Effects of radial temperature gradient on the stability of a narrow-gap annulus flow

JOURNAL

OF MATHEMATICAL

ANALYSIS

AND

152, 156-175

APPLICATIONS

Effects of Radial Temperature Stability of a Narrow-Gap

(1990)

Gradient on the Annulus Flow

H. S. TAKHAR Department of Engineering, University Manchester Ml3 9PL, United

of Manchester, Kingdom

V. M. SOUNDALGEKAR 31AII2,

Brindavan

Society,

Thane 400601,

India

AND

M. A. ALI Gulf Polytechnic,

Bahrain Submitted Received

University,

Bahrain,

by E. Stanley March

Middle

East

Lee

6. 1989

An eigenvalue problem of the stability of Couette flow between two concentric cylinders in relative motion in the presence of a positive ( T2 2 T,) and a negative (T, < T,) radial temperature gradient is solved numerically, where T,, T2 are the temperatures of the inner and outer cylinders, respectively. The numerical values of the critical wave number a, and the critical Taylor number T, are calculated for kp (=Q,/D,, where Q, is the angular speed of the outer cylinder and Sz, is the angular speed of the inner cylinder) and i N (ratio of the Rayleigh and Taylor numbers). The radial eigenfunction and the cell patterns are shown graphically for different values of &p and f N. It is observed that the flow is more stable in the presence of a negative temperature gradient for both +p. The stabilising effect is more promising when the two cylinders are counterrotating. The magnitude of a, also increases steeply in the presence of counterrotating cylinders and a negative radial temperature gradient (T, < T,). 0 1990 Academic Press, Inc.

1. INTRODUCTION Soundalgekar, Takhar, and Smith [ 1] studied the effects of a radial temperature gradient on the stability of flow in a narrow-gap annulus whose inner cylinder was assumedto be rotating. A numerical solution to the disturbance equation was presented and the effect of the parameter N= Ra/T, 156 0022-247X/90

$3.00

Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.

157

RADIAL TEMPERATURE GRADIENT

where Ra is the Rayleigh number and T is the Taylor number, on the stability of the flow was discussed only for positive values on N, i.e., for T2 > T,. This is one of the physical situations. The case of T, < T, has not been studied. Also the effect of N on the stability of flow between two corotating or counterrotating concentric cylinders with narrow gap was studied by Takhar, Smith, and Soundalgekar [2] for N > 0. In both these papers, the cell patterns and the amplitude of the radial eigenfunction were not studied. Hence it is now proposed to solve the disturbance equations of Ref. [ 1 ] to study the effects of f N (T 2 >
2. MATHEMATICAL

ANALYSIS

The governing equations are derived in Ref. [ 11, in which all the assumptions are mentioned, and the numerical procedure is discussed in this paper. We have now solved the disturbance equation d6v z-3a

2 d4v d~4+3a~$-Ubv=

-u~T[~-.x+N(~-~x+x~)]v

(1)

under the boundary conditions 2

v=o,

d"=

dx2

a20= 0,

f!k-a2du=o

dx’

dx

at

x=0,

1.

(2)

Here a = Ad is the wave number, T= ( -4Ad452,/v2) is the Taylor number, Ra = Q:d3R,u(T, - T,)/vk is the Rayleigh number, N = RaJT, R, is the radius of the inner cylinder, A = (52, R2 - Szi R, )/( Rs - R f ), and all the other quantities are defined in Ref. [ 11. The numerical values of the critical wave number a, and the critical Taylor number T, are listed in Table I for both values of + N, where negative values of N correspond to T, < T, ; i.e.,

