Qian Li,Yu-hong Zeng,Yu Bai

State Key Laboratory of Water Resources and Hydropower Engineering Science,Wuhan University,Wuhan 430072,China

Abstract:In combination with a channel bed,suspended vegetationin an open channel can alter flow structureand generate vertically asymmetric flow.This study investigated the mean flow and turbulence structure of an open channel with suspended vegetation through theoretical analysis and laboratory experiments.Three patterns of bionic leaves with different roughness were adopted to imitate suspended vegetation,and three-dimensional velocity field was measured by using an acoustic Doppler velocimeter.The measured data showed that the vertical profile of streamwise velocity obeys a two-power law and that the maximum velocity at the middle depth is close to the smooth boundary (i.e.,thechannel bed in the experiment)under the combined action of vegetation cover and channel bed.Shear stress is linearly distributed along the vertical axis,and the vertical profile of turbulence intensity obeys an exponential law.Then,a two-power law expression was adopted to predict the vertical profile of streamwise velocity.Theoretical models for the vertical distribution of shear stress and turbulence intensity were also established.The predicted results validated by measurements showed that the different magnitudes of vegetation cover and channel bed boundary roughness exert an obvious impact on flow structure.

Key words:Suspended vegetation,streamwise velocity,shear stress,turbulence intensity, boundary roughness

Introduction

Aquatic vegetation,which widely exists in rivers,lakes,wetlands,and artificial floating islands,affects physical,chemical,and biological processes and provides a wide range of ecosystem functions[1].In general,aquatic vegetation can be categorized as submerged,emergent,and suspended vegetation.Suspended vegetation can be primarily found in freshwater systems,which does not root in mud and may float with currents,can mediate trophic interactions,impact phytoplankton and zooplankton biomass[2], absorb nutrientsand heavy metalsto purify waste water,and play a structuring role in ecosystems.Numerous studies have focused on the ecological impact of suspended vegetation[3-4].However,the hydrodynamic properties of open channel flow with suspended vegetation have received little attention.In fact,the existence of vegetation covers alters flow structure and velocity distribution and influences sediment and contaminant transportation.Therefore,the flow structure of open channels covered with suspended vegetation should beinvestigated.

Numerous studies on open channel flow with submerged or emergent vegetation[5-10]have been reported.However,several investigations have only focused on the hydrodynamic interactions of suspended vegetation with developed root canopies that are extended to some distance from water.Plew et al.[11-12]conducted detailed laboratory experiments to investigate flow through and beneath suspended canopies constructed by rigid cylinders and vertically divided the flow field into bottom boundary,canopy shear,and internal canopy layers.Huai et al.[13]developed a three-layer analytical model on the basis of Plew’s experimental data to predict the vertical distribution of streamwise velocity and Reynolds stress.Zhang and Nepf[2]used partial-depth porous obstruction to simulate the root layer,investigated its influences on the vertical distribution of velocity and flux exchange between free water and vegetation layer,and presented a model to predict the magnitude of exchange flow according to energy considerations.Downing-Kunz et al.[14-15]used live Eichhornia crassipes to construct small vegetation rafts,measured wind-induced and water-induced drag forces,and evaluated flow hydrodynamics through and around free-floating root canopies of finite patches.O'Donncha et al.[16]used an improved 3-D hydroenvironmental model to predict the effects of suspended canopies on vertical flow structure.Liu et al.[17]used a modified RNG-k εturbulence model developed in OpenFOAM to simulate the exchange flow between open water and floating vegetation.Zhao et al.[18]incorporated a -k εturbulence model with Delft3D-FLOW module and obtained the mean velocity and Reynolds stress profiles of open channel flow with suspended canopy.

Few studies have explored suspended vegetation forming a thin layer on the water surface without developed roots.Different from suspended canopy with rigid roots penetrating water at a certain depth,suspended vegetation with leaves covering the water surface constitutes the floating body (e.g.,Lemna minor,Spirodela polyrhiza and Salvinia natans),the leaf canopies form suspended canopies with drag elements,and the resistance of their underdeveloped roots for flow is negligible.Suspended vegetation alters flow structure due to its combined action with channel bed roughness and generates vertically asymmetric flow.Flow characteristics with suspended vegetation are comparable to the characteristics of ice-covered flow to some extent[19].Considerable investigations into the flow characteristics in icecovered channels have been conducted[20-22]and can be used as reference in the study of open channel flow with suspended vegetation.

