EFFECT OF A BAFFLE ON PSEUDOSTEADYSTATE
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 PSEUDOSTEADYSTATE
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 PSEUDOSTEADYSTATE
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 NebraskaLincoln 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 PSEUDOSTEADYSTATE
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
Pseudosteadystate 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 constanttemperature 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 midplane. 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 midplane. 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
sisterinlaw 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 PseudosteadyState 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 Pseudosteadystate Nusselt Numbers???????..38
3.4 Closure????????????????????????????...40
CHAPTER 4 EFFECT OF AN INSULATED BAFFLE ON PSEUDOSTEADYSTATE
NATURAL CONVECTION INSIDE SPHERICAL CONTAINERS???????..62
4.1 Mathematical Formulation for PseudosteadyState 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 TimeDependent 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 TimeAveraged Nusselt Number????????????????..76
4.4.3 Strength of Fluctuations of Nusselt Numbers???????????...78
4.4.4 Stream Function ??????????????????????...79
4.5 Variation of the TimeAverage 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 PseudosteadyState 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 TimeDependent 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 TimeAveraged Nusselt Numbers for TimeDependent Cases..????143
5.4.3 Oscillation Strength of the Nusselt Number???????????...144
xiii
5.4.4 Stream Function Field????????????????????.144
5.5 Variation of TimeAverage 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 pseudosteadystate Nusselt numbers ??????..37
Table 3.4 Dimensionless maximum stream function (
*
max
? ) values ??????.38
Table 4.1 Dependence of the timeaveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 (rRMS) for all 60
cases with thin isothermal baffles?????..?????????..150
Table 5.4 Nusselt numbers (
m
Nu ) and relative RMS (rRMS) 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 differentiallyheated 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
userdefined function (UDF)?????????????????..50
Figure 3.11 Momentum boundary conditions in FLUENT??????????...51
Figure 3.12 Residual monitors in FLUENT????????????????..52
xviii
Figure 3.13 Pseudosteadystate 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 3D 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 Pseudosteadystate 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 Pseudosteadystate 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 Pseudosteadystate 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 Pseudosteadystate 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 areaaveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 Pseudosteadystate 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 Pseudosteadystate 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 Pseudosteadystate 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 Pseudosteadystate 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 areaaveraged 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 timeaverage 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 timeaverage 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 timeaverage 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 timeaverage 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 timeaverage 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 timeaverage 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
timeaveraged Nusselt number, qD/k(T
w
 T
c
)
xxxi
Nu
c
(?) areaaveraged Nusselt number
Nu
m
timeaveraged Nusselt number, qD/k(T
w
 T
m
)
Nu
m
(?) areaaveraged Nusselt number
p pressure, Pa
p
0
initial static pressure, Pa
p
*
dimensionless pressuer, (pp
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
? crosscorrelation 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
Buoyancydriven 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 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 pseudosteadystate 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
pseudosteadystate 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 pseudosteadystate 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
timeaveraged
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 timeaveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 pseudosteadystate 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 timeaveraged 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 timeaveraged 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
Pseudosteadystate 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 pseudosteadystate has been achieved will be discussed
in detail later. In short, once the timeaveraged Nusselt numbers (
m
Nu and
c
Nu ) do not
change with time, the pseudosteadystate 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 timedependent flow field streamlines and
temperature contours for Ra=10
7
are plotted and discussed separately. The dependence
of the timeaveraged Nusselt numbers (
m
Nu and
c
Nu ) for all 60 cases are given.
4.3.1 PseudosteadyState 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
pseudosteadystate 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 pseudoconcentric 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 multivortex structure can clearly rearrange the thermal field when compared to the
case of the shortest baffle that exhibited constanttemperature 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 midplane, 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 constanttemperature 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 counterCW 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 constanttemperature 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 CWrotating 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 TimeDependent 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 areaaveraged 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 ah. 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 pseudosteadystate
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 areaweighted 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 surfaceaveraged Nusselt numbers
meaning that
c
T (temperature at the center) is used in the T? expression. Similarly,
subscript ?m? is used for the other surfaceaveraged 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 TimeAveraged 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 pseudosteadystate 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 pseudosteadystate behavior
77
of this system, it was decided that timeaveraging of the Nusselt numbers is sufficient for
this investigation.
A straight line curvefitting equation of the fluctuating Nusselt number can be
employed to determine whether the statistical stationary pseudosteadystate 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 pseudosteadystate. In reality, for statistical
stationary state the value of b can be employed quantitatively to evaluate whether the
pseudosteadystate has been reached. If the coefficient b satisfies the requirement, then
constant a is the timeaveraged Nusselt number that is denoted by
m
Nu .
Considering the statistical distribution error, the leastsquares 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 leastsquares 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 timeaveraged Nusselt number
c
Nu , the details are the same and are
not repeated here.
At this point, the judgment of whether the statistical stationary pseudosteadystate
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 pseudosteadystate
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 timeaveraged 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
pseudosteadystate 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 timedependent. Similar to the Nusslet numbers, the stream function
max
?
is not a constant value but exhibits a fluctuating behavior. Thus, the timeaveraged
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 pseudosteadystate
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 timeaveraged 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 TimeAveraged 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 timeaveraged 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 timeaveraged Nusselt numbers can be found in Section 4.4.
