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Chapter 7 - Establishing an empirical formula
There are finally three parameters which need to be fixed: c c , c (in tanφ∗
) and ξ . The best found
combination of the different parameters is: ξ = 8.6 %, c = 0.843 – 0.272 ⋅ V⋅
R h ⁄ g⋅
B 3 and
c c = 28 ⋅V⋅R h ⁄ g ⋅ B 3 – 9 ⋅ h m ⁄ B – 10 . But the correlation for the maximum scour depth is
only 0.76. For all tested combinations of parameters, the shape of the cross-section fitted badly to
the measured bed level.
g) Conclusions
By using polynomial functions to fit an equation to the cross section shape quite decent correlations
can be obtained. The maximum scour depth can be predicted with good precision, but it is
rather difficult to obtain a good fit to the shape of the cross section. Polynomial functions of the
third degree give the best results. But a radial and vertical adjustment is necessary to obtain a satisfying
scour formula.
Introducing additional terms (e.g. x 3 + x 2 + x instead of x 3 + x ) does not lead to a better prediction
capability but to an important increase of the complexity of the formula. The same observation
applies for polynomes of a higher degree (the fifth degree was analyzed). They only result in
more complicated equations without improving the prediction of the maximum scour depth, nor
the fit to the cross section shape.
The best results were obtained with equation 7.15 and the constant given by equation 7.20. Combining
the two equations and performing some simplifications results in:
(
h s h m φ c 6.23 B r – R c – 8.8B) 3
= + tan ⋅ ⋅ ⋅ –-------------------------------------- + ( r–
R
260 ⋅ B 2
c – 8.8B)
(7.31)
h smax
,
= h m + tanφ ⋅c ⋅[ 0.129 ⋅ B]
(7.32)
where c 290 ⎛1 3.2 h m
– ⋅ -----⎞
V⋅
R h
= ⋅
⎝
(7.33)
B ⎠
⋅ -----------------
g⋅
B 3
The local scour depth depends on the ratio water depth to channel width and the dimensionless
parameter V⋅ R h ⁄ gB 3 . The maximum lateral bed slope is given at the inflection point of the
polynomial function with a value of tanφ∗
= c ⋅ tanφ
. It is important to note that the equation
for the maximum scour depth does not consider the radius of curvature .
R c
page 170 / November 9, 2002
Wall roughness effects on flow and scouring