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Chapter 7 - Establishing an empirical formula
The shape of the last equation gets quite close to the parabolic form of many traditional equations.
To allow an appreciation of the quality of the obtained relation, an equation with corrections
(eq. 7.20) is given on Fig. 7.7.
The so obtained correlation for the maximum scour depth can be compared to the empirical formula
of PETER (1986), who obtains a correlation of 0.87 compared to the complete data set.
Hoping to obtain even higher correlations with other functions, the following two type of functions
were explored: a polynomial function of the 3rd degree, but with additional terms
( y = x 3 + x 2 + x ) and a polynomial function of the 5th degree.
d) Polynomial function of the 3rd degree with additional terms - uncentered with vertical
adjustment
The following function was used with the boundary conditions mentioned in paragraph c).
Solving for the different constants, leads to:
h s = h m + tanφ∗
⋅
c 1
c
h s
---- ⋅ ( r–
c
3 2
) 3 3
=
+ ---- ( r–
c
2 2
) 2 + c 4
⋅ ( r–
c 2
) + c 5
4
--------
3B 2
1
⋅ ----------------
4ξ 2 ⋅ ( r – R
– 1 c – ξB) 3 ---
4 ξ
+ ⋅ ----------------
B 4ξ 2 ⋅ ( r– R
– 1 c – ξB) 2
B
( r – R c – ξB)
---
4ξ 3 – 3ξ
+ + ⋅ -------------------
3 4ξ 2 – 1
(7.21)
(7.22)
dh
------- s
= tanφ∗
dr
⋅
4
-----
B 2
1
⋅ ----------------
4ξ 2 ⋅ ( r – R
– 1 c – ξB) 2 ---
8 ξ
+ ⋅ ----------------
B 4ξ 2 ⋅( r– R
– 1 c – ξB)
+ 1
(7.23)
The final solution has quite a strange behavior as far as the choice of ξ is concerned. At ± 50 %
the curvature (second derivative) has a discontinuity. Therefore this value had to be avoided.
Without any correction factor the correlation is very low (below 0.50) and the order of magnitude
of the predicted scour does not fit at all. The correlation (maximum scour) can be increased up to
0.81 by introducing a correction factor c = 4.45 ⋅ ( 690 ⋅ S eall ,
+ 5.8 ⋅ R c ⁄ B – 79 ⋅ h m ⁄ B)
, but
the shape of the predicted bed does not fit well to the measured one (Fig. 7.8). With other correction
factors, the bed shape fits better, but with lower correlations up to 0.75.
Rendering the function more complicated does not bring the desired effect of a better prediction
of the maximum scour nor a better fit to the transversal bed topography.
page 166 / November 9, 2002
Wall roughness effects on flow and scouring