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A transient natural convection in an annular enclosure
0735-1933/91 $3.00 + .00 Printed in the United States
INT. COMM. HEAT MASS TRANSFER Vol. 18, pp. 373-384, 1991 ©Pergamon Press pie
A TRANSIENT NATURAL CONVECTION IN AN ANNULAR ENCLOSURE Wu-Shung Fu, Yi-Horng Jou Department of Mechanical Engineering National Chiao Tung University, Hsinchu, 30050, Taiwan 1~ O. C. Chien-Hsinng Lee Institute of Nuclear Energy Research, Lung-Tan, 32500, Taiwan K. O. C.
(Communicated by J.P. Hartnett and WJ. Minkowycz) ABSTRACT A transient natural convection in an annular enclosure is investigated numerically. The upper and lower walls are adiabatic, the outer wall is maintained at a low temperature and the temperature of the inner wall is changed to a high value abruptly. During the computing process, the numerical method of S I M P L E R with powerlaw scheme is adopted to solve the axially-symmetric governing equations of the continuity, momentum and energy. The Rayleigh number of 104 and lOs are considered, respectively. Since the area of the annular cross section enlarges with the increasing of the radius, the phenomena occurring in the annular enclosure are different from those occurring in the rectangular enclosure. In addition, the results show that in the initial region the variations of the total Nusselt of the inner wall rebounds under Rayleigh number of lOe case and does not fluctuate under Rayleigh number oflO' case, which are similar to those of the rectangular enclosure.
Introduction A study of transient natural convection in an enclosure is of interest and importance in many engineering applications, such as energy storage system, nuclear reactor equipment and solar energy collector etc. However, most of the studies [1]~[5] were mainly focused on the transient natural convection in the rectangular enclosure. As for the study concerning with the other geometry of the enclosure is comparatively less investigated. Chattee and Sengupta [6] studied effect of bottom slope on laminar natural convection in enclosures. The result showed that the position of the maximum Nusselt number on the heated wall was affected remarkably by the variation of the slope of the 373
374
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
bottom. Huang and Hsieh [7] investigated a natural convection in a cylindrical enclosure. Sun and Oosthuizen [8] investigated a transient natural convection in a cylindrical enclosure numerically and the results showed that when Rayleigh number was larger than 5x10,3 the average Nusselt number of the heated wall fluctuated at a certain time and the peak value decreased with the increasing of the aspect ratio. Aggarwal and Manhapra [9] studied a natural convection in a cylindrical enclosure and found out the Raylelgh number was larger than 5x10 4, the role of heat transfer mechanism was changed from the conduction to convection and the multicellular flow pattern occurred in the convection region. Concerning with the study of natural convection in an annular enclosure, Littlefield and Desal [10] examined temperature-induced buoyant convection in a uniform section annulus of large aspect ratio by similar transformations method and the results showed that the Nusselt number is dependent on both the boundary conditions and the geometry. Lee et.al. [11]~[14] investigated the hydrodynamics stability problem in an annular enclosure, the flow pattern was divided into several types. Weldman and Mehrdadtehranfar [15] conducted an experiment to examine the hydrodynamics stability of large aspect ratio. However, with regard to the study of transient natural convection in an annular enclosure is hardly found out. The aim of this study is to investigate the transient natural convection in an annular enclosure numerically. The numerical method of S I M P L E R with power law and T.D.M.A schemes are adopted to solve the governing equations. Preliminarily, in order to examine the difference of the transient natural convection between in a square enclosure and an annular enclosure, the height and the length (from inner radius to outer radius) of the annular enclosure are equivalent. The Raylelgh numbers of 10 4 and 10 8 are respectively considered to investigate the phenomena of conduction dominant situation and convection dominant situation. In addition, the variations of stream lines, isotherms and local Nusselt number along the walls are also examined in detail.
Vol. 18, No. 3
CONVECTION IN AN ANNULAR ENCLOSURE
375
Physicalmodel and Numerical method Schematic model of an annularenclosureis shown in Fig. 1 The inner and outer radii are r i and ro, respectively, and the height is L. The height L equals to the distance from r i to r o. The upper and lower walls are adiabatic. When the time t=0, the fluid, Prandtl number of which is 0.7, in the enclosure is stationary and at the same temperature T m as those of the inner and outer walls. As t>O, the temperature of the inner wall rises
/
r
/-
~--I'--I.-~,~ I I
I
I
I
I
"~,
\
to a high temperature T w and the temperature of the outer wall is still maintained at T . Due to the
X
w
buoyancy force, the fluid starts to flow. In order to simplify the analysis, the following assumptions are
J
[[=o --
"///////~ ~//////.
made.
