surface irrigation

BOUNDARY SHEAR IN CURVED CHANNEL

WITH SIDE OVERFLOW

By Y. Ramsis Fares)

ABSTRACT: The characteristic changes in the boundary shear stress field of a channelbend at the intersection with a side overflow are investigated in this paper. On the basisof a field survey on bottom topography changes of the meandering river Allan Water atthe cut-off section, a detailed study of boundary shear stresses in an idealized rigid bedmodel was undertaken. Both mathematical and experimental approaches were employedin the idealized model study. The analysis of results showed that continual reductions inshear stresses occurred in the bend at the side overflow region. The maximum reductionof shear stresses was 37% in cases of low side overflows and 82% in cases of high sideoverflows. These reductions were attributed to the development of stagnation and separation zones at the intersection associated with strong lateral outward currents. Basedon the idealized model study, the features observed in the bed topography of Allan Waterat the cutoff section are most likely to develop in cases of combined bend and high sideoverflows.

INTRODUCTION

Flood relief channel's schemes are commonly used to control high flows in natural riversystems. They develop naturally as well as artificially. Natural flood relief channels (known ascutoff channels) develop as the river adjusts to its course during high flood periods. Along ameandering river, zones of riffles and pools are continually formed at river banks as a result ofthe helical motion induced in the flow by the channel curvature. If systematic variations in theerosional and depositional activities occur, the river develops a highly irregular plan form.Depending on the strength of the spiral flow and sediment properties of the channel structure,the meander becomes overgrown and cutoff occurs. Thereafter, normal flows proceed along theriver's meander path and flood flows spill across the neck of the meander loop into the cutoffchannel. The time taken for the meander to reach this critical (cutoff) stage may vary from 2to 10 yr (Gagliano and Howard 1983).

The second type of flood relief channel's scheme is artificial. These are usually constructedat critical locations along the river to avoid long-term destabilization in the river regime. Aspart of a fundamental study of flow mechanisms in meandering rivers during flood periods,using flood relief channel's schemes, a field study was carried out on a meander loop/neck cutoffintersection on the Allan Water, Perthshire, Scotland (Herbertson and Fares 1991). The cutoffwas constructed on the neck of the meander loop in September 1986. At the cutoff section ofthe river, the Allan Water is 15 m wide, drains a catchment area of 210 km2

, and has an averageannual flow of 6 mO/s. The cutoff channel is about 15 m wide, and 20 m long, and has a bedlevel approximately 1 m above the river bed level. The position of the cutoff and the overallgeometry of the meander loop are shown in Fig. 1. Measurements of bed topography profilesin the river were recorded at three sections namely (see Fig. 2): upstream (section A-A), themiddle (section B-B), and downstream (section C-C) of the upstream intersection between theflood channel and the river, from 1987-1990, after the installation of the cutoff. [Figs. I and 2represent profiles after the Herbertson and Fares (1991) study.]

The field investigation carried out on Allan Water showed the partial neck cutoff of a meanderloop results in a marked change in the bed topography from that usually found in a meanderingchannel. The traditional riffle/pool bed profile was completely destroyed, and the followingfeatures were observed (Fig. 1): (1) Development of a longitudinal bar at a distance of about40% of the channel width from the front of the cutoff; (2) formation of a deep scour hole atthe downstream section of the river just beyond the cutoff intersection; and (3) migration ofthe thalweg toward the inside of the bend. To understand the mechanisms by which the abovecomplex features are developed, an investigation was carried out on the characteristic changesto bend flow in an idealized model situation (see Fig. 3). Based on the analysis of the localchanges observed in the secondary flow structure of the idealized bend model, an explanationwas provided for changes in bed topography due to the installation of the cutoff channel (Herbertson and Fares 1991). For the longitudinal bar to form, the typical one-cell structure of thesecondary flow had to be replaced by a two-cell structure rotating in an opposite direction, with

lLect., Dept. of Civ. Engrg., Univ. of Surrey, Guildford GU2 5XH, England.Note. Discussion open until June I, 1995. To extend the closing date one month. a written request must be

filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possiblepublication on July 30, 1993. This paper is part of the Journal ofHydraulic Engineering, Vol. 121. No. I. January.1995. ©ASCE. ISSN 0733-9429/95/0001-0002-0014/$2.00 + $.25 per page. Paper No. 6664.

2 JOURNAL OF HYDRAULIC ENGINEERING

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SECTIONA-A

,,I

: SECTION

-- -

_ .. _ June 1989_______ April 1990

circulation pattern

~--...

~

I I , I

---

6 9 12 15RIVER WIDTH(m)

3o

--- August 1987_._ June 1988

to

O.

0.5

o. \\\\ r\ (

0.5 \, \

cWIn to

circulation pattern

o.(__~_~ i// "'-J9.~'O ) / SECTION

\ - I ->7, ......\ ........" I ~ /.-••-- - " I C • C

0.5 \ r-'.-1..J~." "~::::...~." -... - -/! /\, I ._._J988 ~ '::'::':::"_' !/'-/ . '-'-' \ '-,~'./'_._/ '.,..~j

1.0

-E-...JW>W...J

Longitudinal Bar

NOT TO SCALE

FIG. 1. Allan Water-Meander Loop and CutoffFIG. 2. Bed Topography Profiles in Allan Water at Cutoff Section

one large cell between the bar and the inner bank and a smaller cell between the bar and theoutbank (see Fig. 2, section CoC)o This two-cell-type circulation in front of the cutoff wasresponsible for the deposition of the loose material of the channel and for the initiation of thelongitudinal bar. The formation of the deep scour hole downstream of the intersection wasattributed to the strong downward currents of the outer bank cell, which caused the scouringof bed material and generated local deepening close to the cutoff channel. The thalweg migrationwas attributed to the inner cell of the secondary flow, which produced bank cutting on the innerside of the bend and caused inward shifting of the thalweg. The latter conclusion was qualitativelysupported by measurements of flow patterns in meandering channels with overbank flows (McKeoghand Kiely 1989).