158

TAKHAR,

SOUNDALGEKAR, TABLE Values

AND

ALI

I

of a,, T,

1.0

2.0 1.5 1.0 0.5 0.0 -0.5

3.115 3.114 3.114 3.114 3.115 3.113

569.3 683.1 853.9 1138.6 1707.8 3415.7

-0.5

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 - 1.5

3.208 3.207 3.205 3.200 3.196 3.192 3.183 3.183

3244.1 3701.7 4309.6 5155.7 6413.8 8479.4 12482.9 23342.0

0.5

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0

3.117 3.118 3.118 3.117 3.115 3.114 3.118

902.7 1063.0 1292.6 1648.6 2275.2 3668.4 9401.6

-1.0

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5

3.362 3.452 3.588 3.779 3.999 4.214 4.400 4.570

6918.4 8452.1 10636.6 13828.1 18662.8 26584.4 41845.3 83138.4

0.0

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 - 1.5

3.137 3.136 3.135 3.132 3.129 3.122 3.116 3.184

1598.7 1892.3 2173.4 2649.1 3390.0 4700.8 7623.3 18724.6

the outer cylinder is maintained at a temperature lower than that of the inner cylinder. To check the accuracy of our numerical results, we compare one of our numerical values of a,, T, with those derived by Harris and Reid [14]. If a,, FC are assumed to be the wave number and the critical Taylor number due to Harris and Reid, then CT,, TC are related to our a,, T, by the relations a,=cS,(l

-p)

and

T,= T,(l -p)*.

Hence for ti,= 1.999, and TC= 1166.412, N=O, .D= -1.0, we have a, = 3.999 and T, = 18662.6 and the agreement is very good. The numerical values of T, are plotted in Fig. 35 against p for N= -0.5, 0, 0.5. We observe from this figure that when the outer cylinder is maintained at a temperature lower than that of the inner cylinder, the flow is more stable when the two cylinders are in relative motion. Hence the flow inside the

RADIAL

TEMPERATURE

GRADIENT

159

annular passage of two concentric cylinders in relative motion can be maintained in a laminar state by raising the temperature of the inner cylinder higher than that of the outer cylinder. This conclusion is the most important one from the points of view of industrial applications. To be more precise, we now give percentage changes that take place in the value of T, for different values of p and N. In Table II, these changes are calculated on the basis of those for the case N =O. Thus for corotating cylinders, p = 0.5, T, > T, (N > 0), when N changes from 0 to 0.5, there is a 27.66% rise in the value of T, whereas when N changes from 0 to 1.0, there is a 43% rise in the value of the critical Taylor number. But if T2 < T, (N < 0), when N changes from 0 to -0.5, there is a 61.23% rise in the value of T, and for N changing from 0 to - 1.0 it is observed to rise by 313.23% When p = 0.0 and N ( > 0) changes from 0.0 to 0.5 or 1.0, there is respectively a 21.85 or 35.89% rise in the value of T, and when N (CO) changes from 0.0 to -0.5 or -1.0, there is respectively a 38.64 or 124.86% rise in the value of T,. In the same manner, in the presence of counterrotating cylinders, p = -0.5, when N ( >O) changes from 0.0 to 0.5 or 1.0, there is respectively a 19.61 or 32.13 % rise in the value of T, and when N (~0) changes from 0 to -0.5 or - 1.0, there is respectively a 32.13 or 94.63 % rise in the value of T,. However, for large values of N, say - 1.0, the solutions do not converge in the presence of corotating which lead us to predict that the flow may not remain laminar. TABLE

