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The regime that is still very puzzling is the asymptotic regime for high
Ra
, denoted
in Fig.
4
by
IV
l
, corresponding to “completely” turbulent boundary layers, dubbed
also as the ultimate or Kraichnan regime. Indeed, Kraichnan (
1962
) predicted that,
when the Rayleigh number exceeds some large value (for which he provided prelim-
inary estimates), the boundary layer becomes “completely” turbulent, leading to an
enhancement of heat transport; in this regime, he predicted that the Nusselt number
scales as
Nu
Ra
1
/
2
3
/
2
.
∼
/(
log
Ra
)
Ra
1
/
2
has been attained in configurations with no top
and bottom boundaries, in simulations by Lohse and Toschi (
2003
) and in experi-
ments in a vertical open-ended channel by Cholemari and Arakeri (
2009
) and Gibert
et al. (
2006
). However, the experimental confirmation of the ultimate regime in the
presence of boundary layers on solid boundaries, as one has in the classical RBC,
is still awaited. For contradictory points of view on the ultimate regime, one may
consult Chavanne et al. (
1997
), Niemela and Sreenivasan (
2003
,
2010
), Roche et al.
(
2010
), He et al. (
2012a
), Ahlers et al. (
2012
) and Urban et al. (
2014
), and references
therein. One hopes that this issue will soon be resolved satisfactorily.
Aregimeinwhich
Nu
∼
4 Large Aspect Ratio and Large Scales
Most experiments at high
Ra
are performed so far for
. However, most
natural phenomena are not constrained laterally in this way, with
ʓ
=
0
(
1
)
ʓ
on the order
10
2
. Unfortunately, little is known for high-
Ra
turbulent convection in contain-
ers of large aspect ratio, and, in general, also for different geometries. Theories do
not account for the aspect ratio explicitly; and while geometry effects were discussed
briefly in Grossmann and Lohse (
2003
), only qualitative predictions are known for
ʓ
−
10
larger than unity. Historically, convection theories have been guided by empirical
results, hence the lack of data prevented a proper theoretical formulation of geomet-
rical effects. Just recently data for
significantly different from order unity at high
Ra
are starting to appear both at moderately large aspect ratios (
ʓ
ʓ
∼
10)—see Hogg
and Ahlers (
2013
), du Puits et al. (
2013
)—and small aspect ratios,
10—see
Huang et al. (
2013
). We should call attention to experiments by Niemela and Sreeni-
vasan (
2006
)for
ʓ
∼
1
/
4, covering Rayleigh numbers up to 10
15
, the highest to-date.
Several experiments studied the convection in large aspect ratio,
ʓ
=
100, using
pressurized gas, and using a shadowgraph visualization technique. However, almost
all the studies were done near the onset of convection (see, for example, de Bruyn
et al. (
1996
), Bodenschatz et al. (
2000
)). The reason that the high
Ra
-high
ʓ
∼
para-
meter space is largely unexplored is that it is very challenging: while keeping all
other quantities constant, we find that
Ra
ʓ
∝
ʓ
−
3
.
From available data, it would seem that the dependence of the Nusselt number
Nu
on the aspect ratio
is rather weak. Indeed, there is a strong indication that
the mean wind may become weaker with increasing Rayleigh numbers (Sreenivasan
et al.
2002
). No major change in the Nusselt number scaling was observed in the
process. This points to the likelihood that the mean wind may not play a particularly
ʓ
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