inner curve. However, overall, the performance of the k-ε models, as identified by Shaheed et al. [13], is better than the k-ω SST model.

3.1.4. Variations in Velocity Distribution

Figure 9 illustrates the contour lines representing the distribution and variations in longitudinal and vertical velocities throughout the bend. As expected, the longitudinal velocity initially reached its maximum near the inner bank at the entrance of the bend. This behavior is primarily influenced by longitudinal pressure gradients that develop near the inner bank in sharply curved channels. These pressure gradients arise due to the transition from a straight to a curved channel, leading to corresponding changes in the flow pattern. As the flow continues, the longitudinal velocity gradually decreases as it moves toward the middle of the bend and the bend’s exit. Overall, there is a good similarity in longitudinal velocity distribution between open and closed channels. The vertical velocity reaches its maximum near the center of the bend and close to the inner bank before gradually decreasing towards the bend’s exit. This trend is consistent between open and closed channels.

3.2. Confluent Channel (Closed-Channel (Rigid-Lid Model) Case)

For the confluence channel scenario, the experimental study conducted by Shumate [26] served as a key reference for investigating the flow dynamics within a confluent open-channel system. In this study, Shumate meticulously measured the velocity components using an acoustic Doppler velocimeter (ADV) and determined the water surface elevations with a point gauge to capture detailed flow characteristics. Among the six different discharge scenarios investigated in Shumate’s experiments, the combination where the main channel discharge ($Qm$) was 0.043 m³/s, and the branch channel discharge ($Qb$) was 0.127 m³/s was selected for analysis in this work. This specific discharge configuration was chosen because it exhibited pronounced secondary current structures, making it particularly suitable for highlighting the impact of secondary circulation within the confluence zone. This well-documented experimental setup was adopted to ensure a robust validation framework for evaluating the performance of the numerical model in replicating complex 3D flow patterns and secondary flow structures typically observed at channel confluences. This experimental setup consists of a branch channel (denoted by the subscript b) with a length of approximately 3 m, merging at a 90° angle with a main channel (denoted by the subscript m) that extends for approximately 10 m. The confluence geometry represents a common fluvial feature where flow interaction, turbulence generation, and momentum exchange occur between the merging streams. As depicted in Figure 10, the branch channel introduces flow into the main channel, creating complex hydrodynamic effects such as velocity redistribution, flow separation, shear layer formation, and secondary circulation patterns at the confluence. These flow structures are influenced by parameters such as discharge ratios, momentum flux, turbulence intensity, and bed shear stresses, all of which play a crucial role in determining the overall flow behavior at the junction. The detailed geometric and hydraulic parameters of the confluent channel, including flow rates, velocity distributions, and depth variations, are summarized in

Table 5. The dimensions of the confluent channel and flow conditions [26].

Variable	Symbol	Value
Main channel discharge	$Q_m$	0.043 m3/s
Branch channel discharge	$Q_b$	0.127 m3/s
Downstream water depth	$h$	0.296 m
Channel width	$B$	0.914 m
Chézy factor	$C$	60
Mean velocity at downstream	$U_1$	0.628 m/s
Tailwater Froude number	${Fr}_d$	0.37

. This dataset provides essential benchmark conditions for validating numerical simulations and assessing the performance of turbulence models in replicating real-world confluent flow characteristics.

3.2.1. Boundary Conditions and Mesh Generation

The boundary conditions for the confluent channel, as illustrated in Figure 11, were established to match those applied in the curved channel (closed-channel case), given the similarities in boundary characteristics. These conditions include the following:

• Inlet Boundaries: Defined for both the main and tributary channels where uniform velocity values were imposed.
• Outlet Boundary: A pressure-outlet condition was applied to ensure a smooth flow exit while minimizing numerical reflection effects.
• Wall Boundaries: The channel sidewalls and bottom were treated as no-slip boundaries, incorporating appropriate wall functions to model turbulence effects accurately.
• Surface Treatment: Since the confluent channel was simulated under closed-surface conditions, the water surface was modeled as a symmetrical boundary, assuming negligible vertical velocity gradients.