(I) The fluid is Newtonian, and the flow is laminar.
7/////,/)('///////
(2) Boussinesqapproximationis valid. According to the physical model
"i C, r
and
assumptions, the dimensionlessgoverning equations
Fig.1. A S c h e m a t i c Model
pre expressedas eqs. (1).
~x,(RU)+~-(RV)--0
(la)
~_q.~.[~x~RUV)+~.(RVV)]___SP Pr8 Z •~@V 8V V~ ~r+ ~I-~[~X,( ) - ~8 R •~K)]-P r•R-
(lc)
376
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
(ld)
The dimensionlessvariablesused in Eqs. (I) ate defined as Eq. (2).
X=
x ro '
U = u . ro ~
P"P®
P
Pa3/r2o
Pr = v/a,
a.t
R -r -- ro ' V=
v'l~°
a
r = r-~o , '
(2)
N = ri ro
L A = ro RR =
0 = ( T - T ® ) / ( Tw---T®) ,
g~(Tw-T®)roS v.a
The boundary conditions are shown in Eqs. (3a)~(3d)
r=O,
u=v=s=-0
(3a)
r>O,
R=N,
U=V=0,
R=I,
U=V=0=0
X=O and X = A ,
U=V=O,
0=-1
(3b) (3c)
~,=0
(3d)
The SIMPLE--R algorithm [16] with power law scheme is adopted to solve the governing equations, according to the numerical test, the staggered grid system [17] and 32x32 uniform grids are utilized during the computing process. The time increments for Rayleigh numbers of 104 and 106 case ate 10 -3 and 10 -4, respectively.The convergent condition at every time step is defined as I
$.,
¢~. i
I < 10-3, and the steady state
I < 10"3,in which $ = U, V, P, T. condition for the flow is defined as [ ~r+A ~ r +r--~r Ar
Results and discussion The local Nusselt number NUr,x along the every annular cross section and the
Vol. 18, No. 3
CONVECTION IN AN ANNULAR ENCLOSURE
377
average Nusselt number I~ r on the every annular crosssectionare defined as the following equations.
hr
p-Cp.r-v. (T--T®)-k. r . ~ r
NUr,x = ~-- =
k. {T.--T® )
N'Ur = ~ ~ Nut,x dY
Ca)
The stream function 9 is defined as Eq. (6).
=u,
= v
(8)
The figures shown in Fig. 2 (e) and Fig. 3 (e) are the steady state situation for Ra----104 and 106 cues, respectively. Since the area of the annular cross section increases with the increasing of radius, then the heat flux q (w/m 2) through the annular cross section decreases with the increasing of radius. As a result, the temperature ~adient along the inner wall is larger than that along the outer wall. In addition, the distribution of the isothermal lines is not similar to that, which is symmetric to the diagonal, of the square enclosure [5]. As for the distribution of the stream lines, which is not symmetric to the diagonal too. Fig. 2 (a)~(e) and Fig. 3 (a)~(e) illustrate the transient variations of the isothermal lines and stream lines for Ra=10 4 and 10e cases, respectively. In the initial region, the fluid absorbs the heat transferred from the inner wall, and the isothermal lines not only gather densely near the inner wMl but also are parallel to the inner wall. With increasing of time, the flow of the fluid become quick and the trend of the flow can be predicted approximately by the movement of the position of the maximum stream function value. Due to the domination of heat conduction, the difference of the distributions of the isothermal and stream lines in the an~ular and square enclosures is not remarkable.
