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various types of regular waves were also noted at higher
Taylor numbers.
Jonas
[1981] investigated the influence
of Prandtl number on the incidence of various forms of
vacillation using fluids with Pr ranging from 11 to 74.
He reported that amplitude vacillation in particular was
significantly more widespread at high Prandtl number,
though the onset of “structural vacillation” close to the
transition zone at high Taylor number was less sensitive
to Pr. In most of the published studies so far, however,
the range of Pr investigated has either been limited to rel-
atively high values (using liquids based on water, silicon
oils, or organic fluids such as diethyl ether) or very low Pr
in liquid metals (mercury).
are part of a previous work [
Randriamampianina et al.
,
2006].
Before our numerical investigations of the baroclinic
instability using air as working fluid [
Randriamampianina
et al.
, 2006], there was not yet any available experimental
study devoted to this fluid with Pr =
(
1
)
. However, our
findings have subsequently motivated the installation of a
specific experimental rig at the university of Oxford, UK
[
Castrejón-Pita and Read
, 2007]. Then the measurements
confirmed a posteriori the computed results, especially the
route to obtain the AV regime. Indeed, unlike in previous
experimental works involving liquids, where the onset of
the
m
AV regime was associated with a decrease of the
rotation rate from the established steady wave regime
mS
(defined by a dominant azimuthal wave number
m
), the
AV regime was observed when increasing the rotation rate
in this study with air, during the transition between two
successive steady waves regime, from an azimuthal wave
number
m
to
m
+1.
Hide
[1958] reported that the transition from axisym-
metric to regular wave regimes does not significantly
depend on the value of the Prandtl number but rather
depends on the thermal Rossby number through the
empirical criterion
O
16.3.1. Transition Between Successive Wave
Numbers in Air-Filled Cavity Pr = 0.7
The geometric configuration corresponds to the one
used by
Fowlis and Hide
[1965] in their experimental stud-
ies of liquids, defined by an inner radius
a
= 34.8 mm,
outer radius
b
= 60.2 mm, and height
d
= 100 mm.
The cavity is filled with air, Pr= 0.7, and a tempera-
ture difference
T
=30 K is imposed between the two
cylinders. For the rotation rate values considered, a res-
olution of
N
0.05. In the present
simulation, the first regular steady wave was obtained for
= 1.488
<
c
at
Ta
=1.8
≤
c
= 1.58
±
×
×
×
×
80 was used in the
radial, vertical, and azimuthal directions, respectively,
with a dimensionless time step
δt
= 0.1125. The results
M
K
=64
96
10
5
[
Randriamampianina
et al.
, 2006]. In Figure 16.1 we display the bifurcation
×
0.4
0.35
0.3
0.25
0.2
0.15
QP3
Chaotic
2AV
0.1
US
2S
2MAV
0.05
0
0.17
0.18
0.19
0.2
0.21
0.22
0.23
10
6
]
Ta
[
×
Figure 16.1.
Amplitude of the dominant azimuthal wave number mode at midradius and midheight versus the Taylor number
showing the bifurcation diagram for the transition from the upper symmetric regime to a steady wave and subsequent vacillations
for the
m
= 2 flows in the air-filled cavity.
