River Flow 2012 – Murillo (Ed.)© 2012 Taylor & Francis Group, London, ISBN 978-0-415-62129-8

Higher-order moments of velocity fluctuations in an open channel flowwith mobile bedforms

H. Prashanth Reddy, Vesselina Roussinova, Ram Balachandar & Tirupati BolisettiDepartment of Civil and Environmental Engineering, University of Windsor, Windsor, Canada

ABSTRACT: Detailed two-dimensional particle image velocimetry (PIV) measurements were obtained in arough open channel flow (OCF) over an immobile rough bed and mobile 2-D dunes.The initial flow condition wasin the fully rough regime where the bed was immobile. As the flow velocity is increased, mobile dunes formed.This paper presents preliminary results for the double averaged statistics of mean velocities, turbulence intensitiesand higher-order moments including Reynolds shear stress, skewness and turbulent diffusion coefficients. Theresults obtained from the present mobile bedforms experiments are compared with the previously published datafor higher-order moments of velocity fluctuations on a smooth bed, fixed rough bed, and fixed rough 2-D dunes.It was found that near the moving bedforms, the skewness factors are approximately two orders of magnitudehigher than these obtained on the immobile rough bed at similar Reynolds number.

1 INTRODUCTION

Current knowledge of flow and turbulence over arough sand bed is still unable to explain the effect ofthe mobile sediment on the turbulence structure andthe mechanism that leads to formation of bed formssuch as dunes. Most of the previous experimental andnumerical studies on dunes were based on examiningthe turbulence structures over fixed dunes where therole of mobile sediment and continuous deformationof the bed was ignored. In natural flows, the bed issubjected to sediment transport due to the continu-ously changing flow conditions. Different bed formswere observed with increasing flow velocity. Schindlerand Robert (2005) reported seven distinct stages of thetransition from ripples to dunes in their mobile bedexperiments.

Large–scale vortices such as spanwise vortices,sweeps, ejections and kolk boil vortices contributeto the continuous deformation of the bed. Spanwisevortices are large–scale structures drifting within theboundary layer, with axes normal or nearly normal tothe flow direction. Ejections are responsible for theprocess when a low-speed fluid parcel or streak liftsaway from the wall, oscillates in three dimensions,and then breaks down so that fluid is expelled into theouter flow. The amount of ejected fluid that remainsas a result of retardation is swept away by high-speedfluid that approaches the wall in a process called sweepor in-rush process. Kolk boil vortices are large–scalevertical motions consisting of strong upward vortexmotions that are generated intermittently near the bedand they can transport fine sediment to the free surface.

Measurements of the higher-order moments inopen-channel flows with mobile sand bedforms are

very limited. Third-order moments provide usefulstatistical information on the temporal distributionof the fluctuations around the mean velocity andalso contain information related to coherent structures(Simpson et al. 1981; Gad-el-Hak and Bandyopad-hyay 1994; Keirsbulck et al. 2001). Here, u and vdenote streamwise and vertical velocity fluctuationsfrom mean velocities. Skewness factors defined asSu = u3/u3

rms and Sv = v3/v3rms describe the asymmetry

in the probability density function of turbulent fluctua-tions in streamwise and vertical directions respectively(Nezu and Nakagawa 1993). Here, urms and vrms denotethe root mean square velocities of streamwise and ver-tical components, respectively. Su can be correlatedwith coherent events occurring in the flow and has avalue of zero for a Gaussian distribution. Skewnessother than zero indicates the presence of anisotropicturbulence. The point of change of sign in the skew-ness profile is related to the crossover from sweepto ejection motions (Gad-el-Hak and Bandyopadhyay1994). Nakagawa and Nezu (1977) have shown thatthe relative contributions of ejections and sweeps tothe Reynolds stress at a point are quite well determinedby the third order moments.

The effect of non-porous fixed surface roughnesson the mean velocity profiles is to a large extentwell established (Balachandar and Patel 2002, Jimenez2004). Higher-order moments of velocity fluctuationsof flow on fixed rough dunes in open channels havebeen presented by Balachandar and Bhuiyan (2007).Higher-order moments of velocity fluctuations of flowin open channels with different aspect ratios have beenanalyzed by Roussinova et al. (2008). However, therole of turbulence structures on mobile beds is non-existent. This paper presents higher-order moments of

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velocity fluctuations to study roughness effects of amobile sand bedforms on the higher-order velocitymoments in a turbulent open channel flow. Results ofthe present experiments are compared with the previ-ously published data on flow over an immobile bed(rough porous sand bed without bed movement), flowover a smooth bed and flow over fixed rough duneswhere sand is glued to the dunes surface. Hereafter,flow over porous rough sand bed without bed move-ment is termed as immobile bed and porous roughmobile bedforms is termed as a mobile bed.

