- Email: [email protected]
A transient natural convection in a uniformly heated enclosure under time-dependent gravitational acceleration field
INT. COMM. HEAT MASS TRANSFER Vol. 17, pp. 501-510, 1990 ©Pergamon Press pie
0735-1933/90 $3.00 + .00 Printed in the United States
A TRANSIENT NATURAL CONVECTION IN A UNIFORMLY HEATED ENCLOSURE UNDER TIME--DEPENDENT GRAVITATIONAL ACCELERATION FIELD Wu-Shung Fu and Wen-Jiann Shieh Department of Mechanical Engineering National Chiao Tung University Hsinehu, 30050, Taiwan R. O. C.
(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT A transient natural convection in a square enclosure, in which a uniform heat flux is transferred in from the left wall and the temperature of right wall is maintained at low temperature Tc, under time--dependent gravitational acceleration field is studied numerically. Rayleigh number varies linearly with time from 105 via 106 to -105. The isotherms, streamlines and Nusselt numbers are presented in order to examine heat transfer mechanisms during transient process. The results show that the maximum temperature on left wall increases with decreasing Rayldgh number, and the total Nusselt number on right wall decreases with decreasing Rayleigh number.
Introduction Natural convection in enclosures is important and interested in many engineering application fields, then the topic becomes one of the most active subfields in heat transfer research. A lot of literature have been summarized in reviews and reference book[l~3]. But most of the literature investigated this subject with constant temperature or heat flux on walls under static gravitational field[4~8]. As for the transient natural convection caused by unsteady gravitational acceleration which usually occurs in moving machine or flying vehicle has been scarcely studied so far. In the early time, Chung and Anderson [9] studied the unsteady laminar natural convection along a semi-infinite vertical plate or circular cylinder caused by time--dependent acceleration field. Their results showed the time--dependent deviations of the heat transfer rate from the quasi--steady ones. But the effect of unsteady gravitation on the natural convection in an enclosure with uniformly heated wall has not been studied in the literature to the author's knowledge. 501
502
W.-S. Fu and W.-J. Shieh
Vol. 17, No. 4
Hence, the aim of this study is to investigate the transient natural convection in a uniformly heated square enclosure under time-dependent gravitational acceleration field numerically. A penalty finite element method with backward difference scheme is adopted to solve the governing equations. Rayleigh number varies linearly with time from 105 via 106 to -10 5 . The flow field and heat transfer rate during the transient process are examined. Physical model The physical model sketched in Fig. 1 is a t~O dimensional square enclosure with two insulated horizontal walls. Initially,the two vertical walls and the fluid in the
~9
enclosure are maintained at temperature under 1 go gravitational field. As time t>~, the left wall is suddenly heated by a unif0r ~ heat flux q" and meanwhile, the enclosure begins to be accelerated unsteadily, that is, the gravitational acceleration becomes timedepndent.
T=,Tc
In order to facilitate the analysis, the following assumptions and the dimensionless variables are made, respectively.
0
FIG. 1 Physical Model
1. The Boussinesq approximation is valid. 2. The fluid is Newtonian and the flow is laminar. 3. The radiation effect is neglected.
~-t/(L2/a), U=u/(a/L),
X=x/L, Y=y/L, V=v/(a/L), ~--(T-Tc)/(L'q"/k
t>O
×
t_
)
(1)
P=(p+ pg(t )y )/ (pf'2/L 2), Pr=u/a,
Ra(r)=g(t).#- q". L4/(a- v. k).
In Eq.(1), Ra(r) is time-dependent due to including g(t). The value of g(t) varies linearly with time f~om positive (the direction same as gravity) via zero and stop at negative. The function of Rayleigh number is shown in Eq.(2).
0<~-(0.04236 Ra(~')=1.0~lOS+9.0~105r/0.04236
(2)
Vol. 17, No. 4
CONVECTION UNDER GRAVITATIONAL ACCELERATION FIELD 503
0.04236<40.466
Ra(r)=1.0x !O6-1.1x10sx(r-O.04236)/0.42364. Based upon the assumptions and dimensionless variables, t h e governing equations are expressed as following. ~+
~r = 0
~- + U
dimensionless
(3a)
+ V ~r ....
