Turbulent flow field in a scour hole at a semicircular abutment

Turbulent flow field in a scour hole at asemicircular abutment

Subhasish Dey and Abdul Karim Barbhuiya

Abstract: The three-dimensional turbulent flow field in a scour hole at a semicircular abutment under a clear waterregime was experimentally measured in a laboratory flume using an acoustic Doppler velocimeter. The distributions oftime-averaged velocity components, turbulent intensity components, turbulent kinetic energy, and Reynolds stresses atdifferent azimuthal planes are presented. Upstream, presentation of flow field through vector plots at azimuthal andhorizontal planes shows the existence of a large primary vortex associated with the downflow inside the scour hole. Onthe other hand, downstream, the flow field is irregular. The bed shear stresses are determined from the Reynoldsstresses and velocity gradients. The data presented in this paper would be useful for the development and validation offlow field models, which can be used to determine the strength of the primary vortex that is used to estimate scourdepth at bridge abutments.

Key words: bridge abutments, fluid flow, three-dimensional flow, turbulent flow, open channel flow, scour, sedimenttransport, hydraulic engineering.

Résumé : Le champ de courant turbulent tridimensionnel dans une fosse d’affouillement à une culée de pont semi-circulaire, sous un régime d’eau libre, a été mesuré de façon expérimentale dans un canal d’expérimentation au moyend’un courantomètre à effet Doppler acoustique. Cet article présente les distributions à moyenne temporelle des compo-santes de la vitesse, des composantes de l’intensité de la turbulence, de l’énergie cinétique de la turbulence et des ten-sions de Reynolds selon divers plans azimutaux. Dans la portion amont, la présentation du champ de courant au moyende tracés de vecteurs sur des plans azimutaux et horizontaux démontre l’existence d’un gros vortex primaire associé aucourant descendant à l’intérieur de la fosse d’affouillement. Par contre, dans la portion aval, le champ de courant est ir-régulier. Les contraintes de cisaillement sur le lit sont déterminées à partir des tensions de Reynolds et des gradients devitesse. Les données présentées dans cet article seraient utiles pour développer et valider le modèle du champ de cou-rant, lequel pourrait être utilisé pour déterminer la force du vortex primaire qui est à son tour utilisé pour estimer laprofondeur d’affouillement aux culées des ponts.

Mots clés : culées de pont, écoulement fluide, écoulement tridimensionnel, écoulement turbulent, écoulement dans uncanal à surface libre, affouillement, transport des sédiments, génie hydraulique.

[Traduit par la Rédaction] Dey and Barbhuiya 232

Introduction

Behavior of a river at bridge sites is a challenging prob-lem to the hydraulic engineers because of localized scour atpiers and abutments. Depending on the supply of sedimentby the approaching flow, the localized scour occurs in twoways, clear water scour and live bed scour (Dey 1997). Clearwater scour refers to the situation where no sediment is sup-plied by the approaching flow in the zone of scour. Live bedscour, in contrast, occurs where there is a general transport

of sediment by the approaching flow to the zone subjected toscour.

The flow field at a semicircular abutment, embedded ver-tically in a loose sediment bed, is complex in detail; thecomplexity increases with the development of the scour hole,which involves separation of flow to develop three-dimensional(3-D) vortex flow. Though the flow field at piers has beenwell documented (Hjorth 1975; Melville 1975; Dey 1995;Dey et al. 1995; Graf and Istiarto 2002), research on flowfield at abutments has been limited. Kwan (1988) and Kwanand Melville (1994) measured the 3-D flow field in a scourhole at a wing-wall abutment, using the hydrogen bubbletechnique. They found that a primary vortex, which is similarto the horseshoe vortex at piers, and downflow were primarilyresponsible for the scouring at abutments. Also, Rajaratnamand Nwachukwu (1983) and Ahmed and Rajaratnam (2000)studied the flow fields at groin and abutment, respectively,placed on a planar or unscoured bed. The studies on scour atsemicircular abutments were by Wong (1982) and Kwan(1984), who investigated the effects of flow parameters onscour depth. No attempt has so far been made to study the 3-D turbulent flow field at a semicircular abutment.

Can. J. Civ. Eng. 32: 213–232 (2005) doi: 10.1139/L04-082 © 2005 NRC Canada

213

Received 22 October 2003. Revision accepted 19 August2004. Published on the NRC Research Press Web site athttp://cjce.nrc.ca on 15 March 2005.

