Analysis of the Impacts of Geometric Factors on Hydraulic Characteristics and Pollutant Transport at Asymmetric River Confluences

Abstract

Asymmetrical river confluence zones play a critical role in water quality protection and remediation. This study develops a three-dimensional numerical model to simulate the hydraulic characteristics and contaminant dispersion processes within river channels. The results indicate that variations in the two geometric factors—the confluence angle and elevation difference—can produce a range of effects. Under the combined influence of these factors, the trajectory line at the pollutant-mixing interface follows a “logarithmic” growth pattern. As indicated by the inhomogeneity index, an increase in the junction angle and elevation difference significantly accelerates the mixing rate of pollutants and enhances dispersion. These insights suggest that, in cases with large confluence angles and significant elevation variations, intense mixing of water flow facilitates the rapid transport and extensive dispersion of pollutants, which may help reduce localized pollution loads. These findings are crucial for developing effective water environment management strategies.

1. Introduction

A complex river network system comprising a main river and multiple tributaries integrates artificial channels, natural watersheds, and urban drainage systems [1]. From both theoretical research and practical application perspectives, a thorough investigation of these hydraulic characteristics is undoubtedly highly beneficial. Given their high population densities and intensive economic activities, river confluence zones have increasingly attracted public attention regarding management and development challenges [2]. Near these confluence areas, pollutants transported by tributaries mix with the water flow and subsequently enter the mainstream in these regions [3]. The interaction between mainstreams and tributaries, particularly in the vicinity of confluence zones, results in significant modifications to the flow field, bathymetry, and other hydraulic characteristics due to the mixing and topographic steering effects of the currents, thereby generating a variety of distinct hydraulic phenomena [4]. Confluence estuaries can be primarily classified into two types [5]: the first type consists of symmetrical confluence zones (also referred to as “Y”-shaped confluences), while the second type comprises asymmetrical confluence zones (commonly known as “y”-shaped confluences). As illustrated in Figure 1, statistical analyses indicate that the latter type is more prevalent in natural watersheds [6]. Zones such as (1) the stagnation zone, (2) the flow deflection zone, (3) the flow separation zone, (4) the flow acceleration zone, (5) the flow recovery zone, and (6) the shear layer are usually employed to describe the flow structure at the confluence zone [7]. Figure 2 provides a visual representation of these zones.

Confluence estuaries have increasingly become a focal point in environmental management due to their critical role in retaining pollutant migration [8]. Additionally, these estuaries serve as key locations that significantly impact both navigation regulation and the protection of water environments [9]. Numerous researchers have conducted extensive experimental studies to investigate the hydraulic characteristics of confluence reaches [10,11,12,13,14]. It is widely recognized that various factors can influence the flow structure in these regions; however, geometric factors, including the size of the confluence mouth, the angle of confluence, and the geometry of the confluence reach, play a particularly significant role [15]. These geometric properties have substantial implications for both the flow structure and the dispersion of pollutants. The primary objective of this study is to investigate how geometric factors influence the hydrodynamic characteristics of the confluence zone and the transport pathways of pollutants [16,17]. Specifically, variations in confluence angles and elevations generate distinct flow patterns. These include the distribution of flow velocity, changes in flow direction, and the formation of turbulent eddies [18]. All these factors exert direct and immediate effects on the mixing and dispersion of pollutants [19]. Additionally, they pose significant challenges to the theoretical analysis of flow structures at confluences, which in turn influences the transport and dispersion of contaminants [20,21].

At the confluence of rivers, a complex flow structure is observed due to the merging of tributaries into the main channel, which induces top-supported conditions in the mainstream [22]. The turbulence characteristics of intersecting open channel flows are significantly more complex compared to those of straight open channel flows [23]. This increases flow resistance, accounting for both riverbed resistance and other substantial contributing factors. Taylor was the first to employ the sink test to analyze variations in water depth at different confluence angles [24]. Webber further explored the flow parameters across various confluence angles [25]. For a 90° confluence angle, Ramamurthy and other researchers conducted a comprehensive analysis of multiple confluence ratios and fluid morphologies, as well as the energy loss coefficients, energy exchange mechanisms, distribution of the velocity field in the confluence region, and the variation pattern of the water surface morphology in this zone [26]. Through physical experiments, Best successfully elucidated the mechanism of separation zone formation in the confluence region by investigating the interactions between separation zone size and the confluence and momentum ratios across various confluence angles [27]. Hua Zulin examined the effects of incoming flow characteristics and tributary width ratios on the dimensional properties of the flow separation zone [28]. Yang Zeyi analyzed the impact of riverbed elevation variations on the hydraulic behavior of flows at confluence zones [29].

