Pier and Contraction Scour in Cohesive Soils (2004)

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68 CHAPTER 8 THE SRICOS-EFA METHOD FOR INITIAL SCOUR RATE AT CONTRACTED CHANNELS 8.1 BACKGROUND The existing knowledge on numerical methods for scour studies was presented at the beginning of Chapter 6 (Section 6.2) and the existing knowledge on contraction scour in cohesive soils was presented at the beginning of Chapter 7 (Section 7.1). The initial scour rate is an integral part of the SRICOS Method to predict contraction scour as a function of time because it is one of the two fundamental parameters used to describe the scour depth versus time curve. The other funda- mental parameter is the maximum depth of contraction scour that was studied in Chapter 7. The initial rate of scour for a given contraction scour problem is obtained by first calculat- ing the maximum shear stress τmax existing in the contracted channel before the scour starts (flat river bottom) and then reading the initial scour rate on the erosion function obtained in the EFA test. Therefore, the problem of obtaining the ini- tial rate of contraction scour is brought back to the problem of obtaining the maximum shear stress in the contracted channel before scour starts. This problem was solved by numerical simulations that use the chimera RANS method. This method was described in Section 6.3 and a verification of its reliabil- ity was presented in Section 6.4. This chapter describes the simulations performed and the associated results. The goal was to develop an equation for the maximum shear stress τmax existing in the contraction zone. The equation for the maximum shear stress τmax at the bot- tom of an open channel without contraction is given by (Munson et al., 1990) as where γ is the unit weight of water, n is Manning’s rough- ness coefficient, V is the mean depth velocity, and Rh is the hydraulic radius defined as the cross-section area of the flow divided by the wetted perimeter. The equation obtained for the contracted channel case should collapse to the open channel case when the contraction ratio B2/B1 becomes equal to 1. The objective of the numerical simulations was to obtain correction factors that would introduce the effect of the contraction ratio, the transition angle, and the length of the contracted zone. τ γmax ( . )= − n V R2 2 1 3 8 1h 8.2 CONTRACTION RATIO EFFECT: NUMERICAL SIMULATION RESULTS One of the flume experiments was chosen to perform the numerical simulation. The width of the flume used was 0.45 m. The upstream flow was a steady flow with a velocity V1 of 0.45 m/s and the upstream water depth H was 0.12 m. Three different contraction ratios were chosen: B2/B1 = 0.25, 0.5, and 0.75. In order to reduce the CPU time, a half domain was used based on the symmetry of the problem. For numerical purposes, the characteristic length B was defined as half of the flume width. Based on this definition, the value of the Reynolds Number (Re = VB/ν) was 101250 and the Froude Number was 0.303. The grid was made of four blocks as shown in Figure 8.1. The bed shear stress contours around the abutment and in the contracted zone are shown in Figures 8.2 to 8.4. The maximum bed shear stress is found around the abutment but the maximum contraction bed shear stress τmax is found along the centerline of the channel in the contracted section. As expected, it is found that the magnitude of τmax increases when the contraction ratio (B2/B1) decreases. It is also observed that the distance Xmax between the beginning of the fully contracted section and the location of τmax increases when the contraction ratio (B2/B1) increases. 8.3 TRANSITION ANGLE EFFECT: NUMERICAL SIMULATION RESULTS Again, one of the flume experiments was chosen to per- form the numerical simulation. The width of the flume used was 0.45 m. The upstream flow was a steady flow with a velocity of 0.45 m/s, the contracted length L was such that L/(B1 − B2) = 6.76 and the water depth was 0.12 m. Four dif- ferent transition angles were chosen for the simulations: α = 15, 30, 45, and 90 degrees. The width (B1 − B2) was chosen as the characteristic length B. The Reynolds Number was 101250 and the Froude Number was 0.303. The bed shear stress contours around the abutment and in the contracted zone are shown in Figures 8.5 to 8.7. As can be seen, the magnitude of the maximum bed shear stress τmax along the center of the channel in the contracted section increases when the transition angle θ increases. However the transition angle does not have a major influence on τmax. It is Fr =( )V gB

