Page 1
Acta Mechanica 147, 173 196 (2001)
ACTA MECHANICA
y SpringerVerlag 2001
Natural convection in an annulus
between two rotating vertical cylinders
M. Venkatachalappa, M. Sankar and A. A. Natarajan, Bangalore, India
(Received July 21, 1999; revised September 1, 1999)
Summary. A numerical study is conducted to understand the effect of rotation on the axisymmetric flow
driven by buoyancy in an annular cavity formed by two concentric vertical cylinders which rotate about
their axis with different angular velocities. The inner and outer side walls are maintained isothermally at
temperatures 0c and 0h, respectively, while the horizontal top and bottom walls are adiabatic. The vorti
citystream function form of the NavierStokes equations and the energy equation have been solved by
modified Alternating Direction Implicit method and Successive Line Over Relaxation method. Numerical
results are obtained for a wide range of the Grashof number, Gr, nondimensional rotational speeds
f2~, X?o of inner and outer cylinders and for different values of the Prandtl number Pr. The effects of the
aspect ratio, A, on the heat transfer and flow patterns are obtained for A = 1 and 2. The numerical results
show that when the outer cylinder alone is rotating and the Grashof number is moderate, the outward
bound flow is confined to a thin region along the bottom surface while the return flow covers a major
portion of the cavity. For a given inner or outer cylinder rotation the temperature field is almost indepen
dent of the flow in the annulus for fluids with low Prandtl number, while it depends strongly for high
Prandtl number fluids. At a high Grashof number, with moderate rotational speeds, the dominant flow in
the annulus is driven by thermal convection, and hence an increase in the heat transfer rate occurs. In the
case of unit aspect ratio, the flow pattern is unicellular for the rotation of the cylinders in the same direc
tion, and when they rotate in the opposite direction two or more counter rotating cells separated by a
stagnation surface are formed. The rate of heat transfer at the hot cylinder is suppressed when its speed of
rotation is higher than that of the cooler cylinder. The computed heat transfer and flow patterns are com
pared with the available results of a nonrotating cylindrical annulus, and good agreement is found.
1 Introduction
There has been widespread interest in the study of natural convection of a fluid in a cylindrical
annulus between two vertical concentric cylinders subject to axial rotation. This motivation is
primarily due to the extensive range of practical applications such as electrical machineries
where heat transfer occurs in the annular gap between the rotor and stator, growth of single
silicon crystals and other rotating systems. The flow in a rotating annulus can be considered
as a possible analogue of the motion of a planetary atmosphere. The first numerical study of
free convection in a rotating annulus was conducted by Williams [1]. He made an excellent
analysis of axisymmetrical flows in the annular geometry for different combinations of physi
cal parameters. The accuracy of the numerical study was further verified with experimental
observations [2] for the same geometry and are found to be in good agreement. Samules [3]
has performed a similar type of numerical study for two different aspect ratios and rotational
speeds upto 40 and found that the single convective cell at low rotational speed splits up into
two cellular patterns as the rotational speed increases, de Vahl Davis et al. [4] made a numer
ical study of natural convection in a vertical annular cavity by considering two different cases
Page 2
174 M. Venkatachalappa et al.
in which the inner cylinder as well as the top and bottom rigid surfaces are rotating. In the
first case the inner and lower surfaces are heated and rotating while the other two surfaces are
stationary and cooled. In the second case the inner and upper surfaces are heated and rotat
ing, and other surfaces are cooled and stationary.
The stability of viscous flow between rotating vertical cylinders of infinite length with an
axial flow was investigated by Elliott [5]. Interest in understanding the natural convection fluid
flow and heat transfer in an annulus whose inner cylinder rotates has been motivated by the
desire for additional information on the design of cooling systems of many electrical equip
ments. The heat transfer and its associated flow characteristics of the above geometry have
been demonstrated by experimental visualizations as well as numerical simulations. The experi
ments [6] were conducted for different Prandtl numbers in the presence of axial flow to predict
the influence of various factors on the onset of vortex motion. A finite difference scheme was
used to solve the boundary layer equations of a laminar convective flow in an open ended annu
lus to obtain the tangential boundary layer thickness and tangential, axial and radial velocity
profiles (see [7] and [8]). Langlois [9] made a detailed review on the effect of convective flows
in crystalgrowth melts with numerical simulation of Czochralski bulk flow. It was pointed
out that in many practical situations of materials processing applications, particularly in
Czochralski crystal growth methods, the whole arrangement has cylindrical symmetry and
both crystal and crucible rotate at different speeds about this symmetrical axis either in the
same direction (isorotation) or in opposite direction (counter rotation) (see [10], [11]).
