numerical study of the flow and heat transfer between two horizontal cylinders
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THE NUMERICAL STUDY OF THE FLOW AND
HEAT TRANSFER BETWEEN TWO HORIZONTAL
CYLINDERS
HUO YUNLONG (B. Eng.)
DEPARTMENT OF MECHANICAL ENGINEERING
A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2004
I
ACKNOWLEDGEMENT
The author wishes to record here his indebtedness and gratitude to many who
have contributed their time, knowledge and effort towards the fulfillment of this work.
Particularly, he would like to express his heartfelt gratitude and thanks to Dr. T. S. Lee
and Dr H. T. Low for their invaluable guidance, supervision, encouragement and
patience throughout the course of the investigation.
The technical staffs of the Fluid Mechanics Laboratory are also to be thanked
for their assistance during the phase of the investigation. Special thanks are also due to
my classmates in the Fluid Mechanics Laboratory who give me great help in plotting
the figures and using the Tecplots.
The author also wants to express his appreciation to his last grandfather whose
spirit encourages him, supports him and assists him throughout this period.
Finally, the author wished to express his gratitude to those who have directly or
indirectly contributed to this investigation.
II
SUMMERY
The flow and convective heat transfer in concentric and eccentric horizontal
annuli with isothermal wall conditions are studied numerically using two-dimensional
finite-difference and finite-element models. The Stream-Function Vorticity and
primitive variable formulations are applied to the finite different and finite element
methods respectively. The structure mesh is obtained to simulate the buoyancy driven
flow. Since the complex geometry configuration of the studied cases, the cylindrical
and bipolar coordinates are introduced to solve problems of the finite difference
method. The model is also designed by the Galerkin finite element method with Penalty
Function Approach. The effects of various parameters such as the radius ratio of the
annulus, the eccentricity of the annulus, the Rayleigh number and Reynolds number of
the rotation of the inner cylinder are investigated at the Prandtl number of 0.701.
Overall heat transfer results are obtained. For the case of concentric cylinders, the
numerical results obtained are in good agreement with the similar results of other
investigators. In the extension of the numerical work done here, rotating the inner
cylinder and outer cylinder individually and both are also considered. For the eccentric
cases, comparison with available experimental results with the present two-dimensional
numerical model is good at relatively low Reynolds numbers in the range of 0-800. The
effects of Prandtl number on the flow and heat transfer are also briefly studied in the
present investigations. All of the kidney cells, the stagnant zone and the thermal plume
are changed with different directions and velocities of the rotation, Rayleigh number,
eccentric ratio and Prandtl number.
III
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENT I
ABSTRACT II
NOMENCLATURE V
LIST OF FIGURES VIII
CHAPTER 1 INTRODUCTION
1.1 Background 1
1.2 Literature survey 2
1.3 Flow description 9
1.4 Objectives and scope 12
CHAPTER 2 PROBLEM FORMULATION
2.1 Derivations of the governing equations 14
2.2 Coordinate system for finite difference method 18
2.3 The governing equation and boundary condition in finite element method 19
2.4 Investigated geometric and physical parameters 22
2.5 Boundary conditions in the finite difference method 24
CHAPTER 3 NUMERICAL METHODS
3.1 The finite-difference and finite-element approaches 27
3.2 The solution procedure 28
3.3 Finite difference methods for solving the equations 30 3.4 Finite element methods for solving the equations 37
IV
CHAPTER 4 ANALYSIS OF RESULTS FOR THE FINITE ELEMENT
METHOD
4.1 The effect of Rayleigh numbers 41
4.2 The effect of the radius ratio 42
4.3 The effect of eccentricity 43
4.4 The effect of rotating the inner and/or outer cylinders between concentric cases 45
4.5 The effect of rotating the inner and/or outer cylinders between eccentric
cylinders 47
CHAPTER 5 ANALYSIS OF RESULTS FOR THE FINITE DIFFERENCE
METHOD
5.1 The effects of Rayleigh numbers 51
5.2 The effects of radius ratio 52
5.3 The effect of the inner cylinder rotating 53
5.4 The effects of eccentricity 55
5.5 The effects of rotating inner cylinder in the eccentric annulus 57
CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS 60
REFERENCES 62
FIGURES 68
V
NOMENCLATURE
A Area
C Constant in the transformation equations from Bipolar coordinate systems to
Cartesian Coordinate System
D Diameter
E Magnitude of the eccentricity vector, −e , e = |
−e |
−e Eccentricity Vector,
−e = ( vh ee , )
re Magnitude of the eccentricity ratio vector r
e−
, re = | r
e−
|
re−
Eccentricity ratio vector, r
e−
=−e /L
g Gravity acceleration, g=| −g |
−g Gravity vector
ηξ hh , Metric or scale factors in the Bipolar coordinate system
gi Unit vector in the direction of the gravity vector, gi =−g /g
k Thermal conductivity
eeqk Overall equivalent thermal conductivity
eqlk Local equivalent thermal conductivity
L ‘mean’ clearance between the two cylinders, L= io rr −
DNu Nusselt number based on the diameter of the heated cylinder, kDhNuD /−
=
p, P Pressure
Pr Prandtl number, Pr = αυ /
r radius
VI
DRa Rayleigh number based on the diameter of the heated cylinder,
υαβ 3'DTgRaD∆=
Ral Reyleigh number based on the mean clearance L, υα∆β 3DTg '
====lRa
DRe Reynolds number based on the diameter of the heated cylinder, υ
Ω=
Dr iiDRe
Rel Reynolds number based on the mean clearance L, Rel = υ
Lr iiΩ
t time
T temperature
−U Velocity vector
x, y coordinate variables in the Cartesian coordinate system
Greek
α Thermal diffusivity
β Coefficient of thermal expansion
γ Angle measured clockwise from the upward vertical through the center of the
heat transfer
ε Total emissivity of a surface
ζ Vorticity
ξη , Coordinate variables in the Bipolar coordinate system
θ Angular coordinate in the Bipolar coordinate system
υ Kinematic viscosity
ρ Density
σ Stefan-Boltzmann’s constant
VII
φ Angular position of the gravity vector relative to the negative y-axis measured
in the clockwise direction
ψ Stream function
Ω Angular speed
Subscripts
i stands for inner cylinder; also used as an indexing integer variable for the mesh
points
o stands for outer cylinder
r stands for reference quantity; also used as an indexing integer variable for the
mesh points
w stands for wall
Superscripts
k number of the inner iteration
n number of the time step or global iteration
‘ the ‘prime’ symbol emphasizes the dimensional form of a variable as distinct from its
non-dimensional usage
VIII
LIST OF FIGURES
Table 3.1 Shape Functions of Quadrilateral Elements (4-9) 70 Table 4.1 Parameters used in calculations 70 Fig. 1.1.1 Geometry of annular region and the gravity direction 68 Fig. 2.1.1 Mesh used for the numerical computation 68 Fig. 3.1.1 2-D view of three kinds of shape function 69
Fig. 4.1.1 – 4.5.5 Figures for finite element method Fig. 4.1.1 – 4.1.5 Flow and temperature fields for various Rayleigh numbers of air (71-73) Fig. 4.2.1 – 4.2.2 Flow and temperature fields for various Radius ratio of air (74-75) Fig. 4.3.1 – 4.3.4 Flow and temperature fields for various eccentric ratio of air (76-78) Fig. 4.4.1 – 4.4.4 Flow and temperature fields for various rotations in concentric
annulus (79-82) Fig. 4.5.5 – 4.5.5 Flow and temperature fields for various rotations in eccentric
annulus (83-87)
Fig. 5.0.1 – 5.5.6 Figures for finite difference method Fig. 5.0.1 Overall heat transfer coefficient versus Rayleigh number at the
radius ratio of 2.6, 0=re . (88-88) Fig. 5.1.1 – 5.1.2 Flow and temperature fields for various Rayleigh numbers of air (89-90) Fig. 5.2.1 – 5.2.3 Flow and temperature fields for various Radius ratio of air (91-93) Fig. 5.3.1 – 5.3.2 Flow and temperature fields for various rotations in concentric
annulus (94-95) Fig. 5.4.1 – 5.4.3 Flow and temperature fields for various eccentric ratio of air (96-98) Fig. 5.5.1 – 5.5.6 Flow and temperature fields for various rotations in eccentric
annulus (99-104)
1
CHAPTER 1 INTRODUCTION
1.1 Background
The flow and thermal fields in enclosed space have received much attention
because of theoretical and wide engineering applications, such as thermal energy
storage systems, cooling of electronic components, and transmission cables. Fig. 1.1.1
shows the Geometry of annular region and the gravity direction. Given the problem
illustrated in Fig. 1.1.1, the inner circular is set to high temperature and the outer part
lower temperature under the terrestrial conditions. Due to its simple geometry and well-
defined boundary conditions, the system has been studied extensively by researchers
such as Bishop et al (1968), Kuehn and Goldstein (1976), Farouk and Guceri (1981),
Vafai and Desai (1993), and large number of literatures were published in the past few
decades. For concentric and eccentric cases in a horizontal annulus between two
circular cylinders, the basic and fundamental configuration, the flow and thermal fields
have been well studied. Kuehn and Goldstein (1976) comprehensively studied the
concentric case. The experimental and numerical studies of the eccentric case have also
been conducted by Kuehn and Goldstein (1978), and Guj and Stella (1995). However,
on the whole, little is known about effects of rotation of inner and/or outer cylinders on
the flow and the heat transfer in eccentric annuli. There is much scope for further
investigation in this aspect of the problem. The numerical method of both of finite
element and finite partial difference methods are useful. In present work the author
focuses on two parts. In the first part, based on the previous work, the primary aim is to
extend knowledge into areas that have not been conducted and examined. The main
methods are the finite element method and the finite difference approach. For the finite
element method, original governing equations are used to produce the velocity field and
2
temperature distribution. For the finite difference method, the streamline plot is created
through using Stream-Function Vorticity formulation. Compared with each other, both
of the two methods are capable of solving the flow and the convective heat transfer in
concentric and eccentric horizontal annuli with isothermal wall conditions. Compared
with the results of Kuehn and Goldstein (1976), the computed results of the thesis also
show that the study may be pushed further to enter the second part. The second part is
the knowledge extension that has a little study by other investigators. The primary
objectives of this part focus on natural convection in the eccentric horizontal annuli
where both inner and outer cylinders are rotated. Both rotated inner and outer cylinders
are seldom investigated by other researchers. Different kinds of rotation such as the
same direction of the inner and outer cylinders, and the opposite direction of the inner
and outer cylinders associated with the eccentric ratio will be discussed in the thesis.
