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Chapter 7 - Establishing an empirical formula
which is correct. The ratio friction velocity to mean velocity has no significant influence on the
scour but is almost constant at about 0.1.
7
6
5
h 12,max
Computed
4
3
2
Table 7.5:
1
ID 101
0
0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
Plots of the maximum relative scour depth obtained with GPKernel with macro-roughness
(ID 101 corresponds to eq. 7.58, ID 112 to eq. 7.59, ID 124)
ID 112
Replacing V∗ ⁄ V by a constant of 0.2, leads to almost the same correlation ( R 2 = 0.885 ). If we
regroup the two terms θ to form Cst + θ 2 , and adjust the constant, the correlation is even
increased to R 2 = 0.887 . The constant has a value of 0.002. Since the square of θ cr = 0.047
gives exactly the same result, it is tempting to write:
h
-------------- smax ,
0.95 e s
= ⋅ ----- ⋅( θ2 , (7.62)
h m
R cr + θ 2 ) ⋅Fr
+ 1.83 R 2 = 0.887
h
The problem is that θ2 cr + θ 2 has not directly a physical signification. But if we write
( θ–
θ cr ) 2 = θ 2 – 2 ⋅θ⋅θ cr + θ2
cr , we find a physical explanation (unfortunately with a worse
correlation). Some additional simplifications and optimizations result in:
h
-------------- smax ,
7.7 e s
= ⋅ ----- ⋅ Fr ⋅( 0.001 + ( θ–
θ , (7.63)
h m
R cr ) 2 ) + 1.7 R 2 = 0.876
h
If we check this equation with the data set without macro-roughness, (admitting e s = 11.3 m for
the tests without ribs), we find R 2 = 0.766 . Considering only the data set with macro-roughness,
the correlation is rather poor ( R 2 = 0.537 ) (see Figure 7.12) but still much better compared to
directly computed scour depth (see § 7.4). This poor correlation is due to the fact that the tests
performed with ribs cover only a limited range of the hydraulic parameters. In average, the prediction
capability is quite well (see Figure 7.12).
All parameters in equation 7.63 have a physical meaning. The scour depth grows with increasing
rib spacing, Froude number and Shields parameter. A possible explanation for the constant 0.001
is that the Shields parameter θ is computed in the inlet reach; but in the bend, the erosive power
is more important, which is translated by the constant 0.001.
page 180 / November 9, 2002
Wall roughness effects on flow and scouring