International Journal of Sediment Research 31 (2016) 244–250

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International Journal of Sediment Research

http://d1001-62Elsevier

n CorrE-m

antonio

journal homepage: www.elsevier.com/locate/ijsrc

Original Research

Effect of pier shape and pier alignment on the equilibrium scour depthat single piers

Cristina Fael a,n, Rui Lança b, António Cardoso c

a Faculdade de Engenharia, Universidade da Beira Interior, Edifício II das Engenharias. Calçada Fonte da Lameiro, 6200-358 Covilhã, Portugalb Instituto Superior de Engenharia/Universidade do Algarve, Campus da Penha, 8005-139 Faro, Portugalc Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal

a r t i c l e i n f o

Article history:Received 27 November 2014Received in revised form28 March 2016Accepted 1 April 2016Available online 8 April 2016

Keywords:ScourPier shapePier alignment

x.doi.org/10.1016/j.ijsrc.2016.04.00179/& 2016 International Research and TraininB.V. All rights reserved.

esponding author.ail addresses: cfael@ubi.pt (C. Fael), rlanca@ua.cardoso@tecnico.ulisboa.pt (A. Cardoso).

a b s t r a c t

The equilibrium scour depth at uniform single bridge piers depends on a large number of variables,including the pier horizontal cross-section shape and its alignment angle towards the flow direction. Theinfluence of these variables has been studied by only a few researchers, mostly, on the basis of tests thatwere far from approaching equilibrium. This experimental study aims at revisiting the influence of piers'shape and alignment on local scouring for length–width ratios smaller than or equal to 4, by increasingthe experimental evidence. Fifty five long-duration laboratory tests were run under steady, clear-waterflow, close to the threshold for initiation of sediment motion. Five pier shapes were considered: circular,rectangular square-nosed, rectangular round-nosed, oblong, and zero-spacing (packed) pile-groups; thetested skew-angles were 0°, 30°, 45°, 60°, and 90°. It was concluded that i) the shape factor can be takenas 1.0, for rectangular round-nosed and oblong cross-section piers, and as 1.2, for rectangular square-nosed and packed pile-group cross-section piers, ii) the shape factor does not vary significantly with theduration of tests, this way confirming the robustness of the shape factors reported to date, iii) the effectof shape is present at skewed piers although the associated coefficients remain in the narrow range of1.0–1.2, and iv) for length–width ratios smaller than 4, the shape factor is of the same order of magnitudeas the skew angle factor and should not be neglected.& 2016 International Research and Training Centre on Erosion and Sedimentation/the World Association

for Sedimentation and Erosion Research. Published by Elsevier B.V. All rights reserved.

1. Introduction

Local scour at bridge foundations is a rather common cause ofbridges' failure. Consequently the prediction of the ultimate“equilibrium” scour depth is a key issue in bridge engineering. Inspite of the progresses made on this topic during the last five or sixdecades, it remains a subject of concern in hydraulic engineering.

According to Lança (2013), among others, the scour depth, ds,around uniform single piers depends on the flow depth, d, slope ofthe energy line, S, and acceleration of gravity, g; fluid density, ρ,and kinematic viscosity, ν; median grain size, D50, gradationcoefficient, σD, and density, ρs, of the bed material; pier width, Dp,alignment, and shape of the horizontal cross-section; channelwidth, B, bed slope, S0, cross-section geometry; and time, t. Piers'

g Centre on Erosion and Sedimen

lg.pt (R. Lança),

alignment and shape are usually accounted for through the coef-ficients Kθ and Ks, respectively.

