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where u
(
w
)
is the vorticity component in the
transversal direction (
y
), and
p
is the total pressure. The vorticity field is solenoidal.
Hence, the vortex intensity, defined as:
is the fluid cinematic viscosity,
ð
ð
G
I
v
¼
o
d
A
¼
d
A
(10)
O
is conserved in any section around the cylinder. The term o represents the vector
of the vorticity.
Shen et al. (
1969
) studied the mechanism of local scour near cylindrical piers
and suggest that the horseshoe vortex generated upstream the pier is due to the
concentration of vorticity, induced by the obstacle, already existent in the flow field.
They believe that this horseshoe vortex is the primary mechanism of local scour.
Qadar (
1981
) explored the interaction between horseshoe vortex and the tempo-
ral increase of scour depth. He proposed that the horseshoe vortex is approximated,
in the symmetry plane, by a circle whose radius increases with the depth of the
scour hole, as represented in Fig.
29
.
Muzzammil and Gangadhariah (
2003
) investigated the horseshoe vortex in
the plane of symmetry through a visualization technique that employs suspension
of clay. They observed that the vortex shape is elliptical, instead of circular as
proposed by Qadar (
1981
). The dimensions of the horseshoe vortex were experi-
mentally measured and found to depend on the scour depth, diameter of the cylinder
and Reynolds number. They pointed out the existence of a cusp and two distinct
slopes in the scour hole, in the plane of symmetry, as shown in Fig.
30
.
X
A
u
0
Before
scouring
B
O
D
2
r
0
C
S
¢
G
Velocity profile
on section XX
S
X
H
O
¢
K
At
equilibrium
J
Longitudinal section
Fig. 29 Evolution of the horseshoe vortex with the increase of the scour hole (Adapted from
Qadar
1981
)
