Flow Structure over Bed Irregularities in aStraight Cohesive Open Channel

Dina Vachtman, Ph.D.1; and Jonathan B. Laronne2

Abstract: This paper describes high-resolution in situ three-dimensional (3D) bed topography and 3D flow field measurements in cohesive,straight channel reaches and provides analysis of reciprocal relationships between flow structure and bed topography. Existence of secondaryflows and their structures are shown to be associated with variety of bedforms. Classification of bed microtopography and identification of themajor governing flow mechanisms are described for three different types of bed morphology: single trough, single ridge, and sequences oftroughs and ridges. The maximum mainstream velocity component of secondary cells attains values as high as approximately 20% of theaverage mainstream velocity, and secondary cell width can reach 1=2 bedform width. Analysis of lateral versus vertical fluctuations wasperformed, revealing a good correspondence (r2 ¼ 0:67) between cell form and temporal fluctuations of velocity attributes; with diminishedsize of the secondary flow cell, the number of flow reversals increases. For a channel with given cross-sectional geometry, discharge and meanvelocity, the downflows are almost twice as large and have higher turbulent intensities than upflows. The three-dimensionality has vitalimplications for understanding the flow structure over channel beds, emphasizing the significance of the effect that secondary circulationin addition to lateral and vertical flow perturbations may have on the general flow field and associated bed topography. DOI: 10.1061/(ASCE)HY.1943-7900.0000447. © 2011 American Society of Civil Engineers.

CE Database subject headings: Secondary flow; Velocity distribution; Bed forms; Topography; Streamflow; Open channel flow.

Author keywords: Cohesive bed; Secondary flow; Velocity distribution; Bedform topography; Stream flow-channel bed interactions.

Introduction

Bedforms are found in fluvial, estuarine, and marine environments.They are formed under a wide range of flow patterns and charac-terized by different sediment sizes. Despite significant atten-tion (McLean 1981; Wang et al. 2003, 2004; Wang and Cheng2005, 2006), analysis of flow over bedforms is far from completeand presents a major obstacle to the understanding of flow-forminteractions. Recent advances in computer technologies have ledto remarkable developments in the area of computational fluid dy-namics (e.g., Joung et al. 2007), however, the effect of bed mor-phology on flow direction in natural streams has not been fullyexplored. Few prototype studies have demonstrated the effects ofbed morphology on the mean downstream and cross-stream veloc-ity vectors (Roy and Bergeron 1990; Bridge and Gabel 1992),associated bed-shear stress and consequent entrainment, transport,and deposition of sediment (Dietrich and Smith 1983, 1984).

Existence of secondary flow structures was reported over a va-riety of bed irregularities (Allen 1984, 1985; Nezu and Nakagawa

1984, 1993). Two types of secondary flows were identified: oneinvolves secondary flows owing to a centrifugal force in a curvedchannel, and the other, of interest in the current study, involves sec-ondary flows owing to the anisotropy of turbulence in straight openchannels or ducts (Jin et al. 2004). Additionally, studies have beenperformed to understand the underlying physics of secondary flows(Gessner 1973; Demuren and Rodi 1984; Bradshaw 1987), quan-titative and qualitative descriptions of stream flow-channel bed in-teractions were proposed (Nezu and Nakagawa 1993; Sidorchuk1996; Wang and Cheng 2006), and single and two-phase nume-rical models of flow and initiation of sediment transport have beendeveloped (Du Toit and Sleath 1981; Longuet-Higgins 1981;Huynh-Thanh and Temperville 1990; Blondeaux and Vittori1990; Hansen et al. 1994; Kim et al. 1994; Andersen 1999; Fredsøeet al. 1999; Trouw et al. 2000; Kim et al. 2000) to describe flowover bed irregularities.

Mechanisms by which secondary flows are generated have beeninvestigated (Allen 1984; Hirano and Ohmoto 1988; Karcz 1966;Nezu et al. 1988; Tsujimoto 1989). These studies indicate that sec-ondary flows generated as paired counterrotating cells (or cellularsecondary flows) are responsible for lateral sediment transport,which, in turn, maintains bed topography (McLelland et al. 1996).However, the transformation of secondary flows is influenced bybed topography, which implies a reciprocal relationship betweenflow and bedforms.

It has been suggested that the process of bedform evolution isinitiated by the presence of a corner vortex induced by the sidewalleffect (Nezu and Nakagawa 1989, 1993), generating variations inbed shear stress that further bring rise to additional vortices. Astheir sequence grows, secondary currents eventually occur acrossthe entire cross section. However, the appearance of ridges andtroughs in the central region of a channel, even when the initialnear-wall ridge is poorly developed, implies that the sidewall effectis not required for the generation of such bedforms and cellular

1School of Earth, Atmospheric and Environmental Sciences, Univ. ofManchester, Oxford Rd., Manchester, M13 9PL, UK; formerly, Dept. ofGeography and Environmental Development, Ben Gurion Univ. of theNegev, Beer Sheva, 84105, Israel (corresponding author). E-mail: Dina.Vachtman@manchester.ac.uk

2Professor, Dept. of Geography and Environmental Development,Ben Gurion Univ. of the Negev, Beer Sheva, 84105, Israel; andLaboratoire d’Etude des Transferts en Hydrologie et Environnement-LTHE, Université Josef Fourier, 38041 Grenoble cedex 09, France.

Note. This manuscript was submitted on April 12, 2010; approved onApril 28, 2011; published online on April 30, 2011. Discussion period openuntil April 1, 2012; separate discussions must be submitted for individualpapers. This paper is part of the Journal of Hydraulic Engineering,Vol. 137, No. 11, November 1, 2011. ©ASCE, ISSN 0733-9429/2011/11-1335–1346/$25.00.

