Environmental Engineering Reference
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3 Numerical Results
In a similar way as when the obstacles are solid cylinders, flows past a pair of
magnetic obstacles side by side present different regimes according to the flow con-
ditions. While hydrodynamic regimes are governed only by the Reynolds number
and the dimensionless separation distance
D
(provided three-dimensional effects are
neglected), in the present case flow regimes are controlled by
Q
, in addition to
Re
and
D
. The variation of these parameters leads to steady or time-dependent regimes,
as occurs in flows with a single magnetic obstacle (Honji and Haraguchi
1995
;
Afanasyev and Korabel
2006
). We present numerical results for a pair of magnetic
obstacles side by side with a fixed Reynolds number,
Re
1,000, and investigate
the variation of
Q
and
D
on the flow dynamics. We consider flow conditions where
vortex shedding is present and explore the effect of separation distance
D
on the
coupling of the wakes behind the obstacles. The parameter
Q
is varied in the range
1.5
=
10, and for a given
D
,thevalueof
Q
corresponds to the minimum value
where vortex shedding appears. In turn, four different values of
D
are explored,
namely, 1, 1.5, 2, and 3, which are of interest since results for the hydrodynamic flow
past a pair of solid obstacles are available in the literature for these cases (Peschard
and Gal
1996
; Zdravkovich
1985
).
In hydrodynamic flows, it has been reported that for large distances between the
cylinders, the pair of wakes presents a weak coupling where in phase and out of
phase vortex shedding can appear. In turn, for shorter distances a strong coupling
arises and only in phase shedding is observed which produces a unique von Kár-
mán street (Peschard and Gal
1996
; Zdravkovich
1985
). At intermediate range of
coupling, a bistable regime can emerge which is characterized by a biased flow that
gives two possible values for the vortex shedding frequency. The biased flow is an
intermittent flow between two asymmetric states. That is, through the gap the biased
flow divides asymmetric states with narrow and wide wakes which can intermittently
interchange between the two cylinders (Zdravkovich
1985
), apparently driven by a
random process (Peschard and Gal
1996
). We now show that similar regimes are
observed in the wakes created by a pair of magnetic obstacles side by side.
Figure
3
shows the Lagrangian tracking of flows obtained numerically for different
values of
D
, with the corresponding minimum value of
Q
where vortex shedding
appears. For the smallest separation distance,
D
≤
Q
≤
1 (see Fig.
3
a), the magnets are in
contact and act as a larger magnetic obstacle that gives rise to a single wake similar
to the von Kármán street. If the gap between the obstacles is increased to
D
=
5,
we find a bistable regime where the flow pattern is rather complex, as is observed
in Fig.
3
b. A further increase to
D
=
1
.
2(seeFig.
3
c) leads to a more structured flow
pattern with two interlaced wakes in phase. For the larger gap explored, namely
D
=
3, the separation between the wakes is neatly defined and the in phase behavior
still persists, as observed in Fig.
3
d.
To improve the understanding of the flow behavior and the coupling of the wakes
behind the magnetic obstacles, the velocity component in the
x
-direction is shown
in Fig.
4
as a function of time at two distinct points located on the central line of each
=
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