Characteristics of Horseshoe Vortex in Developing Scour Holes at Piers

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Characteristics of Horseshoe Vortex in DevelopingScour Holes at Piers

Subhasish Dey1 and Rajkumar V. Raikar2

Abstract: The outcome of an experimental study on the turbulent horseshoe vortex flow within the developing �intermediate stages andequilibrium� scour holes at cylindrical piers measured by an acoustic Doppler velocimeter �ADV� are presented. Since the primaryobjective was to analyze the evolution of the turbulent flow characteristics of a horseshoe vortex within a developing scour hole, the flowzone downstream of the pier was beyond the scope of the investigation. Experiments were conducted for the approaching flow havingundisturbed flow depth �=0.25 m� greater than twice the pier diameter and the depth-averaged approaching flow velocity �=0.357 m/s�about 95% of the critical velocity of the uniform bed sand that had a median diameter of 0.81 mm. The flow measurements by the ADVwere taken within the intermediate �having depths of 0.25, 0.5, and 0.75 times the equilibrium scour depth� and equilibrium scour holes�frozen by spraying glue� at a circular pier of diameter 0.12 m. In order to have a comparative study, the ADV measurements within anequilibrium scour hole at a square pier �side facing the approaching flow� of sides equaling the diameter of the circular pier were alsotaken. The contours of the time-averaged velocities, turbulence intensities, and Reynolds stresses at different azimuthal planes �0, 45, and90°� are presented. Vector plots of the flow field at azimuthal planes reveal the evolution of the characteristics of the horseshoe vortex flowassociated with a downflow from intermediate stages to equilibrium condition of scour holes. The bed-shear stresses are determined fromthe Reynolds stress distributions. The flow characteristics of the horseshoe vortex are discussed from the point of view of the similaritywith the velocity and turbulence characteristic scales. The imperative observation is that the flow and turbulence intensities in thehorseshoe vortex flow in a developing scour hole are reasonably similar.

DOI: 10.1061/�ASCE�0733-9429�2007�133:4�399�

CE Database subject headings: Fluid flow; Turbulent flow; Open channel flow; Scour; Sediment transport; Steady flow; Hydraulics.

Introduction

Local scour at bridge piers and abutments in alluvial beds is themajor cause of failure of bridge foundations, and hence the pre-diction of the magnitude of scour at piers and abutments is ofgreat importance to field engineers. Despite a large number ofinvestigations �reviews given by Breusers et al. �1977�; Breusersand Raudkivi �1991�; Sumer and Fredsøe �1992�; Dey �1997�;Hoffmans and Verheij �1997�; Melville and Coleman �2000�; Bar-bhuiya and Dey �2004�� that focused mainly on the prediction ofthe maximum scour depth, an understandable perceptive of thescour mechanism from the viewpoint of the flow and turbulencecharacteristics of the horseshoe vortex during the development ofscour hole is still deficient. Comprehensive understanding of theturbulent flow fields aids prediction of scour magnitude precisely.Whereas characteristics of horseshoe vortex at the junction of thecylinder and the base plate were well explored �reviews put for-ward by Ballio et al. �1998� and Simpson �2001��, only a handful

1Associate Professor, Dept. of Civil Engineering, Indian Institute ofTechnology, Kharagpur 721302, West Bengal, India. E-mail: [email protected]

2Doctoral Research Fellow, Dept. of Civil Engineering, IndianInstitute of Technology, Kharagpur 721302, West Bengal, India.

Note. Discussion open until September 1, 2007. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this paper was submitted for review and pos-sible publication on November 4, 2005; approved on August 28, 2006.This paper is part of the Journal of Hydraulic Engineering, Vol. 133,
No. 4, April 1, 2007. ©ASCE, ISSN 0733-9429/2007/4-399–413/$25.00.
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J. Hydraul. Eng. 2007

of studies focused on the turbulent flow fields within a scour holeat piers �Melville 1975; Dey et al. 1995; Ahmed and Rajaratnam1998; Graf and Istiarto 2002�.

Melville �1975� �also see Melville and Raudkivi �1977�� wasthe pioneer to measure the turbulent flow field within a scour holeat a circular pier with the aid of the hot-film anemometer. Hemeasured the flow field in the upstream axis of symmetry and thenear-bed turbulence intensity for the case of a flat bed, interme-diate, and equilibrium scour hole. He also estimated the bed-shearstresses from the measured near-bed velocities. Dey et al. �1995��see also, Dey �1995�� put forward the three-dimensional velocitydistributions within an equilibrium scour hole at a circular piermeasured by the five-hole Pitot sphere. Ahmed and Rajaratnam�1998� attempted to describe the velocity distributions in the up-stream axis of symmetry within a scour hole at a circular pier byusing a Clauser-type defect method. Graf and Istiarto �2002� �seealso, Istiarto and Graf �2001�� detected velocities, turbulence in-tensities, Reynolds stresses, and bed-shear stresses in differentazimuthal planes within the equilibrium scour hole at a circularpier aided by the acoustic Doppler velocity profiler �ADVP�.