The dynamics of pollutant transport and transformation during the confluence process, wherein tributaries introduce contaminants into the pristine mainstream, significantly influence both human living environments and mainstream water quality [30,31,32]. To safeguard human life activities, it is essential to systematically investigate pollution migration patterns within the confluence zone [33]. Gillibrand and colleagues developed a one-dimensional mathematical model to simulate and analyze water level, salinity, dissolved oxygen, and nitrogen dynamics within the Ythan estuary [34]. This provided a robust basis for understanding pollutant behavior in estuarine environments. However, as research has advanced, the limitations of one-dimensional models have become increasingly evident, highlighting the need for three-dimensional models to more accurately represent the complex factors influencing vertical water flow structures [35]. Particularly in confluence zones with irregular riverbed topography, the significant influence of differing mainstream and tributary bed elevations on pollutant mixing and dispersion has been well documented, especially in regions where the riverbed elevation is non-uniform [36]. In their field study, Sukhodolov observed that, at confluences with non-uniform beds, variations in riverbed elevation enhance the visibility of mixing processes [37].

The confluence angle and elevation difference are critical in determining the hydrodynamic characteristics of river confluence zones, which in turn govern the mixing and dispersion of pollutants. This study investigates asymmetrical river confluence sites to analyze variations in flow structure and patterns of pollutant dispersion under different confluence angles and elevation differences. A detailed examination of the data enables a precise understanding of how these factors influence contaminant mixing and dispersion within the confluence zone.

The findings of this study have significant implications for developing effective strategies to preserve water quality and manage environmental resources. They provide critical insights that can inform future decision-making in water resource management and environmental protection, ensuring the implementation of appropriate measures to safeguard ecological balance and human health in these vital river confluence zones.

2. Methodology

2.1. Control Equations

This study focuses on the turbulence modeling of incompressible fluids. Based on the actual conditions, the RNG k-ε turbulence model and the water quality model are selected, with the corresponding equations presented as follows.

Continuity equations:

$${\displaystyle \frac{\mathsf{\partial }\rho }{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_i}}\rho u_i=0$$

(1)

where ρ is the density, kg/m³; t is the time, s; u_i is the velocity component in the i direction (x, y, or z).

Momentum equation:

$${\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }t}}+u_j{\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }{x}_i}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\mathsf{\partial }p}{\mathsf{\partial }{x}_i}}+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_i}}\left(v{\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }{x}_i}}-u_i{u}_j\right)$$

(2)

$k-\epsilon$ equation:

$${\displaystyle \frac{\mathsf{\partial }k}{\mathsf{\partial }t}}+u_j{\displaystyle \frac{\mathsf{\partial }k}{\mathsf{\partial }{x}_j}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_j}}\left({\alpha }_{kv}v{\displaystyle \frac{\mathsf{\partial }k}{\mathsf{\partial }{x}_j}}+2v_t{S}_{ij}{S}_{ij}-\epsilon \right)$$

(3)

$${\displaystyle \frac{\mathsf{\partial }\epsilon }{\mathsf{\partial }t}}+u_j{\displaystyle \frac{\mathsf{\partial }\epsilon }{\mathsf{\partial }{x}_j}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_j}}\left({\alpha }_{\epsilon }v{\displaystyle \frac{\mathsf{\partial }\epsilon }{\mathsf{\partial }{x}_j}}-R+2c_1{\displaystyle \frac{\epsilon }{k}}{v}_t{S}_{ij}{S}_{ij}-c_2{\displaystyle \frac{{\epsilon }^2}{k}}\right)$$

(4)

Among others:

$$\overline{u_i{u}_j}=2/3k{\delta }_{ij}-\left({\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }{u}_j}}+{\displaystyle \frac{\mathsf{\partial }{u}_j}{\mathsf{\partial }{u}_i}}\right)$$

(5)

$$S_{ij}={\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }{u}_j}}+{\displaystyle \frac{\mathsf{\partial }{u}_j}{\mathsf{\partial }{u}_i}}$$

(6)

$$R=2v{S}_{ij}\overline{{\displaystyle \frac{\mathsf{\partial }{u}_i^{\prime }}{\mathsf{\partial }{x}_j}}}\overline{{\displaystyle \frac{\mathsf{\partial }{u}_j^{\prime }}{\mathsf{\partial }{x}_i}}}={\displaystyle \frac{C_{\mu }{\eta }^3\left(1-\eta /{\eta }_0\right)}{1+\beta {\eta }^3}}{\displaystyle \frac{{\epsilon }^3}{K}}$$

(7)

In the above equations, u_i and u_j denote the velocity of the fluid, respectively; p is the time-averaged pressure; k is the turbulent kinetic energy; ε is the turbulent kinetic energy dissipation rate; ρ is the density of the water; v is the molecular viscosity coefficient; v_i is the vortex viscosity coefficient; δ_ij is the Kronecker function; C_μ = 0.0845, C₁ = 1.42, C₂ = 1.68, α_k = α_ε = 1.39; u^′_i is the impulse velocity; η is the ratio of the turbulence time scale to the mean time scale; S is the strain rate tensor paradigm; η₀ is the typical value of η in uniform shear flow η₀ = 4.38, β = 0.012.