69 -0.4 -0.2 0 Z/(0 .5B1) -2 0 2 4 6 8 X/(0. 5B1 )0 0.25 0.5 0.75 1 Y/(0 .5B1) X Y Z Block 2 Block 1 Block 3Block 4 Center of the channel 0.91 2.11 3.31 4.51 6.92 8.12 6.92 5.714.51 0.91 8.50 9.32 X/(0.5B1) Y/ (0 . 5B 1) -1 -0.5 0 0.50 0.25 0.5 0.75 1 Abutment Flow Center of channel τ = Figure 8.1. Grid system for the simulation in the case of B2/B1 = 0.25. Figure 8.2. Initial bed shear stress distribution (N/m2) for B2/B1 = 0.25 and V = 0.45m/s. also observed that the distance xmax between the beginning of the fully contracted section and the location of τmax increases when θ increases 8.4 CONTRACTED LENGTH EFFECT: NUMERICAL SIMULATION RESULTS Again, one of the flume experiments was chosen to per- form the numerical simulation. The width of the flume used was 0.45 m. The upstream flow was a steady flow with a velocity of 0.45 m/s, the contraction channel ratio (B2/B1) was equal to 0.5, the transition angle was 90 degrees, and the water depth was 0.12 m. Four different contraction lengths were simulated: L/(B1 − B2) = 0.25, 0.5, 1.0 and 6.76. The difference (B1 − B2) was chosen as the characteristic length B. The Reynolds Number was 101250 and the Froude Number was 0.303. The initial bed shear stress distribution around the contracted zone is shown in Figures 114 to 116 for various

70 0.91 1.70 2.49 3.29 3.29 4.08 2.49 1.70 0.91 1.25 3.11 X/(0.5B1) Y/ (0 . 5B 1) -1 -0.5 0 0.50 0.25 0.5 0.75 1 Abutment Flow Center of channel τ = 0.91 2.18 1.29 1.54 1.78 2.03 2.33 2.66 3.00 1.78 2.03 1.290.91 1.54 X/(0.5B1) Y/ (0 . 5B 1) -1 -0.5 0 0.50 0.25 0.5 0.75 1 Abutment Flow Center of channel τ = 1.68 1.68 0.91 1.06 1.22 1.37 1.52 1.68 1.99 2.45 1.83 X/(B1-B2) Y/ (B 1- B2 ) -3 -2 -1 0 1 20 0.25 0.5 0.75 1 Abutment Flow Center of channel τ = Figure 8.3. Initial bed shear stress distribution (N/m2) for B1/B2 = 0.50 and V = 0.45m/s. Figure 8.4. Initial bed shear stress distribution (N/m2) for B2/B1 = 0.75, and V = 0.45m/s. Figure 8.5. Initial bed shear stress distribution (N/m2) for B2/B1 = 0.5, V = 0.45 m/s, L/(B1 – B2) = 6.76, and θ = 15 degrees).

71 0.91 1.10 1.49 1.87 2.06 1.87 2.06 2.25 1.68 1.49 3.17 X/(B1-B2) Y/ (B 1- B2 ) -3 -2 -1 0 1 20 0.25 0.5 0.75 1 Abutment Center of channel Flow τ = 0.91 1.36 1.82 2.02 2.27 2.27 2.02 2.62 3.76 X/(B1-B2) Y/ (B 1- B2 ) -3 -2 -1 0 1 20 0.25 0.5 0.75 1 Center of channel Flow Abutment τ = 0.91 1.44 1.972.49 3.02 0.91 1.97 3.02 3.25 3.25 4.04 3.44 3.75 X/(B1-B2) Y/ (B 1- B2 ) -1 -0.5 0 0.50 0.25 0.5 0.75 1 Flow Center of channel Abutment τ = Figure 8.6. Initial bed shear stress distribution (N/m2) for B2/B1 = 0.5, V = 0.45 m/s, L/(B1 – B2) = 6.76, and θ = 30 degrees). Figure 8.7. Initial bed shear stress distribution (N/m2) for B2/B1 = 0.5, V = 0.45 m/s, L/(B1 – B2) = 6.76, and θ = 45 degrees). Figure 8.8. Initial bed shear stress distribution (N/m2) for B2/B1 = 0.5, V = 0.45 m/s, and L/(B1 – B2) = 0.25).

72 3.27 0.91 1.58 2.49 4.08 2.49 1.58 0.91 3.47 3.27 3.13 3.02 2.83 3.02 3.13 1.22 2.05 2.83 X/(B1-B2) Y/ (B 1- B2 ) -1 -0.5 0 0.50 0.25 0.5 0.75 1 AbutmentFlow Center of channel τ = 0.91 1.55 2.06 2.71 4.05 3.25 3.51 2.06 1.55 0.91 1.13 3.03 3.25 3.03 2.71 3.34 3.81 X/(B1-B2) Y/ (B 1- B2 ) -1 -0.5 0 0.50 0.25 0.5 0.75 1 Center of channel Flow Abutment τ = 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). 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). contraction lengths. As can be seen, the maximum bed shear stress along the center of the channel in the contracted sec- tion is the same for all of the contraction lengths. At the same time, the location of the maximum bed shear stress is not influenced by the contraction length. Therefore, τmax and xmax are independent of the contraction length, and there is no need for any correction factors for contraction length. It was discovered later that in the case of a very thin contraction length (L/(B1 − B2) < 0.33), the maximum shear stress within the contracted length is smaller than the maximum shear stress due to the contraction. The reason is that the maximum shear stress occurs downstream from the contraction. The correction factor will reflect this finding at very small values of the contracted length (L/(B1/B2) < 0.33) (Section 8.6). 8.5 WATER DEPTH EFFECT: NUMERICAL SIMULATION RESULTS It was found that the water depth had no influence on the magnitude or the location of the maximum bed shear stress for the contraction problem. In fact, the water depth is already included in the equation through the hydraulic radius of Equa- tion 8.1. 8.6 MAXIMUM SHEAR STRESS EQUATION FOR CONTRACTION SCOUR The influence of four parameters on the maximum shear stress and its location near a channel contraction was investi-