The present study addresses how the flow and turbulence char-acteristics of the horseshoe vortex change with the developmentof the scour hole at a circular pier providing a comprehensive dataset and proposing some important scaling issues related to theflow and turbulence. In addition, a comparative study on the flowstructures within equilibrium scour holes at circular and square�side facing the approaching flow� piers is put forward. It shouldbe noted that the flow measurements were carried out with afrozen bed, inhibiting possible sediment-fluid interactions �though
presumably small�. However, the flow zone downstream of the
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pier is beyond the realm of this study, as downstream of �beyond90°� the pier, the horseshoe vortex attenuates considerably beinginsignificant, and the shedding of wake-vortices is the predomi-nant mechanism of the flow resulting in an ill-defined flowstructure.

Experimental Setup and ADV Measurements

Experiments were carried out in a glass-sided rectangular flume15 m long, 0.9 m wide, and 0.7 m deep. A perspex made circularcylindrical pier model of diameter b=0.12 m was embedded ver-tically in the middle of a sand recess 2.4 m long, 0.9 m wide, and0.3 m deep, which retained uniform sand �median diameter,d50=0.81 mm; angle of repose, �=30°; and critical bed-shearstress, �c=0.408 Pa�. The geometric standard deviation of the par-ticle size distribution �g �=�d84/d16�0.5� of the sand was 1.34. Foruniformly graded sand, �g is less than 1.4 �Dey et al. 1995�. Thesand recess was positioned at 10 m downstream of the flumeinlet. In order to maintain the same bed level of the sand in therecess, a floor was constructed at an elevation of 0.3 m from theflume bottom throughout the length of the flume. The same uni-form sand was glued over the floor to simulate the turbulent flowover a rough planar sand bed during the experiments. A calibratedV-notch weir, fitted at the inlet of the flume, was used to measurethe flow discharge. In the flume, the flow depth was adjusted by atailgate. During the experiments, the approaching flow depth hwas maintained as 0.25 m �by operating the tailgate�; and thedepth-averaged approaching flow velocity U was set as0.357 m/s, which was about 95% of the critical velocity of theuniform sand bed to satisfy the clear water condition. The depth-averaged approaching flow velocity was determined from themeasured vertical profile of the approaching flow velocity at 2 mupstream the pier �where the presence of the pier did not affectthe approaching flow�. The instantaneous scour depth at a pierwas measured observing the position of the base of the scour holeby sliding a periscope up and down in the pier. An intense lightenabled one to read the scour depth from the graduation on thetransparent body of the pier with an accuracy of 1 mm. Whennegligible �1 mm or less� difference of scour depth was observedat an interval of 2 h after 72 h, it was considered that an equilib-rium stage of the scour hole was attained. However, total durationof the run was 80 h that was adequate for achieving the equilib-rium scour. After the run was stopped, the maximum equilibriumscour depth, observed at the upstream base of the pier, was thencarefully measured by a Vernier point gauge. In addition to theequilibrium scour hole, a number of intermediate scour holes�having intermediate scour depths ds of 0.25, 0.5, and 0.75 timesthe equilibrium scour depth dse� were stabilized for the acousticDoppler velocimeter �ADV� measurements. Once the equilibriumscour hole was obtained, the same run was repeated to achieve theintermediate scour holes for the desired fraction of the equilib-rium scour depth.

After draining out the water, when the bed was reasonably dry,a synthetic resin mixed with water �1:3 by volume� was sprayeduniformly over the scoured bed to freeze it. The bed was suffi-ciently impregnated becoming rock-hard with the resin when itwas left to set for a period of 72 h, facilitating measurements bythe ADV. The maximum equilibrium scour depth dse, located atthe circular pier upstream, was measured as 0.18 m. In order tohave a comparative study, the ADV measurements within an equi-librium scour hole �dse=0.21 m� at a square pier �side facing the
approaching flow� of sides equaling the diameter of the circular
400 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / APRIL 2007


pier were also taken. The contour lines of the scour hole at thecircular and square piers are shown in Figs. 1�a–e�.

