calculation of transverse energy regime in curved channels

Calculation of transverse energy regime in curvedchannels

p

Y.R. Fares*

Department of Civil Engineering, University of Surrey, Guildford, Surrey, GU2 5XH, UK

Received 5 January 1999; received in revised form 6 October 1999; accepted 6 October 1999

Abstract

The development of a quasi two-dimensional model for simulating the transverse and longitudinal energy gradiente�ects on the streamwise ¯ow regime in meandering river channels is reported. The mathematical formulation of the

curved ¯ow is based on the principle of minimum stream power, where the total ¯ow energy rate was divided intotwo components; longitudinal and transverse energy components. The mechanisms by which the energy is dissipatedhas been formulated on the basis of boundary resistance (longitudinal and transverse), internal turbulent friction

and secondary circulation. The resulting increase of the lateral energy dissipation rate in the ¯ow is emphasised. Adirect relationship between the lateral energy dissipation rate and the ensuing excess boundary resistance to ¯ow isproposed. The procedure used in the numerical calculations is described in detail. The model has been applied

successfully to channels with rigid boundaries and found to give accurate predictions for the velocity and shearstress distributions. Finally, the model has been applied to two di�erent case studies representing quasi river channelsituations; a water supply canal and a curved channel with mobile bed. On the basis of the good comparisonsobtained with measurements, it can be concluded that the model can be applied successfully to ¯ow situations in

river systems. # 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Meandering rivers; Secondary circulation; Shear stress; Boundary resistance; Mathematical modelling; Finite di�erence

method

1. Introduction

One of the prime factors for causing the dynamicmorphological changes in river systems is the hydraulic

regime of the river. The ensuing ¯uvial processes inrivers are highly dependent on the patterns of ¯owthat induce these processes and the sediment properties

of the river structure. Because of the strong relation-

ship between the hydraulic conditions in a river and its

formation, many features, such as meander form, bed

topography, bank erosion, lateral migration, do

strongly depend on the dynamics of the turbulent

curved ¯ows. As such, the understanding of curved

¯ow mechanisms, i.e. the three-dimensional spiral ¯ow

pattern associated with strong presence of transverse

circulation induced by centrifugal e�ects due to chan-

nel curvature, has been a subject for many scienti®c in-

vestigations in the past three decades. This is because

such ¯ow mechanisms induce changes to velocities

Computers & Geosciences 26 (2000) 267±276

0098-3004/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.

PII: S0098-3004(99 )00131-4

pCode available at http://www.iamg.org/CGEditor/

index.htm

* Tel.: +44-1483-876629; fax: +44-1483-450984.

E-mail address: [email protected] (Y.R. Fares).

and, hence, to boundary shear stresses which drive

sediment transport in rivers.The determination of the hydraulic geometry of

river channels may, however, be accomplished if the

applicable physical relations are su�cient to describethe degrees-of-freedom in channel morphology, such as

channel width, ¯ow depth, bed and bank slope, andslope of channel banks. Classical investigations on¯ow regime in meandering rivers were con®ned to one-

dimensional simulation models. These types of modelswere ultimately formulated for large river situations,or for situations where the length scale is of the order

of several hundreds of miles. Despite the fact thatthese one-dimensional models simplify the relationship

between the river ¯ow and its induced sedimentmotion, in addition to being relatively straightforwardto handle the data ®les and computational power

requirements, they lack proper evaluation of the lateralcirculation regime. Hence, the lateral energy dissipa-tion rate, or lateral resistance, have always been

accounted for by empirical coe�cients. More recently,quasi two-dimensional models have employed the con-

cept of minimum stream power. The formulation ofsuch models is based on dividing the total energy of¯ow into two components: one is in the longitudinal

(main) direction, and the other is in the transverse (sec-ondary) direction. Each of these components is calcu-lated on the basis of the balance of forces that drive

the ¯ow around the curved path of the channel. Assuch, the dissipation in ¯ow energy is calculated not

only from boundary drag and turbulent shear stresses,but also from the e�ect of transverse circulation. Dueto the nature of their formulation, these models ignore

any lateral heterogeneity of main ¯ow velocities, aswell as of water surface pro®les.In rivers of long courses, however, the use of two

(or higher) dimensional models may not be acceptablefrom the engineering practice viewpoint, especially in

the size of ®eld work involved in collecting the necess-ary river information for the execution of the compu-tational model and hence, they are normally

considered to be time consuming and uneconomical. Itis, therefore, essential to develop models that are rigor-ous in their formulation and yet practical in simulating

accurately the streamwise and crosswise variations ofsuch river regimes.

