The problem of agricultural non-point source pollution has become increasingly serious. How to determine the ecological drainage ditch system is one of the effective methods to solve the agricultural non-point source pollution. This research study focuses on the velocity distribution in a two-stage section ecological channel with ice cover. The results show that the two-stage section channel with ice cover can effectively reduce the flow velocity in the channel and increase the retention time of water in the channel. By comparing with the experimental data, the accuracy of the analytical solution is high, which provides a theoretical reference for the transport of sediment and pollutions in a two-stage section channel with ice cover in the future.

  • Discuss the transverse velocity distribution law in a two-stage channel with ice cover.

  • Give the analytical solution of the transverse velocity distribution.

With the increasingly serious agricultural soil erosion and non-point source pollution, more and more technologies have been applied to agricultural drainage (Dolan & Mcgunagle 2005; Vidon et al. 2012). In recent years, ecological channels have been widely used in practical projects because of their ecological friendliness, and can better solve the problems of agricultural non-point source pollution (Kumwimba et al. 2017). In recent years, the research of two-stage section ecological channel has been widely concerned (Chen 2013; Zeng et al. 2014; Huang et al. 2020).

A two-stage section drainage ditch is a better option than a traditional trapezoidal ditch, because it meets drainage requirements while eliminating sediment and nutrient losses (Västilä & Järvelä 2011; Christopher et al. 2017; Kumwimba et al. 2018). For example, Hodaj et al. (2017) proved that the two-stage ditch can significantly reduce the concentrations of total suspended sediment, nitrate nitrogen, soluble reactive phosphorus and total phosphorus in the drainage. Roley et al. (2016) and Krider et al. (2017) also proposed similar results. At the same time, vegetation on the floodplain has a positive impact on water quality by reducing pollution, improving river bed stability, assisting drainage restoration and controlling flow velocity (Folkard 2011; Davis et al. 2015; Mahl et al. 2015). For a two-stage section channel, the streamwise velocity distribution is more complicated. Due to the different water depth across the channel, it can produce a mixed layer near the interface between the main channel and the floodplain, and secondary flow occurs (Stephenson & Kolovopoulos 1990).

In higher latitude areas, rivers usually freeze in winter, because the river surface icing will lead to great changes in the hydraulic characteristics of the channel, just like adding artificial floating islands on the river surface (Li et al. 2010; Zhou & Wang 2010). It is generally believed that after adding the ice cover, the flow in the channel can be approximately regarded as two layers (Zhu et al. 2011; Zhao et al. 2012; Xavier et al. 2018; Liu et al. 2019), and then the vertical velocity can be calculated (Lau 1982; Urroz & Ettema 1994). However, it is more difficult to calculate the velocity in the channel with two-stage section after adding ice cover, because the lower boundary conditions are more complex. In this paper, the five experiment discharge conditions of a channel with ice cover and two-stage section are conducted, and the transverse velocity is measured. The aims of this paper are: (1) to explore transverse distribution of velocity in the two-stage channel with ice cover; (2) to obtain the analytic solution of the transverse velocity of the two-stage section channel with the ice cover.

Experiments set

The experiment was carried out in the laboratory of Zhejiang University of Water Resources and Hydropower. The width of the laboratory flume is 40 cm, foam board represents ice cover, and plexiglass is used for the two-stage section (Figure 1). The method of combining ice cover with the two-stage cross-section channel was used in the experiment. The slope is 0.0005 and the water depth can be adjusted by the tail gate at the end of the flume. The measuring section with x = 5.1 m is measured by LS1206B current meter, and the measurement error is 1.5% by manufacturer. Five tests have been carried out in this study. The flow parameters are shown in Table 1. In previous works (Wood & Liang 1989; Lin & Shiono 1995; Simoes & Wang 1997; Guan 2003; Farzadkhoo et al. 2018; Gu et al. 2018; Bai & Zeng 2019), the width of floodplain varied from 0.1 m to 2.5 m. Here, the width of the floodplain is 0.1 m, and its height is 10 cm. The width of the main channel is 30 cm.