2 EXPERIMENTS

2.1 Experimental set-up

The experiments were conducted in a horizontal, rect-angular open channel flume 12.0 m long, 0.61 m wide,and 1.0 m high. The sand bed was 5.8 m long and0.15 m thick, and spanning the width of the flume.A ramp was built at the start of the sand bed to mini-mize any disturbances of the approaching flow (Fig. 1).The median size diameter (d50) of the sand grainswas 0.7 mm. A centrifugal pump was used to recir-culate the water in the flume. All experiments wereconducted under a uniform flow conditions. The testsection was located 3.5 m downstream from the startof sand bed. Maximum velocities and Reynolds num-bers based on flow depth and maximum velocities arelisted in Table. 1.

The PIV experiments were conducted in a verticalcentral plane of the channel (to avoid secondary cur-rents effects) and the size of the field of view (FOV)was 0.075 m × 0.075 m. A 300 mJ/pulse dual pulseNd-YAG laser was used to illuminate the seeded flow.The laser sheet was formed through a spherical lensand expanded through a cylindrical lens.The thicknessof the laser sheet was approximately 1 mm and the lasersheet entered the flow through the free surface allow-ing simultaneous measurements of the streamwise and

Figure 1. Schematic of experimental set-up.

Table 1. Summary of the experimental conditions.

DensimetricReynolds Friction Froude

Umax number velocity numberTests (m/s) Re u∗ (m/s) Fo

Immobile bed 0.240 16696 0.010 3.32Mobile bed 0.384 26713 0.024 4.64

wall-normal velocity components.The particle imageswere recorded using a 4 MP camera synchronized withthe laser using a TSI PIV laser pulse synchronizer. Thecamera was fitted with a 28–105 mm Nikon lens andadjusted to give the desired field of view. The cam-era resolution was 2048 × 2048 pixels, which resultsin each pixel representing 0.037 mm of the flow. Theflow was continuously seeded with spherical hollowglass particles of mean diameter 12µm with a specificgravity of 1.13. The same particles were used in previ-ous studies by Shinneeb et al. 2004 and by Roussinovaet al. 2010 who reported that the particles are suitableand follow the flow faithfully. For each experiment,2000 pairs of images were recorded at a frequencyof 1.04 Hz. The time separation between consecutiveimage pairs was �t = 1200 µs and 750 µs for immo-bile and mobile bed experiments, respectively. Thetime separation was maintained in such a way that themaximum particle displacement was 8 pixels (50%of the interrogation area). The images were analyzedwith the TSI Inc Insight 3G software. The velocity atany point was determined from particle displacementcalculated between two consecutive recorded imagesusing a two dimensional cross-correlation technique.Each frame was split up into a small interrogation areas(IA) where the correlation analysis is performed usinga fast FourierTransform. Initially, the particle displace-ments were determined by using an interrogation sizeof 64 x 64 pixels. The peak of the correlation was iden-tified by a Gaussian curve-fitting technique. Using theestimations of prediction, refinement of the velocityvector was obtained in a 32 × 32 pixel interrogationarea.A 50% overlap of the interrogation areas was usedto satisfy the Nyquist criterion.The resulting field con-sists of 127 × 127 vectors in each instantaneous image.Following the correlation analysis, outlier vectors wereidentified and replaced using the cellular neural net-work method with a variable threshold technique, asproposed by Shinneeb et al., (2004). The rejected vec-tors, estimated to be less than 5% and they are primarilylocated at the edges of the fields of view.The PIV yieldinstantaneous vector fields for the whole field of view,for each pair of images were analyzed. Flow and var-ious turbulence statistics were computed by applyingdouble averaging technique (Nikora et al., 2007) andthe results are presented in the following sections.

