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Journal of Hydro-environment Research

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Research papers

An investigation on the outer bank cell of secondary flow in channel bends

Alireza Farhadia,⁎, Christine Sindelara, Michael Trittharta, Martin Glasa, Koen Blanckaertb,Helmut Habersacka

a Institute of Water Management, Hydrology and Hydraulic Engineering, Department of Water, Atmosphere and Environment, University of Natural Resources and LifeSciences (BOKU), Muthgasse 18, 1190 Vienna, Austriab Research Center of Hydraulic Engineering, Institute of Hydraulic Engineering and Water Resources Management, Faculty of Civil Engineering, Technical University Vienna(TUW), 1040 Vienna, Austria

A R T I C L E I N F O

Keywords:Secondary flowsVorticityTurbulenceLaboratory studiesVelocity measurements

A B S T R A C T

Beside the curvature-induced secondary flow in curved channels, often a counter-rotation secondary flow cell isobserved near the outer bank, called outer-bank cell. In the current study, experiments were conducted for threelow subcritical Froude numbers typical for natural streams that complement the Froude number range in-vestigated in previous studies. It was found that for increasing Froude numbers, the main cell of secondary flowexpanded and strengthened. Simultaneously, the outer-bank cell reduced in size and retreated towards theboundary. A term-by-term analysis of the energy production and the vorticity equations was performed tosubstantiate this observation. In the outer-bank zone, for increasing Froude numbers, a reduction of the energyflux from turbulence to the mean flow was discernible; which confirms that the turbulence-induced processesaffect the outer-bank cell. The analysis of the vorticity equation showed the balance between generative anddegenerative terms in the streamwise vorticity equation.

1. Introduction

During the last decades, open channel bends have been investigatedextensively (Shukry, 1950; Rozovskii, 1957; Yen, 1970; Kalkwijk andDe Vriend, 1980; Dietrich, 1987; Odgaard and Bergs, 1988; Rhoads andWelford, 1991; Whiting and Dietrich, 1993; Kawai and Julien, 1996;Boxall et al., 2003; Tritthart and Gutknecht, 2007; Sukhodolov, 2012).In addition to mean flow, considering turbulence parameters is im-portant in order to quantify the channel evolution, alignment mod-ification or migration (Yalin and Karahan, 1979; Bennett and Best,1995; Darby et al., 2010).

Streamwise mean vorticity generated by skewing or vortexstretching is called secondary flow of Prandtl’s first kind (Prandtl, 1942,p.130–134). In a bend, this mechanism generates the curvature-inducedmain cell of secondary flow characteristic for bends (Rozovskii, 1957;Bradshaw, 1987; Nezu and Nakagawa, 1993). In addition to this maincell of secondary flow, a weaker and smaller cell of reverse secondaryflow, called outer-bank cell (OBC), has often been observed in thecorner formed by the water surface and the outer bank both in la-boratory and field experiments (Mockmore, 1943; Einstein and Harder,1954; Rozovskii, 1957; Bridge, 1976; Bathurst et al., 1979; De Vriend,1981; Dietrich and Smith, 1983; Blanckaert and De Vriend, 2004; Kangand Sotiropoulos, 2011; Blanckaert et al., 2012). This OBC is reported

to be more intense and better discernible upstream of the bend apex.Regarding the unknown effects of the OBC on the bank stability, re-searchers have tried to describe its underlying mechanisms. Bathurstet al. (1979) suggested that they appear where the bank is steep. DeVriend (1981) postulated that the OBC is due to the combined effect ofthe centrifugal force and turbulence anisotropy. Streamwise meanvorticity generated by turbulence is called secondary flow of Prandtl’ssecond kind (Prandtl, 1942). Hence, according to de Vriend’s (1981)postulation, the OBC is a combination of Prandl’s first and second kindof secondary flows.

An example of secondary flow cells of Prandtl’s second kind are thesecondary flow cells near the banks in straight channels (Karcz, 1966;Naot and Rodi, 1982, Blanckaert et al., 2010). Nezu and Nakagawa(1993) described their generation mechanism: boundary propertiesgive rise to cross-stream anisotropy and correspondingly to the cross-stream Reynolds stress that drives the secondary flow. In turn, via apositive feedback, the gradient of secondary flow intensifies the cross-stream Reynolds stress. In curved channels, an additional positivefeedback between centrifugal forces and the OBC is reported byBlanckaert and De Vriend (2004). Furthermore, they indicated thatkinetic energy fluxes from turbulence to the mean flow play a role inthe generation of the OBC. These energy fluxes represent an “inversecascade” of the turbulence energy. These generation mechanisms of the

http://dx.doi.org/10.1016/j.jher.2017.10.004Received 30 April 2016; Received in revised form 28 September 2017; Accepted 23 October 2017

⁎ Corresponding author.E-mail address: alireza.farhadi@boku.ac.at (A. Farhadi).

