Page 1

Acta Mechanica 147, 173 196 (2001)

ACTA MECHANICA

y Springer-Verlag 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-

city-stream function form of the Navier-Stokes 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 crystal-growth 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]).

Buoyancy-driven 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 Wu-Shung 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 co-axial 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 + O-xSx2' 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,

0--0;

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 Navier-Stokes 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 vorticity-stream 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

UT------0, --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 two-step 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 co-ordinate system there is an additional

1 Ok~.

term - ~ ~ m the expression for C compared to that in the rectangular co-ordinate 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 no-slip 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. 3a-e. 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. 3d-f. 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 buoyancy-driven 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. 3g-i. 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. 4a-i. 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. 4a-c. 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 d-f) 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

T--0,8

X0.5

10 0,5

R

Fig. 4d-f. 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. 4g-i. 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 vice-versa. 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