Numerical Modeling of the Dispersion Characteristics of Pollutants in the Confluence Area of an Asymmetrical River

Abstract

It is challenging to investigate the transport and dispersion of contaminants in river confluence areas due to the complex flow dynamics. In recent studies on the flow dynamics in river confluence areas, it has been revealed that changes in inflow conditions (discharge ratio, width-depth ratio and concentration difference) can greatly influence pollutant diffusion. In this study, an asymmetric confluence-type river is modeled by a three-dimensional hydrodynamic water quality model, and three hydrodynamic scenarios are numerically simulated. The results show that a higher discharge ratio and width-depth ratio led to an increase in the lateral diffusion area of pollutants, deviation in the trajectory line of the mixing interface towards the opposite bank of the interchange, and an increase in the mixing rate of pollutants. For R = 0.267 and b/h = 3.75, the pollutants at the bottom are completely mixed in the exit end section. However, the difference in the pollutant concentration slightly affects the area, length and shape of the pollutant dispersion zone and notably affects only the concentration in each section.

1. Introduction

Since rivers are typically found in highly populated and economically well-developed areas. Regional development and management (in terms of navigation, ecology, flood control, tourism, etc.) are currently gaining attention and becoming hot topics [1,2].

The confluence area is common among natural rivers, and the natural river system encompasses diverse geographical environments. Therefore, there are many complex flow patterns in the confluence area of natural rivers. In 1988, Best [3] first proposed dividing confluence areas into tributary oblique main stream-type and Y-type confluence areas. In 1998, after analysis, Lanbo [4] divided confluence areas into two types: asymmetric and Y-type confluence areas. Considering the various shapes of the main stream channels of rivers, in 2010, Zhang Qiang and Wang Pingyi [5] divided tributary oblique main stream-type confluence areas into straight main stream-type confluence areas, curved main stream-type confluence areas, and bifurcated main stream-type confluence areas. Regarding confluent rivers, the flow structure [6,7] and riverbed topography [8,9] at the confluence of the main stream and the tributaries are constantly changing, which exerts a considerable impact on the diffusion pattern of pollutants, resulting in complex and variable diffusion and mixing characteristics of pollutants in confluence areas [10]. Therefore, it is very important to study and characterize the transport and diffusion of pollutants in the confluence area to guide the layout of sewage outlets, the selection of the sewage discharge method and the precise formulation of water environment management plans in practical projects.

The diffusion pattern of pollutants in intersecting rivers has been investigated by both domestic and foreign researchers as a result of the increasing importance of water environment issues and the rapid advancement in computer technology. Numerical simulation of the water quality exhibits the characteristics of intuitiveness and rapidity and has become the main tool to study the diffusion pattern of pollutants in the confluence area of rivers [11]. Most scholars have adopted this research method. Gillibrand et al. [12] created a one-dimensional mathematical model for the Ythan estuary and conducted a simulation analysis of the water level, salinity and TON content in the estuary. The pollutant mixing process was simulated in an indoor intersecting open channel experiment by Biron et al. [13] using the numerical simulation approach. It was determined that the upstream flow rate was directly correlated with the degree of pollutant mixing near the intersection. Mixing occurred more quickly with decreasing flow rate. A three-dimensional hydrodynamic model was used by Isabel et al. [14] to analyze the distribution of pollutants in the Douro estuary under different discharge conditions. In his investigation of the hydrodynamics and water quality of two rivers in Chongqing using the two-dimensional model SMA, Liu Xulan [15] observed that high-concentration pollution zones had formed on the river bank on the side of the junction. To more accurately estimate the extent of the pollutant mixing zone, Mao Zeyu et al. [16,17] established a hydrodynamic and pollutant transport model for the intersection of open channels. Wei Juan et al. [18] constructed a mathematical model of pollutant transport and diffusion by combining experiments and numerical simulations under different discharge ratios. By examining a U-shaped confluent channel, Gu Li et al. [19] developed a water vapor two-phase flow model and determined that the longitudinal dispersion coefficient of pollutants exhibited a unimodal structure distribution. Under the lowest width-depth ratio of the open confluent channel, Yuan et al. [20] observed that the higher the fraction of the tributary flow rate, the larger the vortex behind the confluence and the higher the pollutant mixing rate.

In summary, the discharge ratio [21], discharge flow, and width-depth ratio [22] are important factors affecting the hydrodynamic characteristics in the confluence area and notably influence the transport, diffusion and mixing characteristics of pollutants at the confluence of rivers. In this paper, the confluence area of an asymmetric river is studied, and the diffusion pattern and mixing characteristics of pollutants under different hydrodynamic conditions (discharge ratio, width-depth ratio, and concentration difference) are analyzed.

