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Abstract [Objectives] This study was conducted to detect the twodimensional diffusion concentration distribution from sloped wave bank.
[Methods] Diffusion experiments of instantaneous line source discharge were carried out using two sloped wave banks with different inclination angles based on the developed twodimensional diffusion tank device for sloped wave banks by the apex discharge method under grid oscillation frequencies n=15， 20， 40 and 60 r/min. The image acquisition and digital image processing techniques were applied to measure the twodimensional concentration field distribution and to analyze the distribution laws of the pollutant in the angular field.
[Results] The diffusion of the pollutant in the sloped wave bank area became faster with the increase of the grid oscillation frequency， and the pollution range became wider with the diffusion time. The point concentration of the pollutant at the water surface monotonically decreased with the increase of the abscissa， and the vertical concentration distribution decreased with the increase of water depth. The transverse diffusion rate of the pollutant in water was greater than the vertical diffusion rate， and its concentration distribution exhibited a distribution characteristic of farther diffusion in the adjacent area on the water surface. The diffusion experiment area of the sloped wave bank at θ=30° had a higher concentration of the pollutant at each point compared with the diffusion experiment of the sloped bank at θ=45°， under the same experimental conditions. A largescale vortex appeared near the sloped wave bank at θ=45° during the experiment， and the presence of the vortex made the concentration distribution of the pollutant in the direction along the bank slope first decrease and then increase， while no obvious vortex was observed near the sloped wave bank at θ=30°， and the concentration of the pollutant decreased monotonously along the bank slope direction.
[Conclusions] This study is of great significance for the concentration distribution laws and the lateral and vertical diffusion coefficients of side discharge at complex bank slopes and river banks.
Key words Sloped wave bank; Digital image processing; Concentration distribution; Experimental research
In the past， the bank slope was simplified to a sloped bank and some results were obtained when studying side discharge into rivers and reservoirs. Liu et al.[1]derived a formula for the concentration distribution of pollutants in trapezoidal channels under the conditions of instantaneous line source side discharge. Zeng et al.[2]studied the transverse mixing characteristics of pollutants in a complex trapezoidal channel. Holley et al.[3]explored the law of concentration distribution in a trapezoidal channel with depth under side discharge conditions. Li et al.[4]calculated the diffusion coefficient of pollutants in a trapezoidal channel according to the concentration distribution of pollutants in the trapezoidal channel. Boxall et al.[5]analyzed and predicted the transverse mixing coefficient of natural river channels. Chen et al.[6]deduced the longitudinal dispersion coefficient of pollutant diffusion based on the cross sectional velocity distribution in a trapezoidal channel. Huang et al.[2，7-8]analyzed several environmental hydraulics problems in the Three Gorges Project， and calculated its water environmental capacity based on the numerical simulation of the pollutant zone near the discharge outlet in the Three Gorges Reservoir. Under the condition that the angular field mapping coefficient β was even and odd， respectively， Wu[9-10]derived the theoretical formulas for the concentration distribution of an instantaneous line source and a constantintensity continuous point source from the apex of an angular field， and discussed the periodic variation law of the discharge from the apex of the angular field with the odd and even numbers of the angular field mapping coefficient β and a multiple of β=4 on this basis. In real life， natural river and reservoir banks are mostly complex or irregular complex bank slopes， which can be simplified into side discharge models of following complex bank slopes： five sloped wave banks， sloped ladder banks， sloped wave ladder banks， complex ladder banks， and complex trapezoidal banks. In this study， the side discharge at two sloped wave banks with different inclination angles was first investigated based on the developed twodimensional diffusion tank device for sloped wave bank experiments[11]by the apex discharge method when simulating different diffusion coefficients with different grid oscillation frequencies. The image acquisition and digital image processing techniques were used to measure the twodimensional concentration field and to process and analyze the experimental results. Up to now， the research on the concentration distribution pattern of side discharge at complex bank slopes and river banks is still in blank， which affects the development of environmental hydraulics diffusion theory. And the transverse and vertical diffusion coefficients and their change laws cannot be calculated using the concentration distribution simultaneously observed from pollution zones at sewage outlets， making the design of sewage outlets under complex bank slopes of rivers and reservoirs lack theoretical guidance. All these indicate that the concentration distribution laws and the lateral and vertical diffusion coefficients of side discharge at complex sloped river banks are urgently needs to be resolved in theory， and this study is of great significance.
Materials and Methods
Experimental equipment
Experimental devices and apparatuses
The vertical twodimensional diffusion water tank device mainly includes an outer water tank， an inner water tank， a grid oscillation system， a background light box， a pollutant adding system and a drainage and flushing system. The inner water tank consists of a steel frame and plexiglass which form an internal space with the upper part open and other parts closed to provide a water environment for the inner water tank. The inner water tank， i.e.， the diffusion tank， simulates the bank slope boundary. It is made of plexiglass with no cover at the top and an opening at the lower right （connecting with the water in the outer water tank to ensure equal pressure inside and outside the inner water tank）， and the rest is closed. The grid oscillation system consists of a variable frequency motor， an eccentric wheel， a slide rail and a grid， which produces turbulence in the water body with different strengths. The background light box provides uniform light and reduces interference from stray light. The pollutant addition system consists of a drug adding box and a control board. It is located in the upper left corner of the device， and produces transient and lowmomentum discharge. A drain valve is located at the lower right corner of the device. After the experiment is completed， waste water is discharged under water pressure， and the water tanks can be repeatedly flushed with clean water to ensure that the water tanks are smooth and nonstained for the next experiment. The specific picture is shown in Fig. 1. The device is designed to simulate the diffusion process of pollutants in the angular field formed by sloped wave bank under different diffusion coefficient conditions， as shown in Fig. 1. The experimental apparatuses mainly include electronic balance， 1 000 ml measuring cup， 100 ml measuring cylinder， glass rod， thermometer， cuvette， steel tap， timer， digital camera and tripod.