S. Dey.1 Department of Civil Engineering, Indian Institute ofTechnology, Kharagpur 721302, West Bengal, India.A.K. Barbhuiya. Department of Applied Mechanics, NationalInstitute of Technology, Silchar 788010, Assam, India.

Written discussion of this article is welcomed and will bereceived by the Editor until 30 June 2005.

1Corresponding author (e-mail: [email protected]).

The aim of the present study is to investige the 3-D turbulentflow field in a scour hole at a semicircular abutment under aclear water regime, using an acoustic Doppler velocimeter(ADV). The time-averaged velocity components, turbulentintensity components, turbulent kinetic energy, and Reynoldsstresses at different azimuthal planes were measured experi-mentally by the ADV. The flow field is represented using themeasured velocity data through the vector plots at differentazimuthal and horizontal planes. The bed shear stresses aredetermined from the Reynolds stresses and velocity gradi-ents. The data presented in this paper would be useful forthe development and validation of flow field models, whichcan be used for the estimation of scour depth at bridge abut-ments, as was done by Baker (1980) and Dey (1999) in pierscour.

Experimentation

The experiment was carried out in a 20-m-long, 0.9-m-wide,and 0.7-m-deep laboratory flume. A perspex semicircularabutment having a diameter, b, of 0.26 m was embedded in asediment recess 2.4 m long, 0.9 m wide, and 0.3 m deep,which contained uniformly graded sand of a median diame-ter, d, of 0.52 mm. The abutment was attached to the glassside wall of the flume with a protruding length of 0.13 m(that is the transverse length of the abutment l) in the flow.To get the fully developed turbulent flow, the sediment re-cess was located 10 m downstream of the flume inlet. Afalse floor measuring 0.3 m high was constructed along thelength of the flume to maintain the same level of the sedi-ment bed in the sediment recess. To simulate the turbulentflow over a rough planar sediment-bed, uniform sand havingthe same size as that used for the scouring test was gluedover the false floor. Geometric standard deviation, σg, of par-ticle size distribution given by (d84/d16)

0.5, was 1.21; this isusually <1.4 for a uniformly graded sand (Dey et al. 1995).The flow discharge, controlled by an inlet valve, was mea-sured using a calibrated V-notch weir fitted at the inlet of theflume. The flow depth, adjusted by a tailgate, was recordedby a Vernier point gage with an accuracy of ±0.1 mm. Theflume was first slowly filled up with water by a pipe. Oncethe desirable flow depth was reached, the experiment wasrun under clear water scour regime by adjusting the dis-charge to a desired value. The approaching flow depth, h,was set at 0.2 m; and the experiment was run for an averageapproaching flow velocity, U, of 0.294 m/s, which main-tained a clear water scour regime having u*/u*c = 0.95,where u* is the approaching shear velocity, and u*c is thecritical shear velocity of sediment particles. The criticalshear velocity, u*c, of sediment particles was determinedfrom the Shields diagram. The average approaching flow ve-locity, U, was obtained from the measured vertical distribu-tion of approaching flow velocity at the mid-section of theflume, which was located at 2 m upstream of the center ofthe semicircular abutment, where the free flow occurred.The semi-logarithmic plot of the approaching flow velocitydistribution was used to determine approaching shear veloc-ity, u*. The average velocity, U, calculated from the dis-charge (0.0524 m3/s) using the continuity equation is0.291 m/s, close to the average velocity calculated from the

velocity profile. However, a U-value of 0.294 m/s is used inthis study.

An experiment was run under a clear water scour regimefor a period of more than 2 d, at which time an equilibriumstate of scour was attained. The instantaneous maximumdepth of scour at the abutment was measured by observingthe position of the base of the scour hole, using a scale withan accuracy of ±1 mm fitted to the inside of the abutment. Astrong light was used to improve the visibility. When no dif-ference of scour depth was observed at an interval of 2 h af-ter 2 d, it was assumed that an equilibrium state of scourwas reached. Then the inflow rate was slowly reduced tozero, and the water was carefully drained from the scourhole. To stabilize the scour hole, a synthetic resin mixedwith water (1:3 by volume) was uniformly sprayed over thescoured bed when it was reasonably dry. The sand was suffi-ciently impregnated with the resin when it was left to set for12 h. After drying for up to 24 h, the scoured bed profile be-came rock-hard, facilitating flow measurements. The scourdepths at different radial directions were measured by a Ver-nier point gauge to determine the scour profiles at variousazimuthal planes. The contours of the scour hole at a semi-circular abutment are shown in Fig. 1. In this study, the max-imum equilibrium scour depth, ds, located adjacent to theabutment at θ = 45°, was obtained as 0.162 m below theoriginal bed.