The instantaneous three-dimensional velocity componentswere detected by a SonTek made 5 cm downlooking ADV�16 MHz MicroADV Lab Model�, which had a sampling rate andvolume of 50 Hz and 0.09 cm3, respectively. The output datafrom the ADV were filtered using a spike removal algorithm.Depending on the turbulence intensity, the sampling durationswere 3–10 min in order to have a statistically time independentaverage velocity. Near the bed, the sampling durations were rela-tively long. The ADV readings were taken along several verticallines at different azimuthal planes. The lowest vertical resolutionof the ADV measurements was 0.2 cm. On the other hand, thelowest horizontal resolution of the ADV measurements was 1 cm.However, the vertical resolution of the measurements was higherabove the scour hole. �Note: Vector plots representing character-istics of the flow field do not show all the experimental data at thelowest resolution in order to avoid overlapping or congestedplots.� The measurement by the ADV probe was not possible inthe zone 4.5 mm above the bed, because the ADV requires ameasuring volume of 0.09 cm3. However, one may infer the pos-sible magnitude �though it is not always easy� of flow within thiszone from the given flow mappings.

Analyses of Flow and Turbulence Fields

A cylindrical polar coordinate system is used to represent the flowand turbulence fields, as shown in Fig. 1�f�. The time-averagedvelocity components in �� ,r ,z� are represented by �u ,v ,w�,whose corresponding fluctuations are �u� ,v� ,w��. The positivedirections of u, v, and w are counterclockwise, outward, and up-ward, respectively. The velocity and turbulence fields are plottedin an rz plane at different azimuthal angles � ��0, 45, and 90°�.In order to avoid the solid portion of the pier in the diagrammaticrepresentations, the abscissa scale is given by r0 �=r−0.5b�. Itmeans that the ordinate scale at r0=0 represents the circular pierboundary. The resolutions in the abscissa are taken larger thanthose in the ordinate to have a clear representation of the flowfield within the intermediate scour holes, which are smaller indimension. Though it stretches the figures along the horizontal, itmakes it possible to show the flow and turbulence contours. It isimportant to point out that the outer radius of the ADV sensor was2.5 cm having three receiving transducers mounted on short armsaround the transmitting transducer at 120° azimuth intervals,which made it possible to measure the flow as close as 2 cm fromthe pier boundary.

Flow Fields

The contours of the time-averaged tangential velocity u at differ-ent azimuthal planes �0, 45, and 90°� for intermediate scour holes�having scour depths of 0.25dse, 0.5dse, and 0.75dse� at a circularpier and equilibrium scour holes �having equilibrium scour depthsdse� at circular and square piers are shown in Figs. 2�a–c�, whichrepresent the characteristics of the passage of the flow by the sideof the pier. Importantly, it drives the horseshoe vortex toward thepier downstream. At 0° �that is at the upstream axis of symmetry�,the tangential velocity u, which is essentially zero, is negligible aswas detected by the ADV. However, it becomes finite and in-creases with increase in azimuthal angle �. For instance, the mag-nitude of u at 90° is 1.3–1.7 times greater than that of u at 45° at
the corresponding locations. At 45 and 90°, u is relatively strong
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when the dimensions of scour holes are small, but it decreasesprogressively due to increase in flow area as the scour depthincreases. The passage of the downflow flux by the side of thepier results in a considerable enhancement of u near the pier, andthus u decreases with increase in radial distance r0 from the pierand remains almost constant over the flat bed. The magnitude of uincreases with increase in z and the vertical gradient of u �that is�u /�z� within the scour hole �that is z� 0� is more than that abovethe scour hole �that is z� 0�. On the other hand, for a square pier,the magnitudes of u are smaller than those of a circular pier at thecorresponding locations for the equilibrium scour hole, as the sizeof the scour hole at a square pier is larger than that at a circularpier.

The contours of the time-averaged radial velocity v at differentazimuthal planes �0, 45, and 90°� for intermediate scour holes at a

Fig. 1. Scour contours for circular pier with �a� ds=0.25dse; �b� ds=coordinate system

circular pier and equilibrium scour holes at circular and square

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piers are depicted in Figs. 3�a–c�. At 0°, the separation of theapproaching flow is evident just beneath the edge of the scourhole forming a reversal flow inside the scour hole �z�0�. Thus,radial velocity v changes direction on either side of the contourline of v=0, which falls approximately at the depths that are0.5–0.6 times the local depth of the scour hole below the originalbed level. It corroborates that in the pier upstream, a strong horse-shoe vortex exists inside the scour hole. The variations of v in thezones z�0 and z�0 are unlike. In z�0, the variation of v alongz �from z=0 toward the bed� is rapid reducing drastically to be-come zero at the depths that are 0.5–0.6 times the local depth ofthe scour hole. The magnitude of v becomes positive �away fromthe pier� near the bed as the flow returns from the base of the pierresulting in a reversed flow along the sloping bed of the scourhole. The magnitude of the maximum reversed velocity that oc-

