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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).

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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.