Diffusion tank model
The main geometric characteristic parameters of the sloped wave bank are the average bank inclination angle θ， wave height δ， and wave period λ. The inner water tank is made of plexiglass. The main body of the tank is formed by a sloped bottom plate and front， rear and right side walls with strong adhesive. In order to simulate the geometric characteristics of wave bank slopes in rivers， this experiment used 1/4 plexiglass round tubes （R=70 mm）. The 1/4 plexiglass round tubes were adhered to the sloped plexiglass bottom plate. The upper edge of the first 1/4 plexiglass round tube was 100 mm away from the top of the sloped bottom plate， and the span of each 1/4 plexiglass round tube was also 100 mm， so λ was equal to 200 mm. The wave height was the height of the 1/4 circle. It could be known from calculation that δ was equal to 20 mm.
Experimental methods and image processing techniques
Experiment scheme and steps
For the two kinds of water tanks shown in Fig. 2， the diffusion experiments of instantaneous line source discharge were performed respectively according to the grid oscillation frequency n=15， 20， 40 and 60 r/min， respectively， and there were a total of 8 working condition combinations. Under the same conditions， each diffusion experiment were repeated at least once， and the results of the two experiments nearly the same were selected for analysis. The experimental steps were given below.
（1） In order to be able to compare the results of the experimental study with those of the previous study on the discharge from the apex of an angular field under the condition of sloped bank， the concentration of tracer in this experiment was still 2 100 mg/L.A certain amount of rhodamine B （2.1 g） was weighed into a 1 000 mlmeasuring cylinder， and added with about 100 ml of water， followed by stirring and standing for 1-2 h to allow complete dissolution. A camera was arranged at the same time.
（2） After the drug was completely dissolved， water was filled into the water tank until the liquid level was 1-2 cm higher than the bottom of the drug adding box. （3） The motor was turned on and adjusted to the required speed， and the grid reciprocated for 5 min to stabilize the turbulence in the water tank. The temperature of the water in the water tank was measured and recorded.
（4） The drug solution in the measuring cylinder was diluted to 1 000 ml with the same tap water， and the temperature of the drug solution was measured and recorded to ensure that the difference between the temperature of the drug solution and the temperature of the water in the tank did not exceed 0.5 ℃.
（5） The backlight was turned on to give the tank a uniform background light.
（6） The drug solution was slowly added into the drug box at the left side of the water tank， and the partition was then removed to allow the drug solution to flow into the water tank.
（7） When the drug solution flowed into the water tank， timing was started. The first photo of the water tank was taken at 5 s， the second photo at 15 s， and then photos was taken every 15 s until the experiment was completed.
（8） When the liquid spread to the right border of the tank， the experiment was stopped.
（9） The valve was opened to drain the sewage containing the drug solution into the sewer， and the water tank was cleaned with clean water.
（10） Pictures were exported from the camera for digital image processing.
Standardization and measurement of concentration
Because rhodamine B is chemically stable[12]， the color gradients of solutions at different concentrations are large. Furthermore， it has good solubility， so it was selected as the tracer for the experiments. When the rhodamine B solution concentration is small， it is light red， corresponding to a large gray value; and when the concentration is large， it is dark red， corresponding to a small gray value. In order to obtain the quantitative relationship between the rhodamine B solution concentration and the digital image gray， a cuvette （width × height × thickness = 130 mm × 170 mm × 110 mm） was made of the plexiglass the same as the vertical water tank with the same thickness， and rhodamine B standard solutions with eight different concentrations were also prepared as shown in Table 2. And under the same conditions as the diffusion experiments， pictures were taken according to the image acquisition method of concentration field measurement， and then imported into a computer digital image processing system to get the corresponding standard gray values. Zhengtao YANG et al. Experimental Study of Vertical Twodimensional Diffusion Concentration Distribution in Sloped Wave Bank Angular Field
Based on this， the standard equation of the concentration （C） gray （G） fitting curve was obtained as following：
C=12329.07/G-95.67 （1）
Wherein the coefficient of determination （Rsquare） of the fitted curve is 0.998 1， which is very close to 1， indicating that the fitting effect is very good. A digital camera was used to collect tracer distribution images at different times during the diffusion experiments， and then they were input to a computer. The digital image processing technique was applied to measure and analyze the collected concentration fields， and then the concentration field distribution was obtained using the conversion relationship of formula （1）. In this studied， twodimensional concentration field images were collected using a Nikon D700 digital camera. The concentration field collection and measurement are shown in Fig. 3.
Before the start of the diffusion experiment， the camera was mounted 5 m directly in front of the water tank， and the height and focal length were adjusted to make the inner water tank image clear and suitable in the camera. The photographing interval was set to 15 s， and timing was started when the partition of the drug adding box was removed. After each experiment， the pictures were imported into a computer digital processing system and digital image processing was performed to obtain a twodimensional diffusion concentration field distribution.
Digital image processing technique
The digital image processing technique for measuring the concentration field does not change the nature of the original field and can measure the concentration at all positions in the concentration field. It has the advantages of high fidelity and wide coverage， which is a great improvement in environmental hydraulics. Based on the existing results of Ji et al.[13-14]， this processing technique was modified and improved with the help of MATLAB platform. The processing flow is shown in Fig. 4.
The experimental photos were preprocessed， and it was found that the homomorphic filtering algorithm had a good effect on eliminating the influence of uneven illumination. A nonlinear spatial filter was used to deal with the noise interference during the experiment. For the impact of the grid， the open operation processing of erosion first and expansion then was better. To extract the diffusion area， the RGB color space was first converted to HSV， HSI and YCbCr color spaces， which were then compared， and it was found that the Cr component of the YCbCr color space was the best for the description of the red diffusion region.
Results and Analysis
Experimental study on sloped wave bank with the inclination angle θ=45°
According to the experimental steps， the concentration distribution of the sloped wave bank with θ=45° was experimentally studied. Table 2 shows the experimental parameters.