Journal of Hydro-environment Research 18 (2018) 1–11

Available online 24 October 20171570-6443/ © 2017 International Association for Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.

MARK

OBC were confirmed in eddy-resolving numerical investigations by vanBalen et al. (2010), and Kang and Sotiropoulos (2011). Van Balen et al.(2010) detailed the generating effect of the centrifugal force and thedissipating effect of cross-stream turbulence stresses. In a series of la-boratory experiments, Blanckaert et al. (2012) observed an amplifyingeffect of outer bank roughness on all dominant mechanisms with re-spect to the generation of the OBC.

The Froude number is a dominant parameter in open-channel flow.Wei et al. (2016) have investigated the dependence of the main sec-ondary flow cell on the Froude number. The objective of the presentpaper is to investigate, for the first time, the dependence of the OBC andits generating mechanism on the Froude number. Three different sub-critical Froude numbers are investigated in laboratory experiments.These three Froude numbers are representative for natural streams, butlower than Froude numbers in most previous laboratory experimentsreported in literature.

2. Theoretical framework

Nikora and Roy (2012) have summarized and discussed the theo-retical framework for the investigation of secondary flows, including (i)the momentum (balance) equation, (ii) the energy (balance) equationfor mean and turbulence flow, and (iii) the vorticity (balance) equation.

Hinze (1967) explained the importance of the energy equation asfollows. Based on the concept of a turbulence cascade, turbulence en-ergy is generated by the mean-flow energy loss and large eddies (low-frequency structures) disintegrate into smaller eddies (higher-frequencystructures). The energy equation consists of two major parts: First,transit and transport terms that concern the redistribution of kineticenergy and are not responsible for energy production or decay. Second,production and decay terms of the turbulent kinetic energy (TKE) thatimply the exchange of kinetic energy between mean flow and turbu-lence. The production term in the energy equation reads:

= ⎡⎣

⎛⎝

′ − ⎞⎠

+ ⎛⎝

′ − ⎞⎠

+ ⎛⎝

′ − ⎞⎠

+ ′ ′ + ′ ′

+ ′ ′ ⎤⎦

P v 23

k e v 23

k e v 23

k e 2v v e 2v v e

2v v e

s2

ss n2

nn z2

zz s n sn s z sz

n z nz(1)

= ′ + ′ + ′k v v v12

( )s n z2 2 2

(2)

where s, n and z are curvilinear coordinate system componentsalong the streamwise, lateral and vertical axes, ′ =vi(i s,n,z) are fluctuatingvelocity components, ′ ′=v vi j(i,j s,n,z) are turbulent stresses, k representsTKE (Eq. (2)) and =eij(i s,n,z) are strain rates which are defined byBatchelor (1967). The sum of all terms in Eq. (1) is mostly positive;however negative contributions do occur and indicate the return of thekinetic energy from turbulence to the mean flow. This so-called inversecascade is essential for the growth of turbulence in wall-bounded flows(Piomelli et al., 1990).

The vorticity equation provides information about driving and dis-sipating forces of the secondary flow, which is represented by thestreamwise component of the vorticity vector:

=∂∂

− ∂∂

ωvn

vzs

z n

(3)

Blanckaert and De Vriend (2004) and Van Balen et al. (2010) pre-sented the streamwise vorticity balance equation in the following form:

∂∂

= + + + +ωt

ADV DNU DIFF CFG CTCs(4)

The following terms are dismissed in the present study: (i) the ad-vective transport term (ADV) which is not responsible for production ordissipation of the vorticity; (ii) downstream non-uniformity terms(DNU) which are not considered essential for the cross-stream circula-tion mechanism in uniform channels; and (iii) the viscous effect term

(DIFF), which addresses dissipation by molecular viscosity and is con-sidered negligible in open channel problems. The retained terms, i.e.the mean flow and turbulent centrifugal term (CFG) and cross-streamturbulent stress components (CTC) are expressed as follows:

⎜ ⎟= −+

∂∂

⎛⎝

⎞⎠

CFGn R z

vR

11 /

s2

(5)

CTC term is composed of an anisotropy term (ANS) and a homo-geneity term (HOM):

= +CTC ANS HOM (6a)

= ∂∂ ∂

′ − ′ ++

∂ ′∂

ANSz n

v vn R R

vz

( ) 11 /

1n z

n2

2 22

(6b)

= ⎧⎨⎩ +

∂∂

− ∂∂

⎛⎝ +

∂∂

⎞⎠

⎫⎬⎭

+ ′ ′HOMn R z n n R n

n R v v11 /

11 /

[(1 / ) ]n z

2

2 (6c)

3. Experimental setup

The experiments were carried out in an experimental racetrackflume of the Hydraulic Engineering Laboratory of the University ofNatural Resources and Life Sciences, Vienna (Farhadi et al., 2014). Theracetrack flume experimental setup provides constant water depthwhich offers a desirable setting for Froude number studies. In addition,the wide range of applications of racetrack flumes in ecological studiesas well as sedimentology and sediment deposition investigations andlack of detailed experimental studies for this specific setup made theracetrack flume a distinct candidate for the current study. The styr-ofoam flume consisted of two straight sections of 2 meters in length andtwo half circle bends of 180° with a centreline radius of 1.25 meter.Flow was induced using a belt drive; its position, 3D view and dimen-sions are presented in Fig. 1a & b. The flume was characterized bytrapezoidal cross-sections within the investigation area (C1-90 to C2-90; Fig. 1c); in the region of the driving belt, a rectangular cross-sectionwas present which then expanded continuously into the trapezoidalcross-sections (C1-0 to C1-55 and C2-125 to C2-160); the inner banks oftrapezoidal cross-sections were inclined while the outer bank was ver-tical. The discussed results are based on measurements in trapezoidaluniform cross-sections. In addition to the plan of the flume and thealignment of the investigated cross-sections in Fig. 1d, the geometries ofthe investigated cross-sections are provided in Fig. 1e.

The 3D flow velocity field with 1026 points over 18 cross-sections(65–90 points per section) was measured using an Acoustic DopplerVelocimeter (ADV) at 50 Hz sampling frequency. The size of the mea-surement grid per cross-section was 33mm horizontally and 50mmvertically (except for two bottom rows which were distanced by20mm); the lateral distance to the banks was 50mm, the vertical dis-tance to the bottom and to the water surface was 20mm and 61mm,respectively. The latter distance was essentially imposed by the sam-pling volume which is situated 50mm below the transducer probe,which has to be immersed in the water. Nezu and Nakagawa (1993)divided the water column in open-channel shear flow into three re-gions: (i) wall region (z/H ∼ 0–0.15), (ii) intermediate region (z/H ∼0.15–0.6), in which a near equilibrium turbulent energy budget ismaintained and (iii) free surface region (z/H ∼ 0.6–1). Based on thisdivision, the measurement grid used in the current study covers theintermediate region plus small parts of wall and free surface regions (z/H ∼ 0.1–0.70). It was expected that parts of outer bank cells would beresolved within this measuring grid.

Hydraulic properties of the three measured test runs are presentedin Table 1. The water depth was kept constant at 0.20 m in three testruns. This setting provides the opportunity to investigate the flowprocesses for three different Froude numbers: Fr= 0.05, 0.08 & 0.16.Most foregoing laboratory studies (Booij, 2003; Blanckaert and DeVriend, 2004; Abad and Garcia, 2009; Kang and Sotiropoulos, 2011;

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Blanckaert et al., 2012) were characterized by Froude numbers above0.25 (except for Rozovskii, 1957). The Froude number range in thepresent study is representative of most fluvial streams in transfer anddeposition zones (see Fourriere, et al., 2010; Lazarus and Constantine,2013). The measured data comprises three instantaneous velocitycomponents along streamwise, transversal and vertical axes. From theseinstantaneous velocities, the mean velocities (v ,v ,v )s n z and fluctuatingvelocity components ′ ′ ′v v v( , , )s n z were derived. In this paper mean flowand turbulence quantities are presented as values normalized by theaverage streamwise velocity U and the shear velocity ∗u , respectively.Shear velocity was obtained from the mean velocity distribution inconjunction with the logarithmic law, averaged over three investigatedcross-sections. The quality of acquired data is discussed in the Ap-pendix.