Water quality equations:

$$\begin{array}{c}{\displaystyle \frac{\mathsf{\partial }C}{\mathsf{\partial }t}}+{\displaystyle \frac{1}{V_F}}(u{A}_x{\displaystyle \frac{\mathsf{\partial }C}{\mathsf{\partial }x}}+v{A}_y{\displaystyle \frac{\mathsf{\partial }C}{\mathsf{\partial }y}}+w{A}_z{\displaystyle \frac{\mathsf{\partial }C}{\mathsf{\partial }z}})=\\ {\displaystyle \frac{1}{V_F}}\left[{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }X}}(A_{x}D{\displaystyle \frac{\mathsf{\partial }C}{\mathsf{\partial }x}})+R{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}(A_{y}DR{\displaystyle \frac{\mathsf{\partial }C}{\mathsf{\partial }y}})+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}(A_{z}D{\displaystyle \frac{\mathsf{\partial }C}{\mathsf{\partial }z}})\right]+C_{SOR}\end{array}$$

(8)

where C is the substance concentration; D is the diffusion coefficient; R—the coefficients chosen vary in different coordinate systems, and when in Cartesian coordinates, R = 1; C_SOR is the source item.

2.2. Numerical Method

The flume walls are subject to no-slip boundary conditions, and standard wall functions are applied to all surfaces. The model is discretized using the finite difference method, which enhances both the robustness and computational efficiency of the solution process. The volume-of-fluid method is employed to capture the free liquid surface accurately. The control equations are discretized and solved by dividing the flow domain into a cubic structured grid with uniform dimensions.

2.3. Validation Model Overview

The mathematical model is validated using experimental data from physical intersection flume studies as references [38,39]. The validation process involves comparing characteristic cross-sectional flow velocities and pollutant concentrations.

As depicted in Figure 3, a three-dimensional model of the test setup is presented. The model features a height of 0.3 m and a confluence angle of 90°. The inlet flow rate for the main flume is set at 17.34 m³/h, whereas the branch flume has an inlet flow rate of 4.12 m³/h. The initial water levels at both upstream and downstream sections are maintained at 0.16 m. The pollutant concentration at the inlet of the central canal is 0 μg/L, while that at the tributary inlet is 2000 μg/L. The physical model is constructed using a flat-bottomed plexiglass basin.

Model validation.

In this paper, the model results are validated by calculating two goodness-of-fit metrics, the Mean Relative Error (MRE) and the Nash efficiency coefficient (NSE), between the simulated and measured values [40].

Mean Relative Error (MRE):

$$\mathrm{MRE}={\displaystyle \frac{1}{\mathrm{n}}}{\sum }_{\mathrm{s}\mathrm{i}-1}^{\mathrm{n}}|{\displaystyle \frac{\mathrm{C}-{\mathrm{C}}_{\mathrm{si}}}{{\mathrm{C}}_{\mathrm{si}}}}|$$

(9)

Nash–Sutcliffe efficiency coefficient (NSE):

$$\mathrm{NSE}=1-{\displaystyle \frac{\sum _{\mathrm{s}\mathrm{i}-1}^{\mathrm{n}}{(\mathrm{C}-{\mathrm{C}}_{\mathrm{si}})}^2}{\sum _{\mathrm{s}\mathrm{i}-1}^{\mathrm{n}}{{(\mathrm{C}}_{\mathrm{si}}-\stackrel{\mathrm{-}}{\mathrm{C}})}^2}}$$

(10)

where C is the simulated value at each measurement point, mg/L; C_si is the measured value at the measurement point, mg/L; $\overline{\mathrm{C}}$ is the measured average value; n is the number of times the measured and simulated values are valid.

Comparison of flow field characteristics.

Three longitudinal transects, namely, Z1, Z3, and Z5, were selected to compare the near-surface flow velocities, as illustrated in Figure 4. The results indicate a strong correspondence between the simulated and observed values along the flow direction.

Comparison of pollutant concentrations.

The horizontal pollutant concentration changes the calculated curves, and the measured results are compared in Figure 5, where it is clear that the calculated and measured findings are generally in accord.

Figure 4 and Figure 5 illustrate a comparison between the simulated and actual measured data for flow rate and pollutant concentration across different cross-sectional locations.

Table 1. Calculation of indicators for comparison of measured and simulated values of each variable.

Variant	Cross-Section	MRE (%)	NSE
flow rate (m/s)	X = 5 cm	4.13	0.899
	X = 15 cm	3.2	0.998
	X = 25 cm	3.37	0.934
pollutant concentration (μg/L)	Y = 11 cm	4.49	0.992
	Y = 32 cm	4.95	0.983
	Y = 53 cm	4.68	0.962
	Y = 74 cm	3.16	0.959
	Y = 116 cm	4.84	0.975

summarizes the Nash–Sutcliffe efficiency (NSE) and Mean Relative Error (MRE) for each measurement point. The analysis reveals that the MRE and NSE values for all variables across all sections satisfy the evaluation criteria. In this study, all MRE values are below 5%, while the NSE values exceed 0.75 and nearly approach 1. These results confirm that models calibrated with precise measurements can achieve sufficient accuracy to fulfill the simulation requirements.