It can be seen from Fig. 5 that when n= 15 r/min， the grid oscillation frequency was low and the turbulence of the water body was relatively gentle. The tracer diffused along the water surface after entering the water body. Over time， the tracer diffused transversely and vertically， and the transverse diffusion rate was greater than the vertical diffusion. Comparing the figures from Fig. 5 to Fig. 8， it was found that with the increase of the grid oscillation frequency， the diffusion of the tracer in the angular field also became faster. This indicates that the larger the oscillation frequency， the greater the turbulence intensity of the water body， resulting in a larger diffusion coefficient. When we studied the discharge from the apex of the sloped bank angular field， following conclusion was obtained： the discharge from the apex of the angular region of the sloped bank showed a trend of the concentration field gradient decreasing from the source point in the radial direction. However， in this study， we observed that the distribution of the pollutant showed different laws. At different grid oscillation frequencies， vortices appeared in the upper left area of the sloped wave bank angular field during the experiment， and the presence of vortices changed the concentration field distribution in the area. Due to the rotation， the concentration of the pollutant in the vortex area was small， and the concentration of the pollutant at the edge of the vortex was large. The concentration decreased first and then increased in the radial direction from the source， and the larger the grid oscillation frequency， the more obvious the phenomenon. The analysis showed that the reciprocating oscillation of the grid disturbed the interaction between the water body and the sloped wave bank at the bottom to generate a vortex， and the size and intensity of the vortex increased with the increase of the oscillation frequency of the grid.
Fig. 9 shows the concentration contour distribution maps in the pollutant diffusion area of the sloped wave bank with the inclination angle θ=45° under a grid oscillation frequency of 60 r/min at different time conditions. After the pollutant entered the angular water body， it diffused in the horizontal and vertical directions. It could be seen that the transverse diffusion rate was greater than the vertical diffusion rate， and the concentration extended longer at the water surface than at the slope. At t=30 s， due to the short diffusion time， most of the pollutant still accumulated in the nearbank area. The pollutant concentration was high and had large concentration gradient， and the nearbank area was polluted seriously. With the passage of time， the pollutant further diffused in the angular field， leading to expansion in the pollution range and decreases in the high concentration area and concentration. The turbulence of the water body and the action of sloped wave bank at the bottom produced a largescale vortex， which changed the distribution characteristics of the pollutant concentration. The concentration of the pollutant was lower in the vortex area and high in the vortex edge area. It can be seen from Fig. 9c and Fig. 9d that the concentration of the pollutant in the sloped wave bank angular region showed a change raw of decreasing at first and then increasing from the bank along the horizontal direction， which is different from the law that the concentration of the pollutant in the sloped bank angular field decreased with the distance from the apex of the bank slope increasing. Fig. 10 shows the concentration contour distribution maps in the pollutant diffusion area of the sloped wave bank with the inclination angle θ=45° at t=90 s under different grid oscillation frequency conditions. It could be seen from the figure that the oscillation frequency had a great effect on the distribution of the pollutant. The higher the oscillation frequency was， the faster the pollutant diffused， leading to a wider diffusion range. This meant that larger turbulence produced by the grid oscillation caused the diffusion coefficient to increase. From Fig. 10a and Fig. 10b， it can be seen that when the frequency of the grid oscillation was smaller， the turbulence of the water body acted with the bottom bank slope， forming a smaller vortex， which had less impact on the distribution of the pollutant. From Fig. 10c and Fig. 10d， it can be seen that when the oscillation frequency of the grid increased， the size of the vortex generated in the water body became larger， and the effect on the concentration distribution of the pollutant became more obvious. The concentration contours in the figures had a jag
curve. The concentration curve charts were drawn by reading the processed experimental pictures at different oscillation frequencies， and the concentration distribution of the pollutants at different oscillation frequencies were compared and analyzed， as shown in Fig. 11 and Fig. 12.
Fig. 11 showed the transverse concentration distribution of water surface points at different oscillation frequencies. It can be seen from the figure that the point concentrations of the pollutant at the water surface were inversely proportional to the distance from the discharge point， and the curve trends under different oscillation frequencies were similar. With the increase of the oscillation frequency， the abscissa of the water concentration contour also increased， which is because that the transverse diffusion coefficient increases with the oscillation frequency increasing， and the pollutants further diffuse in the transverse direction at a faster rate， leading to a wider range within the same time. It can be seen from Fig. 12 that the concentration of the pollutant decreased monotonically with the increase of water depth， and the curve trends of the concentration of the pollutant with water depth were basically the same at different grid oscillation frequencies. At the same water depth， the concentration of the pollutant increased with the increase of the grid oscillation frequency， which indicates that the larger the oscillation frequency， the larger the vertical diffusion coefficient， leading to faster pollutant diffusion in the vertical direction. Experimental study on the sloped wave bank with the inclination angle θ=30°
According to the experimental steps， concentration distribution of sloped wave bank with θ=30° was experimentally studied. Table 3 shows the experimental parameters.
Fig. 13Fig. 16 are experimental photographs of the pollutant concentration distribution in the angular field of the sloped wave bank at the average bank slope inclination angle θ=30° under different grid oscillation frequencies.
It can be seen from the figure that the pollutant diffused in the horizontal and vertical directions after entering the water body. The diffusion of the pollutant along the adjacent area of the water surface was faster than that on the adjacent area of the sloped bank， and its concentration distribution exhibited the characteristic of expanding farther on the adjacent area of the water surface. When the grid oscillation frequency was small， the diffusion coefficient was small， the pollutant diffused slowly and accumulated in the nearbank area， which made the nearbank seriously polluted. With the increase of the grid oscillation frequency， the diffusion coefficient also became larger， and then the pollutant diffusion distance was longer at the same diffusion time， leading to a decrease in the pollutant concentration in the nearbank area， but an increase in pollution range. Compared with the diffusion experiments of the sloped wave bank at θ=45°， it was found that under the same experimental conditions， the diffusion experiment area of the sloped wave bank at θ=30° had a higher concentration of the pollutant at each point， which was due to the fact that when the slope angle of a bank slope is smaller， the space of the angular field is smaller， and the reflection and staking of pollutants is more frequent， resulting in obvious concentration increase. During this group of experiments， no obvious largescale vortexes appeared near the sloped wave bank， which was a big difference from the phenomenon in the diffusion experiment of sloped wave bank at θ=45°. Because the water depth of the sloped wave bank at θ=30° was smaller， the scale of the vortex generated was smaller， and the disturbance to the water body was not obvious.