2.2 Flow conditions

The nominal depth of flow in the measurement sec-tion was 0.08 m. The channel aspect ratio (=channelwidth/flow depth) is equal to 7.6 which is larger thanthe critical value of 5 (Nezu and Nakagawa, 1993)established as a criteria for the effect of the secondaryflow. The three-dimensional effects are not signifi-cant since the measurements are made in the centerof the channel and the channel aspect ratio is large(≥5.0). The present experiments are carried out on aporous sand bed. A higher velocity and discharge wasmaintained at the mobile bed case compared to theimmobile bed. The Reynolds numbers Re = Umaxh/v

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based on the total depth (h) and maximum veloc-ity (Umax) and the values of the densimetric Froudenumber were shown inTable 1.The Froude number cal-culated for the two tests were <1, which correspondsto subcritical flow regimes.

The friction velocity (u∗) was obtained fromthe measured Reynolds shear stress in the region0.2 < y/h < 0.7 where the contribution from the vis-cous stress is negligible. The value of the Reynoldsshear stress was extrapolated to the wall and this valueis used to estimate the friction velocity. The Reynoldsnumber based on the streamwise distance of measure-ment section defined as Rx = u∗x/v are 3 × 104 and7 × 104 for immobile bed and mobile bed experiments,respectively. These values are similar to that used inprevious studies (Roussinova and Balachandar 2011,Bakken et al. 2005). A summary of the experimentalconditions is shown in Table 1. An estimate for criticalshear stress was obtained from the Shields diagram.The estimated critical shear was τoc = 0.340 N/m2 andcritical u∗c = 0.018 m/s respectively.

Sand bed corresponding to subcritical bed shearstress is called immobile, and bed shear stress τo muchgreater than critical turbulent shear stress τoc is calledmobile bed. In mobile bed experiment the shear veloc-ity is much more than the critical shear velocity whichresulted in formation and movement of dunes overthe bed.

3 MEAN FLOW

The vertical distribution of the mean streamwise veloc-ity, U was computed from the PIV data using adouble averaging procedure (Nikora et al., 2007).The velocity fields are first averaged over time andthen spatially averaged over constant vertical slices.The distributions of the streamwise component of themean velocity in outer variables are shown in Fig. 2.Maximum velocity (Umax) and flow depth (h) areused as scaling parameters for the mean velocity (U )and the wall-normal distance (y), respectively. Theresults from Faruque and Balachandar (2011) are also

Figure 2. Velocity distributions of immobile rough bed(Faruque and Balachandar 2011), present immobile roughbed and mobile bed.

included in Fig. 2. It should be noted that experimentsof Faruque and Balachandar (2011) were performed athigher Reynolds number on a rough bed with sand par-ticles with d50 = 2.46 mm. Near the bed at y/h < 0.12,the velocity profile on mobile bed is shifted to the leftsuggesting increase bed resistance. The shape of theprofile obtained by Faruque and Balachandar (2011)is somewhat different from the present measurementswhich could be due to the different bed and flow con-ditions. With increasing flow velocity, the critical bedshear stress is reached and particles start to move. Inorder for the particles to sustained mobility, energyfrom the mean flow is extracted which reduced theflow velocity near the bed. This is indicated from thevelocity distributions shown in Fig. 2. The velocityprofiles do not start from the same point in mobile andimmobile bed experiments because measurements arenot available very close to the bed in immobile bedexperiment.

4 REYNOLDS SHEAR STRESS

In Figure 3, double-averaged Reynolds shear stress(−uv) profiles scaled in outer variables are shown.For the present experiments, the maximum Reynoldsshear stress on immobile bed occurs near the bed aty = 0.072h whereas the Reynolds shear stress on theimmobile bed reported by Faruque and Balachandar,2011 reaches maximum at y = 0.19h. One should notethat the two experiments are conducted at differentReynolds numbers and at different bed conditions. Inthe present mobile bed experiments, Reynolds stressis reaching maximum at y = 0.3h. This indicates thatthe Reynolds shear stress on immobile beds is highernear the bed compared to that on the mobile bed. Thereduction of the Reynolds shear stress on the mobilebed is due to the accelerated bed particles which tend toreduce the turbulence generated near the bed. Movingparticles create a layer which reduces the vertical fluc-tuations and the maximum value of Reynolds stresson a mobile bed is shifted farther away from the bed

Figure 3. Distribution of Reynolds shear stress (−uv) formobile, immobile and rough beds.