Buoyancydriven convection in vertical annular spaces without rotation, which is a simpli
fied representation of many practical problems, has also been investigated numerically by a
number of researchers. The first extensive study of natural convection in isothermally heated
vertical annuli was reported by de Vahl Davis and Thomos [12]. They obtained the heat trans
fer correlation based on their numerical results in the boundary layer regime as
Nu = 0.286Ra~176176176176 ~
Kumar and Kalman [13] used a numerical technique
based on the false transient method and alternating direction implicit scheme to solve the
similar problem for a wide range of parameters, and their heat transfer correlation is
Nu = 0.18Ra~176176 ~
The other important works on this geometry are by
Satya Sai et al. [14] and WuShung Fu et al. [t5] where a finite element method and a control
volume method respectively are used to obtain the solution.
Most of the earlier studies found in the literature on the combined rotational and buoy
ancy effects in a vertical annulus between two concentric cylinders are concentrated on either
the whole system is rotating or one of the cylinders rotating and the other stationary. Not
much attention has been given to the study of the combined effect of rotation and buoyancy
in a vertical annulus with both cylindrical walls rotating at different speeds, in spite of its
application in crystal growth experiments. Since the flow in the annulus is usually driven by
the buoyant force and rotation of the cylinders, neither pure free convection nor pure forced
convection appears in the annulus. Hence, a complex flow occurs. The present numerical
study investigates the effect of rotation on the flow driven by buoyancy in a cylindrical annu
lus whose side walls rotate about their axis with different angular velocities.
2 Mathematical formulation
Consider an incompressible fluid confined between two vertical coaxial cylinders of inner
and outer radii ri and ro, respectively, as shown in Fig. 1. The inner and outer cylindrical
walls are held at different constant temperatures 0c and Oh, respectively. The bottom wall of
Page 3
Natural convection in an annulus 175
g
gt
~X
~
ot) o
Oc ] Oh
rod
Insulated
Fig. 1. Schematic diagram of the physical configura
tion
the annulus at x = H is a rigid surface, while the top surface x = 0 is assumed to be free. The
top and bottom walls are thermally insulated. Let wi and Wo be the angular velocities of the
inner and outer cylinders, respectively. The fluid is assumed to have constant physical proper
ties but obeys Boussinesq approximation according to which its density is taken constant
except in the buoyancy term of the axial momentum equation. Further, we assume that the
flow is axisymmetric. Employing the above assumptions, the equations governing the conser
vation of mass, momentum and energy for an unsteady laminar flow, after neglecting viscous
dissipation, in cylindrical coordinates (r, r x) with axisymmetry are:
0 (ur)+ O (wr)=O
Or Ox
(1)
'
o~+~ o (~2)+ o (m~)
O~ Or
1 op'
~o Or + " ~71%
[
[ 7~],
u
(2)
Oz
v]
OV t@ 0 (7"UV) @ 0 (WV) @~= Z] ~712V~
ol ~ ~
(3)
Oml_~ 0 (ruw)+ O0
OJC ~r
1 019,
cOo OX ~ /2~712m  g]~(O  0o) , OX (m2) 
(4)
O0
Ot
~_ 0 (~0) + o (m0) = 0~I~7120
r Or Oz
(5)
02 1 0 02
where V12 =~r~r 2 +r Orr + OxSx2' u, v and w are the radial, tangential and axial components
of the velocity, 0 denotes the temperature, p' is the pressure defect defined as
pt = p _ (Po + Qgx), p is the actual pressure at any point and po is the hydrostatic pressure in
the quiescent state, u is the kinematic viscosity, ~ is the fluid density, a is the thermal diffusiv
ity and/3 is the coefficient of volumetric expansion.
Page 4
176 M. Venkatachalappa et al.
The initial and boundary conditions are:
t=0:
t>0:
u=v=w=O,
00;
rl <_r<r20<x<H.
u = w = O , v = riwi , O = Oc ; r = ri ,
u = w  O , V = roWo , 0 = 0t~ ; r = ro ,
Ov O0
ox=O,
u=w=O, =0, x=O,
O0
=0;
Ox
u=w=0,
v=rwo, x=H.
The equations are made nondimensional using
uD vD wD
U , V=, W= ,
l/ l] l ]
0 Oo p' D 2
~o u2 '
tu
'
 ,
T
D2
T  oh _ Oo P
=r X = z ~i =~iri2
R D' D' u '
where 0o  Oh + 0~ and D = ro  ri.
2
The azimuthal component of the NavierStokes equations is replaced by an equation
governing the swirl defined by P = RV which is a physically conserved quantity through the
angular momentum per unit mass, and its importance was demonstrated by Langlois [9].
