Advances in Water Resources 31 (2008) 1662–1673

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Advances in Water Resources

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The effect of geometrical parameters on the discharge capacityof meandering compound channels

M. De Marchis, E. Napoli *

Dipartimento di Ingegneria Idraulica e Applicazioni Ambientali, Universitá di Palermo,Viale delle Scienza, Palermo 90145, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 January 2008Received in revised form 23 April 2008Accepted 31 July 2008Available online 8 August 2008

Keywords:Compound channelsMeandersSinuosityStage—discharge curvesNumerical simulation.

0309-1708/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.advwatres.2008.07.014

* Corresponding author. Tel.: +39 91 6657753; fax:E-mail address: napoli@idra.unipa.it (E. Napoli).

A number of methods and formulae has been proposed in the literature to estimate the discharge capac-ity of compound channels. When the main channel has a meandering pattern, a reduction in the convey-ance capacity for a given stage is observed, which is due to the energy dissipations caused by thedevelopment of strong secondary currents and to the decrease of the main channel bed slope with respectto the valley bed slope. The discharges in meandering compound channels are usually assessed applying,with some adjustments, the same methods used in the straight compound channels. Specifically, the sin-uosity of the main channel is frequently introduced to account for its meandering pattern, although somemethods use different geometric parameters.

In this paper the stage—discharge curves for several compound channels having identical cross-sec-tional area, roughness and bed slope but different planimetric patterns are numerically calculated andcompared, in order to identify which geometric parameter should be efficaciously used in empirical for-mulae to account for meandering patterns. The simulations are carried out using a 3D finite-volumemodel that solves the RANS equations using a k� e turbulence model. The numerical code is validatedagainst experimental data collected in both straight and meandering compound channels.

The numerical results show that the sinuosity is the main parameter to be accounted for in empiricalformulae to assess the conveyance capacity of meandering compound channels. Comparison of thestage—discharge curves in the meandering compound channels with that obtained in a straight channelhaving identical cross-sectional area clearly shows the reduction of discharge due to the presence ofbends in the main channel. The effect of other geometric parameters, such as the meander-belt widthand the mean curvature radius, results very weak.

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1. Introduction

Many rivers are composed of a main channel flanked by one ortwo floodplains. This kind of rivers are called compound channels.When the discharge is low, the current flows in the main channel,while inundating the overbank when the discharge grows, as in theperiod of high rainfall. The flow in compound channels is typicallythree-dimensional, because of the heavy momentum exchangesoccurring between the high-speed current in the main channeland the lower flowing discharge in the floodplains and, frequently,of the different roughness of the main channel and floodplain bed.

A correct estimation of the discharge capacity of compoundchannels is an essential task in designing river floodplains or veri-fying natural channels, since errors in calculating the maximumdepth of the flow or the discharge could involve economic damagesand, in a few cases, also loss of life. Different methods have beenthus proposed to evaluate the conveyance capacity of straight

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compound channels. The conventional formulae are based on theChezy’s or Darcy–Weisbach’s equations, which are applied afterhaving vertically or horizontally divided the cross-section andhaving separately calculated the discharge contribution of eachsub-area (Divided Channel Method, DCM). Wormleaton et al. [23]analysed different type of DCMs, showing that the errors increasewhen the floodplain stages decrease. Knight and Demetriou [11]and Knight and Hamed [12] demonstrated that the DCMs overesti-mate the overall discharge since they do not take into account themomentum exchanges between main channel and floodplains.

In order to obtain a better estimation of the flow rate in straightchannels, Ackers [2] developed a model where the transversemomentum exchanges are accounted for using a coefficient ofcoherence of the cross-section sub-areas, aimed at correcting theconveyance due to the effect of inter-zone interfaces. Lambertand Myers [14] proposed a method based on weighting factors ofeach sub-area contribution, called weighted divided channel method(WDCM), in which the interaction between main channel andfloodplains is neglected but the weighting factors play the role ofa compensation effect. Dey and Lambert [5] proposed an analytical

M. De Marchis, E. Napoli / Advances in Water Resources 31 (2008) 1662–1673 1663

model to estimate the discharge in compound channels startingfrom the end depths in both the main channel and the floodplains.

Most of the natural compound channels exhibit meanderingshapes, which naturally develop because of the adjustments im-posed by the irregular distribution of water shear stresses on theriverbeds, leading to local sedimentation and/or erosion phenom-ena. The complexity of the flow is highly enhanced by the mean-dering pattern, which is responsible for the formation of helicalflows and for the increase of momentum and flow exchanges be-tween the main channel and the floodplains [19]. In fact, althougheven in straight compound channels significant secondary currentsdevelop mainly due to the turbulence anisotropy, as it was clearlydescribed among the others by Tominaga and Nezu [22], in mean-dering compound channels lateral velocities are much higher. Thedevelopment of these stronger secondary currents causes thereduction of the discharge capacity of meandering compoundchannels with respect to that of straight channels having identicalcross-sectional area and hydraulic features (roughness, bed slope,etc.).

