NUMERICAL SOLUTION FOR G-JITTER INDUCED
FREE CONVECTION WITH CONSTANT HEAT
FLUX
SYAHIRA BINTI MANSUR
UNIVERSITI TEKNOLOGI MALAYSIA
NUMERICAL SOLUTION FOR G-JITTER INDUCED FREE CONVECTION
WITH CONSTANT HEAT FLUX
SYAHIRA BINTI MANSUR
A dissertation submitted in partial fulfilment of the
requirements for the award of the degree of
Master of Science (Mathematics)
Faculty of Science
Universiti Teknologi Malaysia
APRIL 2010
iii
ACKNOWLEDGEMENT
Alhamdulillah, my thanks go to Allah SWT for blessing me with the ability
to complete this work.
I would like to extend my deepest gratitude to my supervisor Dr. Sharidan
Shafiee for his guidance throughout the course of this dissertation.
My utmost appreciation to the lecturers of Universiti Teknologi Malaysia for
the knowledge I gained throughout my study (which proves to be useful during
completion of this work).
My special thanks to Universiti Tun Hussein Onn Malaysia and Ministry of
Higher Education for the financial support throughout the course of my study.
Thank you to my colleagues and friends who guided and supported me in the
preparation of the thesis.
My special thanks to my family for their love and support throughout this
entire period.
Thank you.
iv
ABSTRACT
G-jitter characterizes a small fluctuating gravitational field brought about,
among others by crew movements and machine vibrations aboard spacecrafts or in
other low-gravity environments such as the drop-tower and parabolic flights. In this
dissertation, Crank-Nicolson scheme is used to determine the numerical solution of
the g-jitter induced free convection with constant heat flux.
The governing
equations are solved numerically using different values of Prandtl numbers. Results
included are the variations of the skin friction, wall temperature, the velocity and
temperature profiles.
v
ABSTRAK
Ketar-g mencirikan suatu ayunan kecil medan gravity yang terhasil
antaranya oleh gerakan angkasawan dan getaran mesin di dalam kapal angkasa atau
di persekitaran graviti rendah yang lain misalnya menara-jatuh dan penerbangan
parabolik. Dalam disertasi ini, Crank-Nicolson akan digunakan untuk mendapatkan
penyelesaian berangka bagi kesan ketar-g ke atas pemindahan haba di permukaan
sfera.
Persamaan-persamaan
yang diterbitkan akan diselesaikan dengan
menggunakan nilai Prandtl yang berlainan. Keputusan kajian turut digambarkan
secara grafik untuk geseran kulit, suhu serta profil halaju dan suhu.
vi
TABLE OF CONTENTS
CHAPTER
TITLE
PAGE
DECLARATION OF THESIS
SUPERVISOR’S DECLARATION
1
TITLE PAGE
i
DECLARATION PAGE
ii
ACKNOWLEDGEMENT
iii
ABSTRACT
iv
ABSTRAK
v
TABLE OF CONTENTS
vi
LIST OF TABLE
ix
LIST OF FIGURES
x
LIST OF SYMBOLS / NOTATIONS
xi
LIST OF APPENDIX
xiii
INTRODUCTION
1.1 Research background
1
1.2 Significance of research
2
1.3 Objectives of the study
3
1.4 Scope of the study
3
1.5 Thesis outline
3
vii
2
3
4
LITERATURE REVIEW
2.1 Introduction
5
2.2 Microgravity and g-jitter
5
2.3 G-jitter and its effects
7
2.4 The effect of g-jitter on heat transfer
9
THE EFFECT OF G-JITTER ON HEAT
TRANSFER FROM A SPHERE WITH
CONSTANT HEAT FLUX
3.1 Introduction
13
3.2 Basic equations
13
3.3 Solution Procedure
17
METHOD OF SOLUTION IN FINDING THE
NUMERICAL SOLUTION FOR G-JITTER
INDUCED FREE CONVECTION WITH
CONSTANT HEAT FLUX
4.1 Governing Equations in a First-Order System
21
4.2 Crank-Nicolson Scheme
23
4.3 MATLAB Programming in processing
elimination method
5
26
RESULTS AND DISCUSSIONS
5.1 Numerical solution for g-jitter induced free
convection with constant heat flux
5.2 Velocity and temperature profiles
27
31
viii
6
REFERENCES
Appendix A
CONCLUSION
6.1 Summary of research
36
6.2 Suggestions for Future Research
37
38
43 - 53
ix
LIST OF TABLE
TABLE NO.
5.1
TITLE
PAGE
 2 0 s 
Value of skin friction
a , 0  and
 2
wall temperature 0   a , 0  at different
position of  for Pr = 0.7, 1 and 7
s
28
x
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
3.1
Physical model and coordinate system
14
4.1
Net rectangle for difference approximations
23
4.2
MATLAB implementation of naïve Gaussian
elimination
26
Variations of the skin friction with  for different
values of Prandtl numbers, Pr
29
Variations of the wall temperature with  for
different values of Prandtl number, Pr
30
Profiles of the non-dimensional velocity for
different values of  a when Pr = 0.7
32
Profiles of the non-dimensional temperature for
different values of  a when Pr = 0.7
33
Profiles of the non-dimensional velocity for
different values of  a when Pr = 7
34
Profiles of the non-dimensional temperature for
different values of  a when Pr = 7
35
5.1
5.2
5.3
5.4
5.5
5.6
xi
LIST OF SYMBOLS / NOTATIONS
a
-
radius of a sphere
g t  -
g-jitter or residual gravity field
g0
-
magnitude of g-jitter
Gr
-
Grashof number
k
-
unit vector pointing vertically upward
p
-
non-dimensional pressure
Pr
-
Prandtl number
qw
-
wall heat flux
r
-
non-dimensional radial coordinate
Re
-
Reynolds number
t
-
time
T
-
non-dimensional fluid temperature
T0
-
mean temperature
Uc
-
characteristic velocity
u, v
-
velocity components along x and y axes

-
non-dimensional velocity vector
Greek symbols
T
-
thermal expansion coefficient

-
non-dimensional transformed independent variables
a
-
polar angle
c
-
thermal conductivity

-
dynamic viscosity

-
kinematic viscosity
xii

-
density

-
non-dimensional small quantity

-
non-dimensional concentration

-
non-dimensional stream function

-
frequency of g-jitter oscillation
Superscripts

-
dimensional variables
'
-
differentiation with respect to 
s
-
denotes steady part of the solution
u
-
denotes unsteady part of the solution
Subscripts
w
-
condition at the wall

