RESEARCH ARTICLE

David F. Hill Æ Brian D. Younkin

PIV measurements of flow in and around scour holes

Received: 18 November 2005 / Revised: 1 March 2006 / Accepted: 18 April 2006 / Published online: 19 May 2006� Springer-Verlag 2006

Abstract Two sets of experiments related to the scour ofcohesionless sediment by planar turbulent jets are pre-sented and discussed. The first set of experiments mea-sures the growth of the scour hole and downstream duneas a function of time. Measurements reveal a bedformthat is nearly self-similar and whose growth in time isgoverned by a power-law relationship. The bedform iswell represented by three linear segments with slopesnear the angle of repose of the sediment. The second setof experiments uses Particle Image Velocimetry tocharacterize the mean velocity field in the scour hole andabove the dune. For this set of experiments, a series ofsuccessively larger roughened fixed-bed models was usedin place of the mobile bed. The measurements reveal thepresence of strong recirculation in the hole and an at-tached wall jet on the main slope. Discussion of theutility of the present fixed-bed measurements in esti-mating shear stress along the bed and related applicationto predictive modeling of hydraulic scour is provided.Discussion of the technical challenges of similar mobile-bed measurements is also given.

1 Introduction

Hydraulic scour of cohesionless and cohesive soils is aprocess of great significance to many natural and engi-neered flows. In the context of natural river flows, aquantitative understanding of scour is of value in termsof predicting channel migration and the supply of

sediment to estuaries and deltas. With regard to engi-neering applications, practical concerns include theresuspension and transport of contaminated sedimentsand the failure of dams, levees, and bridges due to theundermining of foundations.

The challenges in understanding the processes con-trolling hydraulic scour and developing a reliable pre-dictive capability are myriad. The fluid flow is generallyunsteady and three-dimensional and the sediment char-acteristics are often polydisperse and cohesive. Mostcrucially, the fine details of the fluid flow and sedimentresponse at the water-bed interface have traditionallybeen inaccessible to the researcher. In the words ofBalachandar et al. (2000), describing the state of scourresearch:

A comprehensive understanding of the scourmechanics remains elusive because of the complexnature of the process. The coupling between the shapeof the eroded bed profile and the hydrodynamicscharacteristics of the jet flow increases the complex-ity...Most previous studies have been experimental innature and any attempts to theoretically model thescouring process have been semi-empirical at best.Furthermore, many of the suggested relationshipsprovide a wide range of scour prediction for similarflow conditions.

From an empirical point of view, the scour of cohe-sionless and cohesive soils by horizontal planar andcircular fluid jets has been well-studied in the past(Chatterjee and Ghosh 1980; Rajaratnam 1981; Ali andLim 1986; Chatterjee et al. 1994; Chiew and Lim 1996;Aderibigbe and Rajaratnam 1998; Hamill et al. 1999;Dey and Westrich 2003; Mazurek et al. 2003). A simpleschematic is shown in Fig. 1. Beginning with a hori-zontal bed, the response of the bed to the jet-inducedshear stresses is to develop a scour hole, whose depthand streamwise location are functions of time. The

Communicated by EiF-0269-2005.R1

D. F. Hill (&) Æ B. D. YounkinDepartment of Civil and Environmental Engineering,The Pennsylvania State University, 212 Sackett Building,University Park, PA 16802, USAE-mail: dfh4@psu.eduTel.: +1-814-8637305Fax: +1-814-8637304E-mail: bdy105@psu.edu

Experiments in Fluids (2006) 41: 295–307DOI 10.1007/s00348-006-0156-3

sediment scoured from the hole is deposited at adownstream distance where the shear stress is no longersufficient for mobilization. Eventually, the hole deepensto the point where the scouring process is stopped. Notethat some authors (Balachandar et al. 2000) have re-ported the lack of an equilibrium, even after multipledays of continuous scouring.

Complicating this scour process is the importantinfluence of the tailwater depth, also referred to as thedepth of submergence of the fluid jet. Previous studies(Johnston 1990; Aderibigbe and Rajaratnam 1998;Balachandar et al. 2000) have described two regimes,one where the jet is attached to the bed and anotherwhere the jet is attached to the free surface. In someexperiments, the jet has remained within one of theseregimes while in other experiments, the jet has beenobserved to oscillate or ‘flick’ between the two regimes.As will be discussed later, the present experiments fallwithin the bed jet regime due to the large tailwater depthused.

Most past experimental studies have sought toempirically correlate parameters such as maximumscour depth, maximum dune height, and the stream-wise distances to the trough and dune with forcingparameters such as the jet velocity and height andsediment parameters such as the mean grain size anddensity. While of significant practical use, these studieshave shed limited light on the underlying physics,thereby making extension to different parameter re-gimes difficult.

More recently, greater attention has been paid tomeasuring and simulating the properties of the flow fieldwithin scour holes themselves. For example, Kurniawanand Altinakar (2002) used an acoustic doppler profilerto measure the flow in a scour hole. Measurements wereonly made after equilibrium was reached, unfortunately,yielding no information about the development of thehole. Karim and Ali (2000) carried out some numericalsimulations, using a variety of closure models, of theflow inside an equilibrium scour hole. The simulationsshowed good agreement with limited mean velocitymeasurements, but made no attempt to connect the flowto the scour process itself.