In order to estimate the conveyance capacity of these channels,an improved version of the DCM, the meander-belt method (MBM),was proposed by Greenhill and Sellin [8], where three sub-regions(the main channel area, the meander-belt area and the area outsidethe meander-belt) are identified, as shown in Fig. 1, and the totaldischarge is obtained summing up the contributions from eachzone. The MBM does not directly account for the meandering chan-nel sinuosity, which is the ratio between the actual length of thechannel and its projection on the main flow direction, while usingthe meander-belt width to account for the meandering effects onthe channel conveyance capacity.

Other contributions are based on the introduction of empiricaladjustments of the friction factor f or of the roughness coefficients,which are explicitly related to the channel sinuosity: the Linear SoilConservation Services (LSCS) method [9] prescribes to modify theManning coefficient n for inbank flows in meandering channelsas a function of the sinuosity s, while Shiono et al. [18] suppliedan improved form of the traditional Darcy–Weisbach’s equation,where the channel sinuosity is taken into account, and Toebesand Sooky [21] proposed an empirical formula to modify the

Ba

1

2 2

B

3

b

Fig. 1. Sub-regions used in the meander-belt method to calculate the channeldischarge: (1) main channel; (2) area outside the meander-belt; (3) meander-belt ofwidth B. (a) 3D view. (b) Vertical cross-section.

friction factor f, which is valid for s ¼ 1:1 only. A mixed approachwas proposed by James and Wark [9] which is based on the chan-nel section division into three zones (or four when the lateralfloodplains widths are different) where both the sinuosity s andthe meander-belt width are taken into account.

The main contribution of this paper is to identify which geo-metric parameters should be considered in the empirical formu-lae to estimate the discharge capacity of meandering compoundchannels. To this aim, three-dimensional numerical simulationsare used to obtain the stage—discharge curves for several com-pound channels having different planimetric patterns, coveringa wide range of different sinuosities and meander-belt widths.The calculations are performed using a free-surface finite-vol-ume numerical model which resolves the Reynolds Averaged Na-vier-Stokes equations, adopting the k� e closure model torepresent the turbulent stresses. Some validation tests of thenumerical model are presented first, where the numerical resultsare compared with the experimental measurements performedby [13,18–20] in both straight and meandering compoundchannels.

The paper is organised as follows. Section 2 deals with the gov-erning momentum and continuity equations and briefly describesthe numerical model. In Section 3 the validation with experimentaldata is discussed and in Section 4 the stage–discharge curves ob-tained in several test channels are compared. Finally some conclu-sions are drawn in Section 5.

2. Governing equations and numerical model

The momentum and mass conservation laws (Reynolds Aver-aged Navier–Stokes and continuity equations) in the conventionalsummation approach for an incompressible fluid are:

@ui

@tþ @uiuj

oxj� m

o2ui

oxjoxjþ gdij þ

1q

opoxiþ 1

qosij

oxj¼ 0; i; j ¼ 1; . . . ;3

ð1Þ

and

ouj

oxj¼ 0; j ¼ 1; . . . ;3 ð2Þ

where t is the time, xi is the ith axis (with the axis x3 vertical andoriented upward), ui is the ith component of the Reynolds averagedvelocity, q is the water density, p is the Reynolds averaged pressure,g is the acceleration due to the gravity, m is the kinematic viscosity,dij is the Kronecker delta and sij ¼ qu0iu

0j are the Reynolds stresses.

The pressure p can be decomposed into an hydrostatic and anon-hydrostatic part:

p ¼ c½ðzb þ hÞ � x3� þ qq ð3Þ

where zb is the bed elevation from an horizontal plane of reference,h is the depth of the water column and q is the non-hydrostaticpressure in kinematical units (see Fig. 2). Introducing Eq. (3) into(1), the Reynolds Averaged Navier–Stokes equations can be rewrit-ten as:

oui

otþ ouiuj

oxj� m

o2ui

oxjoxjþ 1

qosij

oxjþ oq

oxiþ g

oðzB þ hÞoxi

¼ 0; i; j ¼ 1; . . . ;3

ð4Þ

where the last term is null for i ¼ 3, since zb and h do not depend onx3.

The turbulent stresses sij are calculated using the k� e turbu-lence model in the standard formulation [15], which is based onthe eddy viscosity type relation:

Fig. 2. Hydrostatic and non-hydrostatic pressures and geometrical definitions.