Fig.17 shows the concentration contour distribution maps of the pollutant diffusion area of the sloped wave bank with the bank slope inclination angle θ=30° under a grid oscillation frequency of 60 r/min at different time conditions. After the pollutant entered the water body， it diffused in the horizontal and vertical directions. It could be seen that the transverse diffusion rate was greater than the vertical diffusion rate， and the concentration extended longer at the water surface than along the sloped wave bank. It also could be seen that within a short diffusion time， most of the pollutant accumulated in the nearbank area， and the pollutant concentration in the pollution area was high and had large concen
tration gradients. With the passage of time， the pollutant gradually diffused in the angular field， leading to expansion in the pollution range and decreases in the high concentration area and concentra
tion. Moreover， the concentration field gradient showed a downward trend from the source point in the radial direction， which is the same as the distribution law of the concentration of sewage discharge from the apex of the sloped bank angular field. The concentration contours in the experimental area had a jagged appearance， which was caused by the uneven local turbulence caused by oscillation， which led to certain difference in the local pulsation speed in the water tank， further resulting in changes in the concentration contours， but this situation could be eliminated by observing the overall trend of the curve.
Fig. 18 is the concentration contour distribution maps in the pollutant diffusion area of the sloped wave bank with the bank slope inclination angle θ=30° at 60 s under different grid oscillation frequencies. It can be seen from the figure that the distribution of the pollutant varied greatly at different grid oscillation frequencies. The greater the grid oscillation frequency was， the faster the pollutant diffused， leading to decreases in high concentration area and contour gradient and an increase in pollution range. This showed that the larger the oscillation frequency， the greater the turbulence intensity of the water body， which caused the diffusion coefficient to increase. The water surface points in Fig. 18 were read to draw the pollutant concentration distribution curves of the water surface points and in the vertical direction.
Fig. 19 shows the transverse concentration distribution of water surface points at different oscillation frequencies. It can be seen from the figure that the point concentration of the pollutant at the water surface decreased monotonically with the increase of the abscissa， and the curve trends were basically the same under different oscillation frequencies. At the same time， the distance of the concentration contour of the same value from the bank increased with the increase of the oscillation frequency， which indicated that the larger the oscillation frequency was， the faster the pollutant spread in the transverse direction. Fig. 20 shows the vertical concentration distribution at y=90 cm and t=90 s. It can be seen from the figure that the pollutant concentration distribution monotonically decreased with the water depth increasing at different oscillation frequencies， and the concentration of the pollutant at the same water depth increased with the increase of the oscillation frequency. This is because the larger the grid oscillation frequency， the larger the vertical diffusion coefficient， leading to faster pollutant diffusion in the vertical direction.
Conclusions and Prospects
In this study， two different sloped wave banks at the bank slope inclination angles θ=30° and θ=45° were experimentally studied for pollutant diffusion. The instantaneous apex line source discharge experiments with different diffusion coefficients were achieved by changing the oscillation frequency of the grid， and the digital image processing method was used to intuitively obtain the tracer concentration distribution. By analyzing the experimental results， conclusions were obtained as below.
（1） Within the same diffusion time， the larger the grid oscillation frequency， the larger the diffusion coefficient， leading to the faster diffusion of the pollutant. At the same oscillation frequency， as the diffusion time increased， the high concentration area gradually decreased， the concentration gradients decreased， and the pollution range increased.
（2） The point concentration of the pollutant at the water surface monotonically decreased with the increase of the abscissa， and the vertical concentration distribution decreased with the increase of water depth.
（3） The transverse diffusion rate of the pollutant in water was greater than the vertical diffusion rate. The diffusion rate of the pollutant in the adjacent area along the water surface was higher than that in the adjacent area on the sloped bank， and its concentration distribution exhibited a distribution characteristic of farther diffusion in the adjacent area on the water surface.
（4） Compared with the diffusion experiment of the sloped bank at θ=45°， under the same experimental conditions， the diffusion experiment area of the sloped wave bank at θ=30° had a higher concentration of the pollutant at each point， which was due to the fact that when the slope angle of a bank slope is smaller， the space of the angular field is smaller， and the reflection and stacking of pollutants is more frequent， resulting in obvious concentration increase.
（5） A largescale vortex appeared near the sloped wave bank at θ=45° during the experiment， and the presence of the vortex made the concentration distribution of the pollutant in the direction along the bank slope first decrease and then increase， while no obvious vortex was observed near the sloped wave bank at θ=30°， and the concentration of the pollutant decreased monotonously alongthe bank slope direction.
In the subsequent work， more angular field angles θ=2π/β （β=8， 9， 10， 11……） and different wave or step heights can be selected for sloped wave banks to conduct hydraulic model diffusion experiments and data analysis， to thereby study the distribution laws of side discharge concentration with the geometric characteristics of each bank slope， the grid oscillation frequency and time. The experimental results of hydraulic model diffusion under the conditions of sloped ladder bank， sloped wave bank and sloped wave ladder bank can be compared with the theoretical analysis and experimental results of the concentration distribution discharged from the apex of sloped bank angular field. For each of sloped wave ladder bank， sloped ladder bank， complex ladder bank slope and complex trapezoidal bank slope， more than 4 groups of geometric scales can be selected to conduct hydraulic model diffusion experiments and data analysis， so as to study the change laws of the concentration distribution of side discharge with the geometric characteristic parameters of each bank slope， grid oscillation frequency and time. The theoretical derivation methods of lateral diffusion coefficient and vertical diffusion coefficient for five types of complex bank slopes can be proposed， respectively， to obtain the change laws and quantitative relationships of the lateral diffusion coefficient and vertical diffusion coefficient with the geometric characteristic parameters of each bank slope， grid oscillation frequency and time. The depthaveraged concentration distribution and average lateral diffusion coefficients for five types of complex bank slopes can be calculated， respectively， to propose a theoretical derivation method for calculating the depthaveraged lateral diffusion coefficient， so as to obtain the change laws and quantitative relationships of the depthaveraged lateral diffusion coefficient with the geometric characteristics of each bank slope， the grid oscillation frequency and time. References
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[Methods] Diffusion experiments of instantaneous line source discharge were carried out using two sloped wave banks with different inclination angles based on the developed twodimensional diffusion tank device for sloped wave banks by the apex discharge method under grid oscillation frequencies n=15， 20， 40 and 60 r/min. The image acquisition and digital image processing techniques were applied to measure the twodimensional concentration field distribution and to analyze the distribution laws of the pollutant in the angular field.