2. Mathematical Models and Verification

The numerical simulation method is an important scientific engineering calculation method based on numerical calculation experiments to obtain an approximate solution of control equations, and mathematical models are constructed for conducting a virtual physical model test, which is widely used in fluid dynamics research. Compared with the traditional physical sink model, the numerical simulation method provides significant advantages, such as the minimal impact on aspects such as resource waste and environmental pollution. At the same time, this method does not require much manpower, material resources, time, or hydropower resources, which reduces the time investment and research costs. Therefore, numerical simulation was applied to simulate the asymmetric river flow structure and pollutant transport and diffusion.

2.1. Control Equations

The model chosen in this study involves turbulent flow of an incompressible fluid, and the continuity equation, N–S equation, and energy equation are used to explain turbulent motion. The RNG k-ε model was created by applying renormalization group mathematics to the instantaneous N–S equation. The water quality equation is used to simulate the diffusion process of pollutants, and the VOF method is used to capture the free surface. The main control equation can be expressed as follows [23,24]:

Continuous equations:

$$\frac{\partial \left({uA}_x\right)}{\partial x}+\frac{\partial \left({uA}_y\right)}{\partial y}+\frac{\partial \left({uA}_z\right)}{\partial z}=0$$

(1)

Momentum equation:

$$\frac{\partial u}{\partial t}+\frac{1}{V_F}\left\{u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right\}=-\frac{1}{\rho }\frac{\partial P}{\partial x}+G_x+f_x$$

(2)

$$\frac{\partial v}{\partial t}+\frac{1}{V_F}\left\{u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right\}=-\frac{1}{\rho }\frac{\partial P}{\partial y}+G_y+f_y$$

(3)

$$\frac{\partial w}{\partial t}+\frac{1}{V_F}\left\{u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right\}=-\frac{1}{\rho }\frac{\partial P}{\partial z}+G_z+f_z$$

(4)

k-ε equation:

$$\frac{\partial \left(\rho k\right)}{\partial t}+\frac{\partial (\rho k{u}_i)}{\partial x_i}=\frac{\partial }{\partial x_j}\left({\alpha }_k{\mu }_{\mathit{eff}}\frac{\partial k}{\partial x_j}\right)+G_k+\rho \epsilon$$

(5)

$$\frac{\partial \left(\rho \epsilon \right)}{\partial t}+\frac{\partial (\rho \epsilon u_i)}{\partial x_i}=\frac{\partial }{\partial x_j}\left({\alpha }_{\epsilon }{\mu }_{\mathit{eff}}\frac{\partial \epsilon }{\partial x_j}\right)+\frac{C_{1\epsilon }^{*}}{k}{G}_k-C_{2\epsilon }\rho \frac{{\epsilon }^2}{K}$$

(6)

where $u, v, \mathrm{a}\mathrm{n}\mathrm{d} w$—fractional velocities along the x, y, and z directions, respectively, m/s;

$A_x, A_y, \mathrm{a}\mathrm{n}\mathrm{d} A_z$—area fractions along the x, y, and z directions, respectively, m²;

$V_F$—volume fraction, m³;

$\rho$—density of the fluid, kg/m³;

$P$—pressure applied to the fluid, N/m³;

$G_x, G_y, \mathrm{a}\mathrm{n}\mathrm{d} G_z$—acceleration due to gravity along the x, y, and z directions, respectively, m/s²; and

$f_x, f_y, \mathrm{a}\mathrm{n}\mathrm{d} f_z$—viscous forces along the x, y, and z directions, respectively, (kg·m)/s² [25].

The VOF method is used to track free surface flow, and the volume fraction continuity equation of two-phase water is solved to determine the position of the free surface. The equation can be expressed as follows:

$$\frac{{\partial \alpha }_w}{\partial t}+u_i\frac{{\partial }_w}{\partial x_i}=0$$

(7)

where α_w is the volume fraction of water. For α_w = 0, the calculation cells all occur in the gas phase; for α_w = 1, the calculation units all occur in the water phase; and for 0 < α_w < 1, the calculation units comprise both the water and gas phases [26].

