Advances in Water Resources 74 (2014) 26–43

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Advances in Water Resources

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An analytical model for lateral depth-averaged velocity distributionsalong a meander in curved compound channels

http://dx.doi.org/10.1016/j.advwatres.2014.08.0030309-1708/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: State Key Laboratory of Hydraulics and MountainRiver Engineering, Sichuan University, Chengdu, Sichuan 610065, China. Tel.: +8628 85401937.

E-mail address: yangkejun@scu.edu.cn (K. Yang).

Chao Liu a,b, Nigel Wright c, Xingnian Liu a,b, Kejun Yang a,b,⇑a State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, Sichuan 610065, Chinab College of Water Resources and Hydropower, Sichuan University, Chengdu, Sichuan 610065, Chinac School of Civil Engineering, University of Leeds, Leeds LS2 9JT, UK

a r t i c l e i n f o

Article history:Received 4 February 2014Received in revised form 28 July 2014Accepted 2 August 2014Available online 19 August 2014

Keywords:Meandering compound channelSecondary flow generation mechanismDivided region methodTwo-dimensional analytical modelBoundary condition

a b s t r a c t

This paper presents an analytical method for modeling the lateral depth-averaged velocity distributionalong a half-meander in a curved compound channel. An equation is derived from the momentumequation and the flow continuity equation which contains a velocity term with both streamwise velocityvariation and lateral secondary flow variation. A velocity variation parameter is proposed in the mainchannel and on the floodplain for a series of test sections. To study the validity of these equations exper-iments were conducted in a large scale meandering compound channel at Sichuan University, China.Based on the experimental data, the generation mechanism of secondary flow in the main channel alonghalf a meander is analyzed. It is shown that the secondary current is enhanced by the centrifugal forceand the floodplain flow. Due to the discontinuity of the flow depth and the effect of meandering in themain channel flow, a region divided method is adopted. A new boundary condition is proposed by intro-ducing the angle between the main channel flow and the floodplain flow, and it is shown that this givesbetter modeling results in cross-over sections. The modeling results indicate that the proposed method,which uses the new boundary condition and includes both the streamwise velocity variation and thelateral secondary flow variation, can model the lateral depth-averaged velocity distributions more accu-rately. Finally, variations in the velocity term between the main channel and floodplain are discussed andanalyzed.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In a river corridor system, rivers play an important role in theprovision of water and habitat to the surrounding fauna and flora.Natural alluvial rivers and streams often exhibit a curved mainriver and one or two corresponding floodplains. When a floodoccurs, the flow depth increases and the floodplains are submergedto convey the extra flow, leading to overbank flow in the meander-ing compound channel. It is important to note that the flowcharacteristics in a curved channel are very different to that in astraight one, especially with regard to the generation mechanismof main channel secondary flows. Accordingly, it is necessary toassess the changes of the velocity, secondary flow, bed shear stressand discharge in the meandering compound channels.

To determine sediment transport, channel morphology andbank erosion, the lateral distributions of depth-averaged velocityand bed shear stress are crucial. In recent decades, the three-dimensional flow structure, turbulence characteristics and second-ary flows in straight compound channels have been extensivelyinvestigated by, amongst others, Knight and Demetriou [14],Pasche and Rouve [25], Knight and Sellin [15], Tominaga et al.[41], Yuen [45], Carling et al. [3] and Yang et al. [43]. Moreover, lat-eral distribution methods (LDM) for the depth-averaged velocityand bed shear stress, along with the secondary flow and the plan-form vorticity, have been developed by Shiono and Knight [31], Caoet al. [2], Rameshwaran and Shiono [29], Huai et al. [9], Hu et al.[8], Liu et al. [20] and Yang et al. [44]. These researchers showedthat the secondary flows affect the predictions of velocity andbed shear stress significantly and ignoring them leads to poorresults. Therefore, Ervine et al. [5], Spooner [38] and Huai et al.[10] proposed new secondary flow expressions which were appliedat the apex sections in the meandering compound channel withnon-mobile bed and mobile bed. However, when the flow depthis discontinuous at the interfaces of main channel and its

Fig. 1. Photograph of the meandering compound channel in SKLH (the co-ordinatesystems in the meandering main channel and on the floodplain are different, exceptat the apex section).

C. Liu et al. / Advances in Water Resources 74 (2014) 26–43 27

corresponding floodplains the boundary condition must be recon-sidered. Knight et al. [16] and Tang and Knight [39] discussed theboundary conditions for the Shiono and Knight Method [31] (here-after referred as SKM) and discovered that improved results wereobtained by using the continuity of depth-averaged shear stressat the interfaces of main channel and floodplain. Furthermore,the turbulent transfer is also an important phenomenon in com-pound channels and based on mathematical integration, Castanedoet al. [4] identified three different forms of the turbulent diffusionterm in the depth-averaged Navier–Stokes equation. Subsequently,these three models and the original SKM were compared by Tangand Knight [40] in both a straight trapezoidal channel and astraight compound channel. Their conclusions show that to obtaingood predictions the four models must contain the secondary flowparameter and this affects the results more than the dimensionlesseddy viscosity for overbank flows.

For overbank flow in a meandering compound channel with afixed bed, experimental work [28,32,33] has shown that secondarycurrent cells in the main channel are much stronger than those onthe floodplain due to the effect of centrifugal force. For overbankflows in a meandering compound channel, discharge assessmentmethods have been proposed by various authors [7,19,26,27,34].Flow characteristics were also discussed in a meandering com-pound channel with non-mobile bed and three different sinuosities(1.093–1.571) by Shiono and Muto [32], who found that the sec-ondary flow for the inbank and overbank flows originated from dif-ferent mechanisms. Their experimental results also showed that astrong intensity of secondary current cells existed mainly in themain channel, especially for the overbank flow. Then, experimentalresearch was extended to curved channels with mobile bed anddifferent floodplain vegetation by Ishigaki et al. [11], Lyness et al.[21], Keevil et al. [13], Shiono et al. [35] and Shiono et al. [36,37].From these studies, some conclusions were obtained: (1) the veg-etation on the floodplain reduced the channel conveyance capabil-ity significantly; (2) the lateral secondary flow distributions in ameandering channel were quite different between the cases withthe non-mobile bed and mobile bed; (3) in the cases with a mobilebed, multiple secondary current cells which cause a series of wavybedforms occur at deeper flow depth along the meandering mainchannel when the floodplain roughness increases. These phenom-ena are usually seen in natural rivers and the conclusions aretherefore valuable for engineering projects.

As well as experimental studies, numerical investigations canalso give insight into flows in meandering channels. A one-dimensional simulation with vegetation in a curved channel waspresented by Martin-Vide et al. [22] and two-dimensional modelswere described by Shao et al. [30] and Zarrati et al. [46], presentinggood modeling results for the velocity distribution, the secondaryflow and the flow depth. Morvan et al. [24] and Jing et al. [12] car-ried out three-dimensional methods to simulate the velocity fieldsin overbank flow. Further, analytical methods have been proposedby Ervine et al. [6], McGahey et al. [23] and Huai et al. [10], basedon the SKM method, to predict the depth-averaged velocity distri-bution. Although velocity patterns in a meandering compoundchannel are highly three dimensional, these researchers neglectedthe streamwise velocity variation to simplify the analytical modelswhich were therefore only applicable at apex sections. Knight et al.[18] pointed out that the modeling capability of SKM is poor inmeandering compound channels because of the methodology isderived for steady flows in straight prismatic channels. Accordingto the research described above, it is hard to find a reasonable ana-lytical method to predict the lateral velocity distributions along ameander, and the lack of detailed experimental data also presentsdifficulties. This forms the motivation for this paper in which a newmodel is presented along with a series of experiments to demon-strate its derivation and validate its results.

