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Chapter 7 - Establishing an empirical formula
As next and final step the parenthesis of the constant A is replaced by the sum of the most
important parameters combined with a factor to weight each one of them:
V⋅
R
A c 1
c 2
c h
R
3
⋅ ----------------- + c 4
⋅ S eall ,
c c h
5
----- c (7.4)
g ⋅ B 3
B
----- m V∗
= ⋅
⎝
⎛ + + ⋅ + 6
⋅ + ------
B V ⎠
⎞ ⋅ Fr d
Introducing the new parameters one by one does not bring an important increase of prediction
capability or only with “strange” equations containing factors at the order of magnitude of 10 3
and more and similar aberrations. Finally the following relation for A was found:
⎛ V⋅
R
A – 1.7 ----------------- h
1.23 R c
----- 16.5 h m
– ⋅ ----- 18 V ∗⎞
= ⎜ + + ⋅
+ ⋅ ------ ⎟ ⋅ Fr , (7.5)
⎝ g⋅
B 3 B B V d R 2 = 0.795
⎠
Cross section in the upper scour hole
Radius from center of the bend [m]
6.5
C01b
C01c
6.3
measured scour
computed scour
6.1
5.9
5.7
5.5
0.0
0.1
0.2
0.3
Water depth (free WS to BL) [m]
WS: free water surface
BL: bed level
C01d
C01b
C01c
C01d
0.5
Maximum scour depth
Hersberger (2002)
80%
60%
40%
0.4
0.5
computed maximum scour depth [m]
0.4
0.3
0.2
0.1
Peter (1986)
sin(beta)
20%
0%
-20%
-40%
computed equation: Kikkawa
upstream scour
0.0
0.0 0.1 0.2 0.3 0.4 0.5
measured maximum scour depth [m]
-60%
0.00 0.02 0.04 0.06 0.08 0.10
h/r
Figure 7.1: Results of the enhanced equation (eq. 7.5) of Kikkawa (see also Fig. 7.3 for explanations)
ON TOP: cross section in the upper scour hole for test C01 (S 0 =0.70%);
BOTTOM LEFT: maximum scour depth; RIGHT: transversal bed slope (without Peter’s data)
If we compute the scour depth with equation 7.5, a correlation of 0.795 can be obtained. With
extensive testing, this result could eventually be refined. A cross-section in the upstream scour for
page 156 / November 9, 2002
Wall roughness effects on flow and scouring