; �c� ds=0.75dse; �d� ds=dse; �e� for square pier with ds=dse; and �f�
0.5dse
curs near the base of the pier is approximately 0.2U. In z�0, v is


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negative �toward the pier� and unidirectional whose variationalong z being relatively less is almost logarithmic due to the in-fluence of the approaching flow, becoming maximum near thefree surface. On the upstream flat bed, v remains essentially loga-rithmic and decreases upon entering the scour hole due to theexposure of larger flow area. Then, it progressively diminishestoward the pier. This is due to the existence of the pier �verticalsolid boundary� and an increase of the scour depth toward thepier. At 45°, though the overall distribution of v is almost similar�having magnitude of v smaller than that at the correspondinglocations of the flow zone at a circular pier at 0°�, the line ofseparation is shallower �approximately 0.7–0.75 times the localscour depth from the original bed level� as compared to that at 0°.The reducing nature of v with increase in � is evident from 0 to90° as a result of horseshoe vortex attenuation, whereas at 0° thedistribution of v is strongest. Dey et al. �1995� and Dey �1995�reported that the horseshoe vortex detaches from the pier at 75°,which was the reason to detect the flow at 90° being out of phasefrom those at 0 and 45°. Nevertheless, the overall flow feature at90° can be summarized as v acts toward the pier within the scourhole �z�0� �though there exists a thin zone of reversed flow closeto the bed� and above the scour hole �z�0� the flow deflectsoutwards by the side of the pier. The size of the separation zone ofv increases with increase in dimension of the scour hole. On theother hand, in case of a square pier, the depth of separation zoneis deeper and is proportional to the scour depth. Though at 0 and45°, the distributions of v at a square pier are similar to those at acircular pier, it is different at 90°, because the flow separationtakes place at 45° from the sharp edge of the square pier. This is

Fig. 2. Contours of time-averaged tangential velocity u �in

the reason why there exists lower v near the pier. However, at 0



and 45°, the nature of flow at a square pier is similar to that at acircular pier having magnitude of v slightly greater than that atthe corresponding locations of the flow zone at a circular pier byapproximately 10%.

Figs. 4�a–c� represent the contours of the time-averaged verti-cal velocity w at different azimuthal planes �0, 45, and 90°� forintermediate scour holes at a circular pier and equilibrium scourholes at circular and square piers. From an examination of thecontours of vertical velocity w at 0°, the separation of the ap-proaching flow below the edge of the scour hole is evident �as itwas in v contours� from the reversal nature of w near the scouredbed, though it is not distinct for the scour hole havingds=0.25dse. While w near the scoured bed is directed upward �thatis positive�, the distribution of w in the majority of the flow zoneis downward �that is negative�. The magnitude of w increases inthe downward direction from the free surface indicating that thereexists a downward negative pressure gradient. There is a core ofthe maximum w �encircled by the higher magnitudes of w� thatoccurs near the pier approximately at a depth of 0.4 times thelocal scour depth below the original bed level and then it de-creases toward the base of the scour hole. This substantiates theexistence of a strong downflow �along the upstream face of thepier� that turns near the base of the pier forming a vortical flowwithin the scour hole. The size of the core of the higher magni-tude of w increases with the development of the scour hole. Themaximum magnitude of w registered was 0.6U at �=0° andz=−0.09 m for the circular pier. �Note: The maximum magnitudeof w very close to the pier upstream, where the flow detection wasnot possible owing to the limitation of ADV, would possibly be

at azimuthal planes: �a� �=0°; �b� �=45°; and �c� �=90°
cm/s�
greater than 0.6U.� This maximum downflow is comparable to

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Fig. 3. Contours of time-averaged radial velocity v �in cm/s� at azimuthal planes: �a� �=0°; �b� �=45°; and �c� �=90°

Fig. 4. Contours of time-averaged vertical velocity w �in cm/s� at azimuthal planes: �a� �=0°; �b� �=45°; and �c� �=90°

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Istiarto and Graf’s �2001� 0.6U, while it is slightly lower than thatof Melville’s �1975� 0.8U, because the ADV measurement at theboundary of the pier was not possible. The magnitude of w de-creases with increase in � showing the attenuation of the horse-shoe vortex toward the downstream. For example, the maximummagnitudes of downflow at 45 and 90° are 0.7 and 0.4 times thatat 0°. On the other hand, for a square pier, the nature of flow issimilar to that of a circular pier having magnitude of downflowgreater than that at the corresponding locations of the flow zone ata circular pier by approximately 40%.