The research presented in this paper explores an analyticalmethod to model the depth-averaged velocity along a meanderin a curved compound channel. A governing equation is derivedfrom the streamwise momentum equation and the flow continuityequation. Its velocity term contains the streamwise velocity varia-tion, ignored by Ervine et al. [6], McGahey et al. [23] and Huai et al.[10], and the lateral secondary flow variation. In order to verify thismodel, two groups of experiments were conducted in a large scalemeandering compound channel at Sichuan University in China. Thethree-dimensional velocities, the flow depth and the Reynoldsshear stress were recorded at seven test sections along half ameander. Based on the experimental data, the generation mecha-nism of secondary flows, including the effects of centrifugal forceand floodplain flow, is analyzed and an expression for the velocityvariation parameter at the apex and cross-over sections is pro-posed. Further, the divided method in half a meander is presentedby considering the effect of meandering main channel flow. A newboundary condition is proposed by introducing the angle betweenthe main channel and floodplain. Finally, the modeling results ofdepth-averaged velocity by this method are compared with theexperimental data. Each part of the velocity term is discussedand analyzed in the meandering main channel and on thefloodplain.

2. Experimental arrangement and apparatus

Two groups of experiments (MN1 and MN2) were conducted ina 35 m long, 4 m wide and 1 m high flume, at State Key Laboratoryof Hydraulics and Mountain River Engineering (SKLH), SichuanUniversity (see Fig. 1). The flow depths of MN1 and MN2 in themain channel are 0.255 m and 0.216 m, respectively. Measure-ments were carried out for the stage-discharge curve, the Reynoldsshear stresses and the three-dimensional velocities. The dischargewas measured by a triangular weir installed in front of the flumeand the flow depth was measured by an Automatic Ultrasonic

Fig. 2. Plan details of geometry and measurement sections for the meanderingcompound channel with the co-ordinate systems in the main channel and on thefloodplain.

Fig. 3. Measurement vertical lines in the meandering main channel.

28 C. Liu et al. / Advances in Water Resources 74 (2014) 26–43

Measurement System (AUMS) produced by the Sinfotek Corpora-tion (www.sinfotek.com). The three-dimensional velocities andReynolds shear stresses were recorded by the three-componentSontek acoustic Doppler velocimeter (ADV), equipped with up-sideand down-side probes. At each point, measurement time and fre-quency were set as 30 s and 50 Hz, respectively.

The shape of the main channel was rectangular with the verticalside slope and the sinuosity of the meandering main channel wasdesigned as 1.381. The total width (B) and main channel width(b) were 4 m and 0.7 m, respectively with the main channel depth(h) set to 0.14 m giving an aspect ratio (b/h) of 5. The bed surfacewas smoothed and covered by a thin layer of concrete. The flumehad a fixed bed slope of 0.001 and the error in the geometry wascontrolled to within 5%. A summary of experimental conditions isshown in Table 1.

Four and a half meanders were constructed and the test mean-der is pointed out in Fig. 2. In the meandering main channel, 13vertical measurement lines were arranged where the lateralinterval was 0.05 m from left (y = 0) to right, denoted 1–13 in eachsection and shown in Fig. 3. On the floodplain, the interval of eachmeasurement line was set as 0.2 m while 17 vertical lines werearranged for CS1 and CS7 and 18 for CS2–CS6. The measurementintervals between two vertical points were arranged as 0.015 min the main channel with 0.015 m and 0.01 m on the floodplainfor MN1 and MN2, respectively.

All experimental data were recorded under the quasi-uniformflow conditions by ensuring that the water surface slope remainedparallel to the bed slope at each meander by manually adjustingthe downstream tailgate. When the deviation of water surfaceand bed slope became less than 5%, the quasi-uniform flow condi-tion was considered to have been attained and the measurementswere started. Using the velocity experimental data, the measure-ment discharges for both cases were back calculated and theirerrors were found to be 2.85% and 3.49%, respectively. In thispaper, the lateral depth-averaged velocity distribution is modeledonly along half a meander because the presence of the flowstructure is periodical.

3. Region divided theory

Knight et al. [17] suggested that more accurate results can beobtained by using more divided panels in lateral distributionmethods (LDM), because of the complex secondary flow distribu-tion in the single channel. In this paper, the proposed model is alsoa two-dimensional LDM method (outlined in Section 4) applied in ameandering compound channel and therefore, each of the seventest sections (CS1–CS7) is divided into several panels to obtainaccurate lateral depth-averaged velocity distributions along ameander. The research by Shiono and Muto [32] has shown thatthe flow behavior is complex and strong momentum exchangeexists in the vicinity of interfaces between the main channel andfloodplains, therefore, the mixing regions have to be separatedout. On the floodplain, the inside floodplain and the outsidefloodplain should be treated separately, because the effect of the

Table 1Experimental conditions.

Total width (m) Meander beltwidth (m)

Wavelength (m) Inner radius (m) Outer ra

4.0 2.99 5.53 0.9 1.6

Series name Main channelaspect ratio (b/h)

Manningroughness n

Dischar

MN1 5 0.015 0.189MN2 5 0.015 0.113

meandering main channel flow to the flow out of the meander beltmay be ignored [7]. Finally, the main channel and two floodplainsmust be separated because of the flow depth discontinuity and thedivided regions in half a meander are shown in Fig. 4. In conclu-sion, the whole sections of CS1 and CS7 are divided into 7 panelswhile that of CS2–CS6 are divided into 9 panels.

4. Theoretical background

The streamwise momentum equation for the quasi-uniformflow may be expressed as follows

@syx

@yþ @szx

@zþ qgS ¼ q U

@U@xþ V

@U@yþW

@U@z

� �ð1Þ

where x, y, z are the streamwise, lateral and vertical directions,respectively, and the co-ordinate systems in the main channel andon the floodplain are as shown in Fig. 1; syx and szx are the Reynoldsstresses on the planes perpendicular to y and z, respectively; q is

dius (m) Cross-overlength (m)

Side slope Valley slope Sinuosity

1.2 90� 0.001 1.381

ge (m3/s) Flow depth (m) Bankfull depth (m) Relative flow depth (Dr)

0.255 0.14 0.4510.216 0.14 0.352

Fig. 4. The scheme of divided region in the meandering compound channel.

Fig. 5. The ratio of transverse velocity to streamwise velocity (VU), based on the

experimental data of case MN1, H = 0.255 m, Q = 0.189 m3/s (a) the centerline ofmeandering main channel (b) the centerline of left floodplain.

Fig. 6. Effect of centrifugal force to the product of streamwise and transversevelocities at CS2.

C. Liu et al. / Advances in Water Resources 74 (2014) 26–43 29

the flow density; g is the local gravitational acceleration; S is thevalley slope; U, V, W are the velocity components correspondingto {x,y,z} directions.

For an incompressible fluid, combining Eq. (1) with the flowcontinuity equation gives

@syx

@yþ @szx

@zþ qgS ¼ q

@U2

@xþ q

@UV@yþ q

@UW@z

ð2Þ

For the velocity term located on the right hand of Eq. (2), Shiono andKnight [31], Rameshwaran and Shiono [29], Huai et al. [10], Hu et al.[8], Liu et al. [20] and Yang et al. [44] ignored the streamwise veloc-

ity variation @U2

@x � 0� �

in a straight compound channel. However,

for the meandering compound channel flow, the streamwisevelocity variation must be considered due to the complex 3D flowbehavior [32,35,42]. Therefore, in the analysis below, thestreamwise velocity variation is included for the curved compoundchannel in contrast to other research [6,10,23] where it was

ignored. Further the secondary flow term q @UV@y

� �is reconsidered

because the secondary flow is enhanced by the centrifugal forceand the floodplain flow, especially at the cross-over sections.

