EFFECT OF A BAFFLE ON PSEUDOSTEADY-STATE  
NATURAL CONVECTION INSIDE  
SPHERICAL CONTAINERS 
 
 
Except where reference is made to the work of others, the work described in this thesis is 
my own or was done in collaboration with my advisory committee.  This thesis does not 
include proprietary or classified information 
 
 
 
 
 
______________________________ 
Yuping Duan 
 
 
 
 
 
 
 
Certificate of Approval: 
 
 
 
___________________________                                    ___________________________ 
Amnon J. Meir          Jay M. Khodadadi, Chair                    
Professor        Professor 
Mathematics           Mechanical Engineering 
 
 
 
___________________________        ___________________________ 
Daniel W. Mackowski          Joe F. Pittman  
Associate Professor          Interim Dean   
Mechanical Engineering         Graduate School 
EFFECT OF A BAFFLE ON PSEUDOSTEADY-STATE  
NATURAL CONVECTION INSIDE  
SPHERICAL CONTAINERS 
 
Yuping Duan 
 
 
A Thesis 
Submitted to  
the Graduate Faculty of  
Auburn University 
in Partial Fulfillment of the  
Requirements for the  
Degree of  
Master of Science 
 
 
 
Auburn, Alabama 
August 4, 2007 
 
   iii 
 
 
EFFECT OF A BAFFLE ON PSEUDOSTEADY-STATE 
NATURAL CONVECTION INSIDE  
SPHERICAL CONTAINERS 
 
 
 
 
 
Yuping Duan 
 
 
Permission is granted to Auburn University to make copies of this thesis at its discretion, 
upon the request of individuals or institutions and at their expense.  The author reserves 
all publication rights. 
 
 
 
 
 
       ____________________________ 
     Signature of Author 
 
        
____________________________ 
     Date of Graduation 
 
 
 
 
 
 
 
 
 
 
 
 
   iv 
 
 
VITA 
 
Yuping Duan, son of Wen Duan and Shouxian Ding, was born on April 20, 1981, 
in Zhangjiayao Village, Shanxi Province, the P. R. China.  He graduated from Kangjie 
High School in 2000 and then was admitted to Zhejiang University.  He studied at 
Zhejiang University for four years and graduated with a Bachelor of Science degree in 
Mechanical and Energy Engineering in June, 2004.  In January 2005, he enrolled at the 
University of Nebraska-Lincoln as a graduate student in Mechanical Engineering.  After 
one year study there, he transferred to Auburn University to complete his MS in 
Mechanical Engineering.      
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   v 
 
 
THESIS ABSTRACT 
EFFECT OF A BAFFLE ON PSEUDOSTEADY-STATE  
NATURAL CONVECTION INSIDE  
SPHERICAL CONTAINERS 
 
Yuping Duan 
Master of Science, August 4, 2007 
(Bachelor of Science, Zhejiang University, 2004) 
226 Typed Pages 
Directed by Jay M. Khodadadi 
 
Pseudosteady-state natural convection within spherical containers with and 
without thin baffles was studied computationally.  Insulated or isothermal baffles were 
considered for passive management of the flow and thermal fields.  For Rayleigh 
numbers of 10
4
, 10
5
, 10
6
 and 10
7
, baffles with 3 lengths positioned at 5 different locations 
were investigated.  Elaborate grid size and time step size independence tests were 
performed.  The solution of the governing equations was obtained by use of a commercial 
computational fluid dynamics (CFD) package.  For the case of no baffle, computational 
results were validated successfully to previous data available in the literature by 
comparing the heat transfer correlations, temperature distribution and streamline patterns.
   vi 
 
 
Both thermally stable and unstable layers are present in this problem and for the higher 
Rayleigh numbers, the onset of instabilities was observed in this system. 
Regardless of the thermal status of the thin baffle, placing it on the inner wall of the 
spherical container directly leads to modification of the velocity field.  It can generally be 
stated that the resulting ?confinement? or ?compartmentalization? causes the fluid above 
the baffle to be characterized by stable constant-temperature layers that are slow moving 
and dominated by heat conduction.  In contrast, the fluid below the baffle is subjected to 
strong natural convection currents.  Regardless of the Ra number, the modifications of the 
flow and temperature fields for short baffles are limited to the vicinity of the baffle and a 
possible interaction with the eye of the primary clockwise rotating vortex.  The 
modifications of the flow and thermal fields were more pronounced for the longest baffle 
for which two clockwise rotating vortices are clearly observed when the baffle is 
positioned at or in the vicinity of the mid-plane.  The Nusselt numbers and maximum 
stream function of the primary vortex were generally lower than the reference cases with 
no baffle.  The degree of degradation of the Nusselt number has a strong dependence on 
the position and length of the insulated baffle.  In contrast to the general reduction of heat 
transfer trends exhibited by the insulated or isothermal baffles, placing a baffle near the 
top of the sphere for high Ra number cases can lead to heat transfer enhancement in 
comparison to the reference case with no baffle.  The extra heat that is brought in the 
fluid through the surface of the sphere is linked to the disturbance of the thermal 
boundary layer by the thin baffle.  Some differences are observed due to the thermal 
status of the baffle.  Due to the extra heating afforded by a thin isothermal baffle, the 
velocity and temperature fields were more complicated than the case with a thin insulated 
   vii 
 
 
baffle.  In addition to confinement, a strong counterclockwise rotating vortex was created 
due to the extra heating of the baffle for high Ra numbers and baffle positions on or 
below the mid-plane.  The hot fluid in this vortex was observed to be transported toward 
the center of the sphere, thus disturbing the stable stratified layers.  In contrast to 
insulated baffles, placing isothermal baffles near the bottom for high Ra number cases 
also gave rise to heat transfer enhancement due to disturbance of the stratified layers by 
the CCW rotating vortex that is energized by the heated baffle. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   viii 
 
 
ACKNOWLEDGMENTS 
 
The author would like to express his special gratitude to his major professor, Dr. 
Jay M. Khodadadi, for his academic guidance, encouragement and patience towards the 
completion of this thesis and all the help he provided during my study at Auburn 
University.  Dr. Khodadadi has offered all the academic help that a major professor could 
possibly provide.  Personally, Dr. Khodadadi has gone way beyond the coverage of a 
major professor.  It is hard to fully express my gratitude in words.   
The author would like to express his gratitude to his other committee members, 
Drs. Mackowski and Meir.  They provided some helpful suggestions to the thesis.  The 
author acknowledges the Department of Mechanical Engineering at Auburn University 
for supporting his graduate assistantship.  Mr. Seyed Farid Hosseinizadeh provided 
valuable technical help early in this project.  Many thanks also go to the Alabama 
Supercomputer Center for their technical support and CPU time.   
Many friends in Auburn also gave me a lot of help. 
Finally, I would like to sincerely thank my parents for their support and 
inspiration.  My siblings, Yanping Duan, Yiping Duan, Liping Duan and Yaping Duan 
are greatly appreciated for their support and encouragement.  I also wish to thank my 
sister-in-law Lifen Zhao and my girl friend Lijin Yao for their encouragement. 
 
 
   ix 
 
 
Style manual or journal used: 
Guide to Preparation and Submission of Thesis and Dissertation 2007 
 
Computer software used: 
MS Word 2003, MS Excel 2003, TECPLOT 9.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   x 
 
 
TABLE OF CONTENTS 
 
LIST OF TABLES???????????????...???????????.xiv 
LIST OF FIGURES??????????????..????????????xvi 
NOMENCLATURE??????????????????????????xxx 
 
CHAPTER 1 INTRODUCTION?????????????????????...1 
 
CHAPTER 2 LITERRATURE REVIEW OF NATURAL CONVECTION INSIDE 
SPHERICAL CONTAINER?????????.?????????..????.....7 
 
CHAPTER 3 COMPUTATIONAL METHODOLOGY AND BENCHMARKING..?21 
 
3.1 Mathematical Formulation for the Pseudosteady-State Natural Convection inside 
A Spherical Container without Baffles????????????????..21 
3.1.1 Modeling Assumptions????????????????????.21 
3.1.2 Governing Equations?????????????????????22 
3.1.3 Boundary and Initial Conditions??????????????.??..23 
3.1.4 Dimensionless Form of Governing Equations???????.????.23 
3.2 Computational Details??????????????????????...26
   xi 
 
 
3.2.1 Mesh Generation ????????????????????.??.26 
3.2.2 FLUENT Configuration????????????????????27 
3.3 Results and Discussion??????????????????????..28 
3.3.1 Definition of the Nusselt Numbers???????????????...28 
3.3.2 Stream Function ??????????????????????...31 
   3.3.3 Fluid Flow and Thermal Fields ?????????????????32 
3.3.4 Code Validation???????????????????????36 
3.3.5 Correlation of the Pseudosteady-state Nusselt Numbers???????..38 
3.4 Closure????????????????????????????...40 
 
CHAPTER 4 EFFECT OF AN INSULATED BAFFLE ON PSEUDOSTEADY-STATE 
NATURAL CONVECTION INSIDE SPHERICAL CONTAINERS???????..62 
 
4.1 Mathematical Formulation for Pseudosteady-State Natural Convection inside 
Spherical Containers with a Thin Insulated Baffle????????????63 
4.1.1 Governing Equations and Boundary Initial Conditions????????63 
4.1.2 Computational Details????????????????????..64 
4.1.3 Code Validation ??????????????????????...65 
4.2 Grid and Time Step Size Independence Study ?????????????.66 
4.3 Results and Discussion ??????????????????????.69 
4.3.1 PSS Fluid Flow and Thermal Fields for Ra=10
4
, 10
5
 and 10
6
?????..70 
4.3.2 Time-Dependent Fluid Flow and Thermal Fields for Ra=10
7
 ??..??...74 
4.4 Nusselt Number Definitions and Other Parameters???????????...75 
   xii 
 
 
4.4.1 Definitions of the Nusselt Numbers?????...?????????...75 
4.4.2 Time-Averaged Nusselt Number????????????????..76 
4.4.3 Strength of Fluctuations of Nusselt Numbers???????????...78 
4.4.4 Stream Function ??????????????????????...79 
4.5 Variation of the Time-Average Nusselt Number and Stream Function ??.?..80 
4.6 Closure ??????????????????????????????87 
 
CHAPTER 5 EFFECT OF AN ISOTHERMAL BAFFLE ON PSEUDOSTEADY-
STATE NATURAL CONVECTION INSIDE SPHERICAL CONTAINERS???...125 
 
5.1 Mathematical Formulation for Pseudosteady-State Natural Convection inside 
Spherical Container with A Thin Isothermal Baffle???????????125 
5.1.1 Governing Equations and Boundary Initial Conditions???????..125 
5.1.2 Computational Details????????????????????127 
5.1.3 Code Validation ??????????????????????.128 
5.2 Grid and Time Step Size Independence Study ????????????...128 
5.3 Result and Discussion ??????????????????????.131 
5.3.1 PSS Fluid Flow and Thermal Fields for Ra=10
4
, 10
5
 and 10
6
?????131 
5.3.2 Time-Dependent Fluid Flow and Thermal Field for Ra= 10
7
 ??..??.136 
5.4 Nusselt Number Definitions and Other Parameters???????????.137 
5.4.1 Definition of the Nusselt Numbers..??????????????...137 
5.4.2 Time-Averaged Nusselt Numbers for Time-Dependent Cases..????143 
5.4.3 Oscillation Strength of the Nusselt Number???????????...144 
   xiii 
 
 
5.4.4 Stream Function Field????????????????????.144 
5.5 Variation of Time-Average Nusselt Numbers and Stream Function????...145 
5.6 Closure ?????????????????????????????..153 
 
CHAPTER 6 CONCLUSIONS AND RECOMMENDATION?????????..188 
 
6.1 Conclusions????????????????????????????188 
6.2 Recommendations for Future Work??????????????????...191 
 
REFERENCES??????????????....................................................?192 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   xiv 
 
 
LIST OF TABLES 
 
Table 3.1 Radial and polar angle positions of the eye of the recirculating  
vortex for the cases with no baffles??????????..???.?34 
Table 3.2 RMS and relative RMS of Nusselt Number???????????..36 
Table 3.3 Comparison of the pseudosteady-state Nusselt numbers ??????..37 
Table 3.4 Dimensionless maximum stream function (
*
max
? ) values ??????.38 
Table 4.1 Dependence of the time-averaged and RMS values of the Nusselt  
numbers on the number of cells for an insulated case with Ra=10
7
,  
Pr=0.7, L=0.25, 
o
b
90=?  and 
4
1057.1
?
?=?? ??????????.68 
Table 4.2 Dependence of the time-averaged and RMS values of the Nusselt 
 numbers on the time step size for an insulated case with Ra=10
7
,  
Pr=0.7, L=0.25, 
o
b
90=?  and 13,868 cells??????????..?..69 
Table 4.3 Nusselt numbers (
c
Nu ) and relative RMS (
c
rNu
RMS | ) for all 60  
cases with thin insulated baffles???????????????....84 
Table 4.4 Nusselt numbers (
m
Nu ) and relative RMS (
m
rNu
RMS | ) for all 60  
cases with thin insulated baffles?????...??????????.85 
 
 
   xv 
 
 
Table 4.5 Maximum stream fucntion (
max
? ) of the primary vortex and  
relative RMS (
max
|
?r
RMS ) for all 60 cases with thin insulated 
baffles??????????????????????.????86 
Table 5.1 Dependence of the time-averaged and RMS values of the Nusselt 
 numbers on the number of cells for an isothermal case with Ra=10
7
, 
Pr=0.7, L=0.25, 
o
b
90=?  and 
4
1057.1
?
?=?? ?????????...130 
Table 5.2 Dependence of the time-averaged Nusselt numbers on the time  
step size for an isothermal case with Ra=10
7
, Pr=0.7, L=0.25,  
o
b
90=?  and 13,868 cells???????????..????..??131 
Table 5.3 Nusselt numbers (
c
Nu ) and relative RMS (r-RMS) for all 60  
cases with thin isothermal baffles?????..?????????..150 
Table 5.4 Nusselt numbers (
m
Nu ) and relative RMS (r-RMS) for all 60  
cases with thin isothermal baffles?????..?????????..151 
Table 5.5 Maximum stream fucntion (
max
? ) of the primary vortex and  
relative RMS (
max
|
?r
RMS ) for all 60 cases with thin isothermal 
baffles?????????????????????...????152 
 
 
 
 
 
 
   xvi 
 
 
LIST OF FIGURES 
 
Figure 1.1 Typical external natural convection flow next to a hot  
 vertical plate at temperature T
wall
, with the motionless  
 fluid faraway at temperature 
?
T ????????????????..4 
Figure 1.2 Internal natural convection in a differentially-heated heated cavity???5 
Figure 1.3       Schematic relations between heat addition/extraction and 
                        bulk temperature trends in a container??????????????.6  
Figure 2.1 Natural convection flow pattern in a sphere for Ra=2x10
3
  
(Pustovolt, 1958)??????????????????????12 
Figure 2.2 Natural convection streamlines in a sphere at dimensionless  
time ?=0.03 (Left) and 0.10 (Right) (Whitley and Vachon, 1972)???13 
Figure 2.3 Natural convection flow pattern in a sphere for Ra=2.8x10
6
  
(Chow and Akins, 1975)????????...??????????.14 
Figure 2.4 Dependence of location of the eye of the recirculation pattern on  
Rayleigh number (Chow and Akins, 1975)???????????...15 
Figure 2.5 Location of eye of recirculation (a) and mean Nusselt number  
variation (b) for natural convection in sphere  
(Hutchins and Marschall, 1989)????????????????.16 
 
   xvii 
 
 
Figure 2.6 Streamline patterns and temperature contours in spherical for  
different Rayleigh number (Hutchins and Marschall, 1989)??.???17 
Figure 2.7 Streamline patterns and temperature contours in spherical for  
 different Rayleigh number (Shen et al., 1995)?????..?????.18 
Figure 2.8 Dependence of the recirculation vortex center position on  
 Rayleigh number (Shen et al., 1995)?..???????????...?19 
Figure 2.9 Streamlines and temperature field contours for composite  
 systems (Zhang et al., 1999)?..????????????????20 
Figure 3.1 Schematic diagram of the problem???????????????41 
Figure 3.2 Hybrid mesh created in GAMBIT???????????????.42 
Figure 3.3 Boundary layer mesh system?????????????????.43  
Figure 3.4 Detailed view of the boundary layer mesh????????????44 
Figure 3.5 Magnified view of part of the adopted hybrid mesh????????.45 
Figure 3.6 Solution controls in FLUENT????????????????...46 
Figure 3.7 Solver settings in FLUENT?????????????????...47 
Figure 3.8 Operating conditions in FLUENT???????????????.48 
Figure 3.9 Fluid properties in FLUENT?????????????????.49 
Figure 3.10 Thermal boundary conditions in FLUENT supplied by a  
user-defined function (UDF)?????????????????..50 
Figure 3.11 Momentum boundary conditions in FLUENT??????????...51 
Figure 3.12 Residual monitors in FLUENT????????????????..52 
 
 
   xviii 
 
 
Figure 3.13 Pseudosteady-state streamline patterns (left half) and  
corresponding temperature contours (right half) for cases with  
no baffles (Ra = 10
4
, 10
5
, 10
6
 and 10
7
)??..???????????53 
Figure 3.14 Thermally stable and unstable structures????????????...54 
Figure 3.15  Nusselt number as a function of dimensionless time for Ra=10
4
  
  with no baffle???????????????????????.55  
Figure 3.16  Nusselt number as a function of dimensionless time for Ra=10
5
  
  with no baffle???????????????????????.56  
Figure 3.17  Nusselt number as a function of dimensionless time for Ra=10
6
  
  with no baffle???????????????????????.57  
Figure 3.18  Nusselt number as a function of dimensionless time for Ra=10
7
  
  with no baffle???????????????????????.58  
Figure 3.19 Nusselt numbers relative differences between two different  
approaches????????????????????????.59 
Figure 3.20 Nusselt number (
c
Nu ) correlations??????????????...60 
Figure 3.21 Nusselt number (
m
Nu ) correlations??????????????...61 
Figure 4.1 Schematic diagram of a spherical container with a thin  
insulated baffle??????????????????????...88 
Figure 4.2 3-D View of the system ???????????????????89 
Figure 4.3 Grid systems with the same baffle (L=0.25) located at (a)
D
30=
b
? ,  
(b) 
D
60=
b
? , (c) 
D
90=
b
?  and (d) 
D
120=
b
?  ??????????...90 
 
   xix 
 
 
Figure 4.4 The Nusselt number 
m
Nu  (based on
mw
TTT ?=? ) as a function  
of grid size????????????????????????..91 
Figure 4.5 The Nusselt number 
c
Nu  (based on
cw
TTT ?=? ) as a function  
of grid size????????????????????????..92 
Figure 4.6 Nusselt number (
mw
TTT ?=? ) as a function of time step size???....93 
Figure 4.7 Nusselt number (
cw
TTT ?=? ) as a function of time step size????.94 
Figure 4.8 Pseudosteady-state streamline patterns and temperature contours  
for three insulated baffles (L = 0.05, 0.10 and 0.25) placed at  
various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for Ra = 10
4
???95 
Figure 4.9 Pseudosteady-state streamline patterns and temperature contours  
for three insulated baffles (L = 0.05, 0.10 and 0.25) placed at  
various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for Ra = 10
5
???96 
Figure 4.10 Pseudosteady-state streamline patterns and temperature contours  
for three insulated baffles (L = 0.05, 0.10 and 0.25) placed at  
various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for Ra = 10
6
???97 
Figure 4.11 Pseudosteady-state streamline patterns and temperature contours  
with an insulated baffle (L = 0.25) placed at various  
locations (
b
? = 30
o
, 90
o
 and 150
o
) for Ra = 10
4
, 10
5
, 10
6
 and 10
7
??.....98 
Figure 4.12 Streamline patterns and temperature contours in one cycle (a?h)  
for case with a thin insulated baffle (L=0.25, 
b
? =60
o
) for Ra=10
7
?...?99 
 
 
   xx 
 
 
Figure 4.13 Cyclic variation of the instantaneous area-averaged Nusselt  
number for case with a thin insulated baffle (L=0.25, 
b
? =60
o
)  
for Ra=10
7
 (Corresponding to Figure 4.12)?????.??????100 
Figure 4.14 Nusselt numbers )(?
c
Nu  oscillation with dimensionless time  
for case (a) with thin insulated baffle (L=0.10,
D
60=
b
? ) and  
case (b) with thin insulated baffle (L=0.05,
D
150=
b
? )?????...?.101 
Figure 4.15 Strength of Nusselt number (
c
Nu ) oscillation with dimensionless  
time for case with thin insulated baffle (L=0.25,
D
120=
b
? )???..?..102 
Figure 4.16 Strength (RMS) of Nusselt number (
c
Nu ) oscillation with  
dimensionless time for case with thin insulated  
baffle (L=0.25,
D
30=
b
? )??????????????????..103 
Figure 4.17 Dependence of the time-averaged Nusselt number (
c
Nu ) on  
 Ra among cases with a fixed thin insulated baffle (L=0.05) at  
 various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) and the  
 case without baffle ???????????????...????...104 
Figure 4.18 Dependence of the time-averaged Nusselt number (
c
Nu ) on  
 Ra among cases with a fixed thin insulated baffle (L=0.10) at  
 various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) and the  
 case without baffle ???????????????...????...105 
 
 
   xxi 
 
 
Figure 4.19 Dependence of the time-averaged Nusselt number (
c
Nu ) on  
 Ra among cases with a fixed thin insulated baffle (L=0.25) at  
 various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) and the  
 case without baffle ???????????????...????...106 
Figure 4.20 Dependence of the time-averaged Nusselt number (
m
Nu ) on  
 Ra among cases with a fixed thin insulated baffle (L=0.05) at  
 various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) and the  
 case without baffle ???????????????...????...107 
Figure 4.21 Dependence of the time-averaged Nusselt number (
m
Nu ) on  
 Ra among cases with a fixed thin insulated baffle (L=0.10) at  
 various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) and the  
 case without baffle ???????????????...????...108 
Figure 4.22 Dependence of the time-averaged Nusselt number (
m
Nu ) on  
 Ra among cases with a fixed thin insulated baffle (L=0.25) at  
 various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) and the  
 case without baffle ???????????????...????...109 
Figure 4.23 Dependence of the Maximum stream function 
max
? on  
 Ra among cases with a fixed thin insulated baffle (L=0.05) at  
 various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) and the  
 case without baffle ????????????????????..110 
 
   xxii 
 
 
Figure 4.24 Dependence of the Maximum stream function 
max
? on  
 Ra among cases with a fixed thin insulated baffle (L=0.10) at  
 various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) and the  
 case without baffle ????????????????????..111 
Figure 4.25 Dependence of the Maximum stream function 
max
? on  
 Ra among cases with a fixed thin insulated baffle (L=0.25) at  
 various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) and the  
 case without baffle ????????????????????..112 
Figure 4.26 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case  
without baffle and the cases with a thin insulated baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
4
???????.113 
Figure 4.27 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among  
case without baffle and the cases with a thin insulated baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
4
???????.114 
Figure 4.28 Dependence of the Maximum stream function 
max
?  on 
b
?   
among case without baffle and the cases with a thin insulated  
baffle of different lengths (L=0.05, 0.10 and 0.25) for Ra=10
4
??...?115 
Figure 4.29 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case  
without baffle and the cases with a thin insulated baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
5
???????.116 
 
 
   xxiii 
 
 
Figure 4.30 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among  
case without baffle and the cases with a thin insulated baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
5
???????.117 
Figure 4.31 Dependence of the Maximum stream function 
max
?  on 
b
?   
among case without baffle and the cases with a thin insulated  
baffle of different lengths (L=0.05, 0.10 and 0.25) for Ra=10
5
??...?118 
Figure 4.32 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case  
without baffle and the cases with a thin insulated baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
6
???????.119 
Figure 4.33 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among  
case without baffle and the cases with a thin insulated baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
6
???????.120 
Figure 4.34 Dependence of the Maximum stream function 
max
?  on 
b
?   
among case without baffle and the cases with a thin insulated  
baffle of different lengths (L=0.05, 0.10 and 0.25) for Ra=10
6
??...?121 
Figure 4.35 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case  
without baffle and the cases with a thin insulated baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
7
???????.122 
Figure 4.36 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among  
case without baffle and the cases with a thin insulated baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
7
???????.123 
 
   xxiv 
 
 
Figure 4.37 Dependence of the Maximum stream function 
max
?  on 
b
?   
among case without baffle and the cases with a thin insulated  
baffle of different lengths (L=0.05, 0.10 and 0.25) for Ra=10
7
??...?124 
Figure 5.1 Schematic diagram of a spherical container with  
a thin isothermal baffle???????????????????154 
Figure 5.2 The Nusselt number 
m
Nu  (based on
mw
TTT ?=? ) as  
a function of grid size...??????????????????...155 
Figure 5.3 The Nusselt number 
c
Nu  (based on
cw
TTT ?=? ) as  
a function of grid size???????????????????..156 
Figure 5.4 The Nusselt number 
m
Nu  (based on
mw
TTT ?=? ) as  
a function of time step size???..??????????????157 
Figure 5.5 The Nusselt number 
c
Nu  (based on
cw
TTT ?=? ) as  
a function of time step size????..?????????????158 
Figure 5.6 Pseudosteady-state streamline patterns and temperature contours  
for three isothermal baffles (L = 0.05, 0.10 and 0.25) placed at  
various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for Ra = 10
4
 ??.159 
Figure 5.7 Pseudosteady-state streamline patterns and temperature contours  
for three isothermal baffles (L = 0.05, 0.10 and 0.25) placed at  
various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for Ra = 10
5
??..160 
Figure 5.8 Pseudosteady-state streamline patterns and temperature contours  
for three isothermal baffles (L = 0.05, 0.10 and 0.25) placed at  
various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for Ra = 10
6
??..161 
   xxv 
 