Since the motion is axisymmetric, the computation time can be reduced if the momentum
equations (2) and (3) can be recast as vorticitystream function formulation. In dimensionless
form, the governing equations can now be written as
Cdoro 2
no  ,
(6)
or+_~
oW
o (Rut)
~
a (wr)
+ U2
~ 1or
R oR ~ axe'
o~r
 oR~
(7)
0
1 2
~rr g7 T,
OT
Or + R ~
1 0 (RUT) + (WT) = (8)
~
L o 2r or
R 30X
( aT
OR '
aT ~ (Re() +
o~
(w()  v2r  + Gr  (9)
OX
~
1 [02~ O2k0 1 Og t]
aX 2
<= ~ [~}
R ~ '
(10)
where
OU OW
OR '
r
(11)
10~
R OX ,
i O~
R U W OR
(12)
and
02 02 1 0
~7 2 
Page 5
Natural convection in an annulus 177
The nondimensional parameters occurring in the above equations are:
Gr
9/3DS(Oh  0~) the Grashof number, Pr
..
2u 2
'
ro
ratio, and ~ = , the radii ratio.
7" i
The initial and boundary conditions in nondimensional form are:
u
a
H
D'
.. , the Prandtl number, A =  the aspect
Ti
~<__~_<~,
To
~=0: U=W=k~=0, F=~=T=0; 0<X<A.
~
OR
ri
D' ~>0: ~
0, F=(x
1)RY2i, T 1, R
cgg/ ~  1 ro
D'
:OR O, F RY2o, T
+1; R
c92 ~
OX 2
OF
OX
cgT
cgX
UT0, 0, 0; X=0,
 OX 0 F= R2f2~ OX
0: X=A (13)
The parameter of practical interest is the rate of heat transfer across the cavity. The overall
heat transfer rate across the cavity is expressed by the average Nusselt number
1 0T
Nu = ~
L0
 A
(14)
The numerical integration of Eq. (14) is performed at each time step after the temperature
field is obtained. Thus, the formal description of the problem is given by the swirl velocity,
vorticity and energy equations together with the definition of vorticity, stream fuction,
Nusselt number and the initial and boundary conditions (7) to (14).
3 Numerical procedure and validation
A numerical technique based on the twostep ADI (Alternating Direction Implicit) method
has been used to solve the vorticity transport, swirl velocity and energy equations. In order to
improve the stability of the numerical scheme and to speed up the convergence, the nonlinear
convective terms in the ADI method are approximated by the second order upwind difference
method. The diffusion terms are approximated using central differences. The elliptic stream
function equation is then solved by employing the Successive Line Over Relaxation (SLOR)
method is preferred here because it converges in fewer iterations than the usual point iteration
methods, and also it immediately transmits the boundary condition information from the
ends of the line to the interior of the domain (see Rudraiah et al. [16]). The discretized alge
braic equations are arranged in tridiagonal matrix form which can be solved readily using the
Thomas algorithm. For the solution of the vorticity equations the values of the vorticity at
the boundary are required. More frequently used second order expressions for the wall vorti
cities, ~, are obtained by expressing the stream function in Taylor series (see Roache [17]).
The form used in this study for ~w is
6,= 2(A~)2 ,
Page 6
178
Table 1. Grid dependence results
NR x NX NU
Gr = 106, Y/i = 2000
NU
Gr = 106, g?o = 2000
21 x 21
4t x 4i
61 x 61
9.3984
9.3145
9.3083
8.0621
8.I099
8.1171
M. Venkatachalappa et al.
Table 2. Comparisons of Nu with published results
Ra
de Vahl Davis and Thomas
[12]
Kumar and Kalam
[]3]
Present study
Aspect ratio (A) = 1
104 4.1734
105 7.559 4
3.3047
6.268 1
3.4078
6.349 0
Aspect ratio (A) = 2
104 3.538 7
105 6.409 8
3.036 8
5.759 9
3.162 9
5.881 8
where w denotes the boundary node and AT is the spatial interval in the direction normal to
the boundary. Here we note that in the cylindrical coordinate system there is an additional
1 Ok~.
term  ~ ~ m the expression for C compared to that in the rectangular coordinate system
(see Roache [17]). However, the form of the vorticity wal condition ~ written above will
remain the same both in the cylindrical and Cartesian coordinate systems, since the additional
term in Eq. (10) vanishes at the boundary due to the noslip condition. Finally, the velocity
components are evaluated using central difference approximation to Eq. (12). Several grid
sizes and time steps are used in order to examine the grid dependency. The uniform meshes
employed in the present study are: 21 x 21,41 • 41 and 61 x 61. The results for the case
Gr = 106 and Y2~ = 2 000, Y2o = 2 000 are given in Table 1. Because of the minor differences
between the 41 • 41 and 61 x 61 grids and the consumption of computer time, a 41 • 41 grid
for A = 1 and 41 x 81 grid for A = 2 are used for further calculations. The above computa
tional cycle is then repeated for each of the next levels, and the steady state solution is
obtained when the following convergence criterion:
n+l n
)/ij  )t~y
. n+l
x~j
I < g
I
is attained. Here X is the primary variable being tested, and c is a prespecified constant usually
set at 10 .2 .
