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Establishment of the scour formula
7.3.3 Approach based on the bed shape of the cross-section
In a typical cross-section profile, the following three important points can be found (Fig. 7.3):
1. In general, the maximum scour depth can be found at the outer side wall. For some tests,
especially the ones with thick vertical ribs (test series E), the maximum scour was sometimes
located close to the outer side wall but not at the wall 1 . For practical reasons, these eventual
depositions (or less scour) were not considered.
2. Somewhere in the outer half of the cross-section, the maximum lateral bed slope can be
found. Since sinβ is an approximation of the lateral bed slope, the highest point on the vertical
axis corresponds to this maximum transversal bed slope (Fig. 7.3).
3. Close to the inner side wall, the highest point in the cross-section can be found (location
with maximum depositions).
The idea explored in this paragraph is to fit a function to the cross-section, passing through the
above mentioned characteristic points. The boundary conditions are given in the following way:
• At the inflection point (steepest transversal bed slope), the first derivative is equal to the
friction slope. The following assumptions are made: (1) the angle of repose of coarse gravel
is the same with dry material as with wet sediments, (2) the maximum transversal bed slope
reaches its maximum possible value φ∗ and (3) this maximal possible value is equal to the
fraction angle multiplied with a factor depending on the main parameters.
• For some functions, the inflection point is assumed being at the center of the channel, for
other ones it is assumed being at a distance ξ ⋅ B from the channel axis.
• Other functions fix the transversal bed slope either at the inner or outer bank equal to zero
(translated by a first derivative of the function being equal to zero).
Looking at the right part of Figure 7.3, the elliptic shape of the curve is obvious. Therefore, functions
fitting that kind of curve were searched. Possible functions are a semi-ellipse, the cosines
hyperbolical (cosh) function or a quadratic polynomial function.
Elliptic and the cosh functions have a solution for y = fct( h s ⁄ r)
, but they cannot be integrated.
Therefore polynomial functions were chosen. The following paragraphs summarize the investigated
functions. Out of a large number of tested parameters, only the best solutions are documented
for each case.
1. This can be explained by some coarse sediments remaining next to the outer wall. In addition,
the secondary current may be less important in the “corner” between the vertical side wall and the
channel bed, leaving coarse sediments in this zone.
EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 159