Longitudinal dispersion coefficients of pollutants in compound channels with vegetated floodplains ＊

Li Gu , Xin-xin Zhao, Ling-hang Xing, Zi-nan Jiao Zu-lin Hua , Xiao-dong Liu

1. Key Laboratory of Integrated Regulation and Resource Development on Shallow Lake of Ministry of Education,College of Environment, Hohai University, Nanjing 210098, China 2. National Engineering Research Center of Water Resources Efficient Utilization and Engineering Safety, Hohai University, Nanjing 210098, China 3. Nanjing Guohuan Science and Technology Co., Ltd., Nanjing 210001, China 4. Changjiang River Scientific Research Institute, Wuhan 430010, China

Abstract: The characteristics of the longitudinal dispersion of pollutants in compound channels remain unclear. This study examines the relationships among the vegetation density, the width of the floodplain, the water depth ratio, the cross-sectional mean velocity, and the longitudinal dispersion coefficient of a symmetrical compound channel with a rigid non-submerged vegetated floodplain. The longitudinal dispersion coefficient is found to increase significantly with the presence of vegetation on floodplains, and is positively correlated with the plant density. When the density of the vegetation on the floodplains exceeds a certain value, the dispersion coefficient no longer changes with the vegetation density. The longitudinal dispersion coefficient is found to increase with the increase of the width of the floodplain. Moreover, the combined effects of the mean velocity and the water depth ratio have a positive correlation with the dispersion coefficient. The effects of the vegetation on the longitudinal dispersion coefficient in the channels with various cross-sections are also compared. The compound channels with a vegetated floodplain are found to differ significantly from the channels with a rectangular cross-section.

Key words: Compound channel, longitudinal dispersion coefficient, relationship, vegetated floodplain

Introduction

The dispersion coefficient is one of the most important parameters used to describe the mixing characteristics of a river, and plays an important role in the transport processes of pollutants. In 1959, Elder showed that the vertical flow velocity follows a logarithmic distribution in an infinitely wide river,with a dimensionless longitudinal dispersion coefficient ofkx/hu*=5.93. The classical three-integral formula was proposed by Fischer to estimate the dispersion coefficient of a straight river. The coefficient for a natural straight river is much larger than that proposed by Elder because he only considered the vertical velocity shear, whereas the transverse velocity shear is more important for the longitudinal dispersion in natural rivers. The dimensionless longitudinal dispersion coefficientkx/hu*of the Mississippi River was found to range from 237.2 to 1 486.45, and even larger dispersion coefficients were recorded by Yotsukura et al. Owing to the irregular cross-sectional shape of natural rivers,their flow structure is complex, and the longitudinal dispersion coefficient is significantly larger than those determined in laboratory flumes and artificial channels with a regular cross-section.

The floodplains are often inundated during the flood period, and thus compound channels come into being[1].A natural river usually comprises a main channel and a floodplain[2]. Vegetation on the floodplain increases the resistance of the bed and reduces the flow velocity, thereby leading to an increase of the non-uniformity of the longitudinal velocity along the transverse direction, which affects the exchange of the mass flow and the momentum in the main channel and the floodplain as well as the overall transport capacity of the river[3]. The complex flow structure of compound channels with vegetation was much studied. The water level and the flow velocity distribution of a compound channel are measured by Ozan[4]at the downstream of a one-line riparian emergent tree along the floodplain edge.Dupuis et al.[5]investigated the mixing layer development in a compound open-channel with submerged and emergent rigid vegetation over the floodplains.The influence of vegetation on the flow structure in a compound channel was also studied by Huai et al.[6].Yuan et al.[7]found that the longitudinal velocity along the vertical direction in a floodplain with submerged vegetation follows an S-distribution,whereas the vertical velocity in the main channel follows a logarithmic distribution. Compound channel flow with a longitudinal transition from wood to meadow were determined by Dupuis et al.[8]. Under steady uniform flow conditions, a model was proposed by Shiono and Knight (SKM) to predict the depth-averaged velocity and the distribution of the bed-shear stress without vegetation by using the equation for the depth-averaged momentum. In 2008,based on the depth-averaged Reynolds equation, a method was proposed by Tang and Knight[9]to predict the depth-averaged velocity of a compound channel with vegetation on the floodplain by adding a drag force term to the momentum equation to consider the effects of vegetation. In 2012, an improved SKM was developed by Jiang et al.[10]by introducing an equivalent resistance coefficient of vegetation into the SKM. This improved SKM model provides a better description of the impact of the floodplain vegetation on the flow characteristics for the compound channel.