[Results] The diffusion of the pollutant in the sloped wave bank area became faster with the increase of the grid oscillation frequency， and the pollution range became wider with the diffusion time. The point concentration of the pollutant at the water surface monotonically decreased with the increase of the abscissa， and the vertical concentration distribution decreased with the increase of water depth. The transverse diffusion rate of the pollutant in water was greater than the vertical diffusion rate， and its concentration distribution exhibited a distribution characteristic of farther diffusion in the adjacent area on the water surface. The diffusion experiment area of the sloped wave bank at θ=30° had a higher concentration of the pollutant at each point compared with the diffusion experiment of the sloped bank at θ=45°， under the same experimental conditions. A largescale vortex appeared near the sloped wave bank at θ=45° during the experiment， and the presence of the vortex made the concentration distribution of the pollutant in the direction along the bank slope first decrease and then increase， while no obvious vortex was observed near the sloped wave bank at θ=30°， and the concentration of the pollutant decreased monotonously along the bank slope direction.
[Conclusions] This study is of great significance for the concentration distribution laws and the lateral and vertical diffusion coefficients of side discharge at complex bank slopes and river banks.
Key words Sloped wave bank; Digital image processing; Concentration distribution; Experimental research
In the past， the bank slope was simplified to a sloped bank and some results were obtained when studying side discharge into rivers and reservoirs. Liu et al.[1]derived a formula for the concentration distribution of pollutants in trapezoidal channels under the conditions of instantaneous line source side discharge. Zeng et al.[2]studied the transverse mixing characteristics of pollutants in a complex trapezoidal channel. Holley et al.[3]explored the law of concentration distribution in a trapezoidal channel with depth under side discharge conditions. Li et al.[4]calculated the diffusion coefficient of pollutants in a trapezoidal channel according to the concentration distribution of pollutants in the trapezoidal channel. Boxall et al.[5]analyzed and predicted the transverse mixing coefficient of natural river channels. Chen et al.[6]deduced the longitudinal dispersion coefficient of pollutant diffusion based on the cross sectional velocity distribution in a trapezoidal channel. Huang et al.[2，7-8]analyzed several environmental hydraulics problems in the Three Gorges Project， and calculated its water environmental capacity based on the numerical simulation of the pollutant zone near the discharge outlet in the Three Gorges Reservoir. Under the condition that the angular field mapping coefficient β was even and odd， respectively， Wu[9-10]derived the theoretical formulas for the concentration distribution of an instantaneous line source and a constantintensity continuous point source from the apex of an angular field， and discussed the periodic variation law of the discharge from the apex of the angular field with the odd and even numbers of the angular field mapping coefficient β and a multiple of β=4 on this basis. In real life， natural river and reservoir banks are mostly complex or irregular complex bank slopes， which can be simplified into side discharge models of following complex bank slopes： five sloped wave banks， sloped ladder banks， sloped wave ladder banks， complex ladder banks， and complex trapezoidal banks. In this study， the side discharge at two sloped wave banks with different inclination angles was first investigated based on the developed twodimensional diffusion tank device for sloped wave bank experiments[11]by the apex discharge method when simulating different diffusion coefficients with different grid oscillation frequencies. The image acquisition and digital image processing techniques were used to measure the twodimensional concentration field and to process and analyze the experimental results. Up to now， the research on the concentration distribution pattern of side discharge at complex bank slopes and river banks is still in blank， which affects the development of environmental hydraulics diffusion theory. And the transverse and vertical diffusion coefficients and their change laws cannot be calculated using the concentration distribution simultaneously observed from pollution zones at sewage outlets， making the design of sewage outlets under complex bank slopes of rivers and reservoirs lack theoretical guidance. All these indicate that the concentration distribution laws and the lateral and vertical diffusion coefficients of side discharge at complex sloped river banks are urgently needs to be resolved in theory， and this study is of great significance.
Materials and Methods
Experimental equipment
Experimental devices and apparatuses
The vertical twodimensional diffusion water tank device mainly includes an outer water tank， an inner water tank， a grid oscillation system， a background light box， a pollutant adding system and a drainage and flushing system. The inner water tank consists of a steel frame and plexiglass which form an internal space with the upper part open and other parts closed to provide a water environment for the inner water tank. The inner water tank， i.e.， the diffusion tank， simulates the bank slope boundary. It is made of plexiglass with no cover at the top and an opening at the lower right （connecting with the water in the outer water tank to ensure equal pressure inside and outside the inner water tank）， and the rest is closed. The grid oscillation system consists of a variable frequency motor， an eccentric wheel， a slide rail and a grid， which produces turbulence in the water body with different strengths. The background light box provides uniform light and reduces interference from stray light. The pollutant addition system consists of a drug adding box and a control board. It is located in the upper left corner of the device， and produces transient and lowmomentum discharge. A drain valve is located at the lower right corner of the device. After the experiment is completed， waste water is discharged under water pressure， and the water tanks can be repeatedly flushed with clean water to ensure that the water tanks are smooth and nonstained for the next experiment. The specific picture is shown in Fig. 1. The device is designed to simulate the diffusion process of pollutants in the angular field formed by sloped wave bank under different diffusion coefficient conditions， as shown in Fig. 1. The experimental apparatuses mainly include electronic balance， 1 000 ml measuring cup， 100 ml measuring cylinder， glass rod， thermometer， cuvette， steel tap， timer， digital camera and tripod.