Fig. 1. Racetrack flume and investigated cross-sections: (a) Isometric view of the physical model, (b) 3D view of the belt drive, (c) Wire-frame view of the flume and measurement cross-sections, (d) Plan of the flume, (e) Geometry of the cross-sections.

Table 1Hydraulic properties of the racetrack flume.

Test Run Q U Re Fr u* Rh R/B R/H B/H

(l s–1) (m s–1) (m s–1) (m)

F – 5 9.9 0.067 7.6×103 0.05 0.009 0.12 1.69 6.25 3.69F - 8 14.5 0.098 11.1× 103 0.08 0.014F - 16 29.1 0.197 22.3× 103 0.16 0.032

Q=channel dischargeU= channel mean velocity ( =U Q/reachArea)Re=Reynolds numberFr= Froude numberu*= shear velocityRh=hydraulic radius of C1-90 to C2-90R=Radius of curvature (1.25m)B=WidthH=Depth (0.20m)

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4. Results

4.1. Patterns of mean and turbulent flow quantities

Among 18 measured cross-sections, three cross-sections (C2-20, C2-55 & C2-90) were selected for further investigations. Data quality con-siderations, which are presented in detail in the Appendix, as well asthe representative positions of the cross-sections – at the entrance, mid-range, and apex of the second bend of the racetrack flume – led to thisselection.

The normalized mean velocity distribution over three selected cross-sections for three runs (F-5, F-8 & F-16) is provided in Fig. 2. In order toavoid experimental scatter and facilitate accurate evaluation; the pro-files are smoothed using a linear convolution with the Gaussian func-tion, where the average deviation between raw and smooth data isnegligible.

The streamwise velocity pattern consists of a core of high velocityencircled by decreasing velocity contours towards the boundaries. Asflow proceeds through the bend, the core of high velocity shifts slightlytoward the outer bank. Simultaneously, the high velocity core sub-merges beneath the surface in the proximity of the outer bank. Thesepatterns are better discernible for the lower Froude number runs.

Fig. 3 shows the normalized vectors of cross-stream velocity com-ponents, and the normalized streamwise vorticity ω H/Us (Eq. (3)). Asexpected, a secondary flow develops as the flow passes through thebend. In the current study, the entire flow domain is affected by theflume’s geometry (i.e. semi-circular plan). After a short straight reachbetween two bends, the secondary flow starts to redevelop from C2-20(near the inner boundaries at approximately n/B ≈ 0.6 and z/H ≈ 0.6)and proceeds to the downstream cross-sections C2-55 and C2-90; thecore of the secondary flow shifts to n/B ≈ 0.5 and z/H ≈ 0.10, i.e.towards the boundaries. Obviously, not the entire secondary flow cell isresolved on the measuring grid. Thus, the main cell vorticity values inC2-90 are underestimated. In Fig. 3 a range of −0.2 to 0.2 representsthe expected uncertainty (approx. 20%; cf. Table A.1) for vorticityprofiles. Therefore, the sense of rotations of cross-stream velocity vec-tors is unambiguously resolved by the measurements, and indicated bythe red contour line of zero vorticity (Fig. 3).

In addition to the classical main secondary flow, a small OBC is alsoobservable in the low Froude number runs (F-5 & F-8) and more pro-nounced in mid-bend (C2-55) and apex (C2-90) sections. The number ofpoints which reveal these OBC zones is between 6 and 12 (approxi-mately 10% of the data points in each section). In previous studies OBCs

were usually detected at higher elevations in the water column, ap-proximately in the upper one-third of the water column.

For the highest Froude number investigated (F-16), the OBC can beperceived visually neither by means of the vorticity parameter nor bycross-stream velocity vectors. This disappearance of the OBC for higherFroude numbers in the present investigation is tentatively attributed toa shift towards the outer bank and water surface whereby the OBCbecomes located outside of the measurement grid. Also, from the zerovorticity line plotted in Fig. 3 (C2-55/F-16), one can observe the traceof an OBC region, which appears to be shifted upwards and beyond themeasurement grid. In addition to visual perception and in order toquantify the observed vorticity in Fig. 3, the normalized mean magni-tudes of the streamwise vorticity ω| |s are presented in Table 2. Themaximum magnitudes of this parameter are observed in section C2-55.In contrast to the OBC zone, the main cell of secondary flow showshigher vorticity magnitudes as the Froude number increases.