Particularly, when the Reynolds number is large, the unstable situation is produced.
1.2 Literature review
Natural convection between horizontal concentric cylinders has been widely
studied experimentally and numerically over the past three decades because of the
importance of this subject in industries, such as transmission cable cooling systems,
latent energy storage systems, nuclear reactor designs, etc. The problem was first
investigated experimentally by Beckmann (1931) with air, hydrogen and carbon
dioxide as the test fluids to obtain overall heat transfer coefficients. A large part of the
experimental work was devoted to finding the overall heat transfer between the
cylinders using the non-dimensional parameter defining the temperature difference
between the cylinders. Particularly Kuehn and Goldstein (1976) investigated the
problem experimentally and numerically. A Mach-Zehnder interferometer was used to
3
determine the temperature distribution and the local heat transfer coefficients in air and
water. With water, they demonstrated that the flow remained steady even though the
Rayleigh number was well over the critical value obtained experimentally with air,
which suggests that the Prandtl number affected the transition characteristics.
Unlike the case of natural convection in concentric annulus, similar
experimental studies for eccentric annulus are few. The effect of vertical and horizontal
eccentricities on the overall heat transfer coefficient was first experimentally
investigated by Zagromov and Lyalikov (1966) using air as the working fluid. Using
optical interferometry, Kuehn and Goldstein (1978) studied the local and overall heat
transfer coefficients for both horizontal and vertical eccentricities of magnitude re up
to about 2/3. They found that although the distribution of the local heat transfer
coefficient was substantially altered by eccentricity, the overall heat transfer coefficient
did not change by more than 10% from the concentric value at the same Rayleigh
number. The effect of moving the inner cylinder downwards is to cause the overall heat
transfer to increase while moving the inner cylinder upwards has the opposite effect.
Yeo (1984) use the same method as Kuehn and Goldstein (1976, 1978) to verify the
overall heat transfer coefficients predicted by the numerical model. His experimental
results are in good agreement with the experimental results of Kuehn and Goldstein
(1978) obtained using nitrogen as test fluid and fit the present numerical curve very
well with deviations typically less than 5%. Lee (1991) performed the numerical
experiments to study rotational effects on the mixed convection of low-Prandtl-number
fluids enclosed between the annuli of concentric and eccentric horizontal cylinders. For
the range of Prandtl numbers considered here, numerical experiments showed the mean
Nusselt number increases with increasing Rayleigh number for both concentric and
eccentric stationary inner cylinders. At a Prandtl number of order 1.0 with a fixed
4
Rayleigh number, when the inner cylinder is rotated, the mean Nusselt number
decreases throughout the flow. Dennis and Sayavur (1998) analytically and
experimentally investigated the flow in eccentric annuli of drilling fluids commonly
used in oil industry. The expression for azimuthal velocity as a function of eccentricity
ratio and rheological parameters of the fluid has been obtained based on the linear
fluidity model. Velocity profiles were measured for a Newtonian glycerol/water
mixture and a non-Newtonian oil field spacer fluid in eccentric annuli using the
stroboscoptic flow visualization method.
Because of the limitations of the analytical approach and encouraged by the
availability of large computing machines, numerical methods were applied to solve the
equations which govern the flow and heat transfer in the annulus.
The first numerical solution was obtained by Crawford and Lemlich (1962)
using a Gauss-Seidel iterative method. Abbot (1962) used a matrix inversion technique
to obtain solutions for the case of narrow annuli. Powe et al. (1971) applied numerical
method to determine the Rayleigh number for the onset instability in the flow at
relatively low radius ratios and obtained reasonably good qualitative agreement with
the earlier experimental results of Powe et al. (1969) on the delineation of the flow
regimes. Their numerical results seem to indicate that the onset of multicellular flow at
low radius ratios does not affect the overall heat transfer significantly. Charrier-Mojtabi
et al. (1980) gave numerical solutions using the alternating direction implicit (ADI)
method for three cases: a wide annulus (R=2.26) for Pr=0.7, a narrow annulus (R=1.2)
for Pr=0.7 and a wide annulus (R=2.5) for Pr=0.02. On treating the problem
numerically at high Rayleigh numbers, Jischke and Farshch (1980) divided the flow
field of an annulus into five regions which include an inner boundary layer near the
inner cylinder, an outer boundary layer near the outer cylinder, a vertical plume region
5
above the inner cylinder, a stagnant region below the inner cylinder and a core region
surrounded by these four regions; they applied the boundary layer approximation to
obtain the temperature distribution and heat transfer rates. A numerical parametric
study was carried out by Kuehn and Goldstein (1980), in which the effects of the
Prandtl number and the radius ratio on heat transfer coefficient were investigated.
Tsui and Tremblay (1984) presented the results of mean Nusselt numbers for
both transient and the steady natural convection. San Andres (1984) found the size of
the separation eddy and the position of the points of separation and reattachment to be
Reynolds number dependent in the numerical study of flow between eccentric
cylinders. The separation point moves in the direction of rotation upon increasing the
Reynolds number, in contradiction of the first-order inertial perturbation theory of
Ballal and Rivlin (1976). The numerical methods employed in their study include
Galerkin’s procedure with B-spline test function. Galpin and Raithby (1986) assessed
the impact of the ‘standard’ treatment of the T-V coupling and proposed an improved
method. Newton-Raphson linearization was investigated as a means of accelerating the
convergence rate of control volume-based predictions of natural convection flow. It is
found that repeated solutions of the Newton-Raphson linear set converge monotonically
for a much wider range of relaxation, and the maximum convergence rate can be
significantly higher than that corresponding to the standard linear set.
Lee and Yeo (1985) developed a numerical model to study the effects of
rotation on the fluid motion and heat-transfer processes in the annular space between
eccentric cylinders when the inner cylinder is heated and rotating. The overall
equivalent thermal conductivity ( eqK ) is obtained for Rayleigh numbers Ra up to 610
with rotational Reynolds number Re variations of 0-1120. Investigation shows that, for
Re up to the order of 102, the numerical model shows promising results when Ra is
6
increased. Numerical solutions for laminar, fully developed, forced convective heat
transfer in eccentric annuli were presented by Manglik and Fang (1995). With an
insulated outer surface, two types of thermal boundary conditions had been considered:
constant wall temperature (T), and uniform axial heat flux with constant peripheral
temperature (H1) on the inner surface of the annulus. Velocity and temperature profiles,
and isothermal Re, Nui,j and Nui,H values for different eccentric annuli ( 6.00 * ≤≤ ε )
with varying aspect ratios ( 75.025.0 * ≤≤ r ) are presented in their paper. The
eccentricity is found to have strong influence on the flow and temperature fields. The
flow trends to stagnate in the narrow section and has higher peak velocities in the wide
section. The flow maldistribution is found to produce greater nonuniformity in the
temperature field and degradation in the average heat transfer. Yoo (1998) numerically
investigated dual steady solution in natural convection in an annulus between two
horizontal concentric cylinders for a fluid of Prandtl number 0.7. It is found that when
the Rayleigh number based on the gap width exceeds a certain critical value, dual
steady two-dimensional (2-D) flows can be realized: one being the crescent-shaped
eddy flow commonly observed and the other the flow consisting of two counter-
rotating eddies and their mirror images. The critical Rayleigh number decreases as the
inverse relative gap width increases.
Mohamed etc. (1998) numerically studied the effect of radiation on unsteady
natural convection in a two-dimensional participating medium between two horizontal
concentric and vertically eccentric cylinders by using a bicylindrical coordinates
system, the stream function, and vorticity. Original results are obtained for three
eccentricities, Rayleigh number equal to 104, 105 and a wide range of radiation-
conduction parameter. Shu and Yeo (2000) applied the global method of polynomial-
based differential quadrature (PDQ) and Fourier expansion-based differential
7
quadrature (FDQ) to simulate the natural convection in an annulus between two
arbitrarily eccentric cylinders. Their approach combined the high efficiency and
accuracy of the differential quadrature (DQ) method with simple implementation of
pressure single value condition. The result confirmed the finding by Guj and Stella
(1995). Escudier et al. (2000) concerned a computational and experimental study of
fully developed laminar flow of a Newtonian liquid through an eccentric annulus with
combined bulk axial flow and inner cylinder rotation. Their results were reported for
calculation of the flow field, wall shear stress distribution and friction factor for a range
of values of eccentricity ε , radius ratio κ and Taylor number Ta. More recently, Lee et
al. (2002) used GDQ method due to the eccentricity of the inner and outer cylinders
studied the nett fluid circulation around the inner cylinder and the effects of rotation of
the inner cylinder with a radius ratio of 2.6.
Discrepancies among the results reported in the literature for narrow annuli are
found by Rao et al. (1985), Fant et al. (1989), Cheddadi et al. (1992), Kim and Ro.
(1994). Large differences are shown not only for the Ra values at which bifurcation
occur but also in regard to a possible existence of hysteresis phenomena. For example,
Kim and Ro (1994) and Fant et al. (1989) found a hysteresis numerically, whereas Rao
et al. (1985) show only one type of multicellular flow. Cheddadi et al. (1992) presented
two numerical solutions at the same Ra that depends on the initial conditions: the
crescent base flow and a multicellular one. However, they failed to obtain multicellular
flows experimentally. Rao et al. (1985) and Kim and Ro (1994) supported numerically
the general trends presented by Powe et al. (1969); that is, the appearance of
multicellular flow patterns in the upper part of narrow annuli. Furthermore, Rao et al.