2.4. Numerical Model Construction

2.4.1. Model Overview

In this study, the main channel was configured with dimensions of 10.6 m in length, 0.9 m in width, and 0.8 m in height, while the tributaries were designed to have dimensions of 2 m in length, 0.24 m in width, and 0.8 m in height (Figure 6a). A structured orthogonal grid was employed for discretization. A single grid block encompassed the entire study area, yielding approximately 2.52 million grid cells (Figure 6b).

2.4.2. Monitoring Section Setup

The study area in this research encompasses mainstem inlet crossings, tributary inlet crossings, and downstream outlet crossings. The longitudinal direction (aligned with the water flow) is defined as the x-axis, the transverse direction as the y-axis, and the vertical direction as the z-axis. Five longitudinal sections—S1, S2, S3, S4, and S5—are strategically located along the x-axis within the junction zone. At the intersection, measurement transects are established every 0.8 m along the y-axis (labeled B1–B12), comprising a total of 12 transects. Figure 7 illustrates the cross-section layout and the boundary of the computational domain.

2.4.3. Calculation and Setting Conditions

Calculation Condition Setup

The primary focus of the physical modeling in this study is to investigate the influence of geometric factors on pollutant mixing characteristics. Two sets of working conditions are designed based on variations in the confluence angle and elevation difference, respectively. Here, h1 denotes the bottom elevation of the main channel, while h2 denotes the bottom elevation of the tributary channel. The difference in river bottom elevation (Δh) is defined as the difference between the bottom elevations of the tributary channel and the mainstream channel, calculated as Δh = h2 − h1, as illustrated in Figure 8. Tributaries exhibit varying elevations relative to the mainstem streambed, influencing flow dynamics at the confluence. The elevation difference causes tributaries to converge with a downward drop, generating distinct mixing effects. In this study, elevation differences of Δh = 0 m, Δh = +0.08 m, and Δh = +0.16 m were investigated. Statistically, confluence angles in natural streams typically range from 30° to 90°. For this study, the selected angles were 45°, 60°, and 90°, encompassing both smaller and larger angles. The working condition settings are presented in

Table 2. Simulated working condition factor table.

Working Condition	Factors to Consider	Serial Number	Confluence Angle	Elevation Difference
			α (°)	Δh (m)
1	confluence angle	1(A)	45	0
		1(B)	60	0
		1(C)	90	0
2	elevation difference	2(A)	90	0
		2(B)	90	0.08
		2(C)	90	0.16

.

Boundary Condition Setting

A specified flow rate is imposed at the inlet boundary for the hydrodynamic field. At the pollutant inlet, a prescribed concentration value is applied. The downstream outlet cross-section employs a pressure outlet boundary condition, where the pollutant is configured as a free outflow with zero gradients for all scalars. The sidewalls and bottom are designated as no-slip boundaries.

3. Results

3.1. Flow Structure in the Confluence Area

3.1.1. Flow Structure Under Different Confluence Angles

The planar flow field in the confluence zone is simulated for test cases 1(A), 1(B), and 1(C) under varying confluence angle conditions. The computational results are presented in Figure 9.

The scenario depicted in Figure 9A indicates that, at a confluence angle of 45°, the tributary discharges into the main channel with an average velocity of 0.145 m/s. In contrast, the average flow velocity at the inlet section of the main channel is 0.193 m/s. Conversely, flow velocities near the right bank are consistently lower than those on the left bank across the entire confluence region, suggesting that the stagnation zone along the right bank significantly affects local flow dynamics. According to the investigation, the average flow velocity in the stagnation zone on the tributary side is 0.112 m/s, which is 42% lower than the incoming flow velocity of the main channel. The confluence of the tributaries creates a flow separation zone at the confluence, deflecting the flow of the mainstream channel. A zone of maximum flow velocity is formed downstream of the confluence area, specifically between X = 6.52 m and X = 9.60 m, located in the center of the channel. This velocity increase is attributed to the reduced cross-sectional area, which facilitates flow acceleration.

Figure 9B illustrates the planar flow field at a confluence angle α = 60°. In the confluence region, the flow on the tributary side is weaker compared to the channel center, exhibiting a slight skew in the opposite direction. The extent of the stagnation zone at α = 60° is notably more significant than that at α = 45°, while the mean flow velocity within this zone remains virtually unchanged. Flow separation is particularly pronounced in the region extending from X = 0.05 m to X = 1.34 m. Moreover, the magnitude of the flow deflection zone, where the flow is redirected toward the side opposite the tributary entrance, exhibits a significant increase. An expansion in the size of the maximum flow zone accompanies this.