Using the experimental data, the ratio of transverse velocity tostreamwise velocity (V/U) at the centerlines of meandering mainchannel and outside floodplain is shown in Fig. 5. From the figure,�0.04 < V/U < 0.04 at the center of left outside floodplain, whoseintensity of secondary flow is almost the same as that in thestraight channel, while �0.6 < V/U < 3.0 at the centerline of themeandering main channel is much larger. Apparently, the trans-verse velocity is enhanced significantly in the meandering mainchannel. Therefore, the generation mechanism of secondary flowsin the curved channel must be analyzed.

The centrifugal force mainly contributes to the super elevationand the transverse velocity V. The proposed model in this paper is atwo-dimensional method whose governing equation is integratedover flow height. According to research by Ervine et al. [6], Shionoand Muto [32], Shiono et al.[33,35–37] and Spooner [38], the flowdepth is regarded as a lateral constant by checking the flow depthat every apex section as done here. Therefore, in this analyticalmodel the lateral flow height is assumed as a constant by ignoringthe local super elevation in the main channel and the mixingregion. With regard to the influence of centrifugal force, the trans-verse velocity V in the main channel increases significantly andaccordingly the UV value increases as well, as seen in Fig. 6.According to dimensional analysis, the centrifugal force and thegradient of original secondary flows have the same dimension.Hence, a direct proportion which represents the impact extent of

centrifugal force may be presented for a unit volume of water bodyin the meandering main channel.

Fc / q@ UV1ð Þ@y

ð3Þ

where V1 is the transverse velocity without the effect of centrifugalforce.

The generation of secondary flows in the meandering main chan-nel is mainly controlled by the centrifugal force and the lateral com-ponent of floodplain flow, especially at the cross-over sections with

30 C. Liu et al. / Advances in Water Resources 74 (2014) 26–43

large angle (h shown in Fig. 2) where the component of floodplainflow contributes significantly. The schematic diagram of the trans-verse velocities at points A (CS2) and B (CS6) in the meanderingmain channel is shown in Fig. 7. The transverse velocity distribu-tions are very different due to the converse rotational directionsof secondary flows at the points A and B. The intensity of the second-ary current above the bankfull level at CS6 is larger than that at CS2because the secondary flow is enhanced by the component of flood-plain flow when their flow directions are the same.

Fig. 7. Vertical distributions of trans

Fig. 8. Generation mechanism of secondary flows in the meandering main chann

Based on the experimental data (case MN1) shown in Fig. 8, itcan be observed that the transverse velocity (V) consists of the sec-ondary current cell (red line), enhanced by the centrifugal force(Fc), and the component of floodplain flow (V2) (blue line). There-fore, the effect of centrifugal force is assumed as qk @ UV1ð Þ

@y , wherek is a dimensionless coefficient reflecting the relation with Fc,and which should be added into Eq. (2). The assumption made heremay be most helpful for understanding the effect of centrifugalforce on original secondary flows, and this influence could be

verse velocities at CS2 and CS6.

el, based on the experimental data of case MN1, H = 0.255 m, Q = 0.189 m3/s.

Fig. 9. Planform of depth-averaged velocity along half a meander for two cases (a)case MN1 (b) case MN2.

Fig. 10. Relation of depth-averaged velocities and the angles in the main channeland on the floodplain.

C. Liu et al. / Advances in Water Resources 74 (2014) 26–43 31

reflected by a dimensionless coefficient k in the velocity term ofgoverning equation. Hence, Eq. (2) may be rewritten by consider-ing the influence of centrifugal force and the component offloodplain flow as follows

@syx

@yþ @szx

@zþ qgS ¼ q

@U2

@xþ qð1þ kÞ @UV1

@yþ q

@UV2

@yþ q

@UW@z

ð4Þ

where V2 is the lateral component of the floodplain flow in the mainchannel, and V = (1 + k)V1 + V2.

From the analysis above, it can be seen that the secondary flowsat different sections along a meander vary. Some new findingsabout the secondary current cells in the meandering main channelare obtained (shown in Fig. 8). For the apex sections (CS1 and CS7),only two secondary current cells (a large one and a small one)rotate conversely in the whole main channel while the componentof floodplain flow can be ignored because the angle between thestreamwise direction and flow direction is almost 0�. For CS2 andCS6, the secondary current and the floodplain flow contribute

together, and the vertical distributions of transverse velocitiesare changed significantly which is demonstrated by the proposedtheory in Fig. 7. For CS3, CS4 and CS5, the floodplain flowdominates the transverse velocity and a new secondary current cellgenerated from the edge of the right wall is compressed below thebankfull level (h). The new secondary current cell is first observedat CS3, and then grows rapidly from CS4 to CS6, reaching its largestat CS7. The periodic phenomenon can also be discovered in theprevious meander and the subsequent one, therefore, the observedconverse rotation of secondary current cells at CS1 may beexplained clearly.

Integrating Eq. (4) over the flow height (H) gives differentresults at various sections

@

@yqkH2

ffiffiffif8

rUd

@Ud

@y

!� q

f8

U2d þ qgS ¼ K ð5Þ

and

Ud ¼1H

Z H

0Udz; sb ¼ q

f8

� �U2

d; syx ¼ qeyx@Ud

@y; eyx ¼ kHU�

ð6Þwhere k is the dimensionless eddy viscosity; f is the Darcy–Weisbach friction factor; Ud is the depth-averaged velocity; K isthe velocity variation parameter whose expression depends onthe test sections, shown in Eqs. (7)–(9); sb is the bed shear stress;syx is the depth-averaged Reynolds shear stress; eyx is the depth-

averaged eddy viscosity; U⁄ is the shear velocity ¼ffiffif8

qUd

� �From the analysis of secondary flows in the main channel, the

transverse velocity (V) is seen to consist of V1 and V2 whose effectsare different at each section, which in turn leads to various expres-sions for K. Therefore,

Fig. 11. Lateral distributions of the angle h1 in the meandering main channel at CS1–CS7 (H = 0.255 m for case MN1 and H = 0.216 for case MN2).

32 C. Liu et al. / Advances in Water Resources 74 (2014) 26–43

For apex sections ðCS1 and CS7Þ;

K ¼ qH@U2

d

@xþ qð1þ kÞH @ðUV1Þd

@yð7Þ

For cross� over sections ðCS2 and CS6Þ;

K ¼ qH@U2

d

@xþ qð1þ kÞH @ðUV1Þd

@yþ qH

@ðUV2Þd@y

ð8Þ

For cross� over sections ðCS3; CS4 and CS5Þ;

K ¼ qH@U2

d

@xþ qð1þ kÞh @ðUV1Þd

@yþ qH

@ðUV2Þd@y

ð9Þ

In order to simplify Eqs. (7)–(9), simple expressions are derivedfor the streamwise variation of U2

d and the lateral variation of ðUVÞdas follows

Kx ¼@U2

d

@x; Ky1 ¼

@ðUV1Þd@y

and Ky2 ¼@ðUV2Þd@y

ð10Þ

where Kx is the streamwise variation coefficient; Ky1 and Ky2 are thesecondary flow coefficients (Ky = Ky1 + Ky2). It should be noted thatthe coefficients of Eq. (10) are not constant in the streamwise andlateral directions and in order to present the analytical solutionsof Eq. (5), they are assumed as constants in each divided panel.Therefore, the velocity term in the meandering main channel maybe rewritten as

For apex sections ðCS1 and CS7Þ;K ¼ qHKx þ qð1þ kÞHKy1ðHÞ; ð11Þ

For cross� over sections ðCS2 and CS6Þ;K ¼ qHKx þ qð1þ kÞHKy1ðHÞ þ qHKy2ðHÞ; ð12Þ

For cross-over sections ðCS3; CS4 and CS5Þ;K ¼ qHKx þ qð1þ kÞhKy1ðhÞ þ qHKy2ðHÞ ð13Þ

Table 2Dimensionless eddy viscosity at each section.