(1985) reported a transition of the steady upper cells to oscillatory motion at moderate
Rayleigh numbers. Using a linear stability analysis of steady two-dimensional natural
8
convection of a fluid layer confined between differentially heated vertical plane walls,
Korpela et al. (1973) reported that the flow is primarily unstable against purely
hydrodynamic steady waves in the limit of zero Prandtl number. These secondary
shear-driven instabilities are crossing cells called “cat’s eyes.” Increases in Prandtl
number lead to the appearance of buoyancy-driven oscillatory instabilities. The critical
value of Pr determining which type of instabilities appears has been numerically
determined which type of instabilities appears has been numerically determined to be
around Pr=12.7 by many authors. In slots of finite ratio of height over width the vertical
temperature gradient is an additional results and linear stability analysis, Roux et al.
(1980) have demonstrated the existence of a zone of limited extent in the (Ra, A)-plane
inside which steady cat’s eyes can develop. This zone is only for aspect ratios larger
than about A=11 for air-filled cavities. This result was confirmed by the numerical
studies of Lauriat (1980), Lauriat and Desrayaud (1985), and more recently by Le
Quere (1990) and Wakitani (1997). As Ra is further increased, a reverse transition from
multicellular flow to unicellular flow occurs and this has been numerically and
experimentally demonstrated by Roux et al. (1980), Lauriat (1980), Desrayaud (1987),
and Chikhaoui et al. (1988). Cadiou et al (1998, 2000) studied numerically the flow
structure which develops both in horizontal and vertical regions of narrow air-filled
annuli and devoted some part of their paper to the thermal instabilities observed in the
top of the annulus and clarify, found in the literature.
Yoo (1998) numerically investigated natural convection in a narrow horizontal
cylindrical annulus for fluids for 3.0Pr ≤ . For 2.0Pr ≤ , hydrodynamic instability
induces steady or oscillatory flows consisting of multiple like-rotating cells. For
Pr=0.3, thermal instability creates a counter-rotating cell on the top of annulus.
9
Until very recently, most numerical studies have been limited to flows in the
steady laminar regime. Farouk and Guceri (1982) applied the ε−k turbulence model
to study the turbulent natural convection for high Rayleigh numbers ranging from
76 10 to10 with a radius ratio of 2.6. A comparison of Nusselt numbers between the
results obtained numerically and those obtained experimentally by other investigators
showed a good agreement. Kenjeres and Hanjalic (1995) studied natural convection in
horizontal concentric and eccentric annuli with heated inner cylinder using several
variants of single-point closure models at the eddy-diffusivity and algebraic-flux level.
Their results showed that the application of the algebraic model for the turbulent heat
flux derived from the differential transport equation and closed with the low-Reynolds
number form of transport equations for the kinetic energy κ , its dissipation rate ε , and
temperature variance 2θ , reproduced well a range of Rayleigh numbers, for different
overheatings and inner-to-outer diameter ratios.
1.3 Flow description
From a theoretical point of view, natural convection in horizontal annuli has
been one of the focuses of research heat transfer on account of the large variety of flow
structures encountered in this configuration according to the value of the radius ratio. A
comprehensive review of steady two-dimensional (2-D) convection was presented in
the work of Kuehn and Goldstein (1976), in which experimental and numerical studies
were performed to determine velocity and temperature distribution and local heat
transfer coefficients for convective flows of air )7.0(Pr ≈ and water )6(Pr ≈ within a
horizontal concentric annulus. In 1978 Kuehn and Goldstein investigated natural
convection heat transfer in concentric and eccentric horizontal cylinders through
experiments. And then parametric study of Prandtl number and diameter ratio effects
10
were done in horizontal cylindrical annuli. Powe et al. (1971) and Rao et al. (1985)
investigated flow patterns for air. They found that free convective flow of air could be
categorized into four basic types: a steady two-dimensional oscillatory flow, a three-
dimensional oscillatory flow, and a two-dimensional multicellular flow. Recently, Yoo
(1998) investigated the existence of dual steady states for a fluid of Pr=0.7.
The basic two-dimensional steady flow that is observed at low Rayleigh
numbers is characterized either by two crescent-shaped or by two kidney-shaped cells
according to the value of the radius ratio R. The first pattern is observed for narrow
annuli whereas the latter is found only for large radius ratios (Bishop and Carley,
1966). These two patterns present symmetry with respect to the vertical centerline. The
main difference between these two basic flow fields is in the shape of the central flow
regions that become istorted into a kidney shape for the second flow structure. From
their experimental work, Powe et al. (1971) depicted flow regime transitions for air-
filled annuli and were the first to present a chart for the prediction of the nature of the
flow according to the Rayleigh number and radius ratio. This chart shows the limit
between the base flow and the two- or three-dimensional flow patterns, stationary or
oscillatory, which follow the named pseudo-conduction regime. The effect of heating
the inner cylinder is the fluid follows an upward stream along the hot inner cylinder and
finally reaches the top of the annular space. The fluid goes then downwards along the
cold cylinder and reaches the almost quiescent bottom portion of the annulus. At low
Rayleigh number, conduction is the major mode of heat transfer between the hot and
cold cylinders. As the Rayleigh number is increased, the center of rotation of the cells
moves upwards and a thermal plume starts to form at the upper part of the annulus with
an impingement region at the outer cylinder. The shape of isotherms shows that the
11
largest part of the heat convected within the annulus is extracted from the lower part of
the inner cylinder.
The buoyancy force is proportional to the temperature difference between
surfaces. Therefore, at a higher temperature difference between the two cylinders, the
‘strength’ or circulation of the convection cells is greater. The rate at which heat is
being transferred or convected by faster moving fluid is therefore increased. The flow
and temperature fields around the inner cylinder greatly resemble that of a heated
cylinder convecting to still ambient air. The position of the inner cylinder relative to the
outer cylinder is an important geometric parameter that deserves studies because it may
either enhance or suppress the development of these flow cells and thus affects the rate
of heat transfer. Such situations where the annular region becomes eccentric are
encountered in actual practice; as in the ‘snaking’ of high voltage underground power
cable caused by thermal expansion of the cable. If eccentricity does affect the heat
transfer in a significant manner, it could be employed as a design factor to either
enhance or reduce the amount of heat transfer between the cylinders as the application
may require. This aspect thus requires detailed investigation. The rotation of inner
and/or outer cylinders will affect the flow in the annular region and thus the heat
transfer.
The flow generated in an annulus due to rotation of the inner cylinder in the
absence of bulk axial flow is one of the most widely investigated topics in the fluid
mechanics. Of the hundreds of papers published to date, the majority have been
concerned with the Taylor vortices which arise above a critical Taylor number cTa .
Lockett (1992) showed that the occurrence of Taylor vortices is inhibited by
eccentricity of the inner cylinder, his numerical calculations being confirmed by the
recent experimental work of Escudier and Gouldson (1997) as well as by earlier
12
experiments reported by Kamal (1966), Cole (1968), Vohr (1968) and Castle and
Mobbs (1968). The flow separation and the recirculating eddy or vortex which occurs
above a critical eccentricity for a given radius has also received widespread attention
(Kamal, 1966; Ballal and Rivlin, 1976; San Andres and Szeri, 1984; Siginer and
Bakhtiyarov, 1998).
In the present study, the effects of various system parameters such as the
temperature difference between the surfaces of the two cylinders, the geometry of the
annulus, the properties of the fluid and the rotation rate of the inner cylinder on the
flow and the heat transfer in the annular spaces are investigated. Because of its
common occurrence in practice, the inner cylinder is considered to be the hotter
cylinder. As a useful idealization, it is further assumed that the two cylinders are kept
isothermal. The author has also restricted his study mainly to flow in the laminar
regime. However, the unstable and turbulence is introduced when the Rayleigh and
Reynolds number is increased. Some cases are analyzed by using analytical method.
Fig. 1.1.1 shows schematically a typical annular region being studied and the physical
quantities involved.
1.4 Objectives and scope
The natural convection phenomena in concentric and eccentric annuli are
numerically studied. The physical behavior of the buoyancy driven flow is investigated
through using the two-dimensional numerical model of the finite element and finite
difference methods. The computational results can provide the important parameters for
the industrial and engineering applications. In present study, both of the methods have
advantage and disadvantage. It is difficult for the finite difference method to solve the
vorticity-stream function formation with the moving boundary condition. Some special
13
approaches such as the single value condition are proposed to deal with the problem.
Corresponding to the finite difference method, the finite element method enables to
combine the boundary condition to the matrix and solve complex geometry more easily
and produces more accurate results. However, the large sparse matrix created by the
finite element method need more compute storage and executed time. Different kinds
of the rotation associated with eccentric ratios and radius ratios are discussed, such as
the same direction of the inner and outer cylinders’ rotation, the opposite direction of
the inner and outer cylinders’ rotation. The overall heat transfer coefficients are
investigated.
14
CHAPTER 2 PROBLEM FORMULATION
The flow and the heat transfer in the annular space between two horizontal
circular cylinders with parallel axes is the main problem that is being studied. The
cylinders are assumed to be isothermal with the inner cylinder being held at a higher
temperature. The annular space may either be concentric and eccentric. Fig. 1.1.1 and
2.1.1 show the geometrical and mesh configuration of a typical problem. In the finite
difference method, effects of such parameters such as radius ratio, eccentricity of the
annular, the Rayleigh number, the Prandtl number and the rotation of the inner cylinder
expressed in the form of a Reynolds number are of interest to the present investigation.
In the finite element method, such primate parameters as the temperature, the radius of
the cylinders, the pressure gradient, the density, the thermal diffusivity and the
coefficients of thermal expansion are investigated by the author. The primate
parameters in the finite element method are compared with the system parameters in
the finite difference method.
In this chapter, the governing equations are put in a form suitable for subsequent
numerical studies. The underlying assumptions of the formulation are stated. The non-
dimensionalization of the governing equations helps to identify the forms of the
relevant parameters in the finite difference method.