Fig. 4 shows the turbulent kinetic energy (TKE) distributions. Withincreasing Froude number, the TKE maximum shift slightly away fromthe flume’s outer bank boundary. This process is more pronounced to-wards the bend’s apex. In addition to the classic pattern of increasingTKE towards the boundaries in the case of lower Froude numbers (F-5 & F-8), a bulge of turbulent kinetic energy is formed in the upper leftregion of the measurement grid. As Froude numbers increase, thispattern tends to retreat to the outer bank boundaries. The more explicitpattern of this kind can be observed in section C2-55. Along the flume,high cross-stream Reynolds stresses − ′ ′v vn z are observed in the vicinityof the inner bank and the bed (Fig. 5). This pattern is similar in all runs(F-5 to F-16).

The maxima of cross-stream Reynolds stress increase starting fromthe bend entrance to the approximate mid-bend section (C2-55) andthen follow a mildly decreasing trend towards the bend apex (C2-90)(Fig. 5). In the same sections, as the Froude number increases, thelargest values of cross-stream Reynolds stress are esn retreating to theboundaries where the OBC shifts to the boundary and is reduced in size.A core of maximum lateral Reynolds stresses in the vicinity of the OBCzone is visible.

4.2. Mechanisms underlying the OBC

The production term in the transport equation (Eq. (1)) was calcu-lated using experimental data. Negligible terms related to downstreamvariations (ess, and e )sz were omitted.

High values of P> 0 (Fig. 6) occur near the core of the main cell of

Fig. 2. Normalized mean streamwise velocity profiles for selected cross-sections.

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secondary flow (cf. Fig. 3 and Table 2); the negative quantities (P < 0)occur in the proximity of the OBC. In general, the negative/positivepatterns of production correspond to regions of higher/lower TKE, re-spectively. This observation is perceptible in C2-20 (bend’s entrance)and especially in the proximity of the vertical outer wall; however, asthe flow evolves towards the middle and apex of the bend (C2-55 & C2-90), the high production magnitudes appear in OBC zones. Along thebend’s path from C2-20 to C2-90, a pattern of intensification of P < 0

is discernible in the proximity of the outer bank; this pattern is nolonger evident when the Froude number is increased; this is more sig-nificant for C2-55 and C2-90. The most perceptible outer bank cells (cf.Fig. 3: C2-90/ F-5 and Table 2) occur where the P < 0 patch is notinterrupted by a patch of P > 0.

Fig. 7 shows the normalized centrifugal term CFG/ UH

22 in the vorticity

equation (Eq. (5)). In section C2-20, small patches of positive values(evidence of velocity profile deformation) can be seen in the uppercorner of the outer banks; these patches fade out by Froude numberincrease. In the next measured cross-section (C2-55), the mentionedpatch in the vicinity of the outer bank is more visible.

The OBC zone corresponds to positive values of the CFG term andhigh clockwise-rotating vorticity magnitudes as provided in Table 2. Inaddition, positive values of the centrifugal term in the middle of C2-90degenerate the already developed vorticity in the previous cross-sectionC2-55. This assertion is underpinned by values from Table 2 and bycomparing magnitudes of the main cell’s vorticity in these two cross-sections (also cf. Fig. 3: C2-55 & C2-90). Despite potential experimentaluncertainties, the positive effect of the centrifugal term on the observedcirculation is significant. Moreover, from Fig. 7 the effect of the Froude

Fig. 3. Vectors of the normalized velocity components in the cross-stream section and normalized vorticityω H/Us . The red and green arrows indicate the approximate location of the OBCand main cell centre respectively. Circular arrows show sense of motion for main secondary flow (counter-clockwise) and OBC (clockwise). Red line represents zero vorticity.

Table 2Normalized mean magnitude of streamwise vorticity ω| |H/Us . −10 2.

C2-20 C2-55 C2-90

F-5 F-8 F-16 F-5 F-8 F-16 F-5 F-8 F-16

MC 0.46 0.44 0.45 0.91 1.02 1.17 0.42 0.57 0.72OBC 0.05 0.04 0.02 0.12 0.14 0.09 0.29 0.20 0.14

U: Average velocity of all data sets F-5, F-8 & F-16.MC: counter clock-wise rotation (cross-stream circulation of main cell of secondary flow)OBC: clock-wise rotation in the cross section (OBC circulation).