However, the aforementioned complex flow mechanism does not directly explain the inevitable changes in the boundary shear stress field as a result of the intersection. This paperinvestigates these changes in the idealized rigid bed model of the intersection (see Fig. 3) inwhich entry to the flood relief (cutoff) channel is controlled by a side overflow. Both numericaland experimental approaches are used in the study, which focuses on changes in the boundaryshear stress field for cases of combined bend and high (or low) side overflows. On the basis ofthe rigid bed model study, an explanation for changes in the bed topography in Allan Waterand an assessment of the river regime is provided.

MATHEMATICAL FORMULATION OF BEND FLOW MODEL

The mathematical formulation is divided into two main stages. In the first stage, the formulation of flow around a gently curved channel bend in undertaken. This is followed, in thesecond stage, by simulating the effects of the side overflow intersection on bend flow characteristics. For curved channel flows, the boundary shear stress field is resolved into longitudinaland radial components because of the effects of secondary flow and boundary resistance. Consequently, two-dimensional (20) models are generally more conductive to the study of boundaryshear distribution in meandering channels. Uncertainties in shear stress calculations are foundto vary with flow depth, but are highly influenced by uncertainties in transverse velocities(Siegenthaler and Shen 1983). Expressions for the boundary shear components are

( 1-3)

JOURNAL OF HYDRAULIC ENGINEERING 3

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Centre ofCurvature

FIG. 3. Sketch for Idealized Rigid Bed Model of Intersection

where r, s, and z = cylindrical coordinates in the radial, longitudinal, and vertical directionsfrom an arbritrary origin at the bottom of the bend channel (see Fig. 3); dz = gradient operatorin z-direction; f.l = turbulent momentum exchange coefficient; U r and u, = velocity componentsin r- and s-directions, respectively; and Tn T, = radial and longitudinal components of the totalshear stress To. In these expressions, shear stress components are explicitly introduced in termsof me vertical gradients of longitudinal and radial flow momentum (by gradients of both U r anduJ. In the formulation, the influence of these gradients on flow structure is introduced by adepth-averaged model in which shear stresses can be calculated from the depth-averaged velocities (u",) rather than from local velocity components. By considering the momentum equationin the longitudinal s-direction, the balance between the vertically integrated forces applied toan elementary area (dr'ds) can be expressed as (see Fares et al. 1992) (where the main flowmomentum and the secondary flow momentum are in balance with the pressure gradient andbed friction)

(4)

Where the functions <1>, and <1>2 are given by

(5. 6)

In (4) dr and d, = gradient operators in r- and s-directions, respectively; h = local flow depth;R = local bend radius; Ss = longitudinal surface slope; U* = V (T)p) = shear velocity, withp being flow density; and g = acceleration due to gravity. The functions <1>1 and <1>2 representthe contribution of the longitudinal and secondary flow momentum to the motion. It is evidentfrom (4) that the convective transport of the longitudinal flow momentum by the secondaryflow is an important cause of redistribution of depth-averaged velocities; hence, of the shearstresses. The evaluation of secondary flow contribution on velocity distributions in channelbends was analytically investigated by Johannesson and Parker (1989). Coefficients (3 and CI. are


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the main and secondary flow convection factors for <P], <P2, respectively. A combination of (5)and (6) gives the following expression:

<P2 h (h) (B)- = R,,~ - = R,,~ - -<PI R", B R",

(7)

Where R,,~ = a/[3 is the ratio between secondary and main momentum coefficients. Eq. (7)shows that <P2/<P I decreases with increasing aspect ratio Blh and bend tightness R",/B, where Rm

= bend mid radius; and B = width. Therefore, the more shallow the flow and the less tightthe bend, the smaller the contribution of the secondary flow for a given channel roughness. Fig.4 gives the linear variation of the <P2/<P I ratio with the hiRm ratio for different bed roughnesscoefficients, in which C is the Chezy coefficient of roughness. In both (7) and Fig. 4 the <P2/<P I

ratio increases with increasing bed roughness and decreases with both the aspect and bendtightness ratios. As expected, this indicates a greater dispersion of flow momentum in thetransverse direction for roughened channels and strongly curved flows.

Calculation Procedure

The solution for the distribution of depth-averaged velocity U m around the bend is numericallyobtained using a finite-difference scheme. The method was previously reported [see Fares (1992)and Fares et at. (1992)]; therefore, the details are not repeated. By substituting (5) and (6) into(4), the latter, after a slight transformation, results in

1 [ h] S, ( gUm) U'" go,U", = -~ R,,~ O,(hu",) + 2r u'" +"2 l3u", - h - 213h C2 (8)

The equation is an explicit version of (4) for the streamwise variation of U m along the bend.In the procedure, the channel bend is divided into a concentric mesh (see Fig. 5) in which ateach grid point [i, j] (where i = r-coordinate index and j = s-coordinate index) U m is computed.The numerical calculations started at the inner bank and proceeded in radius increment, I1r,across the channel to the outer bank. Then the computations proceeded longitudinally in stepsof 11s. For any finite-difference scheme, the consistency and accuracy of computations dependon grid size, which in this case I1r and I1s. Preliminary test runs showed that 11 sections acrossthe channel width and 1.25° angle increments along the tested (60°) bend were sufficient for the

.4 C=20

grtd structure for bendonly conftguretlon

Channel Bend

~b=60·

~·Ili·/./ "

~tl FLOW

----.- dlreeUon 0' computationo grid point• grid point In each strip

Bend Strip

Side Overflow

grtd structure for combinedbend + side overftow

=40~

.3.2.1

.1

.3

.2

.0

FIG. 4. Relation Between Flow Momentum Ratio $2/$, and hiRm Ratio for Different Values of Roughness Coefficient C

FIG. 5. Grid Structure Used in Numerical Calculations of TwoBend Situations


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computed results to approach consistency. The longitudinal surface slope S,[i, j] was evaluatedusing the Chezy flow-resistance equation

S,[i, j]u.,,[i, j]

gC 2h[i, j](9)