 
Figure 5.9 Pseudosteady-state streamline patterns and temperature contours  
with an isothermal baffle (L = 0.25) placed at various locations 
 (
b
? = 30
o
, 90
o
 and 150
o
) for Ra = 10
4
, 10
5
, 10
6
 and 10
7
?????..?162 
Figure 5.10 Streamline patterns and temperature contours in one  
cycle (a?i) for case with a thin isothermal baffle  
(L=0.25, 
b
? =60
o
) for Ra=10
7
????????????????..163 
Figure 5.11 Cyclic variation of the instantaneous area-averaged Nusselt  
number for case with a thin isothermal baffle (L=0.25, 
b
? =60
o
)  
for Ra=10
7
 (Corresponding to Figure 5.10)???????.????164 
Figure 5.12 Detailed drawing of an isothermal baffle surface ????????...165 
Figure 5.13 Nusselt number (
c
Nu ) oscillation with dimensionless time  
for a case (a) with a thin isothermal baffle (L=0.25,
D
60=
b
? ,  
Ra=10
7
) and case (b) with a thin isothermal baffle (L=0.05, 
D
150=
b
? , Ra=10
7
)????????????????????..166 
Figure 5.14 Dependence of the time-average Nusselt number (
c
Nu ) on  
 Ra among cases with a fixed thin isothermal baffle (L=0.05)  
 at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
)  
 and the case without baffle ?????????????????.167 
 
 
 
 
   xxvi 
 
 
Figure 5.15 Dependence of the time-average Nusselt number (
c
Nu ) on  
 Ra among cases with a fixed thin isothermal baffle (L=0.10)  
 at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
)  
 and the case without baffle ?????????????????.168 
Figure 5.16 Dependence of the time-average Nusselt number (
c
Nu ) on  
 Ra among cases with a fixed thin isothermal baffle (L=0.25)  
 at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
)  
 and the case without baffle ?????????????????.169 
Figure 5.17 Dependence of the time-average Nusselt number (
m
Nu ) on  
 Ra among cases with a fixed thin isothermal baffle (L=0.05)  
 at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
)  
 and the case without baffle?????????????????..170 
Figure 5.18 Dependence of the time-average Nusselt number (
m
Nu ) on  
 Ra among cases with a fixed thin isothermal baffle (L=0.10)  
 at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
)  
 and the case without baffle?????????????????..171 
Figure 5.19 Dependence of the time-average Nusselt number (
m
Nu ) on  
 Ra among cases with a fixed thin isothermal baffle (L=0.25)  
 at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
)  
 and the case without baffle?????????????????..172 
 
   xxvii 
 
 
Figure 5.20 Dependence of the Maximum stream function 
max
?  on Ra  
 among cases with a fixed thin isothermal baffle (L=0.05)  
 at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
)  
 and the case without baffle????????????...???.......173 
Figure 5.21 Dependence of the Maximum stream function 
max
?  on Ra  
 among cases with a fixed thin isothermal baffle (L=0.10)  
 at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
)  
 and the case without baffle????????????...???.......174 
Figure 5.22 Dependence of the Maximum stream function 
max
?  on Ra  
 among cases with a fixed thin isothermal baffle (L=0.25)  
 at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
)  
 and the case without baffle????????????...???.......175 
Figure 5.23 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case  
without baffle and cases with a thin isothermal baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
4
??.?????176 
Figure 5.24 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among case  
without baffle and cases with a thin isothermal baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
4
??.?????177 
Figure 5.25 Dependence of the Maximum stream function (
max
? ) on 
b
?   
among case without baffle and cases with a thin isothermal  
baffle of different lengths (L=0.05, 0.10 and 0.25) for Ra=10
4
?...??178 
   xxviii 
 
 
Figure 5.26 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case  
without baffle and cases with a thin isothermal baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
5
??.?????179 
Figure 5.27 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among case  
without baffle and cases with a thin isothermal baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
5
??.?????180 
Figure 5.28 Dependence of the Maximum stream function (
max
? ) on 
b
?   
among case without baffle and cases with a thin isothermal  
baffle of different lengths (L=0.05, 0.10 and 0.25) for Ra=10
5
?...??181 
Figure 5.29 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case  
without baffle and cases with a thin isothermal baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
6
??.?????182 
Figure 5.30 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among case  
without baffle and cases with a thin isothermal baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
5
??.?????183 
Figure 5.31 Dependence of the Maximum stream function (
max
? ) on 
b
?   
among case without baffle and cases with a thin isothermal  
baffle of different lengths (L=0.05, 0.10 and 0.25) for Ra=10
6
?...??184 
Figure 5.32 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case  
without baffle and cases with a thin isothermal baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
7
??.?????185 
 
   xxix 
 
 
Figure 5.33 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among case  
without baffle and cases with a thin isothermal baffle of  
different lengths (L=0.05, 0.10 and 0.25) for Ra=10
7
??.?????186 
Figure 5.34 Dependence of the Maximum stream function (
max
? ) on 
b
?   
among case without baffle and cases with a thin isothermal  
baffle of different lengths (L=0.05, 0.10 and 0.25) for Ra=10
7
?...??187 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   xxx 
 
 
NOMENCLATURE 
 
English Symbols 
a constant in approximated linear equation 
b slope of approximated linear equation 
c constant value in correlation 
c
p
 specific heat at constant pressure, J/(kgK) 
C constant temperature difference between wall and container center, 
o
C 
D diameter of the sphere, m 
g gravitational acceleration, m/s
2
 
g
r 
radial component of the gravitational acceleration, m/s
2
 
g
r
*
 
dimensionless radial component of the gravitational acceleration 
g
? 
polar component of the gravitational acceleration, m/s
2
 
g
?
*
 
dimensionless polar component of the gravitational acceleration 
Gr Grashof number, g??
2
D
3
(T
w
 - T
c
)/?
2
 
h  heat transfer coefficient, W/m
2
K 
k thermal conductivity, W/mK 
l length of baffle, m 
L dimensionless length of baffle, l/D 
Nu
c
 time-averaged Nusselt number, qD/k(T
w
- T
c
)
   xxxi 
 
 
Nu
c
(?) area-averaged Nusselt number 
Nu
m
 time-averaged Nusselt number, qD/k(T
w
- T
m
) 
Nu
m
(?) area-averaged Nusselt number 
p pressure, Pa 
p
0
 initial static pressure, Pa 
p
*
 dimensionless pressuer, (p-p
0
)R
2
/??
2
 
Pr Prandtl number of the fluid, ?/? 
q heat flux, W/m
2
 
r radial coordinate within the sphere, m 
r
*
 dimensionless radial position, defined as r/R 
r
e
 radial position of vortex eye, m 
r
e
*
 dimensionless radial position of vortex eye, r
e
/R 
R radius of the sphere, m 
Ra Rayleigh number, Gr Pr 
t time, s 
T temperature, K 
T
c
 temperature at the center of sphere, K 
T
m
 mean or bulk temperature of the fluid, K 
T
w
 temperature of the spherical container inner surface, K 
T
0
 initial temperature of the fluid, K 
T
*
 dimensionless temperature, defined as (T - T
0
) / (T
w
 ? T
c
)  
V
r
 radial component of the fluid velocity, m/s 
V
?
 polar component of the fluid velocity, m/s 
   xxxii 
 
 
V
r
*
 dimensionless radial component of the fluid velocity, V
r
R/? 
V
?
*
 dimensionless polar component of the fluid velocity, V
?
R/? 
 
Greek Symbols 
? thermal diffusivity of the fluid, m
2
/s 
? coefficient of thermal expansion, K
-1
 
? polar angle in the sphere, degrees 
?
b
 polar angle location of the baffle, degrees 
?
e
 polar angle of the vortex eye, degrees 
? fluid viscosity, kg/(ms) 
? kinematic viscosity, m
2
/s 
? generic variable  
? cross-correlation coefficient 
? density of the fluid, kg/m
3
 
? dimensionless time, ?t/R
2
 
? stream function, m
3
/s 
?
*
 dimensionless stream function, ?/?R 
?
max
 maximum stream function, m
3
/s 
?
*
max
 dimensionless maximum stream function, ?
max
/?R 
 
Subscripts 
b related to baffle 
   xxxiii 
 
 
e related to vortex eye 
r related to the radial direction 
? related to the polar direction 
 
Superscripts 
* dimensionless value 
 
 1
CHAPTER 1 INTRODUCTION 
 
Buoyancy-driven or natural convection is a very important thermal transport 
mechanism that has been studied for many decades.  In natural convection flows, the 
buoyancy effect due to the strong dependence of the fluid density on temperature plays 
the key role.  Natural convection problems can generally be categorized in two broad 
classes, namely: (1) external and (2) internal.  In the external group of problems, an 
infinite amount of fluid is of interest such as the one shown in Figure 1.1.  In this figure, 
an infinite body of fluid is shown next to a vertical wall.  The temperature of the wall 
(T
wall
) and the temperature of quiescent fluid far away (
?
T <T
wall
) are the two distinct 
temperature scales that are generally taken to be constants.  Development of thin 
momentum and thermal boundary layers next to the wall plays the key role in this 
problem.  Accordingly, a variety of analytical/computational and experimental tools have 
been utilized for solving this and problems related to it.   
In contrast, for internal natural convection problems, a finite amount of fluid is 
under consideration.  The idealized problem of a differentially-heated cavity (Figure 1.2) 
is the simplest example of an internal natural convection problem in which a fluid is 
placed between two vertical walls with constant temperatures T
cold
 and T
hot
, whereas the 
two horizontal walls are insulated.  One can generally state that external flows are 
characterized by simpler phenomenon related to uni-directional boundary layer flow, 
whereas internal flows may exhibit further complications such as multiple recirculating
 2
vortices.  In spite of these complications, it is possible to formulate a steady formulation 
for both problems mentioned so far. 
In practice, internal natural convection problems involve geometries which are 
more complicated than the idealized differentially-heated cavity problem.  For instance, 
cylindrical or spherical containers that hold fluids are commonly utilized in industrial 
applications.  Besides the more complicated geometry, these problems are such that one 
can not generally specify a steady formulation for them.  For instance, consider a fluid 
that is held in an arbitrary-shaped container such that it has reached a motionless state 
and has a constant temperature.  If the container is suddenly exposed to a new 
environment such that the temperature of the surface is lowered or increased, a transient 
process will ensue.  This behavior is due to presence of just one temperature scale in the 
problem (i.e. T
wall
).  The temperature difference between that of the wall and the initial 
fluid temperature will give rise to buoyancy-driven convection.  As shown in Figure 1.3, 
any heat that is added or extracted from the fluid will cause the bulk fluid temperature to 
rise or decline, respectively.  In effect, one will be faced to solve an unsteady multi-
dimensional formulation to recover the complex phenomena in this class of problems. 
Given this background, a computational study of pseudosteady-state natural 
convection within spherical containers without and with baffles was performed.  
Spherical containers are widely used in different industries to store, transport and process 
fluids.  The pseudosteady-state condition is distinct from a transient problem.  In a 
transient problem, a motionless fluid will be perturbed early on and will finally settle 
down to a motionless state with a higher or lower temperature.  In a pseudosteady-state 
 3
formulation, the flow and thermal fields will indefinitely evolve with time.  However, 
bulk quantities such as the Nusselt number will be independent of time. 
Following a review of literature in Chapter 2, the mathematical formulation and 
benchmarking are presented in Chapter 3.  Pseudosteady-state natural convection due to 
presence of insulated and isothermal thin baffles placed on the inner wall of a sphere are 
discussed in great detail in Chapters 4 and 5, respectively.  General conclusions are 
finally given in Chapter 6. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4
 
 
 
Figure 1.1 Typical external natural convection flow next to a hot vertical plate at 
temperature T
wall
, with the motionless fluid far away at temperature 
?
T  
 
x
wall
T
?
T
z
Velocity profile
eTemperatur profile
Boundary layer
g
G
0=
?
u
 5
 
 
 
Figure 1.2 Internal natural convection in a differentially-heated cavity 
 
 
 
 
 
 
 
 
cold
T
hot
T g
G
x
z
0=
?
?
z
T
0=
?
?
z
T
 6
 
 
 
 
Figure 1.3 Schematic relations between heat addition/extraction and bulk temperature 
trends in a container 
 
 
 
 
wall
T
g
G
Heat Added
Heat Extracted
m
T
m
T
 7
CHAPTER 2 LITERRATURE REVIEW OF NATURAL CONVECTION INSIDE 
SPHERICAL CONTAINERS 
 
Containment of fluids within spherical vessels is very common in industrial 
applications in relation to storage and transport of cryogenic liquids and propellants.  
Research on natural convection within spherical containers at both applied and 
fundamental levels has been reported for some time.  A review of the relevant literature is 
given in this Chapter.   
Schmidt (1956) can be credited as the first researcher to have reported on 
unsteady heat transfer inside spherical containers subject to a step change in wall 
temperature.  Measurements were carried out for five different diameters ranging from 
100 to 500 mm.  By utilizing water and three alcohols (methyl, ethyl and butyl) with 
different properties, a wide range of Rayleigh numbers (2x10
9
?5x10
11
) was covered.   
Pustovoit (1958) reported results of an approximate method to solve the coupled 
equations governing slowly-varying axisymmetric natural convection inside a spherical 
container by expanding the dependent variables (velocity, pressure and temperature) in 
terms of the Grashof number.  Streamlines and temperature contours were given for water 
(Pr = 6.75) and Gr = 300 for the case of a step change in the wall temperature (Figure 
2.1).  With the initial liquid temperature being higher than the constant surface 
temperature, the theoretical results suggested that the liquid that was being cooled sank 
near the surface replacing the hot liquid that rose vertically in the middle of the sphere.
 8
The upper part of the sphere was at a higher temperature compared to the lower section.  
It was observed that the liquid was cooled least in a region located at about one third of 
the radius above the center, where molecular conduction was dominant.   
Whitley and Vachon (1972) presented their numerical solutions of transient, 
laminar natural convection heat transfer inside spheres (Figure 2.2).  The flow field was 
assumed to be axisymmetric, thus the dependent variables were functions of time and two 
spatial variables (two-dimensional flow).  A finite-difference-based vorticity-stream 
function numerical procedure employing an implicit formulation of the temporal 
derivatives was employed.  Subjecting the wall temperature to a sudden step change, the 
quiescent air in the sphere was set in motion and after a long period of time attained a 
new equilibrium state with zero motion.  In this configuration, air being heated next the 
surface rose upward, replacing the cooler air which sank vertically to the bottom of the 
sphere.   
Chow and Akins (1975) presented their experimental investigation of the 
pseudosteady-state natural convection inside spheres (Figure 2.3).  The main feature of 
the pseudosteady-state treatment is that the driving force for convection is continuously 
maintained, i.e. the temperature outside the sphere was increased so that the temperature 
difference between the outside and the center of the sphere remained constant.  During 
the flow visualization phase of the study, the motion of hollow glass spheres was 
monitored and the location of the eye of the recirculation pattern was quantified (Figure 
2.4).  Comparisons with the analytical results of Pustovoit (1958) and computational data 
of Whitley and Vachon (1972) were attempted.  However, previous mathematical 
analyses treated a transient phenomenon, whereas Chow and Akins (1975) strived to 
 9
maintain pseudosteady-state conditions.  In view of this, direct comparisons should be 
considered unwarranted.  They also provided an empirical correlation for the mean 
Nusselt number in the laminar regime (Rayleigh number below 10
7
).   
Val?tsiferov and Polezhaev (1975) refer to a number of articles directed at 
theoretical and experimental studies of natural convection in spherical vessels filled with 
a liquid.  Among them, only the work of Pustovoit (1958) and Whitley and Vachon 
(1972) are widely available.  They also refer to experimental data for given surface heat 
flux values with relatively high Rayleigh (10
9
?10
11
) and Prandtl numbers (3?1500), for 
which the characteristics of convection were found to be contradictory.  They continued 
to present a computational study of natural convection in a sphere with a thin-walled shell 
for a range of the Rayleigh and Fourier numbers that covered the principal conditions of 
time-dependent laminar convection due to given heat flux on the surface of the sphere.   
Transient natural convection within a thin-walled copper sphere filled with three 
liquids was investigated by Owen and Jalil (1986) both experimentally and 
computationally.  For the experimental study, the sphere was suddenly immersed in a 
bath of boiling water and allowed to reach equilibrium.  Temperature readings along the 
diameter of the sphere at various inclination angles were obtained.  The key observation 
was that the temperature measurements at the bottom of the sphere exhibited oscillating 
behavior indicative of the interaction of sinking cold liquid with the rising hot liquid.   
Detailed computational results for pseudosteady-state, two-dimensional, natural 
convection inside spheres were reported by Hutchins and Marschall (1989).  Similar to 
Whitley and Vachon (1972), they utilized a finite-difference-based vorticity-stream 
function numerical procedure to simulate the flow and temperature fields in the laminar 
 10
regime (10
5
 < Ra < 10
8
) and two different Prandtl numbers of 0.7 and 8.  In order to 
compare to the experimental results of Chow and Akins (1975), a case of Pr = 4 and Ra = 
5.2 ? 10
7
 was also simulated.  The general trends of their computed eye of the 
recirculation zone and mean Nusselt number variation were similar to the findings of 
Chow and Akins (1975) (Figure 2.5).  The heat transfer data were found to be 
independent of the Prandtl number.  Both streamline and temperature contours were 
presented, showing their dependence on the Rayleigh number (Figure 2.6).  Finally, they 
proposed a mean Nusselt number correlation applicable for the Rayleigh number range of 
10
5
 to 10
8
.   
Research at Auburn University concentrated on repeating the work of Hutchins 
and Marschall (1989) with the objective of benchmarking a CFD code in the spherical 
coordinate system (Shen et al., 1995).  Their findings were generally in concert with the 
predictions of Hutchins and Marschall (1989) and experimental observations of Chow 
and Akins (1975) (Figures 2.7 and 2.8).   
Transient natural convection in a spherical composite system that was partially 
filled with a porous medium was studied by Nguyen, Paik and Pop (1997).  Using a 
hybrid spectral method, they numerically simulated the unsteady flow and thermal fields.  
They concluded that heat transfer results were not sensitive to the variation of the Prandtl 
number for the range considered (0.5 to 10).  It was established that the overall heat 
transfer rate was primarily controlled by the transport characteristics in the porous 
medium and the thermal conductivity ratio.   
Zhang et al. (1999) extended the work of Nguyen et al. (1997) by studying the 
pseudosteady-state behavior of composite systems (Figure 2.9).  A parametric study was 
 11
performed for a range of values of the Rayleigh number (Ra), Darcy number (Da) and the 
thermal conductivity ratio.  The local Nusselt number on the surface and interface 
temperature exhibited nearly uniform variations for low Ra and Da numbers, signifying 
little deviation from the limiting pure conduction case.  For high Ra and Da numbers, 
significant heat transfer was observed on the bottom of the sphere.  The interface 
temperature was also seen to deviate from uniform variation for high Ra and Da numbers.  
The intensity of the recirculating flow in the central fluid core region was observed to 
depend on the thermal conductivity ratio.  The thermal conductivity ratio modified the 
time scale of the thermal transport and only the relative magnitudes of the monitored 
quantities were affected.  A recent study by Khodadadi et al. (1999) focused on analysis 
of pseudosteady-state natural convection within rotating spheres.  Their results for the 
limiting case of no rotation were obtained for Ra numbers different than a previous study 
by Shen et al. (1995). 
In extending the breadth of previous work, the present study was initiated to 
explore the utilization of baffles within spherical containers in order to control the flow 
field and consequently heat transfer, through its enhancement and/or degradation.  In 
analyzing the limiting cases for the thermal state of the baffle, the pseudosteady-state 
natural convection within spherical containers due to presence of both insulated and 
isothermal baffles are discussed in this thesis.   
 
 
 
 
 12
 
 
 
Figure 2.1 Natural convection flow pattern in a sphere for Ra=2x10
3
 (Pustovoit, 1958) 
 
 
 
 
 13
 
 
 
 
 
 
Figure 2.2 Natural convection streamlines in a sphere at dimensionless time ?=0.03 (Left) 
and 0.10 (Right) (Whitley and Vachon, 1972) 
 
 
 
 
 
 
 14
 
 
Figure 2.3 Natural convection flow pattern in a sphere for Ra=2.8x10
6
 (Chow and Akins, 
1975) 
 
 
 
 
 15
 
 
Figure 2.4 Dependence of the location of the eye of the recirculation pattern on the 
Rayleigh number (Chow and Akins, 1975) 
 
 
 16
 
 
 
Figure 2.5 Location of the eye of recirculation (a) and mean Nusselt number variation (b) 
for natural convection in a sphere (Hutchins and Marschall, 1989) 
 
 
 
)(a
)(b
 17
 
 
 
 
Figure 2.6 Streamline patterns and temperature contours in a sphere for different 
Rayleigh numbers (Hutchins and Marschall, 1989) 
 
 
 
 
8
10=Ra
6
10=Ra
5
10=Ra
7
10=Ra
 18
 
 
 
 
 
Figure 2.7 Streamline patterns and temperature contours in a sphere for different 
Rayleigh numbers (Shen et al., 1995) 
 
 
 
 
 
6
10258.1 ?=Ra
7
10675.1 ?=Ra
4
10649.1 ?=Ra
5
10879.1 ?=Ra
 19
 
 
 
 
 
 
 
 
 
Figure 2.8 Dependence of recirculating vortex center position on the Rayleigh number 
(Shen et al., 1995) 
 
 
 
 
 
 
 
 
 
 
 
 20
 
 
 
 
 
Figure 2.9 Streamlines and temperature field contours for composite systems (Zhang et 
al., 1999) 
 21
CHAPTER 3 COMPUTATIONAL METHODOLOGY AND BENCHMARKING 
 
The limiting case of pseudosteady-state natural convection within spheres with no 
baffles is presented in this Chapter.  Time-dependent computations were performed for 
the case of a motionless fluid at a constant temperature subjected to a step change of the 
wall temperature at time t = 0.  By varying the wall temperature such that it is always 
greater than the temperature at the center of the sphere by a constant value, the system 
will attain the pseudosteady-state condition after some time period.   
In this Chapter, details of the computational methodology are presented and the 
results corresponding to the pseudosteady-state natural convection within spherical 
containers with no baffles are presented.  The predictions are then compared to the 
available data in literature.   
 
3.1 Mathematical Formulation for the Pseudosteady-State Natural Convection 
inside A Spherical Container without Baffles 
3.1.1 Modeling Assumptions 
Consider a spherical container with a diameter of D = 2R that is completely filled 
with a fluid.  In view of the spherical shape, the orthogonal spherical coordinate system is 
adopted.  A schematic diagram for the posed problem is shown in Figure 3.1 in which a 
typical computational mesh is shown on the left side.  In deriving the governing 
equations, the following assumptions are invoked: 
 22
I. The dependent variables (velocity, pressure and temperature) are symmetric 
with respect to the azimuthal direction (? ).  Therefore, they are functions of 
only two spatial coordinates, i.e. radial coordinate (r) and ? .  Thus, the flow 
field is two-dimensional.  The gravitational acceleration points downward along 
the symmetry axis.  
II. The fluid is viscous and Newtonian with constant properties, except for 
variation of density that is modeled using the Boussinesq approximation. 
III. Viscous heating effects are ignored. 
 
3.1.2 Governing Equations 
Based on these assumptions, the appropriate governing equations in the spherical 
coordinate system can be developed (Bird, Stewart and Lightfoot, 1960).  The continuity 
equation is: 
.sin
sin
0 = )  
V
 (
 r
1
 + ) 
Vr
 (
r
r
1
r
2
2
?
??
?
?
?
?
?
       (3.1) 
The momentum equations in the radial (r) and polar (? ) directions are: 
,
cot
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
 
V
r
2
 - 
V
r
2
 - 
V
r
2
 - 
V
   + )
T
-(Tg - 
r
)p-(p
- =
  
r
V
 - 
V
r
V
 + 
r
V
V
 + 
t
V
  
22
r
2
r0
r
0
2
rr
r
r
?
?
??
?
?
???
?
?
  (3.2) 
and: 
,
sin
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
 
r
V
 - 
V
r
2
 + 
V
   + )
T
-(Tg - 
)p-(p
r
1
- =
  
r
VV
 + 
V
r
V
 + 
r
V
V + 
t
V
  
22
r
2
2 
0
0
r
r
??
???
?
?
?
?
?
?
?????
   (3.3) 
with: 
 23
.sin
sin
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?   
 
r
1
 +  
r
r
 
r
r
1
 = 
2
2
2
?
?
??
     (3.4) 
Finally, the energy equation is: 
.
2
T k = 
T
r
V
 + 
r
T
V
 + 
t
T
 
c r
p
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
       (3.5) 
 
 
3.1.3 Boundary and Initial Conditions 
 
The no-slip boundary condition is applied on the solid wall.  Therefore, on the 
solid surface of the sphere (r = R) the boundary condition can be described as: 
.0 = 
V
 = 
V r?
          (3.6) 
The wall temperature of the sphere (T
w
) was allowed to vary with time so that it was 
always above the temperature of the center (T
c
) of the sphere by a constant value, i.e.:  
.)()( CtTtT
cw
+=          (3.7)  
The boundary conditions on the symmetry axis of the sphere (? = 0 and ? ) are: 
.0 = 
T
 = 
V
 = 
V
r
??
?
?
?
?
?
         (3.8) 
At the beginning of the computations (t=0), the fluid velocity was set to zero and 
the fluid temperature was initialized to a constant value (T
o
).  
 