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secondary flows (Nezu and Nakagawa 1989). The initiation ofcellular secondary flows was argued to be an instability-relatedprocess (Ikeda 1981; Colombini 1993; McLelland et al. 1999),where small disturbances from the bed provoke streamwise vorti-ces, while distance between turbulent structures increases with dis-tance from the wall (Defina 1996).

Bedforms in river beds vary in size, shape, and texture (clay,sand, or gravel beds of different roughness); their orientationmay be longitudinal, spanwise, or irregular (Wang and Cheng2006). Such periodic bedforms have been observed in flume

experiments (Allen 1966; Ikeda 1981; Hirano and Ohmoto1988; Nezu and Nakagawa 1989; McLelland et al. 1999) primarilyin uniform, noncohesive sand beds. A recent study on spatialheterogeneity of flow over a gravel bed (Buffin-Bélanger et al.2006) pointed out that local obstructions and bed microtopographygenerate spatial variations in the properties of the vertical com-ponent of velocity and its alteration over small distances (Danceyet al. 2000). The first exploratory experiment in an open channelflow with rough and smooth strips (Müller and Studerus 1979)reported that upflows occurred over the smooth strips and

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secondary lateral (Vy):right-to-left left-to-right

downstream low

downstream high

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Fig. 1. The studied channel, location of monitored reaches, and their 3D plots: (a) map of the Dead Sea, indicating location of the Ein-fesh’ha naturereserve; (b) digital elevation model (DEM) of the monitored channel, indicating the three studied cases; (c) high-resolution bed topography of thestudied cases and schematic illustration of the corresponding flow patterns

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downflows over the rough strips, forming a pair of counterrota-ting flow cells with a diameter equivalent to the flow depth. Anexperiment over artificial longitudinal ridges in trapezoidal crosssection (Nezu and Nakagawa 1984) distinguished a pair of counter-rotating flow cells, with upflows occurring over the ridge anddownflows over the trough. These investigations, accompaniedby advances in field, laboratory, and numerical techniques, havecontributed to an improved description of the key flow featuresover bedforms. However, all of these studies concerning flowand form interaction have concentrated on two-dimensional (2D)forms (Jin et al. 2004) varying laterally and longitudinally. Becausenatural bedforms are invariably three-dimensional (3D), also vary-ing vertically, the morphological simplification has imposed inher-ent limitations on the interpretation and understanding of bedformand flow dynamics.

The current study aims to contribute to the understanding ofthe interrelationships between bed topography and flow struc-ture in a three-dimensional context. This paper presents resultsfrom a small natural channel, where data obtained in situ focuson processes associated with modification of flow structure overbedform irregularities in a straight channel with cohesive bedand banks, where bed elements vary in shape and size. To providedetailed formulation of the flow structure and bed element inter-dependence, the observed flow structures are described in termsof their 3D form and 3D velocity field. The three-dimensionalityhas vital implications for understanding the flow structure overchannel beds, because 2D demonstration of bedforms may be in-sufficient to understand the significance of the effect that secon-dary circulation and lateral and vertical flow perturbations mayhave on the general flow field, turbulence, momentum trans-fer, shear stress, and their interrelationships with overall plan-form curvature and its dynamics. Additionally, the interactionbetween bedforms and flow pattern studied here is a particu-larly important aspect of bedform dynamics, about which littleis known.

Setting

This study was performed in the Ein-fesh’ha nature reserve locatedin the northern part of the Dead Sea [Fig. 1(a)], where the per-ennial brackish spring water flows into the Dead Sea from amountain aquifer through newly exposed clayey lacustrine land.Ein-fesh’ha springs flow eastward along the wide, up to a 1-kmstrip located west of the present shore. Measurements were carriedout in a cohesive, straight channel [Fig. 1(b)], in which waterdischarge was measured continuously over a two-year period(Vachtman et al. 2006; Vachtman 2009) and was found to besteady (0:15 m3 s�1). The longitudinal bed slope along the mea-sured segment is 0.0013 and varies within �2%. The channelbed is characterized by bedform irregularities classified as troughsand ridges [Fig. 1(c)]. The shapes and location of these are shownin Figs. 1(b) and 1(c) and their geometrical characteristics arelisted in Table 1.

Equipment and Procedure

Side-looking (N-0132) and down-looking (A-815) acoustic Dop-pler velocimeters (ADVs) manufactured by NorTek and SonTek,respectively, were used to measure the 3D velocity distribution.These ADVs measure instantaneous velocities at a single pointand at the location of the sensor relative to the bed/sidewall.The instrument was attached to a device allowing control of its ver-tical and horizontal positioning within an error of 1 mm. The veloc-ities were measured by placing the ADV on a 3-m long and 1-mwide bridge set on the channel banks at the monitored cross sec-tions with no disturbance to the flow. The vertically spaced sam-pling resolution over the cross-section was 5 mm from the bedupward to 20% of the total depth, 1 cm at 20–50% depth, and2 cm at 50–95% of the depth. The horizontal resolution alongthe cross section was 5 cm. The lowest measured spot was located3 mm above the bed and the highest approximately 3 cm below thefree surface (to avoid surface water interference).