Diffusion tank model
The main geometric characteristic parameters of the sloped wave bank are the average bank inclination angle θ， wave height δ， and wave period λ. The inner water tank is made of plexiglass. The main body of the tank is formed by a sloped bottom plate and front， rear and right side walls with strong adhesive. In order to simulate the geometric characteristics of wave bank slopes in rivers， this experiment used 1/4 plexiglass round tubes （R=70 mm）. The 1/4 plexiglass round tubes were adhered to the sloped plexiglass bottom plate. The upper edge of the first 1/4 plexiglass round tube was 100 mm away from the top of the sloped bottom plate， and the span of each 1/4 plexiglass round tube was also 100 mm， so λ was equal to 200 mm. The wave height was the height of the 1/4 circle. It could be known from calculation that δ was equal to 20 mm.
Experimental methods and image processing techniques
Experiment scheme and steps
For the two kinds of water tanks shown in Fig. 2， the diffusion experiments of instantaneous line source discharge were performed respectively according to the grid oscillation frequency n=15， 20， 40 and 60 r/min， respectively， and there were a total of 8 working condition combinations. Under the same conditions， each diffusion experiment were repeated at least once， and the results of the two experiments nearly the same were selected for analysis. The experimental steps were given below.
（1） In order to be able to compare the results of the experimental study with those of the previous study on the discharge from the apex of an angular field under the condition of sloped bank， the concentration of tracer in this experiment was still 2 100 mg/L.A certain amount of rhodamine B （2.1 g） was weighed into a 1 000 mlmeasuring cylinder， and added with about 100 ml of water， followed by stirring and standing for 1-2 h to allow complete dissolution. A camera was arranged at the same time.
（2） After the drug was completely dissolved， water was filled into the water tank until the liquid level was 1-2 cm higher than the bottom of the drug adding box. （3） The motor was turned on and adjusted to the required speed， and the grid reciprocated for 5 min to stabilize the turbulence in the water tank. The temperature of the water in the water tank was measured and recorded.
（4） The drug solution in the measuring cylinder was diluted to 1 000 ml with the same tap water， and the temperature of the drug solution was measured and recorded to ensure that the difference between the temperature of the drug solution and the temperature of the water in the tank did not exceed 0.5 ℃.
（5） The backlight was turned on to give the tank a uniform background light.
（6） The drug solution was slowly added into the drug box at the left side of the water tank， and the partition was then removed to allow the drug solution to flow into the water tank.
（7） When the drug solution flowed into the water tank， timing was started. The first photo of the water tank was taken at 5 s， the second photo at 15 s， and then photos was taken every 15 s until the experiment was completed.
（8） When the liquid spread to the right border of the tank， the experiment was stopped.
（9） The valve was opened to drain the sewage containing the drug solution into the sewer， and the water tank was cleaned with clean water.
（10） Pictures were exported from the camera for digital image processing.
Standardization and measurement of concentration
Because rhodamine B is chemically stable[12]， the color gradients of solutions at different concentrations are large. Furthermore， it has good solubility， so it was selected as the tracer for the experiments. When the rhodamine B solution concentration is small， it is light red， corresponding to a large gray value; and when the concentration is large， it is dark red， corresponding to a small gray value. In order to obtain the quantitative relationship between the rhodamine B solution concentration and the digital image gray， a cuvette （width × height × thickness = 130 mm × 170 mm × 110 mm） was made of the plexiglass the same as the vertical water tank with the same thickness， and rhodamine B standard solutions with eight different concentrations were also prepared as shown in Table 2. And under the same conditions as the diffusion experiments， pictures were taken according to the image acquisition method of concentration field measurement， and then imported into a computer digital image processing system to get the corresponding standard gray values. Zhengtao YANG et al. Experimental Study of Vertical Twodimensional Diffusion Concentration Distribution in Sloped Wave Bank Angular Field
Based on this， the standard equation of the concentration （C） gray （G） fitting curve was obtained as following：
C=12329.07/G-95.67 （1）
Wherein the coefficient of determination （Rsquare） of the fitted curve is 0.998 1， which is very close to 1， indicating that the fitting effect is very good. A digital camera was used to collect tracer distribution images at different times during the diffusion experiments， and then they were input to a computer. The digital image processing technique was applied to measure and analyze the collected concentration fields， and then the concentration field distribution was obtained using the conversion relationship of formula （1）. In this studied， twodimensional concentration field images were collected using a Nikon D700 digital camera. The concentration field collection and measurement are shown in Fig. 3.
Before the start of the diffusion experiment， the camera was mounted 5 m directly in front of the water tank， and the height and focal length were adjusted to make the inner water tank image clear and suitable in the camera. The photographing interval was set to 15 s， and timing was started when the partition of the drug adding box was removed. After each experiment， the pictures were imported into a computer digital processing system and digital image processing was performed to obtain a twodimensional diffusion concentration field distribution.
Digital image processing technique
The digital image processing technique for measuring the concentration field does not change the nature of the original field and can measure the concentration at all positions in the concentration field. It has the advantages of high fidelity and wide coverage， which is a great improvement in environmental hydraulics. Based on the existing results of Ji et al.[13-14]， this processing technique was modified and improved with the help of MATLAB platform. The processing flow is shown in Fig. 4.
The experimental photos were preprocessed， and it was found that the homomorphic filtering algorithm had a good effect on eliminating the influence of uneven illumination. A nonlinear spatial filter was used to deal with the noise interference during the experiment. For the impact of the grid， the open operation processing of erosion first and expansion then was better. To extract the diffusion area， the RGB color space was first converted to HSV， HSI and YCbCr color spaces， which were then compared， and it was found that the Cr component of the YCbCr color space was the best for the description of the red diffusion region.
Results and Analysis
Experimental study on sloped wave bank with the inclination angle θ=45°
According to the experimental steps， the concentration distribution of the sloped wave bank with θ=45° was experimentally studied. Table 2 shows the experimental parameters.