The initial value of S, at the bend entrance (i.e., at 8b 0°) was assumed to match that ofthe uniform flow at the upstream reach of the bend channel. Subsequent values of h[i, j + 1]were then calculated from

h[i, j + 1] = h[i, j] - SJi, j]Li5 ( 10)

Before proceeding towards the calculation of shear stresses, the predicted depth and velocityprofiles were checked against experimental data and found satisfactory [see Fares and Herbertson(1990, 1991, 1992)]. The longitudinal shear stress component T, was calculated from

(11 )

The radial shear stress component Tr was calculated using an approach similar to that adoptedby Bouwmeester (1972) and Jansen et al. (1979). From the output of (8), and by using (1), (5)and (6), a relationship between the depth-averaged velocity um , modified by the radial flowmomentum, and T r was obtained. It reads

~ = -2 ~ U 2 C'2(1 - C)· C = Iip R'" , ..j"kC (l2a,b)

( 13)

(14)

Where k = Von-Karman constant. Eq. (12) shows the lateral shear stress T r depends stronglyon U m , C, and the h/R ratio. For cases of bend flows of similar velocities and roughnesscoefficients, T r increases as flow depth increases and as the degree of bend curvature decreases.The preceding equation is only applicable to the interior region of the channel cross section,away from the bank region. The effects of walls on bend flow behavior is usually limited to thebank region and considered negligible for flows of aspect ratios B/h 2: 6, and for bend tightnessratios R,,/B 2: 3. A relationship can be derived between the ratio of the radial to the longitudinalshear stresses T)Ts and the corresponding ratio <1>2/<1>1 of the depth-averaged flow momentum.By combining (11) and (12), with the use of (5) and (6), the following expression can be obtained:

<1>, k2 (T)

<I>~ = 2(C' - 1) R,,~ ~

This equation shows a direct relationship between the contribution of the secondary flowmomentum and that of the transverse boundary shear stress, normalized by their contributionin the longitudinal direction. It can be deduced from (13) that the order of magnitude of thetransverse shear stress increases by increasing bed roughness and the intensity of secondary flowin the motion. Finally, having determined the shear stress components T, [from (11)] and T r

[from (12)], the total shear stress To was calculated from (3).

Formulation of Bend Flow at the Side Overflow Region

In the case of flow in a combined channel bend and active side overflow, the discharge, dQfR,over a step length, dCw, across the side overflow can be calculated from

dQfR _ A D (h C )1.5dC", - W I' - "

Where A w = weir coefficient, which includes velocity and discharge coefficients; D f = a coefficient, which simulates the drowning condition of flow over the weir (51); C" = weir crestheight; and h = water head in the bend channel. The streamwise variation of bend flow at theintersection is simulated by the spatially varied flow equations, with decreasing discharge, alongthe crest width of the overflow; that is

Where

dh

dC",[So - Sf - ef3g~ 1) (:~:) (u + d~fR) ]/[1 - f3(F2 + ~F)] (15)

(16)(~r =~S.. = bed slope of the channel; Sf = friction slope; u = mean velocity of the bend flow; F =


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Froude number ( = utv(gh»; and A = area of bend flow (= Bh). Eq. (15) = basic equationfor modeling the lateral overflow effects on flow in the main channel. However, since the maindifference between flows in straight and curved channels is the effect of the superelevation andsecondary circulation, the direct application of (15) would lead to unrealistic solutions. Hence,the equation, or rather the approach in which it is applied, has to be modified to include theseeffects. To do this, the bend cross section is divided into a series of concentric strips (subchannels)of equal width (see Fig. 5), each of which has a certain (mean) depth and a certain (mean)velocity. Each strip can be treated separately and the model equation can be applied. Theadvantage of using this approach is that each concentric strip contributes to the overflow according to its mean water depth. The contribution increases as water elevation increases andstrips nearest the bend's outer bank contribute most to the side overflow. This procedure'svalidity depends mainly on the degree of bend curvature rm / B and the aspect ratio of the flowB/h. The more shallow and less curved the bend, the more realistic the predictions obtained.

Calculation Procedure

The depth and velocity profiles at the side overflow were obtained by solving (14) and (15),simultaneously, through a standard numerical method of integration [see Price (1977) and Chapraand Canale (1985)]. For subcritical flow conditions in the bend, which are under downstreamcontrol, computations should start at the downstream end of the intersection and work towardthe upstream end. However, in evaluating the depth and velocity profiles of the approach (bend)flow, as described in the preceding section, the computations needed to be carried out in thedownstream direction. Hence, an assumption for the bend flow at the intersection was necessaryfor the computations to proceed from the downstream end toward the upstream end. Theassumption in the approach was that the specific energy was constant for each bend strip. Thisallowed the values of depth and velocity in each strip to be determined at the downstream end,from which the computations start. This assumption does not imply a constant energy acrossthe bend width, since the radial variations of specific energy are implicitly introduced throughvariations of both depth and velocity in each strip. The assumption of constant specific energywas experimentally confirmed for the case in which the main channel is straight [Ranga Rajuet al. (1979) and Uyumaz and Muslu (1985)]. Thus, this study extends the use of this assumption(in a modified form) to include the case of a gently curved channel.

At the overflow zone, the bend cross section is divided into a concentric mesh (see Fig. 5)in which at each grid point [i, I], i = r-coordinate index and I = s-coordinate index along theintersection. Depth and velocity profiles at each concentric strip were computed (the numberof strips is equal to the number of nodes in the radial direction minus 1). For the 60° bend, theside overflow was located between 25° and 35° to allow significant development of superelevationand secondary circulation in the flow. The initial values of depth and velocity, at each grid pointof the upstream end of the intersection, were fed from the output of the bend flow model [(8)and (10)]. With the assumption of constant specific energy, computations were carried out ineach strip from the downstream end of the intersection toward the upstream end. The predicteddepth and velocity values in each strip were then projected on all grid points of the concentricmesh at which values of shear stress were computed. Similar to the case of flow in the bendonly situation, the predicted depth and velocity profiles were checked against measurementsand were found satisfactory, particularly in the case of combined bend and low side overflows(Qr :::; 0.4 and hr :::; 0.36, where Qr and hr are the ratios of discharge and depth between thebend and side overflow) [see Fares and Herbertson (1990, 1992)]. By using the predicted valuesof depth and velocity, the longitudinal T, and transverse Tr shear stress components were calculated from (11) and (12), respectively; and the total shear stress To was finally calculated from(3). Experience showed that 11 sections (i.e., 10 concentric strips) across the channel width and0.2SO angle increments along the intersections were sufficient to maintain consistency in thepredictions.