378
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
(a) "r=O.O02 O.OIEU
x: --0.088"~
(b) "r=O.010 0.0 x: - - 0 . 4 8 8 9 6
(c) "r=0.068 x : - 1.75236
(d) ~=0.094 0.0
0
x:-1.73159
°:~11/ (e)
X
•7"=0.436
o.o
0.2
(ri)
R
]L.O
(ro)
Isothermal lines
0.2
R
(ri) ×:-1.81393
1 .o
(ro)
~ r e ~ m 1in es
Fig. 2 T r a n s i e n t d e v e l o p m e n t s of i s o t h e r m a l l i n e s a n d s t r e ~ , ' n l t , ~ e s of a n A n ~ u l a r e n o l o s u r e f o r Ra = 10 4
Vol. 18, No. 3
CONVECTION IN AN ANNULAR ENCLOSURE
'r
0.0002
0.0E
I"=0.002 O.C
x: - 8 . 5 9 6 4 1 0.8
(o) "r=O.O06
O.Oll
l x: - g . T g ~ 4
? G)
(d) 1"=0.016 x: -6.51445
(e) "r:0.1459
O.OBL 0.2
(rl)
R
J 1.0
(ro)
0.2
1.0
(ri)
R
(ro)
x: - 7 . r , . ~
Isothermal lines
StroAn'dJne8
Fig. 3 Transient d e v e l o p m e n t s of i s o t h e r m a l l i n e s a n d s t r e R n ' ~ l t ' n e s of a n A n m d ~ e n c l o s u r e for l ~ = 10 e
379
380
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
In Fig. 3, generally the transient variations of the isothermal lines and stream lines for Ra=106 case are rather different from those of Ra=104 case. Since the convection effect is strong for Ra=10e case, the phenomenon of the position of the m a x i m u m stream function
moving from near the inner wall region to near the outer wall region is different from that in the square enclosure [5] significantly. Besides, the low temperature region is much larger than that in the square enclosure,
o.8
I
A
Shown in Fig. 4 (a) and (b), there are the variations of the local Nusselt numbers NUr,x along the inner and outer walls for
X
a.~-O.O02 B:'r-O,OOe C:~'-O.OIO
0.4
D:-r-O.Ol6 F~--O.O~9 F:.'r=O.07§
Ra=104 case. Initially, the conduction is dominant, then the NUr,x (r=ri) (curve A) is almost constant. With increasing of time,
I 0,0
I
0.0
0.8 ~
,
,
,
2.5 Nu r,x (r~ri) (a)
the fluid begins to flow which causes the NUr,x (r=ri) in the lower region to be larger
G."r=St~d~ State (o.4se) 5.0
Z
than that in the upper region. The variations of the NUr,x (r=ro) is opposite to that of the NUr,x (r--ri)" Initially, most of the heat transferred in from the left wall is absorbed
A.1- 0,029 ~v=o.o6o ~--o.ov6 ~-=o, m o
:~ 0.4 ~ I I II / / If/I
by the fluid, then the NUr,x (r=ri) is very minute. With increasing of time, the fluid begins to flow which causes the NUr, x (r=ro) in the upper region to be larger than that of
0.0 ~ 0.0
2.6 Nur.x (r.=ro)
5.0
(b)
Fill. 4 Looal Nmme/t numbe~ ~ both thiner and m~tm" walls of an ,,~M,I.~ ~ fro-Ra = I04 The variations of the local Nusselt number NUr, x along the inner and outer walls for
the lower region.
Ra--106 case are shown in Fig. 5. In general, the variations of the NUr, x for Ra--104 case are similar to those for Ra=104 mentioned above. But the convection is dominant for Ra=106 case, the NUr, x (r=ri) fluctuates in the initialregion which causes the value of the
Vol. 18, No. 3
CONVECTION IN AN ANNULAR ENCLOSURE
381
curve E to be larger than that of the curve D [Pig. 5 (a)]. This phenomenon is also found out in the square enclosure [5]. !1
0.8
10.0
A: 1"-0.000:
I n n e r wall Outer w a n
B:
T-0.00~ C:
1",-0.001 D: T-0.0014
~4 0.4
5.0
g:
T-O.002 le: I"-0.018
(o.14~g)
0.0
'
'
,
,
0.0
x
•
I
8.0
Nur~ (r-rl)
0.0
I
12.0
~ . . . . . . ,,I . . . . . . . . L~,~ ..... 10~ 10"e 10-[
~0.959
Time (T/ r.)
(a)
(")
0.8
15.0 Inner wldl OuLerwell
10.0 ~4
0.4 W'////
V
o.o) |
!
0.0
J
!
B:~"0.010 O: ~ , , 0 . 0 1 8
.
,
]~T-0.029 wT-0.050 I P:.'r-Bt_-m_~yI
5.0
~3.479 /
a
, , , ,
8.0
12.0
Nur~ (~-ro)
(b) le~. 6 Lomd N u m e l t n u m b e ~ elon4j b o t h fnnmr m:d outm- ~ o f aLn ~ u l ~ r mz~immre f o r 1~ = 10 e
0.0
f
...... J , ,,,,,,d , ,,,,,,d , ,,,,
lO-"
lO-' 10-' 10-' Time ('r/~)
1
(b) Fig. 8 Tr,-,,d,-,t developmeat~ of average Nmmelt n u m b e r s f o r P~ ,. 104 a n d 1~ = 101
Shown in Fig. 6 , there are the variations of the average Nnsselt number N~ r on the inner and outer walls for Ra=104 and 106 cases, respectively. In the initial region, the N'~r on the inner wall decreases rapidly, and after the time (r/~'s) equaling to 10 .2 the N~ r is almost invariant. At a certain.time after the beginning, the N~ r on the outer wall appears for both Ra=104 and 10e cases. Both the N~ r on the inner and outer walls are nearly equivalent at about the time (r/rs) of 0.4, which means that the transient phenomenon mainly occurs in the former 40% time of the whole transient time.
