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Establishment of the scour formula
Cross section in the upper scour hole
Radius from center of the bend [m]
6.5
6.3
measured scour
computed scour
6.1
5.9
5.7
5.5
-0.1
0.0
0.1
0.2
Water depth (free WS to BL) [m]
C01b
C01c
C01d
C01b
C01c
C01d
Maximum scour depth
80%
60%
0.3
0.4
0.5
40%
0.5
computed maximum scour depth [m]
0.4
0.3
0.2
0.1
sin(beta)
20%
0%
-20%
-40%
computed equation
upstream scour
0.0
0.0 0.1 0.2 0.3 0.4 0.5
measured maximum scour depth [m]
-60%
-0.02 0.00 0.02 0.04 0.06 0.08 0.10
h/r
Figure 7.10: Results of the uncentered polynomial equation (5th degree), horizontal bed slope at the outer bank,
with correction factor (eq. 7.28) (see Fig. 7.1 and 7.3 for explanations)
f) Polynomial function of the 3rd degree without restricted bed slope at the banks
A last equation based on the shape of the cross-section in radial direction was tested: a polynomial
function of the 3rd degree like the one used in paragraph a) but without fixing the boundary condition
of a horizontal bed slope at the outer or inner bank. The cross-section was adjusted in vertical
direction (average flow depth over the cross-section corresponds to the average water depth).
The constants of equations 7.8 and 7.9 become c 1 = c c ⁄ B 2 (to respect the units),
c 2
= R c + ξB , c 3
= tanφ∗
= c ⋅ tanφ
and c 4
= h m + c 1
⁄ 12 ⋅B 3 ⋅( ξ + 4ξ 3 ) + ξBtanφ
.
This leads to the following equations.
c c
h s = h m + ---------------
12 ⋅ B 2 ⋅[ 4 ⋅ ( r– R c – ξB) 3 + ( ξ + 4ξ 3 ) ⋅ B 3 ] + tanφ∗( r – R c )
dh
------- s
dr
=
c
----- c
B 2 ⋅ ( r– R b – ξB) 2 + tanφ∗
(7.29)
(7.30)
EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 169