In the case shown in Figure 9C, where the confluence angle is 90°, the tributary flow enters the main channel perpendicularly. In this configuration, the average flow velocity within the stagnation zone is lower compared to when α = 60°. Additionally, the width of the stagnation zone observed in the planar flow field expands while its length progressively stretches downstream along the flow direction. Simultaneously, the amplitude of water flow deflection increases, gradually shifting the river channel’s stream toward the left bank on the tributary side. This results in a left bank exhibiting a mean flow velocity more significant than that of the right bank.

In summary, altering the confluence angle between the mainstem and tributary channels can substantially influence the planar flow field within the confluence zone. As the confluence angle increases, the mainstream flow gradually redirects toward the opposite side of the tributary channel. This is attributed to a gradual increase in the area of the separation zone, the amplitude of the deflection zone, and the extent of the stagnation zone. As the flow progresses downstream, the region of maximum flow velocity gradually shifts toward the side farther from the tributary, with its extent progressively expanding.

3.1.2. Flow Structure Under Different Elevation Variations

The effect of elevation variations on the planar flow field in the confluence zone is examined through simulations for test numbers 2(A), 2(B), and 2(C), as illustrated in Figure 10. When Δh = 0 m, the tributary enters the main channel at an angle of 90°. The tributary is in line with the river bottom of the mainstream. The planar flow field for this case is the same as that for the confluence angle α = 90°. When Δh = +0.08 m, the area of maximum flow velocity shifts substantially toward the opposite bank of the tributary. The flow velocity near the tributary side decreases, while the deflection of water flow in the velocity vector diagram increases as it progressively moves toward the opposite bank. Additionally, the range of the maximum flow velocity area decreases. The extent of the stagnation zone in this case is more pronounced compared to when Δh = 0 m. As illustrated in Figure 10C, when Δh = +0.16 m, the riverbed elevation increases, leading to a higher flow velocity in the tributary. Compared with Δh = +0.08 m, the flow velocity bias in the confluence zone on the tributary side increases, the extent of the stagnation zone expands, the maximum velocity zone gradually contracts, and the recovery region of the water flow progressively enlarges until the X = 5.36 m section, where the flow pattern essentially returns to its normal state.

In conclusion, the planar flow field of water in the confluence zone is significantly affected by differences in riverbed elevation between the tributary and the main channel. As the elevation difference increases and the deflection angle of the deflection zone gradually rises, a region of maximum flow velocity forms near the left bank along the direction of flow development. The primary flow of the main channel progressively shifts toward the opposite bank of the confluence. As the elevation difference increases, the maximum flow zone shifts farther from the tributary side and closer to the left bank, with its extent becoming smaller. The area of the stagnation zone expands with the increasing magnitude of the elevation differential, and its position progressively moves toward the opposite bank at the confluence. Overall, as the elevation difference grows, the influence of tributaries on the main channel becomes more pronounced, leading to more intense flow mixing.

3.2. Pollutant Transport and Dispersion in the Confluence Area

3.2.1. Influence of Geometric Factors

To accurately evaluate and estimate the dispersion and mixing of contaminants in river networks, it is essential to predict mixing phenomena downstream of confluence zones, mainly focusing on lateral mixing processes [41]. The study of the extent of the pollution of the water body and the range of influence greatly benefits from examining the maximum width of pollutant diffusion in the lateral direction. This width represents the maximum extent of pollutants transported by the tributaries to mix with the mainstream flow at the confluence. Typically, it is defined as when the pollutant concentration at an edge point reaches 5% of the maximum concentration within the same cross-section; this point is considered the boundary of the pollution zone [42]. The pollution zone is then formed by connecting these boundary points across different sections, with its width denoted as w, as illustrated in Figure 11A.

Effect of confluence angle on lateral dispersion of pollutants.

By simulating condition 1, Figure 11 illustrates the distribution of pollutant concentrations in the surface water body at different confluence angles. The red dashed line delineates the boundary of the pollution zone. As depicted, a wider confluence angle corresponds to a broader diffusion of the pollution zone in the surface water body. Within the confluence zone, the concentration gradient can be divided into three classes: a low-concentration zone (less than 1000 μg/L), medium-concentration zone (1000–1667 μg/L), and high-concentration zone (1667–2000 μg/L). These three classes can be utilized to characterize the concentration gradient within the confluence zone. As shown in Figure 11A, the zone of high concentration is less extensive, with only a tiny portion present at the intersection. The concentration distribution of the pollution zone created by the confluence angle of 45° is primarily concentrated between 333 and 1667 μg/L, which is at low to medium concentrations. At a confluence angle of 60°, both the high-concentration regions (1667–2000 μg/L) and medium-concentration regions (1000–1667 μg/L) were more extensive compared to those at α = 45°.

Compared to α = 45° and α = 60°, a more expansive pollution zone and a much higher proportion of medium- and high-concentration zones are formed at α = 90°. This indicates that, as the area occupied by the high-concentration zone continues to expand, the lateral dispersion distance of the pollutants generated gradually increases as the confluence angle increases. These findings align with those reported by other researchers [43].