The object of this paper is to present a numericalmodel capable of predicting the e�ect of longitudinaland transverse energy gradients on ¯ow regime

changes in meandering channels of arbitrary platformand geometry. A distinct feature of the model formu-

lation will be the inclusion of energy dissipation rates,longitudinal and transverse, resulting from the e�ectsof boundary drag, internal turbulent friction and sec-

ondary circulation in the ¯ow. The variations of watersurface and velocity pro®les on the curved-¯ow lateral

energy regime will also be included. The numericalprocedure involved in the calculations will be reported

in detail. A direct relationship between the excess lat-eral energy dissipation rate and the correspondingchange in channel resistance will be proposed. The ver-

i®cation of the model will be carried out through com-parisons with experimental data reported in theliterature for cases of rigid bed curved channels (Ippen

and Drinker, 1961; Yen, 1965; Tamai et al., 1983).Finally, the model will be applied to two quasi-naturalriver situations for simulating the streamwise distri-

bution of boundary shear stresses. The ®rst represents¯ow in a curved open channel (Chang, 1983), and thesecond represents ¯ow in a curved channel of a mobilechannel bed (Nouh and Townsend, 1979). In general,

good comparisons between the model predictions andmeasurements were obtained.

2. Mathematical formulation

The formulation of the model follows, but in a morerealistic form, the basic principles of that reported ear-lier by Chang (1983, 1984b, 1992). The ¯ow pattern isassumed to be steady, subcritical and fully developed.

The rate of bend ¯ow energy gradient E per unitlength of the channel can be divided into two com-ponents: one in the longitudinal s-direction E ', and

one in the lateral r-direction E0, i.e.

E ��E 0

E 00

��rgQS 0

rgQS 00

��1�

where r is the ¯ow density, Q is the ¯ow discharge, S 'and S0 are the longitudinal and lateral components of

the total energy gradient S, respectively, and g is accel-eration due to gravity. The longitudinal energy gradi-ent S ', is associated primarily with boundary drag,

hence, can be calculated directly from any ¯ow resist-ance equation, e.g. Manning's or Darcy Weisbach'sequation, as follows

S 0 � n2U 2m

H 2r

� U 2mf

8gHr

�2�

where n is the coe�cient for bed roughness, f is thefriction factor, Um is the mean sectional velocity, and

Hr is the hydraulic radius of ¯ow. The determinationof the transverse energy gradient S0 is slightly morecomplicated than S ' since it is associated with lateral

boundary shear, internal turbulent friction and second-ary ¯ow. Accordingly, the balance of forces in the lat-eral direction has to be considered. As previously

reported (Yen, 1965; Varshney and Garde, 1975; Kik-kawa et al., 1976), the ``growth-decay'' cycle of the sec-ondary ¯ow along curved channels is a direct result of

Y.R. Fares / Computers & Geosciences 26 (2000) 267±276268

the balance between convective, centrifugal, pressuregradient and frictional forces. By considering the

momentum equation in the radial direction, the forcesacting upon a ¯uid element (dr, ds, dz ) can beexpressed as follows:

V�r � V� � u2

rÿ gSr � @

@z

�n@v

@z

��3�

where r, s, z are the co-ordinates denoting lateral,longitudinal and vertical directions, respectively,H=(@/@r, @/@s, @/@z ) is the spatial gradient operator,

V=(v, u, w ) is the ¯ow velocity ®eld in the r-, s-, z-directions, respectively, Sr is the lateral surface (press-ure) gradient, and n is the kinematic momentumexchange coe�cient. The non-linear convective terms

of the inertia force on the left-hand-side of Eq. (3), v@v/@r and w @v/@z, were excluded for being much smal-ler than u @v/@s. This simpli®cation is expected to limit

the model applicability to cases of gently curved chan-nels, where lateral momentum exchange at, or near,bank regions has a minor e�ect on the overall ¯ow dis-

tribution in comparison to that in the large middleregion of the channel cross section. Nevertheless, aswill be discussed later, it has also proven to be appli-cable to strongly curved ¯ow cases. The surface gradi-

ent Sr, is calculated from (Yen, 1965; Odgaard, 1981;Chang, 1992):

Sr � @h

@r1b

U 2m

gr�4�

where h is the local ¯ow depth, r is the local bendradius, and b is the main ¯ow convection factor. With

respect to the shear stress term (@/@z )(n @v/@z ) in Eq.(4), it can be expressed by the surface velocity andchannel roughness as (Ippen and Drinker, 1961; Kik-

kawa et al., 1976; Odgaard, 1989):