A 5-cm downlooking ADV, developed by SonTek, wasused to measure the instantaneous 3-D velocity components.The ADV operates on a pulse-to-pulse coherent Dopplershift to provide 3-D velocity components at a rate of 50 Hz,which was adequate for the detection of turbulent fluctuationcomponents or turbulent energy. However, it was not enoughfor the measurement of vertex shedding downstream of theabutment. The acoustic sensor consisted of one transmittingtransducer and three receiving transducers. The receivingtransducers were mounted on short arms around the trans-mitting transducer at 120° azimuth intervals. Acoustic beamswith a frequency of 10 MHz were emitted from the transmit-ting transducer. The beams traveling through the water ar-rived at the measuring point, located 5 cm below thetransducer. They were reflected by the ambient particleswithin the flow being received by the receiving transducers.The processing module performed the digital signal process-ing required to measure the Doppler shifts. The data acquisi-tion software provided real-time display of the data ingraphical and tabular forms. The velocity measurement bythe ADV probe was not possible in the zone, which was5 cm below the free surface. It is important to mention thatthe time-averaged velocity components were taken for a pe-riod of 3 min to ensure that the velocity components wereindependent of time.

The probable error in ADV readings is due to spikes(mainly found near the bed) caused by the phase shift ambi-guities between the outgoing and incoming pulses that lieoutside the range from –180° to 180°, resulting in a spike inthe record. Such a situation can occur when the flow veloc-ity exceeds the upper limit of the ADV probe velocity rangeor when there is contamination from previous pulses re-flected from the boundaries with uneven profiles, for exam-ple, a scoured bed. Also, the spikes may occur because ofhigh turbulence and aerated flows. Therefore, the output data

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from the ADV were filtered using spike removal algorithm,after Wahl (2003).

The ADV readings were taken along several vertical linesat azimuthal planes of θ of 10°, 30°, 45°, 90°, 120°, and170°. Also, to get flow vectors at horizontal planes, ADVreadings were taken along different radial lines at horizontalplanes of z of –0.5ds, 0, and 0.5h. A detailed description ofthe experimental setup and procedure is available inBarbhuiya (2003).

Experimental results

Velocity distributionsIn cylindrical polar coordinates (Fig. 2), the time-averaged

velocity components in θ, r, and z are represented by u, v,and w. All linear dimensions and velocity components arenormalized by the transverse length of the abutment, l, andthe average approaching flow velocity, U, respectively.Hence, the velocity distributions are shown in the ��rz plane at

different azimuthal angles θ, where �r is r/l and �z is z/l. It isimportant to point out that the velocity readings were con-fined to the region 5 cm below the free surface owing to thelimitation of the ADV. Therefore, flow field near the freesurface is beyond the realm of this study.

The vertical distributions of the normalized time-averagedtangential velocity component, û (û = u /U), at different azi-muthal planes are shown in Fig. 3. The tangential velocity,u, shows the characteristic feature of the passage of ap-proaching flow by the side of the abutment as a skewed ve-locity profile. The magnitude of u is relatively low at θ =10°, as it originates beyond 0° and grows with θ. Though novelocity measurement was possible at 0° because of theflume wall, the no-slip results u = 0 at θ = 0°. The distribu-tion of u above the original bed (z ≥ 0) is almost similar tothe turbulent flow distribution on a planar bed and can,therefore, be represented by a power law in the upstream re-gion (10° to 90°) of the abutment. Adjacent to the abutment,the passage of downflow by the side of the abutment en-hances u substantially. The radial distribution of u is notshown here. However, a close examination of the verticaldistribution of u, reveals that there is considerable skewnessin the distribution pattern. This is believed to be due to thepronounced effect of the primary vortex, which is ahelicoidal flow around the abutment. The presence of strongcirculation reduces tangential velocity considerably. This ef-fect is particularly evident from the velocity distribution in-side the scour hole. In fact, the primary vortex, whichoriginates at the upstream of the abutment, is forced by thetangential velocity, u, to drift by the side of the abutment.The magnitude of u increases with an increase in θ, startingwith zero at 0° and becoming a maximum at 90°. Down-stream, the tangential velocity diminishes considerably withfurther increase in θ beyond 90°. At 170°, it shows a flow re-versal (negative) near the abutment as a result of wake dueto the flow separation; and further downstream, the velocitydistributions show a recovery, though it is ill defined. Themaximum magnitude of tangential velocity, u, registered was1.23U at θ = 90°, r = 1.2l, and z = 0.6l.