Case CS1 CS2 CS3 CS4 CS5 CS6 CS7

MN1 kmc 0.075 0.033 0.036 0.032 0.038 0.049 0.070kfp 0.145 0.135 0.101 0.140 0.117 0.148 0.137

MN2 kmc 0.069 0.042 0.040 0.043 0.050 0.049 0.068kfp 0.127 0.117 0.101 0.122 0.119 0.141 0.123

Fig. 12. Lateral distributions of velocity variation parameter K in the meanderingmain channel at CS1–CS7 (a) case MN1, H = 0.255 m, Q = 0.189 m3/s (b) case MN2,H = 0.216 m, Q = 0.113 m3/s.

where the subscripts ‘‘H’’ and ‘‘h’’ are the flow height and thebankfull height, respectively. Due to the different integral heights,the subscript is shown herein to avoid misunderstanding.

On the floodplain, the velocity term without the effect ofcentrifugal force may be simplified as

K ¼ qðH � hÞKx þ qðH � hÞKyðH�hÞ ð14Þ

Fig. 13. Lateral distributions of velocity variation parameter K on the floodplain at CS1–CS7 (a) CS1 (b) CS2 (c) CS3 (d) CS4 (e) CS5 (f) CS6 (g) CS7.

C. Liu et al. / Advances in Water Resources 74 (2014) 26–43 33

Fig. 13 (continued)

34 C. Liu et al. / Advances in Water Resources 74 (2014) 26–43

The analytical solution of Eq. (5) is shown as

Ud ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA1ec1y þ A2ec2y þx

pð15Þ

where c1 ¼ 1H

2k

� �1=2 f8

� �1=4; c1 ¼ �c2; x ¼ 8gHS

f 1� KqgHS

h i, A1 and A2 are

unknown constants.

5. Boundary conditions

To obtain the analytical solutions for Ud [Eq. (15)] the unknownconstants need to be eliminated by appropriate boundary condi-tions. For the apex sections (CS1 and CS7), the boundary conditionsare easily established because the co-ordinate system on the flood-plain is coincident with that in the meandering main channel.However, the boundary condition should be reconsidered for thecross-over sections (CS2–CS6) where the co-ordinate systems ofmain channel and floodplains include an angle (h) (h = 30� forCS2 and CS6; h = 60� for CS3, CS4 and CS5).

The planforms of depth-averaged velocity for MN1 and MN2 inhalf a meander are shown in Fig. 9, and it is noticeable that the trueflow directions in the main channel do not follow the valley direc-tion. This phenomenon was also observed by Shiono et al. [36] andcan be attributed to the influence of centrifugal force and flood-plain flow, especially at the cross-over sections (CS2–CS6) withoverbank flows. The relation of the velocities in the main channeland on the floodplain is shown in Fig. 10, where Ud(1) is thedepth-averaged velocity on the floodplain, Ud(2) is the projectionof Ud(3) in the streamwise direction on the floodplain, Ud(3) is thedepth-averaged velocity in the main channel with the true flowdirection, and Ud(4) is the measured experimental depth-averagedvelocity following the valley direction in the main channel.

When points A and C are infinitely close to point B (i.e. DY ? 0and DL ? 0), the velocity at point A equals that at the point C:

Udð1Þ ¼ Udð2Þ ð16Þ

Due to the relation of velocities shown in Fig. 10, the threevelocities (Ud(2), Ud(3) and Ud(4)) at the point C in the main channelmay be inter-changed.

Udð2Þ ¼Udð3Þ

cos h2¼ Udð4Þ

cos h1 cos h2ð17Þ

where h1 is the angle between Ud(3) and Ud(4); h2 is the anglebetween Ud(2) and Ud(3); h = h1 + h2.

For the cases MN1 and MN2, we derived the lateral distribu-tions of h1 from CS1 to CS7 from the experimental data, shown inFig. 11. From the figure, the h1 values at CS1 and CS7 are almostequal to 0�, h1 values ranges between 12.6� and 24.2� at CS2 andCS6 while h1 = 23.3–37.0� is observed at CS3, CS4 and CS5. There-fore, the assumption of h1 ¼ h2 ¼ h

2 is made at each section for casesMN1 and MN2, leaving h1 = h2 = 0� at CS1 and CS7, h1 = h2 = 15� atCS2 and CS6, and h1 = h2 = 30� at CS3, CS4 and CS5, which almostequals the mean values of the experimental data. Therefore, thevelocity continuity and velocity gradient continuity at the inter-faces of main channel and floodplains have to contain h1 and h2,and new boundary conditions are as follows:

� No-slip condition, i.e. Ud = 0 at remote boundary.� Continuity of velocity at each domain junction, i.e.

UðiÞd ¼Udðiþ1Þ

cos h1 cos h2.

� Gradient continuity of velocity at each domain junction, i.e.@UðiÞ

d@y ¼ 1

cos h1 cos h2

@Uðiþ1Þd@y .

where the superscript (i) is the number of divided panels.Noting h1 = h2 = 0� at the interfaces of divided panels on the

floodplain, therefore, the continuity of velocity and velocity gradi-

ent are simplified as UðiÞd ¼ Uðiþ1Þd and

@UðiÞd

@y ¼@Uðiþ1Þ

d@y .

6. Determination of model parameters

Applying this approach to model the depth-averaged velocity,all parameters remain constant in each divided panel. Beforeapplying this model, four coefficients, i.e. Darcy–Weisbach frictionfactor (f ), dimensionless eddy viscosity (k), velocity parameter (K),need to be determined as follows.

The Darcy–Weisbach friction factor (f ) is usually back calcu-lated by the experimental data from depth-averaged velocity andbed shear stress [29,31,40]. Because there is no means for

measuring the boundary shear stress, the expression f ¼ 8gn2

R1=3 , pro-posed by Knight et al. [17] and Huai et al. [9] was used to predict

Fig. 14. Modeling depth-averaged velocity from CS1 to CS7 with experimental data for case MN1, H = 0.255 m, Q = 0.189 m3/s (a) CS1 (b) CS2 (c) CS3 (d) CS4 (e) CS5 (f) CS6(g) CS7.

C. Liu et al. / Advances in Water Resources 74 (2014) 26–43 35

Fig. 14 (continued)

36 C. Liu et al. / Advances in Water Resources 74 (2014) 26–43

the Darcy–Weisbach friction factor (f ) in the main channel and thefloodplains.

For the dimensionless eddy viscosity (k), in a straight compoundchannel, Shiono and Knight [31] illustrated that k changed slightlyin the main channel and significantly on the floodplain. Later, a

practical relationship for kfp ¼ kmc 1:2Dr�1:44 � 0:2� �

was presented

by Abril and Knight [1] to predict kfp on the floodplain, where Dr isthe relative depth defined as (H–h)/H, and the subscripts ‘‘mc’’ and‘‘fp’’ refer to the main channel and the floodplain, respectively.However, based on the experimental data (shown in Table 2) inthe meandering compound channel, the dimensionless eddy vis-cosity (kmc) changes along a meander, and it is noticeable thatthe kmc values at the apex sections of main channel are almost0.07 which are the same as those in the straight compound channelwhile kmc values at the cross-over sections are about half of 0.07.The mean values of kmc at CS2–CS6 for MN1 and MN2 are 0.038and 0.045, respectively. On the floodplain, the mean values of kfp

are 0.132 and 0.121 for MN1 and MN2, respectively, where the pre-dictive relation proposed in the straight compound channel byAbril and Knight [1] cannot estimate the kfp accurately.