Table 1. Details of Geometrical Characteristics of the Studied Bedforms

Bedform Secondary cell location Shape

Dimensions

Width relative tobedform width (%)

Height relative toflow depth (%)

Trough (Case I) Right and left plains Horizontally skewed 100 100

Ridge (Case II) Crest and right and left plains Horizontally skewed 50 100

Sequences of troughs and ridges

(Case III)

structures are listed from left to right

First trough Diverted to right edge of the bedform Vertically skewed 70 40

First ridge Right edge Vertically skewed 50 100

Second trough Bedform center Vertically skewed 60 50

Second ridge Bedform center Horizontally skewed 50 50

Third trough Diverted to right edge of the bedform Horizontally skewed 80 50

Third ridge Diverted to right edge of the bedform Horizontally skewed 50 50

Case I Case II Case III

Average velocity Vx for cross section (cm s�1)a 50.7 45.4 41.2

Standard deviation (cm s�1) 29.6 22.7 29.3

Secondary cell Vx maximumb (cm s�1) 11.8 (23)c 8.2 (18) 6.3 (15)

Standard deviation (cm s�1) 2.5 2.1 1.2aVx velocity measurements are typically less than �5%.bLateral (Vy) and vertical (Vz) components: the calculated uncertainties are within �10%.cThe magnitude (in percent) relative to the average Vx for the cross section is shown parenthetically.

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Velocity profiles were recorded at a sampling rate of 25 Hzfor a minimal 60 s duration. With more than 500 sampling pointsmonitored for each cross section, this resulted in approximately750,000 measurements of 3D velocity for each cross section.ADV data with a signal correlation greater than 70% were related(Lane and Richards 1998) and the in situ measurements of random3D ADV data were checked for spikes in a 3D phase space (afterGoring and Nikora 2002). On average, approximately 10% of thesamples were removed (Fig. 2).

Data on bed topography were acquired using ADV bed signalsand high-resolution mapping using an electronic theodolite (SokkiaTotal Station model Set4C2) with a �10 mm horizontal and ap-proximately 2 mm height accuracy (Sokkia Co.), including exten-sive topographic surveys of channel slope, morphology, and bedmicrotopography.

Measurements were performed at a 5 cm horizontal spacing.Although the channel bed was invisible owing to the high concen-tration of suspended sediments, the accurate and dense measure-ments allowed extraction of small topographic elements.

The turbulent kinetic energy (TKE) method [Eq. (1)] wasused to calculate the boundary shear stress (τ0) field in the chan-nel, which displayed substantial spatial variability across the bedmorphology:

τ0 ¼ C1½0:5ρð< u02 > þ < v02 > þ < w02 >Þ� ð1Þ

where u0, w0, and v0 = velocity fluctuations of main, lateral, andvertical components of velocity, respectively; ρ = water density;and C1 = proportionality constant in the range 0.18–0.20 (Kim et al.2000).

From the computed turbulent kinetic energy, based on theaverage of the velocity variances for a given depth above the bed(Stapleton and Huntley 1995), the turbulent shear stress wasestimated (after Stapleton and Huntley 1995; Kim et al. 2000;Biron et al. 2004) as

τ ¼ C1TKE ð2Þ

where the specific value of C1 (0.19–0.21) may not apply at alllevels within the water column, because it requires confirmationbefore using the method universally (Kim et al. 2000).

Because the method is sensitive to instrument noise, estimatedinstrument noise is to be subtracted from turbulence statistics.Two procedures have been suggested for removing instrument

noise from the TKE calculation (Stacey et al. 1999; Williamsand Simpson 2004). At some level above the river bottom with-in the water column, turbulent kinetic energy is assumed to bezero. Hence, an error (or noise) of the velocity components wascalculated as

v02err ¼ min

�1n

Xn1

ðv� �vÞ2�

ð3Þ

where v02err = minimum variance of calculated water velocities forall depth cells; �v = average water velocity calculated from allvalid ensembles for a given water cell; v = water velocity for aspecific ensemble; and n = number of valid ensembles includedin the calculation. Subsequently, the error term presented previ-ously (assumed to represent variance owing to instrument noise)was subtracted from the variance as calculated for the verticalvelocity components.

Formulations for y and z components are identical in all coor-dinate directions.

The current in situ research required almost laboratory qualitymeasurements to be obtained under field conditions. Therefore, rec-ognizing that measured data include some bias even after removingthe peaks associated with instrument noise, the writers performeduncertainty analysis for the all values of the measured velocitycomponents and information derived thereof (e.g., mean velocity,turbulence intensity, TKE, τ 0). RMS for each velocity component(u0, v0, and w0) was calculated. Whereas the estimated uncertaintiesin the Vx velocity component measurements are typically less than�5%, for lateral (Vy) and vertical (Vz) velocity components thecalculated uncertainties are within a range of�10%. The uncertain-ties in the turbulent intensity, TKE and bed shear stress (τ b) arewithin �15%. Interestingly, the ADV-derived shear stresses wereoverestimated by more than 10% for low stresses, whereas goodagreement was found for high values. The uncertainties revealedusing the RMS method in this study seem to be more conservativethan those found by Hurther and Lemmin (1998, 2001) for uncer-tainty in acoustic Doppler velocity profiler (ADVP) measurementsand turbulent stresses (where mean-velocity measurements are typ-ically less than 4%, and the uncertainty in turbulent normal stress,turbulent shear stress, and the turbulent kinetic energy as lessthan 10%).

The parameterization used by SURFER software for the delin-eation of isovels [e.g., Figs. 3(b), 4(b), and 5(b)] contributesadditional uncertainty in the patterns of measured data. Thisuncertainty, however, is both complex to estimate and difficultto define.

Results

The results are divided into three cases: Case I, a cross section witha single trough; Case II, a cross section with a single ridge; andCase III, a cross section with sequences of troughs and ridges.In all studied channel segments, the water surface is equivalentto channel width, i.e., the bankfull stage. The hydraulic data asso-ciated with the studied cases are summarized in Table 2.