For fully developed, clear-water uniform flow, in a wide rec-tangular flatbed channel whose bed is composed of uniform, non-ripple forming sand, the non-dimensional scour depth can beshown to read (Lança, 2013):

dsDp

¼φdDp

;UUc

;Dp

D50;UtDp

;Ks;Kθ

� �ð1Þ

In this equation, U is the average velocity of the undisturbedapproach flow and Uc is the approach flow velocity for thethreshold condition of the sediment entrainment. Eq. (1) applies ifscouring is free of viscous effects. Compared, for instance, with theframework suggested Ettema et al. (2011), the above equation doesnot include the Froude number or any Froude-like number sincethis is not compatible with the simultaneous inclusion of flowintensity, U/Uc, and Dp/D50. This is clear, e.g., in Melville (1992). Forconstant U/Uc (usually U/UcE1.0 in laboratory conditions so as to

tation/the World Association for Sedimentation and Erosion Research. Published by

C. Fael et al. / International Journal of Sediment Research 31 (2016) 244–250 245

maximize the scour depth), the non-dimensional equilibriumscour depth becomes:

dseDp

¼ϕdDp

;Dp

D50;Ks;Kθ

� �ð2Þ

since the scour depth no longer evolves at equilibrium.The effect of flow shallowness, d/Dp, on scouring is unan-

imously recognized as inescapable and it has been the subject ofmany studies on scouring. Most of these studies have successivelyassumed that the equilibrium scour depth does not depend onsediment coarseness, Dp/D50, for Dp/D50425–50 and this is stillthe prevailing view in the hydraulics community. However,Sheppard et al. (2004) and Lee and Sturm (2009) have shown thatthe effect of Dp/D50 cannot be discarded for much higher values ofDp/D50. Lança et al. (2013b) have corroborated these findings. Thepredictors issued from these recent studies account for thesimultaneous effects of the flow shallowness, d/Dp, and sedimentcoarseness, Dp/D50. For practical use, such predictors necessarilyrequire the application of multiplying factors to include the effectsof flow intensity, pier shape, pier alignment, gradation coefficientand density of bed the material, flow contraction, cross-sectionshape and time.

The pier shape multiplying factor is defined as the ratiobetween the scour depth at a pier with a given shape of the hor-izontal cross-section and the scour depth at the standard section-shape pier, all the other parameters kept constant. Likewise, thepier alignment or orientation factor is defined as the ratio betweenthe scour depth at a pier aligned at a given angle (angle of attack)with the flow direction and the scour depth at an equal pieraligned with the flow direction (zero angle of attack), all the otherparameters kept unchanged.

The effects of pier shape and alignment have been studied byfew researchers (the most well-known being Laursen & Toch,1956). Yet, a variety of pier shape factors, Ks, were suggested bydifferent researchers (e.g., Melville, 1997 or Melville & Coleman,2000), based on very limited experimental data. With the excep-tion of Laursen and Toch (1956), who have used the rectangularpier as the standard shape, all the others have chosen the circularpier for that purpose. Richardson and Davis (2001) suggested thefollowing expression that fits the multiplying factor for the angleof attack, Kθ, obtained by Laursen and Toch (1956):

Kθ ¼ cos θþL=Dp sin θ� �0:65 ð3Þ

where θ is the angle of attack, Dp stands for the pier width and L isits length.

According to Richardson and Davis (2001), Kθ should only beused if the angle of attack, θ, is higher than 5° and 2rL/Dpr16. Itshould be noted here that, according to Laursen and Toch (1956),Ks and Kθ must not be used together since, in the above domain,the shape effect becomes negligible.

To the authors' best knowledge, no systematic studies otherthan those of Laursen and Toch (1956) were performed on theeffect of angle of attack on local scour. Therefore, improving andupdating the angle of attack factor is essential, in particular, forvalues of L/Dpr4, close the lower validity limit of Eq. (3).

This experimental study aims at revisiting the influence ofpiers' shape and alignment on local scouring, for uniform piersdefined by L/Dpr4, by increasing the experimental evidence. Itshould be stressed here that most of the tests performed byLaursen and Toch (1956) lasted 3 h, under the assumption that,from there on, the pier alignment or orientation factor does notchange. This is equivalent to assume that the scour holes remainself-similar, i.e., that their non-dimensional geometry does notchange in time irrespective of the pier shape and alignment. Thisspecific point is also addressed in this study.