1664 M. De Marchis, E. Napoli / Advances in Water Resources 31 (2008) 1662–1673

sij ¼ �qmtoui

oxjþ ouj

oxi

� �þ 2

3dijqk ð5Þ

with

mt ¼ clk2

eð6Þ

where mt is the eddy viscosity, k and e are the turbulent kinetic en-ergy and its dissipation rate, respectively, and cl ¼ 0:09. The turbu-lent kinetic energy and the dissipation rate e are obtained resolvingthe equations:

okotþ okuj

oxj� o

oxj

mt

rk

okoxj

� �þ P � � ¼ 0; j ¼ 1; . . . ;3 ð7Þ

oeotþ oeuj

oxj� o

oxj

mt

re

oeoxj

� �þ e

kðc1eP � c2eeÞ ¼ 0; j ¼ 1; . . . ;3 ð8Þ

where P is the production of the turbulent kinetic energy given by:

P ¼ � sij

qoui

oxjþ ouj

oxi

� �ð9Þ

and the closure coefficients c1e; c2e;rk and re are given the values of1:44, 1:92, 1 and 1:3, respectively.

The free-surface movements are calculated according to thekinematic boundary condition:

ohotþ u1

oðzb þ hÞox1

þ u2oðzb þ hÞ

ox2� u3 ¼ 0 ð10Þ

which is obtained prescribing that the free-surface is a materialsurface.

Eqs. (2) and (4), as well as the k� � Eqs. (7) and (8) are discre-tised using a collocated finite-volume method, employing curvilin-ear non-orthogonal structured grids consisting of non-overlappinghexahedral cells. First and second spatial derivatives are discre-tised using second-order accurate central formulae (the second-or-der accuracy of the numerical method has been demonstrated inLipari and Napoli [16]).

Time derivatives are calculated using an Euler scheme, provid-ing second-order accuracy at the time level nþ 1=2 (that is at timeðnþ 1=2ÞDt, where Dt is the time step). The terms containing spa-tial derivatives are calculated at the time level nþ 1=2 using an im-plicit discretisation of the vertical turbulent terms based on theCrank–Nicolson scheme, whereas explicitly treating the otherterms by the Adams–Bashfort scheme.

In order to overcome the incompressible pressure–velocitydecoupling, a fractional-step method is applied for the solutiontime-marching (see e.g. [25]). In the predictor step, an intermediatevelocity u�i which is not constrained by continuity is obtained bysolving the free-surface RANS Eq. (4) with q ¼ 0 (that is, assuminga hydrostatic pressure distribution); the correct solenoidal velocityfield is then obtained by summing up an irrotational velocity fielduc whose potential w is obtained solving the discrete form of thePoisson equation:

o2woxjoxj

¼ �ou�joxj

; j ¼ 1; . . . ;3 ð11Þ

Eq. (11) is solved iteratively using an L-SOR algorithm, including adirect implicit solution in the computational columns in a four-col-our type of sequence for the columns. The convergence of the iter-ative method is accelerated using a multi-grid technique [7].

At each time step, the free-surface elevation is recalculatedaccording to Eq. (10), which is discretised using central formulaeas in Eq. (4)

For a detailed description and validation of the numerical code,the reader is referred to [4,16].

3. Comparison with experimental data

In order to validate the numerical code, two different sets ofsimulations have been carried out: in the first the flow in a straightcompound channel has been simulated and the results have beencompared with the experimental measurements obtained by[13,19], while in the second the measurements collected by[18,20] in a compound meandering channel have been used forcomparison. The results obtained in the straight and meanderingchannels are described in Sections 3.1 and 3.2, respectively.

3.1. Experiments and simulations in a straight compound channel

The experiment was conducted at the Hydraulic laboratory ofHR Wallingford in a straight compound open channel (SERC FloodChannel Facility, SERC-FCF) 56-m-long and 10-m-wide. Details ofthe facility can be found in [13]. In this paper the results relativeto one of the best documented case have been used, correspondingto Series 02 in [13], where a trapezoidal cross-section of the mainchannel is considered, with a bottom slope of 1:027� 10�3. Sincethe channel was straight and thus uniform flow conditions wereexpected to develop, the simulations have been performed usingonly 2 cells in the streamwise direction and imposing periodicalboundary conditions at the inflow and outflow sections. A gridmade of 64� 32 computational cells has been used to discretisethe cross-section in the spanwise and vertical directions, respec-tively. In order to obtain a more refined discretisation close tothe momentum transfer regions and to the lateral boundaries, anon-uniform grid-spacing was applied in the spanwise direction,while a uniform discretisation was used in the vertical direction,as shown in Fig. 3.