A significant conceptual step forward in connectingthe flow to the erosion process itself was taken by Hogget al. (1997). Their analysis began by incorporatingexisting empirical data on the mean velocity structure ofwall jets. Next, the streamwise momentum equation wasintegrated in order to derive an expression for theboundary shear. By then making use of a criterion forincipient motion, the steady-state profile of the bed was

derived and showed qualitative agreement with previousmeasurements.

The biggest obstacle to the work of Hogg et al. (1997)was the lack of existing data on boundary shear stresseson a mobile, evolving bed. In the absence of this data,their work made reasonable simplifying assumptions. Bymaking use of classical (flat boundary) wall jet data,their analysis made the implicit assumption that theevolving boundary did not ‘feed back’ in the form ofchanges to the mean flow field. In an effort to heuristi-cally correct for this, their analysis used an arbitraryshape factor in its expression for the streamwise varia-tion of shear stress. A Gaussian profile was used, in lightof the fact that, in their words, ‘there is no availablemodel or data to prescribe the variation of [the shapefactor].’

The intent of the present work is to provide a un-iquely high resolution look at the mean velocity field inand around developing scour holes. The data obtainedin the current study should subsequently help to refine orvalidate the types of assumptions, regarding boundaryshear, that were made by Hogg et al. (1997). In abroader context, it is hoped that the present work willfoster additional experimental studies of scour flows.

The specific scope of the present study can be statedas follows. Particle image velocimetry (PIV) measure-ments have been made downstream of a planar jet,parallel to and flush with an initially flat bed. Followinglive-bed experiments, carried out to chart the evolutionof a two-dimensional scour hole and associated down-stream dune, a series of roughened fixed-bed models ofthe bedform geometry was constructed. These modelsspanned the range from the initially flat bed to a pointnear the end of the bedform evolution. For each model,large ensembles of PIV realizations were acquired atnumerous locations and tiled together into ‘mosaic’images spanning as much of the flow field as practical.

The restriction of the experiments to the simplisticconditions of two-dimensionality and cohesionless sedi-ments is intentional and frames the present study as onethat is primarily conceptual. Future extensions to three-dimensional flows, such as a circular jet, and gradedcohesive sediments are possible and anticipated.

2 Experimental facilities

The experiments were carried out in a custom-built re-circulating flume, illustrated in Fig. 2. Water was with-drawn from one end, passed through a 1 horsepowercentrifugal pump, a valve, and a digital flowmeter, be-fore being discharged at the other end through a gentlytapered planar nozzle, shown in Fig. 3. The nozzle wasconstructed with a variety of flow-conditioning materialsinside, such as coarse fibrous pads and a honeycombstraightener. The height of the nozzle at its exit, b, was1 cm, which, combined with a maximum flowratethrough the system of 140 lpm, placed an upper bound

Fig. 1 Schematic of bed being eroded by horizontal turbulent jet

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of approximately 7,800 on the Reynolds number of theexit flow.

Overall, the flume was 2.4 m in length, 30 cm inwidth, and 45 cm in height. A false floor was thenconstructed in order to create a cavity, or ‘test section’immediately downstream of the nozzle. This cavity had alength of 1 m and a depth of 10 cm, with its upper edgeflush with the bottom of the nozzle. For the live-bedexperiments, the cavity was filled with sediment and forthe PIV experimients, fixed-bed polycarbonate modelswere inserted into the opening, as is illustrated in Fig. 2.Separating the downstream end of the test section andthe pump intake line was a trap, designed to minimizethe amount of sediment passing through the pump. Forall experimental trials, the free surface was maintained atan elevation of 25 cm above the jet axis.

For the first experimental phase, live-bed experi-ments were carried out with three sediment sizes. Glassbeads (Ballotini Impact Beads, from Potters Indus-tries), having a specific gravity of 2.5, were used insteadof natural sediments. The decision to use artificial ra-ther than natural sediments was consistent with theoverall experimental goal of initially removing as muchvariability as possible. Planned future studies will ad-dress the effects of the irregularity in shape and thegradation found in natural sediments. The meandiameters d50, calculations of settling velocity Vs, andmeasured angles of repose h of the glass beads aresummarized in Table 1 and a sample photograph of the‘B’ glass beads is shown in Fig. 4. Note that the specific

gravity of 2.5 is in close proximity to typical values (�2.65) for sandy sediments.

For the second experimental phase, PIV measure-ments were made with a PIV system from TSI, Inc. Thesystem consisted of a four megapixel camera, double-pulsed (120 mJ/pulse) Nd:Yag lasers, a synchronizer tofacilitate system timing, and frame grabbers to facilitateimage acquisition. As shown in Fig. 2, an angled cameramount was used in order to align the camera with themain slope of the bedform model. The laser light sheetwas introduced from above and a shallow glass ‘pan’was used in conjunction with a deep tailwater in order toeliminate refractions from slight free surface distur-bances.

As shown in Fig. 5, a series of fixed-bed models wereconstructed for this experimental phase. Made of poly-carbonate and roughened with sediment grains painted aflat black, the series of models was chosen to span a widerange of the scour hole development. The use of fixed-bed models rather than a live bed was a compromiseaimed at allowing for the acquisition of large ensemblesof data. For each model, ensembles of 1,000 image pairswere obtained at numerous locations in the flow field. Ifa live bed were used, the bed would change noticeably inthe amount of time that it took to fully image the flowfield.

