Geoscience Reference
In-Depth Information
Different experiments have been performed using annuli
of different dimensions. A comparison of these experi-
ments can be used to determine the dependence of the
flow on the aspect ratio
δ
. For example, it is observed that
an increase in
δ
leads to an onset of steady waves with
smaller wave number [see, e.g.,
Fein
, 1973;
Hignett et al.
,
1985]. The effects of the centrifugal buoyancy have been
discussed by
Lewis and Nagata
[2004] and can be related
to the dependence on the rotational Froude number
F
,
which measures the relative strength of centrifugal and
gravitational buoyancy.
Perhaps the most interesting parameter, other than the
thermal Rossby number and Taylor number, is the Prandtl
number Pr. Various experiments have been performed
using fluids of different Prandtl number. Most experi-
ments have used fluids of moderate to high Prandtl num-
ber, e.g., water with Pr
discussion of how the analysis can be extended to study
the flow far from the transition, in particular, in the wave
and vacillation regimes.
2.2.MODEL
As in many previous theoretical investigations of the
annulus experiments, we take the governing equations of
the fluid flow to be the Navier-Stokes equations in the
Boussinesq approximation [see, e.g.,
Williams
, 1971;
Lu
and Miller
, 1997;
Lewis and Nagata
, 2003, 2004;
Lewis
,
2010;
Randriamampianina et al.
, 2006;
Miller and Butler
,
1991]. Here, we take this to mean that all fluid properties
are constant except for the fluid density
ρ
, whose variation
itself is assumed negligible, except for when it is multiplied
by buoyancy forces such as gravity or the centrifugal force.
The equation of state of the fluid is taken to be
7(
Fowlis and Hide
[1965],
Fein
[1973], and others), a water-glycerol mixture with Pr
≈
13
[
Hignett et al.
, 1985] or higher [
Read et al.
, 1992;
Früh
and Read
, 1997], or silicone oil with Pr
≈
−
−
ρ
=
ρ
0
[1
α (T
T
0
)
] ,
(2.3)
63 [
Fein and
Pfeffer
, 1976]. However,
Fein and Pfeffer
[1976] also used
mercury as the working fluid, which has a low Prandtl
number (Pr
≈
where
ρ
0
is the fluid density at a reference temperature
T
0
and
α
, called the coefficient of thermal expansion, is
assumed to be constant and small. Given these assump-
tions, the fluid, to first order, can be considered as incom-
pressible. The heat equation with an advection term is
used to model the evolution of the fluid temperature.
For many fluids, such as water, the Boussinesq approxi-
mation is an accurate model for reasonable thermal forc-
ing. For air in the context of the annulus, the Boussinesq
approximation is still a reasonable approximation; with a
differential heating of 40 K and using a reference tempera-
ture
T
0
= 313 K, the error in the kinematic viscosity
ν
and
the thermal diffusivity
κ
is approximately
0.025), and
Castrejon-Pita and Read
[2007]
performed experiments using air, which has a near-unity
Prandtl number. See also Chapter 3. Many theoretical
studies have looked at fluids of moderate to high Prandtl
number, and some theoretical results exist for Prandtl
number near unity [
Randriamampianina et al.
, 2006;
Read
et al.
, 2008;
Lewis
, 2010]; however, lower Prandtl number
fluids have not been studied extensively.
In this chapter, we focus on the effects that the Prandtl
number has on the dynamics near the primary transition,
i.e., the transition from the steady axisymmetric solution
to rotating waves. In particular, we follow previous work
[
Lewis and Nagata
, 2003;
Lewis
, 2010] and use a dynam-
ical systems approach that uses linear stability analysis,
center manifold reduction, and normal forms to deter-
mine the behavior of the model equations near the pri-
mary transition. The methods will enable us to extend
the results of
Lewis and Nagata
[2003] and
Lewis
[2010]
and compute results over a large range of Prandtl num-
ber, covering values ranging from Pr = 0.025 to Pr = 13.
Results show that for smaller values of the Prandtl num-
ber it is possible to observe a hysteretic primary transition,
and there are regimes in parameter space where stable
mixed-mode flows exist.
In the next section, we describe the model equations.
In Section 2.3, we describe the methods involved in com-
puting the primary transition curve, i.e., the curve in the
parameter space that indicates where baroclinic instability
sets in, and present an example of a regime diagram for
an experiment that uses water as the working fluid. The
nonlinear analysis and results for a wide range of Prandtl
number are presented in Section 2.4. We conclude with a
≈
11%.How-
ever, over this same range, the Prandtl number Pr =
ν/κ
varies by less than 1%. In some investigations [e.g.,
Hignett
et al.
, 1985;
Young and Read
, 2008], an extension to the
Boussinesq approximation is made. In particular, the kine-
matic viscosity
ν
, thermal diffusivity
κ
, and coefficient
of thermal expansion
α
are assumed to vary linearly or
quadratically with temperature. In this case as well, to first
order the fluid is assumed to be incompressible. Such an
approximation allows for a more accurate representation
at larger differential heating.
It is convenient to write the governing equations in a
frame of reference rotating with the annulus at angu-
lar velocity
and in cylindrical polar coordinates, where
the radial, azimuthal, and vertical (or axial) coordi-
nates are denoted
r
,
ϕ
,and
z
, respectively. As such, the
Navier-Stokes equations in the Boussinesq approximation
describing the fluid flow can be written as
∂
u
∂t
=
ν
±
1
ρ
0
∇
2
u
∇
−
p
−
2
e
z
×
u
+
g
e
z
−
2
r
e
r
α (T
−
T
0
)
−
(
u
·∇
)
u
,
(2.4)
