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OCR for page 60
60 200 300 160 250 200 120 Z(t)(mm) Z(t)(mm) Measurement Measurem ent Hyperbola 150 80 Hyperbola 100 40 50 0 0 50 100 150 200 250 0 0 50 100 150 200 250 Time(H) Time(H) Figure 7.6. Maximum contraction scour and hyperbola model for Test 1. Figure 7.7. Uniform contraction scour and hyperbola model for Test 1. configurations that occurred during the tests. If the combina- tion of contraction ratio and transition angle is small, then the traction length (L/B2 > 2, Figure 7.5). The maximum depth of contraction scour is downstream from the contraction inlet contraction scour Zmax is the largest depth that occurs along and the abutment scour exists by itself at the contraction the contraction scour profile in the center of the contracted inlet. If the combination of contraction ratio and transition channel. The uniform contraction scour depth Zunif is the angle is medium, the abutment scour and the contraction scour depth that develops in the contracted channel far from scour occur within the same inlet cross section but do not the transition zone (Figure 7.1). overlap. If the combination of contraction ratio and transition The Reynolds Number (inertia force/gravity force) and the angle is severe, the abutment scour and the contraction scour Froude Number (inertia force/viscous force) were used as overlap but do not add to each other. basic correlation parameters. Both Zmax and Zunif were normal- ized with respect to H1, the upstream water depth. Figure 7.12 shows the attempt to correlate the contraction scour depths to 7.6 MAXIMUM AND UNIFORM CONTRACTION the Reynolds Number defined as V1B2/. As can be seen, the DEPTHS FOR THE REFERENCE CASES Reynolds Number is not a good predictor of the contraction scour depths. Figures 7.13 and 7.14 show the correlation with The reference case for the development of the basic equa- the Froude Number. Although there are only seven points for tion is the case of a 90-degree transition angle and a long con- the correlation, the results are very good. Distance X (mm) -400 0 400 800 1200 1600 0 Angle=15 Relative Scour Depth (Z/Z max) Angle=45 0.25 Angle=60 Angle=90 0.5 0.75 1 Figure 7.8. Contraction scour profile along the channel centerline for transition angle effect.
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61 Distance X (mm) -400 0 400 800 1200 1600 0 L/B2=0.5 Relative Scour Depth (Z/Z max) L/B2=1 0.25 L/B2=1.69 L/B2=6.76 0.5 0.75 1 Figure 7.9. Contraction scour profile along the channel centerline for contraction length effect. Distance X (mm) -500 0 500 1000 1500 0 Relative Scour Depth (Z/Zmax) 0.2 0.4 0.6 0.8 1 1.2 Figure 7.10. Contraction scour profiles along the channel centerline for standard tests. Original Bottom (a) Ridge (b) Valley (c) Plain is the location of maximum contraction Figure 7.11. Overlapping of scour for contraction scour and abutment scour.
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62 2.5 1.5 Max Contraction y = 0.9995x - 4E-06 2.0 1.2 R2 = 0.9558 Unif Contraction 1.5 0.9 Z / H1 Zunif 1.0 H1 0.6 0.5 0.3 0.0 0 50000 100000 150000 200000 0 Re=V1B2/n 0 0.5 1 1.5 1/ 2 B1 c Figure 7.12. Correlation attempt between contraction 1 .31 × V1 B2 scour depth and Reynolds Number. 1 .41 × - 1/ 3 gH 1 gnH 1 The Froude Number was calculated as follows. From sim- Figure 7.14. Normalized uniform contraction scour depth ple conservation, we have V1B1H1 = V2B2H2. Because the water versus Froude Number. depth H2 may not be known for design purposes and because other factors may influence V2, the velocity used for correla- tion purposes was simply V1B1/B2. As will be seen later, a fac- c = (gn 2 Vc2 ) ( H1 )0.33 (7.6) tor will be needed in front of V1B1/B2 for optimum fit. So the critical Froude Number can be written as V * = V1 B1 B2 (7.4) Frc = Vc ( gH1)0.5 = ( c )0.5 ( gnH1)0.33 (7.7) Then the Froude Number was calculated as The contraction scour depths are likely to be proportional Fr* = (V1 B1 B2 ) ( gH1 )0.5 (7.5) to the difference (Fr-Frc). However, as mentioned before, the velocity used to calculate Fr may require a factor . The final The relationship between the critical shear stress and form of the equation sought was the critical velocity for an open channel was established (Richardson et al., 1995). Z H1 = (Fr * - Frc ) (7.8) 2.5 The factors and were obtained by optimizing the R2 value in the regression on Figures 7.13 and 7.14. The pro- posed equations are 2 c 12 1.38 V1 1 B 1.5 Zmax B2 Zmax = 1.90 - 13 (7.9) H1 gH1 gnH1 H1 1 y = 1.0039x - 0.0013 1 2 1.31 V1 1 c R2 = 0.9885 B Z unif B2 0.5 = 1.41 - 13 (7.10) H1 gH1 gnH1 0 where Zmax is the maximum depth of contraction scour; H1 is 0 0.5 1 1.5 2 2.5 the upstream water depth after scour has occurred; V1 is the 1/ 2 B1 c mean depth upstream velocity after the contraction scour has 1 . 38 × V 1 B2 occurred; B1 is the upstream channel width; B2 is the con- 1 .9 × - 1/3 gH 1 gnH 1 tracted channel width; g is the acceleration due to gravity; c is the critical shear stress of the soil (obtained from an EFA Figure 7.13. Normalized maximum contraction scour test); is the mass density of water; and n is the Mannings depth versus Froude Number. Coefficient. Note that V1 and H1 are the upstream velocity