Environmental Engineering Reference
In-Depth Information
Fig. 3 The images show the hexagonal convective cells similar to those observed by Bénard in his
experiments. Bénard was informed that the cells, now known to be due to the temperature depen-
dence of surface tension, resembled the pattern of the solar granulation photographed by Janssen.
Images from Van Dyke ( 1982 ): left photograph by Koschmieder ( 1974 )and right photograph by
Velarde et al. ( 1982 ). The sketch of the hexagonal flow structure in a Rayleigh-Bénard cell is inspired
by Getling ( 1991 )
dimensionless numbers necessary for dynamic similarity, given the geometry, are
the Rayleigh number Ra , and the Prandtl number Pr . Here,
TH 3
ʽʺ
= ʱ
g
ʔ
ʽ
ʺ ,
Ra
,
Pr
=
where
ʱ
is the isobaric thermal expansion coefficient, g is the acceleration of gravity,
ʽ
T is the temperature
difference between the top and bottom plate. Rayleigh found that the fluid starts to
move when the dimensionless temperature difference, expressed by Ra , exceeds the
critical value. For a detailed treatment of the linearized stability problem, see the
classic text by Chandrasekhar ( 1961 ); for a general introduction, see Tritton ( 1988 ).
The RBC flow just beyond the onset of convection has been used as a paradigm
for studying flow instabilities, chaotic systems and pattern formation (e.g., Busse
1978 ; Cross and Hohenberg 1993 ; Bodenschatz et al. 2000 ).
When the convective flow is strong enough, and the Rayleigh number is on the
order of 10 5 , the flow is characterized by thermal plumes, mushroom-shaped features
of hot (cold) fluid detaching from the bottom (top) thermal boundary layer (e.g.
Zocchi et al. 1990 ). At higher Rayleigh numbers, the plumes organize into a large
scale motion linked synergistically with the turbulent flow (Qiu and Tong 2001 ;Xi
et al. 2004 ). At very large Rayleigh numbers, the mean wind may itself be destroyed
(Niemela et al. 2001 ; Sreenivasan et al. 2002 ).
The Rayleigh number, which can be interpreted as the ratio of buoyancy to viscous
and thermal dissipation, is of the order Ra
is the kinematic viscosity,
ʺ
is the thermal diffusivity, and
ʔ
10 18
10 22 in the atmosphere ( Pr
10 20
10 20
10 24
0
.
7), Ra
in the ocean ( Pr
7), Ra
in the Sun ( Pr
10 7
10 20 for most astrophysical phenomena (see Sreenivasan
and Donnelly 2001 ). These values indicate the upper end of the Rayleigh number in
which we are interested.
10 3 ), and Ra
Search WWH ::




Custom Search