3 Live-bed experiments

3.1 Bedform geometry

Visual observations from initial tests revealed that thescour hole and downstream dune that were generatedby the planar jet were highly two-dimensional innature. As a result, it was determined that the shapeof the bedform, at different instances in time, couldadequately be determined by periodically stopping the

Laser

0.45 m

0.1 m

1.0 m

2.4 m

Fig. 2 Schematic of theexperimental facility

Fig. 3 Photograph of the planar nozzle

Table 1 Mean grain diameters, settling velocities, and angles ofrepose of the solid glass spheres used in the present experiments

Sediment d50 (lm) Vs (cm s�1) h (�)

A 725 14.2 23.3B 513 10.3 22.2AC 88 0.94 23.7

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pump and tracing the profile on the sidewall of theflume. As a representative example, Fig. 6 shows thebedform, determined in this manner, for the experi-ment with B sediment and a flowrate of 120 lpm. Theprofiles observed at the different times display a degreeof self-similarity, particularly in the vicinity of thedune.

It should be noted that these profiles represent ‘static’beds. Other authors (Aderibigbe and Rajaratnam 1998)have noted the difference between static and dynamicbeds, a difference that increases with increasing particledensimetric Froude number

F0 ¼U0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gðDq=qÞd50p ; ð1Þ

where U0 is the initial jet velocity, D q the difference indensity between the sediment and fluid and q the fluiddensity. Additionally, Johnston (1990) and Aderibigbeand Rajaratnam (1998) remark on the unsteady natureof the jet in their live-bed experiments, with the jetoscillating between a horizontal position and one whereit is deflected down into the scour hole. To an extent,this behavior was observed in the present live-bedexperiments, although not to the degree reported byJohnston (1990) and Aderibigbe and Rajaratnam (1998).This is due to the generally lower values of F0 and thelarger values of tailwater depth in the present experi-ments. Regardless, it should be acknowledged that thereliance of the present study upon static profiles has theeffect of filtering out the unsteady nature of the bedentirely.

Scaling the bedforms by any characteristic lengthscale, say the distance to the crest, xcrest, confirms thatthe profiles are highly self-similar, with some minordiscrepancies found in the trough and at the down-stream edge of the dune. This is illustrated in Fig. 7,again for the case of B sediment with a flowrate of120 lpm.

With an eye towards a simplistic model of bedformevolution, it was decided to parameterize the bed with asfew parameters as possible. To this end, an attempt wasmade to represent the scaled bedform with three linearsegments of equal/opposite slope and an offset, orintercept, at the origin. This model profile, along withthe dimensionless B sediment bedform, averaged over allflow trials, ranging from 60 to 140 lpm, is shown inFig. 8. Note that this two-dimensional model profileimposes conservation of sediment mass and that theslope and offset parameters were determined through

Fig. 4 Microscopic image of type B sediment in a 1 mm rectan-gular channel

Fig. 5 Fixed-bed models constructed to represent various stages ofdevelopment of the scour hole and downstream dune

Fig. 6 Evolution of the dimensional bedform for the B120experimental trial

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least-squares analysis. This linear approximation wassubsequently used to construct the fixed-bed modelsdescribed in Sect. 2 and shown in Fig. 5.

For the A and B sediments, the scaled profiles eachyielded m=0.395 and b=0.07. The AC sediment, on theother hand, deviated significantly from this simple pro-file. This was due to the fact that the AC sediment wasfine enough to be put into suspension by the fluid jet. Asa result, the sediment was carried and deposited fardownstream of the dune, unlike the A and B trials,where the sediment simply rolled over the crest of thedune. The scope of the present velocity measurementswas limited to these latter cases, characterized by bed-load transport.

It is clear from Fig. 8 that this model representationof the bed is not perfect, even for the bedload transport

cases of the A and B sediment. The discrepancies suggesta lack of conservation of sediment in the experimentaldata. Recalling that bedform profiles were obtained atthe sidewall only, a possible explanation is that slightthree-dimensionality of the bedform was present. Thishypothesis was tested for a single flow rate (100 lpm) forthe A sediment. For this trial, bedform profiles wereobtained at several times both at the sidewall and alongthe flume centerline. The latter measurements were car-ried out with a standard point gage, having a resolutionof 0.025 mm.

Figure 9 shows these two profiles, scaled and aver-aged over the duration of the trial. They indicate thatthere is some degree of three-dimensionality, with thecrest being slightly higher and the trough being slightlyshallower on the flume centerline. This may be due to theinteraction between the sidewall boundary layer and thelower free shear layer of the jet. As the jet discharges,spanwise vorticity is created above and below the jetaxis. Near the flume sidewalls, the presence of a span-wise gradient in streamwise velocity results in stream-wise vorticity being generated through the mechanism ofvortex tilting. This streamwise vorticity, concentratedadjacent to the sidewalls, serves to ‘mine’ small amountsof sediment away from the walls, resulting in a slightlydeeper trough at the sidewall than at the centerline.Given that the centerline measurements are far moretime-intensive, however, than those at the sidewall, itwas decided to use only the latter in determining themodel bed as described above.

