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41 150 a=3 ft a=7 ft a=15 ft 100 t90 in Days D50=1 mm y1/a=2 50 0 0.5 1 1.5 2 2.5 3 V1/Vc Figure 41. Plot of t90 versus V1 / Vc for different pier diameters. 2 This modified equation is referred to here as the Melville/ ij y measured i computed j ys i j a s Sheppard (M/S) equation. - a Note that the original Melville and Chiew equation was SSEn% = 2 × 100 (42) developed for clear-water scour, while the M/S equation ij y measured i j s seems to work equally well for the live-bed scour data. Scour a approaches an equilibrium value asymptotically; thus "time to equilibrium" is misleading at best and has no practical value. where i is the index for the time step and j is the index for the Time to a value like 90% of equilibrium is, however, useful and experiment. Seven different scour evolution predictive equa- thus an expression for t90 is included with the M/S Equation. tions were evaluated (six from the literature review plus the rec- The expression for t90 in terms of the reference time t e and ommended M/S equation). The total and underprediction V1/Vc is given in Equation 40. errors for the various data sets were computed and the results are presented in Figures 42 through 45. Note that the scales V for total and underprediction are substantially different in these t 90 = exp -1.83 1 t e (40) Vc plots in order to emphasize the underprediction. The Mia and Nago (2003) equation errors are not shown in the wide-pier plots (Figures 43 and 45), because they are very large. Plots of t90 versus V1/Vc for one sediment size and normal- The computed scour evolution errors include the differences ized depth and three pier widths are shown in Figure 41. between the equilibrium scour depth that each experiment would achieve and the predicted value using the S/M equation, Final Evaluation of Scour Evolution as well as the errors for the scour rate. However, not all of the Predictive Methods scour evolution tests were conducted for a sufficient duration; therefore, differences between predicted and measured equi- The scour evolution data are reported with different time librium values could not be determined and used to assess the steps. To calculate prediction errors, the following procedure quality of the data. was used. Each time series was divided into 100 equal time steps Based on these results, the M/S equation performed the and the scour depths interpolated to these points in time. All best of the seven in that it has the least total error and nearly of the predictive methods were then evaluated at each of the the lowest underprediction error. However, as can be seen 100 times for the conditions of the experiment and compared from the plots, all of the existing and modified methods have with the measured values. The prediction error was computed relatively large normalized errors. Note that the M/S equation using the following equations: has been optimized using the S/M equilibrium equation. The i j( ij y measured s - ij y computed s ) 2 M/S equation should not be used with any other equilibrium scour equation. SSE% = × 100 (41) i j( ij y me s asured ) 2 It is important to note that the only live-bed scour evolution data in the data set is for small laboratory structures (maximum The normalized SSE then becomes: 1.0 ft for 1.3 > V1/Vc >1.1 and 0.5 ft for V1/Vc >1.3). The
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42 4 Miller 02 3.5 Normalized Scour Error (%) 3 2.5 Kothyari 92 2 1.5 Oliveto 02-07 1 0.5 M/S Melville 97 Kothyari 07 0 Mia 03 0 20 40 60 80 100 Total Normalized Scour Error (% ) Figure 42. Underprediction versus total normalized scour evolution error. 3.5 Oliveto 02-07 3 Normalized Scour Error (%) 2.5 2 1.5 Melville 97 1 0.5 Kothyari 07 Kothyari 92 M/S Miller 02 0 0 5 10 15 20 25 30 35 40 Total Normalized Scour Error (% ) Figure 43. Underprediction versus total normalized scour evolution error for wide piers (defined as y1/a < 0.5 and a/D50 > 100). 7 Kothyari 92 6 5 Miller 02 Error (%) 4 3 Oliveto 02-07 Kothyari 07 2 1 M/S Melville 97 Mia 03 0 0 50 100 150 Total Scour Error (%) Figure 44. Underprediction versus total scour evolution error.
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43 5 Oliveto 02-07 4 Scour Error (%) 3 2 Kothyari 92 1 Melville 97 M/S Kothyari 07 Miller 02 0 80 85 90 95 100 Total Scour Error (%) Figure 45. Underprediction versus total scour evolution error for wide piers (defined as y1/a 100). best-performing equation with existing data, the M/S equa- equilibrium scour depths versus flow velocity for a 30 ft tion, yields, what appears to be, very conservative results for diameter circular pier in 30 ft water depth are shown in large prototype structures (conservative in the sense of pre- Figure 46. dicting scour rates much higher than seems reasonable); An example problem with a large, long skewed pier, however, there are no large-structure data in the live-bed founded in fine sand and subjected to live-bed flow condi- scour range with which to test the predictions. tions is presented in Appendix D. This example illustrates, To illustrate the effects of sediment size on scour rates, among other things, the conservativeness of the scour evo- plots of predicted time to reach 50%, 75%, and 90% of lution equation.
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44 t90 2 Time in Days 10 t75 t50 0 b = 30 ft 10 D50= 0.4 mm y0 = 30 ft 0 1 2 3 4 5 6 7 8 V (ft/s) t90 2 Time in Days 10 t75 t50 0 b = 30 ft 10 D50= 1 mm y0 = 30 ft 0 1 2 3 4 5 6 7 8 V (ft/s) t90 2 Time in Days 10 t75 t50 0 b = 30 ft 10 D50= 3 mm y0 = 30 ft 0 1 2 3 4 5 6 7 8 V (ft/s) Figure 46. Time to 50%, 75%, and 90% of equilibrium scour versus flow velocity for 0.4, 1.0, and 3.0 mm sediment diameters.