ls

. Zn, CChrd,Leva

Received in revised form 29 September2009Accepted 5 October 2009Available online 22 October 2009

found to be different within one sub-section (main channel or oodplain). Experimental measurements

triou [1]. Shiono and Knight [2] showed that three sources of en-ergy losses coexist under uniform ow conditions: bed friction,momentum ux due to both turbulent exchange and secondarycurrents across the total cross-section. Three other ow congura-tions in non-prismatic geometry with constant overall channelwidth were also well investigated: ow in a compound channel

the various experiments was identied: mass transfers betweenows in the main channel and the oodplains generate low varia-tion in ow depth along the longitudinal x-axis. The streamwisevariation in ow depth was even found to be negligible for a fewow congurations.

On the contrary, when the width of the overall channel is vary-ing, the ow depth markedly vary. This was observed for ows insymmetrically converging oodplains [10], in symmetricallydiverging oodplains [9,11], in a compound channel with anabrupt contraction of the oodplain [12], or in presence of a groyneset up on a oodplain perpendicularly to the main ow direction

* Corresponding author. Tel.: +33 4 72 20 86 02; fax: +33 4 78 47 78 75.E-mail addresses: sebastien.proust@cemagref.fr (S. Proust), didier.bousmar@

spw.wallonie.be (D. Bousmar), nicolas.riviere@insa-lyon.fr (N. Rivire), yves.zech@

Advances in Water Resources 33 (2010) 116

Contents lists availab

Advances in Wa

lseuclouvain.be (Y. Zech).For nearly four decades, studies on compound channel ow fo-cused on the momentum exchange between the ow in the mainchannel and the ow in the oodplain, in case where the overallchannel width is constant. The most investigated ow congura-tion was uniform ow in straight geometries. In particular, theshear layer between sub-sections, the secondary currents, theboundary shear stress and the apparent shear stress at the verticalinterfaces between the main channel and the oodplain were de-picted. The interaction between the ow in the main channel andthe ow in the oodplain is investigated, e.g. by Knight and Deme-

two-stage channels [6] or ow in a doubly meandering compoundchannel [7]. These former geometries highlight the role of a previ-ous unstudied source of energy loss: the horizontal shearing lo-cated at the bank-full level in the main channel between theupper ow and the inbank ow. The last case investigated isnon-uniform ow in a straight compound channel [8,9] character-ized by a strong inuence of the upstream discharge distributionbetween sub-sections on the mass transfers along a compoundchannel ume. When the total width of the channel is constantand the ow is non-uniform, a common characteristic betweenKeywords:Compound channelNon-uniform owEnergy lossHead lossMomentum transferTurbulent exchangeMass conservation

1. Introduction0309-1708/$ - see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.advwatres.2009.10.003of the head within the main channel and the oodplain are then analyzed for geometries with constant orvariable channel width. Results show that head loss differs from one sub-section to another: the classical1D hypothesis of unique head loss gradient appears to be erroneous. Using a model that couple 1Dmomentum equations, called Independent Sub-sections Method (ISM), head losses are resolved. Therelative weights of head losses related to bed friction, turbulent exchanges and mass transfers betweensub-sections are estimated. It is shown that water level and the discharge distribution across the channelare inuenced by turbulent exchanges for (a) developing ows in straight channels, but only when theow tends to uniformity; (b) ows in skewed oodplains and symmetrical converging oodplains forsmall relative ow depth; (c) ows in symmetrical diverging oodplains for small and medium relativedepth. Flow parameters are inuenced by the momentum ux due to mass exchanges in all non-pris-matic geometries for small and medium relative depth, while this ux is negligible for developing owsin straight geometry. The role of an explicit modeling of mass conservation between sub-sections is even-tually investigated.

2009 Elsevier Ltd. All rights reserved.

with a diverging left-hand oodplain and a converging right-handoodplain, usually called skewed ows [35], ow in meanderingArticle history:Received 11 June 2009

This paper investigates energy losses in compound channel under non-uniform ow conditions. Using therst law of thermodynamics, the concepts of energy loss and head loss are rst distinguished. They areEnergy losses in compound open channe

S. Proust a,*, D. Bousmar b, N. Rivire c, A. Paquier a, YaCemagref, Hydrology-Hydraulics Research Unit, 3 bis quai Chauveau CP220, 69336 LyobHydraulic Research Laboratory, Service Public de Wallonie, Rue de lAbattoir 164, 6200c Fluid Mechanics and Acoustics Laboratory (UMR CNRS 5509), INSA de Lyon, Bat JacquadCivil and Environmental Engineering Unit, Universit catholique de Louvain, Place du

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

journal homepage: www.ell rights reserved.ech d

edex 09, Francetelet, Belgium20 av A Einstein, 69621 Villeurbanne Cedex, Francent 1, B-1348 Louvain-la-neuve, Belgium

le at ScienceDirect

ter Resources

vier .com/ locate/advwatres

r concerning right-hand oodplain

ate[13]. In this case, the nature of mass transfers is clearly different, asthey are produced by both streamwise changes in the total widthof the channel and in water depth, and consequently become

Nomenclature

Ai sub-section areaBi sub-section widthhi sub-section ow depthh relative ow depth at mid-length of a prismatic channel

or at mid-length of a diverging, converging or skewedreach

Hi sub-section headqin lateral inow per unit lengthqout lateral outow per unit lengthqrm lateral mass discharge between the right-hand ood-

plain and the main channel (algebraic value)qlm lateral mass discharge between the left-hand oodplain

and the main channel (algebraic value)Q total dischargeS0 bed slopeSfi sub-section friction slopeSHi sub-section head loss gradientUd depth-averaged velocity in the longitudinal directionUi sub-section mean velocityUin longitudinal velocity of lateral inow qin at the interfaceUint longitudinal velocity at the interface between one ood-

plain and the main channelUint:l longitudinal velocity at the interface between the left-

hand oodplain and the main channel

2 S. Proust et al. / Advances in Wstronger than when the overall channel width remains constant.As overbank ows with varying width are quite common in the

eld, this paper deals with head losses for ows with constant orvariable channel width. The rst aim of the paper is to identifythe main physical processes responsible for head losses in bothcontexts. The second aim is to estimate the inuence of the headlosses on two hydraulic parameters of interest for engineers, theow depth and the discharge in the oodplain, for various typesof geometry. The last aim is to work out the inuence of an explicitmodeling of mass conservation at the interfaces between sub-sec-tions in the longitudinal evolution of hydraulic parameters. Forthat purpose, we use 1D energy or momentum balances within asub-section (main channel, left-hand or right-hand oodplain),which give a synthetic overview of the predominant physicalphenomena.

First, we develop the equations of energy loss and head loss ap-plied to one sub-section or to the total cross-section of a compoundchannel. An original result is obtained: head loss gradient is equalto energy slope on the total cross-section, but in the sub-sections,head loss gradient differs from the energy slope. Second, we ana-lyze the streamwise evolution of the head in the sub-sectionsand in the total cross-section from various experimental data sets:developing ows in straight compound geometry, ows in sym-metrical diverging or converging compound channels. In particu-lar, we test the validity of a common 1D hypothesis: equal headloss gradients in the main channel and in the oodplain. The exper-imental proles of head in the sub-sections are then compared tothe numerical results of a 1D-improved model, the IndependentSub-sections Method (ISM). This model enables three sources ofhead loss to be accounted for: (1) the classical bed friction on thesolid walls; (2) the momentum ux due to the turbulent exchangesgenerated by the shearing between sub-sections; (3) the momen-tum ux due to mass exchanges between sub-sections. Using theISM simulations, we show the difference between computed headloss gradients and energy slopes in the sub-sections. Afterwards,we examine the relative weights of the three sources of head lossUint:r longitudinal velocity at the interface between the right-hand oodplain and the main channel

Uout longitudinal velocity of lateral outow qout at the inter-face

x longitudinal directiony lateral directionz water level above reference datuma Coriolis coefcient on the total cross-sectionai Coriolis coefcient in sub-section ib Boussinesq coefcient on the total cross-sectionbi Boussinesq coefcient in sub-section id angle between the oodplain lateral walls and the main

channel axis (x-direction)sij shear stress at the vertical interface between two adja-

cent sub-sections i and j along x-axis (depth-averagedvalue)

wt coefcient of turbulent exchange

Subscriptsf concerning oodplaini concerning a sub-section (i = l, r or m)l concerning left-hand oodplainm concerning main channel

r Resources 33 (2010) 116in the various geometries investigated. The inuence of eachsource of head loss on discharge distribution and ow depth isquantied. Lastly, we examine the role of the mass conservationat the interfaces between sub-sections on the ow parameters.