In the laboratory channel the flow was driven by the componentof the gravity force in the streamwise direction, due to the flumeinclination with the slope of 1:027� 10�3. In the numerical simu-lation an equivalent forcing of the current was obtained setting thebottom horizontal and applying a constant mass force (per unitmass) of 1:027� 10�3g in the streamwise direction. A logarithmicwall-law was used near the solid boundaries (at the lateral wallsand at the bottom). The value of the Nikuradse coefficiente ¼ 3� 10�4 m was obtained starting from the value of the Man-ning coefficient n ¼ 0:01 given by [24] and then using the approx-imate relation n ¼ e1=6=ð8:25

ffiffiffigp Þ [1].

0 1 2 3-0.2

-0.15

-0.1

-0.05

0

x

3

2

x

x

Fig. 3. Numerical grid used to discretise the straight compound channel (Series 02 in [13]). Cross-section in the x2–x3 plane (only the left half is shown because of thesymmetry of the domain). Lengths in meters.

00.10.20.30.40.50.60.70.80.9

1

x 2 [m]

< U

> [

m/s

]

Numerical results

Experimental data

Numerical results

Experimental data

a

00.10.20.30.40.50.60.70.80.9

1

0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 3

x 2 [m]

< U

> [

m/s

]

b

Fig. 4. Comparison of the transverse profiles of the depth-averaged streamwisevelocity: (a) Dr ¼ 0:15; (b) Dr ¼ 0:25.

M. De Marchis, E. Napoli / Advances in Water Resources 31 (2008) 1662–1673 1665

In Fig. 4 the transverse profiles of the depth-averaged stream-wise velocity hUi are shown for two different values of the ratioDr ¼ Df=Dc between the water depth on the floodplains (Df ) andin the main channel (Dc): Dr ¼ 0:15 (Fig. 4.a) and Dr ¼ 0:25(Fig. 4.b). The numerical results are in good agreement with theexperimental measurements, showing that the model is able toadequately reproduce the behaviour of flows in straight compoundchannels.

A slight difference between the numerical and experimentalvelocities can be observed in the transition regions between themain channel and the floodplains, particularly in the caseDr ¼ 0:15, similar to the results obtained by Ervine et al. [6] withan analytical model. This difference is mainly due to the use ofthe standard k� e turbulence model, which is known not to be ableto predict the development of secondary currents in straight com-pound channels [10], whose importance on the vertical distribu-tions of horizontal velocity in compound straight channels hasbeen clearly identified by Shiono and Knight [19]. In Fig. 5b andc the isometric plots of the modelled Reynolds stresses s12 ands13 are shown for the case Dr ¼ 0:25. The corresponding valuesmeasured by [19] are plotted too (Fig. 5d and e) for comparison.The modelled values are in very reasonable agreement with the

measured values, although some discrepancy exists in the shear re-gion, which can be again explained with the known limits of thek� e model. Although the use of a more refined turbulence model(e.g. a second-order closure scheme) would have further improvedthe quality of the results, we choose to use the more simple k� emodel since the secondary flows are on the contrary adequatelypredicted in compound meandering channels (to which next sec-tions will be devoted), as will be shown in Section 3.2.

3.2. Experiments and simulations in a meandering compound channel

In order to assess the ability of the numerical model to ade-quately reproduce experimental stage–discharge curves in mean-dering compound channels, we have used the Test Case (2) of theexperiments conducted in the hydraulics laboratory at the Uni-versity of Bradford as reported in [3,18]. The same configurationwas used by Shiono and Muto [20], who performed detailed mea-surements of streamwise velocities and secondary currents usingLaser-Doppler anemometry. In the experiments the main channelmeandered, while the floodplain sides were straight. Shiono et al.[18] carried out some tests where the floodplain banks mean-dered too, but these results were not shown and discussed intheir paper since the reduced practical interest of that configura-tion. The measurements were undertaken under quasi-uniformflow conditions, which were obtained by setting the water surfaceslope parallel to the valley bed slope at each meander wave-length. To this aim a downstream tailgate was suitably manoeu-vred. The slope of the flume bed was 1� 10�3, while the one ofthe main channel was 0:73� 10�3 since the sinuosity was1.372. In order to reproduce the quasi-uniform flow conditionof the test, in the simulations the length of the meandering chan-nel was set to 1.848 m (the meander wavelength L in the testcase) and two 10-m-long straight reaches were added at boththe upstream and downstream ends of the meandering region.Periodical boundary conditions were imposed at the streamwiseextremities of the computational domain.