Finally, Eqs. (7)–(9) define the velocity variation parameter (K)at the apex and cross-over sections due to the different generationmechanisms of secondary flows. These equations may be used tocalculate K in CS1–CS7 if detailed experimental velocities are avail-able. K values apparently change along half a meander because thestrength of secondary flows in the main channel and the effect offloodplain flow are different. Since lateral variation of secondaryflows and longitudinal variation of velocity cannot be easilydescribed at different cross-sections, velocity variation parameters(K) are back calculated at all the seven sections by Eq. (5) which areshown in Figs. 12 and 13. From the figures, it is apparent that thevelocity parameter is a lateral variable, hence, the determination ofvelocity parameter (K) is not straightforward. In this model, thediffering ranges are determined by the experimental data, whichare 1.4–1.8 and 1.2–1.5 in the meandering main channel for MN1and MN2, respectively. On the floodplain, the K values range within0.22–1.25 and 0.21–0.78 for MN1 and MN2, respectively. Theseranges may provide the references for choosing the most suitableK value in each divided region. Therefore, the velocity parameter

(K) is selected as a constant in each panel among the experimen-tally determined ranges and is slightly modified as a calibrationparameter until the best results are obtained. If only depth-averaged velocities are known, the velocity parameter is empiri-cally calibrated in order to give best fit with the experimentalvelocities. In different channels, the proposed model is still usefulin identifying key processes and demonstrating their importanceand once the velocity parameter has been calibrated for each caseit can predict results at other depths for the same geometry. Thismethodology is also applicable in the natural river with muchhigher Manning coefficient due to the mobile bed and flood plainvegetation if the complex cross-sectional shape can be simplifiedas a regular form.

7. Application

In order to plot all the modeling results and experimental datain one figure, the projection of lateral direction (y) in the meander-ing main channel is used for cross-over sections. The experimentaldata along half a meander have been compared with the resultsfrom the proposed method, shown in Figs. 14 and 15. From thesefigures, the method with the slightly modified velocity variationparameters (K) in each divided panel presents accurate resultsfrom CS1 to CS7 for cases MN1 and MN2. It should be noted thatthe modeled lateral depth-averaged velocity distribution followsthe experimental data closely at the cross-over sections, especiallyat the ones with large angle h (CS3, CS4 and CS5), when using thenew boundary condition described in Section 5. The poor modelingresults seen when neglecting the angles [i.e. h1 and h2 in Eq. (17)]indicates that the assumption of the new boundary condition isappropriate, accurate and reliable.

The chosen velocity variation parameters (K) in each dividedregion are shown in Table 3. It is apparent that the K values inthe meandering main channel are much larger than those on thefloodplain, indicating that the centrifugal force and the overtop-ping flow contribute heavily. The largest K values in the three pan-els of the main channel appear at the left mixing region (in CS1,CS2 and CS3), then move to the center region (in CS4 and CS5),finally appearing in the right mixing region (in CS6 and CS7). On

Fig. 15. Modeling depth-averaged velocity from CS1 to CS7 with experimental data for case MN2, H = 0.216 m, Q = 0.113 m3/s (a) CS1 (b) CS2 (c) CS3 (d) CS4 (e) CS5 (f) CS6(g) CS7.

C. Liu et al. / Advances in Water Resources 74 (2014) 26–43 37

the floodplain, the K values on the inside floodplain are 0.6–0.8 and0.4–0.5 which are almost equal to the experimental values at eachsection, as shown in Fig. 13. Eq. (14) shows that only the

streamwise variation of velocity and the lateral variation of sec-ondary flows are considered. Therefore, the K values on the insidefloodplain are much smaller when compared with the ones in the

Fig. 15 (continued)

Table 3Velocity variation parameter in each divided panel.

Case Section Left floodplain Main channel Right floodplain

Outsidefloodplain

Insidefloodplain

Mixingregion

Left Mixingregion

Centerregion

Right mixingregion

Mixingregion

Insidefloodplain

Outsidefloodplain

MN1 CS1 0.4 – – 1.9 1.5 1.4 0.8 0.7 0.4CS2 0.4 – 0.8 1.8 1.6 1.5 0.6 0.8 0.4CS3 0.4 0.7 0.4 1.9 1.7 1.6 0.6 0.8 0.4CS4 0.4 0.7 0.7 1.6 1.8 1.6 0.8 0.8 0.4CS5 0.4 0.7 0.6 1.7 1.8 1.7 0.7 0.8 0.4CS6 0.4 0.7 0.8 1.5 1.7 1.8 0.7 – 0.4CS7 0.4 0.7 0.8 1.5 1.6 1.9 – – 0.4

MN2 CS1 0.2 – – 1.6 1.3 1.2 0.5 0.5 0.2CS2 0.2 – 0.45 1.5 1.4 1.3 0.4 0.5 0.2CS3 0.2 0.5 0.45 1.6 1.4 1.4 0.4 0.5 0.2CS4 0.2 0.5 0.5 1.3 1.6 1.3 0.5 0.5 0.2CS5 0.2 0.5 0.4 1.4 1.5 1.5 0.5 0.5 0.2CS6 0.2 0.5 0.5 1.3 1.4 1.5 0.5 – 0.2CS7 0.2 0.5 0.5 1.2 1.3 1.6 – – 0.2

38 C. Liu et al. / Advances in Water Resources 74 (2014) 26–43

main channel. However, the K values on the outside floodplain areonly 0.4 and 0.2 for MN1 and MN2, respectively, which are abouthalf of the values on the inside floodplain. This phenomenon maybe explained by the fact that the flow on the outside floodplainis affected slightly by the flow in the meander belt, and thehorizontal shear stress generated between the inbank flow andthe overtopping flow would not influence the outside floodplainflow significantly. Greenhill and Sellin [7] proposed a modifiedManning’s equation to predict the stage-discharge curve in themeandering compound channel and found that good results wereobtained by separately considering the meander belt flow andthe outside floodplain flow. Hence, the streamwise variation canbe ignored on the outside floodplain, and Eq. (14) may be simpli-fied as K ¼ qðH � hÞKyðH�hÞ and the K value is absolutely becomingsmall.

For practical engineering purposes, the lateral distributions ofUd along a meander are sufficiently accurate and may be used to

predict the stage-discharge curve by integrating Ud laterally. Thedischarge prediction is appropriately done at the apex cross-section where the horizontal shear stress is small [32]. At thecross-over section with larger angle between the meandering mainchannel and floodplain, the stronger horizontal shear stress leadsto more energy losses. Consequently, a lower local velocity distri-bution is predicted and a smaller discharge is obtained. When thismodel is applied in the natural river to predict the lateral velocitydistribution at different locations, the more complex river geome-tries with more divided panels are really needed. The values of k, fand K are still assumed as constants in each panel, and theirvariations in each panel with increasing flow stage should be rea-sonably systematic. The K value in the natural river will be moredifficult to determine because the secondary flow distributions ina natural river are more complex than that in the experimentalflume with a non-mobile bed. However, fortunately the K valueis the only free parameter in this model, and good results along a

Fig. 16. Streamwise variation coefficient Kx in the meandering main channel atCS1–CS7 (a) MN1 (b) MN2.

Fig. 17. Lateral distributions of secondary flow coefficient Ky in the meanderingmain channel at CS1–CS7 (a) MN1 (b) MN2.

C. Liu et al. / Advances in Water Resources 74 (2014) 26–43 39

meander may be obtained by modifying it in each divided panelwhen the lateral velocity distribution is obtained. In addition, thesediment transport rate may be also calculated using the localboundary shear stress (sb ¼ q f

8 U2d) and using an appropriate trans-

port equation.

Fig. 18. Streamwise variation of Kx/K at centerlines of the inside floodplain and theoutside floodplain (‘‘LOF’’, ‘‘LIF’’, ‘‘RIF’’ and ‘‘ROF’’ indicates ‘‘Left Outside Flood-plain’’, ‘‘Left Inside Floodplain’’, ‘‘Right Inside Floodplain’’ and ‘‘Right OutsideFloodplain’’, respectively).