2. Energy loss, head loss, and momentum

In this section, equations of energy loss, head loss and momen-tum are developed for one-dimensional compound channel ow,relying on the work of Field et al. [14] that deals with energy andmomentum in 1D open channel ow.

Let us consider a uid system in a control volume X bounded bya surface AX presented in Fig. 1a and b. The total energy of this uidsystem is denoted E, and the total energy per unit mass is denotede [J/kg]. We assume that the gravity force is the only volume forcederiving from a potential energy, and we only consider heat trans-fers with the solid walls, the water surface, and with the liquidinterfaces between sub-sections. Under these assumptions, the to-tal energy e is the sum of macroscopic kinetic energy, potential en-ergy of the gravity force and of the internal energy per unit mass,and the rst law of thermodynamics is written (see e.g. [15])

@

@t

ZZZXqedX

ZZAX

qe~v d~AX

~qZZ

AX

pqq~v d~AX

ZZAX

s ~n ~v dAX 1

where ~v is the local velocity vector, ~q is the caloric power ex-changed with the exterior of X [J/s], p is the pressure, s is the tensorof viscous and turbulent shear stresses applied to surfaceAX; d~AX dAX~n; ~n being the unit vector perpendicular to unit sur-face dAX.

X (a

ateY

int.A

XZ

x

Fig. 1. Schematic view of a uid system in a control volume X, bounded by a surface Aparameters.

S. Proust et al. / Advances in WIn absence of other volume force than the gravity force, thepower of external forces applied to AX is the sum of the power ofpressure strengths and the power of viscous and turbulent shearstresses, the second and third terms in the right-hand side of Eq.(1), respectively. We will successively consider a balance on the to-tal compound cross-section and a balance on one sub-section(main channel, left-hand or right-hand oodplain). In both cases,the friction with air is neglected, and we assume no slip of the uidon the solid walls vwall 0.

2.1. Total cross-section

We consider here a control volume X extended to the to-tal cross-section. The ux of total energy, the power of pres-sure strengths and shear forces are equal to zero along thesolid walls (total wetted perimeter), as velocity vwall 0. Onthe contrary, they differ from zero on the entering and exit-ing surfaces A1 and A2 presented in Fig. 1b. Developing Eq.(1) on the total cross-section under steady ow leads to(see Appendix A.1)

ddx

aU2

2g

! dzdx

1gdldx

~qqgQDx

0 2

where ax RR Axu3 dA=U3A is the kinetic energy correction coef-cient, U is the mean velocity on the total cross-section area A, z isthe water level, l is the internal energy per unit mass, Q the totaldischarge with Q = AU. It is proposed in [14] to name energy slope(denoted Se) the sum of the derivative along the x direction of inter-nal energy per unit of distance and unit of weight and of the calo-ric power exchanged with the exterior per unit of distance andunit of weight, which is written

Se 1gdldx

~qqgQDx

3lateralA 2

int.A

A bed1A

A

, b) with AX Alateral [ Abed [ Aint [ A1 [ A2; (c) notations of hydraulic and geometrical

r Resources 33 (2010) 116 3Se is the gradient of energy dissipated into heat by irreversibleprocesses. Eq. (3) is the expression of local phenomena. Recallingthat total head is dened as H z aU2=2g, the two rst termsin Eq. (2) are the opposite of the longitudinal gradient of total head,denoted SH and called below head loss gradient. Hence, Eq. (2) iswritten

SH dHdx @

@xz aU

2

2g

! Se 4

According to Eq. (4), head loss gradient is equal to energy slopeon the total cross-section, similarly to classical open channel owin a single cross-section. This is the expression of the equality be-tween mean parameters of the ow and the integrated value of lo-cal phenomena on the total cross-section.

In addition to Eq. (4), the second equation governing the ow isthe classical equation of momentum conservation, the 1D-Saint-Venant equation. Under steady owwithout inow or outow, thisequation is written on the total cross-section

dzdx

1gA

ddx

bAU2 Sf 0 5

where bx RR Axu2 dA=U2A is the momentum correction coef-cient, and Sf is the friction slope on the solid walls of the overallcross-section, i.e., on the total wetted perimeter.

The link between the concepts of head loss and momentum, orbetween b and a, is obtained by combining Eqs. (4) and (5)

SH Sf 1A@

@xbQ2

gA

! @@x

a12g

U2

6

The head loss gradient on the total cross-section is the sum ofthe friction slope on the total wetted perimeter and of an addi-tional head loss due to the non-uniformity of velocity across the

dx dx 2g dx dx 2g !

ateoverall channel. This non-uniformity is signicant for compoundchannel ow, as the value of kinetic coefcient a may exceed 2according to French [16].

2.2. Sub-section

In the following, we consider compound channels composed ofa main channel and two oodplains as shown in Fig. 1c. Subscriptsm, l, and r are used for mean values of hydraulic parametersin the main channel, the left-hand oodplain, and the right-handoodplain, respectively. Subscript f is used for oodplain ingeneral (left or/and right). Subscript i is used for a sub-sectionin general (i =m, l, r). Two adjacent sub-sections, i.e., parallel withregard to the longitudinal direction, are identied by subscripts iand j.

The head loss gradient SHi and the energy slope Sei are, respec-tively, dened as

SHi dHidx @

@xzi ai U

2i

2g

!7

Sei 1gdlidx

~qiqgQiDx

8

where li; ~qi; Qi; zi; ai, and Ui are sub-section-averagedparameters.

When considering a balance on one sub-section, the verticalsurface at the interface between two adjacent sub-sections has tobe accounted for (denoted Aint in Fig. 1a and b). Eq. (1) applied toone sub-volume leads to (see Appendix A.2)

Sei SHi sij UinthintqgQi 12gQi

qin U2in aiU2i

qout aiU2i U2out

9

with sij = algebraic value of the shear stress acting at the verticalinterface in the x direction between sub-sections i and j; Uint =depth-averaged longitudinal velocity at the interface betweensub-sections i and j, with Uint Uin or Uout , depending on water isentering the sub-section with a lateral discharge qin or leaving thesub-section with a lateral discharge qout; hint = water depth at thevertical interface. Discharges qin and qout are considered positiveand are mutually exclusive.

Eq. (9) shows that energy slope and head slope gradient aredistinct within a sub-section of a compound cross-section.

The second dynamic equation within the sub-section is theequation of momentum conservation, which is written (see [1719]):

Sfi dzidx sijhintqgAi

1gAi

ddx

biAiU2i

Uinqin Uoutqout

gAi10

where Sfi is the friction slope on the solid walls (bed and lateralbanks), and bi is the Boussinesq factor averaged on the sub-section.

On the right-hand side of Eq. (10), the last term is related to themomentum conveyed by the lateral inow qin or the lateral out-ow qout through the interface between sub-sections. The productsUinqin and Uoutqout are the terms of momentum ux (divided by q)due to mass exchange.

Similarly to Eq. (6), a link between SHi and Sfi can be establishedby combining Eqs. (7) and (10):

Sfi SHi sijhintqgAi ddx

aiU2i =2g

1gAi

ddx

biAiU2i

4 S. Proust et al. / Advances in W Uinqin UoutqoutgAi

11 SH dzdxddx

aU2

2g13

with z = mean value of water level across the total channel, zi =mean value of water level across the sub-section, and with totalhead H and the sub-section-averaged head Hi dened as:

Hi zi U2i =2g and H z aU2=2g 14

3.1.2. Correction factors of kinetic energy and momentumUnder non-uniform ow conditions, all the 1D codes assume

that, within one sub-section, ai bi 1. On the contrary, they ac-count for distinct mean velocities in the main channel and in theoodplain (i.e., a b 1). The non-uniformity of velocity betweensub-sections is thus assumed to be larger than the non-uniformityof velocity within one sub-section. Assuming ai bi 1 is actuallyessential for 1D models as they cannot work out the lateral distri-bution of depth-averaged streamwise velocity Ud.