The equation for the water quality can be expressed as:

$$\begin{gathered} \frac{\partial \mathrm{C}}{\partial \mathrm{t}}+\frac{1}{{\mathrm{V}}_{\mathrm{F}}}\left(\mathrm{u}{\mathrm{A}}_{\mathrm{x}}\frac{\partial \mathrm{C}}{\partial \mathrm{x}}+\mathrm{v}{\mathrm{A}}_{\mathrm{y}}\frac{\partial \mathrm{C}}{\partial \mathrm{y}}+\mathrm{w}{\mathrm{A}}_{\mathrm{Z}}\frac{\partial \mathrm{C}}{\partial \mathrm{z}}\right)= \\ \frac{1}{{\mathrm{V}}_{\mathrm{F}}}\left[\frac{\partial }{\partial \mathrm{X}}\left({\mathrm{A}}_{\mathrm{x}}\mathrm{D}\frac{\partial \mathrm{C}}{\partial \mathrm{x}}\right)+\mathrm{R}\frac{\partial }{\partial \mathrm{y}}\left({\mathrm{A}}_{\mathrm{y}}\mathrm{DR}\frac{\partial \mathrm{C}}{\partial \mathrm{y}}\right)+\frac{\partial }{\partial \mathrm{z}}\left({\mathrm{A}}_{\mathrm{z}}\mathrm{D}\frac{\partial \mathrm{C}}{\partial \mathrm{z}}\right)\right]+{\mathrm{C}}_{\mathrm{SOR}} \end{gathered}$$

(8)

C—material concentration;

D—diffusion coefficient;

R—coefficient in the chosen coordinate system, with R = 1 in the Cartesian coordinate system;

CSOR—source item [27].

2.2. Model Validation

2.2.1. Verification Model Overview

To confirm the accuracy of the numerical model in capturing hydrodynamics and pollutant transport, physical experimental data were used for comparison in this research [28,29].

Among them, the intersection angle between the main channel and the branch channel is 90°. The width of the main channel is 0.3 m. The width of the branch channel is 0.1 m. The working conditions are Fr = 0.10 and Reynolds number (Re) = 6357.79. The inlet flow of the main channel is 17.34 m³/h, and the inlet flow of the branch channel is 4.12 m³/h. The initial water level both upstream and downstream is 0.16 m. The pollutant concentration at the inlet of the main channel is 0 μg/L, and the pollutant concentration at the inlet of the branch channel is 2000 μg/L. The model structure is shown in Figure 1a. A structured orthogonal grid is selected. At the same time, considering the computational cost, only a part of the solid sidewall is selected during meshing. The specific grid diagram is shown in Figure 1b.

2.2.2. Numerical Method

The RNG k-ε and water quality models were selected for calculation. All boundary conditions were set according to the physical model parameters. The boundary conditions along the x and y directions are flow boundaries. The boundary condition along the x direction is a pressure boundary condition, and the boundary conditions under the other directions are symmetrical boundaries.

2.2.3. Model Verification

Considering previous research findings, in this paper, the model results are validated by calculating two goodness-of-fit indicators, i.e., the mean relative error (MRE) and Nash–Sutcliffe efficiency coefficient (NSE), between the simulated and measured values [30].

Table 1. NSE value evaluation criteria.

NS Value	Result Evaluation
NSE < 0	Measured values outperform the simulated values
0.5 < NSE < 0.65	Acceptable value
0.65 < NSE < 0.75	Improved simulation results
NSE > 0.75	Excellent simulation results
NSE = 1	Perfect match between the simulated and measured values

shows the NSE coefficient assessment criterion [31], and these indicators can be calculated as follows:

Average relative error (MRE):

$$\mathrm{MRE}=\frac{1}{\mathrm{n}}{\sum }_{\mathrm{si}-1}^{\mathrm{n}}\left|\frac{\mathrm{C} -{ \mathrm{C}}_{\mathrm{si}}}{{\mathrm{C}}_{\mathrm{si}}}\right|$$

(9)

Nash–Sutcliffe efficiency coefficient (NSE):

$$\mathrm{NSE}=1-\frac{\sum _{\mathrm{si}-1}^{\mathrm{n}}{(\mathrm{C} -{\mathrm{C}}_{\mathrm{si}})}^2}{\sum _{\mathrm{si}-1}^{\mathrm{n}}{\left({\mathrm{C}}_{\mathrm{si}} -\overline{\mathrm{C}}\right)}^2}$$

(10)

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

The simulated and measured values of the flow velocity and pollutant concentration for each typical passage are compared in Figure 2 and Figure 3, respectively.

Table 2. Indicators calculated for comparing measured and simulated values of each variable.

Variables	Cross-Section	MRE (%)	NSE
Velocity of flow (m/s)	X = 5 cm	4.13	0.899
	X = 15 cm	3.20	0.998
	X = 25 cm	3.37	0.934
Concentration of pollutant (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

shows the MRE and NSE values at the measurement points of each section.

The outcomes demonstrate that the MRE and NSE values for each variable in the various sections satisfy the assessment requirements. The smaller the MRE value is, the higher the model accuracy, and the MRE values in this study are all less than 5%. The NSE values are all greater than 0.89, suggesting that the model can satisfy the accuracy requirements and can fulfil the simulation criteria when calibrated against the measured data.