II

Percentage Changes in the Values of T, P

N

0.5

0.5 1.0 -0.5 -1.0

21.66 43.20 61.23 313.23

0.0

0.5 1.0 -0.5 -1.0

21.85 35.89 38.64 124.86

0.5 1.0 -0.5 -1.0

19.61 32.13 32.13 94.63

-0.5

%

160

TAKHAR,

SOUNDALGEKAR,

We now study the effects of the radial

AND

ALI

temperature gradient NG 0 of the radial eigenfunction from Figs. l-36. In Fig. 1, the cell pattern for tj = 0.5 is shown for the case of corotating cylinders. We observe from this figure that the cell patterns for N= - 1.0, 0.0, 1.0 coincide and thus the radial temperature gradient has no effect on the cell pattern distribution when both the cylinders are corotating. Figure 2 shows the cell pattern distribution when both the cylinders are corotating. In Fig. 2, the cell pattern for $ = 0.5 is shown for N = kO.5,O.O when the inner cylinder is rotating (p = 0). The effect of a positive and negative radial temperature gradient is seen here. Thus for N> 0, the cell is found to shrink slightly and for NC 0, the cell is found to dilate as compared to the case of N = 0. In Figs. 3 and 4 the cell patterns are shown for N = + 5,0 when the two cylinders are counterrotating. The most important conclusion we have from these figures is that the cells are shifted toward the inner cylinder and when N> 0, these cells are shifted more toward the inner cylinder and when NC 0, the cells are shifted toward the outer cylinder. Thus the radial temperature gradient does affect the cell pattern through the shifting of the cells toward or away from the inner cylinder according as N> or 0. These cells are found to be symmetrically situated in the annular passage between two cylinders. When the temperature of the inner cylinder is raised ( T2 < T,) such that N changes its value from -0.5 to - 1.0, and the two cylinders are corotating, the portion of the cells near the outer cylinder starts contracting toward the inner rotating cylinder and compared to the case of N= 0, the right-hand region is shifted more toward the inner rotating cylinder as shown in Figs. 8 and 9. In Figs. 10-12, the cell patterns are shown for p = 0, i.e., only the inner cylinder rotating with the outer one stationary and for N 3 0 ( T2 2 T,). For N = 0, these cells are almost symmetrical but when the temperature of the outer rotating cylinder is raised such that N changes from 1.0 to 2.0, the edges of the cells near the outer rotating cylinder start to become straightened and thus are slightly contracted toward the center of the rings. In Figs. 13 and 1.4, the cell patterns are shown in the case of only the inner cylinder rotating and N < 0 ( T2 < T,). Thus when the temperature of the inner rotating cylinder is raised such that N changes its value from - 1.0 to - 1.5, the cells start shifting toward the outer cylinder. Physically this is true because the convection currents are moving from the inner to the outer cylinder; thereby the right-hand edges of the cells start to become closer toward the outermost cell. Also the right-hand edge of the outermost cell is straightened and the corners are formed at the upper and the lower ends of the outermost edge of the cell, and if the temperature is raised

(T2 3 T,) on the cell pattern and the amplitude

161

RADIAL TEMPERATURE GRADIENT 05r

:

:

:

:

:

:

:

:

a

0 4 "

03. N. -10, 0, 1.0

01 z O0 -0.1

-02 -0 -05O*:;3 -04

,i_-,;

00

01

02

0.3 04

0.5,06

07

08

:

:;

09

10

FIG. 1. The cell pattern at the onset of instability for I) = 0.5 and p = 1.0.

0 3 ”

0.2 '. 01 zoo -01

.'

"

-02 'L -03"

::;l.i 0.0

01

02 03 04 OS,06

07 08 09

10

FIG. 2. The cell pattern at the onset of instability for I) = 0.5 and p = 0.0. 043

.

.

.

*

0 3 '. N--O5

z

0.1

0

00

0.5

-002:; 1 -02

o-

:;

-0 3 -0.4.1 0.0

FIG.

: 01

02

:

:

:

03

0.4

05

: O6xO7

:

:

T

08

0.9

10

3. The cell pattern at the onset of instability for (I/= 0.7 and p = - 1.0.

162

TAKHAR,

-044 0.0

SOUNDALGEKAR,

:

:

:

:

:

01

0.2

0.3

04

05

AND

:

:

ALI

:

06,07

: f

08

09

1.0

FIG. 4. The cell pattern at the onset of instability for $ = 0.5 and p= -1.0.

ost

:

:

1

;

0.0 01

02

03

04

:

;

:

;

:

07

08

09

t

03 02 01

I

00

z

-0 1 -02 -rl?

FIG.

I

OS,06

10

5. The cell pattern at the onset of instability for p = 1.0 and N= 1.0.

0

04,. 0 3 "

/

’

0 2 '. 0 1 " z 0.0 " -01.. -0.2 -03,.

.5-

-06..

-0

,

00

FIG.

01

0.2

03

OL 0.5,06

0.7 OS 09

1.0

6. The cell pattern at the onset of instability for fi = 0.5 and N= 1.0.

163

RADIAL TEMPERATURE GRADIENT

Ok 1. 03 02

"

01 z

00 " -01

.'