2.1 Derivations of the governing equations
The fluid flow and the heat transfer in the annular region are governed by the
equations of momentum, mass and energy conservation. These equations may be found
in standard texts such as Eckert and Drake (1981) and Parker, Boggs and Blick (1969).
15
2.1.1 Simplifying governing equations for finite element method
These governing equations in their original and complete form are highly
complex. In formulating the actual equations used in this study, several simplifying
assumptions are made:
• The flow is assumed to be effectively invariant along the axial direction of the
cylinders. This leads automatically to a two-dimensional model. The two-
dimensional approximation is a good representation of the real flow in a long
finite length annulus away from the ends provided there are no three-
dimensional instabilities (Kuehn and Goldstein 1976).
• The flow is assumed to be laminar. This is an essential assumption because
unless some form of turbulence modeling is used, the governing equations in
their usual form will break down when the flow becomes turbulent.
• The Boussinesq approximations are adopted.
With these simplifying assumptions above, the governing equations are:
Momentum Conservation Equation:
'2'''''
'
))(1()(−−−
∇+−−+∇−=∇•+∂
∂ UgTTPUUt
Urr
r
υβρ
---------------------------------(2.1)
Continuity Equation
0' =•∇−
U -------------------------------------------------------------------------------------(2.2)
Energy Conservation Equation:
'2'''
'
)( TTUtT
r ∇=∇•+∂∂
−α -------------------------------------------------------------------(2.3)
(Because of the repeated use of some symbols in both dimensional and dimensionless
forms, the prime symbol ' is used to emphasize the dimensional form of the variables
as distinct from its dimensionless usage.)
16
2.1.2 Stream-Function Vorticity formulation for the finite different method
The governing equations (2.1) to (2.3) are recast in the Stream-Function
Vorticity form by taking the curl of equation (2.1) and defining two functions 'ψ and
'ζ called respectively the Stream-Function and the Vorticity by the following relations:
'' ζ=×∇−
U -------------------------------------------------------------------------------------(2.4)
'
''
'' and
xU
yU yx ∂
∂−=∂∂= ψψ -------------------------------------------------------------------(2.5)
The pressure gradient term in equation (2.1) is eliminated because the curl of a gradient
is identically zero. The use of 'ψ , the Stream-Function, ensures that the continuity
condition is automatically met.
The resultant equations in the Stream-Function Vorticity formulation are:
The Vorticity Transport equation:
'2'''''
'
)()( ζνβζζ ∇+−×∇−=⋅∇+∂∂
−− ro gTTUt
--------------------------------------------(2.6)
The Stream-Function Vorticity equation:
''2 ζψ −=∇ -------------------------------------------------------------------------------------(2.7)
which is the definition of Vorticity 'ζ in terms of 'ψ .
The Energy conservation equation:
'2'''
'
)( TUTtT
r ∇=⋅∇+∂∂
−α -------------------------------------------------------------------(2.8)
where the convective terms have been put in the conservative form using the
mathematical identity fUUfUf ∇⋅+⋅∇=⋅∇−−−
)()( and the continuity relation (2.2).
Non-dimensionalization
The governing equations (6) to (8) are made dimensionless by setting
17
rrroi
or LLUU
TTTTT
Ltt
Lxx
αζζ
αψψ
αα 2''
'
''
''
2
''
,,,,, ===−−
=== −
−
Under this scheme of non-dimensionalization, the governing equation (2.6) to (2.8)
assumes the following form:
Vorticity Transport Equation:
ζ∇+×∇••−=ζ•∇+∂ζ∂
−−
2PriTRaPr)U(t g
l -------------------------------------------(2.9)
Stream-Function Vorticity Equation:
ζψ −=∇ 2 -------------------------------------------------------------------------------------(2.10)
Energy Conservation Equation:
TTUtT 2)( ∇=•∇+
∂∂ -----------------------------------------------------------------------(2.11)
where lRaPr and are the dimensionless parameters.
With the non-dimensionalization scheme, a problem is fully specified when the
following parameters are known:
(a) the radius ratio io rr / ,
(b) the eccentricity ratio vector r
e−
or re and angle φ ,
(c) the Prandtl number Pr,
(d) the Rayleigh number Ral, and
(e) the Reynolds number Rel.
(a) and (b) specify the geometry of the two-dimensional solution region. The
dimensionless parameters Pr and Ral appear explicitly in the governing equations. The
Reynolds number Rel, which is a measure of the wall velocity of the inner cylinder, is
implemented through the vorticity boundary conditions.
18
2.2 Coordinate system for finite difference method
For the eccentric cases in the finite difference method, the bipolar coordinate
system, which is a more convenient coordinate system, is adopted. The Bipolar
coordinate system gives a fine mesh around the inner cylinder. The proved to be
essential for the accurate evaluation of temperature gradient in the heat flux calculation.
However, the Bipolar coordinate system can’t be used for the concentric cases owing to
singularity at 0=re . A Polar coordinate system is therefore employed for the
concentric cases.
2.2.1 Concentric geometry
The transformation between the Cartesian x-y coordinate system and the r-θ
Polar coordinate system is given by the following relations:
θθ sin,cos ryrx == )20(),0( πθ ≤≤∞<≤ r ---------------------------------------(2.12)
The governing equations in the θ−r coordinate system are:
)cos(sinPrPr)(1)( 2
θθθζζζ
θζζ
θ ∂∂+
∂∂⋅+∇=+
∂∂+
∂∂+
∂∂ T
rrTRal
rUU
rU
rtr
r -----(2.13)
ζϕ −=∇ 2 -----------------------------------------------------------------------------------(2.14)
TrTUTU
rTU
rtT r
r2)(1)( ∇=+
∂∂+
∂∂+
∂∂
θθ---------------------------------------------(2.15)
where r
Ur
U r ∂∂−=
∂∂= ψ
θψ
θ,1 and ).11( 2
2
22
22
rrrr ∂∂+
∂∂+
∂∂≡∇
θ
2.2.2 Eccentric geometry
The transformation between the Cartesian x-y coordinate system and the
Bipolar ηξ − coordinate system is given by the following relations:
)cos/(coshsincy)cos/(coshsinhcx
ξ−ηξ=ξ−ηη=
)( ∞<<−∞ η )20( πξ ≤≤ ------------------------------(2.16)
19
where c is a scaling factor of the transformation related to the eccentricity ratio re and
the radius ratio of the two cylinders. This transformation is conformal and preserves the
orthogonality of the grid lines.
The Governing Equations in the bipolar coordinate system are:
)](sin)([cosPr
Pr)sinhsin()(1)(1 2
ξηφ
ξηφ
ζηξζζξ
ζη
ζηξξη
∂∂−
∂∂+
∂∂+
∂∂•
+∇=+−∂∂+
∂∂+
∂∂
TATBTBTARal
UUc
Uh
Uht
----------(2.17)
ζψ −=∇ 2 ------------------------------------------------------------------------------------(2.18)
TUUcTTU
hTU
htT 2)sinhsin()(1)(1 ∇=+−
∂∂+
∂∂+
∂∂ ηξ
ξη ηξξη --------------------(2.19)
where )(1,1,12
2
2
2
22
ηξηψ
ξψ
ξη ∂∂+
∂∂≡∇
∂∂=
∂∂−=
hhU
hU
cBcAch /sinhsin,/)coscosh1(),cos/(cosh ηξξηξη −=−=−=
φ is the angle which describes the relation between the eccentricity ratio vector r
e−
and
the gravity vector −g .
2.3 The governing equation and boundary condition in finite element method
2.3.1 The governing equation in finite element method
The governing equations for the thermal and fluid fields are solved using the
Galerkin finite element method. Since details are well documented in many textbooks,
only an outline is given here. In essence, the computational domain is first divided into
small elements. Within each element, the dependent 'u , 'P and 'T are interpolated by
shape functions of θϕφ and , . (In section 2.3, θϕφ and , are shape function for 'u , 'P
and 'T )
)(U),,(u i ttyx Ti φ= -------------------------------------------------------------------------(2.20)
20
)(P),,( ttyxP Tϕ= ---------------------------------------------------------------------------(2.21)
)(T),,( tTtyxT Tr θ=− -----------------------------------------------------------------------(2.22)
Where )(Ui t , P(t) and T(t) are column vectors of elements nodal point unknowns.
Substituting the above equations into the governing equations, we will obtain
the residuals 321 and , RRR which represent the momentum, mass conversion and
energy equations respectively. The Galerkin form of the method of Weighted Residuals
seeks to reduce these errors to zero, and the shape functions are chosen the same as the
weighting functions. Following the procedure, the governing equations for the fluid
flow and heat transfer may be rewritten as
P)(U))/((1
2
^
∫∫ ΩΩ−=+∇⋅ dAdAri T
piT
iT ϕϕεφδφϕ --------------------------------------(2.23)
( pε is the penalty parameter)
0T)(
T)u(T)( T
=∇⋅∇+
∇⋅+
∫
∫∫
Ω
ΩΩ
dAk
dACdtddAC
T
pT
p
θθ
θθρθθρ-----------------------------------------------(2.24)
case)ratation (for or
rotation)non (for 0T)(
U)))(((
U))/2((
P))/((
U)u(U)(
2
^
1
^
^
j^^
i22
2
^
iTi
1
∫∫
∫
∫
∫
∫
∫∫
Ω∂Ω∂
Ω
Ω
Ω
Ω
ΩΩ
⋅+⋅
=⋅
+∇⋅∇⋅+
⋅+∇⋅∇
++∇⋅−
∇⋅+
dsindsin
gi
dAji
dAr
dAri
dAdt
ddA
Y
T
Ti
T
Ti
T
φφ
ρβφθ
φφµ
φφδφφµ
ϕφδφ
φρφρφφ
-------------------------------------------(2.25)
where ^i is unite vector of ith component, pε is the penalty parameter.