3. Program Design

3.1. Overview of the River Model

On the basis of the above model verification, to ensure a closer match between the simulation results and the actual river channel conditions, the length and width of the main and branch rivers are appropriately extended. The main stream of the river designed in this paper is 10.6 m long, 0.9 m wide and 0.8 m high. The tributary river is 2 m long, 0.24 m wide and 0.8 m high. The solid model is shown in Figure 4a. To reduce the computing memory and time, the grid does not encompass the entire sidewall, and each sidewall contains 10 meshes surrounding the identification boundary, finally yielding approximately 6.05 million meshes. The details are shown in Figure 4b.

3.2. Measurement Point Placement and Cross-Sectional Monitoring

The reach area contains three open boundaries: upper boundary 1 in the upstream inlet section of the main stream, upper boundary 2 in the upstream inlet section of the tributary, and the lower boundary in the downstream outlet section of the confluence area of the two rivers. The origin O is located at the boundary at the upstream vertex of the intersection angle. To facilitate the subsequent analysis of the results, five measuring points are established along the y direction in the cross section of the intersection area, at S₁ = 0.15, S₂ = 0.3, S₃ = 0.45, S₄ = 0.61, and S₅ = 0.75. A measurement cross section is set every 0.8 m from the intersection along the water flow direction, located in a total of 12 cross sections of B₁ = 0, B₂ = 0.8, B₃ = 1.6, B₄ = 2.4, B₅ = 3.2, B₆ = 4, B₇ = 4.8, B₈ = 5.6, B₉ = 6.4, B₁₀ = 7.2, B₁₁ = 8, and B₁₂ = 8.8. The calculation area boundary and section settings are shown in Figure 5.

3.3. Calculation and Setting Conditions

3.3.1. Calculation Conditions

In recent years, it has been shown that industrial wastewater is the most important pollution source in rivers, and most of the pollutants in industrial wastewater are persistent substances. Therefore, in this paper, mainly persistent pollutants are examined, and the degradation and adsorption of pollutants can be ignored. To study the three-dimensional distribution of the pollutant concentration in the intersection area of asymmetric river channels and the influence of inflow conditions (discharge ratio, width-depth ratio, and concentration difference), the validated hydrodynamic water quality model was used in this experiment. According to the method of control variables, the pollutant concentration field in the intersection area was studied and simulated. According to the variable discharge ratio, width-depth ratio and pollutant concentration difference, it was divided into three groups of conditions: 1, 2 and 3. The width-depth ratio is defined as b/h, where h is the water level upstream of the main stream, and b is the width of the main channel. Q₁ is the upstream flow of the main stream, Q₂ is the incoming flow of the tributary, and R = Q₂/Q₁ is the discharge ratio, i.e., the ratio of the incoming flow of the tributary to that of the main stream. Among them, the tributary flow is constant at Q₂ = 36.29 m3/h. The concentration difference is the difference between the pollutant concentration at the tributary inlet (C₂) and that at the main stream inlet (C₁), denoted as Cg = C₂ − C₁. A clean water simulation is conducted with this model, with the pollution concentration set to C1 = 0 μg/L in the main stream input section. For more information, refer to

Table 3. Pollutant concentration field simulation group parameters.

Working Conditions	Number	Investigation Factors	Mainstream Flow Q1(m3/h)	Discharge Ratio R	Water Depth h(m)	Width-Depth Ratio b/h	Concentration of Tributary C2 (μg/L)	Concentration Difference Cg (μg/L)
1	1(a)	Discharge ratio	136.08	0.267	0.3	3	2000	2000
	1(b)		194.40	0.187	0.3	3	2000	2000
	1(c)		272.16	0.133	0.3	3	2000	2000
2	2(a)	Width-depth ratio	194.40	0.187	0.24	3.75	2000	2000
	2(b)		194.40	0.187	0.3	3	2000	2000
	2(c)		194.40	0.187	0.36	2.5	2000	2000
3	3(a)	Concentration difference	194.40	0.187	0.3	3	500	500
	3(b)		194.40	0.187	0.3	3	1000	1000
	3(c)		194.40	0.187	0.3	3	2000	2000

.

3.3.2. Establishing the Boundary Conditions

The flow inlets of the main and branch rivers are adopted as boundary conditions, and the flow direction is perpendicular to the entrance section. The pressure outlet boundary condition is defined in the downstream outlet section of the intersection. The water level in the downstream outlet section is consistent with that in the upstream main stream inlet section under each working condition. The pressure in the outlet section satisfies the inlet pressure distribution, and the relative pressure of the liquid surface is 0. The roughness in each section is consistent with the roughness in the verification model.