The channel bottom was set horizontal and, as in the straightchannel case discussed above, a constant mass force was appliedin the streamwise direction to reproduce the flume inclination.The mass force per unit mass was assigned the value of 10�3g.The stage–discharge curve was obtained carrying out several sim-ulations starting from the rest and subsequently setting the waterlevel throughout the channel to different values spanning from0.035 to 0.1 m. Since the height of the floodplains from the bottom(Dc � Df in Fig. 6) in the experiments and numerical simulationswas 0.052 m, we considered both cases with the current flowingin the main channel only and cases where the lateral banks wereinundated.

A curvilinear grid made of 64� 64� 16 cells was used in thestreamwise, spanwise and vertical directions, respectively. In thestreamwise direction 16 cells were altogether used to discretisethe inflow and outflow straight channels, with a clustering of thegrid near the meandering region. A non-uniform grid was used in

a

b c

d e

Fig. 5. Geometrical sketch of the vertical cross-section (a) and isometric plots of the Reynolds stress s12 ((b) numerical results; (d) experimental results) and s13 ((c)numerical results; (e) experimental results) for the case Dr ¼ 0:25. Lengths in meters, stresses in N/m2.

L

d

w

β

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.2

0.4

0.6

0.8

1

1.2c

Wt

a

0 0.050

0.2

0.4

0.6

0.8

1

1.2d

D

D

b

c

f

Fig. 6. Geometrical layout and computational grid of the meandering compound channel used to test the numerical model: (a) plan view; (b) side view; (c) projection of thecomputational grid on the horizontal plane; (d) projection of the computational grid on the vertical plane. Lengths are in meters. The meandering part of the channel only isshowed.

1666 M. De Marchis, E. Napoli / Advances in Water Resources 31 (2008) 1662–1673

Table 1Geometrical parameters of the meandering compound channel used in Section 3.2

L (m) W t (m) w (m) d (m) b Dc (m) Df (m)

1.848 1.2 0.152 0.376 60� 0.035–0.100 0.000–0.048

M. De Marchis, E. Napoli / Advances in Water Resources 31 (2008) 1662–1673 1667

the vertical direction, which was refined near the bottom of themain channel and the floodplains and near the free-surface.The cells were hexahedrical with vertical lateral faces. In Fig. 6the meandering channel configuration and the projection on thehorizontal and vertical planes of the curvilinear numerical gridused to discretise the computational domain in the meanderingregion are shown (the case h ¼ 0:075 m is considered here). Thevalues of the geometrical parameters introduced in Fig. 6 aresummarised in Table 1.

The simulations allowed to obtain the discharges correspond-ing to the different values of the water depth and, thus, thestage–discharge curve for the considered test case, to be com-pared with experimental results. Although the principal aim ofthe simulations was to obtain the stage–discharge curve, we reck-on useful to show the ability of the model to adequately describethe general flow structure and the secondary flows, that are dom-inant features heavily influencing the conveyance capacity of thechannel. To this aim, the numerical results obtained with h ¼ 0:1m have been compared with the experimental measurements gi-ven by [20]. Specifically, the streamwise velocity distributions ob-tained numerically and experimentally in several test sections(whose position is shown in Fig. 7) of the main channel areshown in Figs. 8 and 9, respectively. In the figures the streamwisevelocities are normalised by the section averaged velocityUs ¼ Q=A, where Q is the discharge and A is the cross-sectionalarea at the bend apex, while the lateral distances are measuredfrom the left-hand side of the channel and the heights from thechannel bed. Both lateral distances and heights, finally, are nor-malised by the bankfull level.

Numerical results are in good agreement with the experimen-tal measurements, although some overestimation of the velocitiesabove the bankfull level is observed. Specifically, the numericalmodel correctly predicts the existence of the highest streamwisevelocities above the bankfull level in the test sections 1, 3 and 13and, on the contrary, the development of local maxima belowthat level in sections 5, 7 and 9. In the test section 5 the localmaximum is located near the left-hand sidewall of the channeland then it moves toward the right, until in section 9 an almost

Fig. 7. Cross-sections used to compare streamwise velocities and secondarycurrents obtained with the numerical model and experimentally. Lengths are inmeters.

symmetrical velocity distribution is obtained below the bankfulllevel. The velocity distribution in section 1 is the mirror imageof the distribution in section 13, showing that the flow is fullydeveloped. The agreement of the numerical results in section 1with the experimental measurements is not fully satisfactory,although the existence of two local maxima of the streamwisevelocity near both the left- and right-hand sides of the channelbelow the bankfull level is correctly predicted by the numericalmodel.