Table 1

Experimental parameters of flow

RunChannelDischarge (L/s)Water depth (cm)B (cm)
Two-stage 7.3 15.4 40 0.0005 
Two-stage 9.1 16.7 40 0.0005 
Two-stage 11.2 18.2 40 0.0005 
Two-stage 13.1 19.4 40 0.0005 
Two-stage 14.9 20.7 40 0.0005 
RunChannelDischarge (L/s)Water depth (cm)B (cm)
Two-stage 7.3 15.4 40 0.0005 
Two-stage 9.1 16.7 40 0.0005 
Two-stage 11.2 18.2 40 0.0005 
Two-stage 13.1 19.4 40 0.0005 
Two-stage 14.9 20.7 40 0.0005 

Analytical solution

For steady uniform flow in a compound channel, the resultant surface force on an elementary volume must be equal to the body force in the main flow direction. The simple two-stage flow function can be described by the Shiono and Knight Model (SKM) (Rameshwaran & Shiono 2007). One considers the water body with the length of dx, the width of dy, and the height of H(y) (Yang et al. 2013). The following Equation (1) is a balance equation between shear stresses, and the product is the shear stress exerted by the flow in the flow direction:
formula
(1)
where ρ is the density of water; g is the gravitational acceleration; is the longitudinal bed slope; is the secondary current term; is the boundary shear stress of the channel bed in the plane perpendicular to z-direction; denotes the shear stress in the x-direction in the plane perpendicular to the y direction, K is secondary current coefficient, which expresses the influence of the secondary flow between the floodplain and the main trough on the lateral distribution of velocity; is the depth-averaged velocity in the main flow direction.
After adding ice cover, shear force of ice cover should be added:
formula
(2)
where ; ; is the shear velocity; is the eddy viscosity coefficient; is the water depth (Nie et al. 2017).
The , and are shear velocity of ice cover and channel bed that can be obtained by solving the following equations (Parthasarathy & Muste 1994):
formula
(3)
can be defined as (Rijn 1984)
formula
(4)
Then
formula
(5)
formula
(6)
For shear stress
formula
(7)
where is the Darcy-Weisbach friction factor of channel bed and ice cover.
Equation (2) can be solved as
formula
(8)
where is the depth-averaged velocity.
We define , then it can be changed as
formula
(9)
To solve Equation (9), we can get
formula
(10)
where , , , , , are integral constants determined by boundary conditions.

Boundary conditions

The analytic solutions can be obtained by dividing the compound channel into two subareas (one is the main channel and the other is the floodplain). Boundary conditions are as follows:

  • (i)

    For a symmetric channel, the lateral gradient of velocity at the centerline of the main channel is zero.

  • (ii)

    The joint of the two domains must satisfy the velocity continuity, i.e., , .

  • (iii)

    The depth-averaged streamwise velocity must be zero at the far side of the floodplain, i.e., .

Parameter determination and model application

Friction factor

The friction coefficient can be described by Rameshwaran & Shiono (2007) 
formula
(11)
where is the equivalent roughness height for ice cover and channel bed, and is flow kinematic viscosity.
formula
(12)
where is Manning's number for ice cover and channel bed.

Eddy viscosity coefficient

The lateral eddy viscosity coefficient in a compound channel can be calculated as (Abril & Knight 2004) follows:
formula
(13)
where = 0.4 is the von Karman constant.

Secondary current coefficient

The secondary current coefficient K was empirically calibrated in order to give the best fit with the experimental data.

Error analysis

Error analysis was conducted to determine the difference between the predicted and measured data of depth-averaged velocity. The root mean square error RMSE and coefficient of determination R2 were calculated by the following equations:
formula
(14)
formula
(15)
formula
(16)
formula
(17)
formula
(18)
where N is the number of lateral measuring points; X is the calculated value and Y is the measured value of depth-averaged velocity.

The experimental and predicted lateral distributions of streamwise velocity in a two-stage section channel with ice cover were compared, and results are shown in Figures 26. In general, the predicted data agrees well with the experimental data, and the velocity in the main channel is much larger than that in the floodplain area. The error statistics of the analytic model are shown in Table 2. Each model has a good prediction compared with the measured data, and RMSE of the model range from 0.0062 to 0.0083. Coefficients of determination R2 of the model are over 0.9727.

Table 2

Error statistical analysis and secondary current coefficient

Run12345
 0.0068 0.0062 0.0083 0.0067 0.0083 
 0.9791 0.9832 0.9727 0.9837 0.977 
Kmc (%) 2.34 2.57 2.95 3.03 3.23 
Run12345
 0.0068 0.0062 0.0083 0.0067 0.0083 
 0.9791 0.9832 0.9727 0.9837 0.977 
Kmc (%) 2.34 2.57 2.95 3.03 3.23 
Figure 1

The structure of the experimental channel.