@

@z

�n@v

@z

�� ÿ2vs�s�ku�

h�5�

where vs is the radial surface velocity, k is the Von-Karman constant (=0.41), and u� is the shear velocity.In Eq. (5), the vertical pro®le of lateral velocities v(z )may be assumed to vary linearly with depth, hence,

vs � 2hv�z��z

hÿ 1

2

�for ÿ 1

2R z

hR1

2�6�

where v(z ) is the local lateral velocity at any z co-ordi-

nate. As evident from both laboratory and ®eldmeasurements, the distribution of v(z ) is better rep-resented exponentially rather than by the assumed lin-

ear pro®le (Yen, 1965; Falcon-Ascanio and Kennedy,1983; Tamai et al., 1983). However, in addition to thefact that the assumption presented herein will certainly

simplify the analytical treatment of the problem, it haspreviously been used for fully-developed curved ¯ows

and has found to produce satisfactory results(Odgaard, 1981). The assumed linear approximationfor the v(z ) pro®le will therefore be used. Furthermore,

the vertical distribution of the momentum exchangecoe�cient n(z ) is assumed to follow the classic para-bolic pro®le as previously employed in many investi-

gations; see, for example, Kikkawa et al. (1976) andTamai et al. (1983). By substituting Eqs. (4) and (5)into Eq. (3), the streamwise variation of vs(s ) takes the

form:

@vs

@s� F1� f �vs � F2�h, f �

�Um

rm

��7�

where

F1� f � ��f

2

r "C1 ÿ C2

k

��f

2

r #�8�

F2�h, f � � kh

�d

d� 1

� ��f

2

rwith d � k

��8

f

s: �9�

Here, the terms F1( f ) and F2(h, f ) are analytical func-

tions of friction factor and depth, d is a coe�cient thatdepends on the friction factor and Von-Karman con-stant, and C1 and C2 are parametric constants. Eq. (7)

is a ®rst-order, non-homogeneous ordinary di�erentialequation of vs, for which an exact analytical solutioncan be determined as follows:

vs � eÿ�F1 ds

��F2

�Um

rm

�e

�F1 ds ds� constant

�: �10�

The boundary condition required for its solution canbe obtained from the condition of vs at the entry sec-

tion to the river bend channel. Once the streamwisepro®le of lateral ¯ow ®eld is calculated, the curvaturee�ect on the main ¯ow regime and boundary resistance

can then be obtained.

2.1. Formulation of transverse energy gradient

As mentioned earlier, the excess lateral energy dissi-pation in curved ¯ow is mainly attributed to the e�ectof secondary ¯ow associated with boundary shear. The

determination of the spatial variation of secondary¯ow would allow the changes occurring to the bound-ary resistance and hence, to the velocity and shear

stress to be quanti®ed. The depth-averaged lateralenergy gradient S0 can be expressed as through therate of work done by the induced inertia force; it is

Y.R. Fares / Computers & Geosciences 26 (2000) 267±276 269

E 00 � rgQS 00 ��h0

F 00 � VA dz �11�

where F0 is the force (inertia) vector induced in the

¯ow, and VA is the velocity vector across the channelwidth. For wide river channels, the work done by ver-tical forces is normally much smaller than that doneby lateral forces, so it is ignored (Chang, 1984b, 1992).

In this case, the ¯ow becomes horizontally dominant,and the velocity ®eld becomes essentially two-dimen-sional (u, v ). Hence, only the vertical pro®les of lateral

v(z ) and longitudinal u(z ) velocities would be neededto derive a direct relationship between S0 and vs. Withrespect to the pro®le of u(z ), the power-law equation

is assumed (Falcon-Ascanio and Kennedy, 1983;Odgaard, 1989), i.e.

u�z� � 1� dd

�z

h

�1=d

Um: �12�

By substituting the value of u(z ) from Eq. (12) and theanalytical solution of vs(s ), from Eq. (10), into Eq.(11), an explicit expression for S0 can be found (Fares

and Koefman, 1994); it is

S 00 � Rbc

�h

rm

�2

F 2r

j vs jUm

�13�

Rbc � 1� dd�2� d� and Fr � Um��

ghp �14�

where Fr is the Froude number, Rbc is the bend resist-ance coe�cient, and rm is the mid bend radius. Eq.(13) gives the spatial variation of S0 as a function of

boundary resistance, bend ¯ow geometry and second-ary ¯ow strength expressed by its surface velocity vs(s ).From Eq. (13) it can be deduced that S0 increases with

decreasing roughness Rbc and with increasing h/rmratio, Froude number Fr and lateral surface velocityvs(s ). Therefore, the lateral energy dissipation incurved ¯ows becomes important for situations with

smooth channels and of high depth-to-mid-radiusratios, i.e. for ¯ows of high aspect ratios (B/h < 6,where B is the river surface width) and of low river

tightness ratios (rm/B < 2). Similar conclusions werereportedly obtained by Tamai et al. (1983), Chang(1984a) and Odgaard (1989).