The validation of the present numerical code has been performed by generating heat trans
fer results over a wide range of parameters without rotation, and the results are compared
with the results of de Vahl Davis and Thomas [12], and Kumar and Kalam [13] and are
shown in Table 2. Excellent agreement was found with the numerical solution of Kumar and
Kalam [13]. Their results, obtained using the correlation equation based on the numerical
data, shows good agreement with the experimental results of Prasad and Kulacki [18]. How
ever, the results of de Vahl Davis and Thomas [12] overpredict the experimental results and
the present numerical results.
Page 7
Natural convection in an annulus 179
4 Results and discussion
The natural convection of fluids in a vertical annulus with isothermally heated and cooled ver
tical walls which rotate about their axis with different angular velocities is studied numeri
cally. Computations were carried out for 104 _< Gr _< 106, 500 _< ~?i, S?o _< 2 000, A = 1 & 2,
Pr = 0.01, 0.1 & 1.0, and heat transfer results along with isotherms and flow fields have been
obtained. The range of the dimensionless rotational speeds considered here varies between
5 rpm to 10 rpm for an annulus of inner and outer diameter 8 em and 14 era, respectively, and
for kinematic viscosity u = 0.15 cm2s 1.
Figure 2 shows the flow pattern and temperature distribution in the absence of rotation of
the cylinders for the Grashof numbers Gr = 104 and 106, respectively. For a moderate Gras
hof number (Gr = 104) a simple recirculating pattern with the centre at the middle of the
annulus is obtained. The temperature stratification in the vertical direction is observed inside
the cavity. As the Grashof number increases to 106, the pattern exhibits a strong upward flow
at the outer (hot) cylinder and downward flow at the inner (cold) cylinder, and the fluid move
0
X 0.5
i i
0.5
R
x 0.5
1
(a) Gr = 10 4
0.5
R
X 0.5
0.5
R
1
(b) Gr = 10 6
X 0.5
/
0.4
T=0
, ,
")
0.5
Fig. 2. Streamlines and isotherms for Di = ~2o = 0
Page 8
180
0
X0.5
10
0.5
R
(a) ~i=500
0
X0.5
10
M. Venkatachalappa et al.
0,5
R
1
X0.5
r
0 0.5
R
(b) ~i=1000
X0,5
1 0
0.5
R
X0.5 f~
1; 015
R
(c) ~=2000
Fig. 3ae. Streamlines and isotherms for Gr = 10 4, f2o 0 and A = 1
x0,5 T~ 1
10 0.5
R
ment appears to be very little or stagnant in the central core of the fluid. The isotherms sug
gest that the flow can be considered as the interaction of two thermal boundary layers, one on
the outer cylinder and the other on the inner cylinder. Figure 3 illustrates the effect of rotation
of the inner cylinder on streamline and isothermal patterns for different Grashof numbers. At
low rotational speeds the thermal and flow fields exhibit a structure similar to that of a non
rotating system. It is observed that for a moderate Grashof number (Gr = 10 5) the forced
convection caused by inner cylinder rotation decreases, and hence the flow in the annulus can
be considered as forced convection near the inner cylinder and thermal convection near the
Page 9
Natural convection in an annulus 1 8 1
0
X 0.5
1~ 0.5
(d) ~'~i=500
)11,5
0.5
R
0
31).5
10~
i i
0.5
R
(e) ~i=1000
31).5
10 0.5
R
0
31).5 31}.;
1 0
p.
(f) ~i=2000 R
Fig. 3df. Streamlines and isotherms for Gr = l0 n, Do = 0 and A = 1 (cont.)
outer cylinder (see Fig. 3 d). When the thermal convection dominates over the forced convec
tion by further increasing the Grashof number, the forced convection near the inner cylinder
diminishes, and the major portion of the annulus is occupied by the buoyancydriven convec
tion (Fig. 3 g). At high rotational speeds the flow at low and moderate Grashof numbers
(Gr  10 4 and 10 2) is entirely dominated by centrifugal forces, and when these forces domi
nate, the two cells at moderate Grashof number and low g2i merge to a single cell, and a
strong vertical flow in the cavity is observed (Figs. 3 e, f). The effect of high inner wall rotation
on the isothermal pattern is shown in Fig. 3 b, c, e and f. As a result of the motion in the azi
muthal direction, the fluid adjacent to the rotating inner wall is transported to the stationary
Page 10
182
n
M. Venkatachalappa et al.
0
XD.