-
ambient condition
xiii
LIST OF APPENDIX
APPENDIX
A
TITLE
MATLAB For Numerical Solution For
G-jitter Induced Free Convection With
Constant Heat Flux
PAGE
43
CHAPTER 1
INTRODUCTION
1.1
Research Background
Gravity is identified by physicists as one of the four types of forces in the
universe alongside the strong and weak nuclear forces as well as the electromagnetic
force. Indeed, gravitational attraction is a fundamental property of matter that exists
throughout the known universe [Rogers, Vogt, Wargo [1]].
Nevertheless, there are times when it is not advantageous for scientists to
perform their researches under its full influence. Therefore, these scientists will
conduct their experiments in microgravity environment.
A microgravity
environment is a condition in which the effects of gravity are greatly reduced where
the apparent weight of a system is small compared to its actual weight due to
gravity. The environment where astronauts float in the International Space Station
is one of the many examples of microgravity environment.
Space experiments in accordance with microgravity have revealed unknown
or nonexistent effects on Earth which can be harmful to certain experiments. One of
these effects is g-jitter or residual accelerations phenomena associated with the
microgravity environment.
G-jitter is the inertia effects due to quasi-steady,
oscillatory or transient accelerations arising from crew motions and machinery
vibrations in parabolic aircrafts, space shuttles or other microgravity environments.
G-jitter characterizes a small fluctuating gravitational field, very irregular in
amplitude, random in direction and contains a broad spectrum of frequencies
2
(Schneider and Straub [2], Alexander et. al., [3] Nelson [4]). In an experiment
supported by the NASA Office of Life and Microgravity Sciences and Applications,
g-jitter dominates the spacecraft acceleration environment. It is comprised of a
myriad frequencies and displays no preferred orientation. The g-jitter magnitudes
can be as high as 1 milli-g (10 -3 g) (Ramachandran and Baugher [5]).
For this study, we consider the buoyancy-driven laminar flow around a fixed
sphere of radius a immersed in a viscous and incompressible Boussinesq fluid,
which is at uniform temperature T∞. It is also assumed that the sphere is subjected to
a constant heat flux q ω.
1.2
Significance of research
The effect of g-jitter on experiments, compared to ideal zero gravity
conditions, is largely unknown, especially in quantitative terms. Some researchers
have ventured into this foray, Shafie [6] and Amin [7], to name a few. Thus, it is of
great interest to quantitatively assess acceptable accelerations for a given
experiment.
As noted before, significant levels of g-jitter have been detected during space
missions in which low-gravity experiments were being conducted. Even a relatively
modest acceleration of 10-5 go can have a significant impact on solute segregation
(Pan et al. [8]).
To understand fully the impact of g-jitter, scientists and researchers need to
rely on modelling (Alexander et. al. [3]). Researchers may utilize theoretical models
effectively to predict the experiment’s sensitivity to g-jitter, bearing in mind that the
time-dependent nature of the g-jitter should be properly characterized beforehand
(Alexander et. al. [9]). For materials science experiments conducted in low earthorbit spacecraft, many questions are raised regarding experiment’s sensitivity to
residual acceleration. It is essential to provide the answers for these questions so
3
that the scientific return from such experiments is maximized. Shafie [6] and Amin
[7] have strived to present the much needed answers through their respective
research. Akin to the researches that preceded this particular study, the results of
this study should be helpful in understanding the g-jitter effects on fluid mechanics
process in microgravity conditions and better engineering design could be made in
the future.
1.3
Objectives of the Study
The main objective of this study is to examine theoretically the effect of gjitter on free convection problems.
Specifically, to obtain the numerical
computation for g-jitter induced free convection with constant heat flux.
1.4
Scope of the Study
The study is concerned with the generation of steady streaming due to g-jitter
induced free convection from a sphere, which is subjected to a constant heat flux.
For this study, the governing boundary layer equations are solved numerically using
the Crank-Nicolson method.
1.5
Thesis Outline
This thesis consists of six chapters including this chapter.
In this chapter, which is the introductory chapter, we have presented the
research background, objectives, scope and the significance of this research.
4
Chapter 2 deals with the literature review. We will present numerous studies
that are done on the free convection and discuss them.
Chapter 3 is mainly about the governing equations and the solution
procedure to obtain the equations needed in order to find the solution to the g-jitter
induced free convection.
Chapter 4 is concerned with the method of solution for finding the numerical
solution for g-jitter induced free convection with heat flux. We will provide the
numerical solution to this particular problem by using the Crank-Nicolson method.
Results presented include the streamlines, the isotherms and two physical quantities,
namely the reduced skin friction and the wall temperature, which play important
roles in characterizing the heat transfer. We will consider three different Prandtl
numbers and observe its effects on the reduced skin friction, the wall temperature
and the isotherms. A MATLAB program for this problem is given in the Appendix.
In Chapter 5, results and discussions are portrayed tabularly and graphically.
These results include the variations of skin friction, wall temperatures, and the
profiles of velocity and temperature.
The final chapter, Chapter 6, is the summary of the results. Suggestion and
recommendation for future research is included in this chapter.
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
This chapter consists of a literature review covering various topics. Section
2.2 will explain the microgravity attained on earth as well as the g-jitter that occurs
in microgravity environment. In Section 2.3, studies on the general effects of g-jitter
are presented. The concluding section, namely Section 2.4, will investigate the
effects of g-jitter on heat transfer.
2.2
Microgravity and G-jitter
Despite its name that conveys “weightlessness”, microgravity can also be
attained on Earth. It refers to the condition of free-fall within a gravitational field in
which the weight of an object is significantly reduced compared to its weight at rest
on Earth (Shafie [6]).
Microgravity facilities can be found in several locations around the globe
where scientists conduct their experiments in low gravity. There are five main
microgravity facilities: the Drop Tower, Parabolic Flights, Sounding Rockets,
Orbiting Spacecrafts and International Space Stations.
These facilities possess
different characteristic times which range from a few seconds to several months
(Mell et. al. [11], Yoshiaki et. al. [12]).
6
There are many advantages of conducting experiments under microgravity
environment. One of these advantages, as stated by Yoshiaki et. al. is that the said
environment reduces the effect gravity has on convection and sedimentation. In
short, these unwanted convective flows induced by gravity can be significantly
restrained.
One such example is the directional solidification and melting processes for
crystal growth. A low gravity environment produces conditions in which convection
is decreased to a level at which crystal growth is largely diffusion controlled. Under
microgravity environment, better crystals that own more uniform solute distribution
can be grown. On the whole, this is the assurance needed to develop new crystals
and at the same time, manufacturing techniques (Benjapiyaporn et. al. [13]).
This observation is seconded by Ramachandran and Baugher.
In their
studies, they specifically pointed out that protein crystal growth in microgravity
enjoys the advantages as mentioned by Benjapiyaporn et. al.. In addition, internal
stresses in the complex biological macromolecules are eliminated due to the reduced
hydrostatic pressure environment. This situation helps in improved internal order of
the grown crystal and prevents the collapse of the big complex molecules
(Ramachandran. Baugher [5]).
A microgravity environment provides the basis for a distinguishing
laboratory in which scientists can investigate the three fundamental states of matter:
solid, liquid and gas. The study of the states of matter and their interactions in
microgravity is an exciting opportunity to expand the frontiers of science (Shafie
[6]).
However, past experience has also shown that microgravity environment can
sometimes yield unpredictable results in experiments. Protein crystal growth, for
example, will be affected in that the resulting crystal will show signs of cracking and
stunted growth.
Other experiments conducted under microgravity environment
suffer the same effect. The causative factors for all these effects are not fully
7
understood. Nevertheless, the general consensus is that the g-jitter plays at least
some role in this process. This phenomenon will be presented in the next section.
2.3
G-jitter and its effects
G-jitter is the residual accelerations phenomena associated with the
microgravity environment.
In an experiment supported by the NASA Office of Life and Microgravity
Sciences and Applications,
environment.
orientation.
g-jitter dominates the spacecraft acceleration
It is comprised of myriad frequencies and displays no preferred
The g-jitter magnitudes can be as high as 1 milli-g (10-3 g)
(Ramachandran and Baugher [5]).
Furthermore, it is noted that the typical range of magnitude of g-jitter
occurring in spacecraft is from about 10 -3 g to 10-4 g (Chao [10]). Also, the range of
frequencies of g-jitter occurring in spacecraft is from about 0.1 Hz to 10 Hz.
In earthbound situations, the effects of g-jitter may be negligible. However,
in a low-gravity environment where heat and mass transfer in a fluid medium, in the
absence of radiation, is expected to be affected only by pure diffusion, g-jitter can
give rise to significant convective motions (Shafie [6]).
Pan et. al. [8] and Shu et. al. [14] confirmed that convection in microgravity
is related to the magnitude and frequency of g-jitter and to the alignment of the
gravity field with respect to the growth direction or the direction of the temperature
gradient.
8
Ramachandran and Baugher [5] provided a numerical study on g-jitter
effects in protein crystal growth. According to the study, several proteins have been
flown during past shuttle missions with the aim of growing bigger and better
crystals. However, g-jitter plays a significant role in the crystal degradation process.
Based on their finding, the g-jitter magnitudes can be as high as 1 milli-g ( 10-3 g )
and their calculation shows that the protein crystal growth flow field is susceptible
to 1 – 10 Hz frequency range.
In any protein crystal growth experiment, the solution transport within the
grow medium and the crystal surface attachment kinetics play key roles in
determining the crystal growth rate. The findings of Ramachandran and Baugher [5]
are supported by Pusey et al. [15]. Their experiments, using tetragonal lysozyme,
have shown that forced flow rates of 30 – 40 µm/s will slow and eventually stop the
growth of 10 µm crystals. More experiments by Pusey [16] suggest that this growth
cessation is present even at lower flow rates but the growth declines over a much
larger time than at the higher flow rates.
Furthermore, g-jitter also has large effects on materials processing in space
or in gravity-reduced environment where it interacts with the density gradients and
results in both fluid flow and solute segregation. These effects are reviewed by
Wilcox and Regel [17] in which they concluded that the amount of convection
increases with increasing acceleration and decreasing frequency and hence will
significantly influence some materials processing operations.
In addition, the orientation of the residual gravity is a crucial factor in
determining the suitability of the spacecraft environment as a means to suppress or
eliminate undesirable effects caused by buoyant fluid motion in Bridgman’s crystal
growth experiment.
9
2.4
The effect of g-jitter on heat transfer
The effect of g-jitter can result in significant convective motion, which can
be detrimental to certain experiments. Hence, it is imperative to provide a better
understanding on the relationship between g-jitter induced free convection and fluid
behaviour as well as physical processes.
Merkin [18], Davidson [19] and Haddon and Riley [20] were among those
who pioneered the studies of heat transfer in fluctuating flow situations. Merkin
considered a situation in which the temperature of a circular cylinder fluctuates
about a mean ambient temperature in a constant gravitational field. The unsteady
buoyancy force resulted in a fluid motion, with both steady and unsteady
components, which originated in a boundary layer on the cylinder.
In contrast, Davidson and Haddon and Riley ignored free-convective effects.
They determined the heat transfer from a cylinder which vibrated in a fluid that was
otherwise at rest for unbounded and bounded flows respectively. However, Merkin,
Davidson and Haddon and Riley paid particular attention to the time-averaged heat
transfer.
Langbein [21] once conducted an experimental investigation on the
convection caused by g-jitter in the axial direction in a spherical cavity heated at the
equator and cooled at the poles. He used the spherical cavity as a first order model
of the melting zone in a typical crystal growth experiment under microgravity. The
ensuing shift of the isotherms was calculated.
Heiss et. al. [22] presented numerical solutions of g-jitter induced natural
convection in a cylinder. They demonstrated the increasing and decreasing natural
convective fluid motion caused by gravity pulses of several amplitudes and
durations.
They managed to observe that the fluid motion increases when the
amplitude of the gravity pulse increases.
On the other hand, the fluid motion
decreases when the frequency of the gravity pulse increases.
10
A study on the effect of g-jitter on thermal convection showed that when gjitter is perpendicular to the applied temperature gradient in a liquid layer, the
amplitude of the oscillatory flow reduces as the g-jitter frequency increases (Doi et.
al. [23]).
Okano et. al. [24] had also presented the numerical results on the effect of gjitter in a rectangular cavity filled with liquid. The results showed that the strength
of the convective flow in the liquid is dependent on the frequency of the g-jitter field
when the field is perpendicular to the direction of the induced temperature gradient.
On the other hand, no effects could be observed on the flow field when the field is
parallel to the temperature gradient.
Farooq and Homsy [25] investigated the response of a differentially heated
square cavity to a time-dependent gravitational field where the aspect ratio of the
cavity was fixed at unity and the modulation scale was assumed to be small. The
results showed that the response of the cavity depended strongly on the frequency of
the modulation.
They attested to their earlier results by demonstrating that the modulation
interacts with the natural mode of the system to produce resonance when the
modulation scale is small (Farooq and Homsy [26]).
In addition, their results
illustrated that the modulation has the potential to destabilize the longwave
eigenmodes of the slot problem. These results complemented to those reported by
Biringen and Danabasoglu [27] who had solved the full non-linear, time-dependent
Boussinesq equations for g-jitter in rectangular cavities. Through a weakly nonlinear calculation, Farooq and Homsy [26] were able to explore parametric
dependencies that explain physical mechanisms and scaling.
Several scientists endeavoured to show the effects of g-jitter on RayleighBernard system. Gresho and Sani [28] performed an analysis to show that the g-jitter
effects can significantly affect the stability limits of the said system. They examined
the stability of a horizontal layer of fluid heated from above and below for the case
11
of a time-dependent buoyancy force which is generated by shaking the fluid layer,
hence causing a sinusoidal modulation of the gravitational field.
Biringen and Peltier [29] then produced results that concurred to those of
Gesho and Sani. They, however, investigated the effect of g-jitter on the RayleighBernard convection, by considering the full nonlinear time-dependent problem.
Lord Rayleigh provided the first theoretical analysis of the phenomenon of
streaming, in connection with sound wave (see Farooq and Homsy [25]). This
achievement is crucial in that it was the contributing factor to various transport
phenomena such as heat transfer or the distribution of chemical species in a timeaveraged sense.
Amin [7] performed a theoretical investigation on the effect of g-jitter. Her
investigation is centred upon the heat transfer from a sphere, maintained at a
constant temperature in the presence of g-jitter. For this particular investigation,
Amin considered two cases: the full nonlinear time dependent and the boundary
layer.
These problems are solved analytically using the method of matched
asymptotic expansion and numerically using Crank-Nicolson method.
She
concluded that buoyancy-induced convection due to high-frequency g-jitter cannot
be expected to lead to any significant change in heat transfer characteristics over and
above that due to pure conduction. Nevertheless, for fluids of small kinematic
viscosity and moderately large Prandtl number, it is expected that low frequency gjitter will exert a non-trivial influence on heat transfer.
Shafie [6] has also conducted a study on the effect of g-jitter on heat transfer
from a sphere. However, as opposed to previous study done by Amin [7], Shafie’s
study is maintained at a constant heat flux. The problems which are resulted from
the governing equations of motion are solved analytically and numerically
depending on the Reynolds number, Re. For small Re ( Re << 1), analytical results
are obtained using the method of matched asymptotic expansion. On the other hand,
for Re >> 1, numerical resorts for the boundary layer approximation in the limiting
12
case when Re → ∞ is obtained by using the Keller box method. He observed that
the reduced skin friction along the sphere increases as the angle increases.
However, the surface temperature decreases as the angle increases. Also, both the
reduced skin friction and the wall temperature decrease as the Prandtl number Pr
increases.
Both works are also related to that of Potter and Riley [30].
They
investigated the free convective flow from a heated sphere, in the Boussinesq
approximation at high Gashof number in viscous fluid. They performed numerical
evaluation on the characteristic of the boundary layer close to the surface of the
sphere. The results show that the solution exhibits a singular behaviour where the
boundary layer erupts into the plume which forms above the sphere as the moving
fluid converges onto the upper stagnation point.
CHAPTER 3
THE EFFECT OF G-JITTER ON HEAT TRANSFER FROM A SPHERE
WITH CONSTANT HEAT FLUX
3.1
Introduction
The free convection from a sphere, which is subjected to a constant surface
heat flux in the presence of g-jitter is investigated in this chapter. The governing
equations of motion are first written in dimensionless forms and the resulting
equations obtained after the introduction of the stream function are solved
numerically. Constant heat flux is considered in our studies because an important
practical and experimental circumstance in many convective flows is that generated
adjacent to a sphere dissipating heat uniformly [Shafie and Amin [31]).
3.2
Basic equations
Consider the buoyancy-driven laminar flow around a fixed sphere of radius
a immersed in a viscous and incompressible Boussinesq fluid, which is at uniform
temperature T . We assume that the sphere is placed in a fluctuating gravitational