-02. -03" -Ok. 00

01

02

03

04

05,06

07

08

09

10

FIG. 7. The cell pattern at the onset of instability for ~=0.5 and N =O.O.

00

01

02

03

04

05,0.6

07

0.8

09

1.0

FIG. 8. The cell pattern at the onset of instability for p = 0.5 and N = -0.5.

FIG. 9. The cell pattern at the onset of instability for p = 0.5 and N = - 1.0.

TAKHAR,

00

FIG.

10.

SOUNDALGEKAR,

01

The cell pattern

02

03

04

AND

05

0.6,07

ALI

0.8 09

at the onset of instability

10

for p = 0.0 and N=

2.0.

fLi!l 0.8

0.95

0 fl.0

\

00

FIG.

11.

The cell pattern

01

02

03

04

05,06

07

08

at the onset of instability

09

IO

for p =O.O and N=

1.0.

0 0 41, 0 3 ” 0 2 ” 0 1 ” 2

00

-0 1 ,’ -0 2 ” -03

‘.

-04

,’

-0 .s

FIG.

12.

0.0 0.1 02

The cell pattern

03 0.4 05

06,07

at the onset of instability

08

09

10

for p =O.O and N=O.O.

165

RADIAL TEMPERATURE GRADIENT

o.=+

: . ; :

: :

0.4.. 0 3..

06 08

0 2..

1

0 95

0 1.

=l.0

zo.o-. -0

04

;

0 O!i

02

0

l-.

0

-0 2--0 3.. -0 4.. -0 5c 0.0I

xx

6.1

Q2 03 0.4 0.Sx06 0708

09

-i 10

FIG. 13. The cell pattern at the onset of instability for p = 0.0 and N = - 1.0.

0.4 03 02 01 z 0.0 -01 -0

2

-0 3 -0

4 L

0'0

01

02 d.3 0.4 0.Sx0.6 0.7 0.8 09 I.0

FIG. 14. The cell pattern at the onset of instability for p = 0.0 and N = - 1.5.

DO

0.1

02

03

0.4

05,06

07

08

09

10

FIG. 15. The cell pattern at the onset of instability for p= -0.5 and N = 2.0.

166

TAKHAR,

SOUNDALGEKAR,

AND

AL1

further such that the value of N increases beyond - 1.5, the cells will start breaking through these corners, making the flow unstable. These predictions will help an experimentalist to observe the breaking phenomenon of the cells visually. In the presence of counterrotating cylinders, with N > 0 ( T2 2 T,) the cell patterns are displayed in Figs. 15-18. In this case, the convection currents are flowing from the outer cylinder toward the inner one, and hence the shifting phenomenon is seen from the outer toward the inner cylinder. The formation of corners along the edges of the cells is seen near the inner cylinder and the cells will start breaking near the inner cylinder and the cells will start breaking near the inner rotating cylinder through these corners. The cell patterns displayed in Figs. 19-21 are for the case when the two cylinders are counterrotating, ,U< 0 with NC 0. We observe from these figures that due to an increase in the temperature of the inner cylinder, the cells are shifted toward the inner cylinder and if T, is raised such that N= - 1.5, the cells are reduced in size as compared with those at N= - 1.0 or -0.5. Also, formation of corners of the cell $ = 0.05, near the outer cylinder at N= -0.5, is found to vanish with the increase in the temperature of the inner cylinder such that N= - 1.5. In Figs. 22-26, the cell patterns are displayed when the two cylinders are counterrotating with the same speed and N > 0 is increasing from 0.0 to 2. At N= 0, i.e., in the absence of a radial temperature gradient, cells are shifted toward the inner rotating cylinder and also squeezed in size. But when the radial temperature gradient exists due to T2 > T1, the cells start to become elongated toward the outer cylinder. The spacing between the two consecutive cells is also widened on the right-hand side of cells which lies toward the outer cylinder, and at large values of N> 1, formation of corners is seen. The cell may be broken through these corners if N is increased beyond 2 from which we may conclude that the flow may become unstable. However, the opposite effect is seen when the cylinders are counterrotating with the same speed and when a negative temperature gradient is imposed, i.e., when N < 0 ( T2 < T,). Thus in Figs. 27-29, we observe that the cells start to become squeezed when N takes values from -0.5 to - 1.5, i.e., when the temperature of the inner cylinder is raised above that of the outer cylinder. In Figs. 3CL33, the radial eigenfunction u(x) is shown for N$O and ,U= - 1.0, 0.0, and 1.0. We observe from these figures that for N= 0 due to corotation of the two cylinders, the maximum of U(X) shifts toward the outer cylinder as compared to the case of ,U= 0, whereas in the presence of counterrotating cylinders the maximum of U(X) shifts toward the inner cylinder, but in the presence of a positive radial temperature gradient,