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Dey and Barbhuiya 215

Fig. 2. Schematic diagram of a scour hole at a semicircular abut-ment and coordinate system (θ = 0° at the flume wall).

Fig. 1. The scour hole at a semicircular abutment (contour lines positive for scour and negative for deposition, in millimetres).

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Fig. 3. Vertical distributions of û at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

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Fig. 4. Vertical distributions of �v at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

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Fig. 5. Vertical distributions of �w at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

The vertical distributions of the normalized time-averagedradial velocity component, �v (�v v= /U), at different azimuthalplanes are presented in Fig. 4. It is evident from the radialvelocity, v, distribution upstream of the abutment (10° to90°) that the flow separation at the edge of the scour holeproduces a reversal nature of v inside the scour hole. Thus, vchanges sign in the scour hole on either side of an imaginarylocus of v = 0, which is nearly a median of the approxi-mately triangular sections of scour hole. This confirms astrong helicoidal flow inside the scour hole upstream of theabutment. This may be compared with the radial velocity ina meandering channel (Odgaard 1989) or inside a scour holeat a cylindrical pier (Dey 1995). The variations of v insideand above the scour hole are different. Inside the scour hole,the vertical distribution of v varies almost linearly startingwith zero at the mid-depth of the scour hole and becoming apositive (away from the abutment center) maximum near thebed owing to the reversed flow along the upward directionof the slope and a negative (towards the abutment center)maximum locally at z = 0. Having reached a positive maxi-mum near the bed, v diminishes towards the bed because ofthe effect of bed (solid boundary) in maintaining the no-slipcondition. Above the scour hole, the variation of v along z,which is negative, is almost logarithmic owing to the effectof the approaching flow, starting from the profile at z = 0and becoming a maximum near the free surface. However,outside the scour hole, v remains essentially almost logarith-mic and slightly decreases upon entering the scour hole;then it progressively diminishes over the entire flow depthgoing towards the abutment. Although the radial distributionof v is not shown, a close examination of the vertical distri-bution of v reveals a nonlinear (approximately parabolic)variation of v along r. The magnitude of v decreases with adecrease in r because of the vertical boundary of the abut-ment. Also, the decreasing nature of v with increase in θ for10° to 90° is evident as a result of primary vortex attenua-tion. Downstream, the radial velocity, v, becomes positive,having no circulatory motion inside the scour hole with fur-ther increase in θ beyond 90°. At 170°, it shows a flow rever-sal (positive) above the scour hole adjacent to the abutmentas a result of backflow due to the flow separation. Furtherdownstream, the flow reversal diminishes, and the velocitydistributions show a recovery that is a part of the main flowleaving the scour hole.

The vertical distributions of normalized time-averagedvertical velocity component, �w ( �w = w/U), at different azi-muthal planes are shown in Fig. 5. At the edge of the scourhole, the flow separation is also evident from the reversalnature of w inside the scour hole. The variation of w is non-linear with a negative value over almost the entire depth (in-side the scour hole and above it), whereas close to thescoured bed, w is directed upward and has a positive value.The negative value of w confirms that there is a strongdownflow in front of the abutment in the scoured zone. Inapproaching region r ≥ 2l, w remains negligible but subse-quently grows up towards the abutment. However, upstream(10° to 90°), the downflow that increases towards the abut-ment becomes a maximum near the vertical wall of the abut-ment. The maximum magnitude of w measured was 0.6U atθ = 10° and z = –0.2l. To be more explicit, at the upstreamface of the abutment, an adverse pressure gradient creates a