2. Experimental setup and procedure

Fifty five tests were performed in a 28.00 m long, 2.00 m wideand 1.00 m deep flume of the Universidade da Beira Interior. Six-teen of these tests were already published by Lança et al. (2013a)in a different context: four correspond to cylindrical piers, twelvecorrespond to special (packed) pile-groups, where piles touch eachother (s/b¼1, s¼spacing between pile axes; b¼pile diameter, cf.Table 1).

At the entrance of the flume, two honeycomb diffusers alignedwith the flow direction smoothed the flow trajectories and guar-anteed the uniform cross-wise flow distribution. Immediatelydownstream the diffusers, a 5.00 m long bed-reach was coveredwith small gravel to provide proper roughness and guarantee fullydeveloped rough turbulent flow. The central reach of the flume,starting at 14.00 m from the entrance, included a 3.00 m long,2.00 m wide and 0.60 m deep recess box in the channel bed. Auniform quartz sand (ρs¼ 2650 kg m�3; D50¼0.86 mm; σD¼1.36)was used to fill the recess box. At the downstream end of theflume, a tailgate allows the regulation of the water depth, whichwas kept equal to 0.20 m.

Five different pier shapes were considered in the study,according to Table 1, where the associated values of Dp are sum-marized. All pier, except those circular shaped, were 200 mm longand installed with different angles of attack at 1.0 m from theupstream boundary of the bed recess box: these angles wereθ¼0°, 30°, 45°, 60° and 90° for θ defined according to Table 1.

Prior to each experiment, the sand bed in the recess box wascarefully leveled with the contiguous concrete bed. The arealocated around the pier was covered with a thin metallic plate toavoid uncontrolled scour at the beginning of each experiment. Theflume was then filled gradually, imposing a high water depth and alow flow velocity. The discharge corresponding to the chosenapproach flow velocity, measured by an electromagnetic flowmeter with an accuracy of 70.5% of full scale, was then adjustedto pass through the flume. The flow depth was regulated byadjusting the downstream tailgate and measured with the help ofa point gauge to the accuracy of 71 mm. Once the discharge andflow depth were established, the metallic plate was removed andthe experiment started.

Scour immediately initiated and the depth of scour hole wasmeasured, to an accuracy of 71 mm, with another(adapted) pointgauge, approximately every 5 min during the first hour. After-wards, the interval between measurements increased and, afterthe first day, only a few measurements were carried out each day.In agreement with Simarro et al. (2011), when at least 7 days hadpassed, the experiments were stopped. Fig. 1 shows the scourdepth time evolution for test 50, which is similar to all the others.

The sand bed approach reach located upstream the piers stayedundisturbed through the entire duration of the experiments; thislong term stability ensured that the scour depth was not supple-mented by upstream bed degradation, as documented in Fig. 2. Itshould be noted here that no ripples developed along the testsbecause a practically uniform sand, characterized by D50¼0.86 mm,was used, which is physically incompatible with development ofthose bed forms (see, e.g., Simons & Sentürk, 1992).

3. Results and discussion

3.1. Data characterization

The values of the most important control variables and non-dimensional parameters characterizing the experiments, includingthose reported by Lança et al. (2013a), are summarized in Table 2.It can be concluded that a high relative flow depth (d¼0.200 m;

Table 1Tested pier shapes.

Fig. 1. Time evolution of local scour depth, test 50 (td¼8.17 days).

Fig. 2. Typical sand-bed reach of the piers at the end of the tests.

C. Fael et al. / International Journal of Sediment Research 31 (2016) 244–250246

d/D50¼232.6) was always guaranteed. The average flow intensity,U/Uc, was kept constant and equal to 0.96, with Uc being calculatedthrough the predictor of Neil (1967) as UcE0.31 m/s. The flowFroude number was always E0.22. The aspect ratio B/d was keptequal to 10.0, this way avoiding significant wall effects on theflow field.