4. Analysis of Results

4.1. Analysis of the Horizontal Diffusion Characteristics of Pollutants

The maximum width of the horizontal pollutant diffusion zone reveals the maximum range of the horizontal pollutant distribution, which is important for determining the water pollution level and the radius of effect. In general, when the pollutant concentration at the edge point is set to 5% of the maximum concentration in the same section, the point is defined as the boundary point of the pollution belt [32]. Therefore, the maximum width of the pollution belt w is defined as the maximum width of the pollution belt when the boundary mass concentration of the pollution belt reaches 5% of the mass concentration in the section, as shown in Figure 6b. On this basis, the distribution area of the pollutant concentration and the proportion of the intersection area can be obtained (the intersection area extends from the intersection origin O to the lower boundary downstream outlet section along the flow direction).

Figure 6 shows the distribution of the pollutant concentration in surface water (Z = 0.3 m) in the confluence area under different discharge ratios. The red dotted line in the figure is the boundary of the pollution zone. The results showed that when the tributary flow accounted for a large proportion of the total flow (R = 0.267), the spread width of the surface water pollution zone was the largest (Figure 6a); the spread width of the pollution zone also decreased when the flow proportion of the corresponding tributaries was relatively small (Figure 6b,c). As shown in Figure 6 and

Table 4. Statistics of the pollutant concentration distribution range under different incoming flow conditions.

Number	Pollutant Concentration Distribution Area (m2)	Proportion of the Intersection Area (%)
1(a)	5.25	60.77%
1(b)	3.90	45.16%
1(c)	2.97	34.34%
2(a)	5.08	58.82%
2(b)	3.90	45.16%
2(c)	3.65	42.27%
3(a)	3.77	43.68%
3(b)	3.86	44.66%
3(c)	3.90	45.16%

, the pollution zone in the study area significantly increased with increasing discharge ratio after the confluence of the two rivers. For R = 0.267, the pollutant concentration was 2000–1667, 1667–1333, and 1333–1000 μg/L, and the formed pollution zone was 5.25 m² in size. For R = 0.187, the pollutant concentration was 2000–1667, 1667–1333, 1333–1000 and 1000–667 μg/L, respectively, and the size of the resulting pollution zone was 3.90 m², which is 15.61% smaller than that for R = 0.267. For R = 0.187, the pollutant concentration distribution was the same as that for R = 0.187, but the pollution zone proportion in the 1000–667 μg/L range increased, and the size of the resulting pollution zone was 2.97 m², which is 10.82% smaller than that for R = 0.187. In summary, the discharge ratio greatly impacts the pollution zone. The higher the discharge ratio is, the wider the formed pollution zone and the higher the concentration gradient in each section.

The higher the width-depth ratio is, the shallower the water depth. Under the same flow rate, the higher the relative flow rate is, the lower the pressure gradient. To reflect the influence of different width-depth ratios on horizontal pollutant diffusion more directly, the distribution area of the pollutant concentration and the proportion of the intersection area under each working condition are analyzed, as shown in Table 4, considering Conditions 2(a)–2(c). For b/h = 3.75, the size of the formed pollution zone is 5.08 m²; for b/h = 3, the size of the pollution zone is 3.90 m², which is 13.66% smaller than that for b/h = 3.75. For b/h = 2.5, the size of the pollution zone is 3.65 m², which is 7.43% smaller than that for b/h = 3. In summary, the width-depth ratio exerts an impact on the horizontal diffusion of pollutants. The higher the width-depth ratio is, the wider the pollution zone, but the impact is not as notable as that of the discharge ratio.

In Table 4, 3(a)–3(c) denote the proportions of the area of the pollution zone in the intersection area under three concentration difference conditions, namely, Cg = 500, Cg = 1000 and Cg = 2000 (from top to bottom). There is a concentration difference between the main stream and the tributary, and the distribution of pollutants in the intersection area is only reflected in the concentration in the polluted section. When the concentration difference is larger, the concentration of pollutants in the downstream outlet section of the intersection is higher. However, the concentration difference exerts a negligible effect on the distribution area of the pollutant concentration, and the maximum influence range is 0.98%. It can be concluded that the horizontal diffusion width, length and size of the pollution zone are not affected. In summary, the concentration difference exerts a negligible effect on horizontal pollutant diffusion, which could be almost ignored and only affects the concentration in each section.

4.2. Trajectory Line of Pollutant Mixing Interface Changes along the Path

Based on the idea of Lewis and Rhoads, in this paper, the connection line of the average concentration points in the confluence section is adopted as the mixing interface [33]. Combined with the concept of the jet trajectory line, the above is commonly used to reflect the characteristics of the jet centerline, and the jet trajectory line is the line connecting the maximum concentration points upstream and downstream of the jet center plane [34,35]. With the use of Equation (11), the average concentration C_p of the two rivers after complete mixing can be calculated, as shown in

Table 5. Average concentration of complete mixing in the intersection area.