In conclusion, the confluence angle plays a critical role in influencing the diffusion of pollutant concentrations within the intersection zone. As the confluence angle increases, the maximum width of the pollution zone, the width of the mixing interface, and the proportion of high-concentration areas also increase accordingly.

Effect of elevation differences on lateral dispersion of pollutants.

Figure 12 illustrates the distribution of pollutant concentrations in surface waters under varying elevations. Due to the confluence of tributaries and main streams, the high-concentration zone is mainly located in the 0 m–1.24 m zone. Consequently, there is a significant lateral concentration gradient of pollutants and a shorter and faster transition gap between locations with medium and low concentrations.

The pollutant transverse concentration gradient is more significant when Δh = 0 m because the high-concentration area is primarily situated in the interval X = 0 m–X = 1.24 m, the confluence of the tributary and the mainstream. Additionally, the transition interval from the middle-concentration area to the low-concentration area is shorter and more rapid. When Δh = 0.08 m, the pollution zone extends further in the transverse direction (Y-direction) and exhibits a significantly larger spatial extent. The figure indicates that the high-concentration regions gradually contract and ultimately concentrate exclusively at the intersection. At the same time, the extent of medium-concentration and low-concentration areas expands with increasing elevation differences. This implies that more considerable elevation differences are more favorable for the transport and diffusion of pollutants. Simultaneously, the pollutant concentration at the outlet section of the main channel (X = 9.6 m) has reached 486 μg/L. Under the working condition of Δh = +0.16 m, compared to Δh = +0.08 m, the area of the medium-concentration region decreases rapidly, while the high-concentration region remains essentially unchanged. The pollutant concentration at the outlet cross-section is less than 333 μg/L, and the longitudinal (X-direction) gradient of pollutant concentration gradually decreases from the cross-section at X = 2.56 m. While the resulting pollution zone has expanded in the transverse portion, the opposite side of the confluence zone is still unaffected by pollution.

In conclusion, variations in elevation substantially influence the distribution of pollutants within the confluence zone. The area of the created pollution zone steadily grows as the elevation difference increases, primarily seen in the steady lateral broadening of the pollution zone. As the elevation difference increases, the concentration in the departure region, the spatial extent of the high-concentration zone, and the lateral concentration gradient of pollutants all gradually decrease. This indicates that a more significant elevation difference facilitates the mixing and dispersion of pollutants.

3.2.2. Pollutant Mixing Interface Trajectories Under the Influence of Geometric Factors

Variation in pollutant mixing interface trajectory line under different confluence angles.

Figure 13 illustrates the variation in the motion of the pollutant mixing interface along the trajectory line within the confluence zone under various confluence angles. Using the mainstream river width b = 0.9 m for horizontal (X/b) and vertical (Y/b) coordinates, respectively, the plots are dimensionless for X and Y. The curves represent the fitted curves under examination, whereas the scatter points indicate the actual measured positions of the cross-section points. The fitted curve for the mixed interface’s trajectory exhibits a “logarithmic” growth pattern, consistent with the observations presented in the figure. The range of X/b ≤ 1.4 shows a clear increasing trend, while the growth of Y/b begins to decelerate when X/b > 1.4. It is worth noting that the trajectory line’s starting point does not coincide with the origin due to the stagnation zone formed by tributary inflows at the confluence angle. Additionally, the tributaries exert a top-supporting effect on the mainstream channel flow, which displaces the starting point from the origin. In conclusion, the confluence angle influences the pollutant mixing interface trajectory line; the more significant the confluence angle, the more it extends to the other shore of the intersection.

Variation of pollutant mixing interface trajectory line under different elevation differences.

In the confluence zone, Figure 14 illustrates the along-track variation curves of the pollutant mixing interface trajectory lines under three elevation difference conditions. It is clear from the figure that the trajectory lines exhibit a “logarithmic” growth pattern. As the water flow progressively drives the mixing interface toward the opposite bank of the confluence, effectively displacing the pollutant boundary away from the confluence side, the trajectory line continues to demonstrate a “logarithmic” growth trend. This growth pattern is observable in the region where X/b ≤ 1.79, beyond which the three sets of curves remain nearly parallel. The pollutant mixing interface trajectory line crossed the main channel’s axis at Δh = 0.16 m when Y/b reached its maximum value of 0.54, although it had not yet reached the left bank of the main channel. Overall, the trajectory line of the pollutant mixing interface in the confluence zone is influenced more significantly by variations in elevation than by the confluence angle.