2.3.2 The boundary condition
(1) Temperature boundary condition:
21
The present study is concerned only with isothermal cylinders. The temperature
T at the surfaces of the cylinders for both the concentric and eccentric geometries is the
actual value which is applied to the boundary. For example:
301| =outerwallT K
304| =innerwallT K
(2) The velocity boundary condition:
Case 1: No rotation of the inner or outer cylinders
0|u i =Ω∂ --------------------------------------------------------------------------------------(2.26)
Case 2: with rotation of the inner or outer or both cylinders
Ω∂∈=⋅ 0nu ------------------------------------------------------------------------------(2.27)
2.3.3 Penalty Function Formulation
The penalty function methods can be derived directly from the Stokes viscosity
law (Fukumori and Wake, 1991);
u)32(pp s ⋅∇+−=
−µµ -----------------------------------------------------------------------(2.28)
where sp denotes the thermodynamic or static component of the pressure, p is the mean
pressure and −µ is the second coefficient of viscosity.
The basic idea of the penalty method consists in expressing the pressure through
the pseudoconstitutive relation:
upp s ⋅∇−= pε -------------------------------------------------------------------------------(2.29)
in which pε is a large number. Equation (2.29) is then substituted into the momentum
equations.
P)(U))/((1
2
^
∫∫ ΩΩ−=+∇⋅ dAdAri T
piT
iT ϕϕεφδφϕ
and the continuity equation is no longer necessary.
22
2.4 Investigated geometric and physical parameters
2.4.1. Various parameters for the finite element method
In the present study, finite element method is applied to solve the primitive
variable form of the incompressible Navier-Stokes equations with Boussinesq
approximation. Therefore, the basic parameters:
1) rρ Reference density
2) β Coefficient of thermal expansion
3) 'rT Reference temperature
4) rυ Kinematic viscosity
5) rα Thermal diffusivity
2.4.2. Various parameters for the finite difference method
The vorticity-stream function formulation in the curvilinear coordinate system
is taken as the governing equation. Therefore, the system parameters:
1) The radius ratio irr /0 . ,5/0 =irr 2.6,/0 =irr 1.25/0 =irr are studied so that the
effects of the radius changes can reflect the correct results.
2) The eccentricity ratio vector−re .
−re =
L
e− =(
Le
Le vh , )
−re = 1/3, 2/3 is considered.
For the eccentricity ratio vector−re , L= irr −0 is the ‘mean’ clearance of the
annulus.
3) The Rayleigh parameterνα∆β=
3LTgRa'
l .
For the Rayleigh number, the strength of the buoyancy force relative to the viscous
force is disposed. From the formula it is seen that the Rayleigh number is
proportional to 'T∆ .
23
4) The Reynolds number ν
Ω=
LrRe iil .
The rate of rotation of the inner cylinder is expressed as a Reynolds number based
on the surface velocity of the inner cylinder and the mean clearance L.
5) The fluid properties Pr = αν / .
The relevant fluid properties are ν and α , the kinematical viscosity and the thermal
diffusivity.
Because of its common occurrence in practice, the inner cylinder is considered
to be the hotter cylinder. And the two cylinders are kept isothermal. Different
parameters’ values are studied in the present paper and the effects of various system
parameters are investigated. At last, the heat transfer coefficients are produced
compared with that of Kuehn and Goldstein (1976).
Heat transfer coefficients
One of the primary objectives of the present study is to investigate how heat
transfer, both overall and local are affected by the variations of the above parameters.
The rate of heat transfer is expressed in terms of equivalent thermal conductivities as
defined below:
(i) Overall equivalent thermal conductivity eqK (concentric geometry)
eqK =)/(/2 '
io rrInTK∆πannulusthe of length unit per convertedenergy heat
Where the denominator is the heat transfer by pure conduction of a motionless
medium having the same thermal conductivity k as the fluid in a concentric annulus of
radius ratio io rr / . In this formula the eccentricity of the annulus geometry is ignored;
only the radius ratio is taken into account.
(ii) Overall equivalent thermal conductivity eeqK (eccentric geometry)
24
eeqK =
)(/2 ' SInTK∆πannulusthe of length unit per convertedenergy heat
where S=2222
2222
)()(
)()(
errerr
errerr
ioio
ioio
−−−−+
−−+−+
Here the denominator is the heat transfer by pure conduction of a motionless
medium having the same thermal conductivity as the fluid. The eccentricity of the
annulus is taken into account in the reference conduction term. When the annulus is
concentric, eeqK is equal to eqK .
(iii) Local equivalent thermal conductivity eqlK
Where r is either ir or or depending on the surface considered. The
denominator is the heat flux per unit’s area at the point on the surface if the heat
transfer were by pure conduction through a motionless medium having the same
thermal conductivity as the fluid in a concentric annulus of radius ratio io rr / . The
reference term of this formula, as in (i), does not take into account the annulus
eccentricity.
The heat transfer coefficient eqK was first defined by Beckmann (1931). The
use of reference conduction terms, in the denominators of (i) and (iii), which ignore the
eccentricity of the annulus, facilitates the comparison of heat transfer at different
eccentricities.
2.5 Boundary conditions in the finite difference method
2.5.1 Vorticity boundary conditions
From a given distribution of the Stream-Functionψ , the vorticity boundary condition is
evaluated directly from its definition:
25
wallwall |2ψζ −∇= ----------------------------------------------------------------------------(2.30)
Expression (2.30) in generalized orthogonal curvilinear coordinates and using the non-
slip flow condition at the wall of the cylinders, equation (2.30) becomes
)(112
2
2η
ξ
ηξη ηηψ
ηψζ
hh
hhhwall ∂∂
∂∂+
∂∂−= ----------------------------------------------------(2.31)
where η is constant along the wall and grid lines of constant ξ are perpendicular to the
wall. ξη hh and are the scale factors of the transformation.
(1) Concentric cylinders
For the concentric case, r and θ may be taken as ξη and respectively so that
1== rhhη and .rhh == θξ Equation (2.31) then becomes
wallwall rU
r|)( 2
2θψζ +
∂∂−= ------------------------------------------------------------------(2.32)
(2) Eccentric cylinders
For the eccentric case, from Appendix A, hchh =−== )cos/(cosh ζηηξ in the
Bipolar coordinate system. Equation (2.31) reduces to
wallwall h|1
2
2
2 ηψζ
∂∂−= ------------------------------------------------------------------------(2.33)
2.5.2 Stream-function boundary conditions
The Stream Function value oψ on the outer cylinder may be arbitrarily set to
zero. In the present study iψ is determined using the criterion that the pressure
distribution in the solution region is a single-valued function. Similar criteria were used
by Launder and Ying (1972) and Lewis (1979) for the numerical studies of isothermal
flows in non-simply connected geometries. Mathematically, this criterion implies that
the line integral of the pressure gradient sP
∂∂ along any closed loop circumscribing the
26
inner cylinder is zero i.e. ∫ =∂∂ .0ds
sP
sP
∂∂ can be evaluated from the momentum
conservation equations. The numerical implementation of this criterion is described in
the next chapter. The Stream-Function boundary conditions for both the concentric and
eccentric geometries are of the Dirichlet type as follows:
0| =outerwallψ ----------------------------------------------------------------------------------(2.34)
iinnerwall ψψ =| --------------------------------------------------------------------------------(2.35)
2.5.3 Temperature boundary conditions
The present study is concerned only with isothermal cylinders. The
dimensionless temperature T at the surfaces of the cylinders for both the concentric and
eccentric geometries is as follows:
0| =outerwallT
1| =innerwallT
27
CHAPTER 3 NUMERICAL SOLUTIONS
The finite difference method and finite element method were adopted to solve
the governing equations.
The finite difference method is a well-known method for solving the partial
differential equations. A good introduction to this method may be found in Ames
(1978). The practical applications of the method to problems in incompressible and
compressible fluid dynamics are described in great details in the very important work
of Roache (1972).
It is difficult for the finite difference method to solve the vorticity-stream
function formation with the moving boundary condition. Some special approaches such
as the single value condition are proposed to deal with the problem. Therefore, in the
present study finite element method is applied to solve the primitive variable form of
the incompressible Navier-Stokes equations with Boussinesq approximation.
The finite element method (FEM) is a numerical technique to obtain
approximate solutions to a wide variety of engineering problems. Although originally
developed to study the stresses in frame structures, it has since been extended and
applied to the broad field of engineering. The basic idea about the finite element
method can be found in Heinrich and Pepper (1999).
3.1 The finite-difference and finite-element approaches
3.1.1 The finite-difference method
Separate computer programs were written for the concentric and the eccentric
geometries. The system of mesh used in each case is the one natural to the particular
coordinate system.
28
Second-order accurate finite difference approximations were used for the
discretization of the governing equations whenever possible. The finite difference form
of equations (2.13) to (2.15) for the concentric and equations (2.17) to (2.19) for the
eccentric case were solved in a time-marching manner until satisfactory convergence
was attained. At each time the Stream-Function Vorticity equations must be solved to
convergence. Instead of second-order central differencing, upwind differencing was
used for the convective terms to obtain good stability.
3.1.2 The finite-element method
The basic idea of FEM is that a solution can be analytically modeled or
approximated by replacing it with an assemblage of discrete elements. Since these
elements can be put together in a variety of ways, they can be used to represent
exceedingly complex shapes. The FE discretization procedures reduce continuum
problems to one of a finite number of unknowns by dividing the solution region into
elements and expressing the unknown field variable in terms of assumed approximating
functions (or interpolation functions) within each element. The interpolation functions
(or shape functions) are defined in terms of the values of the field variables at specified
points called nodes (nodal points). The nodal values of the field variable and the
interpolation functions for the elements completely define the behavior of the field
variable within the elements. Once the nodal values are found, the interpolation
functions define the field variable throughout the assemblage of elements.
3.2 The solution procedure
3.2.1 The finite difference method
The solution process begins with the establishment of the necessary initial
values for ψζ , and T at time t=0. Other necessary parameters or constants that are
29
repeatly used in the program are also computed. The governing equations are solved in
a cyclic manner.
At the beginning of any particular cycle the time is increased by t∆ and the
distribution of ζ at the new time t=t+ t∆ is found by solving the Vorticity Transport
equation with boundary conditions obtained from the last known distribution of ψ .