field g t k , where k is the unit vector pointing vertically upward, t is the time and

 
we assume that g t  g 0 cos  t , where g0 is the magnitude of the g-jitter and 
is the frequency of the g-jitter oscillation which is assumed very high   1 . It is
also assumed that the sphere is subjected to a constant heat flux qw .
14
u
r
g t
v


a
qw
Figure 3.1: Physical model and coordinate system
The g-jitter induced free convection is described by the continuity, NavierStokes and energy equations, which can be written in non-dimensional forms as, see
Amin [7].
 3.1
. = 0

 2
   .    p 
   T cos t  k
t
Re
T

   .  T 
 2T
t
PrRe
 3.2 
 3.3
where t is the non-dimensional time,  is the non-dimensional velocity vector, T is
the non-dimensional fluid temperature and p is the non-dimensional pressure. These
non-dimensional quantities are defined as
t  t , r 




T  T
p  p
r

,
, T
, p
Uc
 aqw 
a
 aU c



 c 


 3.4 
15
Further, U c is the characteristic velocity, Re is the Reynolds number, Pr is the
Prandtl number and  is a dimensionless small parameter    1 given by
 aq 
U a
U
v
U c  g 0 T  w  , Re  c , Pr  ,   c


a
  c 
 3.5
with T being the thermal expansion coefficient,  c being the thermal conductivity,
 being the thermal diffusivity and v is the kinematic viscosity. We note that  is a
measure of the ratio of the amplitude of fluid particles fluctuations to the radius of
the sphere and, as we can see below, Re interpreted as a Reynolds number which
characterizes the induced steady streaming. We remark that this Reynolds number
is larger by a factor  1 . Note that  1 is the Strouhal number and the Grashof
3
q aa
Re2
number Gr  g 0 T  w  2 is related to  and Re as Gr 
. With reference

 c  


to spherical polar coordinate r ,  a ,  with a  0 corresponding to the direction of


k, we have for asymmetric flow,   r ,a , 0 . If we define a stream function 
such that
r 
1

1 
, a  
r sin  a  a
r sin  a r
2
equations  3.2  and  3.3 can then be written as
 3.6 
16
 1   , D 2  2
 

2
2
D




D

L

D 4
   r 2   r,   r 2
1 
t
Re


 1   2   L2T  cos t
T    , T 
  2
2

 2

 D T  2 L2T 
t r   r ,   RePr 
r

 3.7 
 3.8
where in  3.7  and  3.8 
2
 2 1     2
  1 


D  2
, L1 

, L2  r  
2
2
2
r
r

1   r r 
r

2
 3.9 
and   cosa . We shall solve these equations assuming the following boundary
conditions

 0 on r  1.
r
  o  r 2  as r  .

T
 1 on r  1.
r
T  0 as r  .
 3.10 
 3.11
 3.12 
 3.13
17
3.3
Solution Procedure
We can obtain equations for the functions 0 s  ,  0u  , T0 s  and T2 u  in the
following form (Amin [7]).

u
s
D 2 0   1   2  L2T0  cos t
t





 3.14 

 s
s
2


1 4 s  1   0 , D  0
2

D 0   2
 2 D 2 0  s  L1 0  s  
Re
  r,  
r
 r


 

2
1  0 ,D 0
 2
r
 r,  
u

u
  2 D    L    1   
2
u
2
r
u 
 s
T2u 
Re   0 , T0
 2
t
r
  r,  
2
0
u
1
0
Re
 L T    cos t
u
2 2

 3.15
3.16 

  0 s  , T0 s 
1  2 s 2
 s   Re
 D T0  2 L2T0   2
Pr 
r
 r,  
 r

 3.17 
It should be noticed that the right-hand side of equation (3.15) consists of the
contribution of the Reynolds-stress and buoyancy due to thermal term.
This
situation is in contrast to the classical one, in which a steady streaming is induced by
vibrations of a solid body in a viscous fluid at rest or with that of free convection
from a circular cylinder whose surface temperature oscillates about a mean ambient
temperature in a constant gravitational field.
In these situations, the dominant
fluctuating flow is non-rotational while in the present problem this fluctuating flow
is rotational.
We consider now the limiting case when Re → ∞, or boundary layer
approximation. This boundary layer has the thickness O  Re 1 3  that encompasses
the much thinner Stokes layer for Re  1 . It is also worth mentioning that for the
corresponding isothermal sphere problem, the thickness of the boundary layer is
18
O  Re 1 2  and it shows clearly the difference between the two problems. The
variables appropriate to this boundary layer region are
  Re 2 3  t ,  ,  , T  Re1 3  t ,  ,  , r  1  Re1 3
 3.18
Substituting (3.18) into equations (3.14) to (3.17) and letting Re   , we obtain the
following boundary layer equations for the corresponding functions  0 s  ,  0u  , 2u 
and 0 s  ,
 