Under this assumption, combining the mass conservation in asub-section

dAiUidx

qin qout 15

with the Eq. (11) simply gives:

SHi Sfi sijhintqgAi qinUi Uin qoutUout Ui

gAi Sfi Sti Smi

16The head loss gradient in one sub-section can be divided into

three parts: the friction slope Sfi, the head loss due to shear stresssij related to interfacial turbulent exchanges (denoted Sti ), and thehead loss due to mass exchanges (denoted Smi ).

3.2. Experimental measurements of head3. Head losses

It should be recalled that, under uniform ow conditions, longi-tudinal gradients dzi=dx, and dz/dx are equal to the bottom slope,denoted S0. Thus, Eqs. (4) and (7) rigorously give:

SHi SHj SH S0 12where subscripts i and j are related to two different sub-sections.

Head loss gradients are equal in the various sub-sections and inthe total cross-section, and are also equal to the bottom slope S0.Eq. (12) also demonstrates that the head loss gradient in a sub-sec-tion is independent of the interfacial dissipation associated withshear stress sij. This is not the case for the energy slope Sei since,according to Eq. (9), Sei S0 sij Uinthint=qgQi.

3.1. Assumptions for non-uniform ow

3.1.1. Head loss gradientsWhen the ow is non-uniform, inspired by Eq. (12), the most of

1D numerical models still considers equal head loss gradients inthe various sub-sections and on the total cross-section. Thisassumption is necessary to solve the water surface prole on thetotal cross-section. It is also assumed that, within a sub-section,ai 1 and bi 1 (see next paragraph). These assumptions lead to:

SHi dzi d U2i

! SHj dzj d

U2j !

r Resources 33 (2010) 116In this section, we will investigate the streamwise evolution ofthe head in the sub-sections and in the total cross-section, Hi and

0.4

0.4

0.4

atex

y

skew angle = 5.1

L = 10 m

S. Proust et al. / Advances in WH, dened according to Eq. (14). In particular, we want to experi-mentally evaluate the validity of Eq. (13) for various types of geom-etries. We use experimental data collected in three differentcompound channel umes: (i) a ume located at Universit Cath-olique de Louvain-la-neuve (UCL), Belgium, with a symmetricstraight geometry (Fig. 2b), with 6 m-long or 4 m-long divergingoodplains (denoted Dv6 and Dv4 in Fig. 2e, respectively), andwith 6 m-long or 2 m-long converging oodplains (Cv6 and Cv2in Fig. 2f); (ii) a ume at Laboratoire de Mcanique des Fluides et

0.7 m

1.5 m

0.4

0.8

2.2 m

L = 9 m

B = 5.6 m

Lefthand floodplain (l)

Main channel (m)

Righthand floodplain (r)

skew angle = 9.2

Relative depth h*

Floodplain (l)Main channel (m)Floodplain (r)

= 3.8

= 5.7

4 m

6 m

4 m2 m

m 2m 2

0.4

0.4

0.4

Dv6

Dv4

Floodplain

Main channel0.8 m

31 m

3 m

Fig. 2. Top view of the compound channels with skewed oodplains (a), straight geometr(f), abrupt oodplain contraction (g).L = 10 m

B = 1.2 m

m

m

mMain channel (m)

r Resources 33 (2010) 116 5dAcoustique (LMFA), with an asymmetric straight geometry(Fig. 2c); and (iii) a ume at Compagnie Nationale du Rhne(CNR), France, with an asymmetric straight geometry presentinga slight curvature (Fig. 2d).

3.2.1. Non-uniform ow in straight compound channelThe rst ow conguration investigated is developing ows in a

straight compound channel (Fig. 2bd). Experimental setup and re-sults are presented and analyzed in [8,9]. In the three umes, the

m

m

L = 8 m

B = 1.2 m

B = 3 m

L = 13 m

Righthand floodplain (r)

Main channel (m)

Main channel (m)

Righthand floodplain (r)

= 11.2

= 3.8

m 2m 6m 2

4 m 2 m 4 m

m

m

m

Cv6

Cv2

= 22

0.7 m1.53 m

.5 m

ies (bd), diverging geometries Dv6 and Dv4 (e), converging geometries Cv6 and Cv2

s of

ateupstream oodplain discharge exceeds the oodplain discharge un-der uniform ow conditions. This imbalance in upstream dischargedistribution creates mass transfers from the oodplain towards themain channel along the ume. The streamwise prole of oodplaindischarge Qf is presented in Fig. 3. The relative ow depth h

, de-ned as the ratio between the ow depth in oodplain and the owdepth in main channel hf =hm, is measured at mid-length of theume. The percentage of increase in oodplain discharge at x 0compared to uniform ow conditions is denoted dQf x 0. Thestreamwise evolution of sub-section head Hi is presented in Fig. 4.Analyzing this gure leads to the results listed below:

As long as the mass transfer is signicant between sub-sections,the head loss gradient in the main channel SHm differs from thehead loss gradient in the oodplain SHf . In this case, SHf is higherthan the bed slope S0, while SHm is lower than S0, and Eq. (13) isnot valid.

If the ow tends to uniformity at the downstream end of theume, the head loss gradients in the sub-sections approachthe value of the bed slope S0 in accordance with Eq. (12).

Given a relative depth h, increasing the imbalance in oodplaindischarge at station x 0 accentuates the difference of evolution

0 2 4 6 810

15

20

Streamwise direction X (m)

Floo

dpla

in d

Geometry

Fig. 3. Developing ows in straight compound channels: experimental measurementumes.25

30

35

40

45

50

55isc

harg

e di

strib

utio

n Q f

/ Q

(%) h* = 0.23 dQf (x=0) = +56%

h* = 0.33 dQf (x=0) = +38%h* = 0.33 dQf (x=0) = +53%h* = 0.40 dQf (x=0) = +32%

6 S. Proust et al. / Advances in Wbetween oodplain head and main channel head (the differencebetween SHm and SHf values is larger).

For a similar upstream imbalance in oodplain discharge, thedifference of evolution between Hm and Hf rises with an increas-ing relative depth h (compare Fig. 4a and d, or Fig. 4c and f).

Head slope gradients are more sensitive to an increase in rela-tive depth (for a given upstream imbalance) than to an increasein upstream oodplain discharge (for a given relative depth), asshown by Fig. 4a and f.

3.2.2. Symmetrically diverging oodplainsThe second ow conguration investigated is ows in symmet-

rically diverging oodplains [9,11], presented in Fig. 2e. Thestreamwise proles of ow depth in the main channel are pre-sented in Fig. 5. The geometry Dv6 presents a diverging semi-an-gle d 3:8, and for Dv4, d 5:7. The relative depth h ismeasured here at mid-length of the diverging reach. The stream-wise evolution of sub-section head Hi and total head H is presentedin Fig. 6. The main results are listed below, the ow congurationbeing identied by Geometry/h*/Q:

In the majority of ow cases, considering equal head loss gradi-ents in the sub-sections is erroneous. Equal head loss gradients in the sub-sections were observed forthe smallest values of total discharge Q, angle d and relativedepth h, i.e., for conguration Dv6/0.2/12, as shown in Fig. 6a.Besides, they are equal to the bed slope: Eq. (12) appears to berelevant in this particular case. It should be noted that for thisow conguration, the streamwise variation in main channelow depth is the smallest of the data set investigated, as shownin Fig. 5a.

Comparing Dv6/0.2/12 and Div4/0.2/12 (Fig. 6a and b) showsthat when increasing angle d from 3.8 to 5.7, the equalitySHm S0 is still valid, but slope SHf becomes lower than S0.