167

RADIAL TEMPERATURE GRADIENT

FIG.

16.

The cell pattern at the onset of instability for p = -0.5 and N= 1.0.

00 01 0.2 03 0.4 OS,06 FIG.

FIG.

07 08 09

1.0

17. The cell pattern at the onset of instability for p = -0.5 and N= 0.5.

18. The cell pattern at the onset of instability for p = -0.5 and ~~0.0.

168

TAKHAR,

SOUNDALGEKAR,

AND

AL1

0 G ” 0.3 " 0 2 .* 0 1 .' 00.

z

-0.1

,'

-0 2,, -03" -0.4.t 00

FIG.

19.

The cell pattern

01

0.2

03

O4O.5xO6

07

at the onset of instability

OS 09

IO

for p = -0.5

and N=

-0.5.

and N=

- 1.0.

and N=

- 1.5.

04 0.3 0.2 0.1 0.0 I -0.1 -0.2 -0.3 -0.4 0.0

FIG. 20.

The cell pattern

00

FIG.

21.

The cell pattern

01

0.2

0 3 04

0 Sx06

07

at the onset of instability

01

02

0.3

0.bx05

0.6

08

for p = -0.5

0.7 0.8

at the onset of instability

09

09

10

for p = -0.5

RADIAL

TEMPERATURE

169

GRADIENT

0.4 ” 0 3 ” 0 2 " 0 1 '. z 0 0 " -01

.'

-0 2 " -03

"

-04..

FIG.

22.

The cell pattern

00

FIG.

23.

The cell pattern

at the onset of instability

01

0.2

03

Of+,05

06

for p = - 1.0 and N = 2.0.

07

08

09

10

at the onset of instability

for p = - 1.0 and N = 1.5.

at the onset of instability

for p = - 1.0 and N = 1.0.

OLl 03" 02 01

"

1 0 0 " -01

"

-02

"

-03. -0 4 "

FIG.

24.

The cell pattern

170

TAKHAR,

SOUNDALGEKAR,

AND

ALI

0 3 .' 0 2.. 0 z

.'

1

0 o-0

.'

1

-02.. -0 3.. 00 FIG.

25.

The cell pattern

00 FIG.

26.

27.

01

The cell pattern

00 FIG.

01

01

The cell pattern

02

03

Oh,05

06

07

08

at the onset of instability

0.2 03

04

05

06

03

01.

OSx 0.6

at the onset of instability

1.0

for p = - 1.0 and N = 0.5.

0.7

0.8 09

at the onset of instability

02

0.9

1.0

for p = - 1.0 and N = 0.0.

07

08

09

10

for p = - 1.0 and N=

-0.5.

RADIAL

00

FIG.

FIG.

28.

29.

01

02

The cell pattern

30.

t

:

:.

00

01

02

The cell pattern

The radial

03

03

04x0.5

06

07

02

eigenfunction

08

09

:

t

:

r

:

04

x 05

06

07

08

09

03

04

u(x)

05

06,07

IO

for p = - 1.0 and N = - 1.0.

:

at the onset of instability

0.1

171

GRADIENT

at the onset of instability

00

FIG.

TEMPERATURE

1.0

for p = - 1.0 and N = - 1.5.

08

09

10

for N = 2.0 and different

values

of p,

172

TAKHAR,

00

FIG.

31.

The radial

SOUNDALGEKAR,

0.2 03

01

eigenfunction

04

u(x)

AND

05,06

0.7

ALI

08

09 10

for N = 1.0 and different

values

of p.

values

of p.