3. Calculation procedure

The method of calculation is divided into two main

stages. In the ®rst stage, an approximate ¯ow surfacepro®le is computed assuming the river channel to havea straight alignment, i.e. the river curvature e�ects on

the ¯ow regime are ignored. This is followed in the sec-ond stage by introducing the second ¯ow features in

the simulation process and modifying the resulting¯ow pattern in accordance with the continuity prin-ciple requirements. The ®nite di�erence approximation

method has been employed in the computations andhas been found to be reliable in producing accuratepredictions.

In the procedure, the river channel is discretized bya step size Ds. By using the standard step method, thestreamwise water surface pro®le is calculated assuming

the channel is straight, i.e. vs(s )=0 and S0=0. In thissituation, the calculations start at the downstream endof the river reach under consideration and proceedtowards the upstream direction. In explicit ®nite-di�er-

ence form, the equations read:

E i � S 0 i�1=2Ds � E i�1 � hi�1=2L �15�

where

Ds � si ÿ si�1 �16�

S 0 i�1=2 � 12 �S 0 i � S 0 i�1� �17�

hi�1=2L � 12 �hiL � hi�1L � �18�

El � hl � a�Um�2l2g

with l � i, i� 1: �19�

Here, hL is the energy head loss and a is the energy

correction factor (=1.3). The determination of an ap-proximate surface pro®le was an important step in thecomputations, as it was essential in calculating the sur-

face lateral velocities (as given next in the second stageof the procedure). Secondly, an optimisation operationis introduced to correct the approximate surface pro-

®les once the lateral surface velocity vs(s ) and trans-verse energy gradient S0(s ) were calculated. For this,the computations started at the upstream section of

channel and proceeded towards downstream. In a®nite-di�erence form, vs(s ) is expressed as:

vi�1s � vis eÿDsFi�1=21 � Fi�1=2

2

U i�1=2m

rm

Ds �20�

where

Fi�1=2l � 1

2 �Fi�1l � Fi

l� with l � 1, 2: �21�

Once the lateral velocity vs is calculated, the correctedlongitudinal and lateral energy gradients are deter-

mined, from which a modi®ed value for the roughnesscoe�cient is assigned. This process completes a one``double-sweep'' iteration of computations. To satisfy


the continuity principle, the computed depth and vel-ocity values were used to calculate the discharge at

each Ds step. The discharge values are then checkedagainst the actual (given) discharge value. If the di�er-ence was found to be r0.01%, then the ¯ow depth is

optimised and the whole ``double-sweep'' calculationsare repeated until the speci®ed accuracy is ®nally re-alised. Following many preliminary test runs, it was

determined that a step length Ds=h/4 was su�cientfor the computed results to converge rapidly and toreach consistency. The numerical algorithm and the

source code used for the computations are given indetail in Fares (1996).

4. Applications and discussion of results

The deliverables from the numerical model are thestreamwise pro®les of ¯ow depth h(s ) and horizontal

velocity ®eld u(s ) and vs(s ), from which the energy gra-dients S ', S0, and boundary shear stresses are calcu-lated. For veri®cation purposes, the model has been

tested against experimental data collected from labora-tory ¯umes. On the basis of its reasonable perform-ance, the model has further been applied to two

di�erent situations representing quasi-natural riverchannels. These situations are: ¯ow in a rigid-bedcurved canal; and ¯ow in a channel bend with amobile bed.