+ Pr (
2+
2)
(3b)
The initial and boundary conditions are as follows. r----0,
U = V - .~--0
r>0,
X=0, U=V=0 a0/ffX=-I (4) X=I, U=V=O=O Y=0 and Y=I,
U=V=a0/~xr=0
Solution method A penalty finite element method with a modified Newton-Raphson algorithm and a backward difference scheme dealing with the time term similar to the one used in Fu et al.[10] are used to solve the governing equations (3a)~(3d) with boundary conditions Eq.(4). •The nonuniform mesh 27x27 is selecte~ in the calculation. The time increments are selected as follows.
0<~_0.0085, 0.0085<~_0.466,
Av=l.0xl0 "4 Av=l.0xl0 "3.
In order to validate the accuracy of tl!i6 method, the glob~J energy balance error Err is used to examine the errors resulting from the method. Err is defined by the following equation:
504
W.-S. Fu and W.-J. Shieh
Vol. 17, No. 4
Err= I QL -QLQR --QEI
(5)
where QL (= 0- k
x--O dy ) is the amount of heat transferred in from the left wall. or
QR (= 0 - k ~
x=L dy ) is the amount of heat transferred out from the right wall.
L L QE (= if0 f0 p Cp fir dx dy )is the increasing rate of the internal energy of the fluid. 0t During the computation,the maximum error is about 1.2%. The local Nusselt number NUx,y on any vertical plane is defined as NUx,y--[pCpU(T-Tc)-k~T~]/[q"]
(6)
The total Nusselt number Nux on any vertical plane is defined as
NUx=~iNUx,ydY
(7)
The dimensionless stream function ¢2 is obtained from solving the following Poisson equation with the boundary conditions 9=0 at all the solid surfaces : 8 2 # + 829
8U
6rv"
(8)
Results and discussion The numbers above the figures (Figs. (2) and (3)) represent the minimum, increment and maximum values of 0 and ¢/, respectively. The sign "x" in the isotherms denotes the location of the maximum temperature. The signs "+*' and "x" in the streamlines denote the occurrence of the maximum clockwise and counterclockwise streamfunction values, respectively. The development of the flow field and temperature distribution are shown in the Fig. (2). From Figs. 2(a) to 2(c), the clockwise--rotating cell develops with stratifying the core region. As time increases, the Rayleigh number becomes small, the clockwise-rotating cell weakens and new cells begin to grow near the corners. At the end of the process (Fig. 2(e)), a counter-clockwise rotating cell forms due to the reverse buoyancy force. During the transient process, the location of the maximum
Vol. 17, No. 4
CONVECTION U N D E R GRAVITATIONAL ACCELERATION FIELD
e.~.o(.o~.a:)o.2,~
p LP IPl
(O) "r~3.15xlO - 2 ,. Ra('r)=7.69x10 '~
X.-0.228
e:o.o(.o22)o.22
,:.-~o.~(-,.o8)o.o
(b) "/--,I..336x10 - 2 . Ro(1")-g.974.x 10 ~
X:0.22 e:o.o(.o321)0.321
M °D Q
(c) I"=0.2594
ea('O-4.,~x I o ~
X.'0.321
,:.-~.eo,(-.~)o.o
(d) I"-0.4274.
X.'0.423
e.-o.o(.o4~)o.4~
*:.o.o9.42,p,~
X:0.423
0
Isotherms
(e) ,r,.o.4~s
Streamlines
Isotherms
1;'IG. 2 and StJ'~a~Lnes
Ro('r)-, - I .Ox 10
s
505
506
W.-S. Fu and W.-J. Shieh
Vol. 17, No. 4
temperature moves from the top to the lower region of the left wall. In order to scrutinize the flow pattern transformation during the variation of Rayleigh number from positive to negative, the omitted detail variations from Fig. 2(d) to 2(e) are shown in Fig. 3. In Fig. 3(a), the counterclockwise-rotating cells begin to grow
(o)
T-0.4405
Ra('r')--3.376x 104
X.~.434 e:O.O 1.0436)0.4.36
~-.142(.131)1.17
(b)
.r-oA435
.
RO('T)--4. I 5x 104
X:0.436 G:0.0(.04366)0.4366
~I,:0.0~.1735)1.735
(c) -rw0.44.55 4 Ra(7")--4..414x 10
o
X:0.4366
e.~.o(.o,m)o.,m
X.~.408 Isotherms
• :.0.0~1.19)1
1.9
0
(d) ~-0.4625 4 Ro('r~,--g.Ogxl 0
Streamlines FIG. 3
Development. of Counterclook'wt~-Rol~.Ur~ Cells
along the left and fight walls due to the reverse buoyancy and the original clockwise-rotating cell is squeezed in the middle region. As time inerea~es, the reverse
Vol. 17, No. 4
CONVECTION UNDER GRAVITATIONAL ACCELERATION FIELD 507
buoyancy becomes strong and the two counterclockwise-rotating cells combine into one cell and the cell merges the original clockwise-rotating cell.
