Environmental Engineering Reference
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Fig. 1 a
Steady state for
R
a
=
2,000. The rolls are
concentric circles
.
b
Steady state for the annular
case for
R
a
=
2,500
=
Fig. 2
Case
R
a
2,500.
a
At the beginning, the
concentric circles
pattern appears and after a
while these circles begin to ripple.
b
The
concentric circles
pattern is totally destroyed
corresponding to
t
28, the initial pattern is lost completely. It is a different shape
with respect to the case shown in the Fig.
1
b. it is important to note that Fig.
2
b has
a similar trend with respect to the experimental results reported by Chandrasekhar
(
1961
).
Figure
3
shows the vertical velocity in the midplane of the container for
R
a
=
=
5,000 at (a)
t
2. We can see that the pattern changes rapidly and
no steady state is reached in this time interval.
The dependence of the growth rate with the Rayleigh number has been well
studied (Charru
2007
). When the Rayleigh number coincides with the critical value
Ra
c
=
3
.
8 and (b)
t
=
4
.
1,708 the growth rate is zero (this is the neutral stability state). It becomes
positive when the Rayleigh number is above
Ra
c
. This is a very important criterion
to test our simulations. We have performed two calculations, one is the time for the
beginning of the pattern formation and the second is the time elapsed to attain the
=
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