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OCR for page 72
72 1 3.02 3.27 2.05 3.02 Center of channel 3.13 2.83 3.13 2.49 3.27 0.75 1.58 2.83 3.47 2.49 = 0.91 1.22 1.58 Y/(B1-B2) 4.08 0.91 0.5 0.25 Abutment Flow 0 -1 -0.5 0 0.5 X/(B1-B2) Figure 8.9. Initial bed shear stress distribution (N/m2) for B2/B1 = 0.5, V = 0.45 m/s, and L/(B1 B2) = 0. 5). contraction lengths. As can be seen, the maximum bed shear 8.5 WATER DEPTH EFFECT: stress along the center of the channel in the contracted sec- NUMERICAL SIMULATION RESULTS tion is the same for all of the contraction lengths. At the same time, the location of the maximum bed shear stress is not It was found that the water depth had no influence on the influenced by the contraction length. Therefore, max and xmax magnitude or the location of the maximum bed shear stress are independent of the contraction length, and there is no for the contraction problem. In fact, the water depth is already need for any correction factors for contraction length. It was included in the equation through the hydraulic radius of Equa- discovered later that in the case of a very thin contraction tion 8.1. length (L/(B1 - B2) < 0.33), the maximum shear stress within the contracted length is smaller than the maximum shear 8.6 MAXIMUM SHEAR STRESS EQUATION stress due to the contraction. The reason is that the maximum FOR CONTRACTION SCOUR shear stress occurs downstream from the contraction. The correction factor will reflect this finding at very small values The influence of four parameters on the maximum shear of the contracted length (L/(B1/B2) < 0.33) (Section 8.6). stress and its location near a channel contraction was investi- 1 3.25 3.25 Center of channel 3.03 3.03 3.34 2.71 2.06 0.75 1.13 3.51 2.71 1.55 3.81 2.06 1.55 Y/(B1-B2) = 0.91 4.05 0.91 0.5 0.25 Flow Abutment 0 -1 -0.5 0 0.5 X/(B1-B2) Figure 8.10. Initial bed shear stress distribution (N/m2) for B2/B1 = 0.5, V = 0.45 m/s, and L/(B1 B2) = 1.0).

OCR for page 72
73 Y and B1/B2 = 2, the value of Xmax was about 0.35 for L/(B1 - B2) varying from 0.25 to 6.76. For engineering design, what we are interested in is scouring around the pier or the abutment B1 and along the contracted section. For small ratios of L/(B1 - B2) (less than 0.33) the maximum shear stress max is past the Xa contracted location and the shear stress of interest is located F lo w Xc at Xc from the beginning of the fully contracted channel. X B2 X m ax The correction factor for a given influencing parameter is defined as the ratio of the max value including that parameter L to the max value for the case of the open channel without any contraction. The results of the numerical simulations were used to plot the shear stress as a function of each influencing parameter. Regressions were then used to obtain the best-fit Figure 8.11. Definition of parameters for contraction equation to describe the influence of each parameter. Figure scour. 8.12 shows the variation of the correction factor kc - R for the influence of the contraction ratio B2/B1. Figure 8.13 shows the correction factor kc - for the influence of the transition angle . Figure 8.14 shows the correction factor kc - L for the gated by numerical simulation. These factors are the contrac- influence of the contracted length L. The correction factor tion ratio (B2/B1), the transition angle (), the length of con- kc - H for the water depth influence was found to be equal to 1. traction (L), and the water depth (H). Figure 8.11 describes the The proposed equation for calculating the maximum shear problem definition for abutment and contraction scour. In this stress within the contracted length of a channel along its cen- figure, B1 is the width of channel, B2 is the width of the con- terline is tracted section, L is the length of abutment, is the transition 1 angle, Xa is the location of maximum bed shear stress due to - max = kc - R kc - kc - H kc - L n 2 V 2 Rh 3 (8.2) the abutment, and Xmax is a normalized distance that gives the location of the maximum bed shear stress along the cen- where terline of the channel (Xmax = X/(B1 - B2)) where X is the actual distance to max. is the unit weight of water (kN/m3); It was found (for certain and B1/B2) that the influence of n is Manning's roughness coefficient (s/m1/3); L on Xmax and max was negligible. In the case of = 90 degrees V is the upstream mean depth velocity (m/s); 6 B1 5 L B2 4 k c-R 3 1.75 B kc - R = 0.62 + 0.38 1 B2 2 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 B1 /B2 Figure 8.12. Relationship between kc R and B1/B2.

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74 2 1.8 1 .5 1.6 k c- =1 + 0 .9 90 1.4 B1 1.2 L k c - 1 B2 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 /90 Figure 8.13. Relationship between kc and /90. 1.75 is the contraction transition angle (in degrees) (Fig- kc - R = 0.62 + 0.38 B1 (Figure 8.12); B2 ure 8.13); Rh is the hydraulic radius defined as the cross-section kc- is the correction factor for the contraction transition area of the flow divided by the wetted perimeter (m); angle, given by kc - = 1 + 0.9 ( ) 1.5 kc-R is the correction factor for the contraction ratio, (Figure 8.13); given by 90 1.2 1 , for k c - L 0.35 2 k c-L L L 1.1 0 . 77 + 1 . 36 -1 . 98 , for k c - L < 0.35 B1 -B 2 B1 -B 2 1 kc-L 0.9 B1 L B2 0.8 0.7 0.6 0 1 2 3 4 5 6 7 L/(B1-B2) Figure 8.14. Relationship between kc L and L/B1 - B2.

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75 kc-L is the correction factor for the contraction length, Note that the last part of the equation (n2V 2R-0.33h) is the given by formula for the bed shear stress in an open channel (Equation 1 , for kc - L 0.35 8.2). Equation 8.2 is consistent with the open channel case since all correction factors collapse to 1 when the parameter kc - L = 2 L L 0.77 + 1.36 B - B - 1.98 B - B , for corresponds to the open channel case. In other words when 1 2 1 2 B1/B2 = 1, = 0, and (B1 - B2)/L = 0, then kc-R = 1, kc-L = 1, kc-L < 0.35 (Figure 8.14); and kc- = 1. kc-H is the correction factor for the contraction water Figures 8.15 and 8.16 give the location of the maximum depth. shear stress along the centerline of the contracted channel. Since the water depth has a negligible influence, then Figure 8.15 shows the influence of the transition angle, and kc-H 1. The influence of the water depth H is in Rh. Figure 8.16 shows the influence of the contraction ratio. 0.4 X 0.34 0.35 = B1 - B2 - 0.42 - 90 0.3 0.11 1+ e B1 0.25 X/(B1-B2) L B2 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 1.2 /90 Figure 8.15. Distance from the beginning of the fully contracted section to the location of the maximum shear stress along the centerline of the channel X, as a function of the normalized transition angle /90. 1.2 B1 1 L B2 0.8 X/(B1-B2) 0.6 0.4 0.2 2 3 X B B B2 = 0.73 2 - 1.48 2 + 2.76 B1 - B 2 B1 B1 B1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 B2/B1 Figure 8.16. Distance from the beginning of the fully contracted section to the location of the maximum shear stress along the centerline of the channel X, as a function of the Contraction Ratio B2/B1.