LABORATORY INVESTIGATION OF PROBLEM

Experimental data collected from a laboratory model are used for quantitative comparisonswith the numerical predictions. Full details of the experimental apparatus and equipments usedin measurements are reported in Fares and Herbertson (1993); hence, only the principal dimensions of the experimental flume will be given. The flume consisted of a 60° bend workingsection lying between two straight channel reaches, see Fig. 6. Both the straight and curvedsections were of rectangular cross sections, 0.5 m wide and 0.12 m deep, with Manning's n valueof 0.01. The bend was specified as a gentle bend (i.e., the mid bend radius to width ratio R",IB was fixed at 3). A gap was left in the outer wall of the bend at the apex (see Fig. 6) betweenbend angles 8b = 25° and 35° to accommodate the side overflow channel. Entry to the sidechannel was controlled by a broad crested weir of varying crest levels to allow changes in waterelevation at the channel entry. The downstream condition of the side channel was uncontrolled


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0.5m

~i

I

~b= 60·

Rm=1.5m

.~m

1.5m

60'

B =Rm =~=

.50

1-0----- '.00 JlL- ~

'ir ._.__.+FLq.YL___.~L. '-- +__.

SIDE WEIR

L r---..---:r'-~--~... I I I.. , , __~~---J...~3O--J varl.bleI. .80m

FIG. 6. Principal Dimensions of Laboratory Flume FIG. 7. Measurement Locations in Bend Section of Flume

TABLE 1. Experimental Program

Channel Bend Data Side Overflow Data

ha Q ua Ch QrR

Test run (mm) (lis) (m/s) hlRm Blh F R (mm) h, (lis) Q,(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11 ) (12)

BAI-BVIBA2-BV2

AI-VI 55.5 5.38 0.196 0.037 9.010 0.266 9,889 25 0.55 2.24 0.42A2-V2 70.0 7.70 0.220 0.047 7.143 0.266 14,000 25 0.64 4.08 0.53A3-V3 60.0 6.00 0.200 0.040 8.333 0.261 10,909 45 0.25 0.76 0.13A4-V4 70.0 7.39 0.211 0.047 7.143 0.255 13,427 45 0.36 1.68 0.23A5-V5 80.0 6.40 0.160 0.053 6.250 0.181 11,636 65 0.19 0.64 (UO

"Mean value upstream of channel bend.

and outflow from the curved channel was controlled by an adjustable sluice gate. The widthratio between the overflow and the bend Cw/B was fixed at 0.6.

The experimental program was divided into two sets of measurements - the first set wastaken with the side channel completely closed (i.e., a bend-only configuration) and the secondset with the bend plus active side overflow configuration. A total of 20 test runs were performedfor water surface profiles, and seven more for point velocities and deviation angles of the resultanthorizontal velocities [Fares and Herbertson (1993)]. The flow conditions used for shear stressm,easurements were the same as those for velocities and deviation angles in the two bendsituations (see Table 1); two test runs for the bend-only situation and five test runs for the bendwith active side overflow. Fig. 7 shows measurement positions along the bend channel upstreamof, at, and downstream of the side overflow.

SHEAR STRESS IN BEND·ONLY CONFIGURATION

Figs. 8 and 9 give comparisons between the predicted and observed shear stress profiles alongthe channel bend at bend radii R/R", = 0.9, 1.0, and 1.1, for test runs BA1-BV1 and BA2BV2, respectively. In the figures, the total shear stress To is normalized by the shear stress atthe upstream straight reach of the bend Tn, (= pghSo). The bands, R/R"" are chosen to correspondwith locations near the inner bank, at the centerline, and near the outer bank of the bend,respectively. In general, the profiles confirm the previous findings reported by, for example,Ippen and Drinker (1962), Yen (1970), Engelund (1974), Varshney and Garde (1975), Zimmermann and Kennedy (1978), Nouh and Townsend (1979), and Chen and Shen (1983). Bothpredicted and observed shear stress profiles show a gradual increase across the width of thechannel, with maximum stresses lie along the inner side of the bend. This trend is induced bythe increase in velocity gradients near the inner bank relative to those near the outer bank. Inthe second half of the bend, the shear stresses at the inner side of the bend (R/R", = 0.9)


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2

1.8 prediction -

•1.6 experiment4...1.4 • • •--1.2

<Il ... ... ... ...0 ... • ... X~ ...... ... ... ... ...\' ... ... ...0.8 ...

0.6• RiRm = 0.9

0.4... RiRm = 1.0

0.2 ... RiRm = 1.1

2

1.8 prediction-•

1.6 experiment...

• ...•1.4 •

1.2 • ... X ...• X ... ... ...§ ... ... ... +

* + ...'0 ... ...\0-

0.8

0.6• RiRm = 0.9

0.4... RiRm = 1.0

0.2 ... RiRm=1.1

0,J....,0'--""I5-'T10-1...5-2,...0-2....5--,3,....0---r35-4""'0-4""5--::5!:0--:'5r:'5"""":'60~·

Sb

FIG. 8. Shear Stress Distribution in Bend-Only Situation, TestRun BA1-BVl

0..1-0....--s.-----,....0-,....S-2Q....--2....S-30....--35....--40c--

45r--

SOr--

SSc--:

60't::·;;l

Sb

FIG. 9. Shear Stress Distribution in Bend-Only Situation, TestRun BA2-Bv2

prediction

•experiment......