3.2.3. Influence of Geometric Factors on the Characteristics of Pollutant Mixing Interfaces

The mixing rate of pollutants at a cross-section is frequently measured by utilizing Dev(x), which is a non-uniform exponent of pollutant concentration at that particular cross-section [44]. Moreover, the inhomogeneity index is an essential indicator for evaluating the efficacy of mixing and the distribution of contaminants downstream of the river. This index measures the degree to which the situation deviates from the ideal mixing condition. The following is the calculating formula:

$$Dev(x)={\displaystyle \sum _y\frac{\left|C_S\left(x,y\right)-C_P\right|}{C_P}}$$

(11)

where C_s (x, y) is the average modeled concentration of the pollutant on the vertical line at (x, y), and C_p is the flow-weighted average predicted concentration, which is defined as follows:

$$C_P={\displaystyle \frac{C_t{Q}_t+C_m{Q}_m}{Q_t+Q_m}}$$

(12)

where C_t and C_m are pollutant concentrations in tributaries and main streams, respectively, and Q_t and Q_m are the upstream flows of tributaries and mainstems, respectively, with lower Dev(x) indicating a more uniform distribution of pollutants (more complete mixing).

Effect of confluence angle on mixing characteristics of pollutant concentrations in the intersection area.

Figure 15 illustrates the variation in the pollutant inhomogeneity index for different confluence angles in the median section at Y = 0.45 m along the flow development direction. For all confluence angles, the inhomogeneity index Dev(x) near the intersection is generally close to 1, suggesting that the pollutant concentrations at the inlet section remain primarily unmixed. Furthermore, it is evident that, as the confluence angle increases, the inhomogeneity index demonstrates a notably pronounced downward trend near the river bottom (Z = 0.05 m). The inhomogeneity index in the exit section X = 9.60 m is close to 10%, especially at α = 90°. However, the flow structure exhibits a higher degree of turbulence at α = 90° compared to 45° and 60°, resulting in a more substantial decrease in the pollutant mixing inhomogeneity index. The figure clearly illustrates that, when the confluence angle is held constant, the water column near the bottom mixes more efficiently than the surface water column.

In conclusion, as the confluence angle increases, contaminants within the bottom plane mix more uniformly and disperse more rapidly. When the confluence angle is held constant, the water column near the riverbed mixes more efficiently compared to the surface water column. These findings highlight the critical role of the confluence angle in influencing pollutant mixing dynamics.

Effect of elevation differences on mixing characteristics of pollutant concentrations.

The pollutant inhomogeneity index distribution along the flow direction at the mid-axis Y = 0.45 m section under various elevation difference conditions is shown in Figure 16. Observations indicate that complete mixing of pollutants is achieved on the intermediate- and high-level planes (Z = 0.17 m, Z = 0.24 m, Z = 0.30 m) when Δh = 0.08 m. This phenomenon indicates that, as the tributary river’s bottom elevation increases, the tributary’s top-supporting effect on the mainstream is greatly enhanced, and the mixing effect of the water flow is intensified. Consequently, the mixing of pollutants in the middle-upper plane is accelerated.

The pollutant inhomogeneity index in the planes at Z = 0.30 m, Z = 0.24 m, and Z = 0.17 m exhibits a decreasing and then increasing trend for the case of ∆h = 0.16 m. This behavior is attributed to the confluence of tributaries within the region between X = 1 m and X = 2.56 m. The enhanced velocity of the tributary flow in this region generates a flow bias zone where velocity vectors are redirected toward the opposite side of the confluence. This phenomenon increases diffusion and promotes the rapid mixing of pollutants in the area. As a result, the pollutant inhomogeneity index initially decreases and then increases, attributed to the water flow gradually stabilizing as it progresses. At the same time, the diffusion rate of pollutants concurrently slows down. The distribution of pollutant mixing velocity is more uniform at Z = 0.05 m and Z = 0.11 m, as illustrated in Figure 16C. This is primarily because the bottom water mass mixes more rapidly and is less influenced by the upper water mass.

In conclusion, variations in elevation play a crucial role in influencing the mixing and dispersion of pollutants within the flow, particularly at the confluences of tributaries and the main channel. More minor elevation differences tend to promote a more uniform mixing of pollutants on mid-to-high planes. In contrast, more considerable elevation differences result in more complex patterns of pollutant mixing.

4. Discussion

The impacts of geometric factors, including the confluence angle and elevation difference, on the hydraulic characteristics and pollutant transport in the confluence zone of asymmetric rivers were investigated through numerical modeling. The data comparison between the physical model and the numerical simulation was assessed using two goodness-of-fit metrics, Mean Relative Error (MRE) and Nash efficiency coefficient (NSE). The results of model validation indicate that the numerical simulation can accurately predict both the flow structure and the pollutant distribution pattern.