Case I: Single Trough

A high-resolution topographic survey along an approximately1.4 m channel segment [Fig. 3(a)] reveals the existence of a singlelongitudinal trough at the center of the channel bed. Flow measure-ments for a cross section are shown in Figs. 3(b) and 3(c). Thestreamwise (Vx) velocity component is generally directed down-stream, with a maximum value located near the water surface at

Fig. 2. Example of data set checked for spikes in an ellipsoid 3D phasespace: data in the darker ellipsoidal ring are excluded

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midsection. A slight upstream flow motion is recorded in proximityto the banks [Fig. 3(b)]. The contour shapes of the velocity com-ponents over the trough are in good correspondence with the bedwavelength and its form, remaining so even at 80% of the depth.The Vx contours over the trough gradually expand when approach-ing the center and skew while approaching the banks.

Flow separation, particularly on the right side of the troughcrest, is illustrated by the secondary velocity vectors [Fig. 3(c)],indicating two counterrotating flow cells, with motion from thecenter of the cross section toward the banks is recorded for a lowerportion of the fluid, and vice versa for the higher portions of thefluid. Whereas downflow occurs above the trough, upflows aregenerated over both plains in proximity to the banks. Upstream[Fig. 3(b)] and upward flow motion accompanied by small counter-rotating cores [Fig. 3(c)] adjacent to the bank indicate the sidewalleffect. This suggests that banks play an important role in generating

secondary currents that may extend to the entire water depth acrossthe entire cross section (Table 1).

Case II: Single Ridge

A high-resolution topographic survey along an approximately1.4 m channel segment [Fig. 4(a)] reveals existence of a singlelongitudinal ridge at the center of the channel bed. Flow measure-ments for Case II are shown in Figs. 4(b) and 4(c). While thestreamwise (Vx) flow component above the plains on both sidesof the ridge is directed downstream, the Vx over the ridge crestis directed upstream [Fig. 4(b)]. It is represented by a multicorestructure of different sizes, where the three cores are roughly lo-cated at the middle of the ridge and at both of its sides. The Vxvelocity over the right and left plains is approximately 20% higherthan the average Vx. The velocity vectors of the secondary flowsalso reveal two counterrotating secondary flow cells on both sidesof the ridge [Fig. 4(c)], where the flow above the left plain exhibits

Table 2. Hydraulic Characteristics of the Studied Cases

Parameter Case I Case II Case III

Discharge (m3 s�1) 0.15 0.15 (�5%) 0.15 (�5%)

Average flow depth (m) 0.29 0.19 (�0:01%) 0.18 (�0:01%)

Maximum flow depth (m) 0.42 (�0:01%) 0.21 (�0:01%) 0.23 (�0:01%)

Width (m) 1.11 (�0:01%) 1.44 (�0:01%) 1.43 (�0:01%)

Average Reynolds number 1:48 × 105 1:16 × 105 1:17 × 105

Discharge values are within 5% measurement error and average flow depth error is �0:01 m.

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Fig. 3. Case I, single trough: (a) DEM of bed topography, where solidline indicates location of 3D flow measurements; (b) contour plots ofmain Vx time-averaged velocity component; positive values indicatedownstream direction; negative values denote upstream flow; down-stream velocities are considerably larger; (c) flow pattern of secondaryflows (Vy and Vz)

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Fig. 4. Case II, single ridge: (a) DEM of bed topography, where solidline indicates location of 3D flow measurements; (b) contour plots ofmain Vx time-averaged velocity component; positive values indicatedownstream direction; negative values denote upstream flow; own-stream velocities are considerably larger; (c) flow pattern of secondaryflow (Vy and Vz)

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bankward motion in a higher part of the fluid and from the banks tothe center of the cross section in its lower part. Vertical flow sep-aration occurs above the right plain, where upflow occurs over theridge and its margins [Fig. 4(c)]. Hence, the multicore structure ofthe streamwise component is presumably generated owing to aneffect of the lateral and vertical division of the flow, i.e., betweenleft and right flow separation in addition to between upflow anddownflow. The highest velocity of the secondary flow (up to20% of Vx) occurred over the interface between ridge and plain,likely resulting from the sudden change in bed roughness.

Case III: Troughs and Ridges of Varying Size andShape

A high-resolution topographic survey along an approximately1.4 m channel segment (Fig. 5) reveals a complex bed topographyconsisting of sequences of troughs and ridges of varying sizeand shape. Figs. 5(b) and 5(c) show contour plots of Vx, and time-averaged secondary velocity components for Case III, respectively.

Over and in proximity to the ridge (65–85 cm) the Vx velocitycomponent is directed upstream [Fig. 5(b)], as it is also in Case II[Fig. 4(b)], while near both banks the flow is directed downstream.Hence, the distribution of Vx consists of alterations of upstream anddownstream moving flow cells. Their cross-sectional distribution ismore variable than, but similar to, that for Case II. The secondaryflow vectors [Fig. 5(c)] demonstrate flow separation over the high-est ridge in midsection. In general, the flow has two counterrotatingcells of a pattern somewhat similar to that of Case II. Very low,

near-bed upflow velocities occur over the central ridge and itsneighboring troughs [Fig. 5(b)]. The maximum downflow occursabove the leftmost trough close to the bank, likely resulting fromthe sidewall effect on secondary flow generation [Fig. 5(c)]. Neg-ligible variations in Vz velocity component occur in proximity tothe water surface.