• RiRm = 0.9

0.6

0.2 ... RiRm = 1.1

0.4... RiRm = 1.0

1.8

Sb

2.------~,----...,.,--------,--i SIDE OVERFLOW :--

1 ZONE :, I

;::~---*--;lIr---'.----r-. • I

~ 1.2 =_~:: t~ : : .~ 0.8 : + r + + +

, ': ,, ,, ,, ': Ii i'-I : limit of numerical~ : model application, ,, ,I I

o 0 5 10 15 20 ;527.53032.5;5 40 45 50 55 60·

prediction

•experiment...

...

•••... ... ...

+ + + +, ...,j,l limit of numericalI model application

I,+ + + +

• •

• RiRm = 0.9

... RiRm = 1.0

... RiRm = 1.1

o 0 5 10 15 20 527.53032.535 40 45 50 55 60·Sb

0.6

0.4

0.2

2...--------...------,-------,

1.8

1.6

1.4

1.2

~o'to' 0.8

FIG. 10. Shear Stress Distribution in Bend with Low Side Overflow Situation, Test Run A3-V3, Q, = 0.13 and h, = 0.25

FIG. 11. Shear Stress Distribution in Bend with Low Side Overflow Situation, Test Runs A5-V5, Q, = 0.10 and h, = 0.19

decrease, and those at the central (R/Rm = 1.0) and outer (R/Rm = 1.1) regions continue toincrease. This pattern can be attributed to the outward flow momentum, which causes outwardshifting to the maximum (inner) velocities toward the outerside of the bend [Kalkwijk and DeVriend (1980) and De Vriend (1981)]. For the central 60% of the channel width, maximumstresses were persistent along the inner bend side (R/Rm = 0.9). This is because the testedlength of the bend (8 b = 10° ~ 51n was insufficient for the secondary flow strength to haveconsiderable effect on shear stress distributions. Nevertheless, some outward skewness in theshear profiles can be seen along the inner region (R/Rm = 0.9) in the second half of the bend(from 8b = 30° onwards).

SHEAR STRESS AT BEND/SIDE OVERFLOW INTERSECTION

The discussion of boundary shear distributions in this bend configuration follow two distinctlines: quantitative and qualitative analyses. Quantitative analysis deals with comparisons betweenthe predicted and observed profiles, and the qualitative analysis focuses on changes to the shearstress distribution as a result of introducing the side overflow. Results from only 3 test runs,namely A3-V3, A5-V5, and AI-VI, are presented to correspond with conditions of high. medium, and low overflow crest levels, respectively. These are plotted in Figs. 10, 11, and 12 forbend radii R/Rm = 0.9,1.0 and 1.1. The full set of results is reported in Fares et al. (1992). Asin the case of the bend-only configuration, the total shear stress To is normalized by the shearstress To.- of the straight upstream reach of the channel bend. For the bend region downstreamof the overflow, only experimental results will be given since the region is outside the scope ofthe numerical model.


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prediction •

experiment ...+

++

I I

~ SIDE OVERFLOW :.-.: ZONE :I II II I

_----~ II:,: limit of numerical: model applicationIIIIIII

: ... ... ...+

2

1.8

1.6

1.4 •1.2 •

en •~

... ...1 ...

~ +0.8 +

+

0.6• RJRm =0.9

0.4... R/Rm = 1.0

0.2 + RJRm = 1.1

•o

o""""r--r----r---r---,r--;----r---r--r--T----T---'p.-.--....--r--,I-!5 10 15 20 2527.53032.535 40 45 50 55 60°

ebFIG. 12. Shear Stress Distribution in Bend with High Side Overflow Situation, Test Run A1-V1, Or =0.42 and hr = 0.55

Combined Bend and Low Side Overflows (Qr :"S 0.4, hr :"S 0.36) (Test Runs A3-V3and A5-V5)

In the region upstream of the intersection (for 6" = 0° ~ 20°, Figs. 10 and 11), similaritybetween the overflow and nonoverflow situations is seen between the predicted and observedshear stress profiles. The predicted profiles behave in a manner similar to those in the bendonly configuration. Progressive development of secondary currents associated with a superelevation at the water surface causes high velocities, and high stresses, to develop along the innerside of the bend, as explained in the preceding section. In general, there is very little effectproduced by the side overflow on shear stress profiles in this region.

Along the intersection (61) = 25° ~ 35°), there is gradual reduction in shear stresses, particularly in the region close to the overflow (R/Rm = 1.1). High stresses continue to occupy theinner side of the bend (R/Rm = 0.9). This mechanism may be explained as follows: as a resultof introducing the side overflow to the curved channel, the outward cross currents combinedwith the preexisting secondary flow in the bend enhance the outward shifting of momentumand diminish the superelevation at water surface. This process increases across the bend widthtoward the overflow and causes more advection of momentum into the side channel. Consequently, a reduction in bend velocities and shear stresses occurs, particularly along the innerside of the bend (R/Rm = 0.9). The reduction in stresses is manifested by the formation of astagnation zone along the inner bank of the bend, opposite the overflow, in addition to the(already existing) separation zone at the upstream corner of the side overflow channel (6" =25°). The effect of the separation zone on the shear stresses at the outer region of the bend (atRIRm = 1.1) can be detected from the noticeable departure of the measured profiles from thepredicted ones, particularly along the first half of the intersection (from bend angle 6" = 25°-> 30°).

To evaluate the local reduction in stresses at the overflow section, the experimental andtheoretical values plotted in Figs. 10 and 11 are presented in Table 2. These are given as thepercent reduction in shear stresses normalized by those at the upstream section of the overflow,i.e., % reduction of To/(-ro)6" = 25°, where (To)6" = 25° is the shear stress To at 6" = 25°, forbend radii RIR", = 0.9, 1.0, and 1.1, respectively. It can be seen that reductions are alwaysgreater at the outer region of the bend (RIRm = 1.1) than at the central (RIR", = 1.0) andinner sides (RIR", = 0.9). For test run A3-V3, the maximum reduction in shear stresses is foundto be 32% (observed) and 37% (predicted). For test run A5-V5, the maximum reduction is20% (observed) and 31 % (predicted). The closer the section to the outer bank, the more theadvection of momentum into the side channel. For test run A3-V3, the predicted values of Tol

(To)6" = 25" are greater than the observed ones by about 4-5% for bend radii RIRI", = 0.9and 1.1, and by about 14% at RIR", = 1.0. For test run A5-V5, the average values of predictedreduction in stresses are higher than those of the measured ones by about 12%.