In recent years, the mixing processes of pollutants in channels with vegetation began to attract research attentions. The primary velocities are much more inhomogeneous in the presence of vegetation,and a large velocity gradient is generated between the area occupied by vegetation and the non-vegetation area, indicating a remarkable exchange of mass and momentum at the junction of these areas, with significantly increased longitudinal dispersion coefficients[11]. In 1997, Nepf et al. studied the influence of vegetation on the longitudinal dispersion in a rectangular flume, and found that the presence of plants reduced the vertical shear but strengthened the turbulence and the vertical diffusion, thereby weakening the overall longitudinal dispersion. However, the longitudinal dispersion was increased by plants to some extent owing to the trapping of pollutants by them. Murphy et al.[12]showed that the momentum exchange between the main channel and the floodplain was slowed by a dense vegetation, with much larger resulting dispersion coefficients than those of sparse canopies in flows with a submerged vegetation. Some small-scale tracer tests were conducted by Rominger[13], who found that the longitudinal dispersion coefficient did not change markedly following the addition of vegetation to the banks. A simplified method for estimating the longitudinal dispersion coefficient in ecological channels with vegetation was proposed by Huai et al.[14]. A quasi-two-dimensional model was used by Perucca et al.[15]to calculate the transverse profile of the depth-averaged velocity in the presence of vegetation,and the longitudinal dispersion coefficient was calculated based on this model. They found that the vegetation enhanced the longitudinal dispersion coefficient in a rectangular section of the river by as much as 70%-100%. In 2012, Schulz et al.[16]found that the gradient of the vertical velocity increases with the bed roughness while the corresponding dispersion effect also increases. Thus, according to past studies,the effects of vegetation on the dispersion coefficients remain unclear, and some conclusions are even contradictory.

The pollutant mixing in a vegetated compound channel is a complex process influenced by many factors, such as the cross-sectional geometry, the water depth, the flow resistance, and the vegetation, as mentioned above[17]. However, only one or two factors were considered in each of previous studies of the dispersion coefficient of a compound channel, and the relationships between various factors and the longitudinal dispersion were not well studied. Therefore,this study investigates the influence of the vegetation density, the relative depth ratio, the mean velocity,and the floodplain width on the dispersion characteristics of compound channels with vegetation.

1. Materials and methods

1.1 Calculation of the velocity profile

A typical cross-sectional drawing of a compound channel and the lateral distribution of the depthaveraged longitudinal velocity are shown in Fig. 1.

The cross-section in Fig. 1(a) is divided into four zones: the main channel, the side slope 1, the floodplain, and the side slope 2.bandHare the width and the depth of the main channel, respectively,Wis the total width,his the difference of the depth between the main channel and the floodplain, the depth ratioDr=(H-h)/H, andsis the side slope.The lateral distribution of the depth-averaged longitudinal velocity is of a two-step type, and a big shear layer appears between the main channel and the vegetated floodplain. According to White and Nepf[18],δis the width of the inner layer. Outside the shear layer includes two steady zones with comparatively steady velocitiesU1andU2,the velocities of the main channel and the floodplain, respectively.

Fig. 1 Cross-sectional drawing of the compound channel and velocity distribution

Under the conditions of a steady uniform flow,the depth-averaged momentum equation for the compound channel without vegetation can be expressed as follows

whereyis the horizontal coordinate,Udis the longitudinal depth-averaged velocity,sis the side slope,S0is the slope of the bed,ρis the flow density,gis the gravitational acceleration,λis a dimensionless eddy coefficient of the viscosity model andfis the Darcy-Weisbach friction coefficient.

The velocity in the compound channel is obtained by solving Eq. (1). The velocities in the main channel and the floodplain domain can be expressed as follows

The velocity along the side slopes 1, 2 is

where

Γis the secondary flow coefficient andA1,A2,A3andA4are the unknown parameters.