The head in the oodplain is clearly more sensitive to a changein parameters angle d, relative depth h or total discharge Q thanis the head in the main channel.

The evolution of oodplain head is very sensitive to an increasein d angle or, to a lesser extent, in total discharge Q. No clear ten-dency was found with an increase in relative depth h.

When increasing angle d from 3.8 to 5.7, and consequentlyincreasing transfers between sub-sections, head loss in theoodplain SHf may be negligible, or become negative as forDv6/0.3/20 and Dv4/0.3/20 (Fig. 6e and f). This phenomenon isalso observed for given angle d and relative depth h when rising

0 2 4 6 8 10 12 1410

Streamwise direction X (m)

Floo

dpla

i

oodplain discharge distribution Qf =Q (%) in (a) LMFA ume and in (b) CNR and UCL20

30

40

50

60

n di

scha

rge

dist

ribut

ion

Q f / Q

(%) CNR: h* = 0.20; dQf (x=0) = +119%CNR: h* = 0.33; dQf(x=0) = +55%

UCL: h* = 0.18; dQf (x=0) = +67%UCL: h* = 0.27; dQf (x=0) = +48%UCL: h* = 0.41; dQf (x=0) = +32%

r Resources 33 (2010) 116up the total discharge (compare Dv6/0.3/16 and Dv6/0.3/20 inFig. 6d and f).

For the highest relative depth h 0:5 (Fig. 6c), both SH and SHiare clearly smaller than the bed slope S0. The energy dissipationis very low for this high relative depth. Besides, Eq. (13) appearsto be valid: head loss gradients are equal in the sub-sections.

In all cases, the evolution of the total head H is very close to theone of the main channel head Hm.

3.2.3. Symmetrically converging oodplainsThe last conguration investigated is ow in symmetrically

converging oodplains, Cv6 d 3:8 and Cv2 d 11:2, studiedby Bousmar et al. [10,17], and presented in Fig. 2f. The streamwiseproles of ow depth in the main channel are presented in Fig. 7.The relative depth h is measured here at mid-length of the con-verging reach. The experimental sub-section-averaged heads inthese converging geometries are presented in Fig. 8. The main re-sults are listed below:

As expected, the head loss in the sub-sections betweenupstream and downstream boundaries rises up with a decreas-ing relative depth h (compare Fig. 8a and b) or with an increas-ing total discharge Q (compare Fig. 8c and d).

ateS. Proust et al. / Advances in W For low and medium relative depth (h 0:2 and 0.3), the headloss gradient in the main channel clearly differs from the headloss gradient in the oodplain. The streamwise variation of headis more signicant in the oodplain, in accordance with the var-iation in width in this sub-section.

For high relative depth h 0:5 and low discharge (Q = 12 l/s),the differences of evolution between oodplain head and mainchannel head are reduced. In this particular case, assumingequal head loss gradients in the sub-sections is less erroneousthan in the other cases.

From the experimental observation of sub-section head Hi inboth contexts with constant or variable overall channel width,we can conclude that ow congurations with equal head loss gra-

Fig. 4. Developing ows in straight compound channel: experr Resources 33 (2010) 116 7dients in the sub-sections are infrequent. As a result, Eq. (13) nec-essary for the 1D models that solve the Bernoulli or Saint-Venantequation on the total cross-section is erroneous in the majorityof cases.

4. Modeling of sub-section head with coupled 1D equations

If the water surface prole is solved within each sub-section,there is no need to assume that the head loss gradients in thesub-sections are equal. This approach is used in the 1D-improvedmodel, called the Independent Sub-sections Method (ISM). TheISM was developed by Proust et al. [9,18] to assess both the waterlevel and the discharge distribution for non-uniform ows in com-pound channel. In this section, the streamwise proles of sub-sec-

imental measurements of sub-section-averaged head Hi .

ate8 S. Proust et al. / Advances in Wtion head computed by the ISM are compared to the experimentalresults.

4.1. Equations of the Independent Sub-sections Method (ISM)

The Independent Sub-sections Method (ISM) solves a set of or-dinary differential equations composed of three coupled 1D equa-tions of water surface prole within each sub-section (mainchannel, left-hand, and right-hand oodplains), and of a mass con-servation on the total cross-section. The method was validatedagainst experimental data for: developing ows in straight com-pound channels (Fig. 2bd), ows in the Flood Channel Facility(FCF) with skewed oodplains (see Fig. 2a), ows in symmetricallyconverging and diverging oodplains (Fig. 2e and f) or in an abruptcontraction of the oodplain (Fig. 2g). The maximum relative er-rors in the calculation of the couple {ow depth,discharge} in theoodplain are {8%;19%} for the 46 ow cases investigated[9,18].

The ISM computes the water surface prole on each sub-sectionby solving an equation that combines Eq. (16) and the mass conser-vation in one sub-section Eq. (15) multiplied by Ui=gAi. By isolat-ing the term that comes from the mass conservation, denoted Mai

Mai qin qoutUi=gAi 17the equation of water surface prole in each sub-section is written

1 U2i

ghi

!dhidx

U2i

gBi

dBidx

S0 Sfi Sti Smi Mai 18

with Sti head loss (or gain) due to interfacial turbulent exchange,Smi head loss (or gain) due to interfacial mass exchanges, and withBi sub-section width (Ai Bi hi, rectangular sub-section).

Fig. 5. Longitudinal prole of experimental ow depth in the main channel fordiverging geometries Dv6(a) and Dv4(b).The friction slope on the solid walls of the sub-section Sfi is com-puting using

Sfi fi4RiU2i2g

19

where Ri is the hydraulic radius accounting for solid walls only, andfi is the DarcyWeisbach coefcient.

As previously dened in Eq. (16), the two sources of interfacialhead loss Sti and S

mi are, respectively, computed using

Sti sijhintqgAi

; Smi qinUi Uin

gAi qoutUout Ui

gAi20

where sij algebraic value of the shear stress between sub-sectionsi and j; and Uin or Uout Uint , the depth-averaged longitudinal veloc-ity at the interface between i and j.

In the following, we dene qrm (resp. qlm) as the lateral mass dis-charge between the right-hand oodplain (resp. the left-handoodplain) and the main channel. Both are positive if mass is leav-ing the oodplains, and negative if mass is entering the oodplains,i.e., in a mathematical way: qout qlm and qin 0 in the left ood-plain; qout qrm and qin 0 in the right oodplain; qout 0 andqin qlm qrm in the main channel.

Using these notations, Table 1 presents the termsMai, Smi , and S

ti

for the three sub-sections. The interfacial velocities are denotedUint:l or Uint:r on the left-hand or right-hand interface, respectively,as shown in Fig. 1c. The interfacial shear stresses slm and srm are theabsolute values of sij on the left-hand and right-hand interfaces,respectively.

It is very important to notice that the head loss due to mass ex-change Smi is due to the inow (or outow) of slower water into fas-ter water, or vice versa. Smi does not exist under uniform owconditions, as the lateral mass discharge is equal to zero in thiscase. As a result, Smi is not related to the secondary currents withinuniform ows observed in [2].

It is also important to notice the difference between the termsMai and S

mi . When the velocity distribution is uniform across the

channel, Ul Uint:l Um Uint:r Ur , there is no momentum uxdue to mass exchanges between sub-sections and consequently,no head loss due to mass exchange Smi 0

. On the contrary, the

Mai term is not negligible with the same uniformity of velocityacross the channel. In this case, mass exchange between sub-sec-tions occurs without transferring momentum.

4.2. Turbulent exchange coefcient and interfacial velocity

The shear stresses slm and srm on the left-hand and right-handinterfaces are modeled using the mixing length model in the hori-zontal plane adopted by Bousmar and Zech [19]

slm qwtUm Ul2; srm qwtUm Ur2 21where wt a constant coefcient of turbulent exchange. When usedin the ISM, wt was calibrated under uniform ow conditions in twosmall-scale compound channel umes, the LMFA and UCL umes,and in the Flood Channel Facility for Series A3, and was found tobe equal to 0.02 [18].