4.1. Flow in a laboratory ¯ume

The veri®cation of the model for examples of rigidbed channels were reported earlier using experimentaldata reported in the literature (Fares and Koefman,

1994; Fares and Spence, 1995). As a sample of theresults, Figs. 1±3 show comparisons between the pre-dicted and measured pro®les for lateral velocities,

longitudinal velocities and shear stresses, respectively.The hydraulic conditions and the channel properties

pertaining to the ®gures are given in Table 1. Also

shown in Figs. 1 and 3 are the numerical predictions,as obtained by Chang's model (1992), for the lateralvelocities vs/Um and the boundary shear stresses, re-

spectively. As can be seen in the ®gures, the longitudi-nal pro®les of the lateral, longitudinal and boundaryshear stresses are strongly inter-related. Upon entering

the channel bend, the rapid development of the sec-ondary ¯ow, associated with the lateral pressure gradi-ent (superelevation at water surface), causes anincrease in ¯ow resistance along the channel path. This

increase of resistance gives rise to the lateral dissipa-tion rate, expressed by the lateral energy gradient S0,and as a consequence to the total energy gradient S.

As such, the boundary shear stresses, which dependstrongly on S, change along the channel in accordancewith that of the lateral S0, and longitudinal S ' energygradients. For a uniform longitudinal gradient S ', thetotal energy gradient S varies solely with that of thelateral energy dissipation rate S0. In general, the modeltends to perform better for smooth and gently curved

channels than for rough and strongly curved channels(Fares and Koefman, 1994). This is because in roughchannels, energy is dissipated mainly by the boundary

drag, and to a lesser extent, by the presence of the sec-ondary ¯ow. The excess resistance to (or energy dissi-pation in) the ¯ow is, therefore, directly related to the

Fig. 2. Longitudinal velocities Ð comparison with Tamai et

al. (1983) experimental data.

Fig. 3. Bed shear stresses Ð comparisons with Ippen and

Drinker (1961) experimental data (points) and Chang (1992)

model predictions (dotted line).

Fig. 1. Lateral surface velocities Ð comparisons with Yen

(1965) experimental data (points) and Chang (1992) model

predictions (dotted line).


continual increase in the transverse ¯ow strength.Maximum resistance normally occurs at the fully-

developed stage of the secondary ¯ow, beyond which itbecomes constant until the bend exits, or until changesto the channel curvature take place.

It should be mentioned that the computed longitudi-nal pro®les of depth-averaged velocities in Fig. 2 areslightly under-predicted. This is found to be associated

with over-predicted depth pro®les (see Koefman,1994), as would be expected in satisfying the continuityprinciple. The consistent di�erence between the com-

puted and the measured pro®les may result from thestrong curvature of the channel (rm/B = 2) ac-companied by short curved reaches of the channel. Inthis situation, the resulting outward shift of momen-

tum causes high lateral and longitudinal velocitiesalong the channel, which accounted for the persistentderivations of the computed pro®les from the measure-

ments. Nevertheless, a similar trend can clearly be seenbetween the computed and observed pro®les.As can also be seen in Figs. 1 and 3, the model pre-

dictions are generally more realistic than those pro-duced by Chang's model. Detailed quantitativecomparisons between this study's predictions and those

predicted by Chang's model are reported earlier inFares and Koefman (1993). On the basis of its reason-able performance in the rigid bed channel situation,the model has further been applied to quasi-natural

river situations, as described in the following two sec-tions.

4.2. Flow in a rigid bed canal

The case study selected for this purpose is takenfrom an investigation carried out on channel bends of

the Poudre Supply Canal near Ft Collins, Colorado,USA (Chang, 1983). The canal is made of concrete

and has a trapezoidal cross-section with bank sideslopes of 1.25:1. Two bend sections along the canalwere considered: bend 6 of central angle yb=73.28,rm=43.66 m; and bend 7 of yb=48.68 andrm=87.33 m. The ¯ow conditions of bends 6 and 7 aregiven in Table 2, while the complete set of compu-

tational experiments are reported in Spence (1994). Asa sample of the results, Fig. 4a and b shows that lat-eral velocity vs/Um and energy S0/S ratios increase

with (h/rm)2. As can be seen in the ®gure, the increase

in vs/Um and S0/S pro®les depends strongly on the h/rm ratio. For h/rm < 0.024, the velocity ratio vs/Um

exceeds that of S0/S. For ratios h/rm r 0.024, S0/Sincreases at a higher rate than that of vs/Um. This isbecause in curved river ¯ows, energy is dissipated pri-marily by boundary friction and to a lesser extent by

secondary ¯ow. Therefore, despite the presence of sec-ondary currents in the ¯ow, their e�ect on energy dissi-pation at shallow depths is relatively minor. The

higher the ratio h/rm the more the lateral dissipation ofenergy S0, and, hence, the more is the excess resistance.This is shown in Fig. 5 by the percentage increase in

the friction factor for di�erent values of the lateralenergy dissipation ratio S0/S. As can be deduced fromthe nearly linear relationship in Fig. 5, an expressionbetween the increase in f and S0/S can be written as:�fb ÿ f

f

�% �

�Ze

S 00

S

�� 100 �22�

where fb is the friction factor in the bend channel, Ze isthe bend energy coe�cient (r1) depends on ¯ow con-dition, channel geometry and planform, and bedroughness. For design purposes, Eq. (22) provides a