31:0,5
10
0.5
R R
(g) ~"~i = 500
0 0
XD.5
)[0,5
10 0.5
R
I 10 0.5
R
(h) ~"~i = 1000
n 0
~. X0.5
10 0,5
R
R
(i) ~~i = 2000
Fig. 3gi. Streamlines and isotherms for Gr = 106, Xgo 0 and A = 1 (cont.)
outer wall opposing the buoyancy motion. Consequently, the isotherms show a considerable
distortion indicating the presence of strong centrifugal acceleration.
The streamlines and isotherms at various rotational speeds of the outer cylinder are shown
in Figs. 4ai. At moderate values of the Grashof number (Gr = 104) the outward bound
flow, that is the flow from the inner cylinder to the outer cylinder, is confined to a small
region along the bottom surface as the rotation of the outer cylinder increases. But the return
flow covers a larger portion of the cavity. This pattern results from the interactions of the
basic natural convection field, centrifugal and Coriolis accelerations caused by the angular
rotation. The outward bound fluid covers less than half of the channel height because the
Page 11
Natural convection in an annulus 183
Table 3. The effect of rotation on the average Nusselt number (Pr = 1.0)
n
A Gr Y2i ~?o
Nu
1 104 0
0
0 2.478 7
1.794 9
2.6494
4.550 5
2.254 7
4.846 2
4.995 8
8.3568
8.324 2
8.109 9
9.235 1
9.314 5
2.202 1
4.045 4
2.2513
2.496 3
500
500 0
105 0
0
500
2 000
500
2 000
0
0
0 10 a 0
0
0
500
2 000
500
2 000
500
2 000
2 000
i 500
0
0
104 500
500
2 000
i 000
2 105 2 000 0 4.549 9
2.781 2 0 2 000
106 I 000
1 500
2 000
1 500
2 000
500
2 000
1 000
7.303 8
7.538 8
3.062 9
3.974 0
fluid flowing towards the outer cylinder encounters a greater resistence due to the rotation of
the hotter cylinder. The return fluid, however, flows into a region without much resistance,
due to the stationary inner wall, and hence occupies the larger portion of the cavity. Also if a
temperature difference exists between any two points in a uniformly rotating fluid, thermal
convection may arise if the resulting density difference is unstable in the gravitational and
centrifugal force fields, and hence an element of fluid with a lighter density than the bulk
will tend to move upwards in the gravitational field and radially inwards in the centrifugal
field. This phenomenon is amplified when the rotational speed further increases, since the out
ward bound flow experiences more resistance at the hotter cylinder while the inward flow
occupies a greater portion of the channel height as there is no resistance at the inner cylinder
(see Fig. 4 b, c). Interestingly this behavior is not exhibited in the case of an inner (cold) rotat
ing cylinder (see Fig. 3 b, c). Also, from the streamlines, it can be seen that the position of the
maximum stream function moves from the outer cylinder to inner cylinder.
The influence of rotation of the outer cylinder on the temperature distribution is apparent
from the isotherms. The isotherms are nearly straight vertical lines with less stratification in
the core, and the temperature gradients are relatively weak as the space between the isotherms
is more, indicating that the strength of the convective motion for the outer cylinder rotation is
suppressed compared to the rotation of the inner cylinder. However, this effect is to be
expected, since the motion induced by the rotation of the outer (hotter) cylinder causes the
fluid near to it to move radially towards the inner cylinder opposing the buoyancy motion,
and this centrifugal motion dominates the flow as the rotational speed increases. As the Gras
hof number is increased to 105 (see Fig. 4 d f) the buoyancy flow strengthens and a more
complex flow occurs. Also, two secondary cells exist with one cell near the hotter cylinder
Page 12
X0.5
0
X0.5
184
10~ .... 0'.5 .....
R
1
(a) f~o = 500
M. Venkatachalappa et al.
0
X0.5 0
10
o
0.5
R
r i ..~ , i , i ,
10 0.5
R
1
(b) f~o = lO00
X0.5
10 0.5
R
XO,5 
1 0 ~ '
015 . . . .