It can be seen from Fig. 5 that when n= 15 r/min， the grid oscillation frequency was low and the turbulence of the water body was relatively gentle. The tracer diffused along the water surface after entering the water body. Over time， the tracer diffused transversely and vertically， and the transverse diffusion rate was greater than the vertical diffusion. Comparing the figures from Fig. 5 to Fig. 8， it was found that with the increase of the grid oscillation frequency， the diffusion of the tracer in the angular field also became faster. This indicates that the larger the oscillation frequency， the greater the turbulence intensity of the water body， resulting in a larger diffusion coefficient. When we studied the discharge from the apex of the sloped bank angular field， following conclusion was obtained： the discharge from the apex of the angular region of the sloped bank showed a trend of the concentration field gradient decreasing from the source point in the radial direction. However， in this study， we observed that the distribution of the pollutant showed different laws. At different grid oscillation frequencies， vortices appeared in the upper left area of the sloped wave bank angular field during the experiment， and the presence of vortices changed the concentration field distribution in the area. Due to the rotation， the concentration of the pollutant in the vortex area was small， and the concentration of the pollutant at the edge of the vortex was large. The concentration decreased first and then increased in the radial direction from the source， and the larger the grid oscillation frequency， the more obvious the phenomenon. The analysis showed that the reciprocating oscillation of the grid disturbed the interaction between the water body and the sloped wave bank at the bottom to generate a vortex， and the size and intensity of the vortex increased with the increase of the oscillation frequency of the grid.
Fig. 9 shows the concentration contour distribution maps in the pollutant diffusion area of the sloped wave bank with the inclination angle θ=45° under a grid oscillation frequency of 60 r/min at different time conditions. After the pollutant entered the angular water body， it diffused in the horizontal and vertical directions. It could be seen that the transverse diffusion rate was greater than the vertical diffusion rate， and the concentration extended longer at the water surface than at the slope. At t=30 s， due to the short diffusion time， most of the pollutant still accumulated in the nearbank area. The pollutant concentration was high and had large concentration gradient， and the nearbank area was polluted seriously. With the passage of time， the pollutant further diffused in the angular field， leading to expansion in the pollution range and decreases in the high concentration area and concentration. The turbulence of the water body and the action of sloped wave bank at the bottom produced a largescale vortex， which changed the distribution characteristics of the pollutant concentration. The concentration of the pollutant was lower in the vortex area and high in the vortex edge area. It can be seen from Fig. 9c and Fig. 9d that the concentration of the pollutant in the sloped wave bank angular region showed a change raw of decreasing at first and then increasing from the bank along the horizontal direction， which is different from the law that the concentration of the pollutant in the sloped bank angular field decreased with the distance from the apex of the bank slope increasing. Fig. 10 shows the concentration contour distribution maps in the pollutant diffusion area of the sloped wave bank with the inclination angle θ=45° at t=90 s under different grid oscillation frequency conditions. It could be seen from the figure that the oscillation frequency had a great effect on the distribution of the pollutant. The higher the oscillation frequency was， the faster the pollutant diffused， leading to a wider diffusion range. This meant that larger turbulence produced by the grid oscillation caused the diffusion coefficient to increase. From Fig. 10a and Fig. 10b， it can be seen that when the frequency of the grid oscillation was smaller， the turbulence of the water body acted with the bottom bank slope， forming a smaller vortex， which had less impact on the distribution of the pollutant. From Fig. 10c and Fig. 10d， it can be seen that when the oscillation frequency of the grid increased， the size of the vortex generated in the water body became larger， and the effect on the concentration distribution of the pollutant became more obvious. The concentration contours in the figures had a jag
curve. The concentration curve charts were drawn by reading the processed experimental pictures at different oscillation frequencies， and the concentration distribution of the pollutants at different oscillation frequencies were compared and analyzed， as shown in Fig. 11 and Fig. 12.
Fig. 11 showed the transverse concentration distribution of water surface points at different oscillation frequencies. It can be seen from the figure that the point concentrations of the pollutant at the water surface were inversely proportional to the distance from the discharge point， and the curve trends under different oscillation frequencies were similar. With the increase of the oscillation frequency， the abscissa of the water concentration contour also increased， which is because that the transverse diffusion coefficient increases with the oscillation frequency increasing， and the pollutants further diffuse in the transverse direction at a faster rate， leading to a wider range within the same time. It can be seen from Fig. 12 that the concentration of the pollutant decreased monotonically with the increase of water depth， and the curve trends of the concentration of the pollutant with water depth were basically the same at different grid oscillation frequencies. At the same water depth， the concentration of the pollutant increased with the increase of the grid oscillation frequency， which indicates that the larger the oscillation frequency， the larger the vertical diffusion coefficient， leading to faster pollutant diffusion in the vertical direction. Experimental study on the sloped wave bank with the inclination angle θ=30°
According to the experimental steps， concentration distribution of sloped wave bank with θ=30° was experimentally studied. Table 3 shows the experimental parameters.
Fig. 13Fig. 16 are experimental photographs of the pollutant concentration distribution in the angular field of the sloped wave bank at the average bank slope inclination angle θ=30° under different grid oscillation frequencies.
It can be seen from the figure that the pollutant diffused in the horizontal and vertical directions after entering the water body. The diffusion of the pollutant along the adjacent area of the water surface was faster than that on the adjacent area of the sloped bank， and its concentration distribution exhibited the characteristic of expanding farther on the adjacent area of the water surface. When the grid oscillation frequency was small， the diffusion coefficient was small， the pollutant diffused slowly and accumulated in the nearbank area， which made the nearbank seriously polluted. With the increase of the grid oscillation frequency， the diffusion coefficient also became larger， and then the pollutant diffusion distance was longer at the same diffusion time， leading to a decrease in the pollutant concentration in the nearbank area， but an increase in pollution range. Compared with the diffusion experiments of the sloped wave bank at θ=45°， it was found that under the same experimental conditions， the diffusion experiment area of the sloped wave bank at θ=30° had a higher concentration of the pollutant at each point， which was due to the fact that when the slope angle of a bank slope is smaller， the space of the angular field is smaller， and the reflection and staking of pollutants is more frequent， resulting in obvious concentration increase. During this group of experiments， no obvious largescale vortexes appeared near the sloped wave bank， which was a big difference from the phenomenon in the diffusion experiment of sloped wave bank at θ=45°. Because the water depth of the sloped wave bank at θ=30° was smaller， the scale of the vortex generated was smaller， and the disturbance to the water body was not obvious.