Figs. 5�a–c� show the time-averaged velocity vectors, whosemagnitude and direction are �v2+w2�0.5 and arctan�w /v�, respec-tively, at different azimuthal planes �0, 45, and 90°� for interme-diate scour holes at a circular pier and equilibrium scour holes atcircular and square piers. The vector plots at 0 and 45° display thecharacteristics of the horseshoe vortex along with the downflowalong the upstream face of the pier. As the length scales of theaxes �ordinate and abscissa� are different for intermediate scourholes, the vortices are apparently stretched. The vortical flow isnot distinct during the initial stage of the scour �ds=0.25dse�. Asthe dimension of the scour hole increases, the size of the horse-shoe vortex core being confined within the scour hole becomesbigger and well defined. The horseshoe vortex is a forced vortextype of flow, as the swirl velocity increases in the outward direc-tion from the center of the vortex. The shape of the vortex isalmost elliptical in cross section with the major axis being ap-proximately the bisector of the angle made by the slope of thescour hole with the horizontal. It is noticeable that the height�length of the minor axis� of the elliptical vortex at 0° is largerthan that at 45°. In general, the length of the minor axis is ap-

Fig. 5. Velocity vectors at azimuthal p

proximately 20–40% of the length of the major axis. While the



strongest vortical flow is observed at 0°, it decreases with increasein �. In z�0, the flow is in general horizontal and toward the pier,but it is downward close to the pier. However, the vortical flow isnot distinct at 90°, because there exists a separated flow. Never-theless, a close observation of the vector field at 90° reveals thata feeble vortical flow prevails near the scoured bed. In case of asquare pier, the size of the horseshoe vortex core is slightly largerthan that of a circular pier for an equilibrium scour hole.

The time-averaged absolute velocity V �=�u2+v2+w2�0.5� con-tours at different azimuthal planes �0, 45, and 90°� for intermedi-ate scour holes at a circular pier and equilibrium scour holes atcircular and square piers are depicted in Figs. 6�a–c�. It may benoted that V being a scalar quantity represents the total velocityintensity. At 0°, the absolute velocity V is solely due to verticalflow, as there exists no tangential velocity u. On the contrary, at90°, tangential velocity u is a predominant flow feature. At 45°,the diminishing nature of the velocity intensity V in the horseshoevortex with � is displayed. The concentration of the contour linesnear the scoured beds refers to the region of rapid change of themagnitude of velocity intensities. With the development of thescour hole, V decreases near the scoured bed. For a square pier,barring at 90°, the distributions of V are almost similar to those ofa circular pier for an equilibrium scour hole. At 90°, the nature ofV is varied, because v and w components are negligible there asthe side of the pier is parallel to the flow and the flow separationtakes place at 45° from the sharp edge of the pier. Therefore, thelower magnitude of V near the square pier is the consequence ofthe flow separation.

Figs. 7�a–c� exhibit the vorticity � �=�v /�z−�w /�r� contoursat different azimuthal planes �0, 45, and 90°� for intermediate

�a� �=0°; �b� �=45°; and �c� �=90°
lanes:
scour holes at a circular pier and equilibrium scour holes at cir-

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Fig. 6. Contours of time-averaged absolute velocity V �in cm/s� at azimuthal planes: �a� �=0°; �b� �=45°; and �c� �=90°

Fig. 7. Contours of vorticity � �in s−1� at azimuthal planes: �a� �=0°; �b� �=45°; and �c� �=90°


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cular and square piers. The vorticity contours are computed fromthe contours of v and w given in Figs. 4 and 5, respectively. InFigs. 7�a–c�, a left hand convention that is positive in the counterclockwise direction is adopted to define the vorticity. The vortic-ity � contours reveals that the vorticity is concentrated within thescour hole. Analysis of the vector plots and the vorticity contoursindicates that it is a forced vortex flow, as was discussed earlier.The region of high vorticity concentration near the center of thehorseshoe vortex indicates the vortex core. The vortex core issmaller during the initial stage of the scour hole and grows withthe development of scour hole. Also, the size of the vortex coredecreases with increase in �. At 90°, the vortex core is relativelysmall and the vorticity is considerably weak. In general, the vor-ticity in equilibrium scour hole at a square pier is marginallystronger than that at a circular pier. The circulation of the horse-shoe vortex is computed from the vorticity contours as shown inFigs. 7�a–c� by using the Stokes theorem. It is

