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Chapter 7 - Establishing an empirical formula
( B⋅ h m ). The ratios e s ⁄ B , e s ⁄ R c and e 2 s ⁄ B⋅
R c shows correlations of R 2 = 0.677 , 0.707
and 0.728 compared to the maximum scour depth. With this modification, equation 7.43 can be
written as:
h
---- s
B
e s
2
R c
(7.44)
Unfortunately it is impossible to get satisfying predictions with this equations or slightly modified
ones, namely by testing other ratios for the macro-roughness term, by introducing adjustment
constants for the whole equation and for one of the terms in brackets. The highest obtained correlation
remains remained below 0.5.
S e
------------- -------
θ
= ⋅ ⎛ – -------⎞
B ⋅ ⎝Fr 2 Fr2⎠
d
2) Without macro-roughness
Without macro-roughness, the drag force D acting on the ribs is replaced with a friction force
acting on the surface of the outer side wall ( ∆φ ⋅R o ⋅ h ). The friction along the inner side wall is
neglected since the water depth is small on the inner bank and the influence on the scour depth
can be neglected. A dimensionless friction coefficient C F is introduced.
D ∝ ρ w ⋅ V2 m ⋅C F ⋅ ∆φ ⋅ R o ⋅ h
(7.45)
Forming the ratios in equation 7.40 by introducing the proportionalities 7.37, 7.38 and 7.45 yields:
1
ρ w ⋅ V2 m ⋅C F ⋅ ∆φ ⋅ R o ⋅h s ρ
----------------------------------------------------------------- w ⋅V∗ 2 ⋅ ∆φ ⋅B⋅
Rc
=
+ -----------------------------------------------------------------
ρ w ⋅g ⋅h m ⋅∆φ ⋅B ⋅R c ⋅ S e ρ w ⋅ g ⋅ h m ⋅ ∆φ ⋅ B ⋅ R c ⋅ S e
(7.46)
After some simplifications (like for the case with macro-roughness), we obtain:
V2 1 m ⋅C F ⋅R o ⋅h s V∗2
= ------------------------------------------
g⋅ h m ⋅B ⋅R c ⋅S + -----------------------
e g⋅
h m ⋅ S e
And finally the following equation for the maximum scour depth is found:
h s 1 R
---- ----- c g ⋅ h
----- ------------- m
S ⎛
d 90
B
V2
e 1 θ ( s – 1)
------ ----
1
⎝
– ⋅ ⋅ ⋅ ⎞
R c S
= ⋅ ⋅ ⋅ ⋅
= ---------------- ⋅ ⎛-------
e
– -------
θ ⎞
⎠
m
⋅ R ⎝Fr 2 Fr2⎠
d
C F
R o
h m
S e
C F
o
(7.47)
(7.48)
This equation has the same layout as the one taking into account the scour reduction due to the
vertical ribs on the outer bank. Furthermore the radius of curvature appears as a parameter in this
equation. The radius to channel width ratio is hidden in the ratio ⁄ ( R o = R c + B ⁄ 2 ).
The last parenthesis in equations 7.43 and 7.48 can also be written as:
since θ = Fr∗ 2 = ( Fr d ⋅ V∗ ⁄ V) 2 .
S e θ
------- – -------
Fr 2 Fr2
d
=
If we compare the computed maximum scour depth (equation 7.48) to the measured values, it is
difficult to obtain satisfying correlations. The best fit was obtained with:
h
---- s
B
R c
S
------- e
Fr 2
– V∗
------
⎝
⎛ V ⎠
⎞2
(7.49)
θ
= ---------------------- ⋅ ⎛------- + 6.3 ⋅ -------⎞
, R (7.50)
0.189 ⋅ R ⎝Fr 2 Fr2⎠
2 = 0.724
d
o
S e
R c
R o
page 174 / November 9, 2002
Wall roughness effects on flow and scouring