pdf, 12 MiB - Infoscience - EPFL
pdf, 12 MiB - Infoscience - EPFL
pdf, 12 MiB - Infoscience - EPFL
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Establishment of the scour formula<br />
Considering the second equation (Appendix <strong>12</strong>.1, ID 2), a correlation of<br />
obtained, after some simplifications:<br />
h<br />
-------------- smax ,<br />
= 1.63<br />
h m<br />
d 90<br />
------<br />
V∗<br />
Fr<br />
h d ------<br />
θ<br />
⋅ ⋅⎛<br />
+ -----⎞<br />
θ 2<br />
m<br />
⎝ V σ 2 ⎠<br />
----- 2 V ∗<br />
------<br />
B<br />
+ + ⋅ ⋅ -----<br />
σ 2 V R c<br />
⋅ ----------------------------------------------------------------------------------------------<br />
⎛<br />
h<br />
----- m ⎞ 2<br />
⎝ B ⎠<br />
– θ 2 V∗<br />
------<br />
B<br />
+ ⋅ -----<br />
V R c<br />
R 2 = 0.825<br />
was<br />
(7.52)<br />
All contained ratio have a physical meaning: d 90 ⁄ h m is the relative roughness, V∗ ⁄ V the influence<br />
of the velocity distribution, θ⁄<br />
σ 2 gives the influence of the non-uniformity of the sediments<br />
on the Shields stress and B ⁄ R c can be interpreted as the influence of the curvature.<br />
But since the correlation is not exceptional, no further analysis of this equation will be performed.<br />
Lets analyze the formulae proposed for the computation of the location of the first scour. After<br />
simplifications, the following equation is obtained (Appendix <strong>12</strong>.1, ID 17):<br />
σ⎛<br />
φ d 90<br />
tan – ------⎞<br />
⎝ h ⎠ σ S e Fr d σ R c<br />
⋅ ⋅ ⋅ ⎛ – ----- + 3.75⎞<br />
α 1 0.94 ----------------------------------- m<br />
⎝ B ⎠ ----- ( Fr<br />
-------------------------------------------------------------------<br />
B d – σ)<br />
Fr<br />
= ⋅ +<br />
⋅ ---------------------------- + ------- d<br />
+ 2.5<br />
----- h m<br />
⋅ ⎛<br />
B<br />
Fr + ------<br />
V∗<br />
σ – Fr<br />
⎞<br />
d<br />
R<br />
----- c V∗<br />
– σ ------<br />
⎝ V ⎠<br />
B V<br />
R c<br />
(7.53)<br />
with a correlation of R 2 = 0.885 . The task of searching a physical signification of the formula is<br />
a big challenge. Some influences like the width of the grain size distribution σ , the densimetric<br />
Froude number are correctly represented. Performing some important simplifications and omissions,<br />
results in the following equation:<br />
B S<br />
α 1 σ 0.58 -----<br />
e<br />
= ⋅ ⎛ ⋅ + <strong>12</strong>.7 ⋅ ---------------- ⎞ + 1.4 ⋅Fr , (7.54)<br />
⎝ h m σ – Fr d<br />
⎠<br />
d ⋅ ------<br />
V<br />
– 6.6 R 2 = 0.829<br />
V∗<br />
For this equation it is much easier to explain the physical phenomena. Like for the previous equation,<br />
the scour shifts downstream with increasing σ . This tendency was also observed by Peter<br />
(1986, see equation 3.119 on page 55). With higher velocities (and consequently higher Froude<br />
numbers), steeper slopes and higher width to depth ratio, the first scour slightly shifts in the<br />
downstream direction. This equation can be recommended for the determination of the first scour<br />
location.<br />
Examining the results for the second scour location (Appendix <strong>12</strong>.1, ID 28), results in the following<br />
“simplified” equation depending on only two parameters, the density Froude number and the<br />
ratio radius of curvature to channel width:<br />
0.5 R c<br />
B<br />
⋅ ----- + 0.15 ⋅ Fr<br />
R<br />
α ⎛<br />
2 9.36 + Fr c<br />
d – ----- ⎞ R<br />
⋅ ⎛9.73<br />
+ ----- c ⎞ B<br />
d ⋅ -----<br />
R<br />
= c<br />
⎝<br />
, (7.55)<br />
B ⎠ ⎝ B ⎠<br />
– --------------------------------------------------------<br />
0.75<br />
Fr d -------- 1.468 R R 2 = 0.676<br />
c<br />
– – ⋅ -----<br />
Fr d<br />
B<br />
<strong>EPFL</strong> Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 177
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