The computational mesh, shown in Figure 12, was carefully refined to improve the resolution of velocity gradients, particularly in critical flow regions such as the following:

• The junction zone, where complex shear layers, velocity mixing, and turbulence structures develop due to the interaction between the main and tributary flows.
• The banks, where velocity gradients and near-wall turbulence require higher resolution to capture flow separation and recirculation effects accurately.
• The transition zones between the branch and main channel, where refined meshing helps reduce numerical diffusion and enhance solution accuracy.

This refined mesh structure ensured that the numerical model could accurately predict velocity redistribution, turbulence generation, and flow mixing at the confluence, providing reliable results for hydrodynamic analysis.

3.2.2. Velocity Patterns and Flow Velocity Properties

Figure 13 presents the velocity magnitude distribution at the top of the confluent channel after the flow reaches a steady-state condition. The velocity field reveals a distinct pattern influenced by the interaction between the branch channel and the main channel. As expected, the key observations in the velocity pattern are as follows:

• Flow Convergence and Acceleration:

  o

  As the flow from the branch channel merges with the main channel, a noticeable increase in velocity occurs at the junction due to the combined discharge.

  o

  The momentum from the branch flow causes a shift in the velocity field, influencing the overall flow dynamics in the main channel.

• Velocity Redistribution at the Confluence:

  o

  The maximum velocity is observed near the outer bank of the main channel, which results from the deflection of the incoming branch flow.

  o

  This redirection occurs due to the formation of a strong shear layer between the merging flows, leading to enhanced turbulence and mixing at the junction.

• Shear Layers and Flow Separation:

  o

  A distinct velocity gradient forms along the interface between the two merging streams, creating zones of high shear stress that contribute to turbulent energy dissipation.

  o

  Flow separation may occur near the inner bank of the main channel, where lower velocities and possible recirculation zones can be observed.

• Downstream Velocity Adjustments:

  o

  Further downstream, the velocity distribution gradually stabilizes as the shear effects weaken, and the flow attains a more uniform structure.

  o

  The influence of secondary currents and transverse velocities may persist, potentially affecting sediment transport and erosion patterns near the banks.

Figure 14 shows the depth-averaged velocity across the width of the channel, where $U$ is the depth-averaged velocity magnitude, which is non-dimensionalized by $U_1$ (the downstream mean velocity); $x^{\ast }$ = $x/B$, with the $x$-coordinate non-dimensionalized by the channel width; $z^{\ast }$= $z/B$, with the $z$-coordinate non-dimensionalized by the channel width; and $B$ is the channel width. The model is capable of simulating the expected flow structure, as explained in the following. As the flow from the branch channel merges into the main channel, the higher velocity region shifts toward the outer bank due to momentum transfer and complex flow interactions between the two streams. To analyze the velocity distribution in this region, cross-sectional profiles were extracted immediately downstream of the junction (Figure 10) and compared with experimental data, revealing reasonable overall agreement. The numerical model effectively captured the primary velocity trends and flow redistribution, but some deviations were noted due to the highly turbulent nature of the confluent zone. A separation zone (or recirculation region) formed near the inner bank of the main channel, generated by the sudden mixing of the two flows, leading to strong velocity gradients, shear layers, and secondary flow structures that introduce computational challenges. These complexities, influenced by turbulence intensity and flow separation, impacted the accuracy of numerical predictions, particularly in areas with strong recirculation and eddy formation. The presence of these dynamic flow features has significant implications for sediment transport, riverbank erosion, and hydraulic stability in natural river confluences. Despite these challenges, the numerical model can reasonably reproduce key velocity characteristics. However, further refinements such as higher-resolution meshing, advanced turbulence modeling, and additional experimental calibration could improve predictive accuracy, especially in highly turbulent and recirculating flow regions. The RMSE and R² values provided in

Table 6. Sectional RMSEs of the resultant velocities in Figure 14.