3.2 Scour rate

One final point of interest regarding the live-bed exper-iments is how the bedform evolves with time. From adimensional analysis point of view, there are a great

Fig. 7 Scaled bedforms for the B120 experimental trial

Fig. 8 Dimensionless bedform (circle) for the B sediment, averagedover all trials, along with linear approximation of the bedform

Fig. 9 Dimensionless bedforms for the A100 trial. Profiles areobtained from the flume sidewall and from the flume centerline

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number of independent variables in the present problemand the literature is characterized by many differentscalings. As many of these variables (e.g. nozzle thick-ness, tailwater depth, fluid and sediment densities, etc.)were not varied in the present experiments, they can beremoved from the analysis. Moreover, if the assumptionof fully turbulent flow is made and a single grain size isconsidered, a dimensional relationship such as

xcrest ¼ f ðU0; g; tÞ; ð2Þ

where U0 is the initial jet velocity and g is gravity, maybe proposed. This yields the equivalent dimensionlessrelationship

xcrestU0t¼ f

gtU0

� �

: ð3Þ

Again, it is noted that other relationships are possible,depending upon the initial choice of parameters.

Figure 10 shows, as an example, the B data, plottedin this fashion, and on logarithmic axes. First, it is clearthat the bedform evolution follows a power-law depen-dence, i.e.

xcrestU0t

� �

¼ agtU0

� �b

: ð4Þ

This indicates that, in a strict sense, the bedforms werenot yet in an equilibrium state at the conclusion of eachtrial. Note that the trials ranged in duration from 4 to20 h, depending upon the combination of sediment sizeand flow rate. The practical desire to perform a large setof trials precluded longer runs and individual trials weretherefore terminated when bedload transport was ob-served to diminish to a minimal level. It is acknowledgedthat this termination point was a subjective measure.

The Reynolds and densimetric Froude numbers, theobtained power-law coefficients, and the correlationcoefficients of the power-law fits for the various trials aresummarized in Table 2. Additionally, the power-lawcoefficients are plotted in the two-dimensional Re � F0

parameter space in Fig. 11.Both Figs. 10 and 11 suggest that, with increasing

Reynolds number, the data are approaching Reynoldsnumber invariance. Additionally, the data indicate aclear variation with grain size. Finally, the values of bgiven in Table 2, in conjunction with (4), which indicatesthat the bedforms grow by t1+b, reveal a growth coef-ficient of �0.05�0.15. Despite the depth of the literatureon hydraulic scour, there appear to be only limitedpoints of comparison. For example, Rajaratnam (1965,Fig. 2b) reports a coefficient of approximately 0.15 for atrial with d50=1.2 mm and F0=O(10). Chatterjee et al.(1994) similarly give coefficients in the range of 0.15–0.20. Ali and Lim (1986) report a higher coefficient of0.33, but it should be noted that their experiments hadvery shallow tailwater depths. It is possible that thelower coefficients being reported in the present studyarise from the highly idealized artificial sediment thatwas utilized. Planned future trials with natural sedimentsmay help to resolve this.

4 Velocity measurements

As discussed in the previous sections, a series of suc-cessively larger fixed-bed models was constructed out ofpolycarbonate. Given the appreciable data storage andprocessing requirements of the PIV measurements, itwas decided to focus on a single sediment–flow ratecombination. The A sediment and a flow rate of 140 lpmwere chosen. Therefore, the sizes of the models werechosen to span the range from (initially) flat bed to theultimate size recorded in the A140 live-bed experiment.

For each model, the camera was positioned at anumber of locations so that the flow field in the vicinityof the bedform was fully covered, as illustrated in

Fig. 10 Variation of dimensionless distance to crest with dimen-sionless time for B sediment experiments. The solid line is a powerlaw fit to the highest Re trial

Table 2 Summary of experimental trials and curve-fit parameters

Trial Re F0 a b Correlationcoefficient

AC40 2,222 6.18 3.71 �0.844 0.9994AC60 3,333 9.26 2.97 �0.846 0.9982AC80 4,444 12.35 2.33 �0.836 1.0000AC100 5,555 15.44 2.14 �0.834 0.9999B40 2,222 2.56 9.49 �0.904 1.0000B60 3,333 3.84 11.7 �0.970 0.9998B80 4,444 5.12 5.31 �0.929 1.0000B100 5,555 6.39 3.53 �0.904 1.0000B120 6,666 7.67 2.72 �0.894 1.0000A60 3,333 3.23 10.07 �0.960 1.0000A80 4,444 4.30 5.87 �0.939 1.0000A100 5,555 5.38 5.05 �0.943 0.9999A120 6,666 6.45 2.77 �0.907 0.9999A140 7,777 7.53 2.17 �0.895 1.0000

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Fig. 12. During post-processing, the individual fields ofview were tiled together into a composite, or mosaic,image of the entire flow field. This was done by usingcalibration images for each field of view, each of whichprovided a unique mapping between raw pixels anddimensional position for that field of view in laboratorycoordinates.

At each individual location, an ensemble of 1,000image pairs was obtained. The storage requirement forthe raw 12-bit images for the five models thereforeamounted to � 400 GB. The image pairs were subse-quently interrogated with PIVSleuth (Christensen et al.2000) and validated using CleanVec, both developed atthe Laboratory for Turbulence and Complex Flow atthe University of Illinois. Images were interrogated with32 · 32 windows, using a 50% overlap. Typical vectorremoval rates ranged from 1 to 5%, with the bulk ofremoved vectors being near the bed or near the left and

right edges of the images. In the case of the bed, mod-erate scattering of laser light from the sediment grainswas responsible, despite efforts to minimize this with flatblack paint.