In order to validate with experimental measurements too thesecondary flow structure obtained numerically, in Figs. 10 and 11the numerical and experimental velocities, respectively, are shownin a two-dimensional vector form in the same sections consideredin Figs. 8 and 9 (with the exception of section 13, which is notshown since the velocity distribution is the mirror image of thatobtained in section 1). In section 1 a cell rotating in the anticlock-wise direction is observed in Fig. 10, consistently with the experi-mental results, that disappears in the latter half of the bend, asobtained by [20]. In section 3, in fact, the current enters into themain channel from the left flowing to the right (the outer wall ofthe bend), while a counter-rotating cell starts to develop in the in-ner part of the bend, to be clearly visible in section 5. The cell thendevelops in size occupying the whole section of the main channelbelow the bankfull level in section 7, to move towards the right(which is now the inner part of the bend) in section 9. Finally, insection 13 (not shown), the counter-rotating cell extends in heightto the whole main channel section, up to the water surface.Although the lateral and vertical velocities obtained numericallyare somewhere slightly underpredicted with respect to the mea-surements of [20], the agreement of the numerically predicted sec-ondary flow structure with experimental results is fairlysatisfactory.

To highlight the occurrence of significant mass exchanges be-tween the main channel and the floodplains in overbank flows,the three-dimensional pattern of some streamlines is shown inFig. 12. The helical current in the main channel, as describedamong the others by [17], is clearly identified, together with thedeviation of some trajectories that, downwind the curve apex,leave the main channel to flow in the lateral floodplain.

The general flow patterns shown in Fig. 10 are obviously di-rectly related to the free-surface position (Fig. 13, where the caseh ¼ 0:075 m is considered), clearly showing a rising near the exter-nal sides of the meandering main channel and a lowering at theopposite wall.

The analysis above demonstrates the ability of the numericalcode to reproduce the complex features of the three-dimensionalflows in meandering compound channels, with very good agree-ment with the experimental measurements obtained by [20]. In or-der to show that the numerical model is able, as well, to correctlypredict the discharges in compound meandering channels, thestage–discharge curves obtained experimentally by [18] and thecorresponding numerical results are shown in Fig. 14. The clearchange in the slope of the experimental curve occurring whenthe water depth exceeds the height of the floodplains(h J 0:052), due to the large increase of the cross-section width(which varies from 0.152 to 1.2 m), is well reproduced by thenumerical results. The agreement of these results with the experi-mental measurements is very satisfactory for all the stages, as canbe seen in Table 2, where the errors (in %) of the numerical pre-dicted discharges for given stages and of the stages for given dis-charges are shown. The errors in the calculated discharges forgiven stages are always below 10%, with a mean value of about8%, while the errors in the stages for given discharges are alwaysbelow 7.5%, with a mean value of about 3%, showing that the modelis able to adequately assess the discharges in meandering com-pound channels.

Section 1

0 0.5 1 1.5 2 2.5

0.5

1

1.5

Section 3

0 0.5 1 1.5 2 2.5

0.5

1

1.5

Section 5

0 0.5 1 1.5 2 2.5

0.5

1

1.5

Section 7

0 0.5 1 1.5 2 2.5

0.5

1

1.5

Section 9

0 0.5 1 1.5 2 2.5

0.5

1

1.5

Section 13

0 0.5 1 1.5 2 2.5

0.5

1

1.5

Fig. 8. Streamwise velocity distributions obtained with the numerical model for h = 0.1 m (Dr ¼ 0:5) in the test sections indicated in Fig. 7. Velocities are normalised by thesection averaged velocity Us , while lateral and vertical distances are normalised by the bankfull level.

1668 M. De Marchis, E. Napoli / Advances in Water Resources 31 (2008) 1662–1673

4. Comparison of stage–discharge curves in different shapemeandering compound channels

Once having validated the numerical model with experimentaldata, a new set of numerical simulations has been performed in or-der to identify which geometric parameters mostly affect the con-veyance capacity of meandering compound channels. A number ofempirical formulae have been proposed in the literature to esti-mate the discharges in meandering channels for given values ofthe water depth. Some of the proposed formulae (among them,those of [18,9]) include the value of the channel sinuosity, whileothers (for instance, the meander-belt method of [8]) estimatethe discharge irrespective of the sinuosity. We have thus analysedseveral meandering compound channels, whose planimetric pat-terns are shown in Fig. 15, to compare their conveyance capacityand relate it to geometric parameters. The considered channelsare assigned the labels Ci (i ¼ 2; . . . 5), while the test channel used