382
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
Conclusion The numerical method of S I M P L E R
is used to investigate transient natural
convection in an annular enclosure. The following conclusions are drawn. (1)
Since the area of annular cross section increases with the increasing of radius, the distributionsof isothermal lines and stream lines are not symmetric to the diagonal.
(2)
After the time 7-/7s = 0.4, the average Nusselt number of both the inner and outer walls are nearly equivalent for Ra=10 4 and 10 s cases.
Nomenclature A
ratio between length of annular enclosure and outer radius (L/to)
Cp
specific heat, J/kg. K
g
gravitational acceleration, m/s 2
k
thermal conductivity, W/re. K
L
length of the annular enclosure, m
N
radius ratio (ri/ro)
Nu
Nusselt number
p
pressure, pa
P
dimensionlesspressure
Pr
Prandtl number
r
radius coordinate, m
R
dimensionlessradius coordinate
Ra
Rayleigh number ( gfl(Tw--T®)roSfl,a )
t
time, s
T
temperature, K
u,v
velocity in x and r direction, m/s
U,V
dimensionless velocity in x and r direction
x
axial coordinate, m
(u/a)
Vol. 18, No. 3
X
C O N V E C T I OIN N AN ANNULAR ENCLOSURE
383
dimensionless axial coordinate thermal diffusivity, m2/s thermal expansion coefficient, 1/K
0
dimensionless temperature
v
kinematic viscosity, m2/s
p
density, kg/m s symbol represents U, V, P, 8 dimensionless stream function
~-
dimensionless time
A~-
increment of dimensionless time
Subscripts i
value at inner radius
o
value at outer radius
s
steady state
w
value at the wall of inner radius
®
ambient value
Superscripts n
iteration number average value
References
1o
J. Patterson and J. Imberger, J. Fluid Mech. ~
2.
G. De Vahl Davis, Int. J. Numer. Mech. Flui& ~, 249 (1983)
3.
V. F. Nieolette, K. T. Yang, and J. R. Lloyd, Int. J. Heat Mass Transfer 28, 1721
.
5.
85 (1980)
(1985) J. D. Hall, A. Bejan and J. B. Chaddock, Int. J. Heat & Fluid Flow 9, 398 (1988) W. S. Fu, J. C. Perng and W. J. Shieh, Numerical Heat Transfer, Part A, 16,
384
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
325 (1989) 6.
M. Chattree and S. Sengupta, Natural Convection in Enclosure - 1986. ASME HTD
-v•l. 63, 33 (1986)
7.
D.Y. Huang and S. S. Hsieh, Numerical Heat Transfer, 12, 121 (1987)
8.
J. Sun and P. H. Oosthuizen, ASME uaDer 88--HT-107. ASME Summer Annual Meeting, 103 (1988)
9.
S.K. Aggarwal and A. Manhapra, Numerical Heat Transfer, Part A, 15, 341 (1989)
10.
D. Littlefield and P. Desai, J. Heat Transfer, 108, 814 (1986)
11.
Yee Lee and Seppo A. Korpela, J. Fluid Mech., 12~, 91 (1983)
12.
Yee Lee, Seppo A. Korpela, and Roland N. Home, PrQ¢. Qf ~b¢ 7th Int. H¢~t Transfer Conference vQ1. 2, 221 (1982)
13.
S.A. Korpela, Yee Lee, and J. E. Drummond, J. Heat Transfer, 104, 539 (1982)
14.
Inn. G. Choi and Seppo A. Korpela, J. Fluid Mech., 99, 725 (1980)
15.
P.D. Weidman and G. Mehrdadtehranfar, Phys. Fluids, 28, 776 (1985)
16.
S.V. Patankax, Numerical Heat Transfer and Fluid Flow, McGraw - Hill, New York (1980)
17.