Recurrence of Secondary Flows

Cross-stream differences in the lateral flow component are plottedfor at four different depths for Case I and Case II (Figs. 6 and 7). Vyprofiles for Case I show a pair of counterrotating flow circulationsat both sides of the trough, where the Vy velocities close to thebottom are of relatively high magnitude [Figs. 6(a) and 6(b)].The cross-stream differences in the spanwise flow velocity at fourdifferent depths for Case I reveal a pair of counterrotating flowcirculations at both sides of the trough, where with increased depthVy velocities rise to almost twice the near-bed Vy values [Figs. 6(c)and 6(d)]. The cross-stream differences in the spanwise flow veloc-ity at four different depths for Case II reveal a pair of counter-rotating flow circulations at both sides of the ridge: Vy velocitiesclose to the bottom (0.5 cm) are of relatively small magnitude inboth left-to-right and right-to-left motions [Fig. 7(a)]. However,at 5 cm above the bed [Fig. 7(b)], the spanwise velocity rises toalmost twice the near-bed Vy values; at 10 and 20 cm above the bed[Figs. 7(c) and 7(d), respectively] the Vy velocity component pro-file is partly in opposite direction, but of similar magnitude relativeto that of the lowest fluid portion. The maximum lateral velocitiesare twice as high on the trough plains as those on the ridge plains(Case II).

The recurrence of a lateral component of cellular secondaryflows (Figs. 6 and 7) suggests a periodic, almost sinusoidal pattern.A similar cellular secondary flow pattern has also been identified influme studies (Colombini 1993; Nezu and Nakagawa 1993; Wanget al. 2003; Wang and Cheng 2006). However, the Vy velocity com-ponent profiles for a case of a single trough revealed in this studyexhibit fewer fluctuations with smaller amplitudes than those inlaboratory experiments for a similar bedform (Wang and Cheng2006). For Cases I and II, Vy varies with bed configuration andappears to be higher at the location above the smooth plains onboth sides of the trough/ridge in the lower portion of the flow.Vy becomes zero above the trough crest zone (Case I) and abovethe ridge and in transition between the ridge and its adjacent plains(Case II). When approaching the free surface [Figs. 6(d) and 7(d)],the variation in amplitude of the Vy values decreases slightly. Be-cause of the direct effect of topography, the near-bed Vy velocitiesabove the trough are less than that above the ridges [Figs. 6(a) ver-sus 6(b) and 7(a) versus 7(b)]. In contrast, the flow in the upperportion exhibits higher velocities above the troughs and smallervelocity above the ridges [Figs. 6(c), 6(d), 7(c), and 7(d)].

The secondary flows appear as single- and multicore cells overthe cross-sectional plane [Figs. 3(b), 3(c), 4(b), 4(c), 5(b), and 5(c)]and their shapes vary from circular to elliptical forms either verti-cally or horizontally skewed. Previous studies of temporal flow per-turbations (e.g., Bennett and Best 1996; Ferguson et al. 1996) havepointed out that vertical and horizontal rotation rates are morefrequent than those in the mainstream direction. In this studywe distinguish the spatial differences of secondary flow cellsthrough temporal analysis of the Vy and Vz velocity components[Fig. 8(a) and Table 3]. Although symmetric flow cells exhibitidentical number of vertical and horizontal reversals, horizontallyskewed cells exhibit more vertical reversals (which may occur onaverage as much as approximately 55% of the time), and vertically

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dept

h [c

m]

width [cm]

width [cm]

50 5010 10 10

30 300 0-1

-1

-3-3

-5downstream

downstream

upstreamupstream

0

-20dept

h [c

m]

(a)

(b)

(c)

Fig. 5. Case III, sequences of troughs and ridges: (a) DEM of bedtopography, where solid line indicates location of 3D flow measure-ments; (b) contour plots of main Vx time-averaged velocity com-ponent; positive values indicate downstream direction; negative valuesdenote upstream flow; downstream velocities are considerably larger;(c) flow pattern of secondary flow (Vy and Vz)

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skewed cells exhibit an opposite pattern, with horizontal reversalsoccurring on average as much as 62% of the time. For both hori-zontal and vertically skewed cells, the number of reversals withinthe cell grows as their size diminishes [Fig. 8(b)].

Following the change in flow structure downstreamwards (fromCase III to Case I), the channel aspect ratio increases from 1∶5 to1∶7, the rotation rate of the secondary flows increases, and thesecondary-flow vortices begin to stretch into the interior of thechannel. Consequently, there is ground to suggest that the Reynoldsnumber is reduced with increase in the number of secondary cells in

a cross section and it increases with increased velocity of the sec-ondary cells (Tables 1 and 2).

Analysis of Bedform Structures and AssociatedFlow Patterns

In the three cases presented, the flow structure is clearly guided bywhat may appear as small lateral changes in topography. The resultsof the in situ flow measurements presented previously clearly dem-onstrate that the flow structure in the presence of secondary flows

distance from left bank [cm]

distance from left bank [cm] distance from left bank [cm]

distance from left bank [cm]

bed

topo

grap

hy [c

m]

bed

topo

grap

hy [c

m]

Vybed topography

Vybed topography

Vybed topography

Vybed topography

bed

topo

grap

hy [c

m]

bed

topo

grap

hy [c

m]

(b)(a)

(d)(c)

Fig. 6. The cross-stream differences in Vy flow velocity component at four different depths for Case I [the ordinate scale of Figs. 6(a) and 6(b) differfrom those of Figs. 6(c) and 6(d)]: (a) 0.5 cm above bed; (b) 5 cm above bed; (c) 10 cm above bed; (d) 20 cm above bed

distance from left bank [cm]

bed

topo

grap

hy [c

m]

distance from left bank [cm]

bed

topo

grap

hy [c

m]

distance from left bank [cm]

bed

topo

grap

hy [c

m]

distance from left bank [cm]

bed

topo

grap

hy [c

m]

Vybed topography

Vybed topography

Vybed topography

Vybed topography

(a) (b)

(d) (d)

Fig. 7. The cross-stream differences in Vy flow velocity component at four different depths for Case II [the ordinate scale of Figs. 7(a) and 7(b) differfrom those of Figs. 7(c) and 7(d)]: (a) 0.5 cm above bed; (b) 5 cm above bed; (c) 10 cm above bed; (d) 20 cm above bed

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differs from that of 2D flows, where the log law is used to representthe profile of the primary velocity component in the form

kVxðyÞ�u�

¼ ln

�yR0

�ð4Þ

where Vx = streamwise velocity component; y denotes lateral co-ordinates;�u� = average shear velocity; k = von Karman coefficient(¼ 0:4); and R0 = zero velocity level or the hydrodynamic rough-ness length (Wang and Cheng 2005).