3.1.4 Dimensionless Form of the Governing Equations 
In order to analyze this natural convection problem, it is necessary to determine 
the dimensionless parameters that influence the solution.  Therefore, the governing 
equations can be rewritten by introducing appropriate dimensionless variables.  The 
appropriate length scale is the radius of the spherical container (R).  There is no free- 
 24
stream velocity in this natural convection problem, thus the reference velocity is taken to 
be 
R
V
?
= .  The resulting dimensionless variables that are denoted by an asterisk 
superscript are listed as follows: 
R
r
r =
*
    
g
g
g
r
r
=
*
    
g
g
g
?
?
=
*
 
2
R
t?
? =     
cw
TT
TT
T
?
?
=
0*
    
?
RV
V
r
r
=
*
 
?
?
?
RV
V =
*
    .
)(
2
2
0
*
??
Rpp
p
?
=     (3.9) 
Substituting these dimensionless variables into equations 3.1-3.5, the dimensionless 
governing equations can be obtained.  These are: 
Dimensionless continuity equation: 
.sin
sin
*
*
*
*
*
*
0 = )  
V
 (
 r
1
 + ) 
Vr
 (
r
r
1
r
2
2
?
??
?
?
?
?
?
     (3.10) 
Dimensionless momentum equation in the radial (r) direction: 
.
cot
PrPr
8
1
*
*
*
*
*
*
*
2
***
*
*
*
2
*
*
*
*
*
*
*
*
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
 
V
r
2
 - 
V
r
2
 - 
V
r
2
 - 
V
   + TgRa - 
r
p
- =
r
V
 - 
V
r
V
 + 
r
V
V
 + 
V
22
r
2
r
r
rr
r
r
?
?
??
?
?
??
  (3.11) 
Dimensionless momentum equation in the polar (?) direction: 
.
sin
PrPr
8
1
*
**
*
*
*
**
*
*
*
***
*
*
*
*
*
*
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
 
r
V
 - 
V
r
2
 + 
V
   +TgRa - 
p
r
1
- =
r
VV
 + 
V
r
V
 + 
r
V
V + 
V
 
22
r
2
2 
r
r
???
??
?
?
?
?????
   (3.12) 
With the dimensionless Laplacian operator defined as:                                  
 25
.sin
sin
*
*
*
*
*
2
*
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?   
 
r
1
 +  
r
r
 
r
r
1
 = 
2
2
2
?
?
??
     (3.13) 
Finally, the dimensionless energy equation is: 
.
*
2
*
*
*
*
*
*
*
*
T =
T
r
V
 + 
r
T
V
 + 
T
r
?
?
?
?
?
?
?
??
?
       (3.14) 
The dimensionless boundary conditions are: 
0 = 
V
 =
V r
**
?
 for ,1
*
=r         (3.15) 
,1)()(
**
+= ??
cw
TT          (3.16) 
0 = 
T
 = 
V
V
r
??
?
?
?
?
?
=
**
*
 for 0=?  and ? .      (3.17) 
Finally, the initial conditions at 0=?  are: 
.0
***
=== TVV
r ?
         (3.18) 
Inspecting the dimensionless governing equations, it is found that the 
pesudosteady-state behavior of this system is characterized by the Rayleigh number 
(
??
? )(
3
cw
TTDg
Ra
?
= ) and Prandtl number (
?
v
=Pr ).  The Rayleigh number is a measure 
of the strength of buoyancy force in comparison to viscous force, whereas the Prandtl 
number is a ratio of momentum diffusivity over thermal diffusivity.  Previous findings 
have established the weak dependence of the results on the Prandtl number.  Therefore, a 
single Prandtl number equal to 0.7 was used.  The Rayleigh number is varied with Ra = 
10
4
, 10
5
, 10
6
 and 10
7
 (mainly in the laminar regime).  
 
 
 
 26
3.2 Computational Details 
The formulated problem above and variations of it were solved by Shen et al. 
(1995) and Khodadadi et al. (1999) who used a research code.  In view of the anticipated 
difficulty associated with presence of baffles in a sphere, the solution of the governing 
equations in the present study was handled by use of a commercial computational fluid 
dynamic (CFD) package.  The computational mesh was generated using GAMBIT 
(version 2.2.30) and the commercial code FLUENT (version 6.2.16) was utilized for 
solving the governing equations.  The computations are based on an iterative, finite-
volume numerical procedure using primitive dependent variables, whereby the time-
dependent, two-dimensional axisymmetric form of the governing continuity, momentum 
and energy equations are solved.  All the computations were performed on the Cray XD1 
supercomputer of the Alabama Supercomputer Authority (Huntsville, Alabama).   
 
3.2.1 Mesh Generation  
In view of strong convective heat transfer that is expected next to the wall, a 
hybrid mesh (Figure 3.2) that consists of a ?boundary layer? mesh (Figure 3.3) and an 
?interior mesh? was employed.  The ?boundary layer? mesh next to the wall is much 
denser than the ?interior? mesh.  The details of the grid system provided next are not 
specific to the cases with no baffles and similar steps with minor changes were taken for 
cases with baffles (Chapters 4 and 5).  A more detailed discussion in relation to the grid 
and time step size independence tests will be given in Chapter 4.  
Combining theoretical estimates of the thickness of a boundary layer next to a 
vertical heated wall and numerical tests, the ?boundary layer? thickness in the radial 
 27
direction is set to be 4% of diameter.  The thickness of the first row of the ?boundary 
layer? mesh in the radial direction is set to be 0.1% of diameter.  In going from the first 
row to last row of the ?boundary layer? mesh, the radial grid spacing expanded with a 
growth factor equal to 1.1.  A total of 17 radial rows are created in the ?boundary layer? 
mesh.  The spacing of the adjacent nodes in the polar direction on the first row of the 
?boundary layer? mesh is set to be 0.98% of diameter and a total of 160 polar nodes are 
laid on the first row (Figure 3.4).  For the ?interior? mesh, unstructured grids were 
created.  A close-up view of both mesh systems is shown in Figure 3.5. The spacing 
between any two adjacent nodes in the ?interior mesh? is less than 1.15% of diameter.  
Thus, there are 6,610 nodes and 10,222 cells in the grid network, whereas there were 
about 900 cells in the entire mesh network of previous studies (e.g. Hutchins and 
Marschall (1989), Shen et al. (1995)).  
 
3.2.2 FLUENT Configuration 
The SIMPLE method (Patankar, 1980) was utilized for solving the governing 
equations.  The QUICK differencing scheme was used for solving the momentum and 
energy equations, whereas the PRESTO scheme was adopted for the pressure correction 
equation.  The under-relaxation factors for the velocity components, pressure correction 
and thermal energy were 0.7, 0.3 and 1, respectively (Figure 3.6).  Second-order accurate 
implicit discretization was utilized for the unsteady term (Figure 3.7).  The operating 
pressure and temperature were 0.1 MPa and 288.16 K, respectively.  The effect of gravity 
is applied using the Boussinesq approximation (Figure 3.8).  Air is selected as the 
working fluid and all the material properties, except density, were assumed to be 
 28
independent of temperature (Figure 3.9).  The Rayleigh number was varied by adjusting 
the thermal expansion coefficient.  A UDF (User-Defined Function) file is utilized to 
apply the thermal boundary condition (Figure 3.10) and the settings for applying the 
momentum boundary conditions are shown in Figure 3.11.  
The time step size is 2 seconds (
4
1057.1
?
?=?? ).  At each time step, the 
continuity and velocity residuals were monitored.  An iteration was terminated and 
solution was forwarded in time when all the three residuals were less than 10
-6
 (Figure 
3.12).  At each time step, the number of iterations varied between 100 and 200.   
  
3.3 Results and Discussion  
3.3.1 Definitions of the Nusselt Numbers   
The Nusselt number can be considered as a dimensionless heat transfer coefficient 
and a significant parameter that is used to describe the strength of the pseudosteady-state 
natural convection inside spherical containers without baffles.  Similar to the work done 
by Hutchins and Marschall (1989), the Nusselt numbers either can be derived from:  
(a) the temperature gradient on the wall (direct approach), or 
(b) performing a lumped energy balance for the fluid within the container 
(indirect approach). 
Theoretically, these two approaches should give the same values when the 
pseudosteady-state is reached.  The accuracy and verification of computational solution 
can be evaluated by inspecting the difference between the Nusselt numbers values that 
are determined by two different approaches. 
(a) Direct Approach 
 29
In order to obtain the instantaneous area-averaged Nusselt number over the entire 
wall surface, the area-averaged heat flux can be expressed as: 
.sin|
2
1
2sin
2sin||
)(
0
0
0
0
0
?
?
?
?
?
=
==
?
?
=
?
?
=
?
?
=
?
?
?
?
?
??
???
???
d
r
T
k
RdR
RdR
r
T
k
dA
dA
r
T
k
tq
Rr
RrRr
     (3.19) 
This quantity might depend on time due to: 
1. transient behavior during the establishment of the flow early on, and/or 
2. instability issues for high Rayleigh number cases. 
The area-averaged Nusselt number can be calculated employing the general definition: 
?
=
?
?
?
=
?
==
?
??
0
.sin|
2
)(
)( d
r
T
T
D
Tk
Dtq
k
hD
tNu
Rr
      (3.20) 
The quantity T?  can be an arbitrary value.  Generally, the term T? is taken 
as
mw
TTT ?=? .  This is a quantity that depends on time for this problem and serves as 
the driving force for natural convection.  The above definition can be simplified by 
introducing expressions for 
*
r  and 
*
T :   
.sin|
)(
)(
0
1**
*
? =
?
?
?
?
??? d
r
T
  
TT
1
 = Nu
r*
*
mw
m
      (3.21) 
Note that a subscript ?m? is used for this area-averaged Nusselt number meaning that 
m
T  
(mean or bulk temperature) is used in the T?  expression. 
However, in this specific pseudosteady-state natural convection work, the 
temperature difference between the wall and center is maintained as a constant value.  
This boundary condition can introduce another logical Nusselt number that is based upon 
a constant driving force
cw
TTT ?=? .  Then, another Nusselt number can be obtained: 
 30
.sin|)(
0
1
*
? =
?
?
?
??? d 
r
T
   =Nu
r*
*
c
       (3.2) 
Note that a subscript ?c? is used for this area-averaged Nusselt number meaning that 
c
T  
(container?s center temperature) is used in the T?  expression. 
(b) Indirect Approach 
As mentioned above, another approach to get the Nusselt number is to perform a 
lumped energy balance for the fluid within the container.  The heat transferred into the 
container must be exactly equal to the enthalpy change of the fluid inside the container, 
i.e.: 
.
34
3
4
)(
2
3
dt
dT
Rc
R
dt
dT
cR
A
dt
dT
mc
tq
m
p
m
p
m
p
?
?
??
===
       (3.23) 
Thus, the Nusselt number can be expressed as: 
.
63
)(
)(
2
dt
dT
T
D
dt
dT
Tk
DRc
Tk
Dtq
k
hD
tNu
mm
p
?
=
?
=
?
==
?
?
     (3.24) 
Similar to the above derivations for the direct approach, term T?  can either be taken as 
)(
mw
TT ?  or )(
cw
TT ? .  Correspondingly, two definitions of the Nusselt numbers are 
obtained by introducing dimensionless time, which are: 
,
)(
1
3
2
)(
*
**
?
?
d
dT
TT
 =Nu
m
mw
m
?
        (3.25) 
.
3
2
)(
*
?
?
d
dT
  =Nu
m
c
         (3.26) 
The two Nusselt numbers that are obtained by the two approaches and are based 
on the driving force of 
mw
TTT ?=?  should equal each other.  Similarly, the two Nusselt 
 31
numbers that are based on the driving force of 
cw
TTT ?=?  should be identical.  These 
four Nusselt numbers were evaluated for each case.  The Nusselt number based on the 
constant driving force 
cw
TTT ?=?  is uniquely proportional to the time rate of change of 
the bulk temperature, while the Nusselt number based on the varying driving force 
mw
TTT ?=?  is also dependent on this driving force besides the time rate of change of 
the bulk temperature.  It is clear that 
c
Nu   can be used as a direct indicator of the 
variation of the bulk temperature with time. 
 
3.3.2 Stream Function  
The difference between the maximum and minimum stream function values of a 
primary vortex is a convenient parameter to measure the strength of a recirculating flow 
field.  This quantity can also be used to monitor the convergence of the pseudosteady-
state natural convection within spheres with no baffles.  If the difference between the 
maximum and minimum time-averaged stream function values remains smaller than 10
-6
, 
then it is declared that pseudosteady-state natural convection is reached.  The minimum 
stream function value is taken by FLUENT as zero at the eye of the vortex.  The 
maximum stream function is the value on the wall.  Both dimensional and dimensionless 
stream function values are presented.  The dimensionless stream function is defined as: 
.
*
R?
?
? =           (3.27) 
 
 
 
 32
3.3.3 Fluid Flow and Thermal Fields  
Composite diagrams of the streamline patterns and temperature contours under 
the pseudosteady-state condition for the Rayleigh numbers of 10
4
, 10
5
, 10
6
 and 10
7
 with 
no baffles are presented in Figure 3.13.  The solution was declared to be the 
pseudosteady-state when the Nusselt number and stream function values satisfied the 
requirement to be discussed in Chapter 4.  Streamlines are shown on the left half of the 
sphere, whereas the temperature contours are shown on the right half of the sphere.  For 
the limiting case of Ra = 0 (pure conduction within the fluid), perfectly concentric rings 
for the temperature contours would have been observed, indicating one-dimensional heat 
diffusion independent of the polar angle.  For the lowest Rayleigh number studied, the 
temperature contours are observed to be skewed from a concentric ring distribution 
suggesting that natural convection effects are important. As buoyancy-induced 
convection effects become more dominant, deviations of the temperature contours from 
concentric ring patterns become more pronounced and the temperature gradients on the 
bottom of the sphere become stronger.  For higher Ra numbers, the region around the 
symmetry axis of the sphere is composed of constant-temperature stratified layers.  The 
spatial extent of this region is observed to increase as the Rayleigh number is raised and 
natural convection is strengthened next to the wall.  This observation on thermal 
stratification does not hold strongly for the highest Ra number studied due to the 
unsteady flow behavior to be discussed below.  In observing the trends of the flow field, 
the fluid that is heated adjacent to the hot wall becomes less dense and rises, replacing the 
colder fluid that sinks slowly downward along the symmetry axis of the sphere.  As the 
Rayleigh number rises, fluid flow is intensified and the velocity gradient next to the 
 33
surface becomes stronger.  The location of the eye of the clockwise (CW) rotating 
recirculating vortex is strongly dependent on the Rayleigh number.  The radial and polar 
angle positions of the eye of the recirculating vortex are summarized in Table 3.1, 
suggesting that the eye travels radially toward the wall of the sphere as the Rayleigh 
number is raised.  At the same time, its polar position is lifted from the bottom half to the 
top half.  There are contradictions between the present results and those of Shen et al. 
(1995) who reported streamline and temperature contours for Ra numbers that were 
higher by factors of 1.3~1.9 (Figure 2.7).  Those data are also summarized in Table 3.1 
showing similar trends for the radial coordinate of the eye of the vortex.  However, 
according to Shen et al. (1995) the polar angle position of the vortex shows almost no 
sensitivity to the change of the Ra.  Whereas the polar angle changes 50 degrees upon 
raising the Ra number by 3 orders of magnitude in the present study, Shen et al. (1995) 
only observed 11 degrees, with the eye always remaining in the lower half.  The 
observations of Shen et al. (1995) were generally supportive of findings of Hutchins and 
Marschall (1989) that reported data on the eye of the vortex for similar Rayleigh numbers 
reported here (Table 3.1).  Interestingly, Khodadadi et al. (1999) reported streamline and 
temperature contours for a higher Rayleigh number of 10
8
, with the vortex eye located at 
824.0
*
=
e
r  and =
e
? 54
o
.  This observation is in concert with the data for the present 
study for a Ra number of 10
7
, whereas the data of Hutchins and Marschall (1989) for Ra 
= 10
8
 exhibits no sensitivity to the Ra number for the polar angle position of the vortex.  
In view of the commonality of the computational procedure used by these investigators, 
the contradictions can be attributed to the differences in spatial grid resolutions among 
 34
the three studies (about 10,000 cells for the present study compared to ~900 cells for the 
earlier studies). 
 
Table 3.1 Radial and polar angle positions of the eye of the recirculating vortex for the 
cases with no baffles 
Ra 10
4
    10
5
       10
6
          10
7
  10
8
  
*
e
r  
 
Present Study  0.605    0.651      0.768       0.814    ----- 
Shen, Khodadadi and Zhang (1995)
*
  0.617    0.733      0.8           0.9            ----- 
Hutchins and Marschall (1989) -----    0.727      0.910       0.850    0.912 
e
? (degrees) 
 
Present Study  104.3    91.5       68.5         54.8     ----- 
Shen, Khodadadi and Zhang (1995)
*
  100    92       89           94     ----- 
Hutchins and Marschall (1989) -----    93       94           92     95 
 
* The reported Rayleigh numbers were higher than the values shown at the top of the 
columns by a factor of 1.3~1.9. Refer to Figure 2.7 for streamlines and temperature 
contours. 
 
Natural convection within spherical enclosures exhibits a variety of flow 
structures that interact constantly.  These include developing momentum and temperature 
boundary layers over concave and convex surfaces and a massive recirculating flow that 
is directly linked to the confinement of the fluid in the vessel.  A very interesting feature 
of natural convection in a spherical container is the simultaneous existence of thermally 
stable and unstable fluid layers in the vicinity of the symmetry axis.  This is illustrated 
schematically in Figure 3.14 that shows enlarged views of these two fluid layers.  For the 
greater portion of the symmetry axis, horizontal stratified fluid layers are observed with 
the hot lighter fluid sitting on top of cooler dense layers.  Such a system is denoted to be 
thermally stable, where fluid motion is negligible and heat conduction dominates.  In 
 35
contrast, the remaining portion of the symmetry axis near the bottom is composed of cold 
fluid layers on top of hotter dense layers.  This type of system is denoted to be thermally 
unstable and will feature recirculating cells.  In the present study, the onset of instabilities 
through creation of a small counter CW (CCW) rotating vortex near the bottom of the 
sphere was observed for the Rayleigh number of 10
7
 (Figure 3.13).  The strength of this 
vortex is dictated by the dynamic interaction of the rising hot fluid along the symmetry 
line with the sinking cooler fluid that originates from the top of the sphere.  In connection 
with this finding, Owen and Jalil (1986) have reported temperature measurements at the 
bottom of the sphere exhibiting an oscillating behavior. 
The instantaneous area-averaged Nusselt numbers ( )(?
c
Nu ) as functions of 
dimensionless time are presented in Figures 3.15-3.18 for Ra=10
4
, 10
5
, 10
6
 and 10
7
, 
respectively.  Marked variations of the values of the Nusselt number is observed during 
the early transient period ( 12.0?? ) as the flow perturbs starting from a motionless state.  
The oscillations observed during the early transient period gain in their frequency as the 
Rayleigh number is raised.  Beyond the early transient period, the Nusselt number tends 
toward its final pseudosteady-state value.  Even then, it is observed that the oscillating 
behavior persists regardless of the Rayleigh number.  The RMS and relative RMS values 
of the Nusselt numbers are given in Table 3.2.  It is clearly observed that the oscillations 
are weak and the pseudosteady-state was established for the cases that were investigated. 
 
 
 
 
 36
Table 3.2 RMS and relative RMS of the Nusselt Number under pseudo-steady state 
condition 
 Ra   
10
4 
1.76x10
-3 
3.06x10
-4 
10
5 
2.09x10
-3 
1.62x10
-4 
10
6 
1.87x10
-3 
7.35x10
-5 
10
7
 3.22x10
-3 
3.22x10
-5
 
 
 
3.3.4 Code Validation  
Due to the steady rise of computing power in recent years, numerical simulation 
of processes and machinery is becoming more important for engineering applications.  
Even so, it is not foreseen that numerical simulation will completely replace experiments.  
However, its low-cost and smart-adoption advantages are elevating CFD to play an 
important role in wide engineering applications.  Ensuring code and solution correctness 
is critical for any computational study.  Code verification and validation allow one to 
quantitatively evaluate the correctness of code and model, respectively.  Before final 
results are obtained, code verification and validation should be carried out to ensure that 
the simulation can represent reality within the required accuracy.  Code verification 
concentrates on the mathematical formulation, while the code validation deals with 
comparison against experimental data (Roy, 2006).  The CFD code FLUENT is a widely-
adopted commercial computational fluid dynamic (CFD) package that has been widely 
tested by others.  Therefore, in this section we mainly focus on code validation that is 
accomplished by comparing the present results with prior known data.     
c
Nu
RMS |
c
rNu
RMS |
 37
Four Nusselt numbers were calculated according to different definitions discussed 
earlier.  The Nusselt numbers obtained from the temperature gradient on the wall were 
computed by FLUENT and this approach is called ?Direct?.  The Nusselt number 
obtained from the lumped energy balance is calculated in Matlab using the temperature 
data file output of FLUENT.  In this thesis, this method is called ?Indirect?.  The 
computed Nusselt number results in comparison to those of Hutchins and Marschall 
(1989) are given in Table 3.3.  First of all, the results of the direct and indirect approaches 
that were adopted for evaluating the specific Nusselt numbers matched almost perfectly.  
Present results are a little higher than the predictions of Hutchins and Marschall (1989).  
The relative differences are within 8 percentage for all the Rayleigh numbers.   
 
Table 3.3 Comparison of the pseudosteady-state Nusselt numbers  
0.7 11.224 5.7556 11.2246 5.7559 ------- -------
0.7 16.129 12.8698 16.1298 12.8705 15.8 11.5
0.7 25.3452 25.4938 25.3465 25.495 23.8 23.5
0.7 40.1974 37.1717 40.1993 37.1735 40.8 36.7
Direct Approach* Indirect Approach** Hutchins and Marschall (1989)
Pr Ra
4
10
5
10
6
10
7
10
c
Nu
m
Nu
c
Nu
m
Nu
c
Nu
m
Nu
 
 
The relative difference of 
m
Nu and 
c
Nu  using two different approaches can be 
employed to examine the accuracy of simulations.  Defining relative differences of 
m
Nu  
and 
c
Nu  as 
*
***
||
m
mm
Nu
NuNu ?
 and 
*
***
||
c
cc
Nu
NuNu ?
, respectively, the relative difference 
values for different Rayleigh numbers are shown in Figure 3.19.  Note that the Nusselt 
number with a single star stands for the ?direct? value, while double stars stand  for the 
?indirect? value.  It can be seen clearly that both relative difference of  
m
Nu  and that of 
 38
c
Nu  are of the order of magnitude of 10
-5
, regardless of the Rayleigh number.  
Considering the truncation error in the discretization, the relative differences can not be 
exactly zero.  Thus, these negligible relative differences indicate the good accuracy of the 
FLUENT in simulating the pseudosteady-state natural convection inside spherical 
containers.   
Another approach for code validation is carried out by comparing the maximum 
dimensionless stream function (
*
max
? ) with those of Hutchins and Marschall (1989).  The 
data of dimensionless stream function reported by Hutchins and Marschall (1989) is for 
Pr=8, while our computational results are for Pr=0.7.  It has been reported that the Prandtl 
number has negligible effect on the solution when Pr lies in the range of 0.7 to 8.  The 
tabulated results can be found in Table 3.4.  In view of the discrepancies that were 
discussed in relation to the location of the eye of vortex, it is shown that the current data 
(
*
max
? ) match the results of Hutchins and Marschall (1989) well.  
 