in the previous section is labelled as C1. All the channels were builtin such a way to have identical cross-sectional area, lengths andbed slopes, while the sinuosities and meander-belt widths weredifferent, as summarised in Table 3. Specifically, three differentvalues of the sinuosity (s ¼ 1:079;1:372 and 1:624) and two differ-ent values of the meander-belt width (B = 0.36 and B = 0.72 m)were considered, resulting in different values of the mean curva-ture radius. A straight channel was considered too (labelled asSt), in order to evaluate the reduction of the conveyance capacitydue to the meandering in the Test Cases Ciði ¼ 1; . . . ;5Þ. As de-scribed in Section 3.2, two 10-m-long straight reaches were addedat both the upstream and downstream extremities of the meander-ing channels, in order to achieve quasi-uniform flow conditions.The boundary conditions, roughness and bed slopes were assignedas in the Test Case C1. The computational grid was made of80� 64� 16 cells for the Test Cases C2; C3 and C4, while76� 64� 16 cells were used in the Test Case C5. In the straight

Fig. 9. Streamwise velocity distributions obtained by [20]. Velocities and distances are normalised as in Fig. 8.

M. De Marchis, E. Napoli / Advances in Water Resources 31 (2008) 1662–1673 1669

channel two cells only were used in the streamwise direction. Foreach channel a set of simulations was performed, subsequentlyassigning water depth values between 0.035 and 0.1 m and run-ning the computations until achieving steady-state conditions.

The effect of the sinuosity in reducing the conveyance capacityof meandering compound channels has been widely recognised inthe past research [18]. Our simulations confirm this result, as canbe seen in Fig. 16, where the stage–discharge curves for the TestCases C2; C4 and C5 are shown. In the figure the stage–dischargecurve for the straight channel is plotted too, to compare the con-veyance capacity of straight and meandering channels. The pro-gressive reduction of the discharges as the sinuosity increases isclearly seen. The percent reduction in the conveyance capacity issummarised in Table 4, which shows that, for the higher value ofthe sinuosity (s ¼ 1:624), values between 15% and 59% have beenobtained. The reduction is much more pronounced for inbankflows, since the main channel only meandered, while the lateralwalls were straight.

In Fig. 17 the calculated discharges are plotted against the sin-uosities, for each value of the water depth. The curves are concavefor h ¼ 0:08 and h ¼ 0:1 m, while being roughly linear in the shal-lower overbank flows (h ¼ 0:060 and h ¼ 0:075 m) and becomingconvex for inbank flows. These results show that, for the higherstages, the sinuosity effect is lower and lower as the sinuositygrows, while the opposite is true for inbank flows.

The importance of the sinuosity is further highlighted by thecomparison of stage–discharge curves obtained for the Test CasesC1, C2 and C3, which are plotted in Fig. 18. In fact, although the geo-metric patterns and meander-belt widths of these channels are en-tirely different (B = 0.72 m for C1 and C3, B = 0.36 m for C2), thecomputed stage–discharge curves are very close one each other(both for inbank and overbank flows), in force of the identical valueof the sinuosity (s ¼ 1:372) shared by them.

The results shown so far indicate on one hand that the convey-ance capacity of meandering compound channels is mainly influ-enced by the sinuosity and, on the other hand, that the

Section 1

0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

Section 3

0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

Section 5

0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

Section 7

0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

Section 9

0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

0.5 Us

Fig. 10. Secondary flow vectors obtained with the numerical model for h = 0.1 m(Dr ¼ 0:5) in the test sections indicated in Fig. 7. Vertical and lateral distances arenormalised by the bankfull level.

Fig. 11. Secondary flow vectors obtained by [20]. Vertical and lateral distances arenormalised as in Fig. 10.

1670 M. De Marchis, E. Napoli / Advances in Water Resources 31 (2008) 1662–1673

correlation of discharges with meander-belt widths is at most veryweak, if existing.

Some effect of the mean curvature radius can be recognised,since in the channels labelled as C4;C2 and C5, with radii of1.107, 0.648 and 0.258 m, respectively, the discharges are higherand higher as the curvature radius grows. The comparison of thechannels C1 and C5, nevertheless, contradicts this result, since

the discharges in the latter are lower than in the former, althoughthey share an almost identical value of the mean curvature radius(see Table 3). The trend observed in the channels C4;C2 and C5 (i.e.the reduction of the discharges when the mean curvature radius isreduced) should be thus explained with the corresponding trend inthe sinuosity (that in the Test Case C5 is higher and in the Test CaseC4 lower than in C2) rather then ascribed to the effects of the cur-vature radius.