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compared to this on the immobile beds. It is observedthat the maximum value of Reynolds stress in mobilebed is slightly less than that observed on the immobilerough bed experiments by Faruque and Balachandar(2011). It is concluded that peak of Reynolds shearstress of immobile bed (Faruque and Balachandar,2011) may be due to lower densimetric Froude numberand higher channel aspect ratio. While both Reynoldsshear stress profiles on the immobile beds approachzero at y = 0.73h, the Reynolds shear stress profile onthe mobile bed does not show the same trend.The smallvalue of Reynolds shear stress observed near the freesurface suggests that the flow is affected by the movingparticles throughout the entire depth.

Fixed rough dune and smooth OCF reported byBalachandar and Bhuiyan 2007 are also comparedwith present experiments as shown in Fig 3. SmoothOCF is agreeing well with immobile bed in presentexperiments. Peak of Reynolds shear stress of fixedrough dune is agreeing with that of immobile bed(Faruque and Balachandar 2011) and more thanpresent mobile bed. It appears that entire flow depthis affected over fixed rough dunes similar to mobilebedforms.

5 HIGHER-ORDER MOMENTS

The skewness factors Su = u3/U 3max and Sv = v3/U 3

maxshown in Figures 4 and 5, provide further informa-tion for the turbulence structures. In Fig. 4a is positiveclose to the bed for all flow conditions. Farther fromthe bed, a rapid decrease in the value of Su wasobserved for the immobile bed experiment. In thecase of the mobile bed, Su is positive near the bedand decreases rapidly attaining a minimum negativevalue at y = 0.2h. The point where the Su becomesequal to zero is also different from that observed onthe immobile bed experiment. The Su for the presentimmobile rough bed experiment compare well with theLDV experiments of Faruque and Balachandar (2011).Distribution of Su of smooth OCF is less than presentimmobile bed. Peak of Su distribution of fixed roughdune is morethan that of immobile bed experiment(Faruque and Balachandar, 2011). Near bed turbulencemeasurements are not available over fixed rough dunesand smooth OCF.

In the case of the mobile bed, it is interesting tonote that the magnitude of the negative and positivevalue of Su are very high (see Figure 4b). To eliminatethe Reynolds number effect, the present mobile bed(Re = 26713) experiment is compared with the LDVdata on immobile rough bed (Faruque and Balachandar2011) performed at similar Reynolds number. It isbelieved that the higher values of Su on the mobile bedare due to the moving particles. The sweep events arevery strong in mobile beds close to bed at y < 0.125h,if we go by Su values then sweep events in mobilebed are 100 times stronger than immobile beds. Theejection events in mobile beds are very strong fromy = 0.125h to 0.28h as compared to immobile beds.

Figure 4. Distribution of triple product u3 for mobile,immobile and rough beds.

Figure 5. Distributions of v3 for mobile, immobile andrough beds.

The variation of Sv is found to be positive through-out the depth for all flow conditions (Fig. 5) when thebed is immobile. A similar observation of positive Sunear the bed and positive Sv throughout the depth wasalso observed by Balachandar and Bhuiyan (2007) inflow over rough bed (sand is glued to the bed). Thevariation of Sv of mobile bed is quite different fromthat of immobile bed.

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Figure 6. Distribution of scaled triple product uv2+

forsmooth, fixed rough dune, immobile and mobile beds.

Close to the bed the variation of Sv of mobile bed isnegative, it reaches peak negative value at y = 0.06h,increases to zero value at y = 0.12h and Sv is foundto be positive for remaining depth. The maximumpositive value of Sv of mobile bed is more than theimmobile bed due to higher turbulence in verticalvelocity component near the bed in the case of mobilebed. The Sv value of mobile bed is decreasing onlyafter y = 0.6h as compared to y = 0.35h of immobilebed experiment, this shows extension of significantturbulence farther from the bed in the case of mobilebed. In mobile and immobile bed experiments, awayfrom near bed, negative value of Su and the positivevalue of Sv indicate a slower moving fluid parcel withan upward transport of u momentum representing anejection type motion throughout the depth. In mobilebeds, near to the bed, positive value of Su and the neg-ative value of Sv indicate that downward transport ofu momentum, which enhances bed movement. Thenear-zero value of Su and Sv of mobile bed aroundy = 0.12h is a cancellation effect of sweep and ejec-tion type events. In mobile beds, away from the wall,y > 0.12h the strength of ejection event increases withincreasing negative value of Su and increasing positivevalue of Sv. The values of Sv distribution of smoothOCF are less than those of immobile bed. The peakvalue of Sv distribution of fixed rough dune is occur-ring close to bed and almost equal to peak of mobilebed. However extension of turbulence over fixed roughdune is similar to immobile bed reported by Faruqueand Balachandar (2011). The rate of decrease of bothSu and Sv of mobile bed are less than the immobilebeds in the outer layer is an indication of extension ofturbulent zone close to the free surface in mobile bedwhere as non-turbulent zone exists close to the freesurface in the case of immobile beds.

