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Chapter 7 - Establishing an empirical formula
b) Polynomial function of the 3rd degree - uncentered
To shift the inflection point towards the outer bank, the coefficient
equations 7.8 and 7.9 by c 2 = R c + ξ ⋅ B :
is replaced in
The boundary conditions become:
1. dh s ( R c + ξ ⋅ B) ⁄ dr = tanφ∗
(maximum bed slope).
2. h s ( R c + ξ ⋅ B) = h m : the water depth at the inflection point is equal to the mean water
depth in the cross-section.
3. dh s ( r o ) ⁄ dr = 0 : the transversal bed slope is equal to zero at the outer side wall.
Equations 7.8 and 7.9 give the following equations after introduction of these boundary conditions:
h s = h m + tanφ∗
⋅
dh
------- s
= tanφ∗
dr
(7.12)
(7.13)
Without any correction factor ( tanφ∗
= tanφ
) and putting ξ = 8 %, a correlation of
R 2 = 0.71 is obtained between measured and computed maximum scour depth (see Fig. 7.5).
Introducing the following correction factor c and putting ξ = 10 %, increases the correlation to
R 2 = 0.82 (compared to the maximum scour over the bend h 12,
max ):
tanφ∗
c ⋅ tanφ
20 ⎛195 ⋅ S eall ,
1.1 h m
+ ⋅ -----⎞
V⋅
R h
= = ⋅ ⋅ ----------------- ⋅ tanφ
(7.14)
⎝
B ⎠
g⋅
B 3
It is obvious (Fig. 7.5) that the obtained equation does not fit well to the observed cross-section.
Especially the inner bank elevation is much too high. There are two reasons for this:
• The fact that the bed slope at the outer wall is fixed to zero avoids a flatter bed slope.
• The fixed water depth ( )at the inflection point makes it impossible to get a vertical
adjustment of the equation.
Therefore tests with the same equation and the same boundary conditions were carried out, but
with a horizontal bed slope at the inner side wall. The coefficient ( – ) 2 in equations 7.12 and
7.13 becomes ( 1 + 2ξ) 2 . Putting ξ = 16 % without any other correction factor gives a correlation
of R 2 = 0.74 . If the correction factor c = 6.8 ⋅ hm ⁄ B – 4.5 ⋅ V ⋅ R h ⁄ g ⋅ B 3 is used with
ξ = 5 %, the correlation can be increased to 0.78, which still is not very satisfying.
c 2
=
R c
4
– ---------------------------------------
3 ⋅ B 2 ⋅ ( 1 – 2ξ) 2 ⋅ ( r– R c – ξB) 3 + ( r – R c – ξB)
⋅
4
– --------------------------------
B 2 ⋅ ( 1 – 2ξ) 2 ⋅ ( r– R c – ξB) 2 + 1
h m
1 2ξ
page 162 / November 9, 2002
Wall roughness effects on flow and scouring