R
1
(c) ~o = 2000
0
X O , 5 1
10 0.5
R
I
Fig. 4ac. Streamlines and isotherms for Gr  105, _Qi = 0 and A = 1
generated by buoyancy motion and another cell generated by a centrifugal force occupying a
portion near to the colder cylinder. The size of the centrifugally generated cell increases as the
rotational speed increases while the size of the buoyancy driven cell decreases. The isotherms
(see Fig. 4 df) indicate the presence of a fairly strong convective motion at lower rotational
speeds. Increasing the outer cylinder rotation leads to a decreased convective strength since
Page 13
Natural convection in an annulus 185
0
X0.5
1~ 0.5
(d) ~o=500
X0.5
10
0,5
R
1
0
xO.5
10 0,5
R
1
(e) ~o=1000
X0.5
10 0,5
R
1
R
(f) ~o=2000
0
T0,8
X0.5
10 0,5
R
Fig. 4df. Streamiines and isotherms for Gr = 105, ~i 0 and A = 1 (cont.)
the flow is dominated by the centrifugal force against the buoyancy motion. As a result, the
rate of heat transfer is reduced considerably. Although this phenomenon is qualitatively
reflected from the less distorted isotherms at higher rotational speeds, a more quantitative
measure of its effect can be seen from the average Nusselt number as shown in Table 3. When
the Grashof number is further increased to Gr = 10 6, the streamline and isothermal pattern
resemble that of a pure buoyancy driven convection which indicates that the small rotational
speeds are not effective at large values of Or. Thus it is clear that the rotation must be suffi
ciently high for the Coriolis force to be effective at large Grashof number.
Page 14
186
0
M. Venkatachalappa et al.
0
X0.5
X0.5
10
0.5
R
1
(g) f~o = 500
i 0
0.5
R
X0.5
X0.,
1 0
0,5
R
1
(h) Qo = 1000 R
0 0
X0,5 X0.5
10 0.5
R
1 10 0.5
R (i) ~o = 2000
Fig. 4gi. Streamlines and isotherms for Gr 106, ~i = 0 and A = 1 (cont.)
The flow pattern and temperature distribution for various combinations of rotation of the
inner and outer cylinders are shown in Figs. 5 and 6. When both the cylinders are rotating at
the same speed, a small subsidiary flow exists in the lower part of the annulus due to the inter
action of two counter flows, from the inner wall to the outer wall and viceversa. The iso
therms show the presence of convective motion at these rotational speeds (Fig. 5 a). As the
rotation of the outer wall increases, the flow generated by it occupies the major portion of the
annulus which is completely encapsulated by a strong eddy, and a secondary flow caused by
rotation of the inner cylinder occurs near the bottom of the cavity. The isotherms show the
Page 15
Natural convection in an annulus
X0.5
10
1
R
(a) ~ ~o = 500
187
0 ~ .8
0.4
X0.5
10 0.5
R
1
X0,5
0
10 0.5
R
0 .4
8
X0.5
T=
I '
(b) ~i = 1000, ~o = 2000 R
0 0
JO,5
1;
X0.5
10
0.5
R
0.5
e
1
(C) ~"~i "= 2000, ~o = 500
Fig. g. Streamlines and isotherms for Gr = 104 and A = 1
conduction like heat transfer in the cavity (Fig. 5 b). When the inner wall rotation is much
larger than the outer wall the fluid near the inner wall undergoes rapid changes in the centrifu
gal acceleration, and hence a strong vertical flow exists. This phenomenon is reflected in iso
therms, too. The isotherms move from the cooler cylinder to the hotter cylinder, in contradic
tion to the buoyancy flow (Fig. 5 c).
When the inner and outer cylinders rotate in the opposite directions two counter rotating
cells separated by a stagnation surface are formed. One is a circulating flow at the bottom of
the annulus caused by the outer cylinder rotation and the other is a circulating flow in the
opposite direction caused by the inner cylinder rotation appearing near the top of the annulus
Page 16
188
0
\
X0.5 ~~_j,
0 0.5
R
M. Venkatachalappa et al.
! o,,
0
X0.5
1
0
0.5
R
~i=2000,~o=2000
x02
11 , ,
0 0.5
R
x05 T/4 8
I
0 0.5
R
1
~i=1500,~o=1000
Fig. 6. Streamlines and isotherms for Gr = 10 4 and A = 1
(see Fig. 6 a). As the rotation decreases, the cells at the top become larger and the bottom cells
decrease in size. The larger "bulge" of isotherms towards the inner cylinder at the middle of
the cavity shows the presence of two cells. Near the hot wall isotherms remain mostly parallel
indicating that conduction is the major mode of heat transfer (see Fig. 6 b).
The temperature distribution and flow pattern in the annulus for various Prandtl numbers
are shown in Fig. 7. For low Prandtl number fluids, (Pr = 0.01) the temperature field is
almost independent of the fluid flow, since the heat is transferred mainly by conduction. The
temperature field and flow pattern remain unchanged for a low Prandtl number irrespective
of whether the inner cylinder is rotating or the outer cylinder is rotating (Fig. 7 a, b, d, e). But
for fluids with higher Prandtl number (Pr = 1.0), the temperature distribution depends
strongly on the fluid flow in the annulus. The temperature distribution in the annulus consists
of two types based on whether the inner wall is rotating or the outer wall is rotating. Inner
wall rotating pushes the isotherms upwards to the outer cylinder while the rotation of the
outer wall draws them downwards (Fig. 7 c, f). Also, for Pr = 1.0, the unicellular flow pattern
for inner wall rotation becomes bicellular for the rotation of the outer walt.