With regards to measurement accuracy, some esti-mates are available from Raffel et al. (1998, Sect. 5.5).Their Monte Carlo simulations suggest that, for a 32 ·32 interrogation window and particle image diameters of1–3 pixels, displacement uncertainty is around 0.02 �0.04 pixels. For the present experiments, computedmean streamwise particle displacements were on theorder of a few pixels, indicating that the uncertainty wassimilarly on the order of a few percent.

The mean velocity fields for the flat bed and the fivemodel bedforms are given in Fig. 13. Both color con-tours of horizontal velocity and velocity vectors areprovided. For the sake of visual clarity, only a very smallpercentage of the total velocity vectors are displayed.Casual observation of the mean velocity fields revealsseveral interesting and important aspects of the flow.First, recirculation of the flow is observed in the scourhole itself. This was observed in the live-bed experimentsand this vortex played a significant role in the erosionalprocess. Sediment would be swept in a clockwise fashionfrom the bottom of the scour hole up into the strong jetflow of the discharge, at which point it would be carrieddownstream.

A more detailed view of the vortex flow in the scourhole is provided in Fig. 14. In this closeup view of themean velocity, the velocity components have been scaledby the initial jet velocity and the horizontal and verticalcoordinates have been scaled by xcrest. As with Fig. 13,only selected vectors are shown for the sake of clarity.As the bedform progresses in scale from small to large(moving from Profile 1 to 5), the most interesting featureis that the vortex progressively moves closer to theleading edge of the bedform. As summarized in Table 3,noting that x/xcrest=0.269 corresponds to the nondi-mensional location of the model trough, the vortex re-sides on the main slope, i.e. between the trough andcrest, of the bedform for the first two models. As thescour progresses to the latter three model sizes, thevortex moves toward the origin and resides on the slopebetween the jet orifice and the trough.

Fig. 11 Contours of power-law coefficients with Re and F0

Fig. 12 Illustration of typical camera fields of view. The solid blackline indicates the upper boundary of the fixed-bed model

301

Fig. 13 Mean velocity fields for the fixed bed, A sediment, 140 lpm trials. Color contours denote horizontal velocity and only selectedvectors are shown for clarity

302

Next, very little flow is observed in the lee of thedune. The flow separates over the crest of the dune andonly an extremely weak recirculation is observed. Recallthat observations of the live-bed experiments revealedthat sediment carried along the main slope simply rolled

down the downstream side of the dune, which wasessentially at the angle of repose of the sediment. Fig-ure 15 shows scaled vertical profiles of streamwisevelocity at streamwise locations of x/xcrest=1 and 1.362.These locations corresponds to the crest itself and thedownstream limit of the downstream slope of the crestwhere the bed returns to horizontal. As indicated in thefigure, the magnitude of the return flow, i.e. that headedup the slope, is limited to approximately 5–10% of theinitial jet velocity while the magnitude of the jet flowthat separates off the crest of the dune and continuesdownstream can reach 50–75% of U0.

Third, the main jet flow is seen to deflect downwardsand then attach to the main slope. Subsequent toattachment, the flow appears to behave like a classic walljet. In the context of the erosion of the bed, the velocityresults help to understand the basic mechanisms atwork. From the point of attachment of the jet on themain slope to the dune crest, sediment transport is dri-ven by shear along the slope. For the A and B sediments,as discussed previously, this transport appeared to belargely bedload. Upstream of the point of attachment,sediment is mined out of the trough by the vortex, liftedinto the flow exiting the nozzle, and swept downstream.As sediment passes over the crest of the dune, it essen-tially rolls down the slope, under the action of gravity.As the bedform increases in size, the attachment pointmoves further and further away from the jet orifice andthe velocity shear along the main slope weakens. Thisleads to a gradual slowing and, presumably, eventualcessation of the scour. When the bedforms and thevelocity results are considered in a scaled format, as inFig. 14, it is interesting to note that, as was the case withthe location of the trough vortex, the point of attach-ment of the jet on the slope migrates toward the jetorigin as the bedform grows. Table 4 gives the locationof the point of reattachment, xr, in nondimensionalterms.

4.1 Streamwise variation of maximum velocity

From the point of view of simple parameterizations ofthe flow, leading to modeling estimates of the erosionrate, the similarity of the flow on the main slope toclassic wall jets will be further explored. Recall thatHogg et al. (1997) used established wall jet power lawequations in deducing expressions for the variation ofboundary shear. Figure 16 shows the variation ofmaximum velocity with distance from the jet orifice.The results in Fig. 16a are for the flat bed case. Themaximum streamwise velocity component is normal-ized with the jet exit velocity U0 and the streamwisedistance is normalized with the initial jet thickness b.The plot is on log–log axes and the straight-line fit forthe region downstream of � 10 b demonstrates thewell-known power law decay of a wall jet. The ob-served decay coefficient of �0.43 is in good agreementwith the results of Tachie et al. (2004) who report