In the presence of secondary cellular flow, the distribution of theprimary velocity Vx deviates from that of Eq. (4), as revealed bythe contours of isovelocity of average Vx for three studied cases[Figs. 3(b), 4(b), and 5(b)]. These flow patterns indicate large var-iations not only in a magnitude of the mainflow component, butalso in its direction. While high speed zones with steep gradientare formed over a trough (Case I), low-speed zones (followedby upstream flows) are formed over a ridge (Case II). The ampli-tude of the changes becomes even more pronounced in a presenceof sequences of troughs and ridges (Case III). However, when

streamwise bed elevation mildly ascends or descends (such as inCase I), cell contours generally follow bed relief.

The secondary flow pattern is characterized by a pair of counter-rotating secondary flow cells, where the interaction between thecircular flows above the trough (Case I) transfers the highermomentum fluid from the surface region to the center of the crosssection; the upflow near the sidewalls brings upward the lower mo-mentum fluid. In contrast, the secondary flow structure for Case IIsuggests transfer of the higher momentum fluid from the surfaceregion to the sidewalls, and the upflow near the center of the crosssection on both sides of the ridge brings the lower momentum fluidupward. The adjacent flow cells over the trough separate at the bot-tom and reattach at the free surface, and detach over the ridge.

Secondary flows can also be generated by a wavy bed with el-evations in the range 7–25% of the total water depth (Case III). Inspite of the different bed configurations of Cases I, II, and III, theVz velocity component regularly exhibits upward motion above

nondimensional w:h form ratio

cell dimensions [cm2]

time

ratio

Vy:

Vz

reve

rsal

s

1.0 1.5

1.5

2.0

2.0

1.0

0.50.5

(a)

90

80

70

600 100 300 400 500200

time

of w

and

v pe

rtur

batio

ns

from

tota

l tim

e [%

]

(b)

Fig. 8. Analysis of spatial differences of secondary flow cells throughtemporal analysis of the Vy and Vz flow components: (a) temporal Vy

and Vz flow reversal ratio versus cell geometry (black dots: horizontallyskewed cells; white dots: vertically skewed cells); (b) time duringwhich vertical and horizontal perturbations occur versus size of second-ary flow cells

Table 3. Spatial Differences in Cell Geometry Associated with Temporal Perturbations in Vy and Vz

Cellwidth

Number ofhorizontalreversals

Cellheight

Number ofverticalreversals

Celldimensionsw∶h ratio

Total numberof reversals

Average time ofreversals fromtotal time

Average timeof horizontalreversals

Average timeof verticalreversals

Average ratiotime w:hreversalscm cm percent percent percent

29.1 380 17.5 404 1.7 784 65.3 48.5 51.5 0.9

18.5 371 15.1 420 1.2 791 65.9 46.9 53.1 0.9

15.7 406 8.2 488 1.9 894 74.5 45.4 54.6 0.8

7.8 530 8.2 450 0.9 980 81.7 54.1 45.9 1.2

14.7 495 18.2 307 0.8 802 66.8 61.7 38.3 1.6

11.8 500 14.3 280 0.8 780 65.0 64.1 35.9 1.8

Note: The number of flow reversals was calculated for 1,200 readings (a measurement period of 48 s). Buffers of 12 s (6 s at the beginning of the measurementand 6 s at the end) were subtracted from the total time to minimize flow perturbations associated with movement of the measurement device.

10

5

-5

0

-10

-15

-20

bedform ID

ridges

troughs

ridges

troughs

00

5

10

15

20

25

0 1 2 3 4

1 2 3 4

turb

ulen

t int

ensi

ty a

bove

bed

form

[%]

Vz

abov

e be

dfor

m [c

m s

ec-1

]

bedform ID

(a)

(b)

Fig. 9.Motion and intensity of secondary component above bedforms:(a) average Vz above bedforms; (b) average turbulent intensity (mea-surement uncertainty �15%) above bedforms (bedform ID: identifica-tion number of ridge/trough of similar size)

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ridges [Fig. 9(a)]. Whereas the Vz velocity component over ridgesis associated with upward flow motion, the flow over troughs ex-hibits downward flow and higher turbulent intensities [Fig. 9(b)],implying higher momentum transfer from the flow toward the bed.When the lateral changes in bed slope are more irregular, such as inCase III, the near-bed flow results in complex hydraulic intercon-nections between ridges and troughs, and the respective secondaryflow cells may appear at the edges of the structures, at their sidesand even in corners [Figs. 5(b) and 5(c)]. This may result from thefact that primary high-velocity cores associated with the dominantdownstream velocity for Case III tend to expand to a greater extentthan the cores associated with Cases I and II owing to the increasein aspect ratio of the cross section up to 1∶7. The increase in aspectratio, therefore, affects the number of roll cells, which increases tofour, where the expanding primary flow cores lead to changes inshape and location of secondary cells. Consequently, the secondaryflow cells for Case III appear to be shifted and distorted, and theirassociated secondary flows are weaker.