Third-order moments of the velocity fluctua-

tions normalized by friction velocity Du = uv2+

and

Dv = u2v+

are shown in Figs. 6 and 7.The results are shown, for flow on smooth bed, flow

over fixed rough dunes (Balachandar and Bhuiyan

Figure 7. Distribution of scaled triple product u2v+

forsmooth, fixed rough dune, immobile and mobile beds.

2007) and present immobile and mobile bed experi-ments. These values express turbulent diffusion in thex-direction (Du), and in the ydirection (Dv), respec-tively. For the mobile bed, Du andDv are positive andas well as negative, i.e., diffusion associated with pos-itive values of v and negative values of u are prevalentfarther away from the bed however negative values ofv and positive values of u are prevalent near the bed.

For immobile bed (Fig. 6), Du remains negativeand decreases with distance from bed, the peak nega-tive value occurs at y/h = 0.2 and Du increases withdistance (y/h > 0.2) towards zero near the free sur-face. This indicates streamwise deceleration of theflow associated with an outward momentum transfernear the bed. For immobile bed, Dv (Fig. 7) increaseswith the distance from the wall and reaches maximumvalue at y/h = 0.20 and remains positive throughout thedepth, however Dv decreases with distance (y/h > 0.2)towards zero near the free surface. For immobile bed,the distribution of Du and Dv are similar to fixedrough dune bed. For immobile bed is concerned, Fig-ures 6 and 7 are consistent with the results presentedby Krogstad and Antonia (1999) and Keirsbulck et al.(2001) for smooth-wall turbulent boundary layers.

For mobile bed (Fig. 6), Du is close to zero verynear the bed, increases away from the bed on positiveside, as observed in zero pressure- gradient boundarylayers (e.g., Keirsbulck et al. 2001) reaches maximumpositive value at y/h = 0.08. The Du value of mobilebed decreases at y/h above 0.08 and crosses zero axisat y/h = 0.20. For y/h > 0.2 variation of Du remainsnegative and reaches peak negative value at y/h = 0.4.Du remains negative and increases with distance fromwall (y/h > 0.4) towards zero near the free surface.Thisindicates streamwise acceleration of the flow associ-ated with an inward momentum transfer near the bed.Diffusion becomes negligible in the outer layer closeto the free surface (y/h > 0.7).

For mobile bed (Fig. 7), Dv distribution is approx-imately symmetrical to Du distribution in Fig. 6,however its sign is opposite to that of Du. In the

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outer region of the flow (y/h > 0.2), Dv is positivewhile Du is negative, implying a positive v motion.This indicates an ejection motion with the transportof u2 momentum away from the wall. The Du andDvcurves of mobile bed are collapsing on fixed roughdune bed for y/h > 0.5. Therefore it is inferred that Duand Dv distributions are similar for fixed rough dunebed and mobile bed in the outer region. The positivepeak value of Dv curve and negative peak of Du curveof mobile bed are less than corresponding peaks offixed rough dune bed but greater than correspondingpeaks of immobile bed and smooth bed. The positivepeak of of Dv and negative peak of Du curve of mobilebed are occurring farther away (y/h = 0.4) from the bedas compared to fixed rough dune bed (y/h = 0.26) thisindicates that diffusion of streamwise turbulence is inthe upward direction. Intense turbulent diffusion is inthe upward direction for flow on fixed rough dune bedand mobile beds as compared to smooth bed, immo-bile bed. Keirsbulck et al. (2001) noted a change overin sign of Dv distribution from negative to positivewith increasing distance from the wall for rough wallflows. The present mobile bed experiment is support-ing the conclusion of Keirsbulck et al. (2001) regardingdistribution of Dv for rough wall flows.