x / xcrest

y/x

cres

t

0 0.05 0.1 0.15 0.2 0.25 0.3 0. 35 0.4 0.45 0.5 0.55 0.6

x / xcrest

0 0.05 0.1 0.15 0.2 0.25 0.3 0. 35 0.4 0.45 0.5 0.55 0.6

x / xcrest

0 0.05 0.1 0.15 0.2 0.25 0.3 0. 35 0.4 0.45 0.5 0.55 0.6

-0.15

-0.1

-0.05

0

0.05

0.1(a)U0

y/x

cres

t

-0.15

x / xcrest

0 0.05 0.1 0.15 0.2 0.25 0.3 0. 35 0.4 0.45 0.5 0.55 0.6

x / xcrest

0 0.05 0.1 0.15 0.2 0.25 0.3 0. 35 0.4 0.45 0.5 0.55

x / xcrest

0 0.05 0.1 0.15 0.2 0.25 0.3 0. 35 0.4 0.45 0.5 0.55

-0.125-0.1

-0.075-0.05

-0.0250

0.0250.05

0.0750.1

-0.15-0.125

-0.1-0.075-0.05

-0.0250

0.0250.05

0.0750.1

(b)U0

y/x

cres

t

-0.15

-0.1

-0.05

0

0.05

0.1

-0.15

-0.1

-0.05

0

0.05

0.1

(c)U0

y/x

cres

t

(d)U0

y/

xcre

st

(e)U0

x / xcrest

0 0.05 0.1 0.15 0.2 0.25 0.3 0. 35 0.4 0.45 0.5 0.55 0.6

x / xcrest

0 0.05 0.1 0.15 0.2 0.25 0.3 0. 35 0.4 0.45 0.5 0.55

x / xcrest

0 0.05 0.1 0.15 0.2 0.25 0.3 0. 35 0.4 0.45 0.5 0.55

Fig. 14 Two-dimensional vector representation of mean velocityfields in the scour hole for the five fixed-bed models. a Through ecorrespond to Profiles 1 through 5, respectively

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�0.455 and who note little difference between roughand smooth boundaries.

Figure 16b shows similar results for the five modelbeds, with some slight differences in presentation. In-stead of considering only the streamwise velocity com-ponent, the maximum magnitude of the velocity vector,i.e. jU jmax is plotted as a function of distance. This ad-dresses the fact that the flow is no longer unidirectional,as was the case for the flat plate. Also, the streamwisecoordinate is now normalized by xcrest so that the data

from the different model sizes can be compared on acommon axis.

It is observed the streamwise variation in maximumvelocity is far more complex for the flow over the modelbeds, although distinct regimes can be identified. Ini-tially, in the range of x/xcrest < 0.3, the jet is basically afree jet and significant decay is observed. Note that, dueto the choice of scaling of the x axis, the different curvesdo not appear to have similar slopes. Plotted against x/binstead, all five curves do indeed have similar slopes inthis initial region. Note as well that the observed decay issomewhat unexpected. The downstream limit of thisregion ranges from about 5b for the smallest model toabout 15b for the largest model. The development regionof free and wall jets is generally at least this large (seeFig. 16a) and does not exhibit appreciable decay ofmaximum velocity. This suggests that perhaps thetrough vortex plays a role in enhancing the observedinitial decay of the jet.

Next, in the range of 0.3 < x/xcrest < 0.6, the velocityincreases as the jet attaches to and is accelerated alongthe slope. Third, in the range of 0.6 < x/xcrest < 1,which corresponds to the region of the fully attachedwall jet, the maximum velocity decays fairly weakly andnot in a definitive power-law fashion. Finally, when 1 <x/xcrest, the jet has separated off of the crest and decaysrapidly with further downstream distance. This complexstructure of the variation of maximum velocity withdistance, in the case of the model beds, appears to limitthe use of classic wall jet relations in determiningboundary shear stress.

5 Boundary shear estimates

To return to one of the main motivations of this work, itis of interest, therefore, to determine whether or notboundary shear stress data can be extracted from highresolution PIV velocity data over bedform. This infor-mation will be of considerable interest and use, given thebroad impact of sediment transport in natural andengineered environments. Generally speaking, there aremany candidate methods for determining boundaryshear.

In the case of a geometrically simple flow, such asuniform open channel flow, theoretical knowledge,obtained from the equations of motion, of the varia-tion of stress can be used to extrapolate boundaryshear from measurements made in the fluid interior.Alternatively, if a flow has been well studied, it ispossible that well accepted equations for mean velocityprofiles exist. These equations may include the shearvelocity, allowing for the determination of boundaryshear through fitting of experimental velocity data. Asa third option, boundary shear can determined ifvelocity measurements in the viscous sublayer areavailable. For the present study, all of these methodsappear to be unsatisfactory. Paying particular atten-tion to the lattermost approach, rough estimates sug-

Table 3 Nondimensional streamwise location of the trough vortex(xv) for the different fixed-bed models

Profile xv/xcrest

1 0.2852 0.2833 0.2544 0.2275 0.175

u / U0

y/x

cres

t

0 0.25 0.5 0.75

0.16

0.18

0.2

0.22

0.24

0.26

0.28Profile 1Profile 2Profile 3Profile 4Profile 5

(a)

(b)

u / U0

y/x

cres

t

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Profile 1Profile 2Profile 3Profile 4

Fig. 15 Vertical profiles of streamwise velocity at a x/xcrest=1 andb x/xcrest=1.362

304

gest that the sublayer should be on the order of a fewtenths of a millimeter. Despite efforts to minimizescattering of laser light by the bed, reflections weresufficiently bright in practice so as to prevent mea-surements this close to the boundary. One possibilityfor future studies is to use particles (such as PTFE)

that have a refractive index very close to water and userefractive matching techniques to adjust the index ofthe water to that of the PTFE.