The modeling of the depth-averaged streamwise velocities Uint:land Uint:r relies on all the experiments presented in Fig. 2. It wasfound that the value of interface velocity strongly depends onthe direction of mass transfer (see [18]). An overview is presentedbelow, considering two adjacent sub-sections i and j:

When the channel is prismatic and mass transfers occur from i

r Resources 33 (2010) 116towards j, as for developing ows in straight geometry

Uint Ui 22

ateS. Proust et al. / Advances in W When the channel is non-prismatic and the total width is con-stant, as for skewed ows

Uint Ui; if dBi=dx < 0 23Uint Uj; if dBi=dx > 0 24

When the channel is non-prismatic and the total width is vari-able, as for diverging or converging geometries (Dv4, Dv6, Cv2,Cv6, and Abrupt oodplain contraction)

Uint:l ulUl 1ulUm andUint:r urUr 1urUm 25

where ul and ur are weighting coefcients depending on thegeometry. In Eq. (25), more weight is given to the oodplain

4.3

comcondentheSHiISMsouchatal

Fig. 6. Experimental sub-section-averaged head Hi and totar Resources 33 (2010) 116 9velocity (Ul or Ur) in converging geometries, while more weightis given to the main channel velocity Um in diverging geometries.

. Comparing numerical and experimental sub-section head

The ows previously investigated in Section 3.2, the ows in apound channel with skewed oodplains [4] or with an abrupttraction of the oodplain [12] were modeled with the Indepen-t Sub-sections Method (ISM) in [18]. To assess the inuence ofdifferent contributions to sub-section head loss (or gain)

Sfi Sti Smi in Eq. (16) (equivalent to Eq. (18)), three types ofsimulations were carried out: (1) accounting for the three

rces of head loss; (2) only taking into account the turbulent ex-nges in themomentumux at the interfaces; (3) ignoring the to-momentum ux at the interfaces, i.e., considering bed friction as

l head H in diverging geometries Dv4 and Dv6.

and experimental values for two ow congurations. The two dia-grams demonstrate that the head loss due to mass exchange Sm is apredominant process in the two geometries, but with notabledifferences.

For the diverging ow (Dv6/0.3/20), the momentum ux asso-ciated with mass exchange enables the head in the oodplain toslightly increase. The ow in the oodplain receives energy fromthe main channel ow thanks to the momentum ux caused bymass exchange. In this context, the Sm term in the oodplain isnegative (head gain) like the head slope gradientSHi Sfi Sti Smi . Besides, we can observe that ISM reproducesdistinct head slope gradients in the two sub-sections mainlythanks to this Sm term in the oodplain. In the main channel, theeffect of Sm term is negligible. Fig. 9a also shows that turbulent dif-fusion at the interfaces is not signicant enough to increase theoodplain-averaged head. Indeed, considering turbulent diffusiononly SH Sf St, leads to positive values of SH (very high headloss at the entrance) that are not in agreement with experimentaldata.

For the converging ow presented in Fig. 9b (Cv6/0.2/10), therole of head loss (or gain) due to mass exchange Sm is also signi-cant, but to a lesser extent than in a channel with diverging ood-plains. However, the inuence of this source of dissipation isdemonstrated here in both the oodplain and the main channel,with positive values in the two sub-sections (head loss).

A general view of the relative weights of the three sources of

10 S. Proust et al. / Advances in Water Resources 33 (2010) 116the only source of dissipation. Simulations (1), (2) and (3) are labeledin the next gures Sf St Sm, Sf St, and Sf , respectively.

To illustrate what can be deduced from ISM simulations, Fig. 9presents a comparison between numerical sub-section head Hi

according to Eq. (9) (with ai 1 in the ISM). Fig. 10 presentsthe streamwise prole of energy slope Sei and head loss gradient

Fig. 7. Longitudinal prole of experimental ow depth in the main channel forconverging oodplains Cv6 (a) and Cv2 (b).

Fig. 8. Experimental sub-section-averaged headSHi in the main channel and the oodplain, calculated for fourow congurations in diverging geometries. According to Eq.head loss will be presented in Section 5 for the various geometriesinvestigated.

4.4. Comparing head loss gradient SHi and energy slope Sei

Using ISM simulations, we computed the energy slope SeiHi in converging geometries Cv2 and Cv6.

(9), the more the turbulent diffusion or/and the momentum uxdue to mass exchange at the interfaces is signicant, the morethe difference between Sei curve and SHi curve is notable.

The ow Dv6/0.3/12 in Fig. 10a presents slight discrepancy be-tween Sei and SHi in the oodplain, but no difference in the main

channel. We can conclude that this ow dissipates little energy be-tween sub-sections due to the interfacial momentum ux. Whenincreasing the discharge from 12 to 20 l/s (compare Fig. 10a and c),discrepancy between Sem and SHm appears in the main channel, andSef stronglydiffers from SHf in theoodplain. Besides, Sef and SHf have

0 2 4 6 8

60

65

70

75

80

Streamwise direction X (m)

Hea

d (su

bsec

tion

avera

ged)

(mm)

Floodplain

Main channel

2 4 6 8 1070

80

90

100

110

120

130

Streamwise direction X (m)

Hea

d (su

bsec

tion

avera

ged)

(mm) Meas. in the Floodplain

Meas. in the Main ChannelSf+S

t+Sm

Sf+St

SfMain channel

Floodplain

Fig. 9. ISM results versus measurements of sub-section head in converging or diverging geometries (Cv6 and Dv6, d 3:8).

Table 1Terms of mass conservation and head loss within each sub-section in Eq. (18).

Terms of Left-hand oodplain Main channel Right-hand oodplain

Mass conservation term Mai qlmUlgAl

qlmUmgAm qrmUmgAm

qrmUrgAr

Head loss due to mass exchange Smi qlmUint:l UlgAl

qlmUm Uint:lgAm

qrmUm Uint:rgAmqrmUint:r Ur

gAr

Head loss due to turbulent exchange Sti slmhlqgAlslmhlqgAm

srmhrqgAm srmhrqgAr

S. Proust et al. / Advances in Water Resources 33 (2010) 116 11Fig. 10. Head slope gradient SHi and energy slope Sei in the sub-sections for ows in diverging geometries Dv4 and Dv6.

opposite signs. In a similarway, for a given total dischargeQ and rel-ativedepthh, increasing theangle d and themass transfers betweensub-sections lead to accentuate the difference between head lossgradient and energy slope (compare e.g. Fig. 10a and b).

5. Main processes responsible for head losses

In this section, we will go further in the understanding of theinuence on the hydraulic parameters of bed friction, turbulentdiffusion, and of momentum ux due to mass exchanges. Indeed,the relative weights of these three sources of head loss varydepending on the geometry in the horizontal plane and the relativedepth h. Using ISM simulations of 46 non-uniform ows witheither constant or variable channel width, the maximum and abso-lute values of Smi =Sfi ratio and S

ti=Sfi ratio were calculated along each

ow in the two sub-sections.

5.1. Coexistence of the three sources of head loss

Fig. 11 presents the maximum absolute values of Smi =Sfi andSti=Sfi ratios for ows with various relative depth h

in (a) the 2-m and 6-m converging reach (Cv2 and Cv6) and in (b) the Flood

Channel Facility with skewed oodplains (see Fig. 2). For theskewed ows, three geometries are investigated: d 5:1, inclinedbanks in the main channel (side slope s 45); d 5:1, verticalbanks s 90; d 9:2; s 45.

According to ISM simulations, the three sources of dissipation,Sm; St , and Sf coexist in these non-prismatic geometries. However,Sm=Sf ratios are generally larger than S

t=Sf ratios. It is also interest-ing to notice that the orders of magnitude of Sm=Sf and S

t=Sf ratiosare comparable in the small-scale ume with converging ood-plains (Cv6 and Cv2) and in the large-scale ume with skewedoodplains.

Indeed, for Cv6 and Cv2, head loss due to mass exchanges Sm isof the same order of magnitude as bed friction Sf in the main chan-nel (Sm 0:3 Sf to 0:85 Sf ), while St can reach 66% of the Sf va-lue in the oodplain. For ows in skewed oodplains, Sm=Sf is inthe range 0.150.9 in the diverging left-hand oodplain and mainchannel, while St=Sf is less than 0.25 in the converging right-handoodplain.