1.0
I
I
I
I
>... 0.5
2i O.O
I
i
0.0
I
0.25
0.50 Q x=o
O) 1.0
0.5
0.0 0.0
0.5
1.0
.7
Nux=l,y (b) FIG. 4 (a) T e m p e r a t u r e D i s t r i b u t i o n Alone Left Wall (b) Local N u s s e l t N u m b e r D i s t r i b u t i o n Alone t h e R i g h t Wall
508
W.-S. Fu and W.-J. Shieh
Vol. 17, No. 4
0.45
l
t
~1.0 z3 Z
0.4
0.3
0.2
0
0.5 0.1
0.0
I
0.0
I
0.0 0.466
t
[w
0.2 "7"
FIG. 5 The Maximum Temperature on the L e f t Wall a n d t h e T o t a l N u s s e l t Number on the Right Tall
Figure 4(a) shows the temperature distribution along the left wall. The indexes 1~5 correspond to the time illustrated in Figs. 2(a)~(e). The location of the maximum temperature for time 1 to 4 occurs at the top of the wall, but that is at Y=0.118 for time 5 due to changing the sign of Rayleigh number. The wall temperature at time 1 is greater than that at time 2, which is the overshoot of thermal boundary layer by transient conduction. Figure 4(b) shows the local Nusselt number NUx=I, Y distribution along the right wall. The time of indexes 1~5 are the same as Fig. 4(a). The location of the maximum local Nusselt number at time 1 to 4 occurs at upper part of the wall, but at time 5, the value of Rayleigh number decreases and the sign is opposite which results in the reduction of Nusselt number and changes the location of the maximum local Nusselt number. Figure 5 shows the variations of the maximum temperature 0max on the left wall and the total Nusselt number NUX=1 on the right wall. The arrows in the figures represent the position of Ra(r)=106 and Ra(T)=0, respectively. Initially, the fluid near the left wall begins to be heated, the transient conduction is a dominant heat transfer mode. Therefore, the left wall temperature increases rapidly to a peak value near time 1, and it drops a little due to convection effect. This overshoot phenomenon is also found out in a semi-infinite vertical heated plate subjected to a step change of heat flux [11]. Afterwards, the maximum
Vol. 17, No. 4
CONVECTION UNDER GRAVITATIONAL ACCELERATION FIELD 509
temperature on the left wall increases with a uniform heat flux transferred in continuously until over time 4 a little. Finally, the 0max descends due to the counterclockwise-rotating flow to be formed. From the definition of Nusselt number Eqs.(6) and (7), the total Nnsselt number NUx= 0 on the left wall which is not affected by Rayleigh number is continuously maintained at one. The total Nusselt number on the right wall NUx= 1 increases due to the heated fluid impinging upon the right wall until time 3. After the time 2 (the maximum Rayleigh number), 1L~yleigh number begins to decrease which causes convection effect to weaken. According to time lag, the effect of the decreasing of Rayleigh number on redudng the value of NUx= 1 appears after time 3. The similar phenomenon is also found out in [9]. The difference of the total Nusselt number between the left and right walls is the energy absorbed by the fluid in the enclosure, which causes the temperature of the fluid to rise. The energy absorbed by the fluid can be represented by the bulk mean temperature 0m, shown in Fig. 5. Conclusion A penalty finite element method is employed to solve the transient natural convection in a uniformly heated square enclosure under time-dependent gravitational acceleration field. Two confusions can be drawn: (1) The temperature overshoot occurs at the end of the one--dimensional transient conduction regime, and the location of the maximum temperature on the left wall shifts as the sign of Rayleigh number is changed. (2) The right wall total Nusselt number NUx= 1 is influenced by 1L~yleigh number, which increases first and decreases due to the decreasing of Rayleigh number which weakens convection effect. Acknowledgement The support of this work by the National Science Council, Taiwan, R.O.C. under contract NSC79--0401-E---009-10 is gratefully acknowledged. Nomenclature g(t) go k L Nu P P Pr
time-dependent gravitational acceleration [m/s 2] standard gravitational acceleration(=9.8 m/s 2) thermal conductivity [W/re. K] length of the enclosure Nusselt number pressure [Pal dimensionless pressure Prandtl number(=v/~)
510
W.-S. Fu and W.-J. Shieh
Vol. 17, No. 4
time-dependent Kayleigh numberC=gCt)-ft, q". L4/(a. v. k)) time [s] temperature [K] velocities of x and y directions [m/s] U,V dimensionless velocities of x and y directions U,V x,y coordinates X,Y dimensionless coordinates thermal diffusivity [m2/s] thermal expansion coefficient [l/K] 0 dimensionless temperature 11 Om dimensionless mean temperature ( : |J0| 0o0 dXdY) dimensionless streamfunction kinematic viscosity [m2/s] v density [kg/m3] P dimensionless time T subscripts c cold wall t T
References 1.
