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Chapter 7 - Establishing an empirical formula
Introducing these boundary conditions in equations 7.8 and 7.9 yields:
4
h s = h corr + tanφ∗
⋅ – ----------------------------------------------------
3 ⋅B 2 ⋅ ( 1 + sig ⋅ 2ξ) 2 ⋅ ( r– R c – ξB) 3 + ( r–
R c – ξB)
(7.15)
dh
------- s
= tanφ∗
dr
⋅
4
– ---------------------------------
B 2 ⋅ ( 1 – 2B) 2 ⋅ ( r– R c – ξB) 2 + 1
(7.16)
where h corr h m ξB φ∗ 1
1 + 4ξ
= + ⋅ tan ⋅ – -----------------------------------------
2
(7.17)
3 ⋅ ( 1 + sig ⋅ 2ξ) 2
The sign sig is positive if the horizontal bed slope is assumed to be at the inner bank and negative
at the outer bank. Looking at the results assuming a horizontal bed slope at the outer bank
( sig = – 1 ) and putting ξ = 13 %, without any correction the correlation is of R 2 = 0.71 . It is
interesting to see (Fig. 7.6) that the cross-section fits already much better to the measured data
than without vertical correction. But the height of the bank is still overestimated whereas the
maximum scour depth is quite well reproduced.
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
C01b
C01c
C01d
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 7.6: Results of the uncentered polynomial equation (3rd degree) with vertical correction and horizontal bed
slope at the outer bank, without correction factor (eq. 7.15) (see Fig. 7.1 and 7.3 for explanations)
page 164 / November 9, 2002
Wall roughness effects on flow and scouring