Geoscience Reference
In-Depth Information
Table 16.1.
Summary of the dimensions of the system, the fluid properties, and the governing
parameters for the liquid-filled cavity: Pr = 16.
Inner radius
a
4.5 cm
Outer radius
b
15. cm
Height
d
26. cm
Gap width
L
=
b
−
10.5 cm
a
Mean temperature
T
0
293 K
Temperature difference
T
=
T
b
−
T
a
2K
Rotation rate
0.25-1.25 rad
/
s
10
−4
K
−1
Volume expansion coefficient
α
3.171
×
10
−2
cm
2
/
s
Kinematic viscosity
ν
2.0397
×
10
−3
cm
2
/
s
Thermal diffusivity
κ
1.2731
×
Aspect ratio
A
=
d/L
2.47619
Curvature parameter
R
c
=
(b
+
a)/L
1.857
Prandtl number
Pr =
ν/κ
16.0215
Ra =
gαTL
3
/(νκ)
10
7
Rayleigh number
2.7735
×
Fr =
2
L/g
10
−4
-1.67
10
−2
Froude number
6.69
×
×
Ta = 4
2
L
5
/(ν
2
d)
10
6
10
7
Taylor number
2.95
×
−
7.37
×
=
gdαT/(
2
L
2
)
Thermal Rossby number
2.3475-0.0939
with azimuthal velocity formulation is introduced. Not
only does it reduce the number of equations to solve, in
comparison with the primitive variable velocity-pressure
formulation, but it also ensures a divergence-free velocity
field irrespective of the mesh used. An influence matrix
technique is implemented to treat the lack of boundary
conditions for the vorticity coupled with the stream func-
tion [
Chaouche et al.
, 1990;
Randriamampianina et al.
,
2001, 2004]. For the three-dimensional solution, the prim-
itive variables are directly solved. We present hereafter
the details of the governing equations and the numerical
method for the latter.
In a frame of reference rotating with the cav-
ity, the resulting dimensionless system is written as
[
Randriamampianina et al.
, 2006]
where
e
r
and
e
z
are the unit vectors in the radial and axial
directions, respectively, and
N(
V
)
represents the nonlin-
ear advection terms. The parameters governing the flow
and the heat transfer are the aspect ratio
A
, the curvature
parameter
R
c
, the Prandtl number Pr, the Rayleigh num-
ber Ra, the Froude number Fr, and the Taylor number Ta
(see the definitions in Table 16.1). For a given fluid within
a fixed geometry, the Taylor number is one of the two
main control parameters traditionally used to analyze this
system, following, e.g.,
Fowlis and Hide
[1965] and
Hide
and Mason
[1975]. The second parameter is the thermal
Rossby number,
gdαT
2
(b
4Ra
Pr Ta
,
=
a)
2
≡
−
introduced by
Hide
[1958] as a stability parameter, which
gives a measure of the buoyancy strength over the Coriolis
term and appears explicitly as coefficient of the advection
terms in equations (16.1) and (16.3).
The “skew-symmetric” form proposed by
Zang
[1990]
was chosen for the nonlinear advection term
N(
V
)
=
[
∂
V
∂t
+
2Ra
A
2
Pr Ta
N(
V
)
+
e
z
×
V
4
A
3
/
2
Ta
1
/
2
∇
2
V
+
F
,
=
−∇
+
(16.1)
∇·
V
= 0,
(16.2)
/
2 in the momentum equation (16.1)
to ensure the conservation of kinetic energy, a necessary
condition for a simulation to be numerically stable in time.
V
.
∇
V
+
∇
.
(
VV
)
]
∂T
∂t
+
2Ra
A
2
Pr Ta
∇
4
A
3
/
2
Pr Ta
1
/
2
∇
2
T
,
.
(
V
T)
=
(16.3)
with
16.2.3. Boundary Conditions
2
ρ
0
2
r
2
ρ
0
gαTd/
2
1
=
p
+
ρ
0
gz
−
,
The boundary conditions are no-slip velocityconditions
at all rigid surfaces,
F
=
1
Fr
4
A
(
r
+
R
c
)
T
e
r
,
2
T
e
z
−
V
=
0
at
r
=
±
1and t
z
=
±
1,