downflow that is returned at the base of the scour hole,showing a strong counter-clockwise vortex, referred to asprimary vortex. The magnitude of w decreases with an in-crease in z above the scour hole. The imaginary locus of w =0 is a curved line meeting the sloping bed of the scour holeat its extreme ends (base of the abutment and edge of thescour hole). On either side of the locus, w changes sign.Thus, there lies a helicoidal flow inside the scour hole up-stream of the abutment. As the magnitude of w decreaseswith an increase in θ, the primary vortex attenuates. In thedownstream flow, the vertical velocity, w, becomes positivewith further increase in θ beyond 90°. At 170°, it shows astrong upward flow adjacent to the abutment as a result ofthe suction that takes place, which helps to dislodge the bedsediments in that region; and further downstream, w dimin-ishes as the flow leaves the scour hole.

Figure 6 exhibits the normalized time-averaged velocityvectors, whose magnitude and direction are ( � � ) /v w2 2 1 2+ andarctan ( � / �)w v , respectively, at different azimuthal planes. Thecharacteristics of the circulatory flow inside the scour holetogether with the strong downflow along the upstream faceof the abutment are depicted. The circulation is strong at theupstream of the abutment and decreases with increase in θ.Above the scour hole, the flow is horizontal for r > 3.5l, andthen the flow gradually curves down towards the abutment.At 90°, the circulatory motion is very feeble. Downstream,the flow is directed outwards with further increase in θ be-yond 90°. At 170°, it shows an upward flow near the abut-ment causing suction, and further downstream, the velocityis a part of the main flow. The normalized time-averagedvelocity vectors, whose magnitude and direction are( � � ) /u v2 2 1 2+ and arctan ( � / �)v u , respectively, at different hori-zontal planes are presented in Fig. 7. The velocity vectorsinside the scour hole (z = –0.5ds) show mainly the passageof flow inside the scour hole. For z = 0, the flow velocity ad-jacent to the abutment is relatively high. The velocity vec-tors at z = 0.5h show the inward motion of the flow in frontof the abutment having a smooth passage by the side of theabutment with a feeble backflow adjacent to the abutment inthe downstream. The normalized time-averaged absolute ve-locity, �V ( �V = ( � � � ) )/u v w2 2 2 1 2+ + , contours at different azi-muthal sections are given in Fig. 8. Having feeble tangentialvelocity, u, at 10°, the absolute velocity is strongly circula-tory, as already shown in Fig. 6a. At 90°, u dominates andcirculatory flow becomes weak. In other azimuthal planes,the diminishing nature of circulatory flow with θ is dis-played. At 45° to 120°, the concentration of contour linesnear the abutment refers to the zone of rapid change of ve-locity magnitudes from low to high velocity. The velocitycontours downstream of the abutment show that the flow isbecoming a part of the main flow as it leaves the scour hole.

The physics of the flow is analyzed with the help of theprevious mentioned results. The main characteristic featuresof the flow at a semicircular abutment are a relatively largeprimary vortex flow and skewed velocity distributions. Theapproaching flow separates at the upstream edge of the scourhole, known to be a line of separation forming a vortex flowinside the scour hole, which acts as a zone of separation.The downflow, developed because of the downward negativestagnation pressure gradient of the differential approachingflow velocity along the adjacent vertical face of the abut-

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ment upstream, is pushed up by the vortex. The limitingstreamlines of the approaching flow along the original bedupstream and the reversed flow along the sloping bed mergeat the edge of the scour hole, forming a separated stream-line. Thus, a surface of separation is produced in the form of

an envelope. In this process, the approaching flow curvesdown into the scour hole and rolls to form a vortex flow,called primary vortex, which migrates downstream by theside of the abutment. The migration of the vortex flow alongthe circumference of the abutment towards the downstream

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Fig. 6. Normalized velocity vectors at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

may partially be compared with the passage of helicoidalflow in a right-angled triangular bend of a channel (Rozovskii1961). However, unlike the flow in the channel bends, thevertical velocity, w, is substantially high because of a strong3-D vortex flow inside the scour hole associated with adownflow in front of the abutment. Downstream, the flowcharacteristics are different. The flow becomes ill defined

and reversed. This is due to the flow separation at the abut-ment side, which produces a wake downstream of it.