8. Discussion

8.1. Components in velocity term

According to Eqs. (11)–(13), the velocity variation parameter (K)contains the streamwise variation of velocity (Kx) and lateral vari-ation of secondary flow (Ky) which merit further discussion. Thetwo parameters are back calculated from the experimental data,and Kx values at each section are shown in Fig. 16. From the figures,the variation pattern at every vertical follows a linear relationwhich is different at vertical line 1–13. The Kx values at the verticalline 1 and 4 remain negative at CS2 and CS3 while positive at CS4,CS5 and CS6. For the vertical line 7 and 10, the Kx values remainnegative at CS2, CS3 and CS4 and positive at CS5 and CS6. Aboutthe vertical line 13, the Kx values are all negative except at CS6.At the apex sections (CS1 and CS7), the Kx values at all vertical linesare almost zero. From the Figs. 14 and 15, the Ud values at the apexsections are the maximal values which can be used to explain theKx values, which are the gradient of U2

d (Eq. (10)), at CS1 and CS7equal 0. Therefore, the velocity term at the apex section (Eq. (7))may be simplified as

For apex sections ðCS1 and CS7Þ; K ¼ qð1þ kÞH @ðUV1Þd@y

ð18Þ

Eq. (18) demonstrates that good prediction at the apex sectioncan be still obtained by ignoring the streamwise velocity variation(Kx). Research by Ervine et al. [5], McGahey et al. [23] and Huaiet al. [10] who ignored the streamwise velocity variation and onlyapplied their models at the apex section in the meanderingcompound channel are, in fact, specific cases of the modelpresented here. The modified secondary parameter proposed

40 C. Liu et al. / Advances in Water Resources 74 (2014) 26–43

by Ervine et al. [5] actually contained the effect of centrifugal force,demonstrated by the parameters in the meandering main channelwhich were much larger than those in the straight main channel intheir paper. In this paper, the K value itself including the k definedin Eq. (3) can reflect the effect of centrifugal force in the meander-ing main channel, where the K values are larger than the ones onthe floodplain (shown in Table 3).

For the secondary flows in the meandering main channel, thesecondary flow variation coefficient (Ky) from CS1 to CS7 has beenalso calculated, and the lateral distribution is shown in Fig. 17.From the figures, the Ky values at CS1 are almost equal the onesat CS7, and remain positive. This can be explained by the definitionof Ky which is the lateral gradient of (UV)d. The rotational directionsof secondary flows at CS1 and CS7 are converse while the intensi-ties are almost the same (see Fig. 8). Therefore, the Ky values arenearly the same at the apex sections. At the cross-over sections(CS2–CS6), the floodplain flows contribute to the secondary flow,and the Ky values at the right side seem larger because the flood-plain flow plunges into the main channel from the right bank inthe half test meander shown in Fig. 2, leading to the stronger trans-verse velocity. The biggest lateral Ky values are shown at CS2, thengradually reduce and reach the lowest values at CS6.

On the floodplain, the K values reduce significantly comparedwith the ones in the meandering main channel due to the absenceof centrifugal force and floodplain flow. The ratios of Kx

K at thecenterline of the outside floodplains are shown in Fig. 18. It can

Fig. 19. Effect of panel divided number on lateral depth-averaged velocity distribution aQ = 0.189 m3/s (a) CS1 (b) CS3 (c) CS5.

be clearly seen that the ratios are very small, indicating that thestreamwise variation of velocity may be ignored. However, at thecenterline of the inside floodplains, the Kx

K values range between0.2 and 2.1 which cannot be ignored during the calculation, repre-senting that the effect of main channel flow to the inside floodplainflow cannot be ignored. The analysis above explains why the K val-ues on the outside floodplains are only a half of the ones on theinside floodplains.

8.2. Sensitivity analysis of panel number

The panel divided theory (shown in Section 3) is important tothe prediction of depth-averaged velocity. In order to explore thesensitivity of the proposed model to the number of divided panels,the recommended panels, (i.e. the outside floodplain, the insidefloodplain, the mixing region next to the inner and outer sidewalls), are all ignored and only three panels (the left floodplain,the main channel and the right floodplain) are used herein. Threetypical cross-sections are selected which are the middle sectionof a meander bend (CS1), the exit of a meander bend (CS3) andthe entrance of a meander bend (CS5). In the three sections, thevelocity parameter on the floodplain is set as 0.7 on both left andright floodplains while in the main channel K is selected as 1.6,1.7 and 1.8 in CS1, CS3 and CS5, respectively. The predictions with3 panels and 7 or 9 panels are shown in Fig. 19. From the figure,with only three panels, the prediction on the floodplain is nearly

t three typical cross-sections, based on experimental data of case MN1, H = 0.255 m,

C. Liu et al. / Advances in Water Resources 74 (2014) 26–43 41

a constant which cannot reflect the increased Ud on the outsidefloodplains. Besides, in the main channel, poor predictions areobtained by ignoring mixing regions, particularly in CS3 and CS5.Therefore, the region divided number affects the accuracy of thismodel. Use of fewer panels may provide convenience for calcula-tion, but leads to poor prediction because in this case some typicalflow characteristics are almost ignored.

8.3. Limitation of the proposed model

It is worth noting the limitations of the proposed model. Allequations were derived and two group experiments were doneunder the hypothesis of quasi-uniform flow. For the other flow con-ditions, the feasibility of governing equation (Eq. (5)) and secondaryflow equations (Eqs. (7)–(9)) along a meander warrant furtherstudy. In the meandering compound channel, the growth and decayof secondary flow in the main channel are clarified in Figs. 7 and 8.From the analysis and discussion of the components in velocityterm, the velocity parameter has been demonstrated as a lateraland longitudinal variable. Setting the velocity parameter in the realranges calculated by experimental data, the good predictions ofdepth-averaged velocity are obtained along half a meander, partic-ularly in cross-over sections, which indicates the proposed model isreasonable. However, in the channels with mobile bed and floodplain vegetation, the generation mechanism and distribution of sec-ondary flow are quite different which lead to significant variationsof the bed form. Further, the bed morphology considerably affectsthe longitudinal distribution of depth-averaged velocity. Thesechanges of bed roughness cause that the velocity parameter (K) ismuch harder to determine and the calibration may be a methodto obtain its value. In addition, the additional resistances of mobilebed and flood plain vegetation also deflect the flow direction, hence,the boundary conditions and the angles have to be reconsidered.

9. Conclusions

This paper presents an analytical model under quasi-uniformflow to model the lateral distributions of depth-averaged velocityfor half a meander in the curved compound channel. This modeldefines the velocity term in the governing equation, which includesboth the streamwise velocity variation and the lateral secondaryflow variation. In the meandering main channel, the generationmechanism of secondary flows is analyzed, finding that the effectof centrifugal force and floodplain flow must be considered, espe-cially at the cross-over sections. New expressions for velocity varia-tion parameters are proposed in the main channel and on thefloodplain. Whilst using the method on each test case requires initialcalibration, once this is done, the method is very useful in identifyingkey processes and demonstrating their importance and it can predictresults at other depths for that same geometry.

Two experiments (MN1 and MN2) were conducted in a largescale meandering compound channel. The three dimensionalvelocity components, the Reynolds shear stress and the stage-discharge were measured at seven designed test sections along halfa meander (CS1–CS7 shown in Fig. 2). The experimental data oftransverse velocity along half a meander demonstrate that theanalysis of the secondary flow’s generation mechanism in themeandering main channel is accurate and reasonable.

The main channel considered here is rectangular, leading to adiscontinuity of flow depth at the interfaces of the main channeland its corresponding floodplains. Further, the effect of the mean-dering main channel flow to the inside floodplain flow must beconsidered while it may be ignored to the outside floodplain flow.Hence, the divided region method for each test section wasproposed here.