In the present study,the impact of suspended vegetation with thin cover and undeveloped roots on flow characteristics was investigated.Theoretical models for the vertical distribution of streamwise velocity,shear stress,and turbulence intensity were established,and the determination of model parameters was discussed.Laboratory experiments were conducted in a flume covered with artificial lotus leaves,and the comparisons between theoretical predictions and experimental data were analyzed to discuss the influences of boundary roughness on flow characteristics.

1.Theoretical analysis

Considering a steady uniform flow along the streamwise direction in an open channel with suspended vegetation (Fig.1),this study adopts a Cartesian coordinate system with the x axis in the streamwise direction and the z axis in the vertical direction.Here,u is the flow velocity in the streamwise ( )x direction.

Fig.1 Schematic offlow with suspended vegetation

1.1 Vertical distribution of streamwise velocity

For open channel flow with suspended vegetation,the increase in wetted perimeter caused by an imposed vegetation cover results in the increase of composite resistance.The roughness of the top vegetation cover and that of the bottom channel bed are generally different,thereby indicating that covered flow is vertically asymmetric.The upper layer of flow is primarily affected by vegetation boundary,and the lower portion is influenced by the channel bed.The maximum velocity is located somewhere near the middle depth rather than near the water surface,where its vertical location is dependent on the roughness of the vegetation cover and channel bed.Thus,the flow region can be vertically divided into two parts,namely,the bed and vegetation layers,on the basis of the vertical distribution of streamwise velocity (Fig.1).

The vertical distribution of streamwise velocity described by a two-power law[23]is

where z is the vertical coordinate counted from the channel bed,H is the flow depth,0K is the flow parameter for a given flow rate per unit width,and bm andvm are the parameters relevant to the boundary roughness of the riverbed and vegetation,respectively.The vertical distribution of streamwise velocity becomes equivalent to the power law expression whenvm approaches infinity,that is,the top vegetation cover disappears.

The velocity gradient in Eq.(1)is

Set / =0u z? ? .Then,the position of maximum streamwise velocity can be deduced as

where umaxis the maximum streamwise velocity,hmis the distance of maximum streamwise velocity from the channel bed.

According to our previous work[19], the depthaveraged streamwise velocities in each layer can be expressed as:

1.2 Shear stress

The mean streamwise momentum equation for a fully developed asymmetric channel flow is reduced to

where hτis the distance of zero shear stress from the channel bed.

1.3 Turbulence intensity

Turbulence intensityrmsu can be determined on the basis of the root mean square of fluctuating velocity in the streamwise direction.The turbulence intensity for free open channel flow monotonically decreases from the channel bed in the vertical direction.For open channel flow with vegetation cover,rmsu first decreases with the z coordinate,reaches a minimum value near the middle of the water depth,and increasestoward the vegetation cover.

Nezu and Rodi[26]proposed a universal exponential function for the vertical distribution of turbulence intensity in open channel flow.The vertical distribution of turbulence intensity for open channel flow with suspended vegetation can be divided into two parts,which are separated at the plane of zero shear stress;it is given by the following exponential expressions:

wherebD ,vD ,bλandvλare empirical constants.

2.Experimental study

Experiments were conducted in a rectangular flume,which measured 9.0 m long,0.3 m wide and 0.5 m deep,with a bed slope of S =0.0002 in the State Key Laboratory of Water Resources and Hydropower Engineering Science in Wuhan University.The water depth was kept constant by adjusting the tailgate located at the end of the flume,and discharge was recorded by using an electromagnetic flowmeter.The experimental setup is shown in Fig.2.The bionic lotus leaves made of ethylene vinyl acetate copolymer materials were used to imitate the floating vegetation with undeveloped roots in nature,three patterns of bionic lotus leaves with different roughness were adopted,and three groups of experiments were conducted correspondingly.The first group covered A1-A3 with a leaf diameter of 0.30 m,the second group covered B1-B3 with a leaf diameter of 0.18 m,and the third group covered C1-C3 with a leaf diameter of 0.10 m.As show in Fig.3,these artificial leaves were connected to one another to constitute a fully covered vegetation canopy,and a vegetation cluster was fixed on the flume by using a fine nylon line to ensure its stability against the flow.The parameters used in the experiments are listed in Table 1.

Fig.2 Schematic of the experimental setup

Fig.3 Arrangementsof thethree patternsof bionic lotusleaves

The length of the vegetation-covered region was 4.5 m,and the vegetated domain started at 3 m from the flume inlet.The measurement section was set at 5.5 m downstream of the flume inlet,where the flow was fully developed (Fig.2).With =0 m x set at the flume inlet,Fig.4 shows the mean streamwise velocity at three longitudinal locations ( =x 5.0 m, 5.5 m and 6.0 m,respectively)in run A2,and one can find that there is little difference between velocity profiles.Hence,the flow can be assumed to be fully developed at =5.5 m x,and the measurement section was set at 5.5 m downstream of the flume inlet.Velocity data ( , , )u v w corresponding to longitudinal,transversal,and vertical directions were recorded by using an acoustic Doppler velocimeter with a sampling frequency of 50 Hz and a sampling time of 120 s for each sample.The measuring line was arranged in the middle of the cross section,and the vertical interval of themeasuring pointsranged from 0.005 m to 0.010 m.