Table 2Water depths (1) and discharges (2) obtained numerically and correspondingpercentage errors on stages (3) and discharges (4) with respect to the results of [18]

Water depth h (mm) (1) Discharges(l/s) (2)

Error on stages(in %) (3)

Error on discharges(in %) (4)

100 22.92 1.94 6.2980 11.11 1.85 7.3775 8.71 2.28 9.3960 3.01 1.26 7.8050 1.75 3.24 9.6635 1.11 7.50 7.35

0

0.5

10

0.05

X

Z

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075[m]

0.5

1

1.5

x

x

x 12

3

Fig. 12. Three-dimensional pattern of streamlines (h = 0.075 m). Colour scale indicates the vertical coordinate x3. Lengths are in meters.

0

0.5

1

1.5

0

0.5

10.0740.0750.076

XY

Z

0.074 0.0744 0.0748 0.0752 0.0756 0.076 0.0764 0.0768 [m]

x

x

x 1

3

2

Fig. 13. Elevation of the free-surface. Lengths are in meters.

0

20

40

60

80

100

120

0 5 10 15 20 25 30Discharge (l/s)

Wat

er d

epth

h (

mm

)

Experimental data

Numerical results

Fig. 14. Comparison of the numerical and experimental stage–discharge curves.Continuous line: experimental data [18]; filled circles: numerical results.

M. De Marchis, E. Napoli / Advances in Water Resources 31 (2008) 1662–1673 1671

It can be thus concluded that, on the basis of the analysis of thechannels considered in this paper, the mean curvature radius andthe meander-belt width should not be included into empirical

Table 3Geometric parameters of the channels used to compare stage–discharge curves. Otherparameters as in Table 1

Label Sinuosity B (m) Mean curvature radius (m)

C1 1.372 0.360 0.252C2 1.372 0.720 0.648C3 1.372 0.720 0.266C4 1.079 0.360 1.107C5 1.624 0.360 0.258St 0.000 ND 1

Table 4Discharge reduction in meandering compound channels with respect to the straightchannel (values in percent)

h (m) C1 C2 C3 C4 C5

0.100 11 13 11 2 190.080 9 14 12 3 150.075 9 10 12 3 170.060 18 21 20 7 360.050 17 15 22 5 540.035 11 15 16 10 59

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

x1

2

C2

x

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1

2

C3x

x

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1

2

C4x

x

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1

2

C5x

x

Fig. 15. Planimetric view and computational grids of the meandering compound channels used to compare stage–discharge curves. Lengths are in meters.

0

5

10

15

20

25

30

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7sinuosity

Q [

l/s]

0.100

0.080

0.075

0.060

0.050

0.035

h [m]

Fig. 17. Dependence of discharges on sinuosity for inbank and overbank flows.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20 25 30Q [l/s]

h [

m]

C4 (s=1.079)C2 (s=1.372)C5 (s=1.624)St

Fig. 16. Comparison of the stage–discharge curves for the cases C2; C4 and C5.

1672 M. De Marchis, E. Napoli / Advances in Water Resources 31 (2008) 1662–1673

formulae to estimate the conveyance capacity of meandering com-pound channels, while the sinuosity is the only geometrical param-eter able to influence the discharges in these channels.

5. Conclusions

A fully three-dimensional finite-volume numerical model hasbeen used to simulate the flow in meandering compound channels.

The numerical model has been validated against experimentalmeasurements collected both in straight and meandering com-pound channels. The comparison with experimental resultsshowed that the numerical model is able to adequately reproduce

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0 5 10 15 20 25 30Q [l/s]

h [

m]

C1 (s=1.372)

C2 (s=1.372)

C3 (s=1.372)

Fig. 18. Comparison of the stage–discharge curves for the cases C1; C2 and C3.

M. De Marchis, E. Napoli / Advances in Water Resources 31 (2008) 1662–1673 1673

the complex behaviour of currents developing in compound chan-nels and to predict with good accuracy the discharges.

The stage–discharge curves calculated in several meanderingcompound channels having the same cross-sectional area, rough-ness and bed slope, but different planimetric layouts showedthat:

� the sinuosity is the main geometric parameter affecting the con-veyance capacity of meandering compound channels. In thesechannels the discharges are lower than in straight channels forthe same water depth and this reduction is more and more pro-nounced as the sinuosity increases;

� the stage–discharge curves obtained for meandering channelshaving the same sinuosity and entirely different planimetricgeometry almost collapse one over the other, showing that thesinuosity is the main geometric parameter to be accounted forin the empirical formulae to calculate the reduction of the chan-nel conveyance due to the meandering;

� the influence of the meander-belt width and of the mean radiusof curvature on the discharges is very weak and no direct corre-lation can be established between the rise or the decrease ofthese parameters and the channel conveyance;

� the results discussed in this paper have been obtained in small-scale meandering channels. A larger set of numerical and exper-imental data would be necessary to extend the conclusionsdrawn in this paper to natural rivers or to laboratory channelswith largely different geometric configurations.

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