Number	Investigation Factors	Mainstream Flow Q1 (m3/h)	Concentration of Mainstream C1 (μg/L)	Tributary Flow Q2 (m3/h)	Concentration of Tributary C2 (μg/L)	Average Concentration Cp (μg/L)
1(a)	R = 0.267	136.08	0	36.29	2000	421.05
1(b)	R = 0.187	194.40	0	36.29	2000	314.61
1(c)	R = 0.133	272.16	0	36.29	2000	235.29
2(a)	b/h = 3.75	194.40	0	36.29	2000	314.61
2(b)	b/h = 3.00	194.40	0	36.29	2000	314.61
2(c)	b/h = 2.50	194.40	0	36.29	2000	314.61
3(a)	Cg = 500	194.40	0	36.29	2000	78.65
3(b)	Cg = 1000	194.40	0	36.29	2000	157.30
3(c)	Cg = 2000	194.40	0	36.29	2000	314.61

, and the trajectory of the mixing interface is visualized.

C_p = (Q₁C₁ + Q₂C₂)/(Q₁ + Q₂)

(11)

where C₁ and C₂ are the tributary and main stream pollution concentrations, respectively, and Q₁ and Q₂ are the tributary and main stream inflows, respectively.

4.2.1. Influence of the Discharge Ratio on Trajectory Line Variation along the Pollutant Mixing Interface

Figure 7 shows the variation curve along the pollutant mixing interface trajectory line in the intersection area under the influence of the discharge ratio. In the figure, X and Y are nondimensionalized with the main stream river width b = 0.9 m for obtaining the abscissa (X/b) and ordinate (Y/b), respectively. The scattered points are the measured points of the section, and the curve is the resulting analysis and fitting curve. The diagram reveals that the mixing interface trajectory shows a logarithmic growth trend. Along the direction of water flow, the higher the X/b value is, the higher the Y/b value. Near the interchange (X/b ≤ 0.89), the growth trend is obvious, and the growth rate gradually decreases when X/b > 0.89. Compared to the three lines, with increasing discharge ratio, Y/b of the trajectory line increases, and this phenomenon is the most obvious near the intersection. For X/b > 0.89, the three curves are approximately parallel, with the same growth rate. In summary, the discharge ratio imposes a certain influence on the trajectory of the pollutant mixing interface. The higher the discharge ratio is, the farther the diffusion to the other side of the intersection.

4.2.2. Influence of the Width-Depth Ratio on Trajectory Line Variation along the Pollutant Mixing Interface

The change curve of the pollutant mixing interface trajectory line in the confluence region is shown in Figure 8 for three sets of width-depth ratio working conditions. With water flow, the mixing interface gradually diffuses toward the other side of the intersection, and the trajectory line still exhibits a logarithmic growth trend. The higher the width-depth ratio, the farther the mixing interface extends from the intersection to the opposite shore and the higher the growth rate. This rapid growth is mainly manifested near the intersection (X/b ≤ 0.86). In summary, the width-depth ratio exerts a certain degree of influence on the mixing interface trajectory, but there is still no significant influence of the discharge ratio.

4.2.3. Influence of the Concentration Difference on the Trajectory along the Pollutant Mixing Interface

Figure 9 shows the trajectory curve of the pollutant mixing interface in the intersection area under different concentration difference conditions. According to the analysis of the horizontal diffusion concentration field of pollutants, the concentration difference between the main stream and tributary slightly affects the shape characteristics of the pollution zone and only greatly influences the maximum concentration value in the pollution zone within the intersection area. Figure 9 can better support this conclusion. The three curves in the figure almost coincide, showing that the trajectory of the mixing interface does not change greatly with increasing concentration difference. However, along the direction of water flow, with increasing concentration difference, there is a slight difference in the vertical coordinates. This may be due to the molecular diffusion caused by the concentration gradient between the two water flows, which is only reflected in diffusion movement of pollutant molecules along the mixing interface, rather than the hydrodynamic force.

4.3. Mixing Characteristics of the Pollutant Concentration

Dev(x), a nonuniformity measure of the pollutant concentration in a given section, is commonly used to assess the pollutant mixing rate in the section [13,36]. Dev(x), also referred to as the full-mixing deviation, represents the deviation of the pollutant concentration after full mixing downstream, which can be calculated as follows:

Dev(x) = (C_i − C_p)/C_p

(12)

where C_i is the average mass concentration of pollutants in the section, and C_p is the weighted average predicted concentration value of pollutant flow, which can be calculated by Equation (11).