5.2. Effect of angle d in non-prismatic geometry

The effect of the d angle between the main channel axis and theoodplain lateral banks is also highlighted in Fig. 11. Arrows indi-

0.10.20.30.40.50.60.70.80.9

Sm / S

f ()

= 5.1 (inclin. banks) Main channel = 5.1 (vertic. banks) Main channel = 9.2 (inclin. banks) Main channel = 5.1 (inclin. banks) Diverging FP = 5.1 (vertic. banks) Diverging FP = 9.2 (inclin. banks) Diverging FP = 5.1 (inclin. banks) Converging FP = 5.1 (vertic. banks) Converging FP = 9.2 (inclin. banks) Converging FP

= 5.1

= 9.2

Geometries

0.10.20.30.40.50.60.70.80.9

Sm / S

f ()

= 3.8 Main channel = 3.8 Flood plain = 11.3 Main channel = 11.3 Flood plain

= 3.8

= 11.3

Geometries

to t

0.6

12 S. Proust et al. / Advances in Water Resources 33 (2010) 1160 0.1 0.2 0.3 0.4 0.5 0.6 0.700.20.4

t0.81

1.21.41.61.8

2

Sm / S

f ()

Cv6: Main channel Cv6: Converging floodplainDv6: Main channel Dv6: Diverging floodplain

Cv6

Dv6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

St / Sf ()

Fig. 11. Effect of the angle d on head loss due to mass exchange Sm and head loss dueArrows indicate the increase in angle d.S / Sf ()

Fig. 12. Converging oodplains versus diverging oodplains for a given d0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

St / Sf ()

urbulent diffusion St: maximum values of Sm=Sf and St=Sf ratios in the sub-sections.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.10.20.30.40.50.60.70.80.9

1

t

Sm / S

f ()

Main channelDiverging left Flood PlainConverging right Flood Plain

Diverging left flood plain

Converging right flood plain S / Sf ()

angle: maximum values of Sm=Sf and St=Sf ratios in the sub-sections.

cate the increase in angle d. With an increasing d angle, the relativeweight of head loss due to mass exchange Sm increases while therelative weight of head loss due to turbulent exchanges St de-creases. Head loss due to mass exchanges appears to occur at theexpense of head loss associated with turbulent diffusion. In theskewed compound channel, the range of values is multiplied by2 from d 5:19:2.

5.3. Distinguishing converging oodplains and diverging oodplains

It is also interesting to analyze values of Sm=Sf and St=Sf ratios

by distinguishing converging oodplains and diverging oodplains.Fig. 12a shows a comparison between the geometries Cv6 and Dv6(same angle d 3:8), and Fig. 12b separates ratios in the divergingleft-hand oodplain and in the converging right-hand oodplain ofthe skewed compound channel. A clear asymmetry is observed inthe momentum ux between converging and diverging oodplainsin the two gures. Diverging oodplains accentuate head loss dueto mass exchange Sm compared to converging oodplains, whileconverging oodplains accentuate head loss due to turbulent ex-change St compared to diverging oodplains.

This asymmetry is apparent in Fig. 12a, when comparing Dv6and Cv6 (same d angle and relative depth at mid-length h). Theseresults are in accordance with the lateral proles of experimentaldepth-averaged longitudinal velocity Ud presented in [10,11,18].

right interface Uint:r equal to the mean velocity in the right ood-plain Ur . Eqs. (23) and (24) used in the ISM are in agreement withthis experimental data.

5.4. Effect of streamwise variation in ow depth

Flows with constant total width can be compared with owswith varying overall width. Considering the diverging oodplainof Dv6 with d 3:8 (Fig. 12a) and the diverging left-hand ood-plain of the skewed channel with d 5:1, side slope s 45 or90 (Fig. 11b), Sm=Sf ratios were compared. In Dv6, Sm=Sf 20:35;1:9 in the diverging oodplain, while for skewed owsSm=Sf 2 0:14;0:31 in the diverging left-hand oodplain. Conse-quently, with a slightly smaller d angle, the diverging oodplainin Dv6 clearly produces higher values of Sm=Sf ratio than doesthe diverging left-hand oodplain of the skewed compoundchannel.

This demonstrates that, in addition to changes in the geometry,the longitudinal variation in ow depth strongly inuences thehead loss due to mass exchange Sm. The Sm=Sf ratios are more sig-nicant when the ow depth is variable.

5.5. Overview of the head losses in compound channel

All the previous results related to head losses are summed up inTable 2 (column 1). We give an overview of the main processes

e w

rren

t ex

6 0the

6 0han

han

6 0

S. Proust et al. / Advances in Water Resources 33 (2010) 116 13For both geometries Cv6 and Cv2, the derivative along the lateraldirection y, dUd=dy is small in the converging oodplain andmarked in the main channel, while gradient dUd=dy is very strongin diverging oodplain and small in the main channel for geome-tries Dv4 and Dv6.

In Fig. 12b, the same asymmetry in the momentum ux is ob-served for the skewed ows, in accordance with the experimentaldepth-averaged velocity Ud. Indeed, data exposed in [4] show thatexperimental gradients dUd=dy are negligible in the convergingright-hand oodplain, and strong in the diverging left-hand ood-plain, leading to a velocity on the left interface Uint:l very close tomean velocity in the main channel Um, and to a velocity on the

Table 2Overbank ows: (1) main physical phenomena responsible for head losses; (2) relativinterfaces in Eq. (18).

Types of overbank ows (1) Head losses: main processes

Uniform ow Bed friction Interfacial turbulent exchange Momentum ux due to secondary cu

Non-uniform ow in straightgeometry:(a) Far from equilibrium (a) Bed friction(b) Slightly destabilized (b) Bed friction and interfacial turbulen

Skewed oodplains (5.1, 9.2) Bed friction Interfacial turbulent exchange for h

Momentum due to mass exchange inoodplain only, for h 6 0:25

Symmetrically converging oodplains(3.8, 11.2)

Bed friction

Interfacial turbulent exchange for h

Momentum transfer due to mass exc

Abrupt contraction of the oodplain(22)

Bed friction

Momentum transfer due to mass exc

Symmetrically diverging oodplains(3.8, 5.7)

Bed friction

Interfacial turbulent exchange for h Strong momentum transfer due to mass

Nb: relative depth h is measured at mid-length of the prismatic or non-prismatic reachresponsible for head losses depending on the relative depth h

for the various types of geometry investigated. We recall that rel-ative depth h is measured at mid-length of the ume for prismaticgeometries, and at mid-length of the non-prismatic reaches forconverging, diverging and skewed geometries.

6. The role of an explicit modeling of the mass conservationbetween sub-sections

In the water prole equation of ISM, Eq. (18), the terms stem-ming from the mass conservation Mal; Mar , and Mam (see Table

eights of terms of mass conservation and of head loss due to mass exchanges at the

(2) Mass exchange: relative weights ofMai term and Smi

term in Eq. (18)

Nil

ts

(a) Conservationmomentum uxchange for h 6 0:27 (b) Conservation > momentum ux

Conservation > momentum ux:25diverging left

Conservation > momentum ux

:2ge for h 6 0:3

Conservation > momentum ux

ge for h 6 0:23

Conservation momentum ux (for 2 ow cases)

:4 Conservation > momentum ux (for 10 ow cases)

exchange for h 6 0:4

.

1) can play a major role in the evolution of quadrupletfz;Ul;Um;Urg, especially when the ow is far from equilibrium.In this case, the values of St and Sm can be of the same order ofmagnitude than Sf but can have a very little inuence on the owparameters, because the mass conservation termsMai are predom-inant. To understand this phenomenon, two cases are treated indetail: developing ows in a straight compound channel and owsin symmetrically converging oodplains Cv6 and Cv2.