Table 3.4 Dimensionless maximum stream function (
*
max
? ) values  
Pr Ra Current Results Hutchins and Marschall (1989)
0.7 1.554137939 -------
0.7 7.64067724 6.01
0.7 19.61292939 14.51
0.7 41.42771552 33.47
4
10
5
10
6
10
7
10
 
 
3.3.5 Correlations of the Pseudosteady-State Nusselt Numbers 
Assume that the dependence of the Nusselt numbers on the Rayleigh number 
takes the form: 
 39
,
b
CRaNu =
?
          (3.28) 
with ?  standing for m or c depending on the use of temperature difference T? .  The 
following correlations were constructed for both 
m
Nu and
c
Nu  using the least-squares 
fitting method: 
,96934.1
185844.0
RaNu
m
=         (3.29) 
272723.0
514756.0 RaNu
c
=   for 10
4
<Ra<10
7
.      (3.0) 
 
The cross-correlation coefficient (? ) is a quantity that describes the divergence between 
the least-squares fitting curve and the original data.  It is defined as: 
,
'
bb=?           (3.1) 
with: 
),ln()ln()ln( RabCNu +=
?
        (3.2) 
that is a rearrangement of Equation 3.28.  Equation 3.28 can also be solved for Ra, thus 
leading to: 
).ln()ln()ln(
''
?
NubCRa +=         (3.3) 
The cross-correlation coefficients are 0.997 and 0.977 for 
m
Nu  and 
c
Nu , respectively.  
These values are very close to unity that indicates good accuracy of the least-squares 
fitting.   
Chow and Akins (1975) reported correlation of their experimental data using the 
least-squares fitting method as: 
3.0
54.0 RaNu
c
=  for 7.74x10
4
<Ra<1.13x10
7
.      (3.4) 
Hutchins and Marschall (1989) reported correlation of their numerical data using 
the least-squares fitting method as: 
 40
,19.1
2215.0
RaNu
m
=          (3.5) 
2521.0
655.0 RaNu
c
=  for 10
5
<Ra<10
8
.       (3.6) 
Shen et al. (1995) reported correlation of their numerical data (using different 
dimensionless form) using the least- squares fitting method as: 
2013.0
78055.0 RaNu
m
=   for 1.649x10
4
<Ra<1.675x10
7
.    (3.7) 
The various correlations of the two Nusselt numbers (
c
Nu  and 
m
Nu ) were drawn in 
Figures 3.20 and 3.21, respectively.  
 
3.4 Closure 
The solution of the pseudosteady-state natural convection in a sphere was 
obtained by using a commercial CFD package.  The code validation was done by 
comparing the Nusselt numbers and stream functions against prior known data.  A 
thermally unstable structure near the bottom of the sphere is observed in a spherical 
container for Ra=10
7
.  The correlations for the Nusselt numbers were established.    
 
 
 
 
 
 
 
 
 
 41
 
 
 
Figure 3.1 Schematic diagram of the problem 
 
 
 
 
 
 
T
c
T
w
?
g
G
D
r
 42
 
 
 
 
 
Figure 3.2 Hybrid mesh created in GAMBIT 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 43
 
 
 
Figure 3.3 Boundary layer mesh system  
 
 
 
 
 44
 
 
 
 
 
Figure 3.4 Detailed view of the boundary layer mesh 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
r
?
First Row
 45
 
 
 
 
Figure 3.5 Magnified view of part of the adopted hybrid mesh 
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
        
 
    
            
       
 46
 
 
 
 
Figure 3.6 Solution controls in FLUENT 
            
            
            
            
            
            
            
            
            
            
            
   
 
 
 
 
 
 
 
 
 
 47
 
 
 
 
Figure 3.7 Solver settings in FLUENT 
 
            
            
            
            
            
         
 
 
 48
 
 
 
 
 
 
Figure 3.8 Operating conditions in FLUENT 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 49
 
 
 
 
 
Figure 3.9 Fluid properties in FLUENT 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 50
 
 
 
 
 
Figure 3.10 Thermal boundary conditions in FLUENT supplied by a user-defined 
function (UDF) 
 
 
 
            
            
            
            
            
            
   
 
 
 
 
 
 
 
 
 
            
     
 
 
 51
 
 
 
 
Figure 3.11 Momentum boundary conditions in FLUENT 
            
            
            
            
            
            
            
            
            
            
            
            
   
 
 
 
 
 
 
 
 
 
 
 
 
 52
 
 
 
 
 
Figure 3.12 Residual monitors in FLUENT 
 
 
 
 
 
 
 
 
 
 
 
 
 53
 
 
 
 
 
Figure 3.13 Pseudosteady-state streamline patterns (left half) and corresponding 
temperature contours (right half) for cases with no baffles (Ra = 10
4
, 10
5
, 10
6
 and 10
7
) 
 
 
 
 
 
5
10=Ra
4
10=Ra
6
10=Ra
7
10=Ra
 54
 
 
 
 
 
 
 
 
 
 
Figure 3.14 Thermally stable and unstable structures 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
FluidHot
FluidCold
UNSTABLE
FluidHot
FluidCold
STABLE
0?V
G
 55
 
 
 
 
 
Figure 3.15 Instantaneous Nusselt number as a function of dimensionless time for Ra=10
4
 
with no baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
?
Nu
c
(
?
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
2
4
6
8
10
12
14
16
18
20
Ra=10
4
Pr=0.7
 56
 
 
 
 
  
Figure 3.16 Instantaneous Nusselt number as a function of dimensionless time for Ra=10
5
 
with no baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
?
Nu
c
(
?
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
6
8
10
12
14
16
18
20
Ra=10
5
Pr=0.7
 57
 
 
 
Figure 3.17 Instantaneous Nusselt number as a function of dimensionless time for Ra=10
6
 
with no baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
?
Nu
c
(
?
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
10
15
20
25
30
35
40
Ra=10
6
Pr=0.7
 58
 
 
 
 
 
Figure 3.18 Instantaneous Nusselt number as a function of dimensionless time for Ra=10
7
 
with no baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
?
Nu
c
(
?
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
5
10
15
20
25
30
35
40
45
50
55
60
65
70
Ra=10
7
Pr=0.7
 59
 
 
Rayleigh Number
Rel
a
ti
ve
D
i
ffe
r
e
nc
e
10
4
10
5
10
6
10
7
4.6E-05
4.8E-05
5E-05
5.2E-05
5.4E-05
5.6E-05
Relative Difference of Nu
m
Relative Difference of Nu
c
 
 
 
Figure 3.19 Relative differences of the Nusselt numbers between two different 
approaches 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 60
 
 
Ra
Nu
c
10
4
10
5
10
6
10
7
10
20
30
40
50
60
70
Present
Hutchins and Marschall (1989)
Chow and Akins (1975)
 
 
 
Figure 3.20 Nusselt number (
c
Nu ) correlations 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 61
 
Ra
Nu
m
10
4
10
5
10
6
10
7
10
15
20
25
30
35
40
45
Present
Shen et al. (1995)
Hutchins and Marschall (1989)
 
 
 
 Figure 3.21 Nusselt number (
m
Nu ) correlations 
 
 62
CHAPTER 4 EFFECT OF AN INSULATED BAFFLE ON PSEUDOSTEADY-
STATE NATURAL CONVECTION INSIDE SPHERICAL CONTAINERS 
 
Pseudosteady-state natural convection inside spherical containers has been studied 
using a computational fluid dynamic package (FLUENT) in Chapter 3.  The phenomenon 
associated with this problem has been widely studied over several decades in relation to 
transport and storage of liquefied natural gas (LNG).  During these operations, it is 
desired that convection be reduced, whereas in other applications, enhancement of 
convection may be sought.  Detailed knowledge related to control of heat transfer and 
management of the flow field due to natural convection inside spherical containers is still 
not available.  At the same time, interest in such knowledge is increasing due to its 
relevance to numerous engineering applications.   
Pseudosteady-state natural convection inside spherical containers is an 
appropriate model to investigate the effect of a baffle inside a container on flow field and 
heat transfer.  Due to the relatively complicated geometry, it is impossible to get an 
analytical solution of the governing equations.  Alternately, conducting proper numerical 
simulation is a good way to study the effect of a baffle on the flow and thermal fields.  
The objective of this Chapter is to investigate how a perpendicular-to-wall insulated thin 
baffle modifies the flow field as well as heat transfer.  Parametric studies were performed 
for a Prandtl number of 0.7.  For Rayleigh numbers of 10
4
, 10
5
, 10
6
 and 10
7
, baffles with
 63
3 lengths positioned at 5 different locations were investigated.  In effect, a parametric 
study involving 60 cases was performed.   
 
4.1 Mathematical Formulation for the Pseudosteady-State Natural Convection 
inside Spherical Containers with a Thin Insulated Baffle 
A thin insulated baffle with length ( l ) is attached on the inside wall of a spherical 
container at polar angle (
b
? ).  It is positioned along the radial direction and points to the 
center.  A schematic diagram of the posed problem is illustrated in Figure 4.1, in which a 
typical computational mesh is also shown on the left half.   As for mathematical 
formulation for this problem, the presence of a thin insulated baffle introduces an extra 
boundary condition, while the modeling assumptions are same as Chapter 3.   
 
4.1.1 Governing Equations and Boundary/Initial Conditions 
Further analysis was performed on the basis of Chapter 3 by adding a thin 
insulated baffle.  The governing equations are same as those stated in Chapter 3 
(Equations 3.1-3.5) and are not repeated here.  Concerning the orientation of the baffle, a 
3-D schematic drawing is shown in Figure 4.2.  The thickness of the insulated baffle is 
exaggerated and in reality the baffle is assumed to be very thin.  Therefore, it can not 
contribute any heat flux to the fluid in the spherical container.   
 The no-slip boundary condition is applied on the wall of the sphere and the two 
sides of baffle, i.e.:   
.
**
0 = 
V
 =
V r?
          (4.1) 
 64
The driving force of natural convection is a constant temperature difference between the 
wall and center: 
1)()(
**
+= ??
cw
TT  for .0??        (4.2) 
On the symmetry axis ( 0=?  and ? ): 
.
**
*
0 = 
T
 = 
V
V
r
??
?
?
?
?
?
=          (4.3) 
On both sides of the thin insulated baffle ( 1)21(
*
<?? rL  and 
b
?? = ) with :
D
l
L =  
.0
*
=
?
?
?
T
          (4.) 
The governing equations are characterized by the Rayleigh and Prandtl numbers, 
dimensionless baffle length (L) and position (
b
? ).  Previous work has shown that the 
Prandtl number has negligible effect on the numerical solution within the 0.7-8 range.  
Therefore, the Prandtl number is not treated as a variable but a constant that is equal to 
0.7.  Various Rayleigh numbers are investigated for different baffle parameters (L and 
b
? ).    
 
4.1.2 Computational Details 
Similar to the cases with no baffles, strong convective heat transfer is expected 
next to the wall of the container due to steep temperature gradients there.  The flow and 
thermal fields next to the baffle are also expected to be highly perturbed.  Specific grid 
generation information for the ?boundary layer? mesh can be found in Chapter 3 and is 
not repeated here.  For cases with baffles (L>0.04), the segment of the baffle outside the 
?boundary layer? mesh was discretized with about 50 or less nodes.  Then, the ?interior? 
 65
grid was generated using an unstructured mesh.  In effect, the number of cells for cases 
with baffles is higher than 10,222 that was utilized in Chapter 3.  For baffles with 
different lengths and locations, the total number of cells varies between 10,476 and 
13,868.  Schematic diagrams of the grid systems for a L=0.25 baffle at different locations 
are shown in Figure 4.3.  Each grid system is very dense next to the inner wall of the 
container, whereas both sides of the baffle are also refined with a dense mesh. 
The solutions of the governing equations were obtained following the same 
procedure outlined in Section 3.2.2 and the pertinent details are not repeated here.  The 
governing equations were solved by combining the commercial codes GAMBIT (version 
2.2.30) and FLUENT (version 6.2.16).  All the computations (60 cases) were performed 
on a Cray XD1 supercomputer of the Alabama Supercomputer Authority (Huntsville, 
Alabama). 
     
4.1.3 Code Validation  
Code verification and validation should be performed to provide confidence in the 
accuracy of numerical simulation.  The code verification procedure is omitted due to the 
use of commercial CFD package (FLUENT) which has been verified repeatedly by 
others.  The code validation procedure of pseudosteady-state natural convection inside 
spherical containers without baffles is discussed in detail in Chapter 3.  Mathematically, 
the presence of a thin insulated baffle introduces an extra boundary condition in the 
problem formulation, while the governing equations are the same as the case without a 
baffle.  Therefore, it is considered that the code validation reported in Chapter 3 has 
provided enough confidence in the appropriateness of the adopted model for the 
 66
pseudosteady-state natural convection inside spherical containers with insulated thin 
baffles.  
 
4.2 Grid and Time Step Size Independence Study  
The accuracy of a numerical solution is highly dependent on both the adopted grid 
density over the physical domain and the proper choice of the time step size.  Numerical 
solutions of the pseudosteady-state natural convection within spheres have been 
presented by Hutchins and Marschall (1989) and Shen et al. (1995).  However, a detailed 
study on grid and time step size sensitivity was not reported.  Considering different 
geometries due to the presence of a baffle and the dynamic strength of convection in the 
system, the specific case with L=0.25 and 
2
?
? =
b
 for an insulated thin baffle for the 
Rayleigh number of 10
7
 is selected for the sensitivity test.  The schematic geometry 
employed for the grid independence study is illustrated in Figure 4.3.c.  Values of the 
time-averaged 
m
Nu  and 
c
Nu  are selected as representative quantities to determine the 
accuracy of the numerical solution.  It was already indicated in Chapter 3 that this case 
with no baffle does exhibit flow patterns suggestive of a thermally unstable fluid layer at 
the bottom of the sphere.  The time-averaged Nusselt number after a long time duration 
(2,000 time steps with 2=?t  seconds, that is equivalent to dimensionless time period 
0.314062213) is observed to be almost constant.  The oscillating strength of a time-
varying dynamic flow can be characterized by the Root Mean Square (RMS) of a 
monitored quantity.  The grid spacing and time step size are two independent variables, 
thus these two studies can not be carried out at the same time.  The grid size 
independence is tested first, while the time step size is fixed to 2 seconds 
 67
(
-4
1057.1 ?=?? ).  The effects of the spacing of the first row of the ?boundary layer? 
mesh, number of nodes on the baffle and the density of the ?interior? nodes are studied.  
For convenience, a parameter referred as the ?number of cells? is utilized to describe the 
overall grid density.  The dependence of the time-averaged Nusselt numbers (
m
Nu  and 
c
Nu ) on the overall grid density are illustrated in Figures 4.4 and 4.5, respectively.  Note 
that the time-averaged Nusselt number values are presented along with the RMS values.  
The tabulated values of these Nusselt numbers are also presented in Table 4.1.  It is 
observed that the time-averaged Nusselt number values are nearly identical when the 
number of cells is greater than 10
4
.  Taking the accuracy of the solution and the 
computational time into account, the grid system with 13,868 cells is considered as a 
?proper? grid system.  All the other grid systems were created based on the parameters of 
this ?proper? mesh.   
       
 
 
 
 
 
 
 
 
 
 
 68
Table 4.1 Dependence of the time-averaged and RMS values of the Nusselt numbers on 
the number of cells for an insulated case with Ra=10
7
, Pr=0.7, L=0.25, 
o
b
90=?  and 
4
1057.1
?
?=??  
Number of Cells
108 21.2572 0.000932596 26.9184 0.001177357
318 25.3077 1.13378496 38.32 0.701256
962 31.0297 3.84147686 44.4273 2.03921307
2,084 31.6797 6.19338135 46.1072 4.25569456
5,200 29.1629 6.52374073 40.7615 6.2120526
6,790 28.7735 6.2438495 39.5513 6.16209254
8,972 28.7477 6.39923802 40.4279 6.28653845
9,500 28.6351 6.52593929 40.3671 6.41029548
10,998 28.5143 6.35298604 40.297 6.3225993
13,868 29.0071 6.671633 40.7759 6.49560087
15,618 28.5983 6.40887903 40.7881 6.12637262
23,584 27.8505 5.737203 40.7732 5.5655418
29,218 28.9628 6.7193696 40.7989 6.51966422
35,744 28.8631 6.73376123 40.7489 6.54019845
c
Nu
m
Nu
c
Nu
RMS |
m
Nu
RMS |
 
 
The grid and time step size independence tests for the pseudosteady-state natural 
convection inside spherical containers without baffles in Chapter 3 were not performed 
separately, and those grid systems were generated using the same parameters mentioned 
in this Section. 
Once the proper mesh was selected, the sensitivity of the solution on the time step 
size can be studied.  The dependence of the time-averaged Nusselt numbers (
m
Nu  and 
c
Nu ) on the time step size for the same insulated thin baffle discussed above are shown 
in Figures 4.6 and 4.7, respectively.  The tabulated data is given in Table 4.2.  The 
predicted Nusselt numbers exhibit little sensitivity when the time step size is less than 6 
seconds (
-4
1071.4 ?=?? ).  Considering both the accuracy of the solution and the 
 69
computational time, a time step size of 2 seconds (
-4
1057.1 ?=?? ) is selected as a proper 
time step size for all 60 cases with thin insulated baffles.   
 
Table 4.2 Dependence of the time-averaged and RMS values of the Nusselt numbers on 
the time step size for an insulated case with Ra=10
7
, Pr=0.7, L=0.25, 
o
b
90=?  and 
13,868 cells 
        (s)
0.1 28.7095 6.5256694 40.7459 5.8307383
0.2 28.4083 6.1930094 40.7709 5.5611508
1 28.9655 6.719996 40.7783 6.2472356
2 29.0071 6.67163 40.7759 6.4956
3 28.3995 6.1655315 40.834 6.0475154
5 28.928 6.1819136 40.9624 6.775181
7 28.0715 5.1735775 40.7907 6.3919027
8 26.2092 3.8081968 40.4022 4.8886662
10 25.2688 2.7745142 40.1154 3.8831707
15 23.9587 1.5453362 39.5984 2.5580566
30 22.3371 0.4668454 38.9356 1.0901968
t?
c
Nu
m
Nu
c
Nu
RMS |
m
Nu
RMS |
 
 
4.3 Results and Discussion  
Pseudosteady-state fluid flow field streamlines and temperature contours are 
presented for 45 cases for fixed Rayleigh numbers (10
4
, 10
5
 and 10
6
).  Through 
examination of the pertinent information for these cases, one will be able to understand 
how the presence of a thin insulated baffle modifies the flow and thermal fields.  The 
criterion used to declare that the pseudosteady-state has been achieved will be discussed 
in detail later.  In short, once the time-averaged Nusselt numbers (
m
Nu  and 
c
Nu ) do not 
change with time, the pseudosteady-state is achieved.  For the Rayleigh number equal to 
10
7
, strong oscillations occurred except the case with dimensionless baffle length of 0.25 
 70
located at 
b
? =150
o
.  Therefore, the time-dependent flow field streamlines and 
temperature contours for Ra=10
7
 are plotted and discussed separately.  The dependence 
of the time-averaged Nusselt numbers (
m
Nu  and 
c
Nu ) for all 60 cases are given.  
 
4.3.1 Pseudosteady-State Fluid Flow and Thermal Fields for Ra=10
4
, 10
5
 and 10
6
 
The composite diagrams of the streamlines and temperature contours under the 
pseudosteady-state condition for three baffles with lengths (L = 0.05, 0.1 and 0.25) 
placed at various polar angle locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for a Rayleigh 
number of 10
4
 are presented in Figure 4.8.  Diagrams in each row correspond to baffles of 
various lengths positioned at a fixed location, whereas for each column the effects of a 
baffle with a fixed length at various positions are given.  By comparing the streamline 
patterns and temperature fields in this figure to the limiting case of no baffle in Figure 
3.13 for the same Ra, the effect of a thin insulated baffle can be elucidated.  Focusing on 
the left column of Figure 4.8, it is clear that the presence of the shortest baffle (L = 0.05) 
does not alter the flow and thermal fields significantly, regardless of the angular position 
of the baffle.  The distortions are restricted to minor alteration of the streamlines next to 
the short baffles, whereas the temperature contours are generally unchanged with minor 
radial shifting of the contours next to the adiabatic baffle.  For the cases corresponding to 
the L = 0.1 baffle that are shown in the middle column, the modifications to the flow and 
thermal fields are a bit more marked than the cases with the shortest baffle, however the 
changes are still observed to be next to the baffle.  In contrast, marked changes to the 
flow field are observed in the right column of Figure 4.8 that correspond to the longest 
baffle (L = 0.25), specially for the cases when the baffle is located at 
b
? = 60
o
, 90
o
 and 
 71
120
o
.  For these cases, two distinct recirculating vortices are observed on both sides of the 
baffle.  These vortices rotate in the clockwise (CW) direction lifting heated fluid next to 
the wall to a higher elevation and bringing down colder fluid.  The temperature fields 
have also been affected by the longer length of the baffle that directly perturbs the flow 
field, however the general pseudo-concentric ring contour patterns are preserved.  It can 
clearly be stated that due to the adiabatic nature of the baffle, this structure does not 
directly participate in perturbation of the thermal field.  The presence of the baffle 
appears to be generally directed at modifying the flow paths for Ra=10
4
 cases. 
For the Rayleigh number of 10
5
, streamlines patterns and temperature contours for 
three baffles with lengths (L = 0.05, 0.1 and 0.25) placed at various locations (
b
? = 30
o
, 
60
o
, 90
o
, 120
o
 and 150
o
) are shown in Figure 4.9.  Diagrams are plotted going from left 
column to right column with the dimensionless baffle length increasing from 0.05 to 0.25. 
Comparing the streamline patterns and temperature fields on the left column of Figure 4.9 
(L = 0.05) with the case of no baffle (Figure 3.13), the presence of the shortest thin 
insulated baffle (L=0.05) modifies the thermal fields to some extent for various angular 
positions of the baffle.  The flow field modifications are not very significant that is 
similar to the cases in Figure 4.8 (Ra=10
4
), while modification of the thermal fields can 
be observed easily.  In general, the flow and thermal fields exhibit features similar to 
those discussed above for Figure 4.8.   
Streamlines and temperature contours for three baffles with lengths (L = 0.05, 0.1 
and 0.25) placed at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for a Rayleigh 
number of 10
6
 are presented in Figure 4.10.  Comparing the streamline patterns and 
temperature fields on the left column of Figure 4.10 (L = 0.05) with the case of no baffle 
 72
(Figure 3.13), the presence of the shortest baffle modifies the flow and thermal fields to 
some extent for various angular positions of the baffle.  The flow modifications are 
confined to streamlines next to the short baffle through its interaction with the eye of the 
vortex that is closer to the wall compared to similar cases in Figures 4.8 and 4.9.  For 
some positions of the short baffle, two eyes within the CW rotating vortex are observed.  
As for the effect of the shortest baffle on the temperature contours, the changes are more 
pronounced in comparison to similar cases of Figures 4.8 and 4.9, however they are still 
localized in the vicinity of the baffle.  The flow modifications are more noticeable for a L 
= 0.1 baffle that are shown in the middle column of Figure 4.10, particularly when the 
baffle is located at 
b
? = 60
o
, 90
o
 and 120
o
.  For these cases, the double CW rotating 
vortex structure is further complicated by appearance of a smaller counter CW rotating 
vortex that is located very near the open end of the baffle.  This vortex does not draw 
energy from the adiabatic thin baffle.  Therefore, it is the lifting of the hot fluid below the 
baffle into the zone above the baffle that is creating this vortex.  One can also note that 
the multi-vortex structure can clearly rearrange the thermal field when compared to the 
case of the shortest baffle that exhibited constant-temperature stratified layers in the 
vicinity of the symmetry axis of the sphere.  In assessing the effect of the long baffle on 
the flow fields in the right column of Figure 4.10, modifications that are very similar to 
the cases of Ra = 10
4
 and 10
5
 in Figures 4.8 and 4.9 are observed.  In general, two CW 
rotating vortices are clearly observed for the cases when the baffle is located at 
b
? = 60
o
, 
90
o
, 120
o
 and 150
o
.  As for the thermal field, when the baffle is positioned such that 
o
b
90?? , the space above the baffle is clearly composed of stable stratified constant-
temperature layers.  This suggests that the flow within the top portion is not strong and 
 73
conduction dominates, whereas the thermal field in the lower half is dominated by natural 
convection.  As a general statement, note that when the long baffle is positioned in the 
bottom half including the mid-plane, the thermal field is clearly divided into two zones.  
Both zones have areas of intense wall heat transfer that are located at the bottom of the 
sphere and angular positions 
b
?? ?? , respectively.  A portion of the top zone is a region 
of stable constant-temperature layers suggesting weak natural convection and dominance 
of heat diffusion, whereas the bottom zone is where natural convection is very prominent. 
The effects of the Rayleigh number (10
4
, 10
5
, 10
6
 and 10
7
) and baffle?s position 
(
b
? = 30
o
, 90
o
 and 150
o
) on streamlines and temperature contours for the case of the 
longest baffle (L = 0.25) are presented in Figure 4.11.  With the baffle positioned near the 
top at 
b
? = 30
o
 (left column), the increase of the Rayleigh number brings about stronger 
convection and fluid flow within the CW rotating vortex as indicated by the denser 
packing of the streamlines next to the surface.  This is accompanied by lifting of the eye 
of the vortex and its migration outward.  At the highest Ra number studied, a weak vortex 
is observed near the top within the cone, part of which is the baffle.  Simultaneously, a 
CCW rotating vortex at the bottom that is driven by the thermally unstable stratified layer 
is still active.  The temperature contours exhibit greater deviation from the concentric 
ring patterns as natural convection strengthens and diffusion is observed to be limited to 
the small space between the baffle and the symmetry axis.  With the baffle located at 
b
? = 
90
o
 (middle column), two CW rotating vortices occupy the two hemispherical regions 
with the lower half of the sphere being the site of stronger convection.  As the Ra number 
is raised, the stronger vortex is observed to penetrate into the top hemisphere and even a 
third counter-CW rotating vortex is created next to the free end of the baffle for Ra=10
6
 
 74
and 10
7
.  The top hemisphere is clearly stratified with stable constant-temperature layers 
occupying it, whereas the thermal field within the bottom hemisphere is heavily affected 
by the stronger rotating vortex that occupies it.  The flow fields for the cases with the 
longest baffle positioned near the bottom at 
b
? = 150
o
 (right column) exhibit many of the 
features with the baffle located at 
b
? = 30
o
, but in reverse.  A CW-rotating recirculating 
vortex that occupies the small space between the baffle and the symmetry line of the 
sphere is clearly observed.  As for the temperature contours, the alterations appear to be 
generally confined to the region between the baffle and the symmetry axis.  The 
remainder of the sphere appears to be generally unaffected by the presence of the longest 
baffle.  The reader is reminded that the last row of Figure 4.11 with Ra=10
7
 is a snapshot 
of the instantaneous flow and thermal fields.  Detailed discussion of a typical case with 
Ra=10
7
 is given in the next Section.    
 