The ratio of channel width to pier diameter, B/Dp, variedbetween 10 and 40. Contraction scour seemed absent since no beddegradation was observed over the contracted cross sections; thisresult agrees with others reported in Breusers and Raudkivi (1991),who have suggested that contraction scour is negligible for valuesof B/Dp as low as 2.0–2.5. Also, according to Ballio et al. (2009),contraction scour is null for B/DpZ10. Thus, it is safe to state thatcontraction effects were not present in the present study.

The pier Reynolds number, Rep¼UDp/ν (ν¼kinematic viscosityof the water), varied between 15,500 (for Dp¼50 mm) and 62,000(Dp¼200 mm). It will be assumed here that viscous effects are

negligible since the condition suggested by Franzetti et al. (1994) –Rep¼UDp/ν47000 – is satisfied.

Since the flow depth was kept constant and equal to 0.20 m,the issuing values of the flow shallowness were d/DpE[1.0, 1.33,2.0, and 4.0]. Sediment coarseness, Dp/D50, took also four values,Dp /D50¼[58.14, 116.28, 174.42, and 232.56], one per Dp. The effectof sediment coarseness reported by Sheppard et al. (1995, 1999,2004), Lee and Sturm (2009) and Lança et al. (2013a) may beexpected to be non-negligible. However, this effect will notinfluence the pier shape and the alignment factors since each pairof scour depths required to define a given factor was obtained forexactly the same sediment coarseness.

Table 2Characteristics control variables and non-dimensional parameters of experiments.

Test θ (°) Shape Dp (mm) L/Dp B/Dp d/Dp Dp /D50 td (day) dsm (mm) dse (mm) dse /Dp