Table 1. Circulations for Different Azimuthal Angles and Scour Depths

Pier type

Circulation �m2/s�

Scour depthds �=0° �=45° �=90°

Circular 025dse 4.35210−3 3.41410−3 6.61310−4

0.5dse 2.58210−2 2.02910−2 4.72210−3

0.75dse 5.34110−2 4.22310−2 1.01810−2

dse 6.65810−2 5.28110−2 1.27210−2

Square dse 8.92310−2 7.07710−2 1.68910−2

Fig. 8. Contours of tangential turbulence intensity u+ �in



=�c

V · ds =� �A

�dA �1�

where V�velocity vector; ds�differential displacement vectorover a closed curve; and A�area enclosed. Table 1 furnishes thecirculations computed for different azimuthal planes and scourdepths. The magnitudes of decrease and increase with increasein � and scour hole size, respectively. The magnitudes of at 45and 90° are 0.75–0.8 and 0.15–0.2 times at 0°, respectively, forall intermediate scour holes at a circular pier and equilibriumscour holes at circular and square piers. For a circular pier, themagnitudes of at all azimuthal planes �0, 45, and 90°� for in-termediate scour depths having ds=0.25dse, ds=0.5dse, andds=0.75dse are 0.05–0.07, 0.37–0.39, and 0.79–0.81 times ofthe equilibrium scour condition, respectively. However, at all azi-muthal planes, the magnitude of for a square pier is 1.33–1.35times for a circular pier in the equilibrium scour condition.

Turbulence Fields

Figs. 8�a–c�, 9�a–c�, and 10�a–c� represent the contours of theturbulence intensities u+ �=�u�u��0.5, where u��fluctuation of u�,v+ �=�v�v��0.5, where v��fluctuation of v� and w+ �=�w�w��0.5,where w��fluctuation of w�, respectively, at different azimuthalplanes �0, 45, and 90°� for intermediate scour holes at a circularpier and equilibrium scour holes at circular and square piers. Theturbulence intensities are in fact the root-mean-square values ofthe velocity fluctuations. Figs. 8�a–c�, 9�a–c�, and 10�a–c� revealthat the distributions of u+, v+, and w+ are almost similar. Ingeneral, there is a core of high turbulence intensity on the scoured

at azimuthal planes: �a� �=0°; �b� �=45°; and �c� �=90°
cm/s�
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Fig. 9. Contours of radial turbulence intensity v+ �in cm/s� at azimuthal planes: �a� �=0°; �b� �=45°; and �c� �=90°

Fig. 10. Contours of vertical turbulence intensity w+ �in cm/s� at azimuthal planes: �a� �=0°; �b� �=45°; and �c� �=90°


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bed as a result of flow separation; and the turbulence intensitiesdecrease with increase in r0 and z. Interestingly, the distributionpatterns of u+, v+, and w+ do not change much with �. Neverthe-less, the sizes of the high turbulence intensity cores increase withthe development of the scour hole. For a square pier, the turbu-lence intensities are slightly greater than those for a circular pierin equilibrium scour hole. Barring �r plane �that is the horizontalplane�, the turbulence characteristics are nonisotropic, because themean values �and standard deviation values� of the ratios ofu+ /v+, v+ /w+, and w+ /u+ are 1.008 �0.216�, 2.462 �0.715�, and0.441 �0.105�, respectively. Also, the distributions of turbulentkinetic energy k �=0.5�u+2+v+2+w2��, whose contours are shownin Figs. 11�a–c�, are similar to those of turbulence intensities.

The contours of the Reynolds stresses �uv�=−��u�v��,where ��mass density of water�, �vw �=−��v�w�� and�wu �=−��w�u�� at different azimuthal planes �0, 45, and 90°� forintermediate scour holes at a circular pier and equilibrium scourholes at circular and square piers are given in Figs. 12�a–c�, 13�a–c�, and 14�a–c�, respectively. In the figures, the Reynolds stressesare shown relative to the mass density � of water. The contours of�uv change sign from negative �above the scour hole� to positive�within the scour hole�. The distributions of �vw and �wu are rea-sonably similar. The Reynolds stresses �vw and �wu vary littleabove the scour hole and increase significantly within the scourhole due to the turbulent mixing of fluid as a result of vorticalflow. It results in a core of higher magnitude of �vw and �wu at thecentral portion of the scour hole. But, close to the scoured bed,

Fig. 11. Contours of turbulent kinetic energy k �in cm2

�vw and �wu reduce due to the lower magnitude �smooth flow� of



upward velocity �below the line of separation� along the inclinedbed of the scour hole. Within the scour hole, the Reynolds stressesat 45° are in general higher than those at 0 and 90°. With thedevelopment of the scour hole, the Reynolds stresses increasewithin the scour hole. In case of a square pier, the Reynoldsstresses are higher than those for a circular pier in an equilibriumscour hole.

