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Chapter 7 - Establishing an empirical formula
Cross section in the upper scour hole
Radius from center of the bend [m]
6.5
6.3
6.1
5.9
5.7
5.5
0.0
3
0.1
Water depth [m]
60%
0.2
2
C01b
C01c
2
40%
0.3
C01d
1 1
20%
0.4
sin(beta)
0%
3
0.5
-20%
Figure 7.3: Characteristic points in a cross-section profile at the (upstream) maximum scour location
on top: cross-section in the upper scour; bottom right: transversal bed slope as a function of h/r
a) Polynomial function of the 3rd degree - centered
First a centered polynomial function is analyzed:
-40%
0.00 0.02 0.04 0.06
h/r
h s
= h s () r =
c
---- 1
⋅ ( r–
c
3 2
) 3 + c 3
( r–
c 2
) + c 4
(7.8)
dh
------- s
= c (7.9)
dr 1
⋅ ( r – c 2
) 2 + c 3
The boundary conditions are:
1. dh s ( R c ) ⁄ dr = tanφ∗
: the maximum bed slope is tanφ∗
, located on the channel axis.
2. h s ( R c ) = h m : the water depth in the center of the channel is equal to the mean water
depth in the cross-section.
3. dh s ( r i ) ⁄ dr = dh s ( r o ) ⁄ dr = 0 : the transversal bed slope is equal to zero at the outer and
inner side wall.
page 160 / November 9, 2002
Wall roughness effects on flow and scouring