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Establishment of the scour formula
b) With macro-roughness
By analyzing the first equation for the maximum scour depth with macro-roughness
(Appendix 12.2, ID 101), it can be seen (after some regroupments and simplifications) that the
first term e d ⁄ e s can be left away without significant changes in the precision of the result. The
relation:
h
-------------- smax ,
h m
=
0.85 ⋅ S e ⋅ Fr
------------------------------------------------------------------------------------- d
3 e Fr∗ φ V ∗
+ + tan ⋅ ------
d
⋅ ---- + tan 2 φ ⋅ Fr∗ – 2 ⋅ tanφ
⋅ Fr∗ 2
V
e s
1
⋅ -------------------- + 0.5
h ----- m
+
V∗
------
B V
(7.58)
has a correlation of R 2 = 0.893 . A plausibility check of the different parameters shows that the
observed tendencies are confirmed by this equation: the relative scour increases with increasing
bed slope, with increasing Froude numbers ( Fr d ≈ 10 ⋅ Fr∗ ), with decreasing roughness depth to
spacing ratio (up to an optimum spacing, afterwards it should decrease again). It was observed that
with increasing mean water depth to width ratio, the scour decreased (Figure 6.3 on page 121). But
the equation is quite complex.
The analysis of another maximum scour formula that obtained a good correlation (Appendix 12.2,
ID 112) results in the following simplified relation:
h
-------------- smax ,
0.862 ------ 1 ⎛ V ∗
------ + θ⎞ e θ Fr s
⋅ ----- σ Fr∗ 2 h m
– -----
(7.59)
h m
1.4⎝
V ⎠ R h
⎝
⎛ B ⎠
⎞ θ Fr h
= ⎛ + – ⎛ ⋅ – ----- m ⎞ Fr 2 + Fr⎞ + 1.33
⎝
⎝ B ⎠ ⎠
Since the third and the fourth term have almost no influence on the correlation they are removed.
h
-------------- smax ,
------
1 ⎛V∗
------ + θ⎞ e s
⋅ ⋅θ ⋅Fr
⋅ ----- σ Fr∗ 2 h m
∝ + ⋅ ⎛ – ⋅ -----⎞
+ 1.2 , R (7.60)
h m 1.4 ⎝ V ⎠ R h
⎝ B ⎠
2 = 0.894
If we continue to remove terms with a small influence we get:
h
-------------- smax ,
0.95 e s
= ⋅ ----- ⋅ ⎛V∗
------ + θ⎞ ,
h m
R h
⎝ V ⎠
⋅ θ ⋅ Fr + 1.83 R 2 = 0.886 (7.61)
This equation is now rather simple. With an increasing ratio rib-spacing to hydraulic radius, the
scour increases. This tendency is due to the fact that a big rib-spacing was associated to the tests
without ribs ( e s = 100 °) 1 ; knowing this, the equation reflects quite well the observations. If the
mean water depth in the channel increases (for the same width), the hydraulic radius increases,
too, but more slowly. Therefore the absolute scour will increase with increasing water depth,
1. This allows the computation of the maximum scour depth of the configuration without ribs
with the same equation by replacing e s
with 100 ⁄ 360 ⋅ 2R o
π = 2π ⁄ 3.6 ⋅ R o
EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 179