Fig. 4. Normalized turbulent kinetic energy ∗TKE u/0.5 2 in selected cross-sections for F-5, F-8 and F-16.

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number on CFG patterns can be observed. With a Froude number in-crease the patch of positive CFG moves out of the measurement gridand probably retreats to the closer proximities of the outer bank. Theprocess is best perceptible with regard to the mid bend’s cross-section(C2-55). This is explicable by stronger main cell secondary flows (seeTable 2).

Fig. 8 provides normalized values of the anisotropy term (Eq. (6b)).Anisotropy profiles show high positive values near the channel’s cen-treline and in the vicinity of the zero vorticity line; this indicates thatthe maximum positive values of the anisotropy term are observedwhere the sense of rotation of the cross-stream flow changes. On theother hand, the maximum negative values were observed in theproximity of the centres of rotation of the main cell and OBC, respec-tively. These observations fit very well the degenerative nature of theanisotropy term in the vorticity equation. In the bend entrance (C2-20),the magnitudes of anisotropy values are lower compared to the onesthat are encountered in the mid (C2-55) and the apex (C2-90) cross-sections. Particularly, the anisotropy term is larger especially near theouter bank area. Regarding the streamwise vorticity equilibrium, this

can be considered as a parallel proof for the existence of a high CFGterm in the corresponding cross-sections. Recalling the increase of theCFG term from the entrance to the apex of the flume in the outer bankzone (Fig. 7), it is clear that the existence of high degenerative terms isnecessary to keep the balance between generation and dissipation of thevorticity. It is not possible to detect any clear dependence of patterns ofthe anisotropy term on the Froude numbers, neither near the outer banknor over mid-sections.

Fig. 9 shows distributions of the homogeneity term HOM (Eq. (6c)).Patches of slightly positive values are observable near the outer banks.As can be seen, this term contains positive and negative values in theOBC zone that approximatively balance. Contrary to the anisotropyterm, the homogeneity term apparently reacts to a Froude number in-crease by pushing zones characterized by positive values near the outerbank out of the measurement grid.

5. Discussion

In all experiments performed in a racetrack flume, an outer bank

Fig. 5. Normalized lateral Reynolds stresses− ′ ′ ∗v v u/n z2 in selected cross-sections. The red and green upward arrows indicate the approximate location of the OBC and main cell centre,

respectively.

Fig. 6. Normalized cross-stream turbulence production rate ∗PκH/u3. The red and green upward arrows indicate the approximate location of the OBC and main cell centre respectively.The red bold line =ω( 0)s shows the change in the sense of direction of the secondary flow.

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cell was observed with opposite sense of rotation with respect to themain cell of secondary flow. The vertical location of the OBCs in thepresent study is in good compliance with wide channel measurementspresented by Booij (2003) (observed in z/h ≈ 0.45 – 1.00) and straightchannel near wall secondary flows (observed in z/h ≈ 0.40 – 1.00)reported by Nezu and Nakagawa (1993). The investigated low Froudenumbers provide similarity with the natural state of rivers. Also, thestreamwise distribution of mean velocity (Fig. 2) is in agreement with anumber of field experiments (e.g. Anwar, 1986; Frothingham andRhoads, 2003). As the Froude number increases, the observed OBCzones shrink (decrease in size and magnitude). In cross-sections wherethe contribution of TKE near the outer bank is higher (lower Froudenumber), the OBC is more pronounced (cf. Figs. 3 and 4). SimilarlyKang and Sotiropoulos (2011) showed that the shear layers are iden-tified by high TKE and are liable to initiate outer bank three-dimen-sional vorticities. As observed in previous studies (e.g. Blanckaert andDe Vriend, 2004) a clear transfer of energy from turbulence to meanflow (P < 0) is also discernible in the vicinity of outer banks (Fig. 6).