Figure 1

The structure of the experimental channel.

Close modal
Figure 2

Comparison between calculated and measured velocity (m/s) of Run 1.

Figure 2

Comparison between calculated and measured velocity (m/s) of Run 1.

Close modal
Figure 3

Comparison between calculated and measured velocity (m/s) of Run 2.

Figure 3

Comparison between calculated and measured velocity (m/s) of Run 2.

Close modal
Figure 4

Comparison between calculated and measured velocity (m/s) of Run 3.

Figure 4

Comparison between calculated and measured velocity (m/s) of Run 3.

Close modal
Figure 5

Comparison between calculated and measured velocity (m/s) of Run 4.

Figure 5

Comparison between calculated and measured velocity (m/s) of Run 4.

Close modal
Figure 6

Comparison between calculated and measured velocity (m/s) of Run 5.

Figure 6

Comparison between calculated and measured velocity (m/s) of Run 5.

Close modal

The results show that the transverse distribution of velocity in the compound channel with ice cover is similar to that in the conventional compound channel, and the main channel is obviously larger than the floodplain (Chen 2013; Zeng et al. 2014; Huang et al. 2020). At the same time, the solution of the transverse velocity distribution is similar to the solution of the transverse velocity distribution of the common compound channel. According to the force balance analysis of the fluid divided into elementary volumes, the solution is divided into sections (Rameshwaran & Shiono 2007). In this paper, the influence of the ice sheet is added to the conventional Equation (2). For the channel with ice cover, the ice cover can significantly increase the water depth and reduce the average velocity of the channel (Bai et al. 2020), and the composite section with ice cover also conforms to this rule. In this paper, the characteristic of the ice cover and the compound channel on the velocity has been taken into account when solving the transverse velocity distribution in the compound channel with ice cover, and a high simulation accuracy is obtained (Table 2).

In the two-stage section channel with ice cover, the water depth is obviously larger than that of the simple two-stage section channel. This is because the ice cover is equivalent to adding a boundary, which increases the channel roughness (Xavier et al. 2018; Liu et al. 2019), and it will increase the retention time of water flow, and is more conducive to retaining sediment and nutrients. For the secondary current coefficient, secondary current coefficient in the floodplain Kfp = 0 is assumed considering the small velocity in that region. For the main channel, a difference from the suggested value could be obtained by Ervine et al. (2000), i.e., secondary current coefficient in main channel Kmc < 0.5% for straight compound channels. The secondary current coefficient of the two-stage section channel with ice cover is more than 2.34%, which proves that the influence factors of the secondary current in the two-stage section channel with ice cover are relatively bigger. The transverse velocity distribution in two-stage section channels with ice cover will be more uniform.

This study aims mainly at the change of hydraulic characteristics of non-vegetated compound section channels after freezing. In some compound channels, vegetation often grows on the floodplains. Vegetation will block the flow on the floodplain, and further reduce the flow velocity on the floodplain (Huai et al. 2008, 2009). Flexible or rigid vegetation (Chapman et al. 2015), different arrangements (Zhang et al. 2018), submerged or non-submerged state (Liu et al. 2013) will have a significant impact on the flow. Hydraulic characteristics of the ice cover compound channels with vegetation growing on the floodplains need to be further studied. Hydraulic characteristics have a great influence on the distribution of sediment and pollutants (Choi & Lee 2014, 2015; Zeng et al. 2014), and the law of sediment transport in winter also needs to be studied. The flume slope is only in 0.0005 condition, more slope conditions should be carried out in the future.

The study on the velocity distribution of a channel with compound sections has been completed, but the addition of ice cover in winter will make the hydraulic conditions of multiple cross sections more complicated. In this paper, the fluid equation under the complex cross section of ice cover is solved by integral solution along water depth. The Darcy-Weisbach friction factor is creatively unified into the sum of the two components of channel bed and ice cover, which is more conducive to the solution of the formula. The main results of this paper are as follows:

  • (1)

    Two-stage section channel with ice cover can better slow down the flow velocity in the channel, increase the retention time of water flow, and is more conducive to retaining sediment and nutrients.