Three lines of analysis were used to examine the impacts of the elevation difference and intersection angle on pollutants: longitudinal, vertical, and horizontal. As indicated by prior studies [45,46], a smaller junction angle results in a pollution zone that is more elongated and narrow, with a steeper gradient in pollutant concentrations. The contaminants are distributed more uniformly in the bottom plane, with a significantly enhanced diffusion rate as the confluence angle increases. This rapid mixing and extensive dispersion of pollutants can be attributed to the combined effects of multiple factors, including the recirculation effect within the separation zone, improved cross-sectional circulation, an expanded mixing interface, and the promotion of three-dimensional flow characteristics at larger angles [29,47]. When the elevation difference is substantial, the significant alteration in flow structure promotes lateral pollutant diffusion [48]. This is evidenced by the gradual decrease in concentration at the outlet cross-section, the marked increase in lateral diffusion distance, and the pronounced reduction in the area occupied by the high-concentration zone. The kinetic energy of the water flow is substantially enhanced as water flows from higher to lower elevations. This generates a distinct drop due to the varying riverbed elevations between the mainstream and its tributaries. The increased turbulence of the water flow [49,50] enables pollutants to mix more rapidly and uniformly in the confluence zone, thereby enhancing mixing efficiency, particularly in intermediate- and high-flow water bodies. Consequently, contaminant mixing efficiency increases with variations in geometric factors.

According to research on the influence of geometric factors on the non-uniformity index of pollutant mixing, the diffusion width of the pollution zone increases with rising intersection angles and elevation differences, leading to enhanced mixing rates and a progressive decrease in the non-uniformity index [51]. This indicates that larger confluence angles and significant elevation differences enhance hydrodynamic conditions, promoting rapid contaminant transport and broader dispersion. The pollutant mixing interface extends beyond the median axis of the main channel due to the substantial elevation difference. Variations in the riverbed’s elevation enhance water velocity and shear force, thereby reducing the time required for complete pollutant mixing. This shear force induces turbulence, disrupting the laminar flow of water and accelerating the mixing of contaminants across different water levels [52]. The offset of the trajectory line of the pollutant mixing interface, the change in pollutant diffusion breadth, and the variation in mixing rate serve as the primary indicators for evaluating the impact of elevation differences and intersection angles on the mixing interface trajectory. These factors determine the confluence zone’s pollutant mixing and diffusion properties. Through the proper adjustment of intersection angles and elevation differences, the water flow structure can be optimized, reducing pollutant retention time in the intersection zone and enhancing the self-purification capacity of the aquatic ecosystem. The elevation difference and intersection angle are key factors influencing the rapid diffusion and uniform distribution of pollutants. This provides a critical theoretical basis and practical guidance for developing effective environmental management and water quality maintenance strategies.

Elevation difference and confluence angles coexist and interact in natural rivers. For example, mountain rivers often exhibit large confluence angles and significant elevation differences due to their steep topography. In contrast, plain rivers typically have gentle beds (with elevation differences of roughly 0 m) and moderate angles. In this study, only single-factor effects were considered to clarify the individual impacts of each component on the confluence area. Future research should further investigate the interactions between these two components. The model might not be immediately relevant to other river confluence sites because it was built under certain geometric constraints. Future research could examine the patterns of pollution dispersion in other confluence zones and consider additional variables that affect pollutant transport and dispersion, such as variations in water velocity, pollutant species, and aquatic vegetation. Furthermore, field monitoring data could be utilized in future research to validate and enhance the predictive capabilities of the model, thereby improving its precision and practical utility.

5. Conclusions

(1)

In this study, the effects of pollutant transport and dispersion within the confluence zone of asymmetric rivers are systematically investigated using numerical simulation techniques. The results reveal significant influences of the confluence angle and elevation difference on the flow structure, lateral dispersion characteristics, trajectory of the pollutant mixing interface, and mixing behavior of pollutant concentrations in the river confluence region.

(2)

The numerical model’s reliability is evaluated and simulated. By comparing the measured and simulated values of flow and pollutant concentration at each section, it is found that the numerical simulation can accurately reproduce the hydrodynamic characteristics of the confluence zone as well as the pollutant transport and diffusion processes. This outcome suggests that the numerical model is reliable and suitable for investigating the impact of geometric factors on the transport and diffusion of pollutants in the confluence zone of asymmetric confluent rivers.

(3)

The trajectory of the pollutant mixing interface exhibits a distinct “logarithmic” growth pattern. In the region near the confluence, the growth rate is markedly accelerated due to the increased influence of various factors. As the current progresses downstream, the trajectory gradually extends toward the opposite bank. Among the geometric factors, increases in both the elevation difference and the confluence angle serve to enhance the rate of pollutant diffusion. Still, the elevation difference exerts a more pronounced influence on the trajectory line of the pollutant mixing interface than the confluence angle. Specifically, under an elevation difference of Δh = 0.16 m, the trajectory line of the pollutant mixing interface has already surpassed the central axis of the main channel. However, under the influence of the confluence angle, the trajectory line has not yet reached the central axis.

(4)

The inhomogeneity index quantifies the mixing characteristics of pollutants within a cross-section. It has been demonstrated that increasing the confluence angle and elevation difference can significantly enhance the rate of pollutant mixing. This leads to a gradual decrease in the inhomogeneity index, promoting more effective mixing and dispersion of pollutants. Notably, elevation differences facilitate faster mixing rates compared to confluence angles.