4.3.2 Time-Dependent Fluid Flow and Thermal Fields for Ra=10
7
 
 In order to illustrate the unsteady nature of the flow, a representative case of 
Ra=10
7
, L=0.25 and 
b
? = 60
o
 was selected.  The instantaneous composite diagrams of the 
streamlines and temperature contour fields for this case during a ?cycle? are shown in 
Figures 4.12 (a)-(h).  In order to aid the reader, a companion diagram showing the 
variation of the instantaneous area-averaged Nusselt number is also shown in Figure 4.13.  
The cyclic nature of the instantaneous Nusselt number is clearly shown and the instants at 
which the composite streamlines and temperature contours of Figure 4.12 were shown are 
marked by letters a-h.  A dynamic flow field is observed within the cycle with distinct 
growth and decay of a multitude of vortices.  The temperature gradients next to the wall 
 75
of the sphere and the baffle vary dramatically during the cycle and are clearly linked to 
the variation of the Nusselt number shown in Figure 4.13.  
 
4.4 Nusselt Number Definitions and Other Parameters 
It is necessary to present and explain the definitions of different parameters that 
are employed in the remainder of this Chapter.  These include two definitions of the 
Nusselt number and stream function.   
 
4.4.1 Definitions of the Nusselt Numbers 
The Nusselt number represents the ratio of convection heat transfer to conduction 
heat transfer.  Therefore, it is employed to evaluate the strength of the pseudosteady-state 
natural convection inside spherical containers with a thin insulated baffle.  Based on the 
results presented so far, there is no doubt that the presence of a thin insulated baffle can 
dramatically change the flow field.  However, the insulated baffle does not contribute any 
heat flux to the fluid within the spherical container.  In the absence of heat addition from 
the baffle, it is logical that the presence of the baffle can lead to increased or reduced heat 
transfer from the surface of the sphere.  For both of these possibilities, the effective heat 
transfer area is the container wall area that is exactly the same as the case without the 
baffle.  The area-weighted heat flux expression is the same as the case without the baffle.  
The specific derivation can be found in Chapter 3.   
The Nusselt numbers based on the temperature gradient at the wall are: 
,sin|
)(
)(
0
1**
*
? =
?
?
?
?
??? d
r
T
  
TT
1
 = Nu
r*
*
mw
m
      (4.5) 
 76
.sin|)(
0
1
*
? =
?
?
?
??? d 
r
T
  =Nu
r*
*
c
        (4.6) 
The Nusselt numbers that are obtained by performing a lumped energy balance are: 
,
)(
1
3
2
)(
*
**
?
?
d
dT
TT
  =Nu
m
mw
m
?
        (4.7) 
.
3
2
)(
*
?
?
d
dT
  =Nu
m
c
         (4.8) 
Note that a subscript ?c? is used for one of the surface-averaged Nusselt numbers 
meaning that 
c
T  (temperature at the center) is used in the T?  expression.  Similarly, 
subscript ?m? is used for the other surface-averaged Nusselt number meaning that 
m
T  
(mean or bulk temperature) is used in the T?  expression.  The validity of these 
expressions were verified by comparing the ?direct? output of FLUENT against post-
processed ?indirect? values. 
 
4.4.2 Time-Averaged Nusselt Numbers  
In general, the flow field is disturbed due to the presence of the insulated baffle 
and the Nusselt number fluctuates with time depending on the location of the baffle, its 
length and the Rayleigh number.  Determining whether the pseudosteady-state natural 
convection inside a spherical container with a thin insulated baffle is reached is a critical 
factor for our analysis.  For some cases, the fitting curve can be easily estimated due to 
the simple oscillating behavior of the Nusselt number (Figure 4.14 (a)), whereas for other 
cases a disorderly and random variation of the Nusselt number was recorded (Figure 4.14 
(b)).  A lengthy analysis can be performed to recover the frequency content of these time-
dependent quantities.  However, in view of our focus on the pseudosteady-state behavior 
 77
of this system, it was decided that time-averaging of the Nusselt numbers is sufficient for 
this investigation.   
A straight line curve-fitting equation of the fluctuating Nusselt number can be 
employed to determine whether the statistical stationary pseudosteady-state natural 
convection inside the spherical container with a thin insulated baffle is reached.  Let us 
focus on )(?
m
Nu  first.  The fitting straight line is assumed to be: 
,)( ?? baNu
m
+=          (4.9) 
where a and b are constants.  The constant b is the slope of the straight line that will be 
equal zero after the system has reached the pseudosteady-state.  In reality, for statistical 
stationary state the value of b can be employed quantitatively to evaluate whether the 
pseudosteady-state has been reached.  If the coefficient b satisfies the requirement, then 
constant a is the time-averaged Nusselt number that is denoted by 
m
Nu .  
Considering the statistical distribution error, the least-squares approach is utilized 
to determine values of coefficients a and b in the linear equation.  The sum of the squares 
of the differences (L(a,b)) between the Nusselt number on the approximated line and the 
discrete Nusselt number is defined as:    
.)}(]{[),(
2
im
N
ki
i
NubabaL ?? ?+=
?
=
       (4.10) 
Upon minimizing the least-squares error L(a,b): 
,0
),(),(
=
?
?
=
?
?
b
baL
a
baL
        (4.1) 
the values of a and b can be found immediately by solving the simple set of linear 
equations (Equations 4.12 and 4.13).   
 78
,)()1(
??
==
=++?
N
ki
N
ki
imi
NubakN ??        (4.12) 
.)(
2
???
===
=+
N
ki
N
ki
imii
N
ki
i
Nuba ????        (4.13) 
For determining the time-averaged Nusselt number
c
Nu , the details are the same and are 
not repeated here. 
At this point, the judgment of whether the statistical stationary pseudosteady-state 
natural convection inside spherical container with a thin insulated baffle is reached can be 
achieved by inspecting the approximated line equation.  For all the cases studied, when 
b is less than 10
-6
, it is considered that the statistical stationary pseudosteady-state 
natural convection inside spherical containers with a thin insulated baffle is 
reached.  For many cases, the slope b can even be as low as 10
-10
, which indicates a 
nearly perfect statistical stationary state.  
 
4.4.3 Strength of Fluctuations of the Nusselt Numbers 
Another parameter should be defined to characterize the strength of the 
fluctuating Nusselt numbers.  There are several ways for doing this, such as the amplitude 
and the Root Mean Square (RMS).  The amplitude is a good choice that indicates the 
largest deviation of the fluctuating Nusselt number from the time-averaged value (Figure 
4.15).  Taking the statistical distribution into account, the Root Mean Square (RMS) may 
be employed to characterize the strength of fluctuations (Figure 4.16).  The Root Mean 
Square for the oscillating Nusselt numbers can be defined as: 
,
))((
|
1
2
N
NuNu
RMS
N
i
i
Nu
?
=
?
=
??
?
?
       (4.14) 
 79
with c=? or m depending on the choice of the T? . 
In order to make comparison among all the cases, the relative RMS for the 
Nusselt numbers can be defined as:  
,
))((
1
|
1
2
N
NuNu
Nu
RMS
N
i
i
rNu
?
=
?
=
??
?
?
?
      (4.15) 
with c=? or m depending on the choice of the T? . 
 
4.4.4 Stream Function  
The difference between the maximum and minimum values of the stream function 
of the primary vortex can be used to characterize the strength of the flow field, while the 
Nusselt number is generally considered for characterization of the thermal field.  In view 
of the effect of the flow on the thermal field, it is appropriate to study the stream function 
fields in relation to the observed heat transfer modifications due to the presence of a thin 
insulated baffle.  The stream function can also be employed to monitor whether the 
pseudosteady-state natural convection inside a spherical container with a thin insulated 
baffle has attained the statistical stationary state.  Defining the minimum stream function 
at the center of vortex as zero, the maximum stream function 
max
?  (value on the wall) 
can be taken as a characteristic quantity for the flow field.  
The computational results indicate that with increase of the Rayleigh number, the 
flow becomes time-dependent.  Similar to the Nusslet numbers, the stream function 
max
?  
is not a constant value but exhibits a fluctuating behavior.  Thus, the time-averaged 
stream function 
max
?  can be utilized. 
 80
For the 60 cases studied in this project, when tolerance of statistical stationary 
state is less than 10
-6
, it is considered that the statistical stationary pseudosteady-state 
natural convection inside spherical containers with a thin insulated baffle is reached.  For 
many cases, the tolerance can even be as low as 10
-10
, which indicates a nearly perfect 
statistical stationary state.  Therefore, there are two different approaches to declare 
whether the statistical stationary state is reached during the course of the computations.   
The time-averaged stream function 
max
?  is defined to describe the mean 
magnitude of the fluctuating stream function )(
max
?? .  The RMS and relative RMS are 
also defined to characterize the strength of the fluctuating stream function )(
max
??  as 
follow: 
,
))((
|
1
2
maxmax
max
N
RMS
N
i
i?
=
?
=
???
?
       (4.16) 
.
))((
1
|
1
2
maxmax
max
max
N
RMS
N
i
i
r
?
=
?
=
???
?
?
      (4.17) 
 
4.5 Variations of the Time-Averaged Nusselt Numbers and Stream Function  
In Section 4.3, the streamline patterns and temperature contours were presented 
for different Rayleigh numbers varying from 10
4
 to 10
7
 with various baffle lengths and 
locations.  The presence of a thin insulated baffle modifies the flow field directly and 
consequently affects the temperature field.  In order to describe these modifications 
quantitatively, variations of the time-averaged Nusslet numbers and maximum stream 
function are studied.  The Nusselt number is directly associated with the temperature 
 81
field, while the stream function can characterize the flow field.  Details about the 
definitions of the time-averaged Nusselt numbers can be found in Section 4.4. 
The variations of the time-averaged Nusselt numbers (
c
Nu  and 
m
Nu ) and 
maximum stream function of the primary vortex (
max
? ) with Ra for a thin insulated baffle  
are presented in Figures 4.17 to 4.25.  In each figure, the baffle length is fixed (L=0.05, 
0.10 and 0.25) and its locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) are identified with 
filled symbols, whereas the reference case with no baffle is shown with an open symbol.  
It must be emphasized that even though variations of the time-averaged Nusselt number 
(
m
Nu ) are presented, they will not be discussed in view of the complexity of their 
relation to the time rate of rise of the bulk temperature (Equation 4.7).  The Nusselt 
numbers and maximum stream function strongly depend on the Rayleigh number and 
clearly exhibit the strengthening trend with the increase of Rayleigh number, irrespective 
of the baffle lengths and locations.  For Ra=10
4
, the flow and temperature fields do not 
change greatly due to presence of the thin insulated baffle.  This is because conduction is 
dominant for this low range of the Rayleigh number and this is .  With increase of Ra, the 
effects of the thin insulated baffle become more noticeable due to the increase of 
convective effects.  For a given Ra, the extent of flow and thermal field modifications is 
directly related to the length of the thin insulated baffle.  It is generally observed (56 out 
of 60 cases) that by adding a thin insulated baffle, the time-averaged Nusselt number Nu
c
 
is lower than the reference case with no baffle.  This Nusselt number is directly 
proportional to the heat that is transferred into the container and the time rate of rise of 
the bulk temperature according to Equation 4.8.  Since there is no heat transfer through 
the surface of the baffle, the presence of the baffle has modified the heat input on the wall 
 82
of the sphere.  This control utility is related to the observed flow and thermal fields 
presented earlier, where it was observed that presence of a baffle can lead to 
?confinement? or ?compartmentalization? of the sphere.  In other words, one can 
generally state that within the stable stratified layers that are formed in the zone above the 
baffle conduction dominates, whereas the lower region is dominated by convection.  
However, it can be observed that the Nusselt numbers and maximum stream function for 
four (4) cases for which the baffle is located at 30
o
 and 60
o
 are unexpectedly higher than 
the reference case for Ra=10
6
 and 10
7
 when L=0.05 and 0.10.  This is explained as 
follows.  For all the cases with no baffle, the thermal boundary layer becomes thicker as 
the fluid from the bottom of the sphere rises along the inner wall of the sphere.  For cases 
with high Rayleigh numbers, the presence of short baffles near the top is beneficial to 
disturbing the boundary layer, thus allowing extra heat to be drawn into the sphere.  This 
effect can not be sustained if the length of the baffle is increased.  Our computations 
show that for the case of L=0.25, the confinement effect of the baffle outweighs the 
disturbance of the boundary layer. 
The behavior of the maximum stream function values that are shown in Figures 
4.23-4.25 match the trends of the Nusselt number (
c
Nu ) that were discussed.  This 
indicates that the presence of thin insulated baffles modifies the temperature field through 
modification of the flow field and it does not directly contribute energy to the system. 
Dependence of the time-averaged Nusselt numbers (
c
Nu  and 
m
Nu ) and 
maximum stream function of the primary vortex (
max
? ) on the position of the baffles 
(
b
? ) are presented in Figures 4.26-4.28, 4.29-4.31, 4.32-4.34 and 4.35-4.37 for Ra= 10
4
, 
10
5
, 10
6
 and 10
7
, respectively.  In each figure, the reference case with no baffle is 
 83
identified with a filled circle, whereas various baffle lengths (L=0.05, 0.10 and 0.25) are 
identified with non-circular filled symbols.  Again, it must be emphasized that even 
though variations of the time-averaged Nusselt number (
m
Nu ) are presented, they will 
not be discussed in view of the complexity of their relation to the time rate of rise of the 
bulk temperature (Equation 4.7).  For a given Ra number, as the location of baffle is 
lowered starting from the top of the sphere and moving toward the bottom, the Nusselt 
number Nu
c
 and maximum stream function exhibit trends suggesting that the 
confinement effect is minimal, when the baffle is placed near the two extremes.  The 
location corresponding to the most marked confinement varies depending on the Ra and 
length of the baffle.  For a fixed location of the baffle and Ra numbers lower that 10
7
, one 
can generally state that as the length of the baffle is raised, the confinement effect 
becomes more enhanced.  This behavior is not observed for the highest Ra studied. 
The tabulated data for the Nusselt numbers (
c
Nu  and 
m
Nu ) and maximum stream 
function (
max
? ) of the primary vortex are listed in Tables 4.3, 4.4 and 4.5, respectively.  
In general, the relative RMS values are low suggesting that the pseudosteady-state is 
established, except for Ra=10
7
, indicating that unsteady effects are promoted as the Ra 
number is raised.   
It is found that the relative RMS values of the Nusselt numbers have strong 
dependence on the Rayleigh number.  This is not unusual, because the larger the Rayleigh 
number, the stronger the convection heat transfer.  The results indicate that relative RMS 
values of the maximum stream function strongly depend on the Rayleigh number.   
 
 
 
 
 84
 
 
Table 4.3 Nusselt numbers (
c
Nu ) and relative RMS (
c
rNu
RMS | ) for all 60 cases with thin 
insulated baffles 
 
30
o
5.7507 3.76E-04 5.6456 3.08E-04 5.336 3.06E-04
60
o
5.5848 3.94E-04 5.2843 3.58E-04 4.6973 4.50E-04
90
o
5.4883 3.21E-04 5.0689 3.32E-04 4.3069 4.93E-04
120
o
5.6264 4.18E-04 5.3355 3.34E-04 4.6715 4.37E-04
150
o
5.7492 4.12E-04 5.6944 4.29E-04 5.4687 4.54E-04
30
o
12.6197 1.68E-04 11.9784 1.77E-04 10.4937 1.75E-04
60
o
11.9103 1.77E-04 10.4538 1.95E-04 8.0405 1.61E-04
90
o
11.9047 1.13E-04 10.5831 1.53E-04 7.8263 2.70E-04
120
o
12.4218 9.36E-05 11.7501 1.65E-04 10.9077 1.74E-04
150
o
12.7907 1.31E-04 12.674 1.57E-04 12.582 1.60E-04
30
o
26.2174 7.97E-05 24.1578 7.88E-05 18.4589 6.33E-05
60
o
22.4345 9.17E-05 19.3234 1.09E-04 13.9664 1.33E-04
90
o
22.1206 9.51E-05 16.641 6.22E-04 14.8931 4.34E-05
120
o
22.8518 9.16E-04 18.8932 0.1405 20.5763 1.47E-04
150
o
24.9339 8.61E-05 24.8963 0.0115 24.9721 9.62E-05
30
o
42.393 0.0958 48.2804 0.0245 33.6157 0.3067
60
o
38.5685 0.0355 35.6409 0.0454 29.1517 0.2621
90
o
31.997 0.1234 33.0963 0.1893 29.0071 0.23
120
o
32.9073 0.0253 28.2387 0.1085 29.5112 0.079
150
o
36.9014 0.0534 35.8857 0.0988 36.6353 0.004
L=0.10 L=0.25
Ra
L=0.05
b
?
4
10
5
10
6
10
7
10
c
Nu
c
Nu
c
Nu
c
rNu
RMS |
c
rNu
RMS |
c
rNu
RMS |
 
 
 
 
 
 
 
 
 
 
 
 
 85
 
 
Table 4.4 Nusselt numbers (
m
Nu ) and relative RMS (
m
rNu
RMS | ) for all 60 cases with 
thin insulated baffles 
 
 
30
o
11.2168 3.16E-04 11.1841 2.33E-04 11.0597 2.53E-04
60
o
11.1416 3.43E-04 11.0078 3.09E-04 10.6936 4.43E-04
90
o
11.0255 2.68E-04 10.7733 2.90E-04 10.3361 4.88E-04
120
o
11.023 3.50E-04 10.7697 2.51E-04 10.4594 4.34E-04
150
o
11.1666 3.41E-04 11.0925 3.45E-04 10.9369 3.50E-04
30
o
16.0937 1.67E-04 16.0216 1.76E-04 15.73 1.75E-04
60
o
15.947 1.76E-04 15.8 1.95E-04 15.0592 1.58E-04
90
o
15.5702 1.13E-04 15.6486 1.53E-04 15.0019 2.69E-04
120
o
15.6737 9.35E-05 15.7196 1.64E-04 15.9574 1.74E-04
150
o
16.0253 1.31E-04 15.9878 1.57E-04 15.994 1.60E-04
30
o
25.4469 7.95E-05 25.0298 7.87E-05 23.5963 6.34E-05
60
o
24.0903 9.16E-05 23.701 1.09E-04 22.6813 1.33E-04
90
o
24.049 9.48E-05 24.1182 2.29E-04 24.3479 4.34E-05
120
o
24.2325 7.07E-04 24.8815 0.0594 26.0726 9.86E-05
150
o
24.9448 8.12E-05 25.4786 0.0078 25.9388 8.25E-05
30
o
41.2438 0.014 42.3179 0.0192 38.6688 0.2812
60
o
41.1094 0.0262 40.7922 0.0381 38.3225 0.2079
90
o
38.6335 0.063 40.1324 0.0941 40.7759 0.1593
120
o
38.655 0.0257 39.787 0.0805 41.7251 0.0352
150
o
40.8409 0.043 40.7907 0.0817 41.8946 0.0065
Ra
L=0.05 L=0.10 L=0.25
b
?
4
10
5
10
6
10
7
10
m
Nu
m
Nu
m
Nu
m
rNu
RMS |
m
rNu
RMS |
m
rNu
RMS |
 
 
 
 
 
 
 
 86
 
 
Table 4.5 Maximum stream fucntion (
max
? ) of the primary vortex and relative RMS 
(
max
|
?r
RMS ) for all 60 cases with thin insulated baffles 
 
30
o
1.52E-05 4.39E-06 1.50E-05 1.57E-05 1.38E-05 5.58E-05
60
o
1.47E-05 2.06E-05 1.35E-05 6.61E-05 1.03E-05 8.74E-05
90
o
1.32E-05 1.75E-05 1.04E-05 1.18E-04 5.46E-06 3.30E-04
120
o
1.36E-05 8.01E-05 1.15E-05 2.04E-04 8.55E-06 2.01E-04
150
o
1.52E-05 2.54E-05 1.49E-05 6.44E-05 1.35E-05 5.71E-05
30
o
7.47E-05 6.93E-06 7.33E-05 1.35E-05 6.82E-05 3.81E-07
60
o
7.48E-05 1.14E-05 7.04E-05 1.18E-05 5.49E-05 8.02E-07
90
o
7.05E-05 1.67E-05 6.38E-05 5.74E-06 3.93E-05 8.68E-06
120
o
6.60E-05 8.90E-07 5.50E-05 1.39E-06 4.96E-05 8.17E-07
150
o
7.29E-05 3.49E-06 7.10E-05 2.86E-06 6.98E-05 9.77E-07
30
o
1.94E-04 6.60E-07 1.93E-04 3.18E-07 1.65E-04 3.76E-07
60
o
1.90E-04 8.50E-08 1.66E-04 6.47E-07 1.41E-04 3.53E-07
90
o
1.66E-04 1.14E-06 1.46E-04 0.0012 1.19E-04 1.31E-06
120
o
1.59E-04 0.0012 1.18E-04 0.1229 1.39E-04 9.05E-05
150
o
1.83E-04 5.64E-05 1.78E-04 0.0214 1.83E-04 5.74E-05
30
o
4.23E-04 0.0818 4.46E-04 0.0164 3.72E-04 0.0454
60
o
4.43E-04 0.0777 3.88E-04 0.1062 3.16E-04 0.0136
90
o
2.84E-04 0.1719 3.00E-04 0.1937 2.84E-04 0.1272
120
o
3.35E-04 0.0284 3.70E-04 0.1516 2.88E-04 0.0698
150
o
4.22E-04 0.0803 3.65E-04 0.0672 3.77E-04 0.0028
Ra
L=0.05 L=0.10 L=0.25
b
?
4
10
5
10
6
10
7
10
max
?
max
?
max
?
max
|
?r
RMS
max
|
?r
RMS
max
|
?r
RMS
 
 
 
 
 
 
 
 
 
 87
4.6 Closure 
The effect of a thin insulated baffle that is attached on the inner wall of a sphere 
on pseudosteady-state natural convection was investigated in this Chapter.  For low 
Rayleigh numbers, psuedosteady-state with minimal fluctuations of the flow and thermal 
fields were observed.  For Ra=10
7
, strong fluctuations were promoted.  In general, 
attaching an insulated baffle degrades the amount of heat that can be added to the stored 
fluid. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 88
 
 
 
 
 
 
 
 
 
Figure 4.1 Schematic diagram of a spherical container with a thin insulated baffle 
            
            
            
            
            
            
            
            
            
D
l
?
b
?
c
T
w
T
g
null
 89
            
       
 
 
 
Figure 4.2 3-D view of the system  
 
            
    
            
            
   
Baffle 
 90
 
 
 
 
Figure 4.3 Grid systems with the same baffle (L=0.25) located at (a)
null
30=
b
? ,  
(b) 
null
60=
b
? , (c) 
null
90=
b
?  and (d) 
null
120=
b
?   
 