The parameters corresponding to the effects of lateral momen-tum transfer, bed friction and secondary flow are introduced. Basedon the experimental data, some new findings about the dimension-less eddy viscosity (kmc) in the main channel are obtained, showingthat its value at the apex section is still equal 0.07, but kmc valueranges from 0.033 to 0.05 at the cross-over sections. On the flood-plain, back calculated values (0.132 and 0.121) are used for bothcases MN1 and MN2, respectively.

The new boundary condition, considering the relation of veloc-ities in the main channel and on the floodplain, was proposedbased on the angles (shown in Fig. 10). At the cross-over sections(CS2–CS6), the angle (h) between the meandering main channeland the floodplain was divided into two angles which are relatedto the velocity continuity and the velocity gradient continuity.From the modeled work, poor results are obtained at the cross-oversections, when neglecting the influence of angles (h1 = h2 = 0�) inEq. (17) demonstrating the importance of the inclusion of theseangles proposed in this paper.

Finally, the modeling depth-averaged velocity distributionsalong half a meander were compared with the experimental data.The modeling capability of the proposed model in both cases (MN1and MN2) is excellent, whether at the apex sections (CS1 and CS7)or the cross-over ones (CS2–CS6), by adopting the new boundaryconditions and considering both the streamwise velocity variationand lateral secondary flow variation. If more experimental dataand field data, including flow depths, bed slopes, meander beltwidths, channel bed conditions and sinuosity can be obtained; fur-ther verification of this model may be made.

Notation

The following symbols are used in this paper:

A1, A2

= i ntegration constants in Eq. (15) B = t otal width b = m ain channel width Dr = r elative depth, (H–h)/H Fc = c entrifugal force f = D arcy–Weisbach friction factor g = l ocal gravitational acceleration H = fl ow depth h = b ankfull level K = v elocity variation parameter, defined by Eqs. (7)–(9) Kx = s treamwise variation coefficient Ky1, Ky2 = s econdary flow coefficients n = M anning’s coefficient R = h ydraulic radius S = v alley slope U, V, W = v elocity components corresponding to x, y, z

directions, respectively

Ud = d epth-averaged streamwise velocity, defined by

Eq. (2)

Ud(1) = d epth-averaged velocity on the floodplain, shown in

Fig. 10

Ud(2) = p rojection of Ud(3) in the streamwise direction on

the floodplain, shown in Fig. 10

Ud(3) = d epth-averaged velocity in the main channel with

the true flow direction, shown in Fig. 10

Ud(4) = m easured experimental depth-averaged velocity

following the valley direction in the main channel,shown in Fig. 10

U⁄

= s hear velocity, defined by Eq. (6) V1 = t ransverse velocity without effect of centrifugal

force

(continued on next page)

42 C. Liu et al. / Advances in Water Resources 74 (2014) 26–43

V2

= c omponent of floodplain flow x, y, z = s treamwise, lateral and vertical coordinates,

respectively

k = d imensionless eddy viscosity x = c oefficient in Eq. (15) eyx = d epth-averaged eddy viscosity, defined by Eq. (6) h = a ngle at the cross-over section (=h1 + h2), shown in

Fig. 10

h1 = a ngle between Ud(3) and Ud(4), shown in Fig. 10 h2 = a ngle between Ud(2) and Ud(3), shown in Fig. 10 sb = b ed shear stress syx = d epth-averaged Reynolds shear stress syx, szx = R eynolds shear stresses on the planes perpendicular

to y and z, respectively; and

q = fl ow density

Subscripts

H = fl ow height h = b ankfull height H–h = fl oodplain mc = m ain channel; and fp = fl oodplain

Acknowledgments

The authors would like to acknowledge the assistance ofDr. Shan Yuqi at Sichuan University in the preparation and revisionof this manuscript. The authors gratefully acknowledge the sup-port of the National Natural Scientific Foundation of China (Nos.51279117, 11072161), the Program for New Century ExcellentTalents in University of China (NCET-13-0393) and the NationalScience and Technology Ministry (No. 2012BAB05B02). Theauthors are also grateful to the anonymous reviewers for their veryhelpful comments and suggestions on this paper.

References

[1] Abril JB, Knight DW. Stage discharge prediction for rivers in flood applying adepth averaged model. J Hydraul Res 2004;42(6):616–29. http://dx.doi.org/10.1080/00221686.2004.9628315.

[2] Cao Z, Meng J, Penderand G, Wallis S. Flow resistance and momentum flux incompound open channels. J Hydraul Eng 2006;132(12):1272–82. http://dx.doi.org/10.1061/(ASCE)0733-9429(2006) 132:12(1272).

[3] Carling PA, Cao Z, Holland MJ, Ervine DA, Babaeyan-Koopaei K. Turbulent flowacross a natural compound channel. Water Resour Res 2002;38(12). http://dx.doi.org/10.1029/2001WR000902. Article Number 1270.

[4] Castanedo S, Medina M, Mendez FJ. Models for the turbulent diffusion terms ofshallow water equations. J Hydraul Eng 2005;131(3):217–23. http://dx.doi.org/10.1061/(ASCE)0733-9429(2005) 131:3(217).

[5] Ervine DA, Willetts BB, Sellin RHJ, Lorena M. Factors affecting conveyance inmeandering compound flows. J Hydraul Eng 1993;119(12):1383–99. http://dx.doi.org/10.1061/(ASCE)0733-9429(1993) 119:12(1383).

[6] Ervine DA, Babaeyan-Koopaei K, Sellin RHJ. Two-dimensional solution forstraight and meandering overbank flows. J Hydraul Eng 2000;126(9):653–69.http://dx.doi.org/10.1061/(ASCE)0733-9429(2000) 126:9(653).

[7] Greenhill RK, Sellin RHJ. Development of a simple method to predictdischarges in compound meandering channels. Proc Inst Civil Eng WaterMaritime Energy 1993;101:37–44. http://dx.doi.org/10.1680/iwtme.1993.22986.

[8] Hu CH, Ji ZW, Guo QC. Flow movement and sediment transport in compoundchannels. J Hydraul Res 2010;48(1):23–32. http://dx.doi.org/10.1080/00221680903568600.

[9] Huai WX, Xu ZG, Yang ZH, Zeng YH. Two dimensional analytical solution for apartially vegetated compound channel flow. Appl Math Mech2008;29(8):1077–84. http://dx.doi.org/10.1007/s10483-008-0811-y.

[10] Huai WX, Gao M, Zeng YH, Li D. Two-dimensional analytical solution forcompound channel flows with vegetated floodplain. Appl Math Mech2009;30(9):1049–56. http://dx.doi.org/10.1007/s10483-009-0906-z.

[11] Ishigaki T, Shiono K, Rameshwaran P, Scott CF, Muto Y. Impact of secondaryflow on bed form and sediment transport in a meandering channel foroverbank flow. Annu J Hydraul Eng JSCE 2000;44:849–54. http://dx.doi.org/10.2208/prohe.44.849.

[12] Jing H, Guo Y, Li C, Zhang J. Three-dimensional numerical simulation ofcompound meandering open channel flow by the Reynolds stress model. Int JNumer Methods Fluids 2009;59:927–43. http://dx.doi.org/10.1002/fld.1855.

[13] Keevil GM, Peakall J, Best JL, Amos KJ. Flow structure in sinuous submarinechannels: velocity and turbulence structure of an experimental submarinechannel. Mar Geol 2006;229:241–57. http://dx.doi.org/10.1016/j.margeo.2006.03.010.

[14] Knight DW, Demetriou JD. Flood plain and main channel flow interaction. JHydraul Eng 1983;109(8):1073–92. http://dx.doi.org/10.1061/(ASCE)0733-9429(1983) 109:8(1073).

[15] Knight DW, Sellin RHJ. The SERC flood channel facility. J Inst Water EnvironManage 1987;41(4):198–204. http://dx.doi.org/10.1111/j.1747-6593.1987.tb01212.x.