For the bend region downstream of the overflow (6" = 40° -> 50°), the maximum shear


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TABLE 2. Percent Reduction of To/(To)6b = 25° at Side Overflow Region

Test Runa Test Run b Test Rune

Bend angle A1-V1 A3-V3 A5-V5

tlbR/Rm R/Rm RiRm

C) 0.9

I1.0

I1.1 0.9

I1.0

I1.1 0.9

I1.0

I1.1

(1 ) (2) (3) (4) (5) (6) (7) (8) (9) (10)

(a) Percent reduction of To/(T")tI,, = 25° (theoretical prediction)

27.5 19.21 23.17 27.00 5.92 8.26 9.65 4.94 6.35 7.7730.0 35.83 42.92 49.54 11.84 15.70 18.99 9.88 12.70 15.5332.5 50.60 59.67 67.95 17.10 23.22 28.04 14.20 19.05 23.2035.0 63.38 73.50 82.20 23.03 30.50 36.68 19.14 25.m; 30.48

(b) Percent reduction of T"/(ToW,, = 25° (experimental measurements)

27.5 10.55 13.05 27.45 5.15 6.02 8.72 0.76 5.88 8.1830.0 22.36 29.43 54.06 9.56 11.94 21.15 5.34 8.40 10.1832.5 39.90 47.90 81.32 16.18 15.75 29.99 7.63 14.29 17.1835.0 67.27 48.57 82.17 19.12 16.55 31.78 7.63 11.76 19.45

"Side overflow condition for high overflow: h, = 0.55; Q, = 0.42."Side overflow condition for medium overflow: h, = 0.25; Q, = 0.13."Side overflow condition for low overflow: h, = 0.19; Q, = 0.10.

stresses at the inner side of the bend (R/R", = 0.9) decrease gradually, and those along thecenter (R/R", = 1.0) and outer side (R/R", = 1.1) remain virtually constant. This patterncontinues steadily until a uniform value of stress, across the bend width, is eventually realized.Maximum shear stresses would be expected along the outer bank for relatively longer channelbends in a manner similar to that of the bend-only configuration.

Combined Bend and High Side Overflows (Qr > 0.4, hr > 0.36) (Test Run A1-V1)

Upstream of the side overflow (6h = 0° ----> 20°), it can be seen from Fig. 12 that, maximumstresses lie along the inner side of the bend. There is a strong tendency toward a uniform stressdistribution across the width of the channel, which indicates the features developed in the flowat the intersection have been propagated upstream. Similar to the pattern of flow along theintersection in cases of low side overflows, the outward advection of velocities, combined witha reduction in the superelevation of the water surface, results in a nearly uniform distributionof shear stress across the width of the bend just upstream of the overflow region (61) = 20°---->25°). This explains the poor performance of the numerical model in this region, since no accountwas taken in the formulation of effects propagating upstream from the intersection.

Along the intersection (61) = 25° ----> 35°), shear profiles are greatly influenced by the strongcross currents generated in the flow and the development of stagnation and separation zones.The decrease in stresses along the inner side is a direct result of the stagnation zone, and thereduction of stresses along the outer side is caused by outward lateral currents. For this testrun, the separation zone is found to occupy almost half the width of the channel bend and abouthalfthe width of the side overflow channel (Fares and Herbertson 1993). The combined reductionin flow depth and velocities in the bend, at the side overflow region, causes a drop in shearstresses, particularly in the second-half of the intersection beyond the separation zone (61) =30° ----> 3SO). Table 2 gives percent values of these reductions along the intersection, expressedby the shear stress ratio To(To)61> = 2SO, at radii R/R", = 0.9, 1.D, and 1.1. As in the case of lowside overflow, the reduction in shear stresses is higher in the outer bend region (R/R", = 1.1)than in the central (R/R", = 1.D) and inner (R/R", = D.9) regions. The observed reductions varybetween ~82% at R/R", = 1.1 and ~67% at R/R", = D.9 for 6h = 35°. In general, the predictionsare overestimated at the central and outer regions of the bend and underestimated at the innerregion. The difference between the predicted and observed shear stress ratio TO<To)6h = 25°,for the inner and central regions at 61> = 3D.no and 32.SO, varies from ~ 11 % to 13%, and theoverall difference is small, ~4%.

Downstream of the overflow (i.e., at 6h = 40° ----> 50°), stresses along the central (R/R", =1.D) and outer (R/R", = 1.1) regions of the bend increase and those along the inner side (R/R",= D.9) diminish substantially. The reduction in shear stresses along the inner side of the bendis a consequence of the continual growth in the size of the stagnation zone downstream of theoverflow. As a result, high shear stresses become concentrated in the outer half of the bendcross section. This concentration of stresses indicates that, in a natural river situation, severebed scour is likely to occur in this region. This result was experimentally justified by observingthe high velocities of high outward deviations near the channel bed [see Fares and Herbertson(1993)].


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BED TOPOGRAPHY CHANGES IN ALLAN WATER

On the basis of the investigation carried out on the rigid bed model, an attempt will be madeto provide an explanation for bed topography changes in the Allan Water river, at the cutoffsection. For combined bend and low side overflow conditions, the river is expected to be governedby the bend flow regime with an occasional spillage of flood flow across the neck of the cutoff.In contrast, changes in bed topography are most likely to develop in cases of bend flows combinedwith high side overflows. The occurrence of both longitudinal bar and local scour hole in thebed topography of the river (Figs. 1 and 2) may be qualitatively explained in terms of shearstress distributions obtained from the idealized model. This can be achieved by matching locations of the features developed in the river, with those of shear stresses in the idealized model.A comparison of the appropriate locations in Figs. 1 and 2 with those in Fig. 13 and Table 2showed a reasonably good correlation between the location of the longitudinal bar and that ofminimum shear stresses along the intersection (at R/Rm = 1.1 and (l" = 32.5" ----> 35°). Byapplying the same analogy for the scour hole location downstream of the intersection, a zoneof high shear stresses in the bend region (l" = 40° ----> 50°, along radii R/Rm = 1.0 and 1.1, wasdetected.