Page 17
Natural convection in an annulus
XO.5
0.5
R
(a) Pr = 0.01
XO.5
189
0 =
14
0.5
R
1
X0.5
0.5
R
(b) Pr = 0.1
0 0,8
X0,5 T
1 r r I
0 0.5
R
0 0
XO.5
XO.5
10 0.5
l~
1 10 0.5
R
1
(c) Pr = 1.0
Fig. 7ac. Streamlines and isotherms for Gr = 105, f2i 1 000, ;20 = 0
The influence of the aspect ratio on the flow pattern and temperature distribution is
shown in Figs. 8, 9 and 10. When the inner cylinder alone is rotating, a bicellular flow pattern
is observed for A = 2 whereas a unicellular flow pattern is found for the case of unit aspect
ratio. From the isotherms, it can be seen that the fluid in the middle of the cavity undergoes a
strong centrifugal acceleration, and the formation of thermal boundary layers at the side
walls indicates the presence of fairly strong convection (Fig. 8 a). As the aspect ratio is
increased to 2 for the outer wall rotation, the size of the main eddies becomes larger than that
of A = 1. Also, the isotherms oriented more towards the colder wall and the gap between iso
thermals are reduced considerably compared to the case of A = 1 which indicates the
enhancement of buoyancy driven convection (Fig. 8 b). In the case of both cylinders rotating
Page 18
190
0
X0.5
1;
0.5
R
1
(d) Pr = 0.01
M. Venkatachalappa et al.
0 ,% "~
X0.5 ~
1 0
0.5
R
1
X0.5
10
0,5
R
(e) Pr = 0.1
0
X0,5
10 O.5
R
1
0 0
X0,5 X0,5
1 ~
R
(f) Pr = 1.0
i0 0,5
R
1
Fig. 7df. Streamlines and isotherms for Gr = 105, X2i = 0, ~?o = 1000 (cont.)
in the same direction for A  2 and Gr = 106, a circulatory motion is set up since buoyancy is
the dominant force, and the flow consists of a symmetric single cell pattern filling the entire
cavity with the thermals spread away from the hot cylinder to the cold cylinder (Fig. 9). As
the cylinders are rotating in the opposite directions, a single cellular pattern with the centre of
rotation at the bottom is observed, and the flow covers only the lower diagonal of the cavity
which is also depicted in the isotherm pattern. The thermals mainly originate from the bottom
of the hotter cylinder and move towards the top adiabatic walls since the centrifugal accelera
tion is more pronounced on the fluid in the upper half of the cavity (Fig. 10 a). As the rota
Page 19
Natural convection in an annulus 191
0 0
X1
X1
i i
0 0.5
R

(a) ~i=2000, ~o=0
0 0
X1
2 2
0 0.5
R
1
(b) ~o=2000, ~"~i0
X1
Fig. 8. Streamlines and isotherms for Gr = 10 5 and A = 2
0 0.5
R
1
0 0.5
R
1
tional speed was decreased, the flow occupies the entire cavity and the thermals are more clo
ser to the hot wall (Fig. 10 b).
To investigate the effects of rotation of the side walls, radial, axial and swirl velocity pro
files at midheight are depicted in Fig. 11. The radial flow near the inner cylinder increases
(decreases) as the inner (outer) cylinder rotation increases. When the rotation of the outer
cylinder increases the radial velocity decreases near the inner cylinder which shows that the
Page 20
192
0
X1
0.5
R
X1
2
0
(a) ~i=1 000, ~o=2000
0
M. Venkatachalappa et al.
0.5
R
x1 x1
2 2
0 0.5
R
1 0 0.5
R
(b) ~i=1500, ~o=500
Fig. 9. Streamlines and isotherms for Gr  106 and A = 2
rate of outward bound flow is less than the rate of inward bound flow (Fig. 11 a). This is also
observed in the streamline pattern of outer cylinder rotation (Fig. 4ac). The suppression
(augmentation) of vertical velocity by the rotation of the outer (inner) cylinder is shown in
Fig. 11 b. Peak velocities are observed only near the vertical boundaries. The vertical velocity
increases near the inner cylinder and decreases along the axis of rotation and regains its velo
city near the outer wall due to buoyancy. The amplification and deamplification of the above
Page 21
X1
lO
0.5
R
Natural convection in an annulus
0
(a) ~i=2000,
X1
2
0
o=2000
0.5
R
193
lO
X1
2 0
0.5
R
X1
(b) i=1500, s
Fig. 10. Streamlines and isotherms for Gr = 10 s and A = 2
2 0
0.5
R
1
phenomena are determined by increase of the inner and outer wall rotation, respectively.