Table 1

Experimental data of rigid-bed curved channels

Channel properties and hydraulic conditions

(1)

Ippen and Drinker

(1961) (2)

Yen

(1965) (3)

Tamai et al.

(1983) (4)

Number of channel bends 1 2 10

Bend central angle Y (8) 60 90 90

Bend mean radius rm (m) 1.524 8.534 0.6

Channel bed width B (m) 0.61 1.83 0.3

Straight reach downstream of (or between) two bends (m) 1.98 4.27 0.3

Manning's coe�cient n 0.009 0.0103 0.013

Bed channel slope S0 (m/m) 0.00064 0.00072 0.001

Bank side slope (m/m) 2:1 (trapezoidal) 1:1 (trapezoidal) 0:1 (rectangular)

Upstream water depth h (m) 0.076 0.156 0.0293

Flow discharge Q (m3/s) 0.024 0.048 0.002

Boundary shear stress (N/m2) 0.382 0.963 0.240

Froude number Fr 0.53 0.58 0.42

Depth/bend radius ratio h/rm 0.05 0.018 0.049


guide for estimating the roughness values in calculatingthe shear stress ®eld in meandering rivers.

4.3. Flow in a curved channel with mobile bed

In this case, the model predictions were compared

with the experimental results obtained from a studyinto the distributions of boundary shear stresses in

bend channels with mobile beds (Nouh and Townsend,1979). In their experiments, two bend channels of 45and 608 central angles were tested under the same ¯ow

condition (i.e. Fr=0.49, B/h=7.5) and the same tight-ness ratio (rm/B = 3). The bed material used in thetests was coarse sand (D50=0.7 mm). A Laser Doppler

Anemometer (LDA) was used in their tests in order tomeasure the instantaneous point velocities and turbu-lence characteristics, from which the boundary shearstresses were determined. In applying the numerical

model to the study, it was necessary to estimate theaverage bed slope and coe�cient of channel roughness,since no data were given concerning these variables.

With respect to the bed slope, an average value wasestimated from the contour lines of bed topographyalong the channel, and subsequently was fed to the nu-

merical code. The coe�cient of bed roughness was cal-culated from the resistance equation, Eq. (3), and waschecked against typical roughness values for channelsof similar bed slope and sediment properties. The

hydraulic conditions used in the computational predic-

Table 2

Field data of rigid-bed curved canal

Channel properties and hydraulic conditions (1) Chang (1983)

Bend 6 (2) Bend 7 (3)

Bend central angle Y (8) 73.2 48.6

Bend mean radius rm (m) 43.66 87.33

Channel bed width B (m) 3.66 3.66

Straight reach downstream of bend (m) 18.29 18.29

Manning's coe�cient n 0.013 0.013

Bed channel slope S0 (m/m) 0.0013 0.0013

Bank side slope (m/m) 1.25:1 1.25:1

Upstream water depth h (m) 0.97±1.98 0.67±1.98

Flow discharge Q (m3/s) 8.6±32.4 5.5±36.7

Boundary shear stress (N/m2) 8.92±22.69 6.62±22.69

Froude number Fr 0.62±0.73 0.76±0.85

Depth/bend radius ratio h/rm 0.021±0.045 0.01±0.023

Fig. 4. (a) Comparisons between predicted and measured lat-

eral surface velocities (vs/Um) and energy (S0/S ) pro®les with

(h/rm)2 for bend 6, yb=73.28 (Chang, 1983). (b) Comparisons

between predicted and measured lateral surface velocities (vs/

Um) and energy (S0/S ) pro®les with (h/rm)2 for bend 7,

yb=48.68 (Chang, 1983).Fig. 5. Relationship between predicted increase in the friction

factor and transverse energy ratio (S0/S ).


tions are given in Table 3. Fig. 6a and b shows thecomparisons between the predicted and measured bed

shear stress pro®les for the two bend cases, at the cen-

treline of the channel. In Fig. 6a, b, the shear stress

values are normalised by the uniform shear stress

(rghS ') for the same ¯ow condition in an in®nitely

wide rectangular channel.