 
 
 
 
)(a
)(b
)(d)(c
 91
 
 
 
Number of Cells
Nu
m
10
3
10
4
25
30
35
40
45
50
55
Proper Grid Size
 
 
Figure 4.4 The time-averaged and RMS values of the Nusselt number 
m
Nu  (based 
on
mw
TTT ?=? ) as a function of grid size for a case with insulated baffle (L=0.25, 
o
b
90=?  and Ra=10
7
) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 92
 
 
 
Number of Cells
Nu
c
10
3
10
4
20
25
30
35
40
45
Proper Grid Size
 
 
Figure 4.5 The time-averaged and RMS values of the Nusselt number
c
Nu  (based 
on
cw
TTT ?=? ) as a function of grid size for a case with insulated baffle (L=0.25, 
o
b
90=?  and Ra=10
7
) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 93
 
 
 
Time Step Size (s)
Nu
m
10
-1
10
0
10
1
34
36
38
40
42
44
46
48
50
Proper Time Step Size
 
 
 
Figure 4.6 The time-averaged and RMS values of the Nusselt number 
m
Nu  (based on 
mw
TTT ?=? ) as a function of time step size for a case with insulated baffle (L=0.25, 
o
b
90=?  and Ra=10
7
) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 94
 
 
 
Time Step Size (s)
Nu
c
10
-1
10
0
10
1
22
24
26
28
30
32
34
36
38
40
Proper Time Step Size
            
            
Figure 4.7 The time-averaged and RMS values of the Nusselt number
c
Nu  (based on 
cw
TTT ?=? ) as a function of time step size for a case with insulated baffle (L=0.25, 
o
b
90=?  and Ra=10
7
) 
 
 
 
 
 
 
 
 
 
            
        
 
 
 
 
 
 
 95
 
 
 
Figure 4.8 Pseudosteady-state streamline patterns and temperature contours for three 
insulated baffles (L = 0.05, 0.10 and 0.25) placed at various  
locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for Ra = 10
4
 
null
150=
b
?
null
120=
b
?
null
90=
b
?
null
60=
b
?
null
30=
b
?
05.0=L 1.0=L 25.0=L
 96
 
 
 
Figure 4.9 Pseudosteady-state streamline patterns and temperature contours for three 
insulated baffles (L = 0.05, 0.10 and 0.25) placed at various  
locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for Ra = 10
5
 
null
30=
b
?
null
60=
b
?
null
90=
b
?
null
120=
b
?
null
150=
b
?
05.0=L 1.0=L 25.0=L
 97
 
 
 
Figure 4.10 Pseudosteady-state streamline patterns and temperature contours for three 
insulated baffles (L = 0.05, 0.10 and 0.25) placed at various  
locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for Ra = 10
6
 
null
30=
b
?
null
60=
b
?
null
90=
b
?
null
120=
b
?
null
150=
b
?
05.0=L 1.0=L 25.0=L
 98
 
 
 
 
 
Figure 4.11 Pseudosteady-state streamline patterns and temperature contours with an 
insulated baffle (L = 0.25) placed at various locations (
b
? = 30
o
, 90
o
 and 150
o
) for  
Ra = 10
4
, 10
5
, 10
6
 and 10
7 
 
 
4
10=Ra
5
10=Ra
6
10=Ra
7
10=Ra
null
30=
b
?
null
90=
b
?
null
150=
b
?
 99
 
 
 
 
 
Figure 4.12 Streamline patterns and temperature contours in one cycle (a?h) for case 
with a thin insulated baffle (L=0.25, 
b
? =60
o
) for Ra=10
7
 
 
 
 
 
 
 
 
 
)(a
)(b )(c
)(d )(e
)( f
)(g )(h
 100
 
 
 
 
 
 
 
 
Figure 4.13 Cyclic variation of the instantaneous area-averaged Nusselt number for case 
with a thin insulated baffle (L=0.25, 
b
? =60
o
) for Ra=10
7
 (Corresponding to Figure 4.12) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
?
Nu
c
(
?
)
0.394 0.396 0.398 0.4 0.402 0.404
15
20
25
30
35
40
45
50
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
?
Nu
c(
?
)
0.36 0.37 0.38 0.39 0.4 0.41 0.42
15
20
25
30
35
40
45
50
 101
 
 
 
 
 
 
Figure 4.14 Dependence of the instantaneous surface-averaged Nusselt number )(?
c
Nu  
with dimensionless time for case (a) with a thin insulated baffle (L=0.10,
null
60=
b
?  and 
Ra=10
7
) and case (b) with a thin insulated baffle (L=0.05,
null
150=
b
? and Ra=10
7
) 
?
Nu
c
(
?
)
0.08 0.082 0.084 0.086 0.088
31
32
33
34
35
36
37
38
39
40
41
?
Nu
c
(
?
)
0.08 0.082 0.084 0.086 0.088
33
34
35
36
37
38
39
)(a
)(b
 102
 
 
 
 
 
?
Nu
c
(
?
)
0.08 0.082 0.084 0.086 0.088
24
26
28
30
32
34
Peak
Valley
Amplitude
            
Figure 4.15 Strength of oscillations of the Nusselt number (
c
Nu ) with dimensionless time 
for a case with a thin insulated baffle (L=0.25,
null
120=
b
? and Ra=10
7
) 
      
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 103
 
 
 
 
 
 
?
N
u
c
(
?
)
0.08 0.082 0.084 0.086 0.088
0
10
20
30
40
50
60
Fluctuating Nusselt Number
Time-Averaged Nusselt Number
Root Mean Square (RMS)
  
Figure 4.16 Strength (RMS) of oscillations of the Nusselt number (
c
Nu ) with 
dimensionless time for case with thin insulated baffle (L=0.25,
null
30=
b
? , Ra=10
7
) 
            
            
            
            
            
            
      
 
 
 
 104
 
 
 
 
 
 
Ra
Nu
c
10
4
10
5
10
6
10
7
5
10
15
20
25
30
35
40
45
No Insulated Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.05
 
 
 
Figure 4.17 Dependence of the time-averaged Nusselt number (
c
Nu ) on Ra among cases 
with a fixed thin insulated baffle (L=0.05) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 105
 
 
 
 
 
 
Ra
Nu
c
10
4
10
5
10
6
10
7
5
10
15
20
25
30
35
40
45
No Insulated Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.10
 
 
 
Figure 4.18 Dependence of the time-averaged Nusselt number (
c
Nu ) on Ra among cases 
with a fixed thin insulated baffle (L=0.10) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 106
 
 
 
 
 
 
Ra
Nu
c
10
4
10
5
10
6
10
7
5
10
15
20
25
30
35
40
45
No Insulated Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.25
 
 
 
Figure 4.19 Dependence of the time-averaged Nusselt number (
c
Nu ) on Ra among cases 
with a fixed thin insulated baffle (L=0.25) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 107
 
 
 
 
 
 
Ra
Nu
m
10
4
10
5
10
6
10
7
10
15
20
25
30
35
40
No Insulated Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.05
 
 
 
Figure 4.20 Dependence of the time-averaged Nusselt number (
m
Nu ) on Ra among cases 
with a fixed thin insulated baffle (L=0.05) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 108
 
 
 
 
 
 
Ra
Nu
m
10
4
10
5
10
6
10
7
10
15
20
25
30
35
40
No Insulated Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.10
 
 
 
Figure 4.21 Dependence of the time-averaged Nusselt number (
m
Nu ) on Ra among cases 
with a fixed thin insulated baffle (L=0.10) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 109
 
 
 
 
 
 
Ra
Nu
m
10
4
10
5
10
6
10
7
10
15
20
25
30
35
40
No Insulated Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.25
 
 
 
Figure 4.22 Dependence of the time-averaged Nusselt number (
m
Nu ) on Ra among cases 
with a fixed thin insulated baffle (L=0.25) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 110
 
 
 
 
 
 
Ra
|
?
max
|
x10
5
10
4
10
5
10
6
10
7
5
10
15
20
25
30
35
40
45
50
No Insulated Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.05
 
 
 
Figure 4.23 Dependence of the maximum stream function 
max
? on Ra among cases with a 
fixed thin insulated baffle (L=0.05) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 
150
o
) and the case without baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 111
 
 
 
 
 
 
Ra
|
?
ma
x
|x
1
0
5
10
4
10
5
10
6
10
7
5
10
15
20
25
30
35
40
45
50
No Insulated Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.10
 
 
 
Figure 4.24 Dependence of the maximum stream function 
max
? on Ra among cases with a 
fixed thin insulated baffle (L=0.10) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 
150
o
) and the case without baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 112
 
 
 
 
 
 
Ra
|
?
ma
x
|x10
5
10
4
10
5
10
6
10
7
5
10
15
20
25
30
35
40
45
50
No Insulated Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.25
 
 
 
Figure 4.25 Dependence of the maximum stream function 
max
? on Ra among cases with a 
fixed thin insulated baffle (L=0.25) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 
150
o
) and the case without baffle  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 113
 
 
 
 
 
 
?
b
(degree)
Nu
c
30 60 90 120 150
4
5
6
7
8
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
4
 
 
 
 
Figure 4.26 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among a case without baffle 
and the cases with a thin insulated baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
4
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 114
 
 
 
 
 
 
?
b
(degree)
Nu
m
30 60 90 120 150
10
11
12
13
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
4
 
 
Figure 4.27 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among a case without 
baffle and the cases with a thin insulated baffle of different lengths (L=0.05, 0.10 and 
0.25) for Ra=10
4
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 115
 
 
 
 
 
 
?
b
(degree)
|
?
ma
x
|x10
5
30 60 90 120 150
0.5
1
1.5
2
2.5
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
4
 
 
 
Figure 4.28 Dependence of the maximum stream function 
max
?  on 
b
?  among a case 
without baffle and the cases with a thin insulated baffle of different lengths  
(L=0.05, 0.10 and 0.25) for Ra=10
4
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 116
 
 
 
 
 
 
?
b
(degree)
Nu
c
30 60 90 120 150
8
10
12
14
16
18
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
5
 
 
 
 
Figure 4.29 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among a case without baffle 
and the cases with a thin insulated baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
5
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 117
 
 
 
 
 
 
?
b
(degree)
Nu
m
30 60 90 120 150
15
16
17
18
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
5
 
 
 
Figure 4.30 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among a case without 
baffle and the cases with a thin insulated baffle of different lengths (L=0.05, 0.10 and 
0.25) for Ra=10
5
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 118
 
 
 
 
 
 
?
b
(degree)
|
?
max
|x1
0
5
30 60 90 120 150
4
5
6
7
8
9
10
11
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
5
 
 
 
Figure 4.31 Dependence of the maximum stream function 
max
?  on 
b
?  among a case 
without baffle and the cases with a thin insulated baffle of different lengths 
(L=0.05, 0.10 and 0.25) for Ra=10
5
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 119
 
 
 
 
 
 
?
b
(degree)
Nu
c
30 60 90 120 150
15
20
25
30
35
40
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
6
 
 
 
Figure 4.32 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among a case without baffle 
and the cases with a thin insulated baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
6
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 120
 
 
 
 
 
 
?
b
(degree)
Nu
m
30 60 90 120 150
22
24
26
28
30
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
6
 
 
 
Figure 4.33 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among a case without 
baffle and the cases with a thin insulated baffle of different lengths (L=0.05, 0.10 and 
0.25) for Ra=10
6
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 121
 
 
 
 
 
 
?
b
(degree)
|
?
ma
x
|x10
5
30 60 90 120 150
10
12
14
16
18
20
22
24
26
28
30
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
6
 
 
 
Figure 4.34 Dependence of the maximum stream function 
max
?  on 
b
?  among a case 
without baffle and the cases with a thin insulated baffle of different lengths 
(L=0.05, 0.10 and 0.25) for Ra=10
6
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 122
 
 
 
 
 
 
?
b
(degree)
Nu
c
30 60 90 120 150
25
30
35
40
45
50
55
60
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
7
 
 
 
Figure 4.35 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among a case without baffle 
and the cases with a thin insulated baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
7
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 123
 
 
 
 
 
 
?
b
(degree)
Nu
m
30 60 90 120 150
38
40
42
44
46
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
7
 
 
 
Figure 4.36 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among a case without 
baffle and the cases with a thin insulated baffle of different lengths (L=0.05, 0.10 and 
0.25) for Ra=10
7
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 124
 
 
 
 
 
 
 
Figure 4.37 Dependence of the maximum stream function 
max
?  on 
b
?  among a case 
without baffle and the cases with a thin insulated baffle of different lengths  
(L=0.05, 0.10 and 0.25) for Ra=10
7
 
 
 
?
b
(degree)
|
?
max
|
x10
5
30 60 90 120 150
25
30
35
40
45
50
55
60
No Insulated Baffle
L=0.05
L=0.10
L=0.25
Ra=10
7
 125
CHAPTER 5 EFFECT OF AN ISOTHERMAL BAFFLE ON PSEUDOSTEADY-
STATE NATURAL CONVECTION INSIDE SPHERICAL CONTAINERS 
 
Pseudosteady-state natural convection inside spherical containers with a thin 
insulated baffle was studied using a computational fluid dynamic package (FLUENT) in 
Chapter 4.  In the absence of addition of extra thermal energy to the fluid by the thin 
insulated baffle, it was shown that the presence of a thin insulated baffle can generally 
degrade heat transfer due to blockage of fluid flow next to the wall of the sphere.  Such a 
knowledge of management of heat transfer and fluid flow is of great interest in 
engineering applications.  The insulated thermal condition of the baffle limits its use in 
some applications.  Considering the other extreme limiting case, further research was 
conducted.  The objective of this Chapter is to investigate the effect of a perpendicular-
to-wall isothermal thin baffle on the flow field as well as heat transfer.  Parametric 
studies were performed for a Prandtl number of 0.7.  For Rayleigh numbers of 10
4
, 10
5
, 
10
6
 and 10
7
, baffles with 3 lengths positioned at 5 different locations were investigated.  
In effect, a parametric study involving 60 cases was performed.   
 
 
 
 
 
 126
5.1 Mathematical Formulation for the Pseudosteady-State Natural Convection 
inside Spherical Containers with a Thin Isothermal Baffle 
A thin isothermal baffle is attached on the inside wall of a spherical container 
along the radial direction and points to the center.  A schematic diagram for the posed 
problem is illustrated in Figure 5.1.  Mathematically, an extra boundary condition is 
introduced due to the presence of a thin isothermal baffle, while the modeling 
assumptions are the same as Chapter 3.   
 
5.1.1 Governing Equations and Boundary/Initial Conditions 
Further research on the effect of a baffle on the flow and thermal fields is reported 
on the basis of Chapters 3 and 4, by switching from a thin insulated baffle to the case of a 
thin isothermal baffle.  The governing equations are same as those formulated in Chapter 
3 (Equations 3.1-3.5) and are not repeated here.   
The no-slip boundary condition is imposed on the wall and two sides of the thin 
isothermal baffle.  The thermal conductivity of the baffle is very high, so that its 
temperature is always same as the container?s wall temperature.  The applicable 
dimensionless boundary conditions on the wall ( 1
*
=r ) and the two sides of the thin 
isothermal baffle ( 1)21(
*
<?? rL  and 
b
?? = ) are:  
,
**
0 = 
V
 =
V r?
          (5.1) 
1)()(
**
+= ??
cw
TT  for .0??        (5.2) 
On the symmetry axis ( 0=?  and ? ): 
.
**
0 = 
T
 = 
V
V
r
??
?
?
?
?
?
=          (5.3) 
 127
A schematic drawing of the system, including the baffle is similar to Figure 4.2.  
Pseudosteady-state natural convection inside spherical containers with a thin isothermal 
baffle is driven by the temperature difference between the solid wall (including baffle) 
and center.  A thin insulated baffle can not bring in any energy into the system, whereas a 
thin isothermal baffle can contribute heat flux to the fluid in the container.  This distinct 
difference indicates that the flow field and heat transfer for cases with isothermal baffles 
can become more complicated than the cases with insulated baffles.     
Pseudosteady-state natural convection inside spherical containers with thin 
isothermal baffles is governed by dimensionless parameters of the Rayleigh number, 
Prandtl number, baffle length and position.  Hutchins and Marschall (1989) and Shen et 
al. (1995) have shown that numerical solution exhibits extremely weak dependence on 
the Prandtl number within the range of 0.7 to 8.  Therefore, the Prandtl number is not 
taken as a variable but a constant equal to 0.7.  Various Rayleigh numbers are 
investigated for different baffle parameters (L and 
b
? ).    
 
5.1.2 Computational Details 
Considering the particular difficulty associated with presence of a thin isothermal 
baffle in a sphere, the solution was handled by use of a commercial computational fluid 
dynamic (CFD) package.  The mesh is generated using GAMBIT (version 2.2.30) and the 
commercial code FLUENT (version 6.2.16) is utilized for solving the governing 
equations. All the computations (60 cases) were performed on the Cray XD1 
supercomputer of the Alabama Supercomputer Authority (Huntsville, Alabama). 
 128
The mesh generation procedures and FLUENT settings are same as the cases with 
thin insulated baffles that were discussed in Chapter 4 and the details are not repeated 
here. 
 
5.1.3 Code Validation  
The computational results were benchmarked against previous data available in 
the literature by comparing the heat transfer correlations, temperature distribution and 
streamline patterns for cases with no baffle.  The code validation and verification 
procedures for the pseudosteady-state natural convection inside spherical containers 
without baffles are discussed in detail in Chapter 3.  As for mathematical formulation, the 
presence of a thin isothermal baffle introduces an extra boundary condition on the baffle, 
while the governing equations are same as the case without baffles.  Therefore, it is 
considered that the code validation reported in Chapter 3 has provided enough confidence 
in the appropriateness of pseudosteady-state natural convection inside spherical 
containers with isothermal thin baffles.  
 
5.2 Grid and Time Step Size Independence Study  
 
The grid density and the time step size exhibit extremely strong effects on the 
accuracy of numerical solution.  The grid density and time step size independence tests 
have been studied in Chapter 4 for the pseudosteady-state natural convection inside 
spherical containers with a thin insulated baffle.  When a thin isothermal baffle replaces a 
thin insulated baffle, the flow is expected to be more dynamic due to extra heating effect 
of the isothermal baffle.  This indicates that the ?proper mesh? suitable for the case with a 
 129
thin insulated baffle may not be suitable for the case with a thin isothermal baffle within 
the required accuracy.  Therefore, independence studies for grid density and time step 
size were performed based on the grid information that is similar to the cases with 
insulated baffles.  The case (L=0.25 and 
2
?
? =
b
) with an isothermal thin baffle for the 
Rayleigh number of 10
7
 is selected for the sensitivity test.  The schematic geometry 
employed for the grid independence study is illustrated in Figure 4.3(c).  Time-averaged 
Nusselt numbers are employed as representative parameters to evaluate the accuracy of 
the numerical solution.  Grid density and time step size are two independent variables, 
thus these two independence studies can not be done at the same time.  The grid size 
independence is tested first, while the time step size is fixed to 2 seconds 
(
-4
1057.1 ?=?? ).  The effect of the spacing of the first row of the ?boundary layer? 
mesh, number of nodes on the baffle and density of the ?interior? nodes are studied.  For 
convenience, a parameter called the ?number of cells? is used to describe the overall grid 
density.  The variations of the time-averaged Nusselt numbers (
m
Nu  and 
c
Nu ) on the 
overall grid density along with the RMS values are illustrated in Figures 5.2 and 5.3, 
respectively.  The tabulated values of these Nusselt numbers are also presented in Table 
5.1.  It is observed that the time-averaged Nusselt number values do not change markedly 
when the number of cells is greater than 10
4
.  Considering both the accuracy of the 
solution and the computational time, the mesh with 13,868 cells is considered as a 
?proper? mesh and is shown in Figure 4.3(c).  All the other grid systems were created 
based on the parameters of this ?proper? mesh.  Hence, it was concluded that the grid 
system suitable for cases with a thin insulated baffle is also suitable for cases with a thin 
isothermal baffle. 
 130
Table 5.1 Dependence of the time-averaged and RMS values of the Nusselt numbers on 
the number of cells for an isothermal case with Ra=10
7
, Pr=0.7, L=0.25, 
o
b
90=?  and 
4
1057.1
?
?=??  
Number of Cells
108 19.8663 5.90E-05 24.8261 2.80E-05
318 27.7329 1.4298 37.0698 0.7274
962 33.5458 3.8704 42.0646 1.7535
2,084 33.9722 7.4818 44.045 4.5685
5,200 32.4959 8.1602 40.3278 6.9394
6,790 32.4986 7.4417 39.3839 6.3753
8,972 31.8137 6.7483 39.9492 5.5549
9,500 31.7294 7.0372 39.9635 5.2822
10,998 32.3284 7.5086 39.7588 6.4909
13,868 32.1272 7.36998 40.2304 6.82308
15,618 32.1188 8.602 40.0673 7.4348
23,584 32.2059 8.5845 40.2089 7.424
29,218 33.0703 8.0383 40.352 6.923
35,744 32.3324 8.1451 40.2915 6.8855
c
Nu
m
Nu
c
Nu
RMS |
m
Nu
RMS |
 
 
Similar to what was discussed in the independence study of Chapter 4, the time 
step size independence was conducted.  The variation of the time-averaged Nusselt 
numbers (
m
Nu  and 
c
Nu ) on the time step size along with the RMS values for the same 
isothermal baffle discussed above is shown in Figures 5.4 and 5.5, repectively.  The 
values of these quantities are also given in Table 5.2.  Taking both the accuracy of the 
solution and the computational time into account, a time step size of 2 seconds 
(
-4
1057.1 ?=?? ) is selected as a proper time step size for all 60 cases with thin 
isothermal baffles.   
 
 
 
 131
Table 5.2 Dependence of the time-averaged and RMS values of the Nusselt numbers on 
the time step size for an isothermal case with Ra=10
7
, Pr=0.7, L=0.25, 
o
b
90=?  and 
13,868 cells 
        (s)
0.1 32.3599 7.76638 40.2526 6.60545
0.2 33.0188 8.28112 40.2959 7.31774
1 32.6375 8.01251 40.2261 7.3654
2 32.1272 7.36998 40.2304 6.82308
3 31.2453 5.74914 40.026 5.29944
5 31.4553 6.79749 40.3208 7.0884
7 33.5314 6.86723 40.903 8.25014
8 33.2931 6.30238 40.8288 7.65948
10 32.3229 5.93448 40.7544 7.39692
15 28.8112 4.5176 39.7598 5.86457
30 25.4682 2.59521 38.4361 3.2171
t? c
Nu
m
Nu
c
Nu
RMS |
m
Nu
RMS |
 
 
5.3 Results and Discussion  
The flow and temperature fields strongly depend on the Rayleigh number, 
isothermal baffle location and length.  Temperature and stream function contours for 45 
cases (Ra=10
4
, 10
5
 and 10
6
) are presented for the pseudosteady-state condition, whereas 
the time-dependent flow and thermal fields of the high Ra cases (Ra=10
7
) are discussed 
separately.     
 
5.3.1 Pseudosteady-State Fluid Flow and Thermal Fields for Ra=10
4
, 10
5
 and 10
6
 
The composite diagrams of the streamlines and temperature contours for three 
isothermal baffles with lengths (L=0.05, 0.1 and 0.25) placed at various polar angle 
locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for a Rayleigh number of 10
4
 are presented 
 132
in Figure 5.6.  Diagrams in each row correspond to baffles of various lengths positioned 
at a fixed location, whereas for each column the effects of a baffle with a fixed length at 
various positions are given.  By comparing the streamline patterns and temperature fields 
in this figure to the limiting case of no baffle in Figure 3.13 for the same Ra, the effect of 
a thin isothermal baffle can be elucidated.  Focusing on the left column of Figure 5.6, it is 
clear that the presence of the shortest baffle (L = 0.05) does not alter the flow and thermal 
fields significantly, regardless of the angular position of the baffle.  The distortions are 
restricted to minor alteration of the streamlines next to the short baffles as well as 
temperature field.  For the cases corresponding to the L = 0.1 baffle that are shown in the 
middle column, the modifications to the flow and thermal fields are a bit more marked 
than the cases with the shortest baffle, however the changes are still observed to be next 
to the baffle.  In contrast, marked changes to the flow field are observed in the right 
column of Figure 5.6 that correspond to the longest baffle (L = 0.25), specially for the 
cases when the baffle is located at 
b
? = 60
o
, 90
o
 and 120
o
.  For these cases, two distinct 
recirculating vortices are observed on both sides of the baffle.  These vortices rotate in 
the clockwise (CW) direction lifting heated fluid next to the wall to a higher elevation 
and bringing down colder fluid.  The temperature fields have also been affected markedly 
by the longer length of the baffle and the pseudo-concentric ring contour patterns have 
been greatly modified in the vicinity of the isothermal baffle.  It can clearly be stated that 
due to the isothermal condition of the baffle, this structure perturbs the thermal field 
dramatically to the extent that a marked temperature gradient next to the baffle is 
observed.  In addition to modifying the flow paths, the isothermal baffle is responsible for 
bringing in heat into the sphere due to availability of a larger surface area. 
 133
For the Rayleigh number of 10
5
, streamlines patterns and temperature contours for 
three baffles with lengths (L = 0.05, 0.1 and 0.25) placed at various locations (
b
? = 30
o
, 
60
o
, 90
o
, 120
o
 and 150
o
) are shown in Figure 5.7.  Diagrams are plotted going from left 
column to right column with the dimensionless baffle length increasing from 0.05 to 0.25. 
Comparing the streamline patterns and temperature fields on the left column of Figure 5.7 
(L = 0.05) with the case of no baffle (Figure 3.13), the presence of the shortest thin 
isothermal baffle (L=0.05) modifies the thermal fields to some extent for various angular 
positions of the baffle.  The flow field modifications are not very significant that is 
similar to the cases in Figure 5.6 (Ra=10
4
), while modification of the thermal fields are 
also observed to be minimal.  In general, the flow and thermal fields exhibit features 
similar to those discussed above for Figure 5.6.   
Streamlines and temperature contours for three isothermal baffles with lengths (L 
= 0.05, 0.1 and 0.25) placed at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) for a 
Rayleigh number of 10
6
 are presented in Figure 5.8.  Comparing the streamline patterns 
and temperature fields on the left column of Figure 5.8 (L = 0.05) with the case of no 
baffle (Figure 3.13), the presence of the shortest baffle modifies the flow and thermal 
fields to some extent for the various angular positions of the baffle.  The flow 
modifications are confined to streamlines next to the short baffle through its interaction 
with the eye of the vortex that is closer to the wall compared to similar cases in Figures 
5.6 and 5.7.  For some positions of the short baffle, two eyes within the CW rotating 
vortex are observed.  In regard to the effect of the shortest baffle on the temperature 
contours, the changes are more pronounced in comparison to similar cases of Figure 5.7, 
however they are still localized in the vicinity of the baffle.  The flow modifications are 
 134
more pronounced for L = 0.1 baffles that are shown in the middle column of Figure 5.8, 
particularly when the baffle is located at 
b
? = 90
o
, 120
o
 and 150
o
.  For these cases, the 
double CW rotating vortex structure is further complicated by appearance of a counter 
CW rotating vortex that is located partly on top of the baffle and very near the open end 
of the baffle.  In contrast to a similar vortex that was observed for the same conditions but 
with an insulated baffle in Chapter 4, this vortex extracts heat from the isothermal baffle 
and is observed to be stronger and has a bigger spatial coverage.  One can also note that 
the multi-vortex structure can clearly rearrange the thermal field when compared to the 
case of the shortest baffle that exhibited constant-temperature stratified layers in the 
vicinity of the symmetry axis of the sphere.  In assessing the effect of the long baffle on 
the flow fields in the right column of Figure 5.8, modifications that are similar to the case 
of Ra = 10
5
 in Figure 5.7 are observed.  In general, two CW rotating vortices are clearly 
observed for all the cases.  In addition, a fresh third counter CW rotating vortex with its 
eye at 
b
?? <?  is created for cases with 
o
b
90?? .  This vortex gains more spatial 
coverage as the baffle is placed deeper into the lower hemisphere and is sandwiched 
between the two CW rotating vortices when the baffle is located at 
b
? = 120
o
 and 150
o
.  
The entire top surface of the baffle is in contact with this counter CW rotating vortex.  
Observing the marked temperature gradient next to the top surface of the baffle, this fresh 
vortex is clearly energized by the extra heat that is brought into the sphere from the top 
surface of the baffle.  As for the effect of the longest baffle on the thermal field, if the 
baffle is located such that ?
b
?  90
o
, parts of the interior of the sphere is composed of 
stable stratified constant-temperature layers.  This suggests that the flow within these 
regions is not strong and conduction mode of transport is dominant.  For other positions 
 135
of the baffle in the lower hemisphere, the remainder of the sphere is greatly affected by 
the extra heat that is brought into the sphere through the extended surface of the baffle.  
The extra heat coming through the baffle energizes a counter CW rotating vortex that 
directs hot fluid toward the center of the sphere, thus reducing thermal stratification.  As 
a general observation, note that the baffle divides the thermal field into two zones.  Both 
zones have areas of intense wall heat transfer that are located at the bottom of the sphere 
and angular positions 
b
?? <? .  The top zone is generally a region of stable constant-
temperature layers suggesting weak natural convection and its spatial coverage 
diminishes as the location of the baffle moves lower.  The bottom zone is generally 
observed to be the site of strong natural convection that is brought about by the extra 
heating afforded by the baffle. 
The effects of the Rayleigh number (10
4
, 10
5
, 10
6
 and 10
7
) and baffle?s position 
(
b
? =30
o
, 90
o
 and 150
o
) on streamlines and temperature contours for the case of the 
longest baffle (L=0.25) are presented in Figure 5.9.  With the baffle positioned near the 
top at 
b
? =30
o
 (left column), the increase of the Rayleigh number brings about stronger 
convection and fluid flow within the CW rotating vortex as indicated by the denser 
packing of the streamlines next to the surface.  This is accompanied by lifting of the eye 
of the vortex and its migration outward.  For the highest Ra number studied, a weak 
vortex is observed near the top within the cone, part of which is the baffle.  The vortex at 
the bottom that is driven by the thermally unstable stratified layers is present.  The 
temperature contours exhibit greater deviation from the concentric ring patterns as natural 
convection strengthens.  With the baffle located at 
b
? = 90
o
 (middle column), two CW 
rotating vortices occupy the two hemispherical regions with the lower half of the sphere 
 136
being the site of stronger convection.  As the Ra number is raised, a third counter-CW 
rotating vortex is created above the top surface of the baffle and next to the free end of 
the baffle.  For the highest Ra number studied, two vortices occupy the top hemisphere.  
For this case, the top hemisphere is partially stratified with stable constant-temperature 
layers occupying it, whereas the thermal field within the bottom hemisphere is heavily 
affected by the stronger rotating vortex that occupies it.  The flow fields for the cases 
with the longest baffle positioned near the bottom at 
b
? = 150
o
 (right column) exhibit 
many of the features with the baffle located at 
b
? = 30
o
, but in reverse in addition to the 
appearance of another counter CW rotating vortex for a wide range of the Rayleigh 
number.  The appearance of this counter CW rotating vortex is directly linked to the 
isothermal surface of the baffle and the resultant extra heat released into the fluid from 
this surface.  A recirculating vortex that occupies the small space between the baffle and 
the symmetry line of the sphere is clearly observed.  As for the temperature contours, the 
alterations appear to be generally confined to the lower hemisphere for all the Rayleigh 
numbers and more dramatically in the vicinity of the symmetry axis for the higher Ra 
numbers.  Extremely pronounced modifications of the temperature fields are observed for 
the high Ra number cases showing that stratifications are generally eliminated and the 
extra heat added to the sphere is transported inside by a counter CW rotating vortex that 
is positioned next to the top surface of the baffle. 
 