Cross-channel variations in bed roughness and elevation cancause perturbations to streamwise bed shear stress (e.g., Colombiniand Parker 1995; Nicholas 2001). This induces lateral imbalance ofthe turbulent stresses of the flow, which correspondingly generatescellular secondary flows. Therefore, the location of secondaryflows cells will be closely related to the maximum difference inbed elevation; such secondary cells will appear in the transition be-tween troughs and ridges. These might explain why in a presence ofmultiple bedforms (Case III), the center of the cell-core is not al-ways located above the center of bedforms. However, interestingly,for all cases the width of secondary flow cell appears to be equal toalmost half of the respective bedform size (Table 1; Fig. 10).

Discussion

The difficulty of providing an in-depth physically based explana-tion for the studied cases has prompted the writers to focus on pro-viding unique information on flow characteristics performed inalmost laboratory-quality measurements in the field, in a naturalchannel exhibiting different bedform configurations. However,where appropriate, the paper refers to and discusses the experi-mental data and physical reasoning suggested previously in theliterature.

To analyze reciprocal relations between secondary flowstructure and bedform configuration, velocity magnitudes anddirections, in addition to average primary, lateral, and verticalvelocities were calculated. The secondary flows associated withvarious bed configurations appear complex, but they do exhibitsome regular patterns, which may be simplified in the followingmanner: the time-mean secondary flows over single trough and

ridge (Cases I and II) appear as paired cells in the cross-sectionalplane. These flow cells may have vertical dimensions up to the en-tire depth and horizontal dimensions up to half the channel width(Table 1). For sequences of troughs and ridges (Case III) secondaryflow cells can be generated by all types and intensities of bedformirregularities, although the details of the flow structure vary withbed geometry. Although flow cells generated by sequences ofbed perturbations tend to be weaker and become attached tothe bed surface, their shape and location may affect the flow struc-ture over the entire channel depth. This further implies that smallchanges in bed configuration do not necessarily generate negligiblesecondary flows as previously suggested by Nezu and Nakagawa(1984), but may be responsible for the initiation of counterrotatingtwin-cell flow over the entire cross section.

The analysis of secondary flows proposed for description of theinterconnection of such flows with different bedforms can hardly beestablished via analytical formulations owing to inherent math-ematical difficulties (Nezu and Nakagawa 1993). However, analyti-cal models for secondary flows (Ikeda 1981; Wang and Cheng2006) reveal results that are similar to those observed in a naturalchannel. For instance, secondary flow structures similar to those inCase I have been found in the computed structures (Nezu andNakagawa 1984; Wang and Cheng 2006). Secondary flow struc-tures for Case II are also analogous to previously reported flowstructures (Nezu and Nakagawa 1984), where the upflows aresomewhat stronger than downflows. The results of the currentstudy, however, indicate that for upflow and downflow componentsover trough and ridge in similar cross-sectional geometry, dischargeand mean velocity, downflows are stronger (p ¼ 0:02) than upflows[Fig. 9(a)]. The difference in the behavior of the Vz velocity com-ponent over ridges and troughs as documented in the current studyimplies different effects of secondary flow mechanisms on bedformmaintenance: whereas upward flows with lower turbulent inten-sities over ridges tend to maintain the given bedform shape, thedownward motion accompanied by higher turbulent intensities overtroughs is more likely to eliminate troughs.

Additionally, lateral flow velocities have large magnitudes evenvery close (approximately 1 cm) to the bed. The trend of velocitydistributions close to the bed is complex owing to the direct effectof bed roughness, often referred to as the roughness layer (approx-imately 0.05 mm) with a thickness 1–4 times the roughness elementsize (Raupach et al. 1991). Analysis of the flow in this layer is be-yond the scope of this study, in part because of the limitation of thenear-bed ADV measurements, which produce an echo from theboundary and bias the velocity reading. To resolve such limitations,joint experimental and numerical investigations should be carriedout. Consequently, the 3D simulation can be used to investigate thespatial variation of the main flow variables, mean velocity, turbu-lence statistics, pressure, and bed-shear stress in much greater detail(Strom et al. 2007). Moreover, the strongest Vy gradient usuallyoccurred over the interface between the trough and ridge (Figs. 6and 7), probably owing to the sudden change in bed topography.

Earlier flume studies of secondary flows have demonstratedthat the highest velocity of a secondary current may reach 3%of the maximum mainstream velocity (Karcz 1973; Nezu andNakagawa 1993). In flow subjected to secondary motion as a resultof bed perturbations, secondary flows are commonly much weaker(< 5%) than the primary flow (e.g., Nezu et al. 1993; Papanicolaouand Hilldale 2002). Velocity profiles describing the spatial variabil-ity of roughness length and the properties of turbulent flow struc-tures along a reach of a gravel bed river have shown that thecomplexity of the bed topography is not reflected in the meancharacteristics of the velocity profiles (Lamarre and Roy 2005).However, results presented in this paper show that not only can

40

30

20

10

00 10 20 30 40 50 60 70 80

bedform width [cm]

seco

ndar

y flo

w c

ell w

idth

[cm

]

troughs

ridges

Fig. 10. Geometry-related relations between secondary flow cell widthand bedform width