F.H. Harlow and J. E. Welch, Phys. Fluids, ~, 2182 (1965)
INT. COMM. HEAT MASS TRANSFER Vol. 18, pp. 373-384, 1991 ©Pergamon Press pie
A TRANSIENT NATURAL CONVECTION IN AN ANNULAR ENCLOSURE Wu-Shung Fu, Yi-Horng Jou Department of Mechanical Engineering National Chiao Tung University, Hsinchu, 30050, Taiwan 1~ O. C. Chien-Hsinng Lee Institute of Nuclear Energy Research, Lung-Tan, 32500, Taiwan K. O. C.
(Communicated by J.P. Hartnett and WJ. Minkowycz) ABSTRACT A transient natural convection in an annular enclosure is investigated numerically. The upper and lower walls are adiabatic, the outer wall is maintained at a low temperature and the temperature of the inner wall is changed to a high value abruptly. During the computing process, the numerical method of S I M P L E R with powerlaw scheme is adopted to solve the axially-symmetric governing equations of the continuity, momentum and energy. The Rayleigh number of 104 and lOs are considered, respectively. Since the area of the annular cross section enlarges with the increasing of the radius, the phenomena occurring in the annular enclosure are different from those occurring in the rectangular enclosure. In addition, the results show that in the initial region the variations of the total Nusselt of the inner wall rebounds under Rayleigh number of lOe case and does not fluctuate under Rayleigh number oflO' case, which are similar to those of the rectangular enclosure.
Introduction A study of transient natural convection in an enclosure is of interest and importance in many engineering applications, such as energy storage system, nuclear reactor equipment and solar energy collector etc. However, most of the studies [1]~[5] were mainly focused on the transient natural convection in the rectangular enclosure. As for the study concerning with the other geometry of the enclosure is comparatively less investigated. Chattee and Sengupta [6] studied effect of bottom slope on laminar natural convection in enclosures. The result showed that the position of the maximum Nusselt number on the heated wall was affected remarkably by the variation of the slope of the 373
374
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
bottom. Huang and Hsieh [7] investigated a natural convection in a cylindrical enclosure. Sun and Oosthuizen [8] investigated a transient natural convection in a cylindrical enclosure numerically and the results showed that when Rayleigh number was larger than 5x10,3 the average Nusselt number of the heated wall fluctuated at a certain time and the peak value decreased with the increasing of the aspect ratio. Aggarwal and Manhapra [9] studied a natural convection in a cylindrical enclosure and found out the Raylelgh number was larger than 5x10 4, the role of heat transfer mechanism was changed from the conduction to convection and the multicellular flow pattern occurred in the convection region. Concerning with the study of natural convection in an annular enclosure, Littlefield and Desal [10] examined temperature-induced buoyant convection in a uniform section annulus of large aspect ratio by similar transformations method and the results showed that the Nusselt number is dependent on both the boundary conditions and the geometry. Lee et.al. [11]~[14] investigated the hydrodynamics stability problem in an annular enclosure, the flow pattern was divided into several types. Weldman and Mehrdadtehranfar [15] conducted an experiment to examine the hydrodynamics stability of large aspect ratio. However, with regard to the study of transient natural convection in an annular enclosure is hardly found out. The aim of this study is to investigate the transient natural convection in an annular enclosure numerically. The numerical method of S I M P L E R with power law and T.D.M.A schemes are adopted to solve the governing equations. Preliminarily, in order to examine the difference of the transient natural convection between in a square enclosure and an annular enclosure, the height and the length (from inner radius to outer radius) of the annular enclosure are equivalent. The Raylelgh numbers of 10 4 and 10 8 are respectively considered to investigate the phenomena of conduction dominant situation and convection dominant situation. In addition, the variations of stream lines, isotherms and local Nusselt number along the walls are also examined in detail.
Vol. 18, No. 3
CONVECTION IN AN ANNULAR ENCLOSURE
375
Physicalmodel and Numerical method Schematic model of an annularenclosureis shown in Fig. 1 The inner and outer radii are r i and ro, respectively, and the height is L. The height L equals to the distance from r i to r o. The upper and lower walls are adiabatic. When the time t=0, the fluid, Prandtl number of which is 0.7, in the enclosure is stationary and at the same temperature T m as those of the inner and outer walls. As t>O, the temperature of the inner wall rises
/
r
/-
~--I'--I.-~,~ I I
I
I
I
I
"~,
\
to a high temperature T w and the temperature of the outer wall is still maintained at T . Due to the
X
w
buoyancy force, the fluid starts to flow. In order to simplify the analysis, the following assumptions are
J
[[=o --
"///////~ ~//////.
made.