When the inner wall is rotating, the swirl velocity decreases rapidly near the inner wall since
the effect of rotation reduces as the fluid moves away from the wall and remains unchanged
for 0.2 < R < 0.8, but again decreases near the hotter cylinder as the buoyancy force causes
vertical flow. For the outer cylinder rotation case, the swirl velocity decreases exponentially
Page 22
194
15
I0
U 5
0'
5
 f2 = 500
 = 1000
~ = 2000 o*~
o
* o
o*
* %
0.5
R
1
(a) Radial Velocity
1
0
1
U2
3
M. Venkatachalappa et al.
4
5 0
& ~
~176
. . . . 015 . . . .
R
200
150
100
V 50
0
50
100 0

..
 fL= 500
= 1000
= 2000
oS**
*
s
so
"""   " "o
%*
L
o~
i
o15
:o.o,o*
R
(b) Axial Velocity
20
10
VO
10
0
~
 ~o = 500

*
= 1000
= 2000
0.5
R
20 I
Is 
15 I s * = 2000
10 {, s
t", **o ..................... ~
F/100
0.5
R
(c) Swirl Velocity
Fig. 11. Velocity profiles at midheight for Gr = 10 ~ and A = 1
10
F/100
2o
so*


~
g2o = 500
= 1000
= 2000 s.
oO
15
o n
o o
o* .''
.o* . ..  """
s .
00 . . . . 0.5
R
from the outer cylinder to the inner cylinder (Fig. 11 c). The temperature profiles for tuner
and outer wall rotation at the midheight are shown in Fig. 12. The temperature gradients
closer to the inner wall decrease and those closer to the outer cylinder are seen to increase
with the increase of inner cylinder rotation. With the increase of outer cylinder rotation the
profiles exhibit a straight line indicating a lower heat transfer rate. The transient variation of
Page 23
Natural convection in an annulus 19 5
0.5
T 0
0.5
~i = 500
 = 1000
o = 2000
DO000000000
 .
0
00000000
0.5
T 0
0.5
1 ~ 1
0 0.5
R
1 0
 &3~ e e'z~
 
o
= 1000
= 2000
~ , ~ /
 ~ o ~
o o
o o~
"'~176 o~176176
/ o o~
,St ~
,o6/~176
0.5
R
Fig. 12. Temperature profiles at midheight for Gr = 10 5 and A = 1
10
8
Nu
6
........ = 2000
. . . . .

= 1000
fJj= 500
10
6
Nu
rio = 500
....
100C
. . . . . . . 2~G
/',% "._. ..........................
0 0.04 0.08 0 0.04 0.08
Fig. 13. Transient variation of average Nusselt number with time
the Nusselt number Nu with the dimensionless time ~ for different rotational speeds is shown
in Fig. 13. The rate of heat transfer increases (decreases) with the increase of the inner (outer)
wall rotations, but in both cases the steady state is attained earlier for lower rotational speeds.
5 Conclusions
A numerical study has been performed to investigate the effects of buoyancy, rotation and
aspect ratio on the axisymmetric flow in a vertical cylindrical annulus with the cylinders rotat
ing at different angular velocities. Computational results reveal that the rate of heat transfer
at the hot cylinder is suppressed when its speed of rotation is higher than that of the cooler
cylinder. In the case of outer cylinder rotation at low speeds the outward bound flow is con
fined to a thin region along the bottom surface while the return flow occupies a major portion
of the annulus for moderate Grashof numbers. The temperature distribution of low Prandtl
number fluids is almost independent of the flow in the annulus, but it depends strongly for
Page 24
196 M. Venkatachalappa et al.: Natural convection in an annulus
high Prandtl number fluids. For the rotation of the inner cylinder, the unicellular flow pattern
for the case of unit aspect ratio changes to a bicellular pattern as the aspect ratio is increased
to 2.
Acknowledgement
The authors are thankful to Prof. N. Rudraiah, INSA Senior Scientist, for his valuable discussions. This
work was supported by UGC under DSA and COSIST programmes. One of the authors (MS) is thankful
to the Council of Scientific and Industrial Research (CSIR), New Delhi for providing financial assistance
through Senior Research Fellowship.
References
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Authors' address: M. Venkatachalappa and M. Sankar, UGCDSA Centre in Fluid Mechanics, Depart
ment of Mathematics, Central College, Bangalore University, Bangalore
Department of Electrical and Electronics Engineering, Vellore Engineering College, Vellore  632014,
Tamil Nadu, India
560001; A. A. Natarajan,
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