The boundary conditions are as follows:

In the center of the main channel

On the domain junction

On the edge

For vegetated floodplains, the Darcy-Weisbach friction coefficientfis replaced by an equivalent resistance coefficient proposed by Jiang et al.[10]

wheref′ is the equivalent resistance coefficient with vegetation,Nis the vegetation density,dis the diameter of the vegetation,Hfpis the depth of the floodplain andCDis the drag coefficient.

Of the parameters used in the formula, the eddy viscosity1λin the domain of the main channel is assumed to be a constant, i.e., it generally takes a value of 0.07, the value of3λof the floodplain isthe value ofλof the side slope domain is approximately two times of that of the main channel[9,19], and the frictional factorfis calculated using the improved Colebrook-White formula (C-Wformula).

This improved SKM model developed by Jiang et al.[10], is used to simulate the velocity distribution in a compound channel with vegetation in this study. As was done by Jiang et al.[10], the experimental data from the UK-FCF (flood channel facility) by Elliott and Sellin are also used in this study to test and verify the conclusions. The experimental flume is 56 m in length and 10 m in width. The floodplain is covered by rigid, non-submerged vegetation with a diameter of 0.25 m,b=0.75 m ,H=0.167 m ,B=3.15 m,h=0.15 m andDr=(H-h)/H=0.10, the side slopes=1, and the bed slopeS0=0.1027%. The cross-sectional shape is the same as that shown in Fig.1(a).

Moreover, the field data of River Senggi located in Kuching, the capital city of Sarawak state, Malaysia,measured by Hin et al.[20]in a vegetated floodplain,are employed in the verification process. The cross-sectional shape of River Senggi is similar to that shown in Fig. 1(a), with a symmetrical floodplain. The velocity distribution in the right part of the river is used, with the bed slopeS0=0.0010%, the depthH=2.278 mandDr=0.43.

A comparison of the calculated and measured depth-averaged longitudinal velocities is shown in Fig.2, which shows the good agreement between the calculated and measured values for the compound channel with vegetated and smooth floodplains.

1.2 Evaluation of the dispersion coefficient

The classical Fischer's triple-integral formula is used to estimate the longitudinal dispersion coefficient

whereKxis the longitudinal dispersion coefficient,Ais the cross-sectional area,yis the coordinate along the transverse direction,h(y)is the local flow depth,uis the deviation of the local depth-averaged velocity from the cross-sectional mean velocity,wis the total width of the river, andtεis the local transverse mixing coefficient.

In the two-step-type velocity distribution as shown in Fig. 1(b), one sees a large difference betweenU1andU2. It is unreasonable to use the velocity along the entire width in the triple-integral formula, which would make the dispersion coefficient too large, even up to an improper value. This is because the depth-averaged longitudinal velocities are very small and slightly vary in most parts of the floodplain, except the inner layerδ(see Fig. 1(b)),and thus the longitudinal dispersion coefficient in this zone (floodplain minus the inner layerδ) is much lower compared with those in the other zones.Furthermore, its influence on the dispersion in other zones is small. Thus, the longitudinal dispersion in this zone may be neglected, as was suggested in a study on the ecological channel with the emergent vegetation along the river bank by Huai et al.[14]. As a result, an improved method to estimate the longitudinal dispersion coefficient is used in this paper,where the velocity in the main channel, the side slope 1, and the inner layerδis used in the Fischer's classical triple-integral formula, whereas the velocity in the remaining part is omitted.

To verify the proposed method, the velocity distribution and the tracer experiment data reported by Hamidifar et al.[21]are applied. The cross-section is in a shape of two-step compound channel, similar to that shown in Fig. 1(a), where two-sided slope domains are absent. If all velocities of the main channel and the floodplain are used, the calculated values are six to nine times of the measured dispersion coefficient.Then, the uniform velocityU2in the floodplain is omitted, and the velocities of only the main channel and the shear layer, including the inner layer, are used.The difference between the calculated and measured values is within 20% (10.3% in the case 4, 15.3% in the case 5, and 2.7% in the case 6), which indicates that the proposed method can be used to estimate the longitudinal dispersion coefficient in compound channels with vegetation.

In this study, the velocity distributions in the typical compound channel (as shown in Fig. 1(a)) with different water depth ratios and mean velocities from the UK-FCF are used. Moreover, the velocity distributions for different vegetation densities and floodplain widths are simulated by the improved SKM model. The longitudinal dispersion coefficient of the compound channels with vegetation is then calculated using the triple-integral formula according to the velocity data.