(I) The fluid is Newtonian, and the flow is laminar.
7/////,/)('///////
(2) Boussinesqapproximationis valid. According to the physical model
"i C, r
and
assumptions, the dimensionlessgoverning equations
Fig.1. A S c h e m a t i c Model
pre expressedas eqs. (1).
~x,(RU)+~-(RV)--0
(la)
~_q.~.[~x~RUV)+~.(RVV)]___SP Pr8 Z •~@V 8V V~ ~r+ ~I-~[~X,( ) - ~8 R •~K)]-P r•R-
(lc)
376
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
(ld)
The dimensionlessvariablesused in Eqs. (I) ate defined as Eq. (2).
X=
x ro '
U = u . ro ~
P"P®
P
Pa3/r2o
Pr = v/a,
a.t
R -r -- ro ' V=
v'l~°
a
r = r-~o , '
(2)
N = ri ro
L A = ro RR =
0 = ( T - T ® ) / ( Tw---T®) ,
g~(Tw-T®)roS v.a
The boundary conditions are shown in Eqs. (3a)~(3d)
r=O,
u=v=s=-0
(3a)
r>O,
R=N,
U=V=0,
R=I,
U=V=0=0
X=O and X = A ,
U=V=O,
0=-1
(3b) (3c)
~,=0
(3d)
The SIMPLE--R algorithm [16] with power law scheme is adopted to solve the governing equations, according to the numerical test, the staggered grid system [17] and 32x32 uniform grids are utilized during the computing process. The time increments for Rayleigh numbers of 104 and 106 case ate 10 -3 and 10 -4, respectively.The convergent condition at every time step is defined as I
$.,
¢~. i
I < 10-3, and the steady state
I < 10"3,in which $ = U, V, P, T. condition for the flow is defined as [ ~r+A ~ r +r--~r Ar
Results and discussion The local Nusselt number NUr,x along the every annular cross section and the
Vol. 18, No. 3
CONVECTION IN AN ANNULAR ENCLOSURE
377
average Nusselt number I~ r on the every annular crosssectionare defined as the following equations.
hr
p-Cp.r-v. (T--T®)-k. r . ~ r
NUr,x = ~-- =
k. {T.--T® )
N'Ur = ~ ~ Nut,x dY
Ca)
The stream function 9 is defined as Eq. (6).
=u,
= v
(8)
The figures shown in Fig. 2 (e) and Fig. 3 (e) are the steady state situation for Ra----104 and 106 cues, respectively. Since the area of the annular cross section increases with the increasing of radius, then the heat flux q (w/m 2) through the annular cross section decreases with the increasing of radius. As a result, the temperature ~adient along the inner wall is larger than that along the outer wall. In addition, the distribution of the isothermal lines is not similar to that, which is symmetric to the diagonal, of the square enclosure [5]. As for the distribution of the stream lines, which is not symmetric to the diagonal too. Fig. 2 (a)~(e) and Fig. 3 (a)~(e) illustrate the transient variations of the isothermal lines and stream lines for Ra=10 4 and 10e cases, respectively. In the initial region, the fluid absorbs the heat transferred from the inner wall, and the isothermal lines not only gather densely near the inner wMl but also are parallel to the inner wall. With increasing of time, the flow of the fluid become quick and the trend of the flow can be predicted approximately by the movement of the position of the maximum stream function value. Due to the domination of heat conduction, the difference of the distributions of the isothermal and stream lines in the an~ular and square enclosures is not remarkable.