1 – C 50 – 40.00 4.00 58.14 7.09 118.00 133.70 2.672 – C 100 – 20.00 2.00 116.28 7.02 199.00 218.00 2.183 – C 150 – 13.33 1.33 174.42 7.02 228.00 252.00 1.684 – C 200 – 10.00 1.00 232.56 10.88 283.00 296.50 1.485 0 RS 50 4.00 40.00 4.00 58.14 6.54 153.10 174.53 3.496 0 RS 100 2.00 20.00 2.00 116.28 7.01 248.00 257.03 2.577 0 RS 150 1.33 13.33 1.33 174.42 7.01 285.00 297.39 1.988 0 RR 100 2.00 20.00 2.00 116.28 6.25 195.60 207.23 2.079 0 RR 150 1.33 13.33 1.33 174.42 6.25 229.80 256.53 1.7110 0 O 50 4.00 40.00 4.00 58.14 7.03 112.40 122.26 2.4511 0 O 100 2.00 20.00 2.00 116.28 7.01 170.80 195.28 1.9512 0 O 150 1.33 13.33 1.33 174.42 7.03 296.00 258.64 1.7213 0 PG 50 4.00 40.00 4.00 58.14 11.01 138.00 153.00 3.0614 0 PG 100 2.00 20.00 2.00 116.28 13.18 239.00 261.00 2.6115 0 PG 150 1.33 13.33 1.33 174.42 8.11 286.00 327.00 2.1816 30 RS 50 4.00 40.00 4.00 58.14 7.04 246.90 278.97 5.5817 30 RS 100 2.00 20.00 2.00 116.28 7.03 269.70 296.68 2.9718 30 RS 150 1.33 13.33 1.33 174.42 7.00 245.50 300.00 2.0019 30 RR 100 2.00 20.00 2.00 116.28 7.01 252.90 264.46 2.6420 30 RR 150 1.33 13.33 1.33 174.42 7.01 263.50 294.17 1.9621 30 O 50 4.00 40.00 4.00 58.14 7.01 195.00 233.17 4.6622 30 O 100 2.00 20.00 2.00 116.28 7.01 231.50 254.27 2.5423 30 O 150 1.33 13.33 1.33 174.42 7.00 259.00 282.54 1.8824 30 PG 50 4.00 40.00 4.00 58.14 6.91 204.00 246.00 4.9225 30 PG 100 2.00 20.00 2.00 116.28 6.74 225.00 316.00 3.1626 30 PG 150 1.33 13.33 1.33 174.42 7.00 272.00 247.00 2.3127 45 RS 50 4.00 40.00 4.00 58.14 7.00 283.30 347.11 6.9428 45 RS 100 2.00 20.00 2.00 116.28 7.00 305.00 328.66 3.2929 45 RS 150 1.33 13.33 1.33 174.42 7.00 319.00 348.55 2.3230 45 RR 100 2.00 20.00 2.00 116.28 7.04 295.40 307.21 3.0731 45 RR 150 1.33 13.33 1.33 174.42 7.05 291.00 322.29 2.1532 45 O 50 4.00 40.00 4.00 58.14 7.07 269.00 302.42 6.0533 45 O 100 2.00 20.00 2.00 116.28 7.93 242.50 264.12 2.6434 45 O 150 1.33 13.33 1.33 174.42 7.08 285.80 308.24 2.0535 45 PG 50 4.00 40.00 4.00 58.14 7.94 283.00 316.00 6.3236 45 PG 100 2.00 20.00 2.00 116.28 11.96 310.00 335.00 3.3537 45 PG 150 1.33 13.33 1.33 174.42 8.95 315.00 359.00 2.3938 60 RS 100 2.00 20.00 2.00 116.28 7.00 326.30 365.77 3.6639 60 RS 150 1.33 13.33 1.33 174.42 7.00 319.00 354.12 2.3640 60 RR 100 2.00 20.00 2.00 116.28 7.06 322.10 349.85 3.5041 60 RR 150 1.33 13.33 1.33 174.42 7.07 305.80 333.57 2.2242 60 O 50 4.00 40.00 4.00 58.14 7.04 300.20 317.60 6.3543 60 O 100 2.00 20.00 2.00 116.28 7.00 253.00 273.50 2.7344 60 O 150 1.33 13.33 1.33 174.42 7.04 290.00 310.91 2.0745 90 RS 200 4.00 10.00 1.00 232.56 7.01 291.70 321.33 1.6146 90 RS 200 2.00 10.00 1.00 232.56 7.01 351.50 391.24 1.9647 90 RS 200 1.33 10.00 1.00 232.56 7.01 338.90 376.56 1.8848 90 RR 200 2.00 10.00 1.00 232.56 6.92 318.80 342.70 1.7149 90 RR 200 1.33 10.00 1.00 232.56 7.00 312.70 348.93 1.7450 90 O 200 4.00 10.00 1.00 232.56 8.17 296.00 314.20 1.5751 90 O 200 2.00 10.00 1.00 232.56 7.00 266.20 286.04 1.4352 90 O 200 1.33 10.00 1.00 232.56 8.17 292.90 308.22 1.5453 90 PG 200 4.00 10.00 1.00 232.56 10.34 322.00 354.00 1.7754 90 PG 200 2.00 10.00 1.00 232.56 8.83 321.00 369.00 1.8555 90 PG 200 1.33 10.00 1.00 232.56 7.39 306.00 328.00 1.64

C – Circular; RS – Rectangular square-nosed; RR – Rectangular round-nosed; O – Oblong; PG – Pile-group.

C. Fael et al. / International Journal of Sediment Research 31 (2016) 244–250 247

According to Franzetti et al. (1994), most of the experimentsshould be expected to have reached equilibrium since Utd/Dp

exceeded 2�106 (td¼test duration, cf. Table 1). However, equili-brium scour does not seem to have been unambiguously reached inany of the present experiments. For this reason, scour depth recordswere extrapolated to infinite time so as to obtain the equilibriumscour depth, dse, as suggested by Lança et al. (2010). Fig. 1 alsoincludes the plot of the 6-parameters polynomial function adjustedto the data so as to extend the scour depth record to infinite time.The equilibrium scour depths obtained through this fitting proce-dure are included in Table 1, together with those measured at end of

each experiment, dsm. For sake of completeness, it should be notedthat dse/dsm¼1.10570.069.

