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19 Table 4. (Continued). Reference Equation Notes No. ys V1 2.5f1f 2 f 3 for 0.47 1.0 a* Vc V1 Vlp V1 1 ys Vc Vc Vc V1 Vl p f1 2.2 2.5f 3 for 1 a* Vlp Vlp Vc Vc 1 1 Vc Vc ys V1 Vl p 2.2 f1 for a* Vc Vc 0.4 y1 a* effective diameter f1 tanh a* projected width * shape factor 2 Sheppard and V1 Shape factor = 1, circular f2 1 1 . 75 l n 4 23 Miller (2006) Vc = 0.86 + 0.97 , square 4 a* = flow skew angle in radians D50 f3 1 .2 0.13 a* a* 0. 4 10.6 D50 D50 Vlp1 = 0.8 g y 0 Vlp2 = 29.31 u *c log10 4y1 D90 Vl p1 for Vl p1 Vl p 2 Vl p Vl p 2 for Vl p2 Vl p1 extensive sets of laboratory data from The University of obtained from small-scale cylinders. Ettema et al. (2006) used Auckland and elsewhere (Chabert and Engeldinger 1956, the largest cylinder size (1.3 ft) as the reference size. It is not Laursen and Toch 1956, Jain and Fischer 1979, Chee 1982, known if this equation can be applied to wider piers and, if Chiew 1984, Ettema 1980, Hancu 1971, Shen et al. 1966). The so, how to select ao. method uses a number of multiplying factors (K-factors) for the effects of the various parameters, which influence scour. Initial Screening of Equilibrium The values of the K-factors were determined from envelope Scour Predictive Equations curves fitted to the data. The method is, therefore, inherently conservative. The method defines wide piers as those having Twenty-three equations were assembled for evaluation and large values of the ratio a/y1 (>5). assessment. These equations are presented in Table 4. Some A similar rationally based method is given by Sheppard and of these equations are a function of critical velocity. If the Miller (2006) equations (numbered 23 in Table 4). The equa- equation did not specify a method to calculate the critical tions are based principally on laboratory data, as well as a few velocity, it was calculated using Equation 24. field measurements. The equations include the important observation that normalized local scour depths' dependence Vc y = 5.75log 1685 1 on a/D50 increases until the value of a/D50 equals approxi- u *c D50 mately 40, at which point dependence begins to decrease. One possible explanation for this behavior was given by u *c = 0.377 ft s + 0.410D1.4 50 0.1 mm < D50 < 1 mm (24) Sheppard (2004). u *c = D50 0.5 - 0.0213 D50 -1 1 mm < D50 < 100 mm Ettema et al. (2006) conducted experiments for local scour at cylindrical piers placed in a sand bed. The authors contend where D50 is in mm, u *c & Vc are in ft s, and y 1 in ft that the experiments show the importance of considering similitude of large-scale turbulence structures when conduct- The first screening procedure consisted of solving all of the ing flume experiments on local scour at cylinders. They pro- equations for a range of input values and comparing the results. posed a correction factor, ao, to adjust scour-depth estimates The values of the parameters used in Figures 13 through 20 are