Finally, with respect to the migration of the thalweg toward the inside of the bend, the idealizedmodel study showed that the shear stresses along the inner side of the bend (RfRm = 0.9)continue to fall rapidly to insignificant values beyond the intersection (Fig. 12 (l" = 40° ----> 5(n.This would indicate that the formation of a bar is likely in this region, which would eventuallylead to the complete blockage of the meander loop and hence, to the development of the oxbowlake (Allen 1965). The observed inward migration of the thalweg may be caused by the changein the secondary flow cell structure, from one cell to two cells, rotating in the opposite sense(Fig. 2). The change in the cell structure of the secondary flow is initiated by the longitudinalbar development in front of the cutoff (expressed by a zone of minimum shear stresses, atR/R I1l = 1.0 and (l" = 32.5° ----> 35° in Fig. 12). The bar, once it reaches a considerable length,splits the bend flow and produces the two-cell pattern. This, in turn, forces the flow to moveaway from the cutoff producing active cutting on the inner side of the bend. It is evident,however, that the existence of zones of high and low shear stresses in the river, at the cutoffsection, causes reduction in the river width and encourages the flow to take the shorter routeacross the cutoff. Further investigations into the time and length scales involved in the mechanismby which the meander is adjusted to the cutoff situation, to provide more rigorous assessmentof the long-term stability of the river regime, are recommended.

CONCLUSIONS

The object of the paper was to determine the changes occurring in the boundary shear stressfield, in a curved channel, resulting from the introductions of a side overflow on the outer bank.Despite the fact that the study focused only on the hydraulic behaviour in an idealized rigidbed model situation, it provided a mechanism for qualitatively evaluating the features developedin the bed topography of a river system at the meander loop/cutoff section. Both numerical andexperimental approaches were employed in the rigid bed model situation. The analysis showedthe shear stress field is strongly dependent on depth and discharge ratios between the twochannels. The following conclusions may be drawn from the rigid bed model study:

A numerical model was suggested for simulating the shear stress distribution in a channelbend with a side overflow. The model was based on the combination of a depth-averaged bendflow model and a spatially varied flow equation. The basic features of flows in curved channels,the superelevation and secondary circulation, were incorporated in the model formulation. Basedon the analysis of results, the overall difference between predicted and measured shear stressvalues at the side overflow region was found to be ~7%.

For all discharge Qr and depth hr ratios tested, the introduction of the side overflow on theouter bank of the channel bend caused a reduction in the boundary shear stresses. A maximumreduction of shear stresses was always found along the outer side of the bend (R/Rm = 1.1),close to the overflow.

For combined bend and low side overflow conditions (Qr :S 0.4 and hr :S 0.36), little effectwas produced by the side overflow on shear stress profiles in the bend region upstream of theintersection. Along the intersection, a gradual reduction was found in stresses, with high stressescontinuing to occupy the inner side of the bend (R/R I1l = 0.9). The maximum reduction of shearstresses in this region was 32% (measured) and 37% (predicted). In the bend region downstreamof the overflow, the stresses along the inner side of the bend (RfR I1l = 0.9) decreased steadily,and those at the central (R/Rm = 1.0) and outer side (R/Rm = 1.1) zones remained virtuallyconstant.

For combined bend and high side overflow conditions (Qr > 0.4 and hr > 0.36), the effectof overflow on the shear stress field was significant in all bend regions of the intersection. Atendency towards a uniform stress distribution was observed in the upstream region as a resultof diminishing the superelevation at the water surface and the outward shifting of bend velocities


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toward the overflow. Along the intersection, the strong lateral outward currents and the formation of a stagnation zone (along the inner bank of the bend) and a separation zone (at theupstream corner of the side overflow) caused substantial reduction in shear stresses. The observed reduction of stresses was 82% in the outer side of the bend (R/Rm = 1.1) and 67% inthe inner side (R/Rm = 0.9). Downstream of the overflow region, the shear stresses continuedto diminish considerably along the inner side of the bend, with an accompanying increase inthose along the outer region of the bend cross section, as a result of the continual stagnationzone growth beyond the intersection.

ACKNOWLEDGMENTS

The numerical calculations of boundary shear stresses were carried out using the central HP-UNIX computerservice at the University of Surrey. The writer wishes to thank Dr. J. G. Herbertson of the University of Glasgowfor his suggestions throughout the study. The writer also wishes to thank the anonymous reviewers for theirhelpful comments.

APPENDIX I. REFERENCES

Allen, J. R. L. (1965). "A review of the origin and characteristics of recent alluvial sediments." Sedimentology,5(2), 89.

Bouwmeester, J. (1972). "Basic principles for the movement of water in natural and artificial water courses."Internal Note, Delft Univ. of Technol., Delft, The Netherlands.

Chapra, S. C, and Canale, R. R. (1985). Numerical methods for engineers with personal computer applications.McGraw-Hili Book Co., Inc., New York, N.Y.

Chen, G., and Shen H. W. (1983). "River curvature-width ratio effect on shear stress." Proc., Int. Conf. onRiver Meandering, ASCE, New York, N.Y., 687-699.

De Vriend, H. J. (1981). "Velocity redistribution in curved rectangular channels." J. of Fluid Mech., Vol. 107,423-439.

Engelund, F. (1974). "Flow and bed topography in channel bends." J. Hydr. Div., ASCE, 100(11), 1631-1648.Fares, Y. R. (1992). Modelling of the cross flow strength in channel bends." Proc., Int. Conf. on Protection and

Development of the Nile and Other Major Rivers, Water Res. Ctr., Nile Res. Inst., Qanater, Egypt, Vol. 1/2,3.2.1-3.2.15.