For developing ows in a straight geometry, the dBi=dx termsvanish in Eq. (18). This context accentuates the role of mass con-servation terms Mai in the ow evolution. Fig. 13 presents the lon-gitudinal proles in the oodplain of head loss terms Sm; Sf ; St andof mass conservation terms Mai for two ow congurations pre-sented in Fig. 3b. At the upstream boundary condition (x = 0), theoodplain discharge value Qf exceeds the oodplain discharge un-der uniform ow conditions by +119% and +48% for the CNR owand UCL ow, respectively.

Fig. 13b shows that the CNR ow is controlled by bed frictionand mass conservation along almost the whole length of the reach

fz;Ul;Um;Urg are the bed friction and the mass conservationbetween sub-sections.

2 4 6 8 100.50

0.51

1.5

2

2.5

33.5

4

4.5

5

Slop

e x

1000

()

Floodplain

Bed slope So

Streamwise direction X (m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

St / Ma ()

Sm / M

a (

)

= 3.8 Main channel = 3.8 Converging floodplain = 11.3 Main channel = 11.3 Converging floodplain

14 S. Proust et al. / Advances in Water Resources 33 (2010) 116concerned here. At the downstream boundary, ow is still far fromequilibrium and consequently the value of Ma term differs fromzero. On the contrary, as shown in Fig. 13a, the UCL ow tends touniformity at mid-length of the reach concerned. The reach canbe divided into three parts: an upstream part where the ow ismainly controlled by bed friction and mass conservation betweensub-sections; a downstream part dominated by bed friction andturbulent diffusion at the interface with jSt=Sf 0:15j in the ood-plain; and a region of transition between the two. Consequently, aslong as the ow is far from equilibrium in a straight compoundchannel, the mass conservation has more inuence on the evolu-tion of the discharge distribution and water level than does theinterfacial momentum ux due to turbulence.

It is also interesting to compare the relative weights of massconservation and head losses at the interfaces in converging geom-etries Cv6 and Cv2. Maximum absolute values of St=Ma and Sm=Maratios are shown in Fig. 14 for the 12 ows investigated. The St=Sfand Sm=Sf ratios were shown in Fig. 11a. Comparing the two g-ures, it is clear that the mass conservation terms reduce the effectof head losses at the interfaces. Indeed, the range of Sm=Sf ratios istwice as extended as the range of Sm=Ma ratios, as the St=Ma ratiosare twice lower than the St=Sf ratios. For instance, the weight of S

t

is negligible relative to the Ma term (one order of magnitude low-er), except for one shallow ow, while St and Sf values are of thesame magnitude order for four shallow ows.Fig. 13. Developing ow in straight geometries: friction slope Sf , head loss due to mass eas modeled by the ISM (values in the oodplain).As a result, we can conclude that for numerous ows, the headloss at the interfaces is not negligible but does not inuence thehydraulic parameters. In this case, the quadruplet fz;Ul;Um;Urgis controlled to a large extent by bed frictions and by the mass con-servation between sub-sections. The interest of accurately model-ing the mass conservation with an explicit law (Eq. (15)) is thushighlighted.

Related to mass exchanges at the interfaces, two distinct pro-cesses were distinguished: the mass conservation symbolized byMai terms in Eq. (18); and the head loss (or gain) due to mass ex-changes Sm. The relative weights of these two processes on hydrau-lic parameters are summed up in Table 2, column 2.

7. Discussion

The comparison between experimental data and the numericalresults of the ISM leads to the results listed below:

The two predominant physical phenomena that inuence thewater level and the sub-section-averaged velocities

Fig. 14. Flows in converging geometries Cv6 and Cv2: maximum values of St=Maand Sm=Ma ratios in the sub-sections.xchange Sm , head loss due to turbulent diffusion St , and mass conservation term Ma

depth is constant.

eM 1=2 uM vM wM gzM l A1

ateor turbulent exchange are not negligible compared to bed fric-tion, but their effect on the quadruplet fz;Ul;Um;Urg is not sen-sitive, because of the predominance of mass conservation terms.

8. Conclusions

Using the rst law of thermodynamics applied to a compoundchannel ow leads to an original result: head loss gradient is equalto energy slope on the total cross-section, but head loss gradientdiffers from energy slope in the main channel or the oodplain.

By examining the experimental head in the sub-sections fordeveloping ows in straight compound channel, or ows indiverging or converging geometries, we tested the validity of acommon 1D hypothesis: equal head loss in the main channeland the oodplain. In the vast majority of cases, the head loss gra-dient differs from one sub-section to another, in both contexts ofconstant or variable channel width. However, in the case of ow indiverging or converging oodplains with high relative depthh 0:5, the energy dissipation is low, and assuming equal headloss gradients in the sub-sections is less erroneous than in theother cases. The evolution of experimental sub-section head isvery sensitive to relative depth h and to the angle d betweenthe axis of the main channel and the oodplain lateral walls; thisevolution is also sensitive, to a lesser extent, to the upstream dis-charge distribution.

Head losses are then resolved with the help of the ISM, a 1Dmodel that solves the water surface prole in each sub-section(main channel, left-hand, and right-hand oodplains). Comparingthe numerical results with the experimental data shows that thehead loss due to mass exchange betweens subsections is notablein non-prismatic geometries, but is negligible in straight geome-tries. The head loss due to turbulent exchange is found to be For numerous ow congurations, the head losses due to mass Mass transfers have far more inuence than turbulent diffusionon the interfacial head losses in all the non-prismatic geometriesinvestigated. The contrary is observed for non-uniform ows instraight geometries.

The ow parameters are inuenced by head loss due to massexchange Sm for: (a) ows in diverging geometries for relativedepth h 6 0:4; (b) ows in converging geometries for h 6 0:3,and (c) ows in an abrupt oodplain contraction or in skewedcompound channel for h 6 0:25.

In symmetrical diverging geometries, Sm plays a specic role.This is a gain of head in the oodplains and this enables theoodplain head to remain constant or to increase for some owcases.

The ow parameters are inuenced by head loss due to turbu-lent diffusion St for: (a) developing ows in straight channelapproaching the equilibrium, for relative depth h 6 0:27; (b)ows in diverging geometries for h 6 0:4 ; (c) ows in converg-ing geometries or in skewed compound channel for h 6 0:25.

For a xed angle d between the oodplain lateral walls and theaxis of the main channel, converging oodplains accentuate thevalues of St compared to diverging oodplains, while divergingoodplains accentuate the values of Sm compared to convergingoodplains.

With an increasing d angle, the relative weight of head loss dueto mass exchange Sm increases while the relative weight of headloss due to turbulent exchange St decreases.

For a xed angle d, the head loss due to mass exchange Sm ishigher when the ow depth is variable than when the ow

S. Proust et al. / Advances in Wimportant in both prismatic and non-prismatic geometries, forsmall and medium overbank ows. Results also demonstrate thatthe bed friction and the terms of mass conservation betweenwhere fu;v ;wg are the velocity components within the orthogonalframe fx; y; zg; l is the internal energy per mass unit [J/kg]; and z isthe level of point M with respect to a reference datum.

Considering one-dimensional ow with mass exchange be-tween sub-sections, we assume in the following that w v < uand that v2 u2.

A.1. Equation on the total cross-section

Under steady ow, Eq. (1) becomes on the total cross-sectionareaZZ

AXA1[A2qe~v d~AX ~q

ZZAXA1[A2

pqq~v d~AX

ZZ

AXA1[A2s ~n ~v dAX A2

Assuming that the vertical distribution of pressure is hydro-static, with pM qgh gM (h being the ow depth abovethe bottom, and gM, the elevation of point M above the bottom),the total energy per unit of mass e is written

eM 1=2 uM2 gzM l A3If the shear tress sxx is assumed to be negligible compared to

pressure p, Eq. (A2) is writtenZZAXA1[A2

qu2

2 gz l

~v d~AXZZsub-sections strongly inuences the water level and the sub-sec-tion-averaged velocities fz;Ul;Um;Urg.

These results imply that traditional 1D approaches could fail orshould be used cautiously in solving engineering problems whenrivers present longitudinal changes in the geometry (with anglesas low as 3.8) or/and in the total wetted area. Indeed, classical1D codes do not explicitly model the mass exchange betweensub-sections, which is required in this context to obtain accurateresults on both water depth and discharge distribution.