 2 0  
2 0

1


sin t


 2

u
 4 0 
 4
s
s
 3.19 
 s  2 0  s  
  0   ,

s
s
 2 
2   2 0    0 



  ,  
1   2  2 
 u   2 0 u  
  0 ,

 2 


  ,  
s
2  2 0    0 

1   2  2

u




u
  0  u  , 0  s 
2  
  Re
t
  ,  
s
 s
s
1  2 0     0 , 0

Pr  2
  ,  
u
s
1    cos t   

2
u
2
Re

 s
 3.20 
 3.21
 3.22
To obtain an equation for the steady stream function 0 s  , we eliminate  0u  and
2u  from equations (3.19) to (3.21) as follows. Equation (3.19) is integrated twice
with respect to  to give

 0u   1   2  sin t  0 s   x,   dx
0
 3.23
19
Substituting this relation into equation (3.21), followed by an integration with
respect to t, we obtain
 cos t 
Re
23
1
0 
 
 2 0


s
2   1   2  0 
u
s
 1   2 
0s 



0
s


0
0   x,   dx
s
0 s 
 x,   dx

 3.24 
We now integrate (3.20) once with respect to  and use (3.19), (3.24) and (3.23) to
obtain the following boundary layer equation for  0 s 
 3 0 s   2 0 s   0 s   0 s   2 0 s 
   0 s  





 3
 
  2 1   2   
2

  1   2  0s 
2
2
 
 3.25
Equation (3.25) is to be solved together with (3.22) for the steady
temperature 0s  subject to the following boundary conditions
 0 
0 
 0,
 1


s
 0 s 
 0 s 
 0, 0s   0

s
as
on
 0
 
 3.26 
 3.27 
We notice again the embodiment of Reynolds stresses and buoyancy in the
effective body-force term in equation (3.25). These terms make equations (3.22)
and (3.25) completely different from the equations which describe the classical
problem of steady free convection from a sphere immersed in a viscous fluid, when
the buoyancy due to the g-jitter effects are absent (Nazar et. al [32]).
20
To start the numerical solution, we need to determine the initial conditions of
equations (3.22) and (3.25). To do this, we notice that the solution develops a
singularity in the vicinity of   1a  0  , i.e. at the pole of the sphere. Thus, we
start the numerical solution near  a  90 , that is, at small values of  and expand
the functions  0 s  and 0s  in the series of small  in the form
s
 0   f0    O   3  , 0 s   h0    O   2 
 3.28
Substituting (3.28) into equations (3.22) and (3.25), we get the following ordinary
differential equations for f 0 and h0
1
f 0  f0 f 0  f 02   h0 2
2
h0  Pr f 0 h0  0
3.29 
 3.30 
subject to the boundary conditions (3.26) and (3.27), which become
f 0  0   f 0  0   0, h0  0   1
f 0     0, h0     0
where prime denotes differentiation with respect to  .
 3.31
 3.32 
CHAPTER 4
METHOD OF SOLUTION IN FINDING THE NUMERICAL SOLUTION
FOR G-JITTER INDUCED FREE CONVECTION WITH CONSTANT
HEAT FLUX
4.1
Governing Equations in a First-Order System
Equations (3.22) and (3.25) subject to boundary conditions (3.26) and (3.27)
were solved numerically using a finite difference scheme.
Previously, a very
efficient finite difference scheme, the Keller-Box method was employed (Shafie
[6]). In this particular case, another scheme, a Crank-Nicolson scheme was used
(Mitchell [33] and Smith [34]). This method is found to be unconditionally stable
and suitable in dealing with nonlinear parabolic problems.
However, before proceeding with the scheme, the governing system of
equations was written in the first-order system. For this purpose, we introduce new
dependent variables y1   ,  , y2   ,  , y3   ,  , y4   ,  and y5   ,  , where
 0 s 
 2 0 s 
0s 
s
y1   0 , y2 
, y3 
, y4  0 , y5 

 2

s
 4.1
22
This means that
y1  y2  0
 4.2 
y2  y3  0
 4.3
y4  y5  0
 4.4 
Using (4.1), equations (3.22) and (3.25) then take the form
y3 

1
y 2   1   2  y42  y2 y2   y3 y1  0
2 2
1 
2
y5  Pr  y2 y4   y5 y1   0
 4.5
 4.6 
Boundary conditions (3.26) and (3.27) becomes
y1   , 0   0, y2   , 0   0, y5   , 0   1
 4.7 
y2   ,    0, y4   ,    0
 4.8
23
4.2
Crank-Nicolson Scheme
With the resulting first-order equations, the “centered-difference” derivatives
and averages at the midpoints of net rectangles are used, as they are required to get
accurate finite difference equations (see Figure 4.1). We consider the net rectangle
on the plane and denote the net points by
1  0,  p 1   p   p ,
p  1, 2,..., P  1
 4.9 
 1  0,  q 1   q   q ,
q  1, 2,..., Q  1
 4.10 
known
unknown
centering

 q 1

q 1
2
 q
q
2 p
 p 1
p
 p 1
Figure 4.1: Net rectangle for difference approximations

24
Here p and q index points on the   ,  plane, and  q and  p define the
steplengths corresponding to the qth and pth intervals, respectively. Given a typical
variable y1 , say, the various quantities in (4.2) to (4.6) are approximated as follows,
q 1
y1 | p
2
q 1
y1 | p
q 1
y1 | p

1 q
y1 p  y1qp1 

2
 4.11
2

1
 y1qp 1  y1qp 1  y1qp11  y1qp11 
4  p
 4.12
2

1
y q 1  y1qp 
q  1p

 4.13
If we substitute (4.11) to (4.13) into (4.2) to (4.6), the resulting finite
difference equations are implicit and nonlinear. Newton’s method is first introduced
to linearize the nonlinear system of equations before constructing a coefficient
matrix of the finite difference equations for all  at given  .
Several methods may be considered to solve the linearized difference
equations.
For example, Jacobi and Gauss-Seidel iterations, block elimination
method and Gaussian elimination method. However, if Jacobi and Gauss-Seidel
method were to be used, convergence will only be guaranteed if the matrix is
diagonally dominant. On the other hand, the block elimination method is also not
suitable for this particular problem because the equations will result in a matrix that
contains blocks of singular matrices. Hence, the naïve Gaussian elimination method
is used to solve the equations.
All the results quoted here were obtained using uniform grids in both
the  and  directions. The convergence was deemed to have taken place when the
maximum absolute pointwise change between successive iterates was 10 -10.
25
Alternatively, we may also another substitution which has the same essence
as Crank-Nicolson,
q 1
y1 | p
2
q 1
y1 | p
q 1
y1 | p