The e�ect of secondary ¯ow can be noted from the

continual increase of stresses along the channel path.

In the downstream region of the bend, reduction in

shear stresses can be observed. The increase in shear

stress along the bend is strongly associated with the

development of secondary ¯ow which causes lateral

energy dissipation in ¯ow. This, in turn, increases the

shear stress along the bend. However, because of the

short length of the bends tested (yb=45 and 608), thesecondary ¯ow did not reach its fully developed stage,

and, hence, continued to give rise to shear stresses

until the bend exit. Beyond the exit, the shear stresses

decrease as the secondary ¯ow starts to decay. As can

be seen in Fig. 6a, b, there is an agreement in the gen-

eral trend of both measured and computed pro®les.

The noticeable discrepancy between the observed and

predicted pro®les is mainly due to the uncertainty con-

cerning the roughness and bed slope values of the

channel that had to be estimated. Furthermore,

because of the short curved path of the channel, the

secondary ¯ow and its associated resistance are maxi-

mum at the bend exit. Beyond the exit, the secondary

¯ow persists at higher values for a considerable dis-

tance in the downstream tangent. The rate of second-

ary ¯ow decay is related to channel roughness and

inversely related to ¯ow depth. As can be seen in Fig.

6a, b, the rate at which the observed stresses decay

downstream of the bend is much faster than that of

the predicted ones. This can be attributed to the shal-

low depths observed in the channel as a result of the

sediment build up in this region.

Table 3

Experimental data of bend channel with mobile bed

Channel properties and hydraulic conditions (1) Nouh and Townsend (1979)

458 Bend (2) 608 Bend (3)

Bend central angle Y (8) 45 60

Bend mean radius rm (m) 0.90 0.90

Channel bed width B (m) 0.30 0.30

Straight reach downstream of bend (m) 2.0 2.0

Manning's coe�cient n 0.0117 0.0117

Bed channel slope S0 (m/m) 0.001 0.001

Bank side slope (m/m) 1.25:1 1.25:1

Upstream water depth h (m) 0.04 0.04

Flow discharge Q (m3/s) 0.0037 0.0037

Boundary shear stress (N/m2) 0.321 0.321

Froude number Fr 0.49 0.49

Depth/mean radius ratio h/rm 0.044 0.044

Fig. 6. (a) Comparison between predicted and measured bed

shear stress in channel bend with mobile bed, for yb=458(Nouh and Townsend, 1979). (b) Comparison between pre-

dicted and measured bed shear stress in channel bend with

mobile bed, for yb=608 (Nouh and Townsend, 1979).


5. Practical signi®cance to river applications

Because of the important e�ects of transverse ¯ow

on the overall resistance and boundary shear ®elds, themodel provides a useful and practical tool for investi-gations of ¯ow patterns in meandering rivers. The gen-

eral trend of agreement between the computed dataand measurements suggests that the geometric regu-larities of river meanders is related to the pattern of

¯ow development in the river. Application of themodel to river situations may certainly result in vari-ations between the predictions and the ®eld measure-ments because of the inevitable local disturbances that

exist in the river course, such as vegetation, variationsin sediments and debris. Nevertheless, the methodspresented here should be used as a ®rst approximation

in describing, qualitatively, the morphological proper-ties in meandering rivers. For example, establishing thecondition at which maximum equilibrium curvature is

attained in a river can be carried out using the energyapproach for the spatial transverse ¯ow, as previouslydiscussed by Chang (1984a). Furthermore, the model-ling of streamwise variation in bed topography in a

river channel requires such a technique for calculatingthe transverse ¯ow distribution along the curved pathof the river (Fares and Lewis, 1998).

6. Conclusions

A numerical model capable of predicting the ¯owcharacteristics in river bends with rigid and mobile

beds is capable of simulating the streamwise variationsof secondary ¯ow, boundary resistance and shear stres-ses along channel bends of arbitrary planform andgeometry. The interaction between longitudinal and

lateral energy gradients is simulated through the con-cept of stream power, from which an expression forthe lateral energy gradient is derived. A direct relation-

ship between the lateral energy dissipation rate andboundary resistance is proposed. The ®nite-di�erencemethod employed in the calculations is e�cient in pro-

ducing consistency as well as rapid convergence in thecomputations. The comparisons between the modelpredictions and measured data for examples of curvedchannels with rigid beds are satisfactory. This is true

for the estimation of boundary resistance, as well asfor the velocity and shear stress distributions. Forexamples of channels with mobile beds, however, the

predictions are less satisfactory. This is mainly attribu-ted to the inaccuracies of estimating the bed slope androughness coe�cient of the channel. In general, the

model has been successful in predicting, qualitatively,the excess ensuing shear stress distribution and bound-ary resistance in the curved channels with moveable

boundaries. On the basis of the satisfactory perform-ance of the model, it may be concluded that the model

can successfully be applied to river ¯ow situations.