With ζ known at all the interior points, the Stream-Function Vorticity equation is
solved in an iterative manner. The boundary value of ψ on the outer cylinder is always
zero. On the inner cylinder, iψ is found through an iterative procedure. From the latest
distribution of ψ , the velocity terms required in the convective terms of the Energy and
the Vorticity Transport equations are calculated. The next step in the cycle is to solve
the Energy equation for the temperature distribution T. Local and overall heat fluxes
may be calculated from the temperature distribution. The last step is to check if the
distributions ψ and T have converged and the energy balance is satisfactory. The
above cycle is repeated with increment in time t until the convergence criteria are met.
3.2.2 The finite element method
There are in general four different routes leading to a FE formulation: direct
approach, variational principle approach, weighted residual approach, and energy
balance approach. The advantage of the direct approach is that an understanding of the
techniques and essential concepts is gained without much mathematical manipulation.
But it can be used only for relatively simple problems. The variational principle relies
on the calculus of variations and involves extremizing a functional. Weighted residuals
approach has an advantage because it becomes possible to extend FEM to the problems
where no functional is available. The most applicable weighted residual approach is
Galerkin’s method. Regardless of the approach used to find the element properties, the
30
solution of a continuum problem by the FEM always follows a step-by-step process
described as follows.
•input properties Materials
ionspecificatcondition boundary and InitialgenerationMesh
processes-Pre
•
SolutionAssemblage
ncalculatioelement Boundary ncalculatioelement Internal
Processes
•solutions ofPlot
variablesderived ofn Computatioprocesses-Post
3.3 Finite difference methods for solving the equations
3.3.1 The detail methods
The governing equations are of two types, the time-dependent non-linear
parabolic type and the linear elliptic type.
(1) Parabolic equation
The time-dependent parabolic equations for the concentric and eccentric cases
are solved using the Alternating Direction Implicit method (abbreviated as ADI). The
ADI method, also commonly known as variable direction method, involves the splitting
of the time step to obtain a multidimensional implicit method which required only the
inversion of tri-diagonal matrices in the case of a rectangular solution region.
In the two-dimensional case, which is the primary concern of the present study,
this involves splitting the time step t∆ into two halves. The advancement of the
solution over t∆ is accomplished in two steps by solving in the θ -direction and then r-
direction. When second-order accurate central difference operators are used, the
method is second-order accurate in space and time. The ADI time-splitting of the
31
Vorticity Transport equation for concentric case, equation and eccentric case, is given
below:
)cos(sinPr)11Pr(
)()(1)(/)(2
21
2
2
22
2
21
21
θθθζ
θζζ
ζζθ
ζζζ θ
∂∂+
∂∂⋅+
∂∂+
∂∂+
∂∂
=+∂∂+
∂∂+∆−
+
++
nnnnn
unn
rnnu
nnr
unn
Trr
TRalrrrr
rUU
rU
rt
---------------(3.1)
And
)cos(sinPr)11Pr(
)()(1)(/)(2
21
2
2
211
2
2
121
121
1
θθθζ
θζζ
ζζθ
ζζζ θ
∂∂+
∂∂⋅+
∂∂+
∂∂+
∂∂
=+∂∂+
∂∂+∆−
+++
+++++
nnnnn
unn
rnnu
nnr
unn
Trr
TRalrrrr
rUU
rU
rt
----------(3.2)
)](sin)([cosPr)(Pr
)sinhsin()(1)(1)(2
2
2
2
5.02
2
5.05.0
ξηφ
ξηφ
ηζ
ξζ
ηξζζξ
ζη
ζζηξξη
∂∂−
∂∂+
∂∂+
∂∂•+
∂∂+
∂∂
=+−∂∂+
∂∂+
∆−
+
++
nnnnnn
nnn
nnu
nnunn
TATBTBTARalh
UUc
Uh
Uht
--(3.3)
And
)](sin)([cosPr)(Pr
)sinhsin()(1)(1)(2
2
12
2
5.02
2
5.015.01
ξηφ
ξηφ
ηζ
ξζ
ηξζζξ
ζη
ζζηξξη
∂∂−
∂∂+
∂∂+
∂∂•+
∂∂+
∂∂
=+−∂∂+
∂∂+
∆−
++
++++
nnnnnn
nnn
nnu
nnunn
TATBTBTARalh
UUc
Uh
Uht
(3.4)
Terms with superscript n are treated as known and taken on its last known
values at time nt . 21
+nζ obtained by solving equation (3.1) and other terms with known
values at nt are substituted into (3.2). The unknown in equation (3.2) are the 1+nζ .
These are solved for in the same manner as the 21
+nζ in equation (3.1). The ADI forms
of the other parabolic equations are similar to above equations, and are not shown.
The superscript u expresses a special form of differencing called upwind
differencing.
(2) Elliptic equations
32
Equation of the concentric case is solved with the Strongly Implicit Procedure
of Stone (abbreviated to SIP) introduced by H.L.Stone (1968). A series of ten
acceleration parameters generated by the method given in was used. The number ten
was arbitrarily selected.
One of the important merits of the SIP is that the rate of convergence is not so
sensitive to the choice of acceleration parameters. This means that suitable parameters
can be more easily and reliably estimated from the coefficient matrix than say in the
corresponding ADI method which requires accurate knowledge of the minimum
eigenvalues of a certain coefficient matrix to obtain good convergence rate.
For the eccentric configuration, the ADI method was favored because the
eigenvalues are allowed to be computed theoretically. See Birkhoff et al (1962) section
9 on the Helmholtz Equation in a rectangle. The Wachspress parameters, which are the
sequence of acceleration parameters used, can hence be obtained with great accuracy.
3.3.2 Boundary conditions
This section describes the numerical implementation of the boundary conditions
stated before. The boundary conditions are all of the Dirichlet type.
(1). Vorticity Transport Equation
First-order and second-order finite difference approximations were obtained
using Taylor expansion out of the walls. The first-order formulae were found to yield
slightly more accurate results. They are as follows:
Concentric Case
w
wwwww r
Ur
rU+
∆∆+−−
= +2
1
)()(2 ψψζ ---------------------------------------------------------(3.5)
Eccentric Case
221
)()(2
ηηψψζ
∆∆+−−
= +
w
wwwww h
hU------------------------------------------------------------(3.6)
33
where subscript w and w+1 refer to the values of the variables at the wall and one mesh
point away from the wall (in the fluid) respectively. wU is the tangential velocity of the
wall, which in the case of the outer cylinder wall is zero. The wU is determined from
the Reynolds number Rey1.
(2). Stream-Function Vorticity Equaiton
0=oψ
fi S=ψ
Assuming that a projected or guessed value of iψ called S is given, the
following steps are used to find a maxP∆ (maximum pressure difference term)
associated with S. This is done by first solving the equation ζψ −=∇ 2 with Si =ψ
and .0=oψ From the solution ψ , the pressure gradient terms θ∂
∂P (concentric case) or
ξ∂∂P are computed at all the interior points using the Momentum Balance equation (2.1)
in the respective coordinate systems. By integrating θ∂
∂P or ξ∂
∂P along all the closed
circumferential loops of constant r or η respectively (from πξθξθ 20 ==== to ), the
pressure difference terms of sP '∆ at all the loops are determined. A P∆ , among the
sP '∆ , which has maximum absolute value is selected as maxP∆ i.e. maxP∆ = P∆ such
that ≥∆ || P the absolute value of any sP '∆ .
A series of such S called ⋅⋅⋅⋅,,, 321 SSS etc. are projected at each time step and
their associated maxP∆ called ⋅⋅⋅∆∆∆ ,,, 3max
2max
1max PPP are computed. The first value 1S in
each time step is projected from Stream Function values 21 −− ni
ni andψψ at the previous
two time-steps. Depending on whether 1S is considered too high or too low a projection
34
(this can be seen from the sign of 1maxP∆ ), 2S is projected linearly from the relation
.||21|| 1
112−−=− n
iSSS ψ Subsequent values say lS are obtained by linear
interpolation between one pS and one qS (p,q<l) whereby pS has negative pPmax∆ and
qS has positive qPmax∆ ; both being of minimum absolute values in their respective sign
categories. In this manner a fS with almost minimum absolute || maxP∆ is obtained
after a sufficient number of projections are made and a convergence criterion applying
to consecutive value of the projection is satisfied. fS is then the value of iψ at time
step nt .
It would be very time-consuming if the Poisson equation ζψ −=∇ 2 is solved
for every projected value of iψ at each time step. Fortunately, by the linearity of the
equation, we need to solve ζψ −=∇ 2 only once at each time step. Using linear
superposition, if 'ψ is the solution to ζψ −=∇ 2 with 1Si =ψ , the solution to
ζψ −=∇ 2 with 2Si =ψ can be obtained by adding "12 )( ψSS − to 'ψ where "ψ is the
solution to the Laplace equation 02 =∇ ψ with 0.1=iψ . Solution to the last equation
are trivial in both coordinate system.
(3). Energy Equation
The two cylinders are maintained at isothermal condition, that is
0.1=iT and 0.0=oT
(4). Progressive built-up of boundary conditions
Solution can become unstable if the full value of a boundary condition is
suddenly imposed. To give a more gradual and stable start, temperature and velocity
boundary conditions are imposed over a number of time steps. The number of steps is
35
roughly proportional to the value of the boundary condition. The final solution is
independent of the number of steps used to introduce the boundary condition.
3.3.3 Convergence criteria
(1). Convergence of the inner iterations
Concentric case
For the concentric case where the SIP is used, a ‘Residue’ term ijsRe is defined
at each interior point as follows:
kijD
nij
kijs ψζ 2Re ∇+= ------------------------------------------------------------------------(3.7)
where ijD ∇ stands for the second-order finite difference operator of the Laplacian, n is
the global iteration number or time step and k is the inner iteration number. kijsRe is a
direct measure of the lack of convergence of kψ . The Stream Function distribution kψ
is considered to have converged sufficiently when
Nsrs
nrs
kij /]||[0005.0Re ∑< ζ ----------------------------------------------------------------(3.8)
for all interior points (i,j). N is the total number of interior points. In words, this means
that the residue at each interior point must be less than 0.0005 times the average
absolute vorticity value.