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secondary cell velocity reach approximately 20% of the averagemainstream velocity, but the presence of roll cells has a consider-able distortional effect on the primary velocity profile. Variation inboth magnitude and direction of the three velocity components areclosely related to the maximum difference in bed elevation: in-crease up to approximately 30% and reduction up to approximately30% in average Vx were found for 10% water column over troughsand ridges, respectively, indicating consequent downstream andupstream flows. This might result from the fact that the size ofthe bed element for channel with aspect ratio of 1∶5 (Cases Iand II: approximately 1=3 of the channel depth; 1=3 width) is suf-ficiently large to lead to a strong asymmetry in the average Vx,where maximum velocity is shifted toward the side walls of thechannel and corresponds to the location of the secondary cells.More precisely, this secondary flow consists of two counterrotatingvortices, which have a spanwise dimension approximately 70% ofthe width of the channel and are somewhat compressed against theupper and lower channel boundaries. This secondary flow affectsthe Vx velocity component profiles of the channel. Their high ro-tation rate causes the secondary cells to stretch into the interior ofthe channel, where it has a distortional effect on the Vx velocitycomponent. This somewhat reduces the flow rate, but increasesthe average velocity of the secondary cell. The separation distancebetween the cells for Case I is greater than for Case II, which isindicative of the fact that the flow over a trough is closer to thestability boundary (Speziale and Thangam 1983).

Increase in aspect ratio to 1∶7 and presence of sequences of bedirregularities (Case III) allows development of four secondary cellsthat may have been generated because of the vertical connectionbetween the adjacent ridges and troughs. The Vx velocity com-ponent distribution for this case, which is highly distorted by thepresence of multiple cells, has a distinct structure that is in reason-able agreement with the numerical study of secondary flows (e.g.,(Speziale and Thangam 1983). It is quite clear that the distortions ofthe Vx profile lead to a substantial reduction in flow (Vx maximum30% less than for Cases I and II). Moreover, the Vx velocity alongthe horizontal centreline of the channel is asymmetric, with maxi-mum velocity shifted bankwards owing to the bankward transfer ofmomentum by secondary flows. The Vx cores stretch toward theinterior of the channel cross section, whereas the smaller flow cellsnear the edges of the bedforms are vertically and horizontallyskewed.

Detection of turbulent structures has been used for quantitativedescription of burst- and sweeplike fluid motion associated withfluctuations of vertical and lateral velocity components (Bennettand Best 1995, 1996; Ferguson et al. 1996; Roy and Buffin-Belanger 1996; Sidorchuk 1996; McLelland et al. 1996). In thisstudy, temporal analysis of vertical and lateral fluctuations was per-formed for the detection of the signature of temporal reversals ofvelocity components in cell geometry. More research is needed tofully clarify the nature of secondary cell geometry in natural openchannel flow. However, as lateral bed perturbations owing to var-iations in bed roughness or bed elevation give rise to lateral gra-dients in bed shear stress (Wang and Cheng 2006), a rough stripmay be equivalent in its impact to a trough, and smooth to a ridge.Considering this similarity based on physical reasoning and exper-imental data, the relation between channel bed and cell configura-tion (Wang and Cheng 2006) may usefully be applied to secondaryflows in natural channels for the prediction of vertically and later-ally skewed cellular patterns of secondary flows for both equal andunequal bedform width.

This study has shown that the size and shape of bedformsstrongly influence the flow field distribution. Following the direc-tion of momentum transfer and distribution of turbulent intensity,

counterrotating flow cells in the cross-sectional plane suggest amechanism for maintaining the ridge-dominated bed topographyand elimination of trough-dominated beds.

Although the driving force for mass transport is gravity, theflow structure presented for the studied cases illustrates that thedirection of gravity differs from the direction of momentum trans-fer. Consequently, the understanding of small- and large-scale bedfeatures may also have important implications for estimation of theassociated diffusion coefficient and, consequently, for understand-ing of sediment transport.

Conclusions

In this study, the mean structures of secondary flows, generatedby bedform irregularities were based on in situ 3D flow measure-ments in a straight open channel. The bedforms included a singletrough (Case I), a single ridge (Case II), and varying ridges andtroughs of smaller width and height (Case III). The generated sec-ondary flows of pairs of counterrotating cells were characterized bycontours that imitate the bed surface and that may be attributed to aslow change in bed roughness, and by complex, multicore struc-tures somewhat diverted from the apex of the bedform. For allcases, cell width is equal to approximately half the associated bed-form width.

The strongest cross-flow usually occurs over the interface be-tween bed elements owing to the sudden change in bed roughness.Over the wavy bed of ridges and troughs, flow cells appear to beboth horizontally and vertically distorted; such geometrical differ-ences are supported by fluctuations in their temporal, lateral andvertical velocity components. With diminished size of the secon-dary flow cell, the number of flow reversals increases. In general,upflows occur over ridges, whereas downflows and higher turbu-lent intensities occur over troughs. In this study, the reciprocal re-lations between time-mean secondary flows and associated bedperturbations could be not been described analytically owing tothe complexity of these relations.

Results of experimental and flume studies of secondary flowsreported in the literature and the in situ derived secondary flow datashown here are generally in good agreement. The secondary flowstructures and patterns associated with different bedforms providean insight into the nature of secondary flow behavior in response tosmall longitudinal and lateral topographic variations occurring onan open channel bed.

Acknowledgments

This study was in part funded by grants from the Israel WaterAuthority, the Dead Sea Drainage Authority and the GermanBMBF-funded SUMAR project. The study has benefited fromwork carried out by students of the Department of Geographyand Environmental Development, the Ben Gurion University ofNegev, particularly Y. Munwes, and I. Schwartz, whose assistancewith data collection was invaluable. We thank the Ein-Fesh’haNature Reserve team for their support, and Tamir Grodek (HebrewUniversity of Jerusalem), Uri Shavit (Department of Civil andEnvironmental Engineering, Technion, Israel) and Roey Egozi(The Soil Erosion Research Station, Israel) for lending the side-looking ADV. We thank two anonymous reviewers whose com-ments helped to improve this manuscript.

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