  • (2)

    According to the equation, the analytical solution of transverse velocity distribution in the two-stage section channel with ice cover is solved. The model has high accuracy, and the analytical solution of velocity also provides a research basis for the distribution of sediment and phosphorus in the two-stage section channel with ice cover, and provides a theoretical reference for the design of the two-stage section channel with ice cover in the future.

Key technology, equipment development and application demonstration of environmental protection and resource comprehensive utilization – Key technology development and application demonstration of comprehensive management and resource utilization of cyanobacteria in Taihu Lake Basin (Key R & D funds of Zhejiang Province: 2021C03196); Zhejiang Basic Public Welfare Research Project (LGF19E090001); Thanks to Qianqian Mao, and Zhicheng Qiu in Zhejiang University of Water Resources and Electric Power for proofreading of the MS.

All relevant data are included in the paper or its Supplementary Information.

Abril
J. B.
&
Knight
D. W.
2004
Stage-discharge prediction for rivers in flood applying a depth-averaged model
.
Journal of Hydraulic Research
42
(
6
),
616
629
.
Chapman
J. A.
,
Wilson
B. N.
&
Gulliver
J. S.
2015
Drag force parameters of rigid and flexible vegetal elements
.
Water Resources Research
51
(
5
),
3292
3302
.
Choi
S. U.
&
Lee
J.
2014
Assessment of total sediment load in rivers using lateral distribution method
.
Journal of Hydro-Environment Research
9
(
3
),
381
387
.
Choi
S. U.
&
Lee
J.
2015
Prediction of total sediment load in sand-bed rivers in Korea using lateral distribution method. JAWRA
.
Journal of the American Water Resources Association
51
(
1
),
214
225
.
Christopher
S. F.
,
Tank
J. L.
,
Mahl
U. H.
,
Yen
H.
,
Arnold
J. G.
,
Trentman
M. T.
,
Kelly-Gerreyn
B. A.
&
Royer
T. V.
2017
Modeling nutrient removal using watershed-scale implementation of the two-stage ditch
.
Ecological Engineering
108
,
358
369
.
Davis
R. T.
,
Tank
J. L.
,
Mahl
U. H.
,
Winikoff
S. G.
&
Roley
S. S.
2015
The influence of two-stage ditches with constructed floodplains on water column nutrients and sediments in agricultural streams. JAWRA
.
Journal of the American Water Resources Association
51
(
4
),
941
955
.
Dolan
D. M.
&
McGunagle
K. P.
2005
Lake Erie total phosphorus loading analysis and update: 1996–2002
.
Journal of Great Lakes Research
31
,
11
22
.
Ervine
D. A.
,
Babaeyan-Koopaei
K.
&
Sellin
R. H.
2000
Two-dimensional solution for straight and meandering overbank flows
.
Journal of Hydraulic Engineering
126
(
9
),
653
669
.
Farzadkhoo
M.
,
Keshavarzi
A.
,
Hamidifar
H.
&
Javan
M.
2018
A comparative study of longitudinal dispersion models in rigid vegetated compound meandering channels
.
Journal of Environmental Management
217
,
78
89
.
Gu
L.
,
Zhao
X. X.
,
Xing
L. H.
,
Jiao
Z. N.
,
Hua
Z. L.
&
Liu
X. D.
2018
Longitudinal dispersion coefficients of pollutants in compound channels with vegetated floodplains
.
Journal of Hydrodynamics
31
(
4
),
740
749
.
Guan
Y.
2003
Simulation of Dispersion in Compound Channels
.
Dissertation
,
École Polytechnique Féderale de Lausanne
,
Lausanne, Switzerland
.
Hodaj
A.
,
Bowling
L. C.
,
Frankenberger
J. R.
&
Chaubey
I.
2017
Impact of a two-stage ditch on channel water quality
.
Agricultural Water Management
192
,
126
137
.
Huai
W. X.
,
Xu
Z. G.
,
Yang
Z. H.
&
Zeng
Y. H.
2008
Two dimensional analytical solution for a partially vegetated compound channel flow
.
Applied Mathematics and Mechanics
29
(
8
),
1077
1084
.
Huai
W. X.
,
Gao
M.
,
Zeng
Y. H.
&
Li
D.
2009
Two-dimensional analytical solution for compound channel flows with vegetated floodplains