5.3.2 Time-Dependent Fluid Flow and Thermal Fields for Ra=10
7 
In order to illustrate the time-dependent nature of the flow for most of the Ra=10
7
 
cases, a representative case of Ra=10
7
, L=0.25 and 
b
? = 60
o
 was selected.  The 
 137
instantaneous composite diagrams of the streamlines and temperature contour fields for 
this case during a ?cycle? are shown in Figures 5.10 (a)-(i).  In order to aid the reader, a 
companion diagram showing the variation of the instantaneous area-averaged Nusselt 
number is also shown in Figure 5.11.  The cyclic nature of the instantaneous Nusselt 
number is clearly shown and the instants at which the composite streamlines and 
temperature contours of Figure 5.10 were shown are marked by letters a-i.  An extremely 
dynamic flow field is observed within the cycle with distinct growth and decay of a 
multitude of vortices.  Most notably, the primary vortex below the baffle decays early in 
the cycle (Figure 5.10 (a)-(c)) and then re-energizes during the remainder of the cycle.  
The vortex that is near the bottom of the sphere and is linked to the unstable stratified 
temperature layers is exhibiting dynamic behavior during the cycle.  The temperature 
gradients next to the wall of the sphere and the baffle vary dramatically during the cycle 
and are clearly linked to the variation of the instantaneous averaged Nusselt number 
shown in Figure 5.11. 
 
5.4 Nusselt Number Definitions and Other Parameters 
Before further discussion, the important parameters used to characterize the flow 
field and heat transfer quantitatively are defined.  
 
5.4.1 Definitions of the Nusselt Numbers 
The Nusselt number represents the ratio of convection heat transfer to conduction 
heat transfer.  Hence, it is used to characterize the strength of pseudosteady-state natural 
convection inside spherical containers with thin isothermal baffles.  The Nusselt number 
 138
either can be derived directly from the temperature gradient at the wall and baffle, or can 
be obtained by performing an energy balance for the fluid within the container, which is 
similar to the work done by Hutchins and Marschall (1989).  Theoretically, these two 
approaches should attain the same values of the Nusselt numbers when the pseudosteady-
state is reached.   
The presence of a thin isothermal baffle does not only modify the flow field but 
also contributes heat flux into the fluid within the spherical container.  Thus, the effective 
heat transfer area is the area of the container?s inner wall plus the surface area of the thin 
baffle.  Obviously, the effective heat transfer area is greater than the case with a thin 
insulated baffle.  The Nusselt number derived from the temperature gradients at the inner 
wall and baffle is illustrated first.  Following that, the Nusselt number obtained by 
performing an energy balance is presented.   
In order to get the area-averaged Nusselt number over the entire wall surface and 
baffle, the area-weighted heat flux should be derived.  In fact, the Nusselt number can 
also be identified as a dimensionless form of the heat flux.   Though the baffle is assumed 
to have zero thickness, it has two identical effective heat transfer surfaces, namely 
baffle?s upper surface and baffle?s lower surface, respectively (Figure 5.12).  The heat 
flow on the baffle?s lower surface (
bl
Q ) is: 
,dA
n
T
kQdQ
R
lR
bl
R
lR
bl
??
?
+
?
?
?
==         (5.4) 
with n standing for the normal-to-surface direction: 
,
1
++
?
?
=
?
?
?
T
rn
T
          (5.) 
.sin2 drrdA
b
??=          (5.6) 
 139
Finally, 
bl
Q  is: 
.sin2 dr
T
kQ
R
lR
bbl
?
?
+
?
?
=
?
??         (5.7) 
The heat flow on the baffle?s upper surface (
bu
Q ) is similarly given by:  
.sin2 dr
T
kdA
n
T
kQdQ
R
lR
b
R
lR
bu
R
lR
bu
???
?
?
?
?
?
?
?
=
?
?
==
?
??      (5.8) 
The total heat flow over the baffle (
b
Q ) can be calculated by combining Equations 5.7 
and 5.8:  
,|sin2)(sin2 dr
T
kdr
TT
kQQQ
R
lR
eb
R
lR
bblbub
??
?
?
?
+
?
?
=
?
?
+
?
?
=+=
?
??
??
??   (5.9) 
where the effective temperature gradient over the baffle is denoted as: 
.|
?+
?
?
+
?
?
=
?
?
???
TTT
e
         (5.10) 
The heat flow over the wall surface of the sphere (
w
Q ) is: 
.sin|2sin2|
0
2
0
??
==
?
?
=
?
?
=
??
?????? d
r
T
kRRdR
r
T
kQ
RrRrw
    (5.1) 
Therefore, the total heat (
t
Q ) conducted into the fluid within the spherical container is: 
??
?
=
?
?
+
?
?
=+=
?
?
?????
0
2
.|sin2sin|2 dr
T
kd
r
T
kRQQQ
R
lR
ebRrbwt
   (5.12) 
The effective heat transfer area (
t
A ) is: 
.sin)2(24
22
bt
lRlRA ???? ?+=        (5.13) 
Hence, the area-averaged heat flux for pseudosteady-state natural convection inside 
spherical containers with a thin isothermal baffle is:  
 140
.
sin)2(2
|sinsin|
)(
22
0
2
b
R
lR
ebRr
t
t
lRlR
dr
T
kd
r
T
kR
A
Q
tq
?
?
???
?
?+
?
?
+
?
?
==
??
?
=
    (5.14) 
Then, the area-averaged instantaneous Nusselt number can be calculated by employing 
the general definition: 
.
sin)2(2
|sinsin|
)(
22
0
2
b
e
R
lR
bRr
lRlR
dr
T
d
r
T
R
T
D
Tk
Dq
k
hD
tNu
?
?
???
?
?+
?
?
+
?
?
?
=
?
==
??
?
=
   (5.15) 
The choice of the temperature difference T?  is arbitrary.  Generally, the term T? is 
taken as
mw
TTT ?=? that serves as the driving force.  The dimensionless form of the 
RHS of Equation 5.15 is preferred.  Hence, the following dimensionless parameters are 
introduced: 
,
0
*
cw
TT
TT
T
?
?
=           (5.16) 
,
*
R
r
r =           (5.17) 
.
D
l
L =           (5.18) 
Substituting these dimensionless parameters into the Nusselt number expression, 
Equation 5.15 is simplified as: 
).|sinsin|(
]sin)(21[
1
)(
1
)(
0
*
1
21
*
1*
*
2**
*
??
?
=
?
?
+
?
?
?+?
=
?
?
???
?
? dr
T
d
r
T
LLTT
Nu
e
L
b
r
bmw
m
 (5.19) 
The term 
b
LL ?sin)(21
1
2
?+
, named shape factor, is related to the location and length of 
a thin isothermal baffle.  If L=0 (i.e. no baffle), the second integral vanishes and this term 
becomes equal to unity and the Nusselt number expression will be identical to Equation 
 141
3.21.  Note that a subscript ?m? is used for the area-averaged Nusselt number meaning 
that 
m
T  (mean or bulk temperature) is used in the T?  expression. 
Considering that in this pseudosteady-state natural convection modeling, the 
temperature difference between the wall and center is maintained as a constant value, 
hence another logical Nusselt number that is based on the constant driving force 
cw
TTT ?=? can be introduced.  This Nusselt number (
c
Nu ) is: 
).|sinsin|(
]sin)(21[
1
)(
0
*
1
21
*
1*
*
2
*
??
?
=
?
?
+
?
?
?+
=
?
?
???
?
? dr
T
d
r
T
LL
Nu
e
L
b
r
b
c
   (5.20) 
Note that a subscript ?c? is used for the surface-averaged Nusselt number meaning that 
c
T  (container?s center temperature) is used in the T?  expression.  If L=0, the second 
integral vanishes and the shape factor equals unity.  Expression 5.20 is then same as 
Equation 3.22. 
As mentioned above, another approach to get the Nusselt number is to perform 
the energy balance for the fluid within the container.  Under the pseudosteady-state 
condition, the heat transferred into the container will be exactly equal to the enthalpy 
change of the fluid inside the container.  Mathematically, this can be written as: 
.
sin)2(2
3
2
sin)2(24
3
4
)(
22
3
22
3
b
m
p
b
m
p
t
m
p
lRlR
dt
dT
cR
lRlR
dt
dT
cR
A
dt
dT
mc
tq
?
?
????
??
?+
=
?+
==      (5.21) 
Taking the dimensionless form of area-averaged heat flux, the mathematical formula of 
the Nusselt number can be obtained: 
.
sin)2(212
1
sin)2(2
3
2
)(
22
4
22
3
b
m
b
m
p
lRlR
dt
dT
D
TlRlR
dt
dT
cR
Tk
D
Tk
Dq
tNu
???
?
?+?
=
?+?
=
?
=  (5.22) 
 142
Similarly, the term T?  can either be taken as 
mw
TTT ?=? or 
cw
TTT ?=? .  
Correspondingly, two definitions of the Nusselt number are obtained as follows via 
introducing dimensionless time: 
,
2
R
t?
? =           (5.23) 
,
)(
1
]sin)(21[
1
3
2
)(
*
**2
??
?
d
dT
TTLL
  =Nu
m
mwb
m
??+
     (5.24) 
.
]sin)(21[
1
3
2
)(
*
2
??
?
d
dT
 
LL
 =Nu
m
b
c
?+
      (5.25) 
When L=0 (i.e. no baffle), the above equations return to the form of Equations 3.25 and 
3.26, respectively.  
The two Nusselt numbers that are based on the driving force of 
mw
TTT ?=? should equal each other. Similarly, the two Nusselt numbers which are 
based on the driving force of 
cw
TTT ?=?  should be identical.  All these four Nusselt 
numbers are evaluated for each case.  The Nusselt number definitions of pseudosteady-
state natural convection inside spherical containers with a thin isothermal baffle is more 
complicated in comparison with cases with no baffle or with a thin insulated baffle.  They 
are not only dependent on dimensionless temperature gradient or time change of rate of 
dimensionless temperature, but also depend on shape factor that varies with the baffle 
length and location.  Note that the shape factor can not be greater than unity.  Therefore, 
there is a chance for some cases with a high time rate of change of bulk temperature that 
the shape factor can lead to a low Nusselt number.  In effect, the presence of a thin 
isothermal baffle brings energy into the system and makes heat transfer more complex.   
 
 143
5.4.2 Time-Averaged Nusselt Numbers for Time-Dependent Cases 
It is shown that for high Rayleigh numbers, the system can not be recognized to 
be stable and therefore is not considered to be strictly at pseudosteady-state.  Determining 
whether the pseudosteady-state natural convection inside a spherical container with a thin 
isothermal baffle is reached is a critical step for our analysis.  The oscillations in some 
cases are quite well-organized (Figure 5.13 (a)), while in other cases the oscillations are 
disorderedly and random (Figure 5.13 (b)).    
Similar to the case with a thin insulated baffle, the time-averaged method is 
employed to monitor whether the flow and thermal fields can be considered to be 
statistically stationary and to calculate the mean or time-averaged value.  The straight-
line fitting curve equation is employed to quantitatively determine the pertinent values 
and judge the statistical stationary state.  The least-squares method is utilized in the 
process.  Equations 4.9 to 4.15 are not repeated here.  More details can be found in 
Chapter 4. 
The determination of whether the statistical stationary pseudosteady-state natural 
convection inside spherical container with a thin isothermal baffle is reached can be 
achieved by the approximated straight line.  The slope of this straight line can determine 
the tolerance and was employed for statistical stationary state judgment.  In the 
simulations when the slope is less than 10
-6
, it is declared that the statistical stationary 
state has been reached.   
 
 
 
 144
5.4.3 Oscillation Strength of the Nusselt Numbers 
The time-averaged Nusselt numbers are defined to characterize the mean 
magnitudes of the fluctuating Nusselt numbers.  However, this can not reflect the strength 
of oscillations of the Nusselt numbers.  Therefore, another parameter should be defined to 
quantitatively describe the strength of the fluctuations which can illustrate the strength of 
flow and thermal fields.  The Root Mean Square (RMS) value is considered an 
appropriate parameter for such characterization and the process is similar to the case with 
a thin insulated baffle.  More details can be found in Chapter 4.   
 
5.4.4 Stream Function Field 
In the previous section, the Nusselt number is introduced to characterize the 
pseudosteady-state natural convection inside spherical containers with a thin isothermal 
baffle.  From its definition, it is known that the Nusselt number is directly related to the 
temperature field but not the flow field.  The stream function is a parameter that is 
directly related to fluid flow.  For example, the difference between the maximum and the 
minimum stream function values of a vortex can be employed to characterize the strength 
of a rotating flow field.  Hence, it is necessary to study the stream function difference to 
determine the possible link to heat transfer due to the presence of a thin isothermal baffle.  
Stream function can also be employed to monitor whether the pseudosteady-state natural 
convection inside spherical containers with a thin isothermal baffle has reached the 
stationary state.  The stream function ?  at the center of vortex is defined as zero.  
Therefore, the maximum stream function 
max
?  (value on the wall) can be taken as a 
characteristic value of the flow field.  
 145
Similar to the Nusslet number, the stream function )(
max
??  is not a constant value 
but is a fluctuating quantity for most cases when the Rayleigh number is as high as 10
7
.  
Thus, the time-averaged stream function 
max
?  is defined for characterization.  The RMS 
and relative RMS values are calculated for each case to describe the strength of the flow 
field.  An approximated straight-line equation is solved to obtain the time-averaged 
stream function 
max
?  and to determine whether the statistical stationary state is reached.  
If the slope of the approximated line is less than 10
-6
, it is considered that the statistical 
stationary state is reached.  Therefore, there are two different ways to monitor the 
determination of statistical stationary state.  In relation to this, equations 4.16 to 4.17 are 
not repeated here and more details can be found in Chapter 4.   
 
5.5 Variations of Time-Averaged Nusselt Numbers and Maximum Stream Function  
In Section 5.3, the streamline patterns and temperature contours were presented 
for different Rayleigh numbers varying from 10
4
 to 10
7
 for different baffle lengths and 
locations.  In addition to affecting the thermal field directly by bringing heat into the 
fluid, the presence of a thin isothermal baffle disturbs the flow field and consequently 
modifies the temperature field indirectly.  In order to describe these modifications 
quantitatively, the variations of the time-averaged Nusselt numbers and maximum stream 
function are studied.  The Nusselt number is uniquely associated with the temperature 
field, whereas the stream function can characterize the flow field.  Details on the 
definitions of the time-averaged Nusselt numbers were given in Section 4.4.2. 
The variations of the time-averaged Nusselt numbers (
c
Nu  and 
m
Nu ) and 
maximum stream function of the primary vortex (
max
? ) with Ra for cases with a thin 
 146
isothermal baffle are presented in Figures 5.14 to 5.22.  In each figure, the baffle length is 
fixed (L=0.05, 0.10 and 0.25) and baffle locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 150
o
) are 
identified with filled symbols, whereas the reference case with no baffle is shown with an 
open symbol.  It must be emphasized that even though variations of the time-averaged 
Nusselt number (
m
Nu ) are presented, they will not be discussed in view of the 
complexity of their relation to the time rate of rise of the bulk temperature (Equation 
5.24).  The Nusselt numbers and maximum stream function strongly depend on the 
Rayleigh number and clearly exhibit the strengthening trend with the increase of 
Rayleigh number, irrespective of the baffle lengths and locations.  For Ra=10
4
, the flow 
and temperature fields do not change greatly due to the presence of the thin isothermal 
baffle.  This is because conduction is dominant for this low range of the Rayleigh 
number.  With increase of Ra, the effects of the thin isothermal baffle become more 
marked due to the increase of convective effects.  For a given Rayleigh number, the 
extent of flow and thermal field modifications is directly related to the length of the thin 
isothermal baffle.  It is evident (51 out of 60 cases) that by adding a thin isothermal 
baffle, the time-averaged Nusselt number (Nu
c
) is lower than the reference case with no 
baffle.  This Nusselt number is directly proportional to the heat that is transferred into the 
container from the surface of the sphere and both sides of the thin baffle.  This quantity 
(Nu
c
) is also very convenient since it is directly related to the time rate of rise of the bulk 
temperature according to Equation 5.25.  This means that even though heat is transferred 
through the surface of the baffle, the presence of the baffle has adversely caused lowering 
of the heat input through the wall of the sphere.  This control utility is related to the 
observed flow and thermal fields presented earlier in Chapter 4, where it was reasoned 
 147
that presence of an insulated baffle can lead to ?confinement? or ?compartmentalization? 
of the sphere.  In other words, it was generally stated that the zone above the insulated 
baffle was dominated by conduction, whereas the lower region was dominated by 
convection. For the majority of the cases investigated in this Chapter (51 out of 60), it is 
clear that the extra heating of the fluid by the baffle is not sufficient to overcome the 
degraded heat transfer through the surface of the sphere due to compartmentalization. 
Among the cases that were investigated, it can be observed that the Nusselt 
numbers and maximum stream function for some cases for which the baffle is located at 
30
o
, 60
o
, 120
o
 and 150
o
 are higher than the reference case for Ra=10
6
 and 10
7
.  This is 
explained as follows.  For all the cases with no baffle, the thermal boundary layer 
becomes thicker as the fluid from the bottom of the sphere rises along the inner wall.  For 
cases with Ra=10
7
, the presence of baffles near the top is beneficial to disturbing the 
thickened boundary layer, thus allowing extra heat to be drawn into the sphere.  This 
effect is even maintained for longer baffles suggesting that the degrading effect of a long 
baffle through compartmentalization is overcome by extra heating.  The enhancement of 
heat transfer that is observed due to presence of an isothermal baffle near the bottom of a 
sphere was seen to be limited to the high Ra number cases of 10
6
 and 10
7
 and the longest 
baffle (L=0.25) that was investigated.  This observation is clearly linked to the creation of 
a counter CW rotating vortex on the upper surface of the baffle that was discussed in 
Figure 5.9 (2 bottom entries on the right column).  Partly due to the orientation of the 
baffle and also its longer length, the counter CW rotating vortex that is created on the 
surface of the baffle due to heating directs the hot fluid toward the center of the sphere 
and penetrates higher into the sphere rising toward the top.  This activity, in turn, 
 148
counteracts the compartmentalization of the baffle by introducing hot fluid into the stable 
stratified fluid layers above the baffle. 
The behavior of the maximum stream function values that are shown in Figures 
5.20-5.22 generally match the trends of Nusselt number (
c
Nu ) that were discussed.  This 
indicates that the presence of thin isothermal baffles alters the temperature field through 
modification of the flow field.  Of course, addition of heat through the surface of the 
baffle is important for high Ra numbers, however this can not be quantified directly from 
the dependence of the maximum stream function values.   
Dependence of the time-averaged Nusselt numbers (
c
Nu  and 
m
Nu ) and 
maximum stream function of the primary vortex (
max
? ) on the position of the baffles 
(
b
? ) are presented in Figures 5.23-5.25, 5.26-5.28, 5.29-5.31 and 5.32-5.34 for Ra= 10
4
, 
10
5
, 10
6
 and 10
7
, respectively.  In each figure, the reference case with no baffle is 
identified with a filled circle, whereas various baffle lengths (L=0.05, 0.10 and 0.25) are 
identified with non-circular filled symbols.  Again, it must be emphasized that even 
though variations of the time-averaged Nusselt number (
m
Nu ) are presented, they will 
not be discussed in view of the complexity of their relation to the time rate of rise of the 
bulk temperature (Equation 5.24). 
For a given Ra number, as the location of baffle is lowered starting from the top 
of the sphere and moving toward the bottom, the Nusselt number Nu
c
 and maximum 
stream function exhibit trends suggesting that the confinement effect is minimal, when 
the baffle is placed near the two extremes.  The location corresponding to the most 
marked confinement varies depending on the Ra and length of the baffle.  For a fixed 
location of the baffle and Ra numbers lower that 10
7
, one can generally state that as the 
 149
length of the baffle is raised, the confinement effect becomes more enhanced.  This 
behavior is not observed for the highest Rayleigh number studied. 
The tabulated data for the Nusselt numbers (
c
Nu  and 
m
Nu ) and maximum stream 
function (
max
? ) of the primary vortex are listed in Tables 5.3, 5.4 and 5.5, respectively.  
In general, the relative RMS values are low suggesting that the pseudosteady-state is 
established, except for Ra=10
7
, indicating that unsteady effects are promoted as the Ra 
number is raised.   
It is found that the relative RMS values of the Nusselt numbers have strong 
dependence on the Rayleigh number.  This is not unusual, because for higher Rayleigh 
numbers, convection heat transfer is stronger.  However, there are four exceptions when 
Ra=10
7
. These are
o
b
L 30|05.0 == ? , 
o
b
L 30|10.0 == ? , 
o
b
L 30|25.0 == ?  and 
o
b
L 150|25.0 == ? , which are shaded in grey in Table 5.3.  This indicates that the 
oscillation strength does not only depend on the Rayleigh number, but also depends on 
the combination of thin isothermal baffle location and length.   
The results indicate that relative RMS values of the maximum stream function 
strongly depend on the Rayleigh number.  When Ra=10
7
, for most cases the oscillation 
effect is strong except three cases with relative RMS being less than 10
-6
. These 
are
o
b
L 30|05.0 == ? ,
o
b
L 30|10.0 == ?  and 
o
b
L 30|25.0 == ? which are shaded in 
grey in Table 5.5.  It can be concluded that the relative RMS value does not only depend 
on the Rayleigh number but also depends on the combination of the baffle length and 
location.   
 