Fig. 2 Comparison of measured and calculated values under different conditions in a compound channel with flood-plain : (a) a nd (b) r efer to the cas e Dr =0.10 with vegetatedandsmoothfloodplains,respectively,withdata from UK-FCF, (c) refers to the field measured data from the River Senggi, Dr =0.43

2. Results

2.1 Comparison of smooth and vegetated floodplains

The velocity distributions of the smooth floodplain and the vegetated floodplain are different, as shown in Figs. 2(a), 2(b). Compared with the case of the smooth floodplain, the velocity is decreased because of the resistance due to the vegetation on the floodplain, and the variations of the longitudinal velocity along the transverse direction are increased.The effect of the transverse shear in the case with the vegetation is greater than that without the vegetation.Hence, the dispersion coefficient is also increased.The velocity distribution and the longitudinal dispersion coefficient are calculated for the case withH=0.20 m ,h=0.15 m ,b=0.75 m ,B=3.15 m ,Dr=(H-h)/H=0.25, the vegetation densityN=12 stems/m2, ands=1. The vegetation diameterd=0.025 m and the vegetation densityN=1/(Δx*Δy), where Δxand Δyare the longitudinal and transverse spacings of the simulated vegetation,respectively. The results show that the longitudinal dispersion coefficient is increased in a compound channel with a vegetated floodplain by approximately 1.06 times, as compared with the case without the vegetation. Hamidifar et al.[21]obtained the same conclusion for an asymmetric compound channel, and showed that the cross-sectional dispersion coefficient was enhanced by 82% with the vegetation on the floodplain.

2.2 Longitudinal dispersion coefficient in cases of different vegetation densities

The experimental conditions under different floodplain vegetation densities are shown in Table 1.In all cases in Table 1,Handhare kept constant at 0.20 m and 0.15 m, respectively,Dr=0.25, andBandbare 3.15 m and 0.75 m, respectively. The side slopes are assumed to be unity. The flume bed slopeS0is 1.072 × 10-3.

Table 1 Experimental conditions for different floodplain vegetation densities

The equivalent resistance coefficient for the floodplain (Eq. (4)) is increased with the increase of the vegetation density (Table 1). The variations of the longitudinal dispersion coefficient and the velocity in the compound channel of different vegetation densities are shown in Fig. 3.

The velocity of the floodplain is first decreased significantly as the vegetation density is increased whenN＜100stems/m2(Figs. 3(a), 3(b)), but remains almost constant with further increase of the density (N＞100stems/m2). To clearly see the changes of the velocity, only four conditions are shown in Fig. 3(a). The mean cross-sectional velocity assumes similar variations as the velocity in the flood-plain (Fig. 3(b)) . The decrease of the mean crosssectional velocity reduces the longitudinal dispersion coefficient. However, Fig. 3(c) shows a positive correlation between the dispersion coefficient and the vegetation density. Clearly, the increase of the vegetation density strengthens the shear effect of the longitudinal velocity along the transverse direction,which exceeds the influence of the reduced mean cross-sectional velocity, thereby the dispersion coefficient is increased. However, when the vegetation density in the floodplains is greater than a certain value, the dispersion coefficient no longer changes with the increase of the vegetation density.

Fig. 3 Variations of velocity and longitudinal dispersion coefficient in a compound channel of different vegetation densities: (a) Lateral distribution of depth-averaged longitudinal velocity, (b) Mean velocities of the main channel,the floodplain, and the entire cross-section, and (c) Variations of the longitudinal dispersion coefficient with the vegetation density

2.3 Longitudinal dispersion coefficients for cases of different floodplain widths

The variations of the longitudinal dispersion coefficient are analyzed by increasing the floodplain width with the main channel width, depth, and other hydraulic conditions kept constant. The geometrical parameters of the channel are shown in Table 2. For all cases in Table 2,H=0.20 m ,h=0.15 m ,Dr=0.25,b=0.75 m andS=1.072× 10-3. The side

0slopes are assumed to be unity, and the vegetation densityN=12stems/m2.