Acknowledgements

This study is part of a research programme into the¯ow mechanisms of natural streams at the Department

of Civil Engineering, University of Surrey, UK. Thenumerical experiments were carried out using the Uni-versity's central HP-UNIX computer service. Some of

the numerical experiments were carried out by L. M.Koefman and G. J. Spence as part of their MEng.research projects.

References

Chang, H.H., 1983. Energy expenditure in curved open chan-

nels. Hydraulics Division of the American Society of Civil

Engineering 109 (7), 1012±1022.

Chang, H.H., 1984a. Regular meander path model.

Hydraulics Division of the American Society of Civil

Engineering 110 (10), 1398±1411.

Chang, H.H., 1984b. Variation of ¯ow resistance through

curved channels. Hydraulics Division of the American

Society of Civil Engineering 110 (10), 1772±1782.

Chang, H.H., 1992. Fluvial Processes in River Engineering,

1st ed. Krieger Publishing Co, Florida (423 pp).

Falcon-Ascanio, M., Kennedy, J.F., 1983. Flow in alluvial-

river curves. Fluid Mechanics 133, 1±16.

Fares, Y.R., 1996. The TRENGIC Program Ð A

FORTRAN code for simulating ¯ow mechanisms and

¯ow resistance in curved open channels. University of

Surrey, Guildford, Surrey, UK (12 pp).

Fares, Y.R., Koefman, L.M., 1994. Lateral distribution of

energy dissipation in mildly-curved channels. In:

Proceedings of the International Conference on

Hydrodynamics, Wuxi, Jiangsu, China, pp. 315±322.

Fares, Y.R., Spence, G.J., 1995. Spatial variation of ¯ow re-

sistance in meandering channels. In: Proceedings of the

2nd International Conference on Hydroscience and

Engineering, Beijing, China, pp. 230±238.

Fares, Y.R., Lewis, M.R.S., 1998. Simulation of bed level

changes in erodable channels. In: Proceedings of the 3rd

International Conference on Hydrodynamics, Seoul,

Korea, pp. 307±313.

Ippen, A.T., Drinker, P.A., 1961. Boundary shear stress in

curved trapezoidal channels. Hydraulics Division of the

American Society of Civil Engineering 88 (HY5), 143±179.

Kikkawa, H., Ikeda, S., Kitagawa, A., 1976. Flow and bed

topography in curved open channels. Hydraulics Division

of the American Society of Civil Engineering 102 (HY9),

1327±1342.

Koefman, L.M., 1993. Flow resistance and energy dissipation

in curved channels. MEng. thesis, University of Surrey,

Guildford, Surrey, UK (119 pp).

Nouh, M.A., Townsend, R.D., 1979. Shear-stress distribution


in stable channel bends. Hydraulics Division of the

American Society of Civil Engineering 105 (HY10), 1233±

1245.

Odgaard, A.J., 1981. Transverse bed slope in alluvial channel

bends. Hydraulics Division of the American Society of

Civil Engineering 107 (HY12), 1677±1694.

Odgaard, A.J., 1989. River-meander model. I: Development.

Hydraulic Engineering Division of the American Society

of Civil Engineering 115 (11), 1433±1450.

Spence, G.J., 1994. Determination of ¯ow energy expenditure

in a curved channel. MEng. thesis, University of Surrey,

Guildford, Surrey, UK (107 pp).

Tamai, N., Ikeuchi, K., Yamazaki, A., Mohamed, A.A.,

1983. Experimental analysis on the open channel ¯ow in

rectangular continuous bends. Hydroscience and

Hydraulic Engineering 1 (2), 17±31.

Varshney, D.V., Garde, R.J., 1975. Shear distribution in

bends in rectangular channels. Hydraulics Division of the

American Society of Civil Engineering 101 (HY8), 1053±

1066.

Yen, B.C., 1965. Characteristics of subcritical ¯ow in a mean-

dering channel. Institute of Hydraulic Research, The

University of Iowa, Iowa (155 pp).


calculation of transverse energy regime in curved channels

Documents