An option that remains is using a quadratic frictionlaw to estimate the boundary shear based upon the meanvelocity data. As discussed in numerous papers on wall-bounded shear flows from the very early (Rajaratnam1965) to the very recent (Tachie et al. 2004), theboundary shear s0 may be expressed as

s0 ¼1

2cfqu2m; ð5Þ

where q is the fluid density, um is the maximumstreamwise velocity at a given streamwise location, andcf is a friction coefficient. Studies such as Rajaratnam(1965) provide experimental data on the variation of cfwith local Reynolds number for smooth boundaries.Tachie et al. (2004) pay specific attention to the case ofrough boundaries and their data indicate that cf is fairlyindependent of Reynolds number and has a value ofroughly 0.008.

Using the above information, the boundary shearstress was determined for each of the five model bed-forms. Figure 17 shows the variation of stress with anormalized alongslope distance, x¢. Here, a value of 0corresponds to the trough and a value of 1 to the crest.The maximum velocity um corresponds to the maximumalongslope velocity found by scanning in the slope-normal direction at different values of x¢. Results areonly given for the range 0.5�1, which roughly corre-sponds to the portion of the slope where the jet is at-tached.

The results confirm that, as the scour hole gets larger,the shear along the slope, which is driving the sedimenttransport, weakens. Additionally, it is seen that thevariation in boundary shear along any given slope isonly moderate.

As a rough check on the results, note that the Shieldsparameter for incipient sediment motion is defined as

s�c ¼sc

ðcs � cÞd ; ð6Þ

where sc is the critical bed stress required for motion, csand c are the specific weights of the sediment and fluid,and d is the grain size. This parameter has been shown tobe a function of a Reynolds number based upon shearvelocity and grain size, Re*.

In the transitional regime, roughly 1< Re* < 70, therelationship between sc

* and Re* is given by the empiricalfit (Yalin and Karahan 1979)

s�c ¼X

4

i¼0AiðlogRe�Þi; ð7Þ

where A0=0.10000, A1=�0.13610, A2=0.05977,A3=0.01984, and A5=�0.01134. Thus, for a given grainsize and fluid and sediment densities, the critical stress isreadily determined. For the A sediment used in the

Table 4 Nondimensional streamwise location of jet reattachment

Profile xr/xcrest

1 0.4252 0.4233 0.3904 0.3595 0.319

x / b

Um

ax/U

0

10 20 30 400.5

0.6

0.7

0.8

0.9

1

1.1

x x x x xx x x x x

x xx x

x xx xx

xxxx

xx

xxxxxxxx

xxx

xxxx

xxxxxxxxx

xxxxxxxxxxxxxxxx

xxxxxxx

xxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxx

x / xcrest

|U| m

ax/U

0

0.5 10.1

0.2

0.3

0.4

0.5

0.6

0.70.80.9

1

Profile 1Profile 2Profile 3Profile 4Profile 5

x

(b)

(a)

Fig. 16 a Variation of maximum streamwise velocity with distancealong the flat plate. The trendline indicates a power-law fit to thedata. b Variation of maximum velocity vector magnitude withscaled streamwise distance. Profiles 1 through 5 denote the smallestto the largest model beds

305

roughened fixed-bed trials, a critical stress of 0.35 Pa isobtained. The nondimensional Shields parameter, thedimensional critical shear stress, and the critical grainsize Reynolds number for the three sediment sizes aresummarized in Table 5.

Next, it must be noted that this critical stress is rel-evant only to the case of a flat bed. Chiew and Parker(1994) report on experiments conducted with non-hori-zontal beds and express the ratio of critical shear on anon-horizontal bed ðscnhÞ to that on a horizontal bedðschÞ as

scnh

sch¼ cos/ 1� tan/

tan h

� �

; ð8Þ

where / is the bed angle and h the sediment angle ofrepose. For the present experiments, /=�21.5� (nega-tive since adverse), which leads to scnh ¼ 1:78sch ¼0:62 Pa:

From an idealized conceptual point of view, the fluidjet in a live-bed experiment strikes the slope someplaceslightly upstream of the midpoint between the troughand crest. Sediment is dislodged and swept up the slopeas bedload transport. As long as the shear stress at thepoint of jet impingement on the boundary exceeds thecritical value required for motion, the scouring processwill proceed. As the bedform grows, the point ofimpingement moves further away from the jet origin andthe shear stress along the slope steadily declines. Even-tually, the maximum shear along the boundary will fallbelow the critical value and the scouring process shouldcease. As Fig. 17 shows, the maximum shear along theslope of Profile 5, which represents a near-equilibriumstate, is approximately 0.6 Pa. Given the large error barstypically associated with incipient motion criteria, thisappears to be in reasonable agreement with the abovecalculations.