1 q
 y1 p 1  y1qp11  y1qp 1  y1qp11 
4
 4.14 
2

1
y1qp 1  y1qp 1  y1qp11  y1qp11 

4  p
 4.15
2

1
 y1qp11  y1qp1  y1qp11  y1qp 1 
2  q
 4.16 
If we simplify (4.14) and (4.16), equations (4.11) and (4.13) will be obtained
respectively.
The solution procedure to solve numerically equations (3.29) and (3.30)
subject to boundary conditions (3.31) and (3.32) are the same as solving equations
(3.22) and (3.25).
26
4.3
MATLAB Programming in Processing Elimination Method
The following code is the MATLAB implementation of naïve Gaussian
elimination where there will be no zeros in the diagonal:
% A – (n x n) matrix
% rr – column vector of length n
m = size(A,1); % get numbers of rows in matrix a
n = length(rr); % get length of b
if(m~=n)
error('A and rr do not have the same number of rows')
end
% Step 1: Form (n,n+1) augmented matrix
A(:,n+1) = rr;
for i = 1:n
% Step 2: Make diagonal elements into 1.0
A(i,i+1:n+1) = A(i,i+1:n+1)/A(i,i);
% Step 3: Make all elements below diagonal into 0
for j = i+1:n
A(j,i+1:n+1)=A(j,i+1:n+1)-A(j,i)*A(i,i+1:n+1);
end
end
% Step 4: Begin back substitution
for j = n-1:-1:1
A(j,n+1) = A(j,n+1) - A(j,j+1:n)*A(j+1:n,n+1);
end
% Return solution
x = A(:,n+1);
Figure 4.2: MATLAB implementation of naïve Gaussian elimination
CHAPTER 5
RESULTS AND DISCUSSIONS
5.1
Numerical Solution for G-Jitter Induced Free Convection with Constant Heat
Flux
We have solved numerically the two systems of the steady-state boundary
layer equations (3.22), (3.25), (3.29) and (3.30) due to g-jitter flow for some values
of the Prandtl number, Pr and at some positions  a around the sphere between
a  0 and a  90 .
Figures (5.1) and (5.2) represent the results of the reduced skin friction,
 2 0 s 
a , 0  and the wall temperature, 0s   a , 0  respectively for the steady part
 2
of the solution induced by g-jitter for Pr = 0.7 (air), 1 and 7(water at 21°). They
show that the surface temperature decreases almost continuously from the value at
the upper pole  a  0  to a finite value at a  90 . However, the reduced skin
friction first increases from the upper pole a  0 to a maximum value, then it
decreases indefinitely after a  45 .
The peak of these profiles decreases as the values of Pr decreases which may
be attributed due to the g-jitter effects. Table 5.1 shows some values of the reduced
skin friction for the three value of Pr and several values of the angle  a . As shown
by Shafie[6], it is seen that the reduced skin friction along the sphere increases as a
increases. On the other hand, the surface temperature decreases with the increment
28
of  a .
Both the reduced skin friction and the wall temperature decrease as the
Prandtl number, Pr increases.
 2 0 s 
 a , 0  and wall temperature 0s   a , 0  at
 2
different position of  for Pr = 0.7, 1 and 7
Table 5.1: Value of skin friction
Pr
0.7
a
 2 0 s 
a , 0 
 2
90°
88°
86°
84°
82°
80°
78°
76°
74°
72°
70°
65°
60°
45°
30°
20°
10°
1°
0.175850
0.270748
0.362206
0.186446
0.158858
0.351699
0.340919
0.541015
0.622781
0.426139
0.499184
0.734152
0.748340
0.928840
0.814968
0.553013
0.250557
0.011053
1
0   a , 0 
 2 0 s 
 a , 0 
 2
1.567517
1.720088
1.769240
1.815499
1.938704
1.976573
2.002909
2.022443
2.036500
2.047051
2.055781
2.067097
2.080737
2.130953
2.241005
2.374511
2.600668
3.121831
0.115820
0.149095
0.265887
0.124494
0.072313
0.231639
0.224658
0.401929
0.472928
0.301018
0.365025
0.563293
0.577728
0.730255
0.636973
0.418746
0.165380
0.001846
s
7
0   a , 0 
 2 0 s 
 a , 0 
 2
0   a , 0 
1.524204
1.688328
1.739379
1.787112
1.905031
1.937479
1.959070
1.973429
1.983360
1.990425
1.994797
2.001141
2.006866
2.044312
2.140936
2.261278
2.466925
2.944446
0.047961
0.114475
0.213849
0.091408
0.056449
0.095921
0.052454
0.143772
0.186582
0.102871
0.142060
0.204713
0.223693
0.281268
0.252034
0.176517
0.091700
0.005245
1.348028
1.587294
1.633221
1.677724
1.728277
1.728713
1.685515
1.691792
1.695411
1.701707
1.707348
1.679052
1.686740
1.688661
1.736194
1.803043
1.921184
2.206789
s
s
29
s 
2
 0
 a , 0 
2

1
Pr = 0.7
Pr = 1
0.8
0.6
0.4
Pr = 7
0.2
0
10
20
30
40
50
60
70
80
position ,  a
Figure 5.1: Variations of the skin friction with  for different values of Prandtl
numbers, Pr
90
30
0 s   a , 0
3
2.4
Pr = 0.7
Pr = 1
1.8
Pr = 7
1.2
0.6
0
10
20
30
40
50
60
70
80
position ,  a
Figure 5.2: Variations of the wall temperature with  for different values of Prandtl
number, Pr
90
31
5.2
Velocity and Temperature Profiles
Figures (5.3), (5.4), (5.5) and (5.6) show the velocity and temperature
profiles.
It is shown that near the pole, the thickness of both boundary layer
increases considerably due to the singularity in equation (3.25) at a  0 . But, these
velocity and temperature profiles do not increase monotonically with the increase of
the position along the sphere of the wall,  a , with the notable exception of the
temperature profiles for Pr = 0.7 (Figure 5.4). This situation concurs with the
behavior of the wall temperature for Pr = 7 (Figure 5.2) and the reduced skin friction
for Pr = 0.7 and Pr = 7 (Figure 5.1). Such a singularity at the pole has also been
observed by Potter and Riley [30] for the same boundary-layer problem without the
g-jitter effect.
32
  0 s 

0.02
a  90,80, 70, 60,30,10,5
0.015
0.01
0.005
0
0.19
0.38
0.57
0.76
Figure 5.3: Profiles of the non-dimensional velocity for different values of a
when Pr = 0.7
0.95

33
0 s 
2
a  90,80, 70, 60,30,10,5
1.5
1
0.5
0
0.19
0.38
0.57
0.76
Figure 5.4: Profiles of the non-dimensional temperature for different values of  a
when Pr = 0.7
0.95

34
  0 s 

0.01
a  90,80, 70, 60,30,10,5
0.008
0.006
0.004
0.002
0
0.19
0.38
0.57
0.76
Figure 5.5: Profiles of the non-dimensional velocity for different values of a when
Pr = 7
0.95

35
0 s 
2
1.5
a  90,80, 70, 60,30,10,5
1
0.5
0
0.19
0.38
0.57
0.76
Figure 5.6: Profiles of the non-dimensional temperature for different values of  a
when Pr = 7
0.95