Table 2 Experimental conditions with different floodplain widths

Fig. 4 Variations of velocity and longitudinal dispersion in a compound channel of different floodplain widths: (a)Transverse distribution of depth-averaged longitudinal velocity, (b) Variation of the longitudinal dispersion coefficient with floodplain width

The variations of the velocity and the longitudinal dispersion coefficient in the compound channel with different floodplain widths are shown in Fig. 4. As the width of the floodplain increases, the inhomogeneity of the velocity distribution is streng-thened in the entire cross-section, and thus the longitudinal dispersion coefficient increases accordingly. According to Eq. (5), as the width increases with other conditions unchanged, the flow area also increases. The dispersion coefficient is inversely proportional to the flow area, so the value of the longitudinal dispersion coefficient is reduced with the increase of the flow area. However, as shown in Table 2, Fig. 4(b), the longitudinal dispersion coefficient increases with the increase of the floodplain width.Therefore, it can be concluded that the effects of the width and the velocity gradient on the dispersion coefficient are greater than the effect of the flow area,thus the dispersion coefficient increases with the increase of the floodplain width.

Table 3 Experimental conditions in cases of different main channel/floodplain depth ratios

2.4 Longitudinal dispersion coefficient in cases of different main channel/floodplain depth ratios and mean velocity

The transverse distribution of the depth-averaged longitudinal velocity is varied with the changes of the water depth. The velocity distributions in cases of different main channel/floodplain depth ratios measured from the experimental compound channel of the UK-FCF are used to calculate the longitudinal dispersion coefficient. The geometric parameters of the channel are shown in Table 3 (for all cases,h=0.15 m ,b=0.75 m ,B=3.15 m ,N=12 stems/m2andS=1.027× 10-3).

0

The variations of the velocity and the longitudinal dispersion coefficients in cases of different main channel/floodplain depth ratios and mean velocitiesUare shown in Fig. 5. To make a clear picture, we show only three runs forDr=0.1, 0.2 and 0.3 in Fig. 5(a) to see the characteristics of the velocity distribution. AsDrincreases, the velocity of the main channel decreases, whereas the velocity of the floodplain increases (Fig. 5(a)). Hence, the velocity gradient and the inhomogeneity in the main channel and the floodplain are reduced, and thus the longitudinal dispersion coefficientKxdecreases accordingly (Fig. 5(b)).Drthus seems to have a negative correlation with the dispersion coefficientKxif onlyDra ndKxare co nsider ed.

Interestingly,Hamidifaretal.[21]foundthatthe longitudinal dispersion coefficient increased with the main channel/floodplain depth ratio in a flume experiment, which differs from the above conclusions.Of many influential factors, the mean velocity is dominant, and has a positive correlation with the longitudinal dispersion coefficient. In the experiment by Hamidifar et al., the mean velocity increases with the increase ofDr, whereas it decreases with the increase ofDrin the experiments from the UK-FCF adapted to this study.

Fig. 5 Variations of velocity and longitudinal dispersion coefficient in a compound channel: (a) Transverse distribution of depth-averaged longitudinal velocity, (b) Variation of the longitudinal dispersion coefficient with the water depth ratio and mean velocity

In this study, the decrease of the dispersion coefficient is due to the combined effects of the mean velocity and the water depth ratio. Then, multiple regression is used to derive the relation amongDr,the dimensionlessmean velocityU/u*, and the dimensionless dispersion coefficientkx/hu*. The results show that whenDrandU/u*are considered together,Drhas a positive effect on the dimensionless dispersion coefficient in the multiregression model, rather than a separate negative correlation, because of the interaction of these two or more factors.

Table 4 Longitudinal dispersion coefficients for different cross-sections

Table 5 Values of the longitudinal dispersion coefficient (m2/s) estimated by different models