Fares, Y. R., and Herbertson, J. G. (1990). "Partial cut-off of meander loops-a comparison of mathematicaland physical model results." Proc., Int. Conf. on River Flood Hydr., Wiley and Sons, Chichester, England,289-297.

Fares, Y. R., and Herbertson, J. G. (1991). "Formulation of the flow characteristics in a channel bend with aside overflow." Proc., 24th Congr. of IAHR, IAHR Publ., Delft, The Netherlands, Vol. A, AI733-A740.

Fares, Y. R., and Herbertson, J. G. (1992). "Performance of side channel overflows at river bends. " Proc., Int.Conf. on Protection and Development of the Nile and Other Major Rivers, Water Res. Ctr., Nile Res. Inst.,Qanater, Egypt, Vol. 2/2, 8.3.1-8.3.15.

Fares, Y. R., and Herbertson, J. G. (1993). "Behaviour of flow in a channel bend with a side overflow (floodrelief) channel." J. Hydr. Res., 31(3), 383-402.

Fares, Y. R., Laufs, W., and Herbertson, J. G. (1992). "Boundary shear changes in a channel bend at flood relief(cut-off) channel intersection." Rep. No. CE-FM/FWL/92.1, Dept. of Civ. Engrg., Univ. of Surrey, England.

Gagliano, S. M., and Howard, P. C (1983). "The neck cut-off oxbow lake cycle along the lower Mississippiriver." Proc., 1nt. Conf. on River Meandering, ASCE, New York, N.Y., 147-158.

Herbertson, J. G., and Fares, Y. R. (1991). "Bed topography changes produced by partial cut-off of a meanderloop." Proc., Eur. Conf. on Adv. in Water Resour. Technol., A. A. Balkema, Rotterdam, The Netherlands,113-120.

Ippen, A. T., and Drinker, P. A. (1962). "Boundary shear stresses in curved trapezoidal channels." 1. Hydr.Div., ASCE, 88(5), 143-179.

Jansen, P. P. et al. (1979). Principles of river engineering, the non-tidal alluvial river. Pitman Publishers Ltd.,London, England.

Johannesson, H., and Parker, G. (1989). "Secondary flow in mildly sinuous channel." J. Hydr. Engrg., ASCE,115(3), 289-308.

Kalkwijk, J. P. T., and De Vriend, H. J. (1980). "Computation of the flow in shallow river bends." 1. Hydr.Res., 18(4),327-342.

McKeogh, E. J., and Kiely, G. K. (1989). "Experimental study of the mechanisms of flood flow in meanderingchannels." 23rd Congr. of 1AHR, IAHR Publ., Delft, The Netherlands, Vol. B, B/491-B/498.

Nouh, M. A., and Townsend, R. D. (1979). "Shear stress distribution in stable channel bends." J. Hydr. Div..ASCE, 105(10), 1233-1245.

Price, R. K. (1977). "A mathematical model for river flow - Theoretical development." Rep. No. INT 127,Hydr. Res. Wallingford Ltd., Oxfordshire, England.

Ranga Raju, K. G., Prasad, B., and Gupta, S. K. (1979). "Side weir in rectangular channel." J. Hydr. Div.,ASCE, 105(5),547-554.

Siegenthaler, M. C, and Shen, H. W. (1983). "Shear stress uncertainties in bends from equations." Proc., Int.Conf. on River Meandering, ASCE, New York, N.Y., 662-674.

Uyumaz, A., and Muslu, Y. (1985). "Flow over side weirs in circular channels." J. Hydr. Engrg., ASCE, 111 (1),144-160.

Varshney, D. V., and Garde, R. J. (1975). "Shear distribution in bends in rectangular channels." J. Hydr. Div.,ASCE, 101(8), 1053-1066.

Yen, C. L. (1970). "Bed topography effect on flow in a meander." J. Hydr. Div., ASCE, 96(1), 57-73.Zimmermann, C., and Kennedy, J. F. (1978). "Transverse bed slopes in curved alluvial streams." 1. Hydr. Div.,

ASCE, 104(1),33-48.


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APPENDIX II. NOTATIONS

The following symbols are used in this paper:

n -Q

Q1RQ,

R,R",R

Ra~r, s, Z

SoSIS,U

Um

U,., U.\

u*a, !3~C"

~r, ~s

an an az

81>61>JJP

T,., T.\., To

(To)61> = 25"

area of bend flow ( = Bh);weir coefficient [in (14)];total width of bend;Chezy coefficient of roughness;nondimensional coefficient of roughness [ = y/gl(kC)];crest height of side overflow;total width of side overflow;drowned flow reduction factor (s1);Froude number of bend flow [=u/Y/(gh)];acceleration due to gravity;mean and local flow depth;water head ratio (=hw/h);water head above weir crest ( = h - CI»;indices for r- and s-coordinates, respectively;Von Karman constant (=0.41);s-coordinate index at overflow region of bend;Manning's coefficient of roughness (= 0.01);discharge in bend channel;discharge in flood (side overflow) channel;discharge ratio (= QfR/Q);local and mid bend radii, respectively;Reynolds number of bend flow;ratio between secondary and main momentum coefficients ( = a/!3);radial, tangential, and vertical cylindrical coordinates, respectively;channel bed slope;friction slope;longitudinal surface slope in channel bend;mean velocity in bend;depth-averaged velocity in bend;velocity components in r- and s-directions, respectively;shear velocity (=Y/(T)p»;coefficients for secondary and main flow momentum, respectively;step size along s-direction at overflow zone;step sizes along r- and s-directions, respectively;gradient operators in r, s, and z directions;local bend angle;total bend angle ( = 60°);turbulent momentum exchange coefficient;flow density;radial, longitudinal, and total shear stress components;bed shear stress at 61> = 25°;bed shear stress upstream of bend ( = pghSo); andfunctions for main and secondary flow momentum.

Subscriptsb bend;

fR flood relief (side overflow) channel;r ratio; and

w weir


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surface irrigation

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