Turbulent intensityThe vertical distributions of the normalized tangential tur-

bulent intensity component u+ (u+ = (u′2)1/2/U, where u′ is

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Fig. 7. Normalized velocity vectors at sections: (a) z = –0.5ds, (b) z = 0, and (c) z = 0.5h.

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Fig. 8. Normalized absolute velocity �V contours at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°,and (f) θ = 170°.

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Fig. 9. Vertical distributions of u+ at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

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Fig. 10. Vertical distributions of v+ at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

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Fig. 11. Vertical distributions of w+ at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

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Fig. 12. Vertical distributions of �k at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

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Fig. 13. Vertical distributions of uw+ at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

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Fig. 14. Vertical distributions of vw+ at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

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Fig. 15. Vertical distributions of uv+ at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

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Fig. 16. Variations of �τb with �r at azimuthal sections: (a) θ = 10°, (b) θ = 30°, (c) θ = 45°, (d) θ = 90°, (e) θ = 120°, and (f) θ = 170°.

the fluctuation of u) at different azimuthal planes are pre-sented in Fig. 9. The tangential turbulent intensity remainsessentially unchanged above the scour hole, having a distri-bution more or less linear up to z = 0. Upon entering thescour hole, the flow encounters the incline in the bed in theform of a reversal flow. Hence, inside the scour hole, u+

increases towards the scoured bed. A most distinguishablefeature of the distribution is the pronounced bulges immedi-ately below the z = 0 line, as a result of flow separation in-side the scour hole. It is also evident that u+ slightlydecreases with an increase in θ up to 90°. Then the flow sep-arates from the abutment, forming the wake in the down-stream, where u+ increases significantly. The spikes near thescoured bed in the upstream and downstream of the abut-ment are due to the shuddering effect of the primary vortexand the shedding of the wake vortices. The vertical distribu-tions of the normalized radial turbulent intensity componentv v v U+ + = ′( ( ) //2 1 2 , where v′ is the fluctuation of v) and the

vertical turbulent intensity component w w w U+ + = ′( ( ) //2 1 2 ,where w ′ is the fluctuation of w) at different azimuthalplanes are shown in Figs. 10 and 11, respectively. The distri-bution characteristics of v+ and w+ are almost similar to thatof u+. However, it is apparent that v+ distributions are rela-tively dominant ones and w+ distributions do not show anyspike near the scoured bed, as there is no shuddering effector shedding of primary or wake vortex in the vertical direc-tion. The cumulative effect of u+, v+, and w+ is observed inthe vertical distributions of normalized turbulent kinetic en-ergy, � ( � / )k k k U v w= = + ++ + +2 2 2 20.5 (u , where k is the tur-bulent kinetic energy), at different azimuthal planes and isshown in Fig. 12. The bulges result from the flow separationinside the scour hole, and the kinetic energy gets very strongover the entire depth behind and close to the abutment. It isimportant to mention that in the wake region the measure-ments did not have a resolution that takes care of chaoticflow with shedding of wake vortices. This is the possiblecause of spikes in the distributions.

Reynolds stressesThe vertical distributions of normalized Reynolds stresses

uw+ ( / )*uw u w u+ = − ′ ′ 2 , vw+ ( / )*vw v w u+ = − ′ ′ 2 , and uv+

( / )*uv u v u+ = − ′ ′ 2 at different azimuthal planes are shown inFigs. 13 to 15. Upstream, the Reynolds stresses, being rea-sonably linear, are unchanged above the scour hole and showdistinguishable bulges inside the scour hole. However, closeto the scoured bed Reynolds stresses become small. In theentire region, uw+ and vw+ are essentially positive and nega-tive, respectively, and change sign near the scoured bed ow-ing to the reversal flow. On the other hand, uv+ is relatively

weak, though a few bulges are present near the scoured bed.Downstream, uw+, vw+, and uv+ grow substantially, havinginconsistent distributions (no conclusive trend) due to theflow separation and vortex shedding. However, above thescour hole, uw+ and uv+ change signs, and in the vicinity ofscoured bed, they are small in magnitude.