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20 ys /a ys /a In In 0 1 2 3 4 5 0 1 2 3 4 5 gl gl Ah is 1 Ah is 1 La La ur ma 949 ur ma 949 se d se d n 19 n 19 1 5 1 5 C 958 3 C 958 3 hi hi ta , 6 ta , 6 le 3 le 3 La 1 La 1 9 9 Br rras 62 Br rras 62 eu 1 eu 1 se 96 se 96 3 3 B rs B rs Sh len 196 Sh len 196 en ch 5 en ch 5 e 1 e 1 C t a 969 C t a 969 ol l. ol l. em 19 em 19 live-bed scour condition. a 69 a 69 H n1 H n1 V1/Vc=3, V1/Vc=1, an 97 an 97 Br cu 1 Br cu 1 eu 1 eu 1 se Ne 97 se Ne 97 rs ill 1 rs ill 1 et 19 et 19 al 73 al 73 .1 .1 M J 97 M J 97 a a Br y & Fro ain 7 Br y & Fro ain 7 y1/a=0.33, y1/a=0.33, eu W eh 19 eu W eh 19 se il lic 81 se il lic 81 rs lou h rs lou h & g h 19 & g h 19 clear-water to live-bed scour conditions. R by 88 R by 88 au au d 1 d 1 An Ga kiv 990 An Ga kiv 990 sa o e i 1 sa o e i 1 ri 9 ri 9 & t al. 91 & t al. 91 Q Q ad 199 ad 199 a 3 a 3 D50=0.2 mm D50=0.2 mm R W r1 R W r1 ic ils 99 ic ils 99 ha r M n 4 o ha r M n 4 o Sh dso el 19 Sh dso el 19 ep n & ville 95 ep n & ville 95 predictions using 22 different methods for a particular pa 1 pa 1 Figure 14. Comparison of normalized local scour depth Figure 13. Comparison of normalized local scour depth rd Dav 99 rd Dav 99 & is 7 & is 7 Note: The pier width is large compared to the water depth, and the sediment is fine sand. Note: The pier width is large compared to the water depth, and the sediment is fine sand. M 20 M 20 predictions using 22 different methods for transition from ille 0 ille 0 r2 1 r2 1 00 00 6 a= 3 ft 6 a= 3 ft a= 2 in a= 2 in a= 33 ft a= 33 ft

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ys /a ys /a In In 0 1 2 3 4 5 0 1 2 3 4 5 g gl A lis Ah is 1 La hm 19 La ur a 49 ur ma 949 se d se d n 19 n 19 1 5 1 5 C 958 3 C 958 3 hi hi ta , 6 ta , 6 le 3 le 3 La 1 La 1 9 9 Br rras 62 Br rras 62 eu 1 eu 1 se 96 se 96 3 3 B rs B rs Sh len 196 Sh len 196 en ch 5 en ch 5 e 1 e 1 C t a 969 C t a 969 ol l. ol l. em 19 em 19 live-bed scour condition. a 69 a 69 H n1 H n1 an 97 an 97 V1/Vc=3, V1/Vc=1, Br cu 1 Br cu 1 eu 1 eu 1 se Ne 97 se Ne 97 rs ill 1 rs ill 1 et 19 et 19 al 73 al 73 .1 .1 M J 97 M J 97 a a y1/a=3, y1/a=3, Br y & Fro ain 7 Br y & Fro ain 7 eu W eh 19 eu W eh 19 se il lic 81 se il lic 81 rs lou h rs lou h & gh 19 & g h 19 clear-water to live-bed scour conditions. R by 88 R by 88 au au d 1 d 1 An Ga kiv 990 An Ga kiv 990 sa o e i 1 sa o e i 1 ri 9 ri 9 & t al. 91 & t al. 91 Q Q D50=0.2 mm D50=0.2 mm ad 199 ad 199 R a 3 R a 3 ic W r1 ic W r1 ils 99 ils 99 ha r M n 4 o ha r M n 4 o Sh dso el 19 Sh dso el 19 Note: The water depth is deep relative to pier width, and the sediment is fine sand. Note: The water depth is deep relative to pier width, and the sediment is fine sand. ep n & ville 95 ep n & ville 95 predictions using 22 different methods for a particular pa 1 pa 1 Figure 16. Comparison of normalized local scour depth Figure 15. Comparison of normalized local scour depth rd Dav 99 rd Dav 99 & is 7 & is 7 M 20 M 20 predictions using 22 different methods for transition from ille 0 ille 0 r2 1 r2 1 00 00 6 6 a= 3 ft a= 3 ft a= 2 in a= 2 in a= 33 ft a= 33 ft 21