Table 1 Summary of experimental parameters

Fig.4 The mean streamwise velocity at three locations in run A2

3.Model parameters

Exponential parameter m,friction coefficient f ,and flow parameter0K should be determined to obtain the vertical profiles of streamwise velocity and shear stress in an open channel flow covered with suspended vegetation.

3.1 bm and vm

Equation (31)can be iteratively solved to obtain bh with knownbn ,vn and H ,andbm andvm can becalculated by using Eq.(28).

(2)Manning’s roughness coefficient n

For a fully developed vertically asymmetric channel flow, the logarithmic expressions of velocity based on the two-layer hypothesis shows some drawbacks,such as the discontinuous velocity gradient at the location of maximum velocity.However,the vertical profiles of streamwise velocity in each sublayer satisfy the log law well.A previous study[27]stated that the turbulent boundary layer has a composite layer consisting of inner and outer regions and that the thickness of the inner region represents 10%-20% of the entire boundary layer thickness.The velocity distribution in the region close to the boundary can be described as logarithmic.In this study,the velocity distributions of the inner region in the bed and vegetation layers are analyzed by using a well-known log law to estimate the boundary roughnesscoefficients.

The log law can be described as[21]

The depth-averaged velocities for the bed and vegetation layers can be calculated by integrating velocity from the boundary to the location of the maximum velocity.Thus,Manning’s roughness coefficient for each sublayer can be calculated as:

wherebU andvU are the depth-averaged velocities of the bed and vegetation layers,respectively,which are determined on the basis of the measured velocity data.

3.2 K0

K0is a flow parameter for a given flow rate per unit width.Substituting Eq.(4)intoU = Q /BH gives

4.Resultsand discussion

4.1 Manning’s roughness coefficient

The results of the calculated Manning’s roughness coefficient for each test are shown in Table 2.The values ofbn remain relatively constant in the range of 0.0124-0.0135,and the mean value is 0.0130 with a standard deviation of 0.000377.The values of vn vary with different vegetation covers.For runs A (A1-A3),the mean value forvn is 0.0145 with a standard deviation of 0.000170.For runs B(B1-B3),the mean value forvn is 0.0167 with a standard deviation of 0.000205.For runs C(C1-C3),the mean value forvn is 0.0185 with a standard deviation of 0.000216.The Manning roughness coefficient of vegetation cover varies with the variation of vegetation boundary roughness.Thus,bn is set to 0.0130 for all runs,and 0.0145,0.0167 and 0.0185 are adopted forvn in runs A,B and C,respectively.

Table 2 Summary of Manning’s roughness coefficient

4.2 Velocity distributio n

The measured and predicted velocity data are compared in Fig.5.The calculated data agree well with the measurements,except for cases A2 and C1,in which small deviations between the measured and calculated data are found in the region close to the fixed boundaries.These deviations are probably due to the measurement of velocity near the boundary being susceptible to interference.

The parameters used to calculate velocity are listed in Table 3.In the research of ice-covered channel flow[28],the values of m were in the range of 1.5< <8.5m,with the preponderance of the values being 3.0-4.0.In this study,m approximately ranges between 4.0 and 7.0.

Correlation coefficient r and average relative error of velocityvΔare listed in Table 3 to clearly determine the difference between the calculated and measured data.

The average relative error of velocityvΔ is defined as

Fig.5 Comparison between the measured and calculated velocity profiles

wherecv is the calculated velocity,mv is the measured velocity and N is the number of measured pointsfor each case.

The correlation coefficients between the measured and calculated velocity data for nine runs range from 0.930 to 0.970,with an average value of 0.953.Average relative errorvΔ varies from 2.71%to 6.31%.The error statistics confirm the reliability and authenticity of the proposed theoretical model.

The shape of the velocity profile depends on exponentsbm andvm .The ratio ofbm andvm can beobtained on the basisof Eq.(28).

4.3 Shear stress profiles

Reynolds shear stress for open channel flow is maximized near the channel bed and gradually decreases along the z coordinate.With the presence of vegetation cover,the local maximum values of Reynolds stress occur in the vicinity of the vegetationcover and channel bed.Reynolds stresses are negative for the upper flow region,whereas they are positive for the lower flow region,and Reynolds shear stress showsa linear distribution along thevertical axis.