3.2. Shape factor

In this study, the standard-shape pier was chosen to bethe circular pier since the most spread scour predictors havebeen derived for this pier shape. This is the most common option(see, e.g., Melville, 1997). The values of Ks at equilibrium arepresented in Fig. 3. It should be emphasized that all the non-dimensional parameters controlling the process, including thesediment coarseness and the flow shallowness, are exactly the

Fig. 3. Shape factor at the equilibrium, Ks.

Fig. 4. Ks factor for: (a) rectangular square-nosed piers; and (b) rectangular round-nosed piers.

Fig. 5. Ks factors for: (a) oblong piers; and (b) pile-group piers.

C. Fael et al. / International Journal of Sediment Research 31 (2016) 244–250248

same for each Ks value. Fig. 3 includes 710% error bands aroundKs¼1.2 (continuous horizontal lines) and around Ks¼1.0 (thindashed lines). It is clear that the influence of the length–widthratio, L/Dp, is minor or non-existent within the experimentalrange, i.e., for 1.33oL/Dpo4.00. For practical engineering appli-cations it may be assumed that Ks¼1.0 for rectangular round-nosed and oblong cross-section piers, and Ks¼1.2 for rectangularsquare-nosed and pile-group cross-section piers.

Compared with the findings reported in the literature, namelyby Melville (1997), it may be concluded that the oblong cross-section piers behave as circular cylinders or round nosed pierssince Ks is identically equal to 1.0, with the 10% error band. Thevalues of Ks for round-nosed (Ks¼1.0) piers as well as for rectan-gular square-nosed piers (Ks¼1.2) also confirm the values sug-gested by Melville (1997). The value referring to packed pilegroups (Ks¼1.2) could not be compared to others since they werenot found in the literature. It should be noticed here that theincreased shape coefficient for packed pile groups may derive fromthe increase of vorticity associated with the wake vortices shedfrom the successive recesses between contiguous piles.

Fig. 4 summarizes the values of the shape factor for rectangularsquare-nosed and the rectangular round-nosed piers after 3 h,1 day, 3 days and 7 days and at equilibrium time (infinite time);the corresponding values for the oblong and the packed pile-groupcross-section piers are included in Fig. 5. From Figs. 4 and 5, itbecomes clear that, apart from two cases for t¼3 h, the shapefactors are constant in time, falling within the 710% error bandaround the suggested equilibrium values, which means that thescour holes remain practically self-similar, this way validatingmost of the results and explaining the success of the experimentalstudy reported by Laursen and Toch (1956), whose tests lastedonly 3 h.

As stated in the Introduction, it is commonly assumed,according to Richardson and Davis (2001), that Ks¼1 for θ45°irrespective of the pier shape and L/Dp, implicitly assuming thatthe shape effect is hidden by the skew angle effect. This hypothesisis checked next.

Since the oblong pier shape factor can be taken as Ks¼1.0 forθ¼0°, suggesting that this pier behaves mostly as a circularcylinder, the oblong shape was adopted, in this study, as thestandard pier shape to derive the values of Ks for θa0. Since theshape coefficient does not evolve in time, the results shown hereinrefer only to equilibrium. The values of Ks are given in Fig. 6 as afunction of L/Dp for different skew angles, including θ¼90°. Forthis angle, L and Dp are interchanged, as if θ were 0°, leading tovalues of L/Dp¼[0.25, 0.50, and 0.75], represented as triangles inFig. 6. In this case, the scaling scour depth was obtained at thecylindrical pier defined by Dp¼200 mm.