3. Discussions

The effect of the vegetation density on the longitudinal dispersion coefficient is related to the channel forms and the position of the vegetation. In 2011, Wang et al. studied the influence of the rigid non-submerged vegetation on the longitudinal dispersion coefficient in a rectangular flume, and found that the longitudinal dispersion coefficient decreases as the vegetation density increases. This can be explained by the high resistance of a fully vegetated channel, such that the flow velocity and the longitudinal dispersion of pollutants both decrease, as is consistent with the results reported by Nepf et al. In 1997, Nepf et al. suggested that the vegetation increases the turbulence intensity, enhances the vertical diffusion, and reduces the vertical shear,which decreases the longitudinal dispersion coefficient. However, the transport of the material flux is limited and delayed by the circulation area behind each piece of vegetation, and this increases the longitudinal dispersion to a certain extent. In general,the effects of the vegetation to increase the dispersion are smaller than their effects to reduce the dispersion.In 2008, White and Nepf[18]showed that the dispersion coefficient of vegetation is significantly higher when the vegetation is planted on one side of a rectangular flume compared with that without the vegetation. The same conclusion was obtained by Perucca et al.[15]using the vegetation planted on both sides of a rectangular flume. According to this study,the dispersion coefficient for a vegetated floodplain was significantly larger than that without the vegetation in a compound channel, and the longitudinal dispersion coefficient was positively correlated with the vegetation density of the floodplain. When the vegetation density increases in the floodplain, the resistance of the plants to the flow reduces the velocity on the floodplain, which increases the transverse difference of the longitudinal velocity and,thus, the longitudinal dispersion coefficient. Moreover,it should be noted that when the vegetation density on the floodplains exceeds a certain value, the dispersion coefficient no longer changes with the vegetation density. As discussed above, in partially vegetated rectangular channels and compound channels with floodplain vegetation, the velocities of the vegetated and non-vegetated zones can generate a strong shearing to increase the lateral differences and the dispersion coefficients. However, in a fully vegetated rectangular channel, the resistance of the channel is much higher, and the vegetation reduces the crosssectional flow velocity and reduces the velocity gradient, thereby decreases the dispersion coefficient.

Moreover, the dispersion coefficients of a river of various cross-sectional shapes are listed in Table 4.It is evident that the dispersion coefficients of this study are larger than those of the other two studies,owing to many factors. In addition to the crosssectional shapes, the mean velocity, the width, and the depth are also influential factors. Furthermore, a comparison between the longitudinal dispersion coefficients estimated by different models from the literature and those estimated by our method is made in Table 5.

Several models for estimating the longitudinal dispersion coefficient in the river were developed in various studies. Some empirical formulae proposed by Fischer et al. (1979), Seo and Cheong (1998), Deng et al.[22]and Zeng and Huai[23]are used in a comparison with the longitudinal dispersion coefficient determined in this study.

Fischer et al. (1979)

Zeng and Huai[23]

The results of the estimated dispersion coefficient in cases of various depth ratios are shown in Table 5. Considering the large difference in the water depth in the compound channel, the dispersion coefficients are calculated for the main channel, the side slope zone, and the floodplain separately for each model, and the values shown in Table 5 are the mean values of different zones. It is evident that the values calculated by the different formulas are scattered. The model proposed by Seo and Cheong (1998) comparatively overestimates the dispersion coefficient,whereas the model proposed by Fischer et al. (1979)underestimates it. Moreover, the values obtained by the proposed method are comparable with those calculated by Deng et al.[22].

4. Conclusions

In this study, the variations of the longitudinal dispersion coefficient with consideration of different influence factors in the compound channel are explored. The vegetation has an important effect on the dispersion of contaminants. In the compound channels with a floodplain, the velocity distribution gradient for a vegetated floodplain is significantly larger than that for a smooth floodplain, and the increase in the transverse shear effect caused by plants increases the longitudinal dispersion coefficient.Moreover, the longitudinal dispersion coefficient is positively correlated with the vegetation density on the floodplain, where it is significantly increased at a low density, but when the vegetation density on the floodplains exceeds a certain value, the dispersion coefficient no longer changes with the increase of the vegetation density.

The mean velocity is the dominant influence factor, and has a positive correlation with the longitudinal dispersion coefficientKx. However,Dris negatively correlated with the dispersion coefficientKxif onlyDrandKxare considered. The multiple regression forDr, the mean velocity, and the dimensionless dispersion coefficientshows thatDrhas a positive effect on the dimensionless dispersion coefficient, rather than a separate negative correlation, because of the effect of the interaction of these two or more factors. Moreover, as the width of the floodplain increases, the longitudinal dispersion coefficient increases.

Acknowledgements

This work was supported by the Fundamental Research Funds for the Central Universities (Grant No.2016B06714), the PAPD Project and the National Public Research Institutes for Basic R&D Operating Expenses Special Project (Grant No.CKSF2015050/SL).