10

09 08 07 06 05 04 03 02 01

00 -0 1

FIG.

32.

The radial

01

02 03 04

eigenfunction

u(x)

OSx06

07

for N = 0.0 and different

10

09 08 07 06 05 u Ir :I_ o4 03 02 01 00 -0 1

FIG.

33.

The radial

eigenfunction

u(x)

for N = -0.5

and different

values

of P

RADIAL

TEMPERATURE

173

GRADIENT

10 09 08 07 06 U(Xl

05 04 03 02 01 00 -01

-02 FIG.

34.

The radial

eigenfunction

u(x)

for p = - 1.0 and different

values

of N.

N > 0, the maximum of U(X) shifts more and more toward the outer cylinder as N increases. In the presence of a negative temperature gradient, N= -0.5, the maximum of u(x) shifts more toward the inner cylinder. In Fig. 34, U(X) is shown for p = - 1.0 and N = + 1.0, 0.0, and - 1.0. The variation of u(x) with N is quite remarkable when x > 3.0 and U(X) is found to increase with increasing N. In Fig. 36, the critical wave number a, is plotted against p for different values of N. It is interesting to note that when the two cylinders are corotating there is a rise in the value of a, with increasing N > 0 (T, > T,), i.e., when the temperature of the outer cylinder is higher than that of the 30

FIG.

35.

Critical

T

Taylor

number.

174

TAKHAR,SOUNDALGEKAR,

ANDALI

-1

-A

-1 0

-05 FIG.

1.5

0 36.

Critical

/1 wave

0.5

ZN

10

number.

inner cylinder. But when N < 0 ( T2 < T, ), i.e., when the temperature of the inner cylinder is higher than that of the outer cylinder and the two cylinders are counterrotating, there is a steep rise in the value of the critical wave number and the steepness is more when the temperature of the inner cylinder is higher.

3.

CONCLUSIONS

It is observed that the flow is more stable in the presence of a negative radial temperature gradient (T2 < T,), when the two cylinders are either corotating or counterrotating. However, this stabilizing effect is more promising when the cylinders are counterrotating. The critical wave number also increases steeply in the presence of a negative temperature gradient and when the two cylinders are counterrotating.

RADIAL

TEMPERATURE

GRADIENT

175

REFERENCES 1. V. M. S~UNDALGEKAR, H. S. TAKHAR, AND T. J. SMITH, Warme Stoffubertragung 15 (1981), 233. 2. H. S. TAKHAR, T. J. SMITH, AND V. M. SOUNDALGEKAR, .I. Mafh. Anal. Appl. 111 (1985), 349. 3. H. AOKI, H. NOHIRA, AND H. ARAI, Bull. JSME 10 (1967), 523. 4. S. L. BAHL, Trans. ASME Ser. E J. Appl. Mech. 39 (1972), 593. 5. K. M. BECKER AND J. KAYE, Trans. ASME Ser. C J. Heat Transfer 84 (1962), 1061. 6. I. S. BJORKLUND AND W. M. KAYS, Trans. ASME Ser. C J. Hear Transfer 81 (1959), 175. 7. S. CHANDRASEKHAR, Arch. Rational Mech. Anal. 3 (1954), 181. 8. F. C. HAAS AND A. H. NISSAN, “Third International Heat Transfer Conference Boulder, Colorado, 1961.” 9. C. Y. Ho, J. L. NARDACCI, AND A. H. NISSAN, Amer. Inst. Chem. Eng. 10 (1964), 194. 10. A. H. NISSAN, J. L. NARDACCI, AND C. Y. Ho, Amer. Inst. Chem. Eng. 9 (1963), 620. 11. C. S. YIH, Phys. Fluids 4 (1961), 806. 12. R. D. SHARMAN, I. CATTON, AND P. AYYASWAMY, Amer. Inst. Chem. Eng. Symp. Ser. 69 (1973), 118. 13. J. WALOWIT, S. TSAO, AND R. C. DIPRIMA, Trans. ASME Ser. E J. Appl. Mech. 31 (1964), 585. 14. D. L. HARRIS AND W. R. REID, J. Fluid Mech. 20 (1964), 95.