The variations of the timeaveraged 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 timeaveraged 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 timeaveraged 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
4.234.25 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 timeaveraged 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 4.264.28, 4.294.31, 4.324.34 and 4.354.37 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 noncircular filled symbols. Again, it must be emphasized that even
though variations of the timeaveraged 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 pseudosteadystate 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 3.76E04 5.6456 3.08E04 5.336 3.06E04
60
o
5.5848 3.94E04 5.2843 3.58E04 4.6973 4.50E04
90
o
5.4883 3.21E04 5.0689 3.32E04 4.3069 4.93E04
120
o
5.6264 4.18E04 5.3355 3.34E04 4.6715 4.37E04
150
o
5.7492 4.12E04 5.6944 4.29E04 5.4687 4.54E04
30
o
12.6197 1.68E04 11.9784 1.77E04 10.4937 1.75E04
60
o
11.9103 1.77E04 10.4538 1.95E04 8.0405 1.61E04
90
o
11.9047 1.13E04 10.5831 1.53E04 7.8263 2.70E04
120
o
12.4218 9.36E05 11.7501 1.65E04 10.9077 1.74E04
150
o
12.7907 1.31E04 12.674 1.57E04 12.582 1.60E04
30
o
26.2174 7.97E05 24.1578 7.88E05 18.4589 6.33E05
60
o
22.4345 9.17E05 19.3234 1.09E04 13.9664 1.33E04
90
o
22.1206 9.51E05 16.641 6.22E04 14.8931 4.34E05
120
o
22.8518 9.16E04 18.8932 0.1405 20.5763 1.47E04
150
o
24.9339 8.61E05 24.8963 0.0115 24.9721 9.62E05
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 3.16E04 11.1841 2.33E04 11.0597 2.53E04
60
o
11.1416 3.43E04 11.0078 3.09E04 10.6936 4.43E04
90
o
11.0255 2.68E04 10.7733 2.90E04 10.3361 4.88E04
120
o
11.023 3.50E04 10.7697 2.51E04 10.4594 4.34E04
150
o
11.1666 3.41E04 11.0925 3.45E04 10.9369 3.50E04
30
o
16.0937 1.67E04 16.0216 1.76E04 15.73 1.75E04
60
o
15.947 1.76E04 15.8 1.95E04 15.0592 1.58E04
90
o
15.5702 1.13E04 15.6486 1.53E04 15.0019 2.69E04
120
o
15.6737 9.35E05 15.7196 1.64E04 15.9574 1.74E04
150
o
16.0253 1.31E04 15.9878 1.57E04 15.994 1.60E04
30
o
25.4469 7.95E05 25.0298 7.87E05 23.5963 6.34E05
60
o
24.0903 9.16E05 23.701 1.09E04 22.6813 1.33E04
90
o
24.049 9.48E05 24.1182 2.29E04 24.3479 4.34E05
120
o
24.2325 7.07E04 24.8815 0.0594 26.0726 9.86E05
150
o
24.9448 8.12E05 25.4786 0.0078 25.9388 8.25E05
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
1.52E05 4.39E06 1.50E05 1.57E05 1.38E05 5.58E05
60
o
1.47E05 2.06E05 1.35E05 6.61E05 1.03E05 8.74E05
90
o
1.32E05 1.75E05 1.04E05 1.18E04 5.46E06 3.30E04
120
o
1.36E05 8.01E05 1.15E05 2.04E04 8.55E06 2.01E04
150
o
1.52E05 2.54E05 1.49E05 6.44E05 1.35E05 5.71E05
30
o
7.47E05 6.93E06 7.33E05 1.35E05 6.82E05 3.81E07
60
o
7.48E05 1.14E05 7.04E05 1.18E05 5.49E05 8.02E07
90
o
7.05E05 1.67E05 6.38E05 5.74E06 3.93E05 8.68E06
120
o
6.60E05 8.90E07 5.50E05 1.39E06 4.96E05 8.17E07
150
o
7.29E05 3.49E06 7.10E05 2.86E06 6.98E05 9.77E07
30
o
1.94E04 6.60E07 1.93E04 3.18E07 1.65E04 3.76E07
60
o
1.90E04 8.50E08 1.66E04 6.47E07 1.41E04 3.53E07
90
o
1.66E04 1.14E06 1.46E04 0.0012 1.19E04 1.31E06
120
o
1.59E04 0.0012 1.18E04 0.1229 1.39E04 9.05E05
150
o
1.83E04 5.64E05 1.78E04 0.0214 1.83E04 5.74E05
30
o
4.23E04 0.0818 4.46E04 0.0164 3.72E04 0.0454
60
o
4.43E04 0.0777 3.88E04 0.1062 3.16E04 0.0136
90
o
2.84E04 0.1719 3.00E04 0.1937 2.84E04 0.1272
120
o
3.35E04 0.0284 3.70E04 0.1516 2.88E04 0.0698
150
o
4.22E04 0.0803 3.65E04 0.0672 3.77E04 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 pseudosteadystate natural convection was investigated in this Chapter. For low
Rayleigh numbers, psuedosteadystate 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 3D 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 Pseudosteadystate 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 Pseudosteadystate 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 Pseudosteadystate 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 Pseudosteadystate 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 areaaveraged 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 surfaceaveraged 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
TimeAveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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 timeaveraged 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
Pseudosteadystate 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
towall 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 PseudosteadyState 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 3.13.5) and are not repeated here.
The noslip 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(
*