CHAPTER 6
CONCLUSION
6.1
Summary of research
The problem discussed is focused on the generation of steady streaming. We
solved numerically the equations involved.
The numerical results of the two
 2 0 s 
physical quantities, namely the reduced skin friction,
 a , 0  and the wall
 2
temperature, 0   a , 0  were discussed.
s
The reduced skin friction is significant because it controls the heating of the
body due to the shear stress on the body. For example, when the skin friction equals
zero, the flow will be separated from the body surfaces and the boundary layer
equations are available only up to this separation point and after this point, the full
Navier-Stokes equations have to be solved.
In our results, we observed that the reduced skin friction and wall
temperature oscillate. This is in contrast to the case without g-jitter effects (Shafie
[6]). The reduced skin friction and the wall temperature is higher than that of the
case without g-jitter effects (Shafie [6]). In addition, the reduced skin friction and
wall temperature decrease as Pr increases.
37
5.2
Suggestions for Future Research
This research pertaining g-jitter induced free convection shows that all the
physical quantities, namely the reduced skin friction, heat and mass transfer as well
as the wall temperature exhibit oscillatory behavior due to the g-jitter effects. Due
to the importance of the frequency and the amplitude of the g-jitter in determining
the convective flow behaviour of the system, the study of these flow field and heat
transfer is crucial in a manufacturing process because the quality of the final product
depends heavily on the skin friction and the surface heat transfer rate.
Various
studies
have
been
made
regarding
the
g-jitter
effects.
Ramachandran and Baugher [5] conducted the research on the protein crystal
growth. Merkin [18], Davidson [19], Haddon and Riley [20] were among those that
pioneered the study of heat transfer in fluctuating flows.
Other studies include those of Shafie [6], Amin [7], Nazar et. al. [32],
Langbein [21], Potter and Riley [30], Farooq and Homsy [25], Okano et. al. [24]
and Doi et. al. [23] which covers various scopes and aspects.
This research only pertains to the case of g-jitter induced free convection
with constant heat flux. It will be beneficial to further the study on the effect of gjitter on double diffusion with constant heat flux.
38
REFERENCES
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2.
Schneider, S. and Straub, J. Influence of the Prandtl number on laminar natural
convection in a cylinder caused by g-jitter. Journal of Crystal Growth 1989. 97. n.1:
235-242.
3.
Alexander J. I. D., Amirondine, S., Ouzzani, J. and Rosenberger, F., Analysis of the
low gravity tolerance of Bridgman-Stockbarger crystal growth II; Transient and
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Nelson, E. S. An examination of anticipated g-jitter in Space Station and its effects
on materials processes. NASA TM 103775, 1991.
5.
Ramachandran, N. and Baughar, C. D. G-Jitter effects in Protein Crystal Growth – A
Numerical Study. Universities Space Research Institution
6.
Shafie, S. The effects of G-Jitter on Induced Free Convection Models. Universiti
Teknologi Malaysia, 2005.
7.
Amin, N. The effect g-jitter on heat transfer. Proc. R. Soc. Lond. A 419, 151-172
(1988).
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8.
Pan, Bo, Shang, D-Y., Li, B. Q. and de Groh, H. C. Magnetic field effects on g-jitter
induced flow and solute transport. Int. J. Heat Mass Transfer, 2002. 45: 125-144.
9.
Alexander, J. I. D., Garandet J.P., Favier, J.J. and Lizee, A. G-Jitter effects on
segregation during directional solidification of tin-bismuth in the MEPHISTO
furnace facility. J. Crystal Growth, 1997. 178: 657-661.
10.
Chao, L. Lateral g-jitter effects on liquid motion and thermocapillary convection in
an open square container under weightless condition. Case Western Reserve
University, 1991.
11.
Mell, W. E., Mc Grattan, K.B., Nakamura, Y., and Baum, H.R. Simulation of
combustion systems with realistic g-jitter. Sixth International Microgravity
Combustion Workshop. May 22-24, 2001. NASA Glenn Research Center,
Cleveland, OH, CP-2001-210826, 2001. 333-336.
12.
Yoshiaki, H., Keisuke, I., Toru., M., Satoshi, M., Shinichi, Y. and Kyoichi, K.
Numerical analysis of crystal growth of an InAs-GaAs binary semiconductor under
microgravity conditions. J. Phys. D: Appl. Phys., 2000. 33: 2508-2518.
13.
Benjapiyorn, C., Timchenko, V., Leonardi, E. and davis, G.D.V. Effects of space
environment on flow and concentration during directional solidification. Int. J. Fluid
Dynamics, 2000. 4(3).
14.
Shu, Y., Li, B.Q., de Groh, H.C. Numerical study of g-jitter induced double
diffusive convection in microgravity. Numerical Heat Transfer B: Application,
2001. 39: 245-265.
40
15.
Pusey, M., Witherow, W. and Naumann, R. Preliminary investigations into solutal
flow about growing tetragonal lysozyme crystals. Journal of Crystal Growth, 1988.
90: 105-111.
16.
Pusey, M. Fourth International Conference on Crystallization of Biological
Macromolecules, 1991.
17.
Wilcox, W.R. and Regel, L.L. Microgravity effects on material processing: A
Review. Conference Proceedings of EOROMAT 2001, Rimini, Italy. July 10-14,
2001. Associazione Italiana di Metallurgia, 1-200121 Milano, 2001. 1-9.
18.
Merkin, J.H. Oscillatory free convection from an infinite horizontal cylinder. J.
Fluid. Mech, 1967. 30: 561-575.
19.
Davidson, B.J. Heat Transfer from a vibrating circular cylinder. Int. J. Heat Mass
Transfer, 1973. 16: 1703-1727.
20.
Haddon, E.W. and Riley, N. The heat transfer between concentric vibrating circular
cylinders. Q. Jl Mech. appl. Math., 1981. 34: 345-359.
21.
Langbein, D. Oscillatory Convection in a Spherical Cavity Due to G-jitter. ESA
Mater. Sci. under Microgravity. International Organization, 1983: 359-363.
22.
Heiss, T., Schneider,S. and Straub, J. G-jitter effects on natural convection in a
cylinder: A three-dimensional numerical calculation. ESA, Proceedings of the South
European Symposium on Material Sciences under Microgravity Conditions. 1987.
International Organization. 1987. 517-523.
41
23.
Doi, T., Prakash, A., Azuma, H., Yoshihara, S. and Kawahara, H. Oscillatory
convection induced by g-jitter in a horizontal liquid layer. AIAA, Aerospace Sciences
Meeting and Exhibit, 33rd. Jan 9-12, 1995. Reno, NV, United States, 1995.
24.
Okano, Y., Umemura, S. and Dost, S. G-jitter effect on the flow in a threedimensional rectangular cavity. Journal of Materials Processing and manufacturing
Science. 2001. 10(1): 3-6.
25.
Farooq, A. and Homsy, G.M. Streaming flows due to g-jitter induced natural
convection. J. Fluid Mech., 1994. 271: 351-378.
26.
Farooq, A. and Homsy, G.M. Linear and nonlinear dynamics of a differentially
heated slot under gravity modulation. J. Fluid Mech., 1996. 313: 1-38.
27.
Biringen, S. and Danabasoglu, G. Computation of convective flow with gravity