Fig. 6 shows that, apart from the exception defined byL/Dp¼0.25 for the rectangular round-nosed pier, the shape factor,Ks, is E1.2 for θ¼90° irrespective of the pier horizontal sectionshape. For θa90°, Fig. 6a and c shows that the shape factor doesnot appreciably vary with the skew angle for the rectangular

square-nosed and the packed pile-group section piers, respec-tively, the exception being the latter for L/Dp¼4.0. It remainspractically equal to 1.2 as for θ¼0°. On the contrary, Fig. 6b indi-cates that the shape factor increases for rectangular round-nosedpiers as compared with the aligned equivalent (where KsE1.0), forL/Dp¼[1.33, 2.00] and θ¼[45°, 60°], where the effect of the cornersroundness seems to vanish and the rectangular nature of thecross-section seems to emerge. The explanation for these beha-viors can only be speculative and deserves further investigationthrough the mapping of the velocity field, turbulence structureand vorticity around different pier shapes and alignment angles.

In short, contrary to what is usually assumed, the effect ofshape is present at skewed piers although the associated coeffi-cients remain in the narrow range of 1.0–1.2.

Fig. 6. Ks factors at equilibrium: (a) rectangular square-nosed piers; (b) rectangularround-nosed piers; and (c) pile-group piers.

Fig. 7. Kθ factors at equilibrium: (a) L/Dp¼4.0; and (b) L/Dp¼[1.33, 2.0].

C. Fael et al. / International Journal of Sediment Research 31 (2016) 244–250 249

3.3. Pier alignment factor

The pier alignment (or skew-angle) factor, Kθ, is the ratiobetween the scour depth at a pier with a given shape, implanted ata given skew angle, θ, and the scour depth at the same pier forθ¼0°.

The pier alignment factors derived in this study are presentedin Fig. 7 together with the predictions of Eq. (3). Fig. 7a refers toL/Dp¼4.0 while Fig. 7b applies to L/Dp¼[1.33, 2.0]. It must bereminded that θ¼90° becomes, finally, θ¼0°, which means thatthis configuration may be treated through appropriate shapefactors.

It is clear that Eq. (3) constitutes a precise predictor of Kθ forL/Dp¼4.0, while it tends to over-estimate Kθ for L/Dp¼[1.33, 2.0].For the latter cases, the following linear equations are suggested(see continuous lines in Fig. 6b) on the basis of the presentexperimental results:

Kθ ¼ 1þ 4θ1000

for L=Dp ¼ 1:33 ð4Þ

Kθ ¼ 1þ 8θ1000

for L=Dp ¼ 2:0 ð5Þ

As shown above, the pier shape is the same for any θ includingθ¼0°. Consequently, the scour depth at a skewed pier necessarilyreflects the effects of both shape and alignment and the shapefactor – like the alignment factor – is conceptually inescapable.This is relevant whenever equilibrium scour is to be calculated at

piers defined by small L/Dp on the basis of predictors derived forcylindrical piers. In this case, Ks may be of the same order ofmagnitude as Kθ, as shown in Figs. 5 and 7b.

4. Conclusions

The present study focus on the effects of pier shape andalignment on the equilibrium scour depth at uniform single piers.It covers four different pier shapes. Experiments were run forclear-water flatbed uniform flow, in a wide rectangular channelwhose bed is composed of uniform, non-ripple forming sand. Fromthe previous discussion, it can be concluded that, for L/Dpr4:

1. The shape factor, Ks, can be taken as Ks¼1.0, for rectangularround-nosed and oblong cross-section piers, and Ks¼1.2, forrectangular square-nosed and pile-group cross-section piers.

2. Ks does not vary significantly with the duration of tests, this wayconfirming the robustness of the experimental results reportedby Laursen and Toch (1956) obtained for short experiments.

3. The effect of shape is present at skewed piers although theassociated coefficients remain in the narrow range of 1.0–1.2.

4. For values of L/Dpo4, Ks are of the same order of magnitude ofKθ and should not be neglected.

Acknowledgments

This work was partially funded by projects RECI/ECM-HID/0371/2012 (FEDER, program COMPETE, Portuguese Foundation forScience and Technology, FCT) and PTDC/ECM/101353/2008 (FCT).

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