Eccentricity Case
Because the Stream Function frequently attains values close to zero, the
criterion that
δψψψ ≤≤≤≤−−−−++++ ||/|| kij
kij
kij
1 ------------------------------------------------------------------------(3.9)
(where δ is a small real number) holds at all interior points (i,j) is not practical. An
alternative criterion is used which involves first finding kpq∆ , the maximum absolute
36
difference between successive iterates kψ and 1+kψ , and the grid point (p,q) where is
occurs:
||max 1
),(
kij
kijji
kpq ψψ −=∆ + --------------------------------------------------------------------(3.10)
The criterion for the convergence of the inner iterations is that
0001.0)/|]|/([ <∆ ∑rs
krs
kpq Nψ --------------------------------------------------------------(3.11)
i.e. the maximum deviation between successive iterates is less than 0.0001 times the
average absolute Stream Function value. This criterion is satisfactory as long as ψ is a
reasonably well-behaved function which does not have highly localized peak values.
(2). Overall convergence
A run is deemed to have converged sufficiently when the following three
criteria are met:
a) 001.0)/|]|/([ <∆ ∑rs
nrs
npq Nψ
b) 001.0/|| 1 <−+ nij
nij
nij TTT
at all interior points (i,j).
c) 985.0/ ≥io QQ
where Q stands for heat flux through the cylinders.
The convergence of the Vorticity distribution is not included as a convergence
criterion because it is directly related to the Stream Function through its definition. The
convergence of the Stream Function ψ implies the convergence of the vorticity
distribution through the criterion may be different.
The degree of heat balance is included as a convergence criterion because it is a
fairly good barometer of the accuracy of the solution and the sufficiency of the mesh
size. It was found that the temperature solution usually converged more slowly than the
37
Stream Function. The exceptions were runs at high Reyleigh numbers where the lack of
convergence of the Stream Function indicated the presence or onset of flow instability.
It is not always possible to satisfy the heat balance criterion though qualitatively
it is clear the solution has already converged to a far greater extent than required by
both criteria (a) and (b). This happens predominantly in cases where the inner cylinder
is moved eccentrically downwards. In these cases, the mesh at the top of the annulus is
very coarse, while at the same time the temperature gradient at the top of the outer
cylinder is very high because of the impinging thermal plume. For such cases, even
interpolation is not sufficient to resolve the extremely high gradient encountered.
3.4 Finite element methods for solving the equations
3.4.1 The details solving procedures
Once the form of shape functions of θϕφ and , is specified, the integrals
defined in the equations (2.23) to (2.25) can be expressed by the matrix equations. The
momentum and energy equations may be combined into a single global matrix
equation.
=
×
+
+++
•
•
0F
TU
LU)(D0
BEEM1KA(U)
TU
N00M
TT
T1-p
pT
ε ------------------------------(3.12)
Note that in constructing the above element matrix equation, the penalty formulation
has been applied, and P in the momentum equation is substituted by T1-p EEM1
pε. The
coefficient matrices in the above equation are defined by:
∫Ω= dATϕϕPM ∫Ω= dATρφφM
38
∫Ω ∇⋅∇= dAk TθθTL ∫Ω ∇⋅= dAC pT
T uU)(D θθρ
∫∫ ΩΩ∇⋅∇⋅+⋅+∇⋅∇= dAjidAr T
ijT
iT ))(())/2((K
^^2
2ij φφµδφφδφφµ
∫Ω= dAC TpθθρTN ∫Ω +∇⋅= dAri T
i ϕφδφ )/(E 2
^
i
∫Ω ∇⋅= dATuA(U) φρφ ∫Ω ⋅= giT^
B ρβφθ
∫ Ω∂⋅= dsn F φτ
where U is a global vector containing all nodal values of u and v. The assembled global
matrix equations are stored in the skyline form and solved using the Gaussian
elimination method. The successive substitution method is applied for nonlinear
iteration and the time derivatives are approximated using the implicit finite difference
scheme.
3.4.2 The shape function
The element can be chosen from any of the four node, eight node and nine node
elements (fig. 3.1.1). The shape function of quadrilateral elements (4-9) nodes may be
described in Table 3.1.
39
CHAPTER 4
ANALYSIS OF RESULTS FOR THE FINITE ELEMENT METHOD
The numerical results from the finite element method will be analyzed and
discussed in this chapter. The numerical model from the finite element method
produces the computational results of the velocity and temperature fields, which show
the detailed flow fields and temperature distributions.
The computational algorithm developed above is capable of predicting the
temperature distribution, the streamline distribution and internal convection in the
horizontal annular between two circular concentric or eccentric cylinders. A set of
computed results is shown for air, whose physical properties are given in Table 4.1.
The mesh independence testing procedure for the computation is considered here and
the final mesh used for the computation is determined so that any further refinement of
the mesh produces an error smaller than 0.1%. Finally, 221 nine-node elements are
adopted to calculate the thermal and fluid flow for half mode, which is used to simulate
the natural convection without both inner and outer cylinders’ rotation, and 578 nine-
node elements for whole mode with the cylinders rotating. The penalty formulation is
used for the pressure field computation. A convergence criterion of 1×10-3 is set for
relative error associated with unknowns for temperature and velocity fields.
The effects of Rayleigh number, radius ratio and eccentric ratio on the flow and
heat transfer are separately discussed in sections 4.1, 4.2 and 4.3. Section 4.4 and 4.5
emphasized the effect of the rotation. Section 4.4 focuses on the effect of rotating the
inner, outer and both cylinders between the concentric cases and section 4.5 is set for
the eccentric cases. The air is selected as the working fluid, which is used for the
numerical studies of the present investigation. The present finite element method is
40
validated by comparing its numerical results with the available data in the literature and
very good agreement has been achieved.
41
4.1 The effect of Rayleigh numbers
The effect of different Rayleigh number is shown in Fig 4.1.1 – 4.1.5. Half
model is adopted given the symmetry of the concentric annuli. In Fig. 4.1.1, a
symmetric pair of counter-rotating kidney is set in the annuli. However, the temperature
distribution (see Fig. 4.1.1(a)) seems like that of the natural convection from a heated
circular cylinder. The temperature contour suggests that thermal transport in the annuli
is primarily due to conduction. Temperature inversion appears to occur when the
Rayleigh number is above 104. Fig. 4.1.2-4.1.5 show that the inversion is gradually
becoming greater with increasing Rayleigh number.
There are two approaches to increase Rayleigh number, enlarging the radius of
cylinders or temperature difference between inner and outer cylinders. Fig. 4.1.2 and
4.1.3 separately depict the temperature, streamline and velocity at the Rayleigh number
of 9×104. Compared with each other, little difference is found so that Rayleigh number
is the major parameter in the low Rayleigh number.
The section also discussed when the unstable flow set in as a form of two-
dimensional oscillatory flow about the longitudinal axis. The numerical solution is
reasonable in Rayleigh number of 3.09×105. However, the streamline plots and the
temperature contours yield oscillation, therefore, there is some hint that the unstable
flow appears. When the Rayleigh number is 3.85×105, one zero streamline is yielded in
the old stagnant area. It may be deduced that for Rayleigh numbers about 1.1×105, the
time-averaged flow behavior and heat transfer of the actual physical flow between the
two cylinders are not very different from those of a hypothetical two-dimensional flow
without the oscillatory motion, and this lends credence to the possibility that the
numerical method may yet yield a reasonable solution. From Fig. 4.1.4 and Fig. 4.1.5,
42
the highly distorted form of the isotherms and streamlines manifests the unstable nature
of the flow at so high Rayleigh numbers.
The experimental investigation of Powe et al. (1969) described that for radius
ratios greater than about 1.8, instability at high Rayleigh number initially sets in as a
form of two-dimensional oscillatory flow about the longitudinal axis. The flow in a
plane perpendicular to the longitudinal axis exhibits the same kidney-shaped pattern
and the patterns in adjacent planes oscillates out of phase with each other resulting in
an apparent wave motion along the axis of the annulus. At the Rayleigh number of
5103× , these oscillations were still only intermittent and presumably of small
amplitude. Above the Rayleigh number of about 6105.1 × there was no definite
frequency in the oscillation and the plume became turbulent at the Rayleigh number of
about 7102 × .
In present the numerical results show that unstable phenomena began to be
yielded between Rayleigh number 3.09×105 and 3.85×105. One circular flow at the
stagnation area of the low Rayleigh number is shown from Fig. 4.1.5 (c). The present
numerical results agree well with experimental results obtained by Powe et al.
4.2 The effect of the radius ratio
This section focuses on the effect of the radius ratio. This part only emphasizes
the effect of the radius ratio of 2.6 and 5.0 from the finite element method in the high
Rayleigh number. The computed results for radius ratio of 2.6 and 5.0 are shown in Fig.
4.2.1 – 4.2.2.
The radius ratio has a significant effect on the flow and temperature distribution
between horizontal cylinders. Fig. 4.2.1 depicted temperature and flow fields of radius
ratio 2.6 and 5.0 cases at the Rayleigh number of 1.5×105. From Fig. 4.2.1 (b) and (e),
43
the center of kidney cell moved down and the stagnant zone becomes larger when the
radius ratio is enlarged from 2.6 to 5.0 at the same Rayleigh number.
It is easy for the higher radius ratios to yield temperature inversion. In
comparison of computed results between Figure 4.2.1 (a) and (d) it is apparent that the
higher radius ratio has a relatively greater temperature inversion. The stagnant
temperature zone is increased further for the larger radius ratios of 5.0. The major
reason is the boundary layer flow at high radius ratios has a higher momentum which
enables it easily to distort the isotherms to create the effect of temperature inversion.
The main conclusion is that the convective heat transfer is favored at higher radius
ratio. The convective flow around the inner cylinder increases rapidly as the radius ratio
is increased.