modulation in rectangular cavities. AIAA J. Thermophys heat transfer, 1990. 4: 357365.
28.
Gresho, P.M. and Sani, R.L. The effects of gravity modulation on the stability of a
heated fluid layer. J. Fluid Mech., 1970. 40: 783-806.
29.
Biringen, S. and Peltier, L.J. Computational study of 3-D Benard convection with
gravitational modulation. Phys. Fluids, 1990. A 2: 279-283.
30.
Potter, J. M. and Riley, N. Free Convection From a Heated Sphere at Large Grashof
Number. J. Fluid Mech., 1980. 100: 769-783.
42
31.
Shafie, S. and Amin, N. Numerical solution for g-jitter induced free convection
boundary layer from a sphere using Keller-box method. Universiti Teknologi
Malaysia, 2005.
32.
Nazar, R., Amin, N. and Pop, I. free convection boundary layer on an isothermal
sphere in a micropolar fluid. International Communications in Heat and Mass
Transfer, 2002. 29(3): 377-386.
33.
Mitchell, A. R. Computational Methods in Partial Differential Equations. John
Wiley & Sons Ltd., 1969.
34.
Smith, G. D. Numerical Solution of Partial Differential Equations. Oxford
University Press, 1985
43
APPENDIX A
MATLAB FOR NUMERICAL SOLUTION FOR G-JITTER INDUCED FREE
CONVECTION WITH CONSTANT HEAT FLUX
% Data Input
bl_thick = input ('Enter the boundary layer thickness: ');
deleta = input ('Enter the step size of boundary layer
thickness: ');
angle = input ('Enter the position: ');
delmju = input ('Enter the size interval of position: ');
meta = (bl_thick/deleta) + 1; nmju = (angle/delmju) + 1;
Pr = input ('Enter the Prandtl number: ');
stop = 1.0 ; i = 1; epselon = 0.00001;
while stop > epselon
eta(1,1) = 0.0;
for m = 2:meta
eta(m,1) = eta(m-1,1) + deleta;
end
etametaq = eta(meta,1)/12;
etametar = eta(meta,1)/2;
etametas = 1/eta(meta,1);
for m = 1:meta
etab = eta(m,1)/eta(meta,1);
etab1 = etab^2;
etab2 = (1-etab)^2;
y1(m,1) = etametaq*etab1*(6 - 8*etab + 3*etab1);
y2(m,1) = etab*etab2;
y3(m,1) = etametas*(1 - 4*etab + 3*etab1);
y4(m,1) = 1 - etab;
y5(m,1) = -1/eta(meta,1);
end
for m = 2:meta
44
cntr_y1(m,i) = 0.5*(y1(m,i) + y1(m-1,i));
cntr_y2(m,i) = 0.5*(y2(m,i) + y2(m-1,i));
cntr_y3(m,i) = 0.5*(y3(m,i) + y3(m-1,i));
cntr_y4(m,i) = 0.5*(y4(m,i) + y4(m-1,i));
cntr_y5(m,i) = 0.5*(y5(m,i) + y5(m-1,i));
sqr_cntr_y2(m,i) = (cntr_y2(m,i))^2;
sqr_cntr_y4(m,i) = (cntr_y4(m,i))^2;
cntr_y1y3(m,i) = cntr_y1(m,i)*cntr_y3(m,i);
cntr_y1y5(m,i) = cntr_y1(m,i)*cntr_y5(m,i);
derv_y1(m,i) = (1/deleta)*(y1(m,i) - y1(m-1,i));
derv_y2(m,i) = (1/deleta)*(y2(m,i) - y2(m-1,i));
derv_y3(m,i) = (1/deleta)*(y3(m,i) - y3(m-1,i));
derv_y4(m,i) = (1/deleta)*(y4(m,i) - y4(m-1,i));
derv_y5(m,i) = (1/deleta)*(y5(m,i) - y5(m-1,i));
% Coefficients
a1(m,i)
a2(m,i)
a3(m,i)
a4(m,i)
a5(m,i)
a6(m,i)
a7(m,i)
a8(m,i)
=
=
=
=
=
=
=
=
0.5*cntr_y3(m,i);
-cntr_y2(m,i);
-(1/deleta) + 0.5*cntr_y1(m,i);
0.5*cntr_y4(m,i);
a1(m,i);
a2(m,i);
a3(m,i) + (2/deleta);
a4(m,i);
% Coefficients
b1(m,i)
b2(m,i)
b3(m,i)
b4(m,i)
=
=
=
=
0.5*Pr*cntr_y5(m,i);
-(1/deleta) + 0.5*Pr*cntr_y1(m,i);
b1(m,i);
b2(m,i) + (2/deleta);
% Values of r
r1(m,i)
r2(m,i)
r3(m,i)
r4(m,i)
=
=
=
=
-derv_y1(m,i) + cntr_y2(m,i);
-derv_y2(m,i) + cntr_y3(m,i);
-derv_y4(m,i) + cntr_y5(m,i);
-derv_y3(m,i) - cntr_y1y3(m,i) +
sqr_cntr_y2(m,i) - 0.5*sqr_cntr_y4(m,i);
r5(m,i) = -derv_y5(m,i) - Pr*cntr_y1y5(m,i);
end
% Constructing block-tridiagonal matrix
a{2,i} = [0 0 (1/deleta) 0 0; -0.5 0 0 -0.5 0; 0 -(1/deleta)
0 0 -0.5; a3(2,i) a4(2,i) a5(2,i) a7(2,i) 0; 0 0
b3(2,i) 0 b4(2,i)];
for m = 3:meta
45
a{m,i} = [-0.5 0 (1/deleta) 0 0; -(1/deleta) 0 0 -0.5 0;
0 -(1/deleta) 0 0 -0.5;
a2(m,i) a4(m,i)
a5(m,i) a7(m,i) 0; 0 0 b3(m,i) 0 b4(m,i)];
b{m,i} = [0 0 -(1/deleta) 0 0; 0 0 0 -0.5 0; 0 0 0 0 0.5; 0 0 a1(m,i) a3(m,i) 0; 0 0 b1(m,i) 0
b2(m,i)];
end
for m = 2:meta
c{m,i} = [-0.5 0 0 0 0; (1/deleta) 0 0 0 0; 0 (1/deleta)
0 0 0; a6(m,i) a8(m,i) 0 0 0; 0 0 0 0 0];
end
% Block-Tridiagonal Algorithm
% Factorisation
alpha{2,i} = a{2,i}; gamma{2,i} = inv(alpha{2,i})*c{2,i};
for m = 3:meta
alpha{m,i} = a{m,i} - (b{m,i}*gamma{m-1,i});
gamma{m,i} = inv(alpha{m,i})*c{m,i};
end
% Forward substitution
for m = 2:meta
rr{m,i} = [r1(m,i); r2(m,i); r3(m,i); r4(m,i); r5(m,i)];
end
ww{2,i} = inv(alpha{2,i})*rr{2,i};
for m = 3:meta
ww{m,i} = inv(alpha{m,i})*(rr{m,i} - (b{m,i}*ww{m1,i}));
end
% Backward substitution
dely1(1,i) = 0.0; dely2(1,i) = 0.0; dely5(1,i) = 0.0;
dely2(meta,i) = 0.0; dely4(meta,i) = 0.0;
dell{meta,i} = ww{meta,i};
for m = meta-1:-1:2
46
dell{m,i} = ww{m,i} - (gamma{m,i}*dell{m+1,i});
end
dely3(1,i) = dell{2,i}(1,1); dely4(1,i) = dell{2,i}(2,1);
dely1(2,i) = dell{2,i}(3,1);
dely3(2,i) = dell{2,i}(4,1); dely5(2,i) = dell{2,i}(5,1);
for m = meta:-1:3
dely2(m-1,i)
dely4(m-1,i)
dely1(m,i)
dely3(m,i)
dely5(m,i)
=
=
=
=
=
dell{m,i}(1,1);
dell{m,i}(2,1);
dell{m,i}(3,1);
dell{m,i}(4,1);
dell{m,i}(5,1);
end
% Newton's Iteration
for m = 1:meta
y1(m,i+1)
y2(m,i+1)
y3(m,i+1)
y4(m,i+1)
y5(m,i+1)
=
=
=
=
=
y1(m,i)
y2(m,i)
y3(m,i)
y4(m,i)
y5(m,i)
+
+
+
+
+
dely1(m,i);
dely2(m,i);
dely3(m,i);
dely4(m,i);
dely5(m,i);
end
% Checking for convergence
stop = abs(dely3(1,i)); imax = i; fprintf('i = %f\n',i);
i = i + 1;
end
% Shift Profile
for m = 1:meta
yyy1(m) = y1(m,imax); yyy2(m) = y2(m,imax);
yyy3(m) = y3(m,imax); yyy4(m) = y4(m,imax);
yyy5(m) = y5(m,imax);
end
mju(1) = 0.0; AA(1) = 0.0; BB(1) = 0.0;
for n = 2:nmju
47
mju(n) = mju(n-1) + delmju;
delD(n) = delmju;
end
for n = 2:nmju
if n==nmju
AA(n) = 0.0; BB(n) = 0.0;
else
AA(n) = mju(n)/(1-(mju(n))^2);
BB(n) = 0.5*mju(n)*(1-(mju(n))^2);
end
end
for n = 2:nmju
stop = 1.0; i = 1; epselon = 0.00001;
while stop > epselon
eta(1,1) = 0.0;
for m = 2:meta
eta(m,1) = eta(m-1,1) + deleta;
end
% To generate initial value of velocity and temperature
profile
etametaq = eta(meta,1)/12;
etametar = eta(meta,1)/2;
etametas = 1/eta(meta,1);
for m = 1:meta
deta(m,i) = deleta;
etab = eta(m,1)/eta(meta,1);
etab1 = etab^2;
etab2 = (1-etab)^2;
y1(m,1,n) = etametaq*etab1*(6 - 8*etab + 3*etab1);
y2(m,1,n) = etab*etab2;
y3(m,1,n) = etametas*(1 - 4*etab + 3*etab1);
y4(m,1,n) = 1 - etab;
y5(m,1,n) = -1/eta(meta,1);
end
for m = 2:meta
% Previous station
48
cy1(m,n) = yyy1(m); cy2(m,n) = yyy2(m);
cy3(m,n) = yyy3(m); cy4(m,n) = yyy4(m);
cy5(m,n) = yyy5(m);
cyy2(m,n) = cy1(m,n)^2; cyy4(m,n) = cy4(m,n)^2;
cy1y3(m,n) = cy1(m,n)*cy3(m,n);
cy2y4(m,n) = cy2(m,n)*cy4(m,n);
cy1y5(m,n) = cy1(m,n)*cy5(m,n);
% Present station
y2y2(m,i,n)
y1y3(m,i,n)
y2y4(m,i,n)
y1y5(m,i,n)
=
=
=
=
y2(m,i,n)^2; y4y4(m,i,n) = y4(m,i,n)^2;
y1(m,i,n)*y3(m,i,n);
y2(m,i,n)*y4(m,i,n);
y1(m,i,n)*y5(m,i,n);
% Coefficient of the differenced momentum equation
a1(m,i)
a2(m,i)
a3(m,i)
a4(m,i)
a5(m,i)
=
=
=
=
=
-delD(n);
deta(m,i)*(y3(m,i,n) + cy3(m,n));
-2*deta(m,i)*y2(m,i,n)*(1 + delD(n)*AA(n));
delD(n) + deta(m,i)*(y1(m,i,n) - cy1(m,n));
2*deta(m,i)*delD(n)*BB(n)*y4(m,i,n);
% Coefficients of the differenced energy equation
b1(m,i)
b2(m,i)
b3(m,i)
b4(m,i)
b5(m,i)
=
=
=
=
=
-delD(n);
deta(m,i)*Pr*(y5(m,i,n) + cy5(m,n));
-deta(m,i)*Pr*(y4(m,i,n) - cy4(m,n));
-deta(m,i)*Pr*(y2(m,i,n) + cy2(m,n));
delD(n) + deta(m,i)*Pr*(y1(m,i,n) - cy1(m,n));