[16] Knight DW, Omran M, Abril JB. Boundary conditions between panels in depth-averaged flow models revisited. River flow. In: Proceedings of 2ndInternational Conference on Fluvial Hydraulics, vol. 1, 23–25 June, Napoli,Italy; 2004. p. 371–380.

[17] Knight DW, Omran M, Tang X. Modeling depth-averaged velocity andboundary shear in trapezoidal channels with secondary flows. J HydraulEng 2007;133(1):39–47. http://dx.doi.org/10.1061/(ASCE)0733-9429(2007)133:1(39).

[18] Knight DW, Tang X, Sterling M, Shiono K, Mc Gahey C. Solving open channelflow problems with a simple lateral distribution model. In: River flow 2010,proceedings of the international conference on fluvial hydraulics,Braunschweig, Germany. p. 41–48.

[19] Lambert MF, Sellin RHJ. Velocity distribution in a large-scale model of a doublymeandering compound river channel. Proc Inst Civil Eng Water MaritimeEnergy 1996;118(1):10–20. http://dx.doi.org/10.1680/iwtme.1996.28161.

[20] Liu C, Luo X, Liu XN, Yang KJ. Modeling depth-averaged velocity and bed shearstress in compound channels with emergent and submerged vegetation. AdvWater Resour 2013;60:148–59. http://dx.doi.org/10.1016/j.advwatres.2013.08.002.

[21] Lyness JF, Myers WRC, Cassells JBC, O’sullivan JJ. The influence of planform onflow resistance in mobile bed compound channels. Proc Inst Civil Eng WaterMaritime Eng 2001;148:5–14. http://dx.doi.org/10.1680/wame.2001.148.1.5.

[22] Martin-Vine JP, Moreta PJM, Lopez-Querol S. Improved 1-D modeling incompound meandering channels with vegetated floodplains. J Hydraul Res2008;46(2):265–76. http://dx.doi.org/10.1080/00221686.2008.9521860.

[23] McGahey C, Samuels PG. Methodology for conveyance estimation in two-stagestraight, skewed and meandering channels. In: Proceedings of the XXXcongress of the international association for hydraulic research, vol. C1,Thessaloniki, Greece; 2003. p. 33–40.

[24] Morvan H, Pender G, Wright NG, Ervine DA. Three-dimensionalhydrodynamics of meandering compound channels. J Hydraul Eng2002;128(7):674–82. http://dx.doi.org/10.1061/(ASCE)0733-9429(2002) 128:7(674).

[25] Pasche E, Rouve GM. Overbank flow with vegetatively roughened flood plains.J Hydraul Eng 1985;111(9):1262–78. http://dx.doi.org/10.1061/(ASCE)0733-9429(1985) 111:9(1262).

[26] Patra KC, Kar SK. Flow interaction of meandering river with floodplains. JHydraul Eng 2000;126(8):593–604. http://dx.doi.org/10.1061/(ASCE)0733-9429(2000) 126:8(593).

[27] Patra KC, Kar SK, Amartya KB. Flow and velocity distribution in meanderingcompound channels. J Hydraul Eng 2004;130(5):398–411. http://dx.doi.org/10.1061/(ASCE)0733-9429(2004) 130:5(398).

[28] Rameshwaran P, Spooner J, Shiono K, Chandler JH. Flow mechanisms in two-stage meandering channel with mobile bed. In: Process 28th congress of IAHR,vol. 256, Graz, Austria; 1999. p. 259–266.

[29] Rameshwaran P, Shiono K. Quasi two-dimensional model for straight overbankflows through emergent vegetation on floodplains. J Hydraul Res2007;45(3):302–15. http://dx.doi.org/10.1080/00221686.2007.9521765.

[30] Shao X, Wang H, Chen Z. Numerical modeling of turbulent flow in curvedchannels of compound cross-section. Adv Water Resour 2003;26:525–39.http://dx.doi.org/10.1016/S0309-1708(03)00008-3.

[31] Shiono K, Knight DW. Turbulent open channel flows with variable depth acrossthe channel. J Fluid Mech 1991;222:617–46. http://dx.doi.org/10.1017/S0022112091001246.

[32] Shiono K, Muto Y. Complex flow mechanisms in compound meanderingchannel for overbank flow. J Fluid Mech 1998;376:221–61. http://dx.doi.org/10.1017/S0022112098002869.

[33] Shiono K, Muto Y, Knight DW, Hyde AFL. Energy losses due to secondary flowand turbulence in meandering channel for overbank flows. J Hydraul Res1999;37(5):641–64. http://dx.doi.org/10.1080/00221689909498521.

[34] Shiono K, Al-Romaih JS, Knight DW. Stage-discharge assessment in compoundmeandering channels. J Hydraul Eng 1999;125(1):66–77. http://dx.doi.org/10.1061/(ASCE)0733-9429(1999) 125:1(66).

[35] Shiono K, Spooner J, Chan TL, Rameshwaran P, Chandler JH. Flowcharacteristics in meandering channels with non-mobile and mobile bedsfor overbank flows. J Hydraul Res 2008;46(1):113–32. http://dx.doi.org/10.1080/00221686.2008.9521848.

[36] Shiono K, Chan TL, Spooner J, Rameshwaran P, Chandler JH. The effect offloodplain roughness on flow structures, bedforms and sediment transportrates in meandering channels with overbank flows: Part I. J Hydraul Res2009;47(1):5–19. http://dx.doi.org/10.3826/jhr.2009.2944-I.

[37] Shiono K, Chan TL, Spooner J, Rameshwaran P, Chandler JH. The effect offloodplain roughness on flow structures, bedforms and sediment transport

C. Liu et al. / Advances in Water Resources 74 (2014) 26–43 43

rates in meandering channels with overbank flows: Part II. J Hydraul Res2009;47(1):20–8. http://dx.doi.org/10.3826/jhr.2009.2944-II.

[38] Spooner J. Flow structures in a compound meandering channel with flat andnatural bedforms [Ph.D. thesis]. Loughborough University, UK; 2001.

[39] Tang X, Knight DW. A general model of lateral depth-averaged velocitydistributions for open channel flows. Adv Water Resour 2008;31:846–57.http://dx.doi.org/10.1016/j.advwatres.2008.02.002.

[40] Tang X, Knight DW. Analytical models for velocity distributions in openchannel flows. J Hydraul Res 2009;47(4):418–28. http://dx.doi.org/10.1080/00221686.2009.9522017.

[41] Tominaga A, Nezu I, Ezaki K, Nakagawa H. Three-dimensional turbulentstructure in straight open channel flows. J Hydraul Res 1989;27(1):149–73.http://dx.doi.org/10.1080/00221688909499249.

[42] Wormlemeaton PR, Sellin RHJ, Bryant T, Loveless JH, Hey RD, Catmur SE. Flowstructure in a two-stage channel with a mobile bed. J Hydraul Res2004;42(2):145–62. http://dx.doi.org/10.1080/00221686.2004.9728378.

[43] Yang KJ, Cao SY, Knight DW. Flow patterns in compound channels withvegetated floodplains. J Hydraul Eng 2007;133(2):148–59. http://dx.doi.org/10.1061/(ASCE)0733-9429(2007) 133:2(148).

[44] Yang KJ, Nie RH, Liu XN, Cao SY. Modeling depth-averaged velocity andboundary shear stress in rectangular compound channels with secondaryflows. J Hydraul Eng 2013;139(1):76–83. http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000638.

[45] Yuen KWH. A study of boundary shear stress, flow resistance and momentumtransfer in open channels with simple and compound trapezoidal crosssections [Ph.D. thesis]. University of Birmingham, UK; 1989.

[46] Zarrati AR, Tamai N, Jin YC. Mathematical modeling of meandering channelswith a generalized depth averaged model. J Hydraul Eng 2005;131(6):467–75.http://dx.doi.org/10.1061/(ASCE)0733-9429(2005) 131:6(467).