Bed-Shear Stress

The bed-shear stress �b �time-averaged� acting within the scourhole is estimated using the distributions of the Reynolds stresses,as was done by Dey and Barbhuiya �2005�, as follows:

�b = ��2 + ��r cos � + �z sin ��2�at scoured bed �2�

where ��=�wu+�uv ; �r=�uv+�vw ; �z=�wu+�vw; and ��localangle of the scoured bed with the horizontal. Figs. 15�a–c� showthe variations of bed-shear stress �b with radial distance r0 atdifferent azimuthal planes for intermediate scour holes at a circu-lar pier and equilibrium scour holes at circular and square piers.The bed-shear stress �b increases with increase in � becomingmaximum at 45° and then it decreases. It is in conformity with theresults of Melville �1975� and Dey and Bose �1994�. The bed-shear stress �b also increases with increase in radial distance fromthe pier becoming maximum in the middle portion of the scour

azimuthal planes: �a� �=0°; �b� �=45°; and �c� �=90°
/ s2� at
hole and then it reduces to take the value of �b on the flat bed.

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This is the average trend, though the profiles have undulations asa result of bed slope irregularities. It is noticeable that beyondds=0.5dse, the magnitudes of the distributed �b decrease with thedevelopment of the scour hole, while the magnitudes of the dis-tributed �b at ds=0.5dse are greater than those at ds=0.25dse. Thecritical bed-shear stresses �bc �time-averaged� on the sloping bed,estimated using the method proposed by Dey �2003� for adversesloping beds with oblique near-bed flow, are also plotted alongwith �b for the purpose of comparisons. To estimate �bc using themethod proposed by Dey �2003�, one requires information oncritical bed-shear stress �c on a horizontal bed that was deter-mined from the Shields diagram. At 0°, it is obvious that themagnitudes of �bc are greater than �bc in intermediate scour holesindicating further scouring. On the other hand, in equilibriumscour holes, the values of �b are in general either almost equal orlower than �bc suggesting the equilibrium of scour. But at 45 and90°, the higher values of �b than �bc indicate the effects of adverseslope and oblique near-bed flow. Melville �1975� also had similarobservations. Importantly, Dey �2003� identified that there existsa discrepancy between the experimental and estimated criticalbed-shear stress on adverse sloping beds. This is the probablereason that �b values in some locations remain greater than �bc

values. However, in case of a square pier, the magnitudes of �b

are greater than those for a circular pier in an equilibrium scourhole.

Similarity Characteristics of Flow and Turbulence

To group all the flow data together in the horseshoe vortex flow at

Fig. 12. Contours of −u�v� �=�uv /�, in cm2/s2� a

the piers, two length scales d and h are introduced for the flow

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zone within �z�0� and above �z�0� the scour hole, respectively.Here, d is the local depth of the scour hole measured from theoriginal bed level. The scales of the velocities u, v, and w areconsidered as �u�max, �v�max, and �w�max for the individual profiles,respectively. For analyzing the profiles, the flow data for the scourhole with ds=0.25dse are not considered due to the smallness ofthe scour hole dimension. Also, for v and w profiles, the data at�=90° are not considered, as the vortex flow is not prominentthere. Figs. 16�a–c� exhibit that the proposed length and velocityscales facilitate all the flow data to collapse reasonably on a singleband. For w profiles, emphasis is given on the downflow profilesat the pier upstream. For turbulence intensities, consideration ofthe corresponding maximum turbulence intensity for the indi-vidual profiles makes it possible to bring down all the data ap-proximately on a single band �Figs. 17�a–c��. However, it is notpossible to collapse the Reynolds stress data on a single band. Thepossible reasons are partially attributed to the fact that there is arigorous mixing of fluid due to the vortex flow, and hence theReynolds stresses being very sensitive to the turbulent fluctua-tions subject to uncertain attenuation. Nevertheless, the flow andthe turbulence intensities in the horseshoe vortex flow in a devel-oping scour hole are reasonably similar.