The instability of the two-dimensional oscillatory type sets in at higher Rayleigh
number for higher radius ratio. From Figure 4.2.2 (b), the strong oscillation appears
between the two horizontal cylinders, however, the temperature distribution has no
great change. Therefore, it may be deduced that for Rayleigh numbers between 5×105
and 1×106, the heat transfer of the actual convective flow between the two cylinders are
not different from those of a hypothetical two-dimensional flow without the oscillatory
motion, and this lends credence to the possibility that the finite element method may
yet yield a reasonable solution for the radius ratio 5.0.
4.3 The effect of eccentricity
The effects of eccentricity between the two horizontal cylinders were
numerically studied for eccentric ratio 0.75 and 0.5. In this case of a natural convection
in annulus between eccentric cylinders, the cylinder’s intracenter line is defined as the
line to connect the centers of the outer cylinder and the inner cylinder as shown in Fig.
44
1.1.1. For the case of eccentric ratio 0.75, Fig. 4.3.1 and 4.3.2 show that the
temperature and fluid flow between eccentric cylinders, which intracenter line is
parallel to the gravity vector. This case is very similar to the concentric case in such a
way that the flow is symmetric with respect to the intracentral line, and only half of the
field can be considered as the computational domain. The computational study of the
problem when the intracentral line is inclined with respect to the gravity vector is
relatively complicated so that Fig. 4.3.3 depicted two kinds of situation whose
intracentral line has degree 45o and –450 against the horizontal line.
The immediate effect of moving the inner cylinder upwards (Fig. 4.3.2) is to
enlarge the size of the stagnant zone and reduce the size of the of the kidney flow cell.
At high eccentric ratio 0.75, Fig. 4.3.2(b) depicted that one circular flow cell appeared
to occur at the bottom of the annuli and the kidney cell is squashed and connected to
the circular flow. In Fig. 4.3.2(a), the thermal plume is squashed to create a pair of
symmetrical humps. This indicates that the thermal boundary layer on the inner
cylinder has begun to separate from the cylinder before reaching the top of the annulus.
There is no district thermal plume as in other cases. A reduction of convective activities
can be expected. Moving the inner cylinder downwards (Figure 4.3.1) has the quite
opposite effect. The stagnant zone is reduced and convection is increased.
In Fig. 4.3.3, the Rayleigh number is about 4.95×104 and the eccentricity ratio is
0.5 with 6 k temperature difference between inner and outer cylinders. Figure 4.3.3 (a-
c) shows temperature, streamline and velocity plots for moving the inner cylinder to
angular 450 against the horizontal line, which has characteristics like that of moving the
inner cylinders both right and up. Fig. 4.3.4 (1) and (3) depict the temperature
distribution along the shortest and longest lines between the inner and outer cylinders
for the case moving the inner cylinder to angular 450 in the anti-clockwise direction.
45
Compared with the concentric case, the thermal plume is a little shifted to the left side
and reduced; the kidney shaped flow cell on the left side is enlarged and that on the left
side is squeezed to very small size; A large area where the temperature gradient is very
small appears in the left down side, which can be viewed in Fig. 4.3.3 (a) and found in
Fig. 4.3.4 (3). Corresponding to the small temperature gradient close to the center of
the left kidney cell, a small fluid flow is shown in the velocity plot in Fig. 4.3.3 (c).
Fig. 4.3.4 (1) describes the temperature distribution in the shortest distance line
between inner and outer cylinders, along which a sharp temperature gradient exists.
On the other hand, moving the inner cylinder to angular –450 produced a larger
thermal plume and smaller stagnation zone compared to that to angular 450. Fig. 4.3.4
(4) and (2) also shows separately a flat temperature distribution and a large
temperature gradient along the longest distance line and the shortest distance line
between inner and outer cylinders.
4.4 The effect of rotating the inner and/or outer cylinders between concentric cases
This section focuses on investigation of the effects of the rotation between
horizontal concentric cylinders, which include inner cylinder rotating, outer cylinder
rotating and both of them rotating in the same and opposite direction. The Rayleigh
number is kept 4.95×104 and the eccentricity ratio is 0.5 with the inner and outer
cylinders’ temperature being separately 307 K and 301 K. The tangential velocity is set
to 0.1 m/s for only one cylinder rotating and 0.05 m/s for both of them rotating. In this
section, the numerical results and the physics of the flow as revealed by the numerically
obtained streamline and isothermal plots and temperature distribution along the selected
lines are discussed.
46
The numerical results in Fig. 4.4.1, show pictorially the flow and temperature
fields when the inner cylinder is rotated (a-b) and the outer cylinder is rotated (c-d).
The rotation considered is both in the clockwise direction. The immediate effect of
rotating the inner cylinder is to cause the fluid adjacent to the surface of the rotating
cylinder to move with it by the virtue of viscous drag as clearly illustrated in Fig. 4.4.1
(b) and (d). For the non-rotating concentric case, the hot fluid adjacent to the inner
cylinder rise upwards because of its reduced density and then it is cooled at the top of
the annulus so that the cooled fluid falls sideways along the outer cylinder wall. When
the inner cylinder is rotated in the clockwise direction, the left-hand hot-fluid flow’s
velocity has the same velocity direction as the rotating cylinder. Therefore, the left-
hand cell is elongated in size while the right-hand cell is reduced for the inner rotating.
The circulation, defined as the integral ∫ ⋅ dsu , of the right-hand is thus reduced while
that in the left-hand is increased. On the other hand, rotating the outer cylinder in the
clockwise induces the right-hand cooled fluid with the same velocity direction
increasing so that it has a large right-hand cell and small left-hand cell. The isotherm
plots Fig. 4.4.1 (a) and (c) show that the thermal plume and the stagnant zone are
shifted in the direction of the rotating. Corresponding to the non-rotation case, the
thermal plume with the inner rotating is driven to an angular position of Φ = 800
(clockwise) from the highest point of the inner cylinder while the lower temperature
stagnant zone is only changed a little; the outer cylinder rotating caused the stagnant
zone approaching to Φ = 900 (clockwise) from the lowest point of the outer cylinder
with slightly shifted thermal plume, which is clearly depicted in Fig. 4.4.1 (a) and (c).
The temperature distribution along the line with different angle against the horizontal
line from the inner cylinder to the outer cylinder is shown in Figure 4.4.2. The higher
temperature along line 1 (Φ = 00) compared with the lower line 2 (Φ = 900) and the
47
nearly same temperature of line 3 and 4 numerically reveals the large thermal plume
shifting and small lower temperature stagnant zone shifting for inner cylinder rotating.
On the other hand, line 5, 6, 7 and 8 described the outer cylinder rotating case.
The effects of rotating both the inner and outer cylinders with the same and
opposite directions on the flow and heat transfer for the concentric configuration is
pictorially and numerically shown in Fig. 4.4.3 and 4.4.4. Fig. 4.4.3 (a-c) illustrate the
temperature, the streamline and the velocity with the same rotating direction, which
temperature distribution seems like that of only inner cylinder rotating except that the
low temperature stagnant zone move further away from the bottom part. The hot fluid
flow on the left-hand cell impinge into the right side with the help of the rotating inner
cylinder; the cooled fluid flow on the right-hand cell spreads around most area of the
outer cylinder (see Fig. 4.4.3 (b)). Lines 1-4 in Fig. 4.4.4 show the temperature
distribution along lines shooting from the inner to the outer cylinder with angular
position Φ = 00, 900, 1800 and 2700 against the horizontal line. Various parameters’ plot
for the inner and outer cylinder rotating in the opposite direction is shown in Fig. 4.4.3
(d-f). The root of the thermal plume dives into the right-hand part because of the drag
force from the rotating inner cylinder while the top of the thermal plume driven by the
viscosity of the rotating outer cylinder is shifted to the left-hand side. The low
temperature stagnant zone is moved slightly up to the right-hand. Fig. 4.4.3 (e) shows
that the left-hand cell is largely elongated that it will engulf the right cell because the
viscosity of both the inner and outer cylinders enlarges the left-hand cell.
4.5 The effect of rotating the inner and/or outer cylinders between eccentric cylinders
This section numerically simulates the natural convection in the annulus
between the vertically eccentric cylinders (the cylinder’s intracenter line is parallel to
48
the gravity vector) with both inner and outer cylinders rotating. The Rayleigh number is
kept at 4.95×104 and the inner and outer cylinders’ temperature is respectively at 307 K
and 301 K. The tangential velocity is set to 0.05 m/s for both of them rotating. Several
cases with different eccentric ratios (0.5 and 0.7) are discussed in the section. The close
proximity of the outer cylinder at the narrow throat exerts a great influence on the fluid
flow and temperature distribution in the annulus.
In the case of moving the inner cylinder up at the eccentric ratio of 0.5, the
narrowness of the throat region makes it difficult for the thermal plume to go through it.
Fig. 4.5.1 (a) and (d) respectively show the temperature of both cylinders rotating with
the same and opposite directions. Since the narrowness of the throat region reduces the
thermal plume, only the low temperature stagnant zone is moved left or right with the
outer cylinder rotating. Fig. 4.5.2 shows that the temperature along the shortest line and
longest shooting from the inner and outer cylinder with the same cylinders’ rotation
direction is similar to that with opposite direction. Therefore, the right-hand cell evaded
and occupied the stagnant zone when both cylinders rotate in the same direction and
shrank and left the stagnant zone when both cylinders rotate in the opposite direction. It
is interesting to note that the flow in the right-hand cell (see Fig. 4.5.1 (b)) is cut off
from the outer cylinder when the inner cylinder has different velocity as the outer
cylinder and the left-hand cell (see Fig. 4.5.1 (e)) is cut off from the inner cylinder
when both cylinders rotate in the same direction.
Similar temperature and fluid flow fields at the eccentricity ratio of 0.75 are
given in Fig. 4.5.3. The velocity of both inner and outer cylinder is 0.1 m/s. The narrow
throat plays a further important role on both the flow and temperature in the annulus.
From Fig. 4.5.3, the rotation of the outer