gated by numerical simulation. These factors are the contrac- tion ratio (B2/B1), the transition angle (θ), the length of con- traction (L), and the water depth (H). Figure 8.11 describes the problem definition for abutment and contraction scour. In this figure, B1 is the width of channel, B2 is the width of the con- tracted section, L is the length of abutment, θ is the transition angle, Xa is the location of maximum bed shear stress due to the abutment, and Xmax is a normalized distance that gives the location of the maximum bed shear stress along the cen- terline of the channel (Xmax = X/(B1 − B2)) where X is the actual distance to τmax. It was found (for certain θ and B1/B2) that the influence of L on Xmax and τmax was negligible. In the case of θ = 90 degrees 73 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 and along the contracted section. For small ratios of L/(B1 − B2) (less than 0.33) the maximum shear stress τmax is past the contracted location and the shear stress of interest is located at Xc from the beginning of the fully contracted channel. The correction factor for a given influencing parameter is defined as the ratio of the τmax value including that parameter to the τmax 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 equation to describe the influence of each parameter. Figure 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 − θ for the influence of the transition angle θ. Figure 8.14 shows the correction factor kc − L for the influence of the contracted length L. The correction factor kc − H for the water depth influence was found to be equal to 1. The proposed equation for calculating the maximum shear stress within the contracted length of a channel along its cen- terline is where • γ is the unit weight of water (kN/m3); • n is Manning’s roughness coefficient (s/m1/3); • V is the upstream mean depth velocity (m/s); τ γθmax ( . )= − − − − − k k k k n V RLc R c c H c h2 2 1 3 8 2 B 1 B 2 L X a X m ax X c X Y F lo w θ 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 B1/B2 k c-R B2 B1 L 751 2 1380620 . Rc B B ..k += − Figure 8.11. Definition of parameters for contraction scour. Figure 8.12. Relationship between kc
R and B1/B2.

• θ is the contraction transition angle (in degrees) (Fig- ure 8.13); • Rh is the hydraulic radius defined as the cross-section area of the flow divided by the wetted perimeter (m); • kc-R is the correction factor for the contraction ratio, given by 74 • (Figure 8.12); • kc−θ is the correction factor for the contraction transition angle, given by • (Figure 8.13);kc− = + ( )θ θ1 0 9 90 1 5. . k B Bc R− = +  0 62 0 38 12 1 75 . . . 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 θ /90 k c - θ B2 B1 L 5.1 9 0 9.01 += − θ θck 0.6 0.7 0.8 0.9 1 1.1 1.2 0 1 2 3 4 5 6 7 L/(B1-B2) kc-L B2 B1 L    < − − − + − − − 3509 813 617 70 3501 2 2121 .kfor, BB L . BB L .. .kfor, k Lc Lc Lc Figure 8.13. Relationship between kc
 and /90. Figure 8.14. Relationship between kc
L and L/B1 − B2.

• kc-L is the correction factor for the contraction length, given by • for kc-L < 0.35 (Figure 8.14); and • kc-H is the correction factor for the contraction water depth. Since the water depth has a negligible influence, then kc-H ≈ 1. The influence of the water depth H is in Rh. k for k L B B L B B c L c L − − = ≥ + −   − −     1 0 35 0 77 1 36 1 98 1 2 1 2 2 , . . . . , 75 Note that the last part of the equation (γn2V 2R−0.33h) is the formula for the bed shear stress in an open channel (Equation 8.2). Equation 8.2 is consistent with the open channel case since all correction factors collapse to 1 when the parameter corresponds to the open channel case. In other words when B1/B2 = 1, θ = 0, and (B1 − B2)/L = 0, then kc−R = 1, kc−L = 1, kc−θ = 1. Figures 8.15 and 8.16 give the location of the maximum shear stress along the centerline of the contracted channel. Figure 8.15 shows the influence of the transition angle, and Figure 8.16 shows the influence of the contraction ratio. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.2 0.4 0.6 0.8 1 1.2 /90 X/ (B 1- B 2) B2 B1 L − − + = − 110 420 9021 1 340 . . e . BB X  θ 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 B2/B1 X/ (B 1-B 2) B2 B1 θ L 3 1 2 2 1 2 1 2 21 762481730 +−= − B B . B B . B B . BB X 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. 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 θ/90.