6 CONCLUSIONS

The objective of this paper was to experimentallyinvestigate the turbulence characteristics of open chan-nel flows under conditions of porous rough mobilebedforms and immobile bed conditions. For thispurpose, vertical distributions of the higher-ordermoments of streamwise and vertical velocities andReynolds stress have been investigated using PIV mea-surements.The present data is compared with previousdata obtained on immobile rough bed, smooth bed andfixed rough dunes available in the literature. It wasfound that for the mobile bedforms and immobile bedsthe ejection events were prevalent through most of thedepth while the sweep events were significant onlynear the bed. Interesting observation in the presentexperiment is the higher values of skewness factorsobserved in the case of the mobile bedforms. At sim-ilar Reynolds number, the skewness factors obtainedon mobile bedforms are 100 times higher than thisobserved on the immobile rough bed. Near the mobilebed, the shapes and magnitudes of turbulent diffu-sion coefficients are also different from these obtainedon immobile rough bed, smooth and fixed rough 2-Ddune. Finally, the results prove that the PIV techniquecan be successfully applied to investigate turbulencein open channel flow over mobile bedforms.

REFERENCES

Balachandar, R., Blakely, D., Tachie, M. F. and Putz, G.(2001). “A study of turbulent boundary layer on a smoothflat plate in an open channel,” J Fluids Eng., 123, 394–400.

Balachandar R., Patel V. (2002). Rough wall boundary layeron plates in open channels. J Hydraul Eng, 128:947–951.

Bakken, O.M.,Krogstad, P.-A.,Ashrafian,A., andAndersson,H.I.,(2005) Reynolds number effects in the outer layer ofthe turbulent ?ow in a channel with rough walls, Phys.Fluids 17 , p. 065101.

Faruque A. A., and Balachandar R., (2011). “Seepageeffects on turbulence characteristics in an open chan-nel flow”,Canadian Journal of Civil Engineering, Volume38(7), pp. 785–799(15).

Gad-el-Hak, M., and Bandyopadhyay, P. (1994). “Reynoldsnumber effects in wall-bounded turbulent flows.” Appl.Mech. Rev., 47(8), 307–365.

Jimenez J (2004)Turbulent flows over rough walls.Annu RevFluid Mech 36:173–96.

Keirsbulck, L., Mazouz, A., Labraga, L., and Tournier, C.(2001). “Influence of the surface roughness on the third-order moments of velocity fluctuations.” Exp. Fluids, 30,592–594.

Nakagawa, H. and Nezu, I. (1977). “Prediction of the con-tributions to the Reynolds stress from bursting eventsin open-channel flows,” J Fluid Mech., April, 80, part.l,99–128.

Nezu I., Nakagawa H. (1993). Turbulence in open channelflows. IAHR Monograph, Balkema, Rotterdam.

Nikora, V. I., McEwan, I. K., McLean, S. R., Coleman, S. E.,Pokrajac, D., and Walters, R. (2007). “Double-averagingconcept for rough-bed open-channel and overland flows:Theoretical background.” J. Hydraul. Eng., 133(8), 873–883

Roussinova, V., and Balachandar, R. (2011). “Open channelflow past a train of rib roughness.” J. turbulence., Vol. 12,No. 28, 2011, 1–17.

Roussinova, V., Biswas, N., and Balachandar, R. (2008).Revisiting turbulence in smooth uniform open channelflow. Journal of Hydraulic Research, 46(sup1), 36–48.

Roussinova, V., Shinneeb,A. M., and Balachandar, R. (2010).Investigation of Fluid Structures in a Smooth Open-Channel Flow Using Proper Orthogonal Decomposition.Journal of Hydraulic Engineering, 136(3), 143–154.

Shinneeb, A. M., Bugg, J. D., and Balachandar, R. (2004).“Variable threshold outlier identification in PIV data.”Meas. Sci. Technol., 15, 1722–1732.

Simpson, R., Chew, Y., and Shivaprasad, B. (1981). “Thestructure of a separating turbulent boundary layer. Part2: Higher-order turbulence results.” J. Fluid Mech., 113,53–73.

Schindler, R. J., and A. Robert (2005), Flow and turbulencestructure across the ripple-dune transition: An experi-ment under mobile bed conditions, Sedimentology, 52,doi:10.1111/j.1365-3091.2005.00706x.

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