pdf, 12 MiB - Infoscience - EPFL

Chapter 7 - Establishing an empirical formula
7.6.2 Conclusions
Without macro-roughness, the best correlation was obtained by an approach based on the shape of the
cross-section in the scour holes. A polynomial function of the third degree combines an excellent
correlation ( R 2 = 0.86 ) with a good fit over the cross-section 1 .
(
h s h m φ c 6.23 B r – R c – 8.8B) 3
= + tan ⋅ ⋅ ⋅ –-------------------------------------- + ( r–
R (7.31)
260 ⋅ B 2
c – 8.8B)
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
Unfortunately, the application to results of scale model tests and field data shows that the predicted
scour depth is too small compared to the measured one. Therefore the modified formula of
Bridge (eq. 7.7) is proposed for the computation of the maximum scour depth without macroroughness.
Despite a slightly lower correlation, this formula gives a much better prediction if
applied to field data. It further takes into account the bend geometry ( R c , B ) the flow conditions
( ) and the bed characteristics ( tanφ
)
h m
sinβ
0.394 11 23 h m
= ⋅ ⎛ – ⋅ ----- ⎞ ----- , (7.7)
⎝ B ⎠
⋅ ⋅ tanφ
⋅ ---- R
B r
2 = 0.817
In the presence of vertical ribs, equation 7.63 is recommended for the computation of the maximum
scour depth ( R 2 = 0.876 compared to the whole dataset, with Peter’s tests):
R c
h s
h
-------------- smax ,
7.7 e s
= ⋅ ----- ⋅ Fr ⋅( 0.001 + ( θ–
θ
h m
R cr ) 2 ) + 1.7
h
2 ) (7.63)
This formula is very simple, depending only on the Froude number and the difference between
the Shields stress and the critical Shields stress. The ratio rib spacing to hydraulic radius accounts
for the influence of the ribs. It is important to use an optimum rib spacing to compute the maximum
scour depth since the scour depth decreases linearly with the reduction of the rib spacing. If
the rib spacing is reduced sufficiently (below an optimum spacing), the scour depth can become
even more important than the scour depth obtained without the presence of ribs. This tendency
cannot be found in equation 7.63. Figure 6.5 on page 119 gives some indications on the reduction
of the scour depth as a function of the rib spacing. Furthermore it needs to be mentioned that the
result h smax ,
⁄ h m is close to 1.7 and that the different parameters in equation 7.63 have only a
reduced impact on the result.
1. The main parameters in this equation are the friction slope , a ration combining a flow characteristic
(velocity) with the channel geometry ( R h
and B ) and the flow depth to width ratio. These
ratios determine the scour depth compared to the mean bed level. The expression in the brackets
“fits” the 3rd-degree polynome on the cross-section with a steepest slope of tanφ ⋅ c . The term
r– R c
– 8.8B stands for the radius corrected by the radial shift of the inflection point of the cross
section (point for which the radial bed slope is maximum).
2. for θ >
θ cr
page 190 / November 9, 2002
Wall roughness effects on flow and scouring