Bed shear stressThe shear stress, τb, acting on the scoured bed is estimated

using the Reynolds stresses as follows:

[1] τ τ τ φ τ φθb r b z b= + +2 2( cos sin )

where τθ is − ′ ′ + ′ ′ρ( )w u v u , τr is − ′ ′ + ′ ′ρ( )u v w v , τz is− ′ ′ + ′ ′ρ( )u w v w , φb is the local angle of the scoured bedwith the horizontal, and ρ is the mass density of water. Thenormalized bed shear stress �τb at the scoured bed can be ex-pressed as

[2] � ( cos sin )τ ττ τ

τ τ φ τ φο ο

θbb

r b z b= = + +1 2 2

where τo is the bed shear stress of approaching flow, that is,ρu*

2. The shear stress on the scoured bed is also calculatedfrom the velocity gradients, as was done by Melville andRaudkivi (1977). The normalized equation of bed shearstress is

[3] �

( )τ ρ ν ετ ηο

bdd

= + q

where ν is the kinematic viscosity of water (assumed as10–6 m2/s), ε is the eddy viscosity, q is the velocity parallelto the scoured bed at a normal distance η of 4 mm above thebed, and η is the distance normal to the bed. Since the eddyviscosity, ε, is a function of circulatory flow (Odgaard1986), it is a variable in the scour hole for different azi-muthal angles θ. Therefore, the value of �τb obtained fromeq. [2] is used to evaluate ε using eq. [3]. Table 1 shows theaverage values of ε estimated for different θ. Figure 16shows the variations of normalized bed shear stress �τb with �rfor different θ obtained using eqs. [2] and [3]. The bed shearstress decreases with an increase in θ up to 90°. At �r = 2 to2.5, �τb becomes greater than unity, as the reversed flow en-counters the adverse slope of the scour hole. Downstream,the values of �τb at different locations are also >1 for thesame reason.

Conclusions

In a laboratory flume, experimental detections were madeby using an ADV to determine 3-D turbulent flow field in ascour hole at a semicircular abutment under a clear water re-gime. The distributions of time-averaged velocity compo-nents, turbulent intensity components, turbulent kineticenergy, and Reynolds stresses at different azimuthal planeshave been presented. Upstream, the flow vectors at azimuthaland horizontal planes show the existence of a large primaryvortex associated with the downflow inside the scour hole.On the other hand, downstream, the flow field is essentiallyill defined, comprising irregularities due to the vortex shed-ding. The bed shear stresses have been determined from the

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θ (°) ε (m2/s)

10 1.576 × 10–4

30 3.575 × 10–5

45 6.000 × 10–5

90 3.246 × 10–5

120 4.562 × 10–5

170 1.210 × 10–4

Table 1. Values of eddy viscosityfor different azimuthal angles.

Reynolds stresses and velocity gradients. The data presentedin this paper would be useful for the development of mathe-matical models of flow field in a scour hole at bridge abut-ments. Thus, the accurate estimation of scour depth wouldbe possible using the flow field model.

References

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List of symbols

b streamwise length or diameter of semicircular abutmentd median diameter of bed sediment

ds depth of scourh approaching flow depthk turbulent kinetic energy�k normalized turbulent kinetic energyl transverse length or radius of semicircular abutmentq velocity parallel to scoured bed at a normal distance

4 mm above bed�r normalized radial distance

U average approaching flow velocityu tangential velocity�u normalized time-averaged tangential velocity component

u* shear velocity of approaching flowu+ normalized tangential turbulent intensity component

′u fluctuation of uu*c critical shear velocity of bed sedimentuv+ normalized Reynolds stress in θr-plane

u v w, , time-averaged velocities in cylindrical polar coordinatesuw+ normalized Reynolds stress in θz-plane

�V normalized time-averaged absolute velocityv radial velocity�v normalized time-averaged radial velocity component

v+ normalized radial turbulent intensity component′v fluctuation of v

vw+ normalized Reynolds stress in rz-planew vertical velocity�w normalized time-averaged vertical velocity component

w+ normalized vertical turbulent intensity component′w fluctuation of w�z normalized vertical distanceε eddy viscosity

φb local angle of scoured bed with horizontalη distance normal to bed

θ, ,r z cylindrical polar coordinatesv kinematic viscosity of waterρ mass density of water

σg geometric standard deviationτb local bed shear stress in scour hole�τb normalized bed shear stressτο bed shear stress of approaching flowτ r radial bed shear stressτz vertical bed shear stressτθ tangential bed shear stress

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232 Can. J. Civ. Eng. Vol. 32, 2005

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Turbulent flow field in a scour hole at a semicircular abutment