Gaudet and Roy defined complete mixing as a Dev(x) value less than 10% [36]. This demonstrates that the closer Dev(x) is to 10%, the more uniform and adequate the pollutant mixture, and conversely, the less complete the mixture. The mixing interface trajectory line above shows that under the three groups of hydrodynamic circumstances, because the majority of the pollutant mixing interface is focused at the central axis point, the mixing characteristics at the central axis position are considered in this section.

4.3.1. Effect of the Discharge Ratio on the Pollutant Mixing Characteristics

The fluctuation in the pollutant nonuniformity index along the flow development direction in the central axis section (Y = 0.45 m) under varied discharge ratios is shown in Figure 10. Figure 10 shows that the most notable declining tendency of the nonuniformity index is observed near the ground (Z = 0.05 m). For R = 0.267, Dev(x) = 10% at X = 3.2 m suggests that the underlying pollutants have reached total mixing in this area and that the contaminants in the bottom plane swiftly permeate and are uniformly distributed. Comparing the nonuniformity index values in the same plane under the different discharge ratios, it can be found that the nonuniformity index of all planes decreases with increasing discharge ratio, and the decreasing trend in the surface section is greater than that in the bottom section. The higher the discharge ratio is, the lower Dev(x) in the junction area exit section and the clearer the declining trend of Dev(x) at the intersection. In general, the pollutant nonuniformity index in the junction area is strongly influenced by the discharge ratio. Dev(x) in each section of the intersection region decreases with increasing discharge ratio and tends to indicate full mixing, especially for R = 0.267.

4.3.2. Effect of the Width-Depth Ratio on the Mixing Characteristics of Pollutants

Figure 11 shows the mixing nonuniformity index of pollutants in the intersection area for three width-depth ratios. To observed the change trend more directly and clearly under the varied operating conditions, all measurement sites along the z direction are rendered dimensionless by utilizing the main stream upstream water depth h, i.e., Z′ = Z/h. The findings reveal that at the origin O, the nonuniformity index is the same. The change trend and discharge ratio are the same, and Dev(x) exhibits a declining trend along the direction of water flow development. However, the magnitude of the change is not as large as that of the change in the discharge ratio. For b/h = 3.75, Dev(x) changes at the highest rate. Dev(x) of the bottom plane (Z′ = 0.17) is reduced to less than 10% from the junction to the exit part of the intersection area. The contaminants have been combined well. Finally, the higher the width-to-depth ratio is, the better the mixing of pollutants.

4.3.3. Effect of the Concentration Difference on the Mixing Characteristics of Pollutants

Figure 12 shows the difference in the pollutant mixing inhomogeneity index under the influence of the concentration difference between the main stream and tributary inputs. As shown in the figure, Dev(x) decreases with increasing x. The fluctuation trend under the three working conditions, however, is comparable, with only slight deviations. Notably, with increasing concentration difference, the nonuniformity index in each section decreases, but it does not reach complete mixing uniformity. This shows that the pollutant concentration difference slightly affects the nonuniformity index of pollutant mixing, which is only the slight difference caused by molecular diffusion due to the concentration gradient.

5. Discuss

The rule of pollution dispersion in the asymmetric river confluence region is numerically explored to better understand its application and limitations. Some key aspects of the research process, i.e., (1) research methods and (2) data analysis, are further examined below.

The most important aspect of river environmental governance is precisely identifying the distribution pattern of contaminants to develop suitable sewage discharge and treatment programs. The features of river pollutant transmission, diffusion, and mixing in confluence areas have long been a hot issue in water environment management research. A gas—liquid two-phase flow mixing model was utilized by Wei Wenli et al. [37] to numerically simulate the three-dimensional hydraulic properties of equal-width open channel interchange at various junction angles, and the velocity nonuniformity coefficient distribution trend in each downstream cross section was determined. Han S. et al. [13] numerically simulated the mixing characteristics of pollutants in indoor converging open channels. The mixing speed of pollutants at the interchange was mostly governed by the upstream input flow, with a higher mixing speed when the inflow was minimal. Lan Bo [4] and numerous international scholars [38,39,40,41,42,43] analyzed the full features of confluence estuaries using prototype observation data.