 
 150
 
 
Table 5.3 Nusselt numbers (
c
Nu ) and relative RMS (
c
rNu
RMS | ) for all 60 cases with thin 
isothermal baffles 
 
30
o
5.5733 3.30E-04 5.4652 2.59E-04 5.8663 2.03E-04
60
o
5.304 2.56E-04 5.012 3.62E-04 5.425 2.15E-04
90
o
5.1474 3.62E-04 4.6801 3.03E-04 4.5801 3.30E-04
120
o
5.2647 2.41E-04 4.8007 1.76E-04 4.2544 3.75E-04
150
o
5.4984 3.53E-04 5.248 3.64E-04 4.6944 2.15E-04
30
o
12.3137 1.36E-04 11.807 1.62E-04 11.5104 1.52E-04
60
o
11.476 1.71E-04 10.2345 1.79E-04 8.9583 1.73E-04
90
o
11.2919 1.71E-04 9.9437 1.47E-04 7.7898 9.57E-05
120
o
11.7066 1.57E-04 10.6524 1.54E-04 8.6384 1.85E-04
150
o
12.2407 1.36E-04 11.6042 1.59E-04 10.07 1.64E-04
30
o
25.1747 8.05E-05 23.9844 5.01E-05 20.4738 7.40E-05
60
o
22.136 3.95E-05 20.3639 8.68E-05 17.777 7.00E-05
90
o
21.641 7.93E-05 16.8075 1.32E-04 16.0448 0.0283
120
o
22.1494 0.004 16.3173 0.182 15.8997 9.80E-05
150
o
23.9819 0.002 18.3277 0.2927 29.2267 4.37E-05
30
o
37.8193 5.11E-05 39.1508 3.89E-05 40.6807 3.07E-05
60
o
39.1656 0.0216 38.995 0.0077 34.7519 0.0617
90
o
34.0939 0.1523 31.987 0.1609 32.1272 0.2294
120
o
32.8959 0.0172 27.506 0.2894 38.9514 0.3485
150
o
36.7836 0.0535 33.2905 0.4207 74.5868 8.20E-04
L=0.05 L=0.10 L=0.25
Ra
4
10
5
10
6
10
7
10
c
Nu
c
Nu
c
Nub
?
c
rNu
RMS |
c
rNu
RMS |
c
rNu
RMS |
 
 
 
 
 
 151
 
 
Table 5.4 Nusselt numbers (
m
Nu ) and relative RMS (
m
rNu
RMS | ) for all 60 cases with 
thin isothermal baffles 
 
 
30
o
10.8826 3.00E-04 10.835 2.59E-04 11.4643 2.02E-04
60
o
10.6079 2.56E-04 10.539 3.61E-04 12.1179 2.15E-04
90
o
10.4532 3.62E-04 10.3222 3.02E-04 12.2986 3.29E-04
120
o
10.5258 2.41E-04 10.3532 1.76E-04 11.8568 3.75E-04
150
o
10.8477 3.53E-04 10.7599 3.63E-04 11.2307 2.14E-04
30
o
15.5981 1.36E-04 15.4156 1.62E-04 15.7231 1.52E-04
60
o
15.1604 1.70E-04 14.8853 1.79E-04 15.8039 1.72E-04
90
o
14.8112 1.70E-04 14.7301 1.47E-04 16.0063 9.57E-05
120
o
14.9991 1.56E-04 14.7917 1.53E-04 15.6979 1.84E-04
150
o
15.5322 1.36E-04 15.2921 1.58E-04 15.3046 1.64E-04
30
o
24.597 8.03E-05 24.1461 4.99E-05 23.6414 7.38E-05
60
o
23.1803 3.95E-05 22.7097 8.66E-05 23.2855 6.98E-05
90
o
23.0235 7.91E-05 23.2071 1.09E-04 23.6109 0.0201
120
o
23.3297 0.0033 23.4176 0.0749 24.9867 9.77E-05
150
o
24.2325 0.0035 23.1952 0.1939 25.8978 4.03E-05
30
o
39.2061 5.15E-05 39.0203 3.87E-05 39.316 3.07E-05
60
o
39.8818 0.0192 39.2148 0.0076 39.0198 0.0507
90
o
37.5667 0.1124 38.5192 0.102 40.2304 0.1696
120
o
37.5437 0.0181 38.715 0.2009 42.7318 0.2304
150
o
40.427 0.044 39.5723 0.3388 47.0156 5.52E-04
Ra
L=0.05 L=0.10 L=0.25
m
Nu
m
Nu
m
Nu
4
10
5
10
6
10
7
10
b
?
m
rNu
RMS |
m
rNu
RMS |
m
rNu
RMS |
 
 
 
 
 
 
 
 
 
 152
 
 
Table 5.5 Maximum Stream function (
max
? ) and relative RMS (
max
|
?r
RMS ) for all 60 
cases with thin isothermal baffles 
 
 
30
o
1.52E-05 2.91E-06 1.48E-05 1.81E-07 1.37E-05 1.22E-06
60
o
1.48E-05 1.08E-07 1.37E-05 1.11E-06 1.06E-05 3.46E-06
90
o
1.36E-05 8.87E-08 1.12E-05 6.58E-07 5.90E-06 3.59E-06
120
o
1.38E-05 9.44E-07 1.19E-05 1.25E-06 8.64E-06 8.10E-08
150
o
1.52E-05 7.16E-07 1.48E-05 1.37E-06 1.26E-05 1.21E-06
30
o
7.43E-05 2.17E-14 7.23E-05 1.81E-07 6.61E-05 3.12E-07
60
o
7.44E-05 4.62E-08 7.02E-05 6.96E-08 5.36E-05 1.28E-07
90
o
7.10E-05 4.61E-08 6.63E-05 3.04E-08 3.92E-05 1.10E-08
120
o
6.94E-05 4.68E-08 6.07E-05 4.70E-08 5.01E-05 3.20E-08
150
o
7.34E-05 3.50E-08 7.04E-05 5.43E-08 6.00E-05 2.66E-08
30
o
1.93E-04 2.24E-06 1.88E-04 3.56E-07 1.69E-04 1.24E-07
60
o
1.87E-04 1.02E-07 1.72E-04 7.43E-08 1.39E-04 6.24E-06
90
o
1.71E-04 1.49E-06 1.41E-04 2.07E-04 1.25E-04 0.0236
120
o
1.67E-04 0.0051 1.62E-04 0.1201 1.28E-04 4.79E-08
150
o
1.83E-04 0.0062 1.80E-04 0.1809 1.84E-04 1.66E-07
30
o
4.12E-04 1.22E-07 4.06E-04 3.82E-08 3.77E-04 2.51E-07
60
o
4.34E-04 0.0802 3.95E-04 0.0284 3.18E-04 0.0465
90
o
4.34E-04 0.1523 3.27E-04 0.1874 2.59E-04 0.2029
120
o
3.63E-04 0.019 3.74E-04 0.1705 4.30E-04 0.1296
150
o
4.21E-04 0.0733 3.43E-04 0.1393 4.96E-04 0.0352
Ra
L=0.05 L=0.10 L=0.25
max
?
max
?
max
?
4
10
5
10
6
10
7
10
b
?
max
|
?r
RMS
max
|
?r
RMS
max
|
?r
RMS
 
 
 
 
 
 
 
 
 
 153
5.6 Closure 
The effect of an isothermal baffle that is attached on the inner wall of a sphere on 
the pseudosteady-state natural convection was investigated in this Chapter.  For low 
Rayleigh numbers, psuedosteady-state with minimal fluctuations of the flow and thermal 
fields was observed.  For higher Ra, strong fluctuations were promoted.  In general, 
because of compartmentalization, attaching an isothermal baffle degrades the amount of 
heat that can be added to the stored fluid.  When an isothermal baffle is placed at the top 
or bottom of a sphere, the disturbance of the boundary layer next to the wall and extra 
heating were shown to overcome the compartmentalization. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 154
 
 
 
 
 
 
Figure 5.1 Schematic diagram of a spherical container with a thin isothermal baffle 
 
 
 
 
 
 
 
 
 
 
 
 
 
l
?
b
?
c
T
w
T
g
G
D
 155
 
 
 
Number of Cells
Nu
m
10
2
10
3
10
4
25
30
35
40
45
50
55
Proper Grid Size
 
 
 
Figure 5.2 The time-averaged and RMS values of the Nusselt number 
m
Nu  (based 
on
mw
TTT ?=? ) as a function of grid size for a case with isothermal baffle (L=0.25, 
o
b
90=?  and Ra=10
7
) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 156
 
 
 
Number of Cells
Nu
c
10
3
10
4
20
25
30
35
40
45
Proper Grid Size
 
 
 
Figure 5.3 The time-averaged and RMS values of the Nusselt number
c
Nu  (based 
on
cw
TTT ?=? ) as a function of grid size for a case with isothermal baffle (L=0.25, 
o
b
90=?  and Ra=10
7
) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 157
 
 
 
Time Step Size (s)
Nu
m
10
-1
10
0
10
1
32
34
36
38
40
42
44
46
48
50
52
Proper Time Step Size
 
 
 
Figure 5.4 The time-averaged and RMS values of the Nusselt number 
m
Nu  (based on 
mw
TTT ?=? ) as a function of time step size for a case with isotehrmal baffle (L=0.25, 
o
b
90=?  and Ra=10
7
) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 158
 
 
 
Time Step Size (s)
Nu
c
10
-1
10
0
10
1
24
26
28
30
32
34
36
38
40
42
44
46
Proper Time Step Size
 
 
 
Figure 5.5 The time-averaged Nusselt number 
c
Nu  (based on
cw
TTT ?=? ) as a function 
of time step size 
 159
 
 
Figure 5.6 Pseudosteady-state streamline patterns and temperature contours for three 
isothermal baffles (L = 0.05, 0.10 and 0.25) placed at various locations (
b
? = 30
o
, 60
o
, 
90
o
, 120
o
 and 150
o
) for Ra = 10
4
  
D
30=
b
?
D
60=
b
?
D
120=
b
?
D
90=
b
?
D
150=
b
?
05.0=L 10.0=L 25.0=L
 160
 
 
 
Figure 5.7 Pseudosteady-state streamline patterns and temperature contours for three 
isothermal baffles (L = 0.05, 0.10 and 0.25) placed at various locations (
b
? = 30
o
, 60
o
, 
90
o
, 120
o
 and 150
o
) for Ra = 10
5
 
D
30=
b
?
D
60=
b
?
D
120=
b
?
D
90=
b
?
D
150=
b
?
05.0=L 10.0=L 25.0=L
 161
 
 
 
Figure 5.8 Pseudosteady-state streamline patterns and temperature contours for three 
isothermal baffles (L = 0.05, 0.10 and 0.25) placed at various locations (
b
? = 30
o
, 60
o
, 
90
o
, 120
o
 and 150
o
) for Ra = 10
6
 
D
30=
b
?
D
60=
b
?
D
120=
b
?
D
90=
b
?
D
150=
b
?
05.0=L 10.0=L 25.0=L
 162
 
 
 
Figure 5.9 Pseudosteady-state streamline patterns and temperature contours with an 
isothermal baffle (L = 0.25) placed at various locations (
b
? = 30
o
, 90
o
 and 150
o
)  
for Ra = 10
4
, 10
5
, 10
6
 and 10
7
 
 
 
 
 
 
 
4
10=Ra
5
10=Ra
6
10=Ra
7
10=Ra
D
30=
b
?
D
90=
b
?
D
150=
b
?
 163
 
 
 
 
 
Figure 5.10 Time-dependent streamline patterns and temperature contours in one cycle 
(a?i) for a case with a thin isothermal baffle (L=0.25, 
b
? =60
o
) for Ra=10
7
 
 
 
 
 
 
 
 
 
 
)(a
)(b
)(c
)(d
)(e )( f
)(g
)(i)(h
 164
 
 
 
 
 
Figure 5.11 Cyclic variation of the instantaneous area-averaged Nusselt number for a 
case with a thin isothermal baffle (L=0.25, 
b
? =60
o
) for Ra=10
7
 (Corresponding to Figure 
5.10) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
?
Nu
c
(
?
)
0.395 0.4 0.405 0.41
30
32
34
36
38
40
42
(a)
(b)
(c)
(d)
(e)
(g)
(h)
(i)
(f)
?
Nu
c
(
?
)
0.36 0.37 0.38 0.39 0.4 0.41 0.42
30
32
34
36
38
40
42
 165
 
 
 
 
 
Figure 5.12 Detailed drawing of an isothermal baffle surface  
 
 
 
 
 
 
Baffle upper surface 
Baffle lower surface 
 166
 
 
Figure 5.13 Instantaneous area-averaged Nusselt number oscillation with dimensionless 
time for case (a) with a thin isothermal baffle (L=0.25, 
D
60=
b
? , Ra=10
7
) and case (b) 
with a thin isothermal baffle (L=0.05, 
D
150=
b
? , Ra=10
7
) 
 
 
 
)(a
?
Nu
c
(
?
)
0.48 0.49 0.5 0.51
34
35
36
37
38
39
40
41
42
)(b
?
Nu
c
(
?
)
0.48 0.49 0.5 0.51
31
32
33
34
35
36
37
38
 167
 
 
 
Ra
Nu
c
10
4
10
5
10
6
10
7
5
10
15
20
25
30
35
40
No Isothermal Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.05
 
 
 
Figure 5.14 Dependence of the time-average Nusselt number (
c
Nu ) on Ra among cases 
with a fixed thin isothermal baffle (L=0.05) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle  
 168
 
 
 
Ra
Nu
c
10
4
10
5
10
6
10
7
5
10
15
20
25
30
35
40
No Isothermal Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.10
 
 
 
Figure 5.15 Dependence of the time-average Nusselt number (
c
Nu ) on Ra among cases 
with a fixed thin isothermal baffle (L=0.10) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 169
 
 
 
Ra
Nu
c
10
4
10
5
10
6
10
7
10
20
30
40
50
60
70
No Isothermal Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.25
 
 
 
Figure 5.16 Dependence of the time-average Nusselt number (
c
Nu ) on Ra among cases 
with a fixed thin isothermal baffle (L=0.25) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 170
 
 
 
Ra
Nu
m
10
4
10
5
10
6
10
7
10
15
20
25
30
35
40
No Isothermal Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.05
 
 
 
Figure 5.17 Dependence of the time-average Nusselt number (
m
Nu ) on Ra among cases 
with a fixed thin isothermal baffle (L=0.05) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 171
 
 
 
Ra
Nu
m
10
4
10
5
10
6
10
7
10
15
20
25
30
35
40
No Isothermal Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.10
 
 
 
Figure 5.18 Dependence of the time-average Nusselt number (
m
Nu ) on Ra among cases 
with a fixed thin isothermal baffle (L=0.10) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 172
 
 
 
Ra
Nu
m
10
4
10
5
10
6
10
7
10
15
20
25
30
35
40
45
50
No Isothermal Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.25
 
 
 
Figure 5.19 Dependence of the time-average Nusselt number (
m
Nu ) on Ra among cases 
with a fixed thin isothermal baffle (L=0.25) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 
and 150
o
) and the case without baffle 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 173
 
 
 
Ra
|
?
max
|
x10
5
10
4
10
5
10
6
10
7
0
5
10
15
20
25
30
35
40
45
No Isothermal Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.05
 
 
 
Figure 5.20 Dependence of the maximum stream function 
max
?  on Ra among cases with 
a fixed thin isothermal baffle (L=0.05) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 
150
o
) and the case without baffle 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 174
 
 
 
Ra
|
?
max
|
x10
5
10
4
10
5
10
6
10
7
0
5
10
15
20
25
30
35
40
45
No Isothermal Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.10
 
 
 
Figure 5.21 Dependence of the maximum stream function 
max
?  on Ra among cases with 
a fixed thin isothermal baffle (L=0.10) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 
150
o
) and the case without baffle 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 175
 
 
 
Ra
|
?
max
|x1
0
5
10
4
10
5
10
6
10
7
0
5
10
15
20
25
30
35
40
45
50
No Isothermal Baffle
?
b
=30
o
?
b
=60
o
?
b
=90
o
?
b
=120
o
?
b
=150
o
L=0.25
 
 
 
Figure 5.22 Dependence of the maximum stream function 
max
?  on Ra among cases with 
a fixed thin isothermal baffle (L=0.25) at various locations (
b
? = 30
o
, 60
o
, 90
o
, 120
o
 and 
150
o
) and the case without baffle 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 176
 
 
 
?
b
(degree)
Nu
c
30 60 90 120 150
4
5
6
7
8
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
4
 
 
 
Figure 5.23 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case without baffle 
and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 177
 
 
 
?
b
(degree)
Nu
m
30 60 90 120 150
10
11
12
13
14
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
4
 
 
 
Figure 5.24 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among case without baffle 
and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 178
 
 
 
?
b
(degree)
|
?
max
|x10
5
30 60 90 120 150
0.5
1
1.5
2
2.5
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
4
 
 
 
Figure 5.25 Dependence of the Maximum stream function (
max
? ) on 
b
?  among case 
without baffle and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 
and 0.25) for Ra=10
4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 179
 
 
 
?
b
(degree)
Nu
c
30 60 90 120 150
8
10
12
14
16
18
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
5
 
 
 
Figure 5.26 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case without baffle 
and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 180
 
 
 
?
b
(degree)
Nu
m
30 60 90 120 150
14.5
15
15.5
16
16.5
17
17.5
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
5
 
 
 
Figure 5.27 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among case without baffle 
and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 181
 
 
 
?
b
(degree)
|
?
ma
x
|x10
5
30 60 90 120 150
4
5
6
7
8
9
10
11
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
5
 
 
 
Figure 5.28 Dependence of the Maximum stream function (
max
? ) on 
b
?  among case 
without baffle and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 
and 0.25) for Ra=10
5
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 182
 
 
 
?
b
(degree)
Nu
c
30 60 90 120 150
15
20
25
30
35
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
6
 
 
Figure 5.29 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case without baffle 
and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 183
 
 
 
?
b
(degree)
Nu
m
30 60 90 120 150
23
24
25
26
27
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
6
 
 
 
Figure 5.30 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among case without baffle 
and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 184
 
 
 
?
b
(degree)
|
?
max
|
x10
5
30 60 90 120 150
12
14
16
18
20
22
24
26
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
6
 
 
 
Figure 5.31 Dependence of the Maximum stream function (
max
? ) on 
b
?  among case 
without baffle and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 
and 0.25) for Ra=10
6
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 185
 
 
 
?
b
(degree)
Nu
c
30 60 90 120 150
30
35
40
45
50
55
60
65
70
75
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
7
 
 
 
Figure 5.32 Dependence of the Nusselt number (
c
Nu ) on 
b
?  among case without baffle 
and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 186
 
 
 
?
b
(degree)
Nu
m
30 60 90 120 150
37
38
39
40
41
42
43
44
45
46
47
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
7
 
 
 
Figure 5.33 Dependence of the Nusselt number (
m
Nu ) on 
b
?  among case without baffle 
and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 and 0.25) for 
Ra=10
7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 187
 
 
 
?
b
(degree)
|
?
max
|
x10
5
30 60 90 120 150
25
30
35
40
45
50
55
60
No Isothermal Baffle
L=0.05
L=0.10
L=0.25
Ra=10
7
 
 
 
Figure 5.34 Dependence of the Maximum stream function (
max
? ) on 
b
?  among case 
without baffle and cases with a thin isothermal baffle of different lengths (L=0.05, 0.10 
and 0.25) for Ra=10
7
 
 
 
 
 
 
 
 
 
 
 188
CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS 
 
In order to passively manage fluid flow and natural convection heat transfer under 
the pseudosteady-state condition inside spherical containers, a thin baffle (insulated or 
isothermal) was attached to the inner wall of the container.  The baffle is positioned such 
that it points along the radial direction.  For Rayleigh numbers of 10
4
, 10
5
, 10
6
 and 10
7
, 
baffles with 3 lengths positioned at 5 different locations were investigated.  Elaborate 
grid size and time step size independence tests were performed.  The solution of the 
governing equations was obtained by use of a commercial computational fluid dynamic 
(CFD) package. 
 
6.1 Conclusions 
Based on the results of the computational study conducted during the course of 
this research, the main conclusions are: 
1. Code validation was carried out by comparing the simulated results for the flow 
and thermal fields with no baffle to previous experimental and computational data.  The 
comparison among the various approaches was satisfactory.  For the four Rayleigh 
numbers that were studied, the psuedosteady-state condition was clearly established and 
no noticeable fluctuations of the Nusselt numbers were observed.  Specifically, two 
correlations for the Nusselt number (
m
Nu  and 
c
Nu ) were proposed.  Both thermally 
stable and unstable layers are present in this problem and for the higher Rayleigh
 189
numbers, the onset of instabilities was observed in this system.  Within the thermally 
unstable layer near the bottom of the spherical container, an active small vortex was 
formed for Ra=10
7
.  This flow pattern has not been reported to date by others. 
2. In the absence of extra heating, it was observed that introduction of a thin 
insulated baffle on the inner wall of the spherical container directly leads to modification 
of the velocity field, which in turn affects the temperature field.  It can generally be stated 
that an insulated baffle gives rise to ?confinement? or ?compartmentalization?, by which 
the fluid above the baffle is generally characterized by stable constant-temperature layers 
that are slow moving and dominated by heat conduction.  In contrast, the fluid below the 
baffle is subjected to strong natural convection currents.  Regardless of the Ra number, 
the short baffle (L = 0.05) alters the flow field and temperature contours to a minor extent 
in comparison to the corresponding Ra number cases with no baffle.  The modifications 
are limited to the vicinity of the baffle and a possible interaction with the eye of the 
primary CW rotating vortex.  The changes in the flow and thermal fields are observed to 
be more dramatic for the L = 0.1 baffle.  Multi-vortex structures rotating in opposite 
directions were observed.  The modifications of the flow and thermal fields were more 
pronounced for the longest baffle (L = 0.25) for which two CW rotating vortices are 
clearly observed when the baffle is positioned at or in the vicinity of the mid-plane.  The 
Nusselt numbers and maximum stream function of the primary vortex were generally 
lower than the reference cases with no baffle.  In general, placing a baffle in the mid-
plane of the container can lead to the lowest time rate of change of the bulk temperature, 
that is equivalent to 
c
Nu .  The degree of degradation of the Nusselt number has a strong 
dependence on the position and length of the insulated baffle.  In contrast to the general 
 190
reduction of heat transfer trends exhibited by the insulated baffles, placing it near the top 
of the sphere for high Ra number cases can lead to heat transfer enhancement in 
comparison to the reference case with no baffle.  The extra heat that is brought in the 
fluid through the surface of the sphere is linked to the disturbance of the thermal 
boundary layer by the thin baffle. 
3. In light of the extra heating afforded by a thin isothermal baffle, the velocity and 
temperature fields were more complicated than the case with a thin insulated baffle.  
Modified expressions for the Nusselt numbers were derived by introducing a ?shape 
factor? that depends on the length and position of the isothermal baffle.  The effect of the 
baffle?s length on the flow and thermal fields was observed to be similar to the trends 
exhibited by the insulated baffles.  In addition to confinement, a strong counterclockwise 
(CCW) rotating vortex that was created due to the extra heating of the baffle was present 
for high Ra numbers and baffle positions on or below the mid-plane.  The hot fluid in this 
vortex was observed to be transported toward the center of the sphere, thus disturbing the 
stable stratified layers.  The Nusselt number and maximum stream function values were 
dependent on Ra, baffle length and baffle location.  In comparison to the cases with no 
baffles, most Nusselt numbers and maximum stream function values were lower 
indicating that the confinement effect of the baffle still persists for the isothermal baffles 
in spite of the extra heating.  By placing an isothermal baffle near the top of the sphere 
and with high Ra numbers, heat transfer enhancement can be realized that is due to 
disturbance of the thermal boundary layer by the thin baffle.  In contrast to insulated 
baffles, placing isothermal baffles near the bottom for high Ra number cases also gave 
 191
rise to heat transfer enhancement due to disturbance of the stratified layers by the CCW 
rotating vortex that is energized by the heated baffle. 
4. Regardless of the thermal state of the baffle, its length and position, the 
pseudosteady-state condition was achieved for the three lower Ra numbers that were 
investigated.  For the highest Rayleigh number case, strong fluctuations of the velocity 
and temperature fields were clearly recorded.  Even though the instantaneous Nusselt 
numbers exhibit marked fluctuations, the corresponding time-averaged values were 
shown to exhibit the pseudosteady-state behavior. 
 
6.2 Recommendations for Future Work 
The results obtained in this study greatly enrich the knowledge base of 
management of flow field and natural convection heat transfer in spherical containers.  
The recommendations for future work are summarized as follow: 
I. Explore the effect of an oscillating thin insulated and isothermal baffle, 
II. Study the effect of multiple baffles pointing to the center, 
III. Investigate a single insulated or isothermal baffle pointing away from the 
center. 
 192
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Chow, M. Y. and Akins, R. G., 1975, ?Pseudosteady-State Natural Convection 
inside Spheres,? Transactions of the ASME, Journal of Heat Transfer, Vol. 97, pp. 54-59.  
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 193
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