378
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
(a) "r=O.O02 O.OIEU
x: --0.088"~
(b) "r=O.010 0.0 x: - - 0 . 4 8 8 9 6
(c) "r=0.068 x : - 1.75236
(d) ~=0.094 0.0
0
x:-1.73159
°:~11/ (e)
X
•7"=0.436
o.o
0.2
(ri)
R
]L.O
(ro)
Isothermal lines
0.2
R
(ri) ×:-1.81393
1 .o
(ro)
~ r e ~ m 1in es
Fig. 2 T r a n s i e n t d e v e l o p m e n t s of i s o t h e r m a l l i n e s a n d s t r e ~ , ' n l t , ~ e s of a n A n ~ u l a r e n o l o s u r e f o r Ra = 10 4
Vol. 18, No. 3
CONVECTION IN AN ANNULAR ENCLOSURE
'r
0.0002
0.0E
I"=0.002 O.C
x: - 8 . 5 9 6 4 1 0.8
(o) "r=O.O06
O.Oll
l x: - g . T g ~ 4
? G)
(d) 1"=0.016 x: -6.51445
(e) "r:0.1459
O.OBL 0.2
(rl)
R
J 1.0
(ro)
0.2
1.0
(ri)
R
(ro)
x: - 7 . r , . ~
Isothermal lines
StroAn'dJne8
Fig. 3 Transient d e v e l o p m e n t s of i s o t h e r m a l l i n e s a n d s t r e R n ' ~ l t ' n e s of a n A n m d ~ e n c l o s u r e for l ~ = 10 e
379
380
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
In Fig. 3, generally the transient variations of the isothermal lines and stream lines for Ra=106 case are rather different from those of Ra=104 case. Since the convection effect is strong for Ra=10e case, the phenomenon of the position of the m a x i m u m stream function
moving from near the inner wall region to near the outer wall region is different from that in the square enclosure [5] significantly. Besides, the low temperature region is much larger than that in the square enclosure,
o.8
I
A
Shown in Fig. 4 (a) and (b), there are the variations of the local Nusselt numbers NUr,x along the inner and outer walls for
X
a.~-O.O02 B:'r-O,OOe C:~'-O.OIO
0.4
D:-r-O.Ol6 F~--O.O~9 F:.'r=O.07§
Ra=104 case. Initially, the conduction is dominant, then the NUr,x (r=ri) (curve A) is almost constant. With increasing of time,
I 0,0
I
0.0
0.8 ~
,
,
,
2.5 Nu r,x (r~ri) (a)
the fluid begins to flow which causes the NUr,x (r=ri) in the lower region to be larger
G."r=St~d~ State (o.4se) 5.0
Z
than that in the upper region. The variations of the NUr,x (r=ro) is opposite to that of the NUr,x (r--ri)" Initially, most of the heat transferred in from the left wall is absorbed
A.1- 0,029 ~v=o.o6o ~--o.ov6 ~-=o, m o
:~ 0.4 ~ I I II / / If/I
by the fluid, then the NUr,x (r=ri) is very minute. With increasing of time, the fluid begins to flow which causes the NUr, x (r=ro) in the upper region to be larger than that of
0.0 ~ 0.0
2.6 Nur.x (r.=ro)
5.0
(b)
Fill. 4 Looal Nmme/t numbe~ ~ both thiner and m~tm" walls of an ,,~M,I.~ ~ fro-Ra = I04 The variations of the local Nusselt number NUr, x along the inner and outer walls for
the lower region.
Ra--106 case are shown in Fig. 5. In general, the variations of the NUr, x for Ra--104 case are similar to those for Ra=104 mentioned above. But the convection is dominant for Ra=106 case, the NUr, x (r=ri) fluctuates in the initialregion which causes the value of the
Vol. 18, No. 3
CONVECTION IN AN ANNULAR ENCLOSURE
381
curve E to be larger than that of the curve D [Pig. 5 (a)]. This phenomenon is also found out in the square enclosure [5]. !1
0.8
10.0
A: 1"-0.000:
I n n e r wall Outer w a n
B:
T-0.00~ C:
1",-0.001 D: T-0.0014
~4 0.4
5.0
g:
T-O.002 le: I"-0.018
(o.14~g)
0.0
'
'
,
,
0.0
x
•
I
8.0
Nur~ (r-rl)
0.0
I
12.0
~ . . . . . . ,,I . . . . . . . . L~,~ ..... 10~ 10"e 10-[
~0.959
Time (T/ r.)
(a)
(")
0.8
15.0 Inner wldl OuLerwell
10.0 ~4
0.4 W'////
V
o.o) |
!
0.0
J
!
B:~"0.010 O: ~ , , 0 . 0 1 8
.
,
]~T-0.029 wT-0.050 I P:.'r-Bt_-m_~yI
5.0
~3.479 /
a
, , , ,
8.0
12.0
Nur~ (~-ro)
(b) le~. 6 Lomd N u m e l t n u m b e ~ elon4j b o t h fnnmr m:d outm- ~ o f aLn ~ u l ~ r mz~immre f o r 1~ = 10 e
0.0
f
...... J , ,,,,,,d , ,,,,,,d , ,,,,
lO-"
lO-' 10-' 10-' Time ('r/~)
1
(b) Fig. 8 Tr,-,,d,-,t developmeat~ of average Nmmelt n u m b e r s f o r P~ ,. 104 a n d 1~ = 101
Shown in Fig. 6 , there are the variations of the average Nnsselt number N~ r on the inner and outer walls for Ra=104 and 106 cases, respectively. In the initial region, the N'~r on the inner wall decreases rapidly, and after the time (r/~'s) equaling to 10 .2 the N~ r is almost invariant. At a certain.time after the beginning, the N~ r on the outer wall appears for both Ra=104 and 10e cases. Both the N~ r on the inner and outer walls are nearly equivalent at about the time (r/rs) of 0.4, which means that the transient phenomenon mainly occurs in the former 40% time of the whole transient time.
