Environmental Engineering Reference
In-Depth Information
)
reproduces the velocity profile in the layer thickness (Lara
2013
). Substituting Eq. (
4
)
in Eqs. (
1
)-(
2
) and averaging along the height of the fluid layer, the governing equa-
tions in the Q2D approximation are found to be (Figueroa et al.
2009
;Lara
2013
)
(
,
,
where
u
and
v
are the mean velocity components in the
x
-
y
plane, while
f
x
y
z
∂
u
x
+
∂
v
y
=
0
,
(5)
∂
∂
∂
u
u
∂
u
v
∂
u
y
=−
∂
p
u
˄
,
2
u
t
+
x
+
x
+∇
+
(6)
∂
∂
∂
∂
∂
v
∂
u
∂
v
v
∂
v
y
=−
∂
p
v
˄
2
v
Re
∗
B
z
,
t
+
x
+
y
+∇
+
−
ʱ
(7)
∂
∂
∂
where, for simplicity, the overline in the velocity components was dropped. The fac-
tor
2
1
e
−
ʳʵ
in Eq. (
7
) represents the attenuation of the magnetic field
in the normal direction while the linear terms that appear in both Eqs. (
6
) and (
7
)
account for the friction at the bottom of the container. The dimensionless time scale
˄
ʱ
=
ʳʵ
(
1
−
)
for the decay of vorticity due to viscous effects is given by
2
e
−
ʳʵ
ʳ(
1
−
)
˄
−
1
=
2
.
(8)
4
)
+
ʳʵ
1
2
2
e
−
ʳʵ
e
−
ʳʵ
ʳ
(
1
−
−
ʵ
2
The governing equations (
5
)-(
7
) were solved using a Finite Volume Method and the
SIMPLEC algorithm (Versteeg and Malalasekera
1995
), with no-slip boundary con-
ditions at the walls of the rectangular frame and a motionless fluid as initial condi-
tion. Once the velocity field is calculated, the advection equations (Figueroa et al.
2014
) are solved to perform the Lagrangian tracking.
4 Comparison Between Numerical
and Experimental Results
Let us now compare the numerical calculations with experimental visualizations.
For each array of magnets, different applied current intensities are explored and the
associated Reynolds number is calculated from the maximum velocity reached by the
numerical solution,
Re
. Figure
4
shows the experimental and numerical
results for the steady flow produced with an array of four magnets with alternating
orientation and a current of 50mA, five times larger than the one used in Fig.
2
b.
The corresponding Reynolds number is
Re
=
U
max
L
/ʽ
69. The experimental visualization is
showninFig.
4
a while Fig.
4
b, c correspond to the streamlines and Lagrangian track-
ing obtained from the numerical solution, respectively. The comparison of Eulerian
and Lagrangian results is very illustrative. While the streamlines capture the sym-
metry of the flow field, they do not show the characteristic features of the advec-
=
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