Fig.17 shows the concentration contour distribution maps of the pollutant diffusion area of the sloped wave bank with the bank slope inclination angle θ=30° under a grid oscillation frequency of 60 r/min at different time conditions. After the pollutant entered the water body， it diffused in the horizontal and vertical directions. It could be seen that the transverse diffusion rate was greater than the vertical diffusion rate， and the concentration extended longer at the water surface than along the sloped wave bank. It also could be seen that within a short diffusion time， most of the pollutant accumulated in the nearbank area， and the pollutant concentration in the pollution area was high and had large concen
tration gradients. With the passage of time， the pollutant gradually diffused in the angular field， leading to expansion in the pollution range and decreases in the high concentration area and concentra
tion. Moreover， the concentration field gradient showed a downward trend from the source point in the radial direction， which is the same as the distribution law of the concentration of sewage discharge from the apex of the sloped bank angular field. The concentration contours in the experimental area had a jagged appearance， which was caused by the uneven local turbulence caused by oscillation， which led to certain difference in the local pulsation speed in the water tank， further resulting in changes in the concentration contours， but this situation could be eliminated by observing the overall trend of the curve.
Fig. 18 is the concentration contour distribution maps in the pollutant diffusion area of the sloped wave bank with the bank slope inclination angle θ=30° at 60 s under different grid oscillation frequencies. It can be seen from the figure that the distribution of the pollutant varied greatly at different grid oscillation frequencies. The greater the grid oscillation frequency was， the faster the pollutant diffused， leading to decreases in high concentration area and contour gradient and an increase in pollution range. This showed that the larger the oscillation frequency， the greater the turbulence intensity of the water body， which caused the diffusion coefficient to increase. The water surface points in Fig. 18 were read to draw the pollutant concentration distribution curves of the water surface points and in the vertical direction.
Fig. 19 shows the transverse concentration distribution of water surface points at different oscillation frequencies. It can be seen from the figure that the point concentration of the pollutant at the water surface decreased monotonically with the increase of the abscissa， and the curve trends were basically the same under different oscillation frequencies. At the same time， the distance of the concentration contour of the same value from the bank increased with the increase of the oscillation frequency， which indicated that the larger the oscillation frequency was， the faster the pollutant spread in the transverse direction. Fig. 20 shows the vertical concentration distribution at y=90 cm and t=90 s. It can be seen from the figure that the pollutant concentration distribution monotonically decreased with the water depth increasing at different oscillation frequencies， and the concentration of the pollutant at the same water depth increased with the increase of the oscillation frequency. This is because the larger the grid oscillation frequency， the larger the vertical diffusion coefficient， leading to faster pollutant diffusion in the vertical direction.
Conclusions and Prospects
In this study， two different sloped wave banks at the bank slope inclination angles θ=30° and θ=45° were experimentally studied for pollutant diffusion. The instantaneous apex line source discharge experiments with different diffusion coefficients were achieved by changing the oscillation frequency of the grid， and the digital image processing method was used to intuitively obtain the tracer concentration distribution. By analyzing the experimental results， conclusions were obtained as below.
（1） Within the same diffusion time， the larger the grid oscillation frequency， the larger the diffusion coefficient， leading to the faster diffusion of the pollutant. At the same oscillation frequency， as the diffusion time increased， the high concentration area gradually decreased， the concentration gradients decreased， and the pollution range increased.
（2） The point concentration of the pollutant at the water surface monotonically decreased with the increase of the abscissa， and the vertical concentration distribution decreased with the increase of water depth.
（3） The transverse diffusion rate of the pollutant in water was greater than the vertical diffusion rate. The diffusion rate of the pollutant in the adjacent area along the water surface was higher than that in the adjacent area on the sloped bank， and its concentration distribution exhibited a distribution characteristic of farther diffusion in the adjacent area on the water surface.
（4） Compared with the diffusion experiment of the sloped bank at θ=45°， under the same experimental conditions， the diffusion experiment area of the sloped wave bank at θ=30° had a higher concentration of the pollutant at each point， which was due to the fact that when the slope angle of a bank slope is smaller， the space of the angular field is smaller， and the reflection and stacking of pollutants is more frequent， resulting in obvious concentration increase.
（5） A largescale vortex appeared near the sloped wave bank at θ=45° during the experiment， and the presence of the vortex made the concentration distribution of the pollutant in the direction along the bank slope first decrease and then increase， while no obvious vortex was observed near the sloped wave bank at θ=30°， and the concentration of the pollutant decreased monotonously alongthe bank slope direction.
In the subsequent work， more angular field angles θ=2π/β （β=8， 9， 10， 11……） and different wave or step heights can be selected for sloped wave banks to conduct hydraulic model diffusion experiments and data analysis， to thereby study the distribution laws of side discharge concentration with the geometric characteristics of each bank slope， the grid oscillation frequency and time. The experimental results of hydraulic model diffusion under the conditions of sloped ladder bank， sloped wave bank and sloped wave ladder bank can be compared with the theoretical analysis and experimental results of the concentration distribution discharged from the apex of sloped bank angular field. For each of sloped wave ladder bank， sloped ladder bank， complex ladder bank slope and complex trapezoidal bank slope， more than 4 groups of geometric scales can be selected to conduct hydraulic model diffusion experiments and data analysis， so as to study the change laws of the concentration distribution of side discharge with the geometric characteristic parameters of each bank slope， grid oscillation frequency and time. The theoretical derivation methods of lateral diffusion coefficient and vertical diffusion coefficient for five types of complex bank slopes can be proposed， respectively， to obtain the change laws and quantitative relationships of the lateral diffusion coefficient and vertical diffusion coefficient with the geometric characteristic parameters of each bank slope， grid oscillation frequency and time. The depthaveraged concentration distribution and average lateral diffusion coefficients for five types of complex bank slopes can be calculated， respectively， to propose a theoretical derivation method for calculating the depthaveraged lateral diffusion coefficient， so as to obtain the change laws and quantitative relationships of the depthaveraged lateral diffusion coefficient with the geometric characteristics of each bank slope， the grid oscillation frequency and time. References
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