Conclusions

The turbulent horseshoe vortex flow within the developing �inter-mediate stages and equilibrium� scour holes at a circular pier andequilibrium scour holes at a square pier was measured by an

uthal planes: �a� �=0°; �b� �=45°; and �c� �=90°
t azim
acoustic Doppler velocimeter. The contours of time-averaged ve-


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Fig. 13. Contours of −v�w��=�vw /�, in cm2/s2� at azimuthal planes: �a� �=0°; �b� �=45°; and �c� 0=90°

Fig. 14. Contours of −w�u� �=�wu /�, in cm2/s2� at azimuthal planes: �a� �=0°; �b� �=45°; and �c� 0=90°


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locities, turbulence intensities, and Reynolds stresses at differentazimuthal planes for developing and equilibrium scour holes havebeen presented. The velocity is reversal within the scour holeforming a horseshoe vortex. The maximum downflow at pier up-stream �2 cm from the pier boundary� registered was 0.6U at�=0° and z=−0.09 m for a circular pier in an equilibrium scourhole. However, the maximum downflow very close to the pierupstream, where the measurements were not possible due to thelimitation of ADV, would be greater than 0.6U. The vector plotsof the flow field provide a good understanding of the evolution ofthe horseshoe vortex within the developing scour hole. With thedevelopment of the scour hole, the size of the horseshoe vortexcore �having the shape of an ellipse�, which is a forced vortextype of flow being confined within the scour hole, becomes larger.The horseshoe vortex circulations, which decrease and increasewith increase in azimuthal angle and scour hole size, respectively,have been computed from the vorticity contours by using theStokes theorem. Within the scour hole, there exists a core ofhigher magnitude of turbulence intensities and Reynolds stressesthat increase with the development of the scour hole. The bed-shear stresses have been computed from the Reynolds stress dis-tributions. The magnitudes of the distributed bed-shear stressbeyond ds=0.5dse reduce with the development of scour hole,while magnitudes of the distributed bed-shear stress at ds

=0.5dse are greater than those at ds=0.25dse. The magnitudes ofbed-shear stress at 0° are generally greater and lower �or almostequal� than critical bed-shear stress in intermediate and equilib-rium scour holes, respectively, though there exist some discrep-ancies due to near bed flow over adverse slope. In general, for asquare pier, all the aforementioned flow, turbulence, and stressparameters are greater than those for a circular pier in an equilib-

Fig. 15. Distributions of �b and �bc �in Pa� at a
zimuthal planes: �a� �=0°; �b� �=45°; and �c� �=90°
rium scour hole. The flow and turbulence characteristics of the

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Fig. 16. Similarities of �a� u profiles; �b� v profiles; and �c� w profiles


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horseshoe vortex flow have been analyzed from the point of viewof the similarity proposing the possible velocity and turbulencecharacteristic scales. The most important observation is that theflow and turbulence intensities in the horseshoe vortex flow in adeveloping scour hole are reasonably similar.

Notation

The following symbols are used in this paper:A � area �L2�;b � pier diameter or width �L�;d � local depth of scour hole �L�;

ds � scour depth measured from initial bed level�L�;

dse � equilibrium scour depth �L�;ds � differential displacement vector �L�;

d16 � 16% finer sand diameter �L�;d50 � median diameter of sand �L�;d84 � 84% finer sand diameter �L�;

h � approaching flow depth �L�;k � turbulent kinetic energy �L2 T−2�;r � radial distance �L�;

r0 � r−0.5b �L�;U � depth-averaged approaching flow velocity

�L T−1�;−1

Fig. 17. Similarities of �a� u+ profiles; �b� v+ profiles; and �c� w+

profiles

u � time-averaged tangential velocity �L T �;



u� � fluctuation of u �L T−1�;u+ � �u�u��0.5�L T−1�;V � time-averaged absolute velocity �L T−1�;V � time-averaged velocity vector �L T−1�;v � time-averaged radial velocity �L T−1�;

v� � fluctuation of v �L T−1�;v+ � �v�v��0.5 �L T−1�;w � time-averaged vertical velocity �L T−1�;

w� � fluctuation of w �L T−1�;w+ � �w�w��0.5 �L T−1�;

z � vertical distance �L�;� � local angle of scoured bed with horizontal

�—�; � circulation �L−2 T−1�;� � azimuthal angle �—�;� � mass density of water �M L−3�;

�g � geometric standard deviation �—�;�b � local bed-shear stress in scour hole

�M L−1 T−2�;�bc � critical bed-shear stress on adverse sloping

beds with oblique near-bed flow �M L−1 T−2�;�c � critical bed-shear stress on horizontal bed

�M L−1 T−2�;�r � radial bed-shear stress �M L−1 T−2�;

�uv � −��u�v�� M L−1 T−2�;�vw � −��v�w�� M L−1 T−2�;�wu � −��w�u�� M L−1 T−2�;

�z � vertical bed-shear stress �M L−1 T−2�;�� tangential bed-shear stress �M L−1 T−2�;� � angle of repose �—�; and� � vorticity �T−1�.

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Characteristics of Horseshoe Vortex in Developing Scour Holes at Piers

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