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22 ys /a ys /a In In 0 1 2 3 4 5 0 1 2 3 4 5 gl gl Ah is 1 Ah is 1 La La ur ma 949 ur ma 949 se d se d n 19 n 19 1 5 1 5 C 958 3 C 958 3 hi hi ta , 6 ta , 6 le 3 le 3 La 1 La 1 9 9 Br rras 62 Br rras 62 eu 1 eu 1 se 96 se 96 3 3 B rs B rs Sh len 196 Sh len 196 en ch 5 en ch 5 e 1 e 1 C t a 969 C t a 969 ol l. ol l. e m 19 em 19 live-bed scour condition. a 69 a 69 H n1 H n1 an 97 an 97 V1/Vc=1, V1/Vc=3, Br cu 1 Br cu 1 eu 1 eu 1 se Ne 97 se Ne 97 rs ill 1 rs ill 1 et 19 et 19 al 73 al 73 .1 .1 M J 97 M J 97 a a Br y & Fro ain 7 Br y & Fro ain 7 eu W eh 19 eu W e h 19 y1/a=0.33, y1/a=0.33, se il lic 81 se il lic 81 rs lou h rs lou h & g h 19 & gh 19 clear-water to live-bed scour conditions. R by 88 R by 88 au au d 1 d 1 An Ga kiv 990 An Ga kiv 990 sa o e i 1 sa o e i 1 ri 9 ri 9 & t al. 91 & t al. 91 Q Q D50=3 mm ad 199 ad 199 D50=3 mm a 3 a 3 R W r1 R W r1 ic ils 99 ic ils 99 ha r M n 4 o ha r M n 4 o Sh dso el 19 Sh dso el 19 ep n & ville 95 ep n & ville 95 predictions using 22 different methods for a particular pa 1 pa 1 Figure 18. Comparison of normalized local scour depth Figure 17. Comparison of normalized local scour depth rd Dav 99 rd Dav 99 & is 7 & is 7 M 20 M 20 predictions using 22 different methods for transition from ille 0 ille 0 r2 1 r2 1 Note: The pier width is large relative to the water depth and the sediment is very coarse sand. 00 00 6 6 a= 3 ft Note: The pier width is large compared to the water depth, and the sediment is very coarse sand. a= 3 ft a= 2 in a= 2 in a= 33 ft a= 33 ft

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ys /a ys /a In In 0 1 2 3 4 5 0 1 2 3 4 5 gl gl Ah is 1 Ah is 1 La La ur ma 949 ur ma 949 se d se d n 19 n 19 1 5 1 5 C 958 3 C 958 3 hi hi ta , 6 ta , 6 le 3 le 3 La 1 La 1 9 9 Br rras 62 Br rras 62 eu 1 eu 1 se 96 se 96 3 3 B rs B rs Sh len 196 Sh len 196 en ch 5 en ch 5 e 1 e 1 C t a 969 C t a 969 ol l. ol l. e m 19 em 19 live-bed scour condition. a 69 a 69 H n1 H n1 an 97 an 97 Br cu 1 Br cu 1 V1/Vc=1, V1/Vc=3, eu 1 eu 1 se Ne 97 se Ne 97 rs ill 1 rs ill 1 et 19 et 19 al 73 al 73 .1 .1 M J 97 M J 97 a a y1/a=3, y1/a=3, Br y & Fro ain 7 Br y & Fro ain 7 eu W eh 19 eu W eh 19 se il lic 81 se il lic 81 rs lou h rs lou h & g h 19 & g h 19 clear-water to live-bed scour conditions. R by 88 R by 88 au au d 1 d 1 An Ga kiv 990 An Ga kiv 990 sa o e i 1 sa o e i 1 ri 9 ri 9 & t al. 91 & t al. 91 Q Q D50=3 mm D50=3 mm ad 199 ad 199 R a 3 a 3 W r1 R W r1 ic ils 99 ic ils 99 ha r M n 4 o ha r M n 4 o Sh dso el 19 Sh dso el 19 ep n & ville 95 ep n & ville 95 predictions using 22 different methods for a particular pa 1 pa 1 Figure 20. Comparison of normalized local scour depth Figure 19. Comparison of normalized local scour depth rd Dav 99 rd Dav 99 & is 7 & is 7 M 20 M 20 predictions using 22 different methods for transition from ille 0 ille 0 r2 1 r2 1 Note: The water depth is deep relative to the pier width, and the sediment is very coarse sand. Note: The water depth is deep relative to the pier width, and the sediment is very coarse sand. 00 00 6 6 a= 3 ft a= 3 ft a= 2 in a= 2 in a= 33 ft a= 33 ft 23