S. Ostrach, Adv. Heat Transfer 8, 161 (1972)
2.
I. Carton, Proc. of the 6th Int. Heat Transfer Conference Vol. 6, 13 (1978)
3.
K.T. Yang, Natural Convection in Enclosures, In S. KaKac, R.K. Shah and W. Aung (eds.) Handbook of _Single Phase Convective Heat Transfer Chap. 13. Wiley-Interscience, New York (1987)
.
5. .
J. Patterson and J. Imberger, J. Fluid Mech. 100, 65 (1980) V.F. Nicolette, K.T. Yang, and J.R. Lloyd, Int. g. Heat Mass Transfer 28, 1721
(1985)
J.D. Hall, A. Bejan and J.B. Chaddock, Int. J. Heat and Fluid Flow 9, 396 (1988)
7.
S. Kimura and A. Bejan, J. Heat transfer 106, 98 (1984)
8.
W.S. Fu, J.C. Perng and W.J. Shieh, W~rme--Stoffiibertrag (In press)
9.
P.M. Chung and A.D. Anderson, J. Heat Transfer 83, 473 (1961)
i0,
W.S. Fu, J.C. Perng and W.J. Shieh, Numer. Heat Transfer Part A 16 325 (1989)
11.
Y. Jaluria, Natural Convection Heat and Mass Transfer, Chap. 5. Pergamon Oxford
(198o)
0735-1933/90 $3.00 + .00 Printed in the United States
A TRANSIENT NATURAL CONVECTION IN A UNIFORMLY HEATED ENCLOSURE UNDER TIME--DEPENDENT GRAVITATIONAL ACCELERATION FIELD Wu-Shung Fu and Wen-Jiann Shieh Department of Mechanical Engineering National Chiao Tung University Hsinehu, 30050, Taiwan R. O. C.
(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT A transient natural convection in a square enclosure, in which a uniform heat flux is transferred in from the left wall and the temperature of right wall is maintained at low temperature Tc, under time--dependent gravitational acceleration field is studied numerically. Rayleigh number varies linearly with time from 105 via 106 to -105. The isotherms, streamlines and Nusselt numbers are presented in order to examine heat transfer mechanisms during transient process. The results show that the maximum temperature on left wall increases with decreasing Rayldgh number, and the total Nusselt number on right wall decreases with decreasing Rayleigh number.
Introduction Natural convection in enclosures is important and interested in many engineering application fields, then the topic becomes one of the most active subfields in heat transfer research. A lot of literature have been summarized in reviews and reference book[l~3]. But most of the literature investigated this subject with constant temperature or heat flux on walls under static gravitational field[4~8]. As for the transient natural convection caused by unsteady gravitational acceleration which usually occurs in moving machine or flying vehicle has been scarcely studied so far. In the early time, Chung and Anderson [9] studied the unsteady laminar natural convection along a semi-infinite vertical plate or circular cylinder caused by time--dependent acceleration field. Their results showed the time--dependent deviations of the heat transfer rate from the quasi--steady ones. But the effect of unsteady gravitation on the natural convection in an enclosure with uniformly heated wall has not been studied in the literature to the author's knowledge. 501
502
W.-S. Fu and W.-J. Shieh
Vol. 17, No. 4
Hence, the aim of this study is to investigate the transient natural convection in a uniformly heated square enclosure under time-dependent gravitational acceleration field numerically. A penalty finite element method with backward difference scheme is adopted to solve the governing equations. Rayleigh number varies linearly with time from 105 via 106 to -10 5 . The flow field and heat transfer rate during the transient process are examined. Physical model The physical model sketched in Fig. 1 is a t~O dimensional square enclosure with two insulated horizontal walls. Initially,the two vertical walls and the fluid in the
~9
enclosure are maintained at temperature under 1 go gravitational field. As time t>~, the left wall is suddenly heated by a unif0r ~ heat flux q" and meanwhile, the enclosure begins to be accelerated unsteadily, that is, the gravitational acceleration becomes timedepndent.