For eld cases such as oods in valleys with variable width, thecalibration of the coefcients representing the head losses in clas-sical 1D (manning roughness, coefcients of contraction andexpansion) is expected to be tricky, notably for small and mediumoverbank ows h 6 0:3. Erroneous results on both water depthand ow distribution are also expected. The more erroneous re-sults are expected when using a 1D code that does not explicitlymodel the mass conservation between sub-sections, and that doesnot consider both turbulent exchange and momentum ux due tomass exchanges at the interface.

Acknowledgments

Experiments in CNR ume, UCL ume, and LMFA ume werefunded by research programmes PNRH 99-04 and ECCO-PNRH05CV123. D. Bousmar, N. Rivire, and S. Proust travel costs weresupported by the Tournesol programme grant 02947VM, fundedby EGIDE, France, and CGRI, Communaut franaise de Belgique.

Appendix A

The total energy e per unit mass is the sum of macroscopic ki-netic energy, potential energy of the gravity force and of the inter-nal energy per unit mass. Energy e is dened at point M as [15]

2 2 2

r Resources 33 (2010) 116 15AXA1[A2

qgh g~v d~AX ~q 0 A4

The development of Eq. (A4) gives

qddx

aAU3

2

!dx qg d

dxzbedQdx qg ddx hQdx

q ddx

lQdx ~q 0 A5

where a is the Coriolis coefcient on the total cross-section, andzM zbed gM.

Under steady ow, the total discharge Q = AU is constant on thetotal cross-section, and Eq. (A5) is written

qQddx

aU2

2

!dx qgQ dh

dx S0

dx qQ dl

dxdx ~q 0 A6

with S0 dzbed=dx.

Introducing the head loss gradient SHi and the energy slope Sei in onesub-section (Eqs. (7) and (8)), and dividing Eq. (B3) by qgQidxleads to Eq. (9).

References

[1] Knight DW, Demetriou JD. Floodplain and main channel ow interaction. JHydraul Eng, ASCE 1983;109(8):107392.

[2] Shiono K, Knight DW. Turbulent open channel ows with variable depth acrossthe channel. J Fluid Mech 1991;222:61746.

[3] Elliot SCA, Sellin RHJ. SERC ood channel facility: skewed ow experiments. JHydraul Res 1990;28(2):197214.

[4] Sellin RHJ. SERC ood channel facility: experimental data Phase A Skewedoodplain boundaries. Department of Civil Engineering, University of Bristol;1993.

[5] Chlebek J, Knight DW. Observations on ow in channels with skewedoodplains. In: Altinakar, Kokpinar, Aydin, Cokgor, Kirkgoz, editors.

16 S. Proust et al. / Advances in Water Resources 33 (2010) 116By dividing Eq. (A6) by qgQdx, it leads to Eq. (2).

A.2. Equation on a sub-section

Under steady ow conditions, Eq. (1) is written on the sub-section:ZZ

AXA1[A2[Aintqe~v d~AX

~qZZ

AXA1[A2[Aint

pqq~v d~AX

ZZ

AXA1[A2[Aints ~n ~v dAX B1

which develops in:

qddx

aiAiU

3i

2

!dx qg d

dxzbedQidx qg

ddx

hiQidx

q ddx

liQ idx ~q qgqdxzbed h

qqindx U2in2

qqoutdx U2out2

qqlidx sijUinthintdx 0 B2

where subscript i is related to a sub-section i, q qin qout dQi=dx, and where it is also assumed that sxx p in the sub-section.

Eq. (B2) develops in

qQiddx

aiU2i2

!dx qai U

2i

2qin qoutdx qgQi

dhidx

S0

dx

qQidlidx

dx ~q qqindx U2in2

qqoutdx U2out2

sij Uinthintdx 0 B3Proceedings of the fourth international conference on uvial hydraulics, riverow 2008, vol. 1, Cesme-Izmir, Turkey, September 35, 2008. p. 51927.

[6] Shiono K, Muto Y. Complex ow mechanisms in compound meanderingchannels with overbank ow. J Fluid Mech 1998;376:2216.

[7] Islam GMT, Kawahara Y, Tama N. Unsteady ow pattern in a doublymeandering compound channel. In: Altinakar, Kokpinar, Aydin, Cokgor,Kirkgoz, editors. Proceedings of the fourth international conference on uvialhydraulics, river ow 2008, vol. 1, Cesme-Izmir, Turkey, September 35, 2008.p. 499507.

[8] Bousmar D, Rivire N, Proust S, Paquier A, Morel R, Zech Y. Upstream dischargedistribution in compound-channel umes. J Hydraul Eng, ASCE2005;131(5):40812.

[9] Proust S. Ecoulements non-uniformes en lits composs: effets de variations delargeur du lit majeur/Non-uniform ow in compound channel: effect ofvariation in channel width. PhD thesis, INSA Lyon, no. 2005-ISAL-0083, Lyon,France; 2005. 362pp. .

[10] Bousmar D, Wilkin N, Jacquemart JH, Zech Y. Overbank ow in symmetricallynarrowing oodplains. J Hydraul Eng 2004;130(4):30512.

[11] Bousmar D, Proust S, Zech Y. Experiments on the ow in a enlarging compoundchannel. In: Proceedings of the international conference on uvial hydraulics,river ow 2006, Lisbon, Portugal, September 68, 2006. p. 32332.

[12] Proust S, Rivire N, Bousmar D, Paquier A, Zech Y, Morel R. Flow in compoundchannel with abrupt oodplain contraction. J Hydraul Eng, ASCE2006;132(9):95870.

[13] Peltier Y, Proust S, Bourdat A, Thollet F, Rivire N, Paquier A. Physical andnumerical modeling of overbank ows with a groyne on the oodplain. In:Altinakar, Kokpinar, Aydin, Cokgor, Kirkgoz, editors. Proceedings of theinternational conference on uvial hydraulics, river ow, vol. 1, Cesme-Izmir, Turkey, September 35, 2008. p. 44756.

[14] Field WG, Lambert MF, Williams BJ. Energy and momentum in onedimensional open channel ow. J Hydraul Res 1998;36:2942.

[15] Perez JP, Romulus AM. Thermodynamique, fondements etapplications. Paris: Masson; 1993. 564pp..

[16] French RH. Open channel hydraulics. New York: McGraw-Hill; 1985.[17] Bousmar D. Flow modelling in compound channels/momentum transfer

between main channel and prismatic or non-prismatic oodplains. PhDthesis, Unit de Gnie Civil et Environnemental, Universit Catholique deLouvain, Facult des Sciences Appliques; 2002. 306pp.

[18] Proust S, Bousmar D, Rivire N, Paquier A, Zech Y. Non-uniform ow incompound channel: a 1D-method for assessing water level and dischargedistribution. Water Resour Res, in press. doi:10.1029/2009WR008202.

[19] Bousmar D, Zech Y. Momentum transfer for practical ow computation. JHydraul Eng 1999;125(7):696706.

Energy losses in compound open channelsIntroductionEnergy loss, head loss, and momentumTotal cross-sectionSub-section

Head lossesAssumptions for non-uniform flowHead loss gradientsCorrection factors of kinetic energy and momentum

Experimental measurements of headNon-uniform flow in straight compound channelSymmetrically diverging floodplainsSymmetrically converging floodplains

Modeling of sub-section head with coupled 1D equationsEquations of the Independent Sub-sections Method (ISM)Turbulent exchange coefficient and interfacial velocityComparing numerical and experimental sub-section headComparing head loss gradient {S}_{Hi} and energy slope {S}_{ei}

Main processes responsible for head lossesCoexistence of the three sources of head lossEffect of angle \delta in non-prismatic geometryDistinguishing converging floodplains and diverging floodplainsEffect of streamwise variation in flow depthOverview of the head losses in compound channel

The role of an explicit modeling of the mass conservation between sub-sectionsDiscussionConclusionsAcknowledgmentsAppendix AEquation on the total cross-sectionEquation on a sub-section

References