Cross-Sections	Models
	k-ω SST Model (RLM)	Standard k-ε Model (RLM) [13]	Realizable k-ε Model (RLM) [13]	Standard k-ε Model (FSM) [25]	Realizable k-ε Model (FSM) [25]
−1.0	0.1994	0.1775	0.1798	0.3110	0.2075
−1.667	0.2387	0.2688	0.2610	0.3518	0.3182
−3.0	0.3571	0.1795	0.1700	0.3199	0.2051
−4.0	0.1631	0.0708	0.0687	0.2271	0.1543
Mean RMSE	0.2395	0.1741	0.1698	0.3024	0.2212

and

Table 7. R² values of the resultant velocities in Figure 14.

Cross-Sections	Models
	k-ω SST Model (RLM)	Standard k-ε Model (RLM) [13]	Realizable k-ε Model (RLM) [13]	Standard k-ε Model (FSM) [25]	Realizable k-ε Model (FSM) [25]
−1.0	0.6879	0.5872	0.6081	0.9612	0.9293
−1.667	0.9336	0.9231	0.9301	0.9773	0.9148
−3.0	0.6354	0.9583	0.9707	0.925	0.9176
−4.0	0.7422	0.9578	0.9647	0.9806	0.943
Mean R2	0.7497	0.8566	0.8684	0.961	0.9261

. indicate that the numerical model demonstrates a satisfactory level of agreement with the experimental data. This suggests that the k-ω SST model can reasonably capture key flow characteristics, including velocity distributions. The overall consistency between numerical predictions and experimental observations highlights the reasonable reliability and effectiveness of the model in simulating flow dynamics. Again, the overall accuracy of the k-ε models Shaheed et al. [13] is better than that of the k-ω SST model.

4. Discussion and Remarks

This study presents a numerical investigation of turbulent flow dynamics in river bends and confluences using the k-ω SST turbulence model within the OpenFOAM framework, considering both free- and closed-surface conditions. The numerical results demonstrate that the model effectively captures key flow features, including velocity redistribution, secondary currents, flow separation, and recirculation zones, and shows good agreement with experimental data. However, some deviations were observed, particularly near the inner and outer banks of both curved and confluent channels, where high turbulence intensity, secondary flow interactions, and shear layers introduce additional complexities. These discrepancies highlight the challenge of accurately modeling highly turbulent flows, especially in regions affected by flow separation, velocity gradients, and anisotropic turbulence structures. This study also emphasized the impact of surface constraints, where free-surface variations in open channels introduce additional flow instabilities, wave effects, and depth-dependent turbulence interactions, making them more complex to simulate compared to rigid-lid approximations.

Overall, the model was reasonably able to simulate the essential features of the flow, such as the formation of a secondary flow due to centrifugal forces. This flow redistribution, due to the secondary currents, plays a crucial role in sediment transport, bank erosion, and bed morphology changes. Similarly, the model was reasonably able to simulate the expected key features of flow in confluent channels. As the tributary flow merges with the main channel, a high-velocity core develops, accompanied by a separation zone (in the recirculation region) near the inner bank downstream of the confluence. This recirculation zone, driven by the sharp mixing of two streams with different velocities, introduces strong turbulence, vortex generation, and increased energy dissipation, impacting flow stability and sediment deposition patterns. While the numerical models successfully captured these primary flow structures, finer-scale turbulent interactions, particularly those associated with anisotropic eddy formation, vortex shedding, and three-dimensional turbulence effects, were more challenging to resolve due to modeling limitations and mesh resolution constraints.