T=,Tc
In order to facilitate the analysis, the following assumptions and the dimensionless variables are made, respectively.
0
FIG. 1 Physical Model
1. The Boussinesq approximation is valid. 2. The fluid is Newtonian and the flow is laminar. 3. The radiation effect is neglected.
~-t/(L2/a), U=u/(a/L),
X=x/L, Y=y/L, V=v/(a/L), ~--(T-Tc)/(L'q"/k
t>O
×
t_
)
(1)
P=(p+ pg(t )y )/ (pf'2/L 2), Pr=u/a,
Ra(r)=g(t).#- q". L4/(a- v. k).
In Eq.(1), Ra(r) is time-dependent due to including g(t). The value of g(t) varies linearly with time f~om positive (the direction same as gravity) via zero and stop at negative. The function of Rayleigh number is shown in Eq.(2).
0<~-(0.04236 Ra(~')=1.0~lOS+9.0~105r/0.04236
(2)
Vol. 17, No. 4
CONVECTION UNDER GRAVITATIONAL ACCELERATION FIELD 503
0.04236<40.466
Ra(r)=1.0x !O6-1.1x10sx(r-O.04236)/0.42364. Based upon the assumptions and dimensionless variables, t h e governing equations are expressed as following. ~+
~r = 0
~- + U
dimensionless
(3a)
+ V ~r ....
+ Pr (
2+
2)
(3b)
The initial and boundary conditions are as follows. r----0,
U = V - .~--0
r>0,
X=0, U=V=0 a0/ffX=-I (4) X=I, U=V=O=O Y=0 and Y=I,
U=V=a0/~xr=0
Solution method A penalty finite element method with a modified Newton-Raphson algorithm and a backward difference scheme dealing with the time term similar to the one used in Fu et al.[10] are used to solve the governing equations (3a)~(3d) with boundary conditions Eq.(4). •The nonuniform mesh 27x27 is selecte~ in the calculation. The time increments are selected as follows.
0<~_0.0085, 0.0085<~_0.466,
Av=l.0xl0 "4 Av=l.0xl0 "3.
In order to validate the accuracy of tl!i6 method, the glob~J energy balance error Err is used to examine the errors resulting from the method. Err is defined by the following equation:
504
W.-S. Fu and W.-J. Shieh
Vol. 17, No. 4
Err= I QL -QLQR --QEI
(5)
where QL (= 0- k
x--O dy ) is the amount of heat transferred in from the left wall. or
QR (= 0 - k ~
x=L dy ) is the amount of heat transferred out from the right wall.
L L QE (= if0 f0 p Cp fir dx dy )is the increasing rate of the internal energy of the fluid. 0t During the computation,the maximum error is about 1.2%. The local Nusselt number NUx,y on any vertical plane is defined as NUx,y--[pCpU(T-Tc)-k~T~]/[q"]
(6)
The total Nusselt number Nux on any vertical plane is defined as
NUx=~iNUx,ydY
(7)
The dimensionless stream function ¢2 is obtained from solving the following Poisson equation with the boundary conditions 9=0 at all the solid surfaces : 8 2 # + 829
8U
6rv"
(8)
Results and discussion The numbers above the figures (Figs. (2) and (3)) represent the minimum, increment and maximum values of 0 and ¢/, respectively. The sign "x" in the isotherms denotes the location of the maximum temperature. The signs "+*' and "x" in the streamlines denote the occurrence of the maximum clockwise and counterclockwise streamfunction values, respectively. The development of the flow field and temperature distribution are shown in the Fig. (2). From Figs. 2(a) to 2(c), the clockwise--rotating cell develops with stratifying the core region. As time increases, the Rayleigh number becomes small, the clockwise-rotating cell weakens and new cells begin to grow near the corners. At the end of the process (Fig. 2(e)), a counter-clockwise rotating cell forms due to the reverse buoyancy force. During the transient process, the location of the maximum
Vol. 17, No. 4
CONVECTION U N D E R GRAVITATIONAL ACCELERATION FIELD
e.~.o(.o~.a:)o.2,~
p LP IPl
(O) "r~3.15xlO - 2 ,. Ra('r)=7.69x10 '~
X.-0.228
e:o.o(.o22)o.22
,:.-~o.~(-,.o8)o.o
(b) "/--,I..336x10 - 2 . Ro(1")-g.974.x 10 ~
X:0.22 e:o.o(.o321)0.321
M °D Q
(c) I"=0.2594
ea('O-4.,~x I o ~
X.'0.321
,:.-~.eo,(-.~)o.o
(d) I"-0.4274.