.
Applied Mathematics and Mechanics
30
(
9
),
1121
1130
.
Krider
L.
,
Magner
J.
,
Hansen
B.
,
Wilson
B.
,
Kramer
G.
,
Peterson
J.
&
Nieber
J.
2017
Improvements in fluvial stability associated with two-stage ditch construction in Mower County, Minnesota
.
JAWRA Journal of the American Water Resources Association
53
(
4
),
886
902
.
Kumwimba
M. N.
,
Meng
F.
,
Iseyemi
O.
,
Moore
M. T.
,
Zhu
B.
,
Tao
W.
,
Liang
T. J.
&
Ilunga
L.
2018
Removal of non-point source pollutants from domestic sewage and agricultural runoff by vegetated drainage ditches (VDDs): design, mechanism, management strategies, and future directions
.
Science of the Total Environment
639
,
742
759
.
Lau
Y. L.
1982
Velocity distributions under floating covers
.
Canadian Journal of Civil Engineering
9
(
1
),
76
83
.
Lin
B.
&
Shiono
K.
1995
Numerical modelling of solute transport in compound channel flows
.
Journal of Hydraulic Research
33
(
6
),
773
788
.
Mahl
U. H.
,
Tank
J. L.
,
Roley
S. S.
&
Davis
R. T.
2015
Two-stage ditch floodplains enhance N-Removal capacity and reduce turbidity and dissolved P in agricultural streams
.
JAWRA, Journal of the American Water Resources Association
51
(
4
),
923
940
.
Nie
S.
,
Sun
H.
,
Zhang
Y.
,
Chen
D.
,
Chen
W.
,
Chen
L.
&
Schaefer
S.
2017
Vertical distribution of suspended sediment under steady flow: existing theories and fractional derivative model
.
Discrete Dynamics in Nature and Society
2017
,
5481531
.
Parthasarathy
R. N.
&
Muste
M.
1994
Velocity measurements in asymmetric turbulent channel flows
.
Journal of Hydraulic Engineering
120
(
9
),
1000
1020
.
Rijn
L. C. V.
1984
Sediment transport, part II: suspended load transport
.
Journal of Hydraulic Engineering
110
(
11
),
1613
1641
.
Roley
S. S.
,
Tank
J. L.
,
Tyndall
J. C.
&
Witter
J. D.
2016
How cost-effective are cover crops, wetlands, and two-stage ditches for nitrogen removal in the Mississippi River Basin?
Water Resources and Economics
15
,
43
56
.
Simoes
F. J.
&
Wang
S. S. Y.
1997
Numerical prediction of three-dimensional mixing in a compound open channel
.
Journal of Hydraulic Research
35
(
5
),
619
642
.
Stephenson
D.
&
Kolovopoulos
P.
1990
Effects of momentum transfer in compound channels
.
Journal of Hydraulic Engineering
116
(
12
),
1512
1522
.
Urroz
G. E.
&
Ettema
R.
1994
Application of two-layer hypothesis to fully developed flow in ice-covered curved channels
.
Canadian Journal of Civil Engineering
21
(
1
),
101
110
.
Västilä
K.
&
Järvelä
J.
2011
Environmentally preferable two-stage drainage channels: considerations for cohesive sediments and conveyance
.
International Journal of River Basin Management
9
(
3–4
),
171
180
.
Vidon
P.
,
Hubbard
H.
,
Cuadra
P.
&
Hennessy
M.
2012
Storm phosphorus concentrations and fluxes in artificially drained landscapes of the US Midwest
.
Agricultural Sciences
3
(
04
),
474
.
Wood
I. R.
&
Liang
T.
1989
Dispersion in an open channel with a step in the cross-section
.
Journal of Hydraulic Research
27
(
5
),
587
601
.
Zeng
L.
,
Zhao
Y. J.
,
Chen
B.
,
Ji
P.
,
Wu
Y. H.
&
Feng
L.
2014
Longitudinal spread of bicomponent contaminant in wetland flow dominated by bank-wall effect
.
Journal of Hydrology
509
,
179
187
.
Zhang
S.
,
Liu
Y.
,
Zhang
J.
,
Liu
Y.
&
Wang
Z.
2018
Study of the impact of vegetation direction and slope on drag coefficient
.
Iranian Journal of Science and Technology, Transactions of Civil Engineering
42
(
4
),
381
390
.
Zhao
F.
,
Xi
S.
,
Yang
X.
,
Yang
W.
,
Li
J.
,
Gu
B.
&
He
Z.
2012
Purifying eutrophic river waters with integrated floating island systems
.
Ecological Engineering
40
,
53
60
.
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