% Definition of Rm
clb = -delD(n)*(cy3(m,n) - cy3(m-1,n)) +
deta(m,i)*delD(n)*(AA(n-1)*cyy2(m,n) - BB(n1)*cyy4(m,n)) - deta(m,i)*cyy2(m,n) +
deta(m,i)*cy1y3(m,n);
cmb = -delD(n)*(cy5(m,n) - cy5(m-1,n)) deta(m,i)*Pr*cy2y4(m,n) +
deta(m,i)*Pr*cy1y5(m,n);
% Definitions of rm
r1(m,i) = -y1(m,i,n) + y1(m-1,i,n) - cy1(m,n) + cy1(m1,n) + deta(m,i)*(y2(m,i,n) + cy2(m,n));
r2(m,i) = -y2(m,i,n) + y2(m-1,i,n) - cy2(m,n) + cy2(m1,n) + deta(m,i)*(y3(m,i,n) + cy3(m,n));
r3(m,i) = -y4(m,i,n) + y4(m-1,i,n) - cy4(m,n) + cy4(m1,n) + deta(m,i)*(y5(m,i,n) + cy5(m,n));
r4(m,i) = -delD(n)*(y3(m,i,n) - y3(m-1,i,n)) +
deta(m,i)*delD(n)*(AA(n)*y2y2(m,i,n) BB(n)*y4y4(m,i,n)) + deta(m,i)*y2y2(m,i,n)
- deta(m,i)*y1y3(m,n) +
deta(m,i)*cy1(m,n)*y3(m,i,n) deta(m,i)*cy3(m,n)*y1(m,i,n) + clb;
49
r5(m,i) = -delD(n)*(y5(m,i,n) - y5(m-1,i,n)) +
deta(m,i)*Pr*(y2y4(m,i,n) - cy4(m,n)*y2(m,i,n) +
cy2(m,n)*y4(m,i,n) - y1y5(m,i,n) +
cy1(m,n)*y5(m,i,n) - cy5(m,n)*y1(m,i,n)) + cmb;
end
% Obtain the elements of the matrix
% Obtaining the elements of the coefficient matrix A
for m = 2:meta
d{m,i} = [0 0 0 0 0; 0 0 0 0 0; 0 0 0 0 0; 0 0 0 0 0; 0
0 0 0 0];
end
a{2,i} = [a1(2,i) 0 a2(2,i) a4(2,i) 0; 0 -1 0 0 -deta(2,i); 0
0 1 0 0; 0 0 0 -deta(2,i) 0; 0 0 b2(2,i) 0 b5(2,i)];
c{2,i} = [a3(2,i) a5(2,i) 0 0 0; 0 1 0 0 0; -deta(2,i) 0 0 0
0; 1 0 0 0 0; b3(2,i) b4(2,i) 0 0 0];
for m = 3:meta
a{m,i} = [-1 0 0 -deta(m,i) 0; 0 -1 0 0 -deta(m,i); 0 0
1 0 0; 0 0 a2(m,i) a4(m,i) 0; 0 0 b2(m,i) 0
b5(m,i)];
b{m,i} = [0 0 0 0 0; 0 0 0 0 0; 0 0 -1 0 0; 0 0 0
a1(m,i) 0; 0 0 0 0 b1(m,i)];
end
for m = 3:meta-1
c{m,i} = [1 0 0 0 0; 0 1 0 0 0; -deta(m,i) 0 0 0 0;
a3(m,i) a5(m,i) 0 0 0; b3(m,i) b4(m,i) 0 0 0];
end
A1 = [a{2,i} c{2,i} d{2,i} d{2,i} d{2,i} d{2,i} d{2,i} d{2,i}
d{2,i} d{2,i} d{2,i} d{2,i} d{2,i} d{2,i} d{2,i} d{2,i}
d{2,i} d{2,i} d{2,i} d{2,i}];
A2 = [b{3,i} a{3,i} c{3,i} d{3,i} d{3,i} d{3,i} d{3,i} d{3,i}
d{3,i} d{3,i} d{3,i} d{3,i} d{3,i} d{3,i} d{3,i} d{3,i}
d{3,i} d{3,i} d{3,i} d{3,i}];
50
A3 = [d{4,i} b{4,i} a{4,i} c{4,i} d{4,i} d{4,i} d{4,i} d{4,i}
d{4,i} d{4,i} d{4,i} d{4,i} d{4,i} d{4,i} d{4,i} d{4,i}
d{4,i} d{4,i} d{4,i} d{4,i}];
A4 = [d{5,i} d{5,i} b{5,i} a{5,i} c{5,i} d{5,i} d{5,i} d{5,i}
d{5,i} d{5,i} d{5,i} d{5,i} d{5,i} d{5,i} d{5,i} d{5,i}
d{5,i} d{5,i} d{5,i} d{5,i}];
A5 = [d{6,i} d{6,i} d{6,i} b{6,i} a{6,i} c{6,i} d{6,i} d{6,i}
d{6,i} d{6,i} d{6,i} d{6,i} d{6,i} d{6,i} d{6,i} d{6,i}
d{6,i} d{6,i} d{6,i} d{6,i}];
A6 = [d{7,i} d{7,i} d{7,i} d{7,i} b{7,i} a{7,i} c{7,i} d{7,i}
d{7,i} d{7,i} d{7,i} d{7,i} d{7,i} d{7,i} d{7,i} d{7,i}
d{7,i} d{7,i} d{7,i} d{7,i}];
A7 = [d{8,i} d{8,i} d{8,i} d{8,i} d{8,i} b{8,i} a{8,i} c{8,i}
d{8,i} d{8,i} d{8,i} d{8,i} d{8,i} d{8,i} d{8,i} d{8,i}
d{8,i} d{8,i} d{8,i} d{8,i}];
A8 = [d{9,i} d{9,i} d{9,i} d{9,i} d{9,i} d{9,i} b{9,i} a{9,i}
c{9,i} d{9,i} d{9,i} d{9,i} d{9,i} d{9,i} d{9,i} d{9,i}
d{9,i} d{9,i} d{9,i} d{9,i}];
A9 = [d{10,i} d{10,i} d{10,i} d{10,i} d{10,i} d{10,i} d{10,i}
b{10,i} a{10,i} c{10,i} d{10,i} d{10,i} d{10,i} d{10,i}
d{10,i} d{10,i} d{10,i} d{10,i} d{10,i} d{10,i}];
A10 = [d{11,i} d{11,i} d{11,i} d{11,i} d{11,i} d{11,i}
d{11,i} d{11,i} b{11,i} a{11,i} c{11,i} d{11,i} d{11,i}
d{11,i} d{11,i} d{11,i} d{11,i} d{11,i} d{11,i} d{11,i}];
A11 = [d{12,i} d{12,i} d{12,i} d{12,i} d{12,i} d{12,i}
d{12,i} d{12,i} d{12,i} b{12,i} a{12,i} c{12,i} d{12,i}
d{12,i} b{12,i} d{12,i} d{12,i} d{12,i} d{12,i} d{12,i}];
A12 = [d{13,i} d{13,i} d{13,i} d{13,i} d{13,i} d{13,i}
d{13,i} d{13,i} d{13,i} d{13,i} b{13,i} a{13,i} c{13,i}
d{13,i} d{13,i} d{13,i} d{13,i} d{13,i} d{13,i} d{13,i}];
A13 = [d{14,i} d{14,i} d{14,i} d{14,i} d{14,i} d{14,i}
d{14,i} d{14,i} d{14,i} d{14,i} d{14,i} b{14,i} a{14,i}
c{14,i} d{14,i} d{14,i} d{14,i} d{14,i} d{14,i} d{14,i}];
A14 = [d{15,i} d{15,i} d{15,i} d{15,i} d{15,i} d{15,i}
d{15,i} d{15,i} d{15,i} d{15,i} d{15,i} d{15,i} b{15,i}
a{15,i} c{15,i} d{15,i} d{15,i} d{15,i} d{15,i} d{15,i}];
A15 = [d{16,i} d{16,i} d{16,i} d{16,i} d{16,i} d{16,i}
d{16,i} d{16,i} d{16,i} d{16,i} d{16,i} d{16,i} d{16,i}
b{16,i} a{16,i} c{16,i} d{16,i} d{16,i} d{16,i} d{16,i}];
51
A16 = [d{17,i} d{17,i} d{17,i} d{17,i} d{17,i} d{17,i}
d{17,i} d{17,i} d{17,i} d{17,i} d{17,i} d{17,i} d{17,i}
d{17,i} b{17,i} a{17,i} c{17,i} d{17,i} d{17,i} d{17,i}];
A17 = [d{18,i} d{18,i} d{18,i} d{18,i} d{18,i} d{18,i}
d{18,i} d{18,i} d{18,i} d{18,i} d{18,i} d{18,i} d{18,i}
d{18,i} d{18,i} b{18,i} a{18,i} c{18,i} d{18,i} d{18,i}];
A18 = [d{19,i} d{19,i} d{19,i} d{19,i} d{19,i} d{19,i}
d{19,i} d{19,i} d{19,i} d{19,i} d{19,i} d{19,i} d{19,i}
d{19,i} d{19,i} d{19,i} b{19,i} a{19,i} c{19,i} d{19,i}];
A19 = [d{20,i} d{20,i} d{20,i} d{20,i} d{20,i} d{20,i}
d{20,i} d{20,i} d{20,i} d{20,i} d{20,i} d{20,i} d{20,i}
d{20,i} d{20,i} d{20,i} d{20,i} b{20,i} a{20,i} c{20,i}];
A20 = [d{21,i} d{21,i} d{21,i} d{21,i} d{21,i} d{21,i}
d{21,i} d{21,i} d{21,i} d{21,i} d{21,i} d{21,i} d{21,i}
d{21,i} d{21,i} d{21,i} d{21,i} d{21,i} b{21,i} a{21,i}];
A = [A1; A2; A3; A4; A5; A6; A7; A8; A9; A10; A11; A12; A13;
A14; A15; A16; A17; A18; A19; A20];
rr{2,i} = [r4(2,i); r3(2,i); r1(2,i); r2(2,i); r5(2,i)];
for m = 3:meta
rr{m,i} = [r2(m,i); r3(m,i); r1(m,i); r4(m,i); r5(m,i)];
end
rrr = [rr{2,i}; rr{3,i}; rr{4,i}; rr{5,i}; rr{6,i}; rr{7,i};
rr{8,i}; rr{9,i}; rr{10,i}; rr{11,i}; rr{12,i}; rr{13,i};
rr{14,i}; rr{15,i}; rr{16,i}; rr{17,i}; rr{18,i}; rr{19,i};
rr{20,i}; rr{21,i}];
% Performing naive Gaussian elimination
p = size(A,1); % get number of rows in matrix A
q = length(rrr); % get length of rrr
if(p~=q)
error('A and rrr do not have the same number of rows')
end
% Step 1: Form (q,q+1) augmented matrix
A(:,q+1) = rrr;
52
for k = 1:q
% Step 2: Make diagonal elements into 1.0
A(k,k+1:q+1) = A(k,k+1:q+1)/A(k,k);
% Step 3: Make all elements below diagonal into 0
for j = k+1:q
A(j,k+1:q+1) = A(j,k+1:q+1) - A(j,k)*A(k,k+1:q+1);
end
end
% Step 4: Begin back substitution
for j = q-1:-1:1
A(j,q+1) = A(j,q+1) - A(j,j+1:q)*A(j+1:q,q+1);
end
% Return solution
x = A(:,q+1);
% Estimations
dely1(1,i) = 0.0; dely2(1,i) = 0.0; dely5(1,i) = 0.0;
dely2(meta,i) = 0.0; dely4(meta,i) = 0.0;
dely3(1,i) = x(1,1); dely4(1,i) = x(2,1);
for m = 1:meta-1
dely1(m+1,i) = x((5*m)-2,1);
dely3(m+1,i) = x((5*m)-1,1);
dely5(m+1,i) = x(5*m,1);
end
for m = 1:meta-2
dely2(m+1,i) = x((5*m)+1,1);
dely4(m+1,i) = x((5*m)+2,1);
end
% Newton's Iteration
53
for m = 1:meta
y1(m,i+1,n)
y2(m,i+1,n)
y3(m,i+1,n)
y4(m,i+1,n)
y5(m,i+1,n)
=
=
=
=
=
y1(m,i,n)
y2(m,i,n)
y3(m,i,n)
y4(m,i,n)
y5(m,i,n)
+
+
+
+
+
dely1(m,i);
dely2(m,i);
dely3(m,i);
dely4(m,i);
dely5(m,i);
end
% check for convergence of the iterations
stop = abs(dely3(1,i)); imax(n) = i; fprintf('n = %f ',n);
fprintf('i = %f\n',i);
i = i + 1;
end
% Shift Profile
for m = 1:meta
yyy1(m) = y1(m,imax(n),n); yyy2(m) = y2(m,imax(n),n);
yyy3(m) = y3(m,imax(n),n); yyy4(m) = y4(m,imax(n),n); yyy5(m)
= y5(m,imax(n),n);
end
clear a b c d a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 A1 A2 A3 A4 A5 A6
A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A20 rr A rrr
cy1 cy2 cy3 cy4 cy5 cyy2 cyy4
end