X.'0.423
e.-o.o(.o4~)o.4~
*:.o.o9.42,p,~
X:0.423
0
Isotherms
(e) ,r,.o.4~s
Streamlines
Isotherms
1;'IG. 2 and StJ'~a~Lnes
Ro('r)-, - I .Ox 10
s
505
506
W.-S. Fu and W.-J. Shieh
Vol. 17, No. 4
temperature moves from the top to the lower region of the left wall. In order to scrutinize the flow pattern transformation during the variation of Rayleigh number from positive to negative, the omitted detail variations from Fig. 2(d) to 2(e) are shown in Fig. 3. In Fig. 3(a), the counterclockwise-rotating cells begin to grow
(o)
T-0.4405
Ra('r')--3.376x 104
X.~.434 e:O.O 1.0436)0.4.36
~-.142(.131)1.17
(b)
.r-oA435
.
RO('T)--4. I 5x 104
X:0.436 G:0.0(.04366)0.4366
~I,:0.0~.1735)1.735
(c) -rw0.44.55 4 Ra(7")--4..414x 10
o
X:0.4366
e.~.o(.o,m)o.,m
X.~.408 Isotherms
• :.0.0~1.19)1
1.9
0
(d) ~-0.4625 4 Ro('r~,--g.Ogxl 0
Streamlines FIG. 3
Development. of Counterclook'wt~-Rol~.Ur~ Cells
along the left and fight walls due to the reverse buoyancy and the original clockwise-rotating cell is squeezed in the middle region. As time inerea~es, the reverse
Vol. 17, No. 4
CONVECTION UNDER GRAVITATIONAL ACCELERATION FIELD 507
buoyancy becomes strong and the two counterclockwise-rotating cells combine into one cell and the cell merges the original clockwise-rotating cell.
1.0
I
I
I
I
>... 0.5
2i O.O
I
i
0.0
I
0.25
0.50 Q x=o
O) 1.0
0.5
0.0 0.0
0.5
1.0
.7
Nux=l,y (b) FIG. 4 (a) T e m p e r a t u r e D i s t r i b u t i o n Alone Left Wall (b) Local N u s s e l t N u m b e r D i s t r i b u t i o n Alone t h e R i g h t Wall
508
W.-S. Fu and W.-J. Shieh
Vol. 17, No. 4
0.45
l
t
~1.0 z3 Z
0.4
0.3
0.2
0
0.5 0.1
0.0
I
0.0
I
0.0 0.466
t
[w
0.2 "7"
FIG. 5 The Maximum Temperature on the L e f t Wall a n d t h e T o t a l N u s s e l t Number on the Right Tall
Figure 4(a) shows the temperature distribution along the left wall. The indexes 1~5 correspond to the time illustrated in Figs. 2(a)~(e). The location of the maximum temperature for time 1 to 4 occurs at the top of the wall, but that is at Y=0.118 for time 5 due to changing the sign of Rayleigh number. The wall temperature at time 1 is greater than that at time 2, which is the overshoot of thermal boundary layer by transient conduction. Figure 4(b) shows the local Nusselt number NUx=I, Y distribution along the right wall. The time of indexes 1~5 are the same as Fig. 4(a). The location of the maximum local Nusselt number at time 1 to 4 occurs at upper part of the wall, but at time 5, the value of Rayleigh number decreases and the sign is opposite which results in the reduction of Nusselt number and changes the location of the maximum local Nusselt number. Figure 5 shows the variations of the maximum temperature 0max on the left wall and the total Nusselt number NUX=1 on the right wall. The arrows in the figures represent the position of Ra(r)=106 and Ra(T)=0, respectively. Initially, the fluid near the left wall begins to be heated, the transient conduction is a dominant heat transfer mode. Therefore, the left wall temperature increases rapidly to a peak value near time 1, and it drops a little due to convection effect. This overshoot phenomenon is also found out in a semi-infinite vertical heated plate subjected to a step change of heat flux [11]. Afterwards, the maximum
Vol. 17, No. 4
CONVECTION UNDER GRAVITATIONAL ACCELERATION FIELD 509