Overall, it was observed that the k-ω SST model is a robust tool but generally less accurate than the realizable k-ε model. Despite the reasonable predictive capabilities of the k-ω SST model, there are several areas for improvement in future research. Implementing higher-order turbulence models, Large Eddy Simulation (LES), and Detached Eddy Simulation (DES) could improve the accuracy of turbulence predictions, particularly in regions with intense turbulence, coherent structures, and transient eddies. Further experimental validation using high-resolution flow measurements (e.g., Particle Image Velocimetry (PIV)) could also help refine model calibration, improve agreement with the observed data, and provide a better understanding of flow physics in complex fluvial environments.

Moreover, future work should investigate the influence of sediment transport, bedform evolution, and morphological changes in curved and confluent channels, as sediment dynamics significantly impact flow characteristics, secondary current behavior, and long-term river stability. The presence of variable bed roughness, vegetation effects, and channel irregularities should also be incorporated into simulations to enhance their real-world applicability in river engineering, flood risk assessment, and hydraulic design. Investigating the effects of unsteady flow conditions, such as flood waves and dam operations, could further improve the predictive capabilities of numerical models for river and watershed management. Finally, extending these simulations to natural river systems with complex topographies, multiple confluences, and varying boundary conditions could provide deeper insights into large-scale hydrodynamic processes, aiding in the development of sustainable river restoration strategies, erosion control measures, and sediment management practices.

5. Conclusions

This study presents a numerical simulation of turbulent flow in river bends and confluences using the k-ω SST turbulence model within the OpenFOAM framework, incorporating both free-surface and closed-surface conditions. The results demonstrate reasonable agreement with experimental data, capturing essential hydrodynamic features, including velocity redistribution, secondary currents, flow separation, and recirculation zones. In the curved channel, the numerical model reasonably reproduced the centrifugal forces and secondary flow structures. This velocity redistribution is crucial for understanding sediment transport dynamics, riverbank erosion, and channel stability. Similarly, in the confluent channel, the simulation reasonably predicted the high-velocity core development and recirculation zone near the inner bank downstream of the confluence, which is primarily influenced by the momentum exchange between the tributary and main channel as well as turbulence-induced mixing and flow separation at the junction. Despite the overall reliability of the numerical model, some discrepancies were observed in velocity distributions, particularly near channel boundaries, where localized turbulence effects and three-dimensional vortical structures were more complex to resolve. Although the k-ω SST model can simulate the secondary flow to some extent, overall, the results of the standard and realizable k-ε models Shaheed et al. [13] were found to be more accurate than the k-ω SST model.

A key focus of this study was examining the impact of free-surface variations versus closed-surface assumptions on turbulent flow behavior. The rigid-lid model (RLM), which assumes a closed surface, provided computational efficiency but introduced simplifications that may not fully capture surface deformation effects. In contrast, the Free-Surface Model (FSM) required a higher computational cost and finer grid resolution to maintain numerical stability. These findings emphasize the need for the careful selection of surface boundary conditions in numerical modeling, particularly in applications where wave dynamics and surface instabilities significantly influence the overall flow structure. Minor discrepancies in velocity distributions near the boundaries suggest the need for higher resolution meshes or advanced turbulence models such as LES (Large Eddy Simulation) or DES (Detached Eddy Simulation) methods with refined grids in the recirculation zones, zones of flow separation and zones of enhanced secondary flow. Dynamic boundary conditions can be employed in such models to overcome the limitations of the time-averaged methods used in this study and reproduce the coherent structures observed in such complex flow conditions as the outcome of geometric constraints, including curved boundaries or confluences. These advanced approaches could help to better capture anisotropic turbulence effects, vortex shedding, and three-dimensional secondary flow structures, which are particularly dominant in sharp bends and confluence zones. From an application perspective, the findings of this research contribute to a reasonably reliable simulation of the main flow features with practical implications for river engineering, sediment transport modeling, and hydraulic infrastructure design. A reasonably reliable prediction of secondary flow structures, recirculation zones, and velocity gradients is essential for designing erosion control measures, optimizing dredging strategies, and improving flood management techniques. Additionally, the numerical methodologies used in this study can be extended to investigate more complex fluvial environments, including meandering rivers, braided streams, and estuarine confluences.