One of the most fundamental forms of confluence rivers is the asymmetric river type. Because of the complex flow characteristics, contaminants flowing from tributaries into the main stream can hardly be immediately transferred and modified at the confluence location. Some researchers have discovered that mathematical models may be used to estimate the river water quality; therefore, it is widely considered that the water quality model can best simulate the diffusion trend of contaminants in water. Streeter and Phelps [44] developed the first oxygen balancing model, known as the S-P model, which paved the way for mathematical models of aquatic environments. With the advancement of research, two-dimensional models can no longer satisfy the development needs, and a three-dimensional model can be used after accounting for the influence of vertical water flow. Signell [45], Ng [46], Isabel [14], Azevedo [14], and Xing et al. [47] developed three-dimensional hydrodynamic water quality models to investigate the transport mechanism of contaminants in estuaries and obtained useful findings. Compared with previous studies, in this paper, a fast numerical simulation method is used, and a three-dimensional hydrodynamic water quality model is employed considering vertical inflow to systematically study the diffusion pattern of pollutants in the intersection area. By adjusting the inflow conditions (discharge ratio, width-depth ratio, and concentration difference), the lateral diffusion characteristics, mixing interface trajectory and mixing characteristics of pollutants were analyzed from horizontal, vertical and vertical perspectives. The study of typical confluence rivers could provide a reference for the reasonable layout and management of water environments.

The horizontal dispersion of pollutants is directly connected to river pollution and determines whether water on the opposite side of sewage discharge is contaminated. Yang Zhishan [48], Mi Tan [30], and Lu Weigang [49] investigated the horizontal diffusion distribution of contaminants using numerical simulation or theoretical deduction methods. The findings revealed that the horizontal diffusion distribution of contaminants varies depending on the hydrodynamic circumstances. The discharge ratio and the width-depth ratio are proportional to the horizontal distribution of pollutants, while the pollutant concentration exerts a minimal impact on the horizontal distribution. Compared to the discharge ratio, the aspect ratio yields a smaller effect. When the discharge ratio was decreased from R = 0.267 to 0.133, the size of the polluted zone in the intersection area decreased by 26.43%, and when the aspect ratio was decreased from 3.75 to 2.50, the size of the polluted zone in the intersection area decreased by 16.53%. According to the three operating conditions, when the concentration difference was decreased from 2000 to 500 μg/L, the polluted area formed in the intersection area was only reduced by approximately 1%.

The path and features of the mixing interface of the polluted zone directly influence the design of the water environment remediation scheme. A comprehensive understanding of the movement trajectory and mixing rate at the pollutant mixing interface may clarify the pollutant concentration distribution gradient in each segment, and different treatment methods can be deployed in each section. In the achievement of water environment governance, this may also reduce governance costs. Biron [13] numerically modeled pollutant mixing in an indoor intersecting open channel and investigated the link between the mixing rate and incoming flow. Yuan Hang [50] used a physical model of the junction of water and grass between the oblique branch and main stream to investigate the effect of different flow ratios on the mixing characteristics of pollutants. Hua [51] investigated the mixing properties of contaminants under various junction scenarios using a physical model of a multi-tributary river. Based on the coupling model created after secondary development, Wu Dean et al. [52] investigated the aggregate concentration fluctuations of dissolved pollutants in the water body movement process and provided the migration trajectory of substances in the water body. Based on previous research, in this work, the trajectory and change trends of the mixing characteristics and the mixing interface of pollutants were investigated under various hydrodynamic circumstances, and the variation patterns under the different working conditions were summarized.

In this article, only the transport and diffusion of contaminants are investigated under hydrodynamic circumstances. Other influencing factors must be further studied.

6. Conclusions

(1)

The dependability of the numerical model is simulated and assessed. According to the verification findings, the flow rate and pollutant content in each section occur within tolerable limits. The numerical model may be used to explore the diffusion mechanism of contaminants in the asymmetric river junction region and to better capture hydrodynamic and water quality changes in the river intersection area.

(2)

The three-dimensional properties of the pollutant concentration distribution in the junction area are analyzed. The findings indicate that the discharge ratio and the aspect-to-width-depth ratio significantly impact the distribution of pollutants in the junction region. This is mostly manifested in the horizontal distribution of pollutants, the trajectory of the mixing interface, and the degree of mixing homogeneity. The horizontal diffusion range of pollutants increases with increasing discharge ratio and width-depth ratio, and the mixing homogeneity in each section increases. Pollutants in the bottom plane Z = 0.05 m are totally mixed in the downstream exit section for R = 0.267 and b/h = 3.75. In general, the trajectory line of the mixing interface of pollutants in the junction region exhibits a logarithmic growth tendency, and it progresses along the direction of water flow development. The mixing interface expands to the center axis point after progressively moving to the other side of the interchange.

(3)

The concentration difference affects the horizontal distribution and mixing degree of pollutants. The degree of influence, however, is not as high as that of the discharge ratio or width-depth ratio, with only a slight impact. However, the mixing interface trajectory line still exhibits a logarithmic development pattern, and with increasing concentration difference, the mixing interface slightly deviates to the opposite side of the interchange. Molecular diffusion due to concentration variations causes subtle changes in the mixing interface and inhomogeneity index. In summary, the concentration difference only affects the concentration in the pollution belt, but does not influence its width, length, or size.