temperature on the left wall increases with a uniform heat flux transferred in continuously until over time 4 a little. Finally, the 0max descends due to the counterclockwise-rotating flow to be formed. From the definition of Nusselt number Eqs.(6) and (7), the total Nnsselt number NUx= 0 on the left wall which is not affected by Rayleigh number is continuously maintained at one. The total Nusselt number on the right wall NUx= 1 increases due to the heated fluid impinging upon the right wall until time 3. After the time 2 (the maximum Rayleigh number), 1L~yleigh number begins to decrease which causes convection effect to weaken. According to time lag, the effect of the decreasing of Rayleigh number on redudng the value of NUx= 1 appears after time 3. The similar phenomenon is also found out in [9]. The difference of the total Nusselt number between the left and right walls is the energy absorbed by the fluid in the enclosure, which causes the temperature of the fluid to rise. The energy absorbed by the fluid can be represented by the bulk mean temperature 0m, shown in Fig. 5. Conclusion A penalty finite element method is employed to solve the transient natural convection in a uniformly heated square enclosure under time-dependent gravitational acceleration field. Two confusions can be drawn: (1) The temperature overshoot occurs at the end of the one--dimensional transient conduction regime, and the location of the maximum temperature on the left wall shifts as the sign of Rayleigh number is changed. (2) The right wall total Nusselt number NUx= 1 is influenced by 1L~yleigh number, which increases first and decreases due to the decreasing of Rayleigh number which weakens convection effect. Acknowledgement The support of this work by the National Science Council, Taiwan, R.O.C. under contract NSC79--0401-E---009-10 is gratefully acknowledged. Nomenclature g(t) go k L Nu P P Pr
time-dependent gravitational acceleration [m/s 2] standard gravitational acceleration(=9.8 m/s 2) thermal conductivity [W/re. K] length of the enclosure Nusselt number pressure [Pal dimensionless pressure Prandtl number(=v/~)
510
W.-S. Fu and W.-J. Shieh
Vol. 17, No. 4
time-dependent Kayleigh numberC=gCt)-ft, q". L4/(a. v. k)) time [s] temperature [K] velocities of x and y directions [m/s] U,V dimensionless velocities of x and y directions U,V x,y coordinates X,Y dimensionless coordinates thermal diffusivity [m2/s] thermal expansion coefficient [l/K] 0 dimensionless temperature 11 Om dimensionless mean temperature ( : |J0| 0o0 dXdY) dimensionless streamfunction kinematic viscosity [m2/s] v density [kg/m3] P dimensionless time T subscripts c cold wall t T
References 1.
S. Ostrach, Adv. Heat Transfer 8, 161 (1972)
2.
I. Carton, Proc. of the 6th Int. Heat Transfer Conference Vol. 6, 13 (1978)
3.
K.T. Yang, Natural Convection in Enclosures, In S. KaKac, R.K. Shah and W. Aung (eds.) Handbook of _Single Phase Convective Heat Transfer Chap. 13. Wiley-Interscience, New York (1987)
.
5. .
J. Patterson and J. Imberger, J. Fluid Mech. 100, 65 (1980) V.F. Nicolette, K.T. Yang, and J.R. Lloyd, Int. g. Heat Mass Transfer 28, 1721
(1985)
J.D. Hall, A. Bejan and J.B. Chaddock, Int. J. Heat and Fluid Flow 9, 396 (1988)
7.
S. Kimura and A. Bejan, J. Heat transfer 106, 98 (1984)
8.
W.S. Fu, J.C. Perng and W.J. Shieh, W~rme--Stoffiibertrag (In press)
9.
P.M. Chung and A.D. Anderson, J. Heat Transfer 83, 473 (1961)
i0,
W.S. Fu, J.C. Perng and W.J. Shieh, Numer. Heat Transfer Part A 16 325 (1989)
11.
Y. Jaluria, Natural Convection Heat and Mass Transfer, Chap. 5. Pergamon Oxford
(198o)