6 Concluding remarks

The present study was undertaken in order to inves-tigate the application of PIV to hydraulic scourproblems. The main objectives were twofold: to obtainhigh-resolution images of the flow field within scourholes and above their associated dunes, and to use thevelocity data to estimate the shear stress along theboundary. To assist in the execution of the study,several simplifying compromises were made. First,artificial cohesionless sediment, in the form of solidglass spheres, was used. Second, and based upon live-bed experiments, roughened fixed-bed models of thetwo-dimensional scour hole geometry were used duringthe PIV measurements. This ensured that the bedwould not change during the amount of time that ittook to image the entire domain.

The velocity data illuminated the recirculating flow inthe scour hole and the nature of the jet flow as it emergesfrom the nozzle and eventually attaches to the slopingbed. While slope-normal profiles of alongstope velocityresemble a classic wall jet, it was found that thestreamwise variation of maximum velocity was quitecomplex in structure. Due to the complexity of the flowand the measurements, the options for determiningboundary shear stress from the velocity data appear tobe limited. The use of a quadratic friction law illustratedthe weakening boundary shear as the bedform pro-gressed in size. Additionally, the predicted values ofshear for the largest bedform, which represents the stageat which bedform evolution had nearly stopped, were inreasonable agreement with standard incipient-motioncriteria.

The authors conclude that PIV measurements have avaluable role to play in sediment transport research. Inthe future, the use of either artificial sediment, close inrefractive index to water, or fluorescent PIV seed parti-cles and camera filters will improve the ability of the PIVmeasurements to resolve the flow details very close to thebed. Additionally, steady improvements in camera datatransfer rates will result in the time scale of imageacquisition becoming much less than the time scale ofbed evolution. As a result, PIV measurements with livebeds, particularly at the early stages of scour hole evo-lution, will become increasingly more feasible.

x x x x x xx

x x x xx x x x x x x x x x x x x x x

x x x x x x x x x x x x x x xx

x x x x

x

x x x

x’

τ(P

a)

0.5 0.6 0.7 0.8 0.9 10

0.25

0.5

0.75

1

1.25

1.5

Profile 1Profile 2Profile 3Profile 4Profile 5

x

Fig. 17 Variation of boundary shear with normalized alongslopedistance

Table 5 Summary of incipient motion parameters for the studiedsediments

Sediment sc* sc (Pa) Re*

A 0.0326 0.349 13.5B 0.0328 0.248 8.10AC 0.100 0.129 1.00

306

References

Aderibigbe O, Rajaratnam N (1998) Effect of sediment gradationon erosion by plane turbulent wall jets. J Hydraul Eng124(10):1034–1042

Ali K, Lim S (1986) Local scour caused by submerged wall jets. In:Proceedings of Instn. Civil Engineers 81:607–645

Balachandar R, Kells J, Thiessen R (2000) The effect of tailwaterdepth on the dynamics of local scour. Can J Civil Eng 27:138–150

Chatterjee S, Ghosh S (1980) Submerged horizontal jet overerodible bed. J Hydraul Div 11:1765–1782

Chatterjee S, Ghosh S, Chatterjee M (1994) Local scour due tosubmerged horizontal jet. J Hydraul Eng 120(8):973–992

Chiew YM, Lim SY (1996) Local scour by a deeply submergedhorizontal circular jet. J Hydraul Eng 122(9):529–532

Chiew Y, Parker G (1994) Incipient sediment motion on non-horizontal slopes. J Hydraul Res 32(5):649–660

Christensen K, Soloff S, Adrian R (2000) PIV Sleuth—integratedacquisition, interrogation, and validation software for particleimage velocimetry. Tech. Rep. 943, University of Illinois,Department of Theoretical and Applied Mechanics

Dey S, Westrich B (2003) Hydraulics of submerged jet subject tochange in cohesive bed geometry. J Hydraul Eng 129(1):44–53

Hamill G, Johnston H, Stewart D (1999) Propeller wash scour nearquay walls. J Waterw Port Coast Ocean Eng 125(4):170–175

Hogg A, Huppert H, Dade W (1997) Erosion by planar turbulentwall jets. J Fluid Mech 338:317–340

Johnston A (1990) Scourhole development in shallow tailwater. JHydraul Res 28(3):341–354

Karim O, Ali K (2000) Prediction of flow patterns in local scourholes caused by turbulent water jets. J Hydraul Res 38(4):279–287

Kurniawan A, Altinakar M (2002) Velocity and turbulence mea-surements in a scour hole using an acoustic doppler velocityprofiler. In: Proceedings of the third international symposiumon ultrasonic Doppler methods for fluid mechanics and fluidengineering, Lausanne, Switzerland, pp 37–43

Mazurek K, Rajaratnam N, Sego D (2003) Scour of a cohesive soilby submerged plane turbulent wall jets. J Hydraul Res41(2):195–206

Raffel M, Willert C, Kompenhans J (1998) Particle image veloci-metry. Springer, Berlin Heidelberg New York, Sect. 5.5

Rajaratnam N (1965) The hydraulic jump as a wall jet. J HydraulDiv 91(5):663–673

Rajaratnam N (1981) Erosion by plane turbulent jets. J HydraulRes 19:339–358

Tachie M, Balachandar R, Bergstrom D (2004) Roughness effectson turbulent plane wall jets in an open channel. Exp Fluids37:281–292

Yalin M, Karahan E (1979) Inception of sediment transport. ASCEJ Hydraul Div 105(11):1433–1443

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