382
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
Conclusion The numerical method of S I M P L E R
is used to investigate transient natural
convection in an annular enclosure. The following conclusions are drawn. (1)
Since the area of annular cross section increases with the increasing of radius, the distributionsof isothermal lines and stream lines are not symmetric to the diagonal.
(2)
After the time 7-/7s = 0.4, the average Nusselt number of both the inner and outer walls are nearly equivalent for Ra=10 4 and 10 s cases.
Nomenclature A
ratio between length of annular enclosure and outer radius (L/to)
Cp
specific heat, J/kg. K
g
gravitational acceleration, m/s 2
k
thermal conductivity, W/re. K
L
length of the annular enclosure, m
N
radius ratio (ri/ro)
Nu
Nusselt number
p
pressure, pa
P
dimensionlesspressure
Pr
Prandtl number
r
radius coordinate, m
R
dimensionlessradius coordinate
Ra
Rayleigh number ( gfl(Tw--T®)roSfl,a )
t
time, s
T
temperature, K
u,v
velocity in x and r direction, m/s
U,V
dimensionless velocity in x and r direction
x
axial coordinate, m
(u/a)
Vol. 18, No. 3
X
C O N V E C T I OIN N AN ANNULAR ENCLOSURE
383
dimensionless axial coordinate thermal diffusivity, m2/s thermal expansion coefficient, 1/K
0
dimensionless temperature
v
kinematic viscosity, m2/s
p
density, kg/m s symbol represents U, V, P, 8 dimensionless stream function
~-
dimensionless time
A~-
increment of dimensionless time
Subscripts i
value at inner radius
o
value at outer radius
s
steady state
w
value at the wall of inner radius
®
ambient value
Superscripts n
iteration number average value
References
1o
J. Patterson and J. Imberger, J. Fluid Mech. ~
2.
G. De Vahl Davis, Int. J. Numer. Mech. Flui& ~, 249 (1983)
3.
V. F. Nieolette, K. T. Yang, and J. R. Lloyd, Int. J. Heat Mass Transfer 28, 1721
.
5.
85 (1980)
(1985) J. D. Hall, A. Bejan and J. B. Chaddock, Int. J. Heat & Fluid Flow 9, 398 (1988) W. S. Fu, J. C. Perng and W. J. Shieh, Numerical Heat Transfer, Part A, 16,
384
W.-S. Fu, Y.-H. Jou and C.-H. Lee
Vol. 18, No. 3
325 (1989) 6.
M. Chattree and S. Sengupta, Natural Convection in Enclosure - 1986. ASME HTD
-v•l. 63, 33 (1986)
7.
D.Y. Huang and S. S. Hsieh, Numerical Heat Transfer, 12, 121 (1987)
8.
J. Sun and P. H. Oosthuizen, ASME uaDer 88--HT-107. ASME Summer Annual Meeting, 103 (1988)
9.
S.K. Aggarwal and A. Manhapra, Numerical Heat Transfer, Part A, 15, 341 (1989)
10.
D. Littlefield and P. Desai, J. Heat Transfer, 108, 814 (1986)
11.
Yee Lee and Seppo A. Korpela, J. Fluid Mech., 12~, 91 (1983)
12.
Yee Lee, Seppo A. Korpela, and Roland N. Home, PrQ¢. Qf ~b¢ 7th Int. H¢~t Transfer Conference vQ1. 2, 221 (1982)
13.
S.A. Korpela, Yee Lee, and J. E. Drummond, J. Heat Transfer, 104, 539 (1982)
14.
Inn. G. Choi and Seppo A. Korpela, J. Fluid Mech., 99, 725 (1980)
15.
P.D. Weidman and G. Mehrdadtehranfar, Phys. Fluids, 28, 776 (1985)
16.
S.V. Patankax, Numerical Heat Transfer and Fluid Flow, McGraw - Hill, New York (1980)
17.
F.H. Harlow and J. E. Welch, Phys. Fluids, ~, 2182 (1965)