Based on the observations the flux of TKE towards mean flow kineticenergy is smaller for higher Froude numbers. In the current study, thebalance between centrifugal force and turbulence terms in the vorticityequation (Eq. (6b)b & c) was investigated. It was found that the balanceis only discernible between anisotropy (ANS) and centrifugal terms(CFG) (see: Figs. 7 and 8); the homogeneity term (HOM) was con-siderably smaller (Eq. (4)). This result complies with Blanckaert et al.(2012). Near the outer bank where the OBC occurs, the HOM termchanges sign and shows positive (i.e. vortex generating) values. Forhigher Froude numbers both the positive values of HOM and themagnitude of OBC vorticity (see Table 2) decrease. In summary, theincrease in Froude number is followed by a size-reduction of nearboundary zones of high TKE. In addition, negative values of the pro-duction equation (i.e. the inverse cascade), show weaker fluxes of thekinetic energy from turbulence to the mean flow. Consequently, OBCs(as turbulence related structures) will be constrained to the closeproximity of the outer bank and near to the surface. It is where bothwall and surface boundaries affect the flow and lead to an increase of

Fig. 7. Normalized profiles of the centrifugal term CFG/ U2

H2 of the vorticity equation .The red and green upward arrows indicate the approximate location of the OBC and main cell centre,

respectively. The red bold line =ω( 0)s shows the change in the sense of direction of the secondary flow.

Fig. 8. Normalized profiles of the anisotropy term ANS/ U2

H2 of the vorticity equation .The red and green upward arrows indicate the approximate location of the OBC and main cell centre

respectively. The red bold line =ω( 0)s shows the change in the sense of direction of the secondary flow.

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turbulence anisotropy. In the proximity of the outer wall and near thesurface conditions are in favour of OBCs and mean flow characteristicshave the least influence on these counter-rotating cells. Particularly, asthe Froude number increases, OBCs are migrating to the proximity ofboundaries where higher amounts of turbulent energy exist.

6. Conclusions

This study reported velocity measurements for experiments at threelow Froude numbers (F-5, F-8 and F-16) in a racetrack laboratoryflume. The observation focused on three cross-sections (C2-20, C2-55and C2-90) in the second bend of the flume. In addition to the existingmain cell of secondary flow, a counter-rotating cell (OBC) was observedin the outer bank region of the investigated cross-sections. For F-5 andF-8 run tests (low Froude numbers), the near bank reduction of TKE inC2-55 & C2-90 coincides with the obvious presence of the OBC. It canbe seen that an increase of the Froude number in the bend gives rise tothe extension of the main cell of secondary flow over a larger area of thecross-section; meanwhile, the OBC is pushed towards the boundaries.Based on the experimental results the main cell of the secondary flow isdriven by mean flow properties, the exchange between mean flow andturbulence (see Fig. 6: turbulence production rate Pκ) shows a gradient

toward the main cell of secondary flow; therefore, the OBC which ispartially driven by the turbulence energy, loses its energy to the maincell of secondary flow. The observed inverse flux of kinetic energy fromturbulence to the mean flow in the vicinity of the OBC, confirms thatturbulence-induced vorticity generation contributes to the generationof the OBC. This process is dependent on the Froude number, corre-sponding to weaker inverse fluxes of energy. As a result OBCs shrink insize and migrate to the zones where a higher amount of turbulent ki-netic energy exists. Moreover, the centrifugal term (mean flow vortexgenerator term) favours the OBC. It reacts to the increase of the Froudenumber by decreasing values in the proximity of the outer bank. Con-sidering the streamwise vorticity equations, the anisotropy and homo-geneity terms oppose the centrifugal term, but their reaction to theFroude number does not form a clear pattern. The homogeneity termcontains positive and negative contributions that are approximativelyin balance. When investigating whether the OBC has a stabilizing effecton the banks or not, it should be taken into account that its position andmagnitude, in addition to the flow regime, are affected by the state ofthe mean flow. Further investigations would be desired in order to re-veal possible effects of the OBCs on bank erosion and channel stabilityin different flow regimes.

Appendix

The quality of the measured data is crucial for a proper data analysis. Considering the mean flow measurement, both the signal–noise-ratio (SNR)and the correlation coefficient (COR) were observed continuously during the course of measurement and in the analysis phase. The SNR and COR forthe ADV receiver were always higher than 84% and 18 dB, respectively. These values meet the manufacturer’s criteria; which suggests SNR and CORalways higher than 75% and 15 dB Also, the convergence time of mean velocities with a 95 percent confidence interval was considered in order toverify whether the measurement time for each data point was long enough. For all three experiments time convergence control was applied and therewere not any points where the measurement time (90 s) was lower than the maximum convergence time, determined to be 80 s. Fig. A.1 provides avisual depiction of this analysis for the test run F-8. Hence, concerning the ADV measurements, acoustic Doppler noise error does not affect the meanvalues, unless the error is caused by strong velocity gradients. This kind of error is recognized as the only source of error in averaging procedures(Abad and Garcia, 2009) and was not found to be the case in the current measurements.In order to assess the quality of the measured turbulenceparameters, such as power spectrum and time scales, a method proposed by García et al. (2005) was applied to the data series. This method is basedon calculating a dimensionless parameter (F) which should be higher than a critical value of 20. Previously, Nezu and Nakagawa (1993) based on anempirical analysis proposed a threshold value of 16.67 for open channels. The parameter is calculated as follows:

= ×LF fUcon (A1)

where f is frequency, L the energy containing eddy length (channel depth for open channels) and Ucon is the convective velocity:

Fig. 9. Normalized profiles of the homogeneity term HOM/ U2

H2 of the vorticity equation .The red and green upward arrows indicate the approximate location of the OBC and main cell

centre respectively. The red bold line =ω( 0)s shows the change in the sense of direction of the secondary flow.

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= ⎡

⎣⎢ + + +

′+

′+

′ ⎤

⎦⎥U v 1 2

vv

2vv

vv

2vv

2vvcon

2s2 n

2

s2

z2

s2

s2

s2

n2

s2

z2

s2

(A2)

A detailed description of the parameter calculation and analysis is provided by García et al. (2005). This method was applied to differentexperimental studies (Rodríguez and García, 2008; Abad and Garcia, 2009; Bhuiyan et al., 2010). In the current study, the belt drives’ shovels cangenerate surface waves. These affect the turbulence measurement quality of ADV measurements. Therefore, a general examination using the Fparameter was performed to confirm the quality of the applied data. Fig. A.2 depicts a plan of the F parameter for the racetrack; the minimum valuesof F were selected in each measurement column as the representative value of the corresponding column. Areas with very poor (F < 10) to poor(F < 20) quality are mostly discernible in the vicinity and downstream of the belt drive. Based on this observation, the second bend’s cross-sectionscharacterized by the same geometry (i.e. cross-sections C2-20 through C2-90) were selected for the current investigation. The calculated F para-meters for these selected cross-sections are visualized in Fig. A.3. It can be seen that in all cases F values are beyond 20 (F > 20), indicating thatsmall errors could be introduced, but important parts of the power spectrum are resolved. The high F values in most of the selected cross-sections aredue to the high sampling frequency (50 Hz) and relatively low-velocity range (6.7–19.7 cm/s).

Experimental data sets contain measurement uncertainty (UNC). Determination of this parameter in order to investigate hydrodynamic processesis crucial. The UNC addresses a p N robability interval in which a measured property lies within. Systematic (bias) and random (precision) errorscontribute to this parameter. The UNC is determined (at 95% confidence level) using the following equations (Tavoularis, 2005):

= +UNC p b2 2 (A3)

where b and p are the systematic and random errors, respectively. The random error is defined as:

Fig. A1. Convergence time in seconds for mean streamwise, lateral and vertical velocity components (second experimental run F-8; measurement points are marked with black dots).

Fig. A2. F parameter plan for the racetrack flume.

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=p σN

2(A4)

where σ is the standard deviation and the number of the data points. One should take into account that a proper data point has to be anindependent variable. Therefore, in order to consider variable independency the following equation is used:

∑= ×

+ −=p σ

N

ρ k

U2

1 2 [(1 ) ( )]k

NkN

1

(A5)

where ρ (k) is an autocorrelation function with time step k. The systematic error (b) includes a large part of the uncertainty for current data series;its corresponding value is provided by the ADV device manufacturer: b= ±0.5% of measured value (Nortek, 2009).

Random error is classified as spatial random errors (related to the grid coverage) and temporal random errors (related to the length of themeasurement time series and their standard deviation). In the current measurement the spatial random error is limited to a few tenth of one percent(0.1–0.4%); therefore it is ignored. Using Eq. (A.3) the uncertainty for the measured and derived quantities is provided in Table 2. For uncertaintyestimations of vorticity and other derived quantities, Blanckaert and De Vriend (2004: Appendix) provided a formulation which was adopted here.

As mentioned, in this study a large portion of the measurement uncertainty is introduced by systematic errors. It is already known that systematicerrors are not prohibitive for hydrodynamic investigations as long as they remain constant in space (Blanckaert et al., 2010). As shown in Table A.1,measurement uncertainties are in a similar range for each of the three runs investigated.

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