Table 3 Summary of results from calculated velocity and shear stress

The experimental and theoretical results of the vertical distribution of shear stress are compared in Fig.6.For the entire water depth,the straight line profile fits the Reynolds stress measurements well,and the correlation coefficients for nine runs vary from 0.949 to 0.976.Deviations are observed in the region close to the bed and the vegetation boundary because viscous shear stress,which may be large near the boundary,is ignored.Nevertheless,the result is acceptable.

Fig.6 Comparison between the measured and calculated shear stress profiles

For vertically asymmetric flow,the rough boundary is associated with a large production of shear stress and turbulent kinetic energy.The ratios of boundary shear stressτv/τbfor different boundary roughness and different Reynolds numbers are shown in Fig.7.τv/τbincreases with the increase of nb/nv,and the value of τv/τbis approximately equal to ( mb/ mv)2.Although the boundary roughness remains constant,τv/τbhas no remarkable change with the increase of the Reynolds numbers,that is,the influences of the Reynolds numbers on the ratio of boundary shear stress can be ignored.

Fig.7 Effect of boundary roughnesson shear stress

4.4 Variationsof hm and hτ

The comparison of the results of hm/H and hτ/H from runs A to runs C shows that hm/H decreases with the increase of nb/nvand that the vertical distribution of streamwise velocity appears asymmetric.τv/τbincreases with the increase of nb/nv,resulting in the decrease of hτ/H ,and the location of zero shear stress is located close to the channel bed.When the roughness of the opposite boundaries remains the same,no remarkable changes are observed on the ratios of hm/H and hτ/H with theincrease of the Reynolds number.

4.5 Turbulence intensity

Fig.8 Turbulence intensity profiles for streamwise velocity component

Table 4 Parameters used to predict turbulence intensity

Fig.9 Relationship between the minimum value of turbulence intensity (u rms )min and total shear velocity u*

4.6 Effects of the leaf diameter

In this study,three patterns of bionic lotus leaves with different diameters were adopted in experiments.Different-sized leaves cause different surface roughness,and the corresponding vertical distributions of velocity and shear stress varied.Differentsized leaves have different cover rate,i.e.,here the cover rates of runs A,B and C are 74.3%,78.4%and 77.5%,respectively,and the difference is not obvious because vegetation leaves are arranged in regular rows instead of a staggered or random arrangement.The gaps between the adjacent vegetation leaves were affected by the leaf diameter too.For runs C with smaller diameter vegetation,there are more small and scattered gaps instead of few and big ones for runs A with larger diameter vegetation.According to our previous work[19],the presence of gaps can change the path of flow,and affect the dispersive fluxes and shear stress distribution,and the momentum exchanges are intense at the canopy–gap interface.Further quantitative analysis is required to investigate the effects of the leaf diameter and arrangement pattern on the featureof vegetation-covered open channel flow.

5.Conclusions

In this study,the structure of open channel flow covered by suspended vegetation was experimentally investigated,and theoretical models were presented to predict the vertical distribution of streamwise velocity,shear stress,and turbulence intensity.The suspended vegetation investigated in this study forms a thin leaf canopy on the water surface and has no developed root system,hence,it is different from rigid suspended canopies that penetrate water at a certain depth.The results showed that the existence of suspended canopy alters flow structure with the combined action of channel bed and generates vertically asymmetric flow.

We adopted a two-power law expression to describe the vertical distribution of streamwise velocity by dividing the flow into a vegetationaffected layer and a channel bed-affected layer.The results showed that the maximum velocity is located near the middle depth close to the smooth boundary.Analysis showed that shear stress is linearly distributed along the vertical axis,with negative values in the upper side and positive values in the lower side.Moreover,zero shear stress is constantly located closer to the smoother surface than the plane of maximum velocity.The turbulence intensity first decreases with coordinate z ,reaches a minimum valueat thelocation of zero shear stress,and increases with coordinate z,its profile can be divided into two parts and can be described as exponential expressions.The results of the theoretical solution agree well with the available laboratory experimental data,which indicate that the theoretical model is efficient and feasible.

This study expands our understanding of the effects of suspended vegetation on the hydraulic characteristics in open channels.Future studies will involve detailed experiments to reveal flow characteristics and the impact of different vegetation coverage ratios on flow structure.Acknowledgements

This work was supported by the CRSRI Open Research Program (Program SN:CKWV2017503/KY),Hubei Natural Science Foundation (Grant No.2018CFA010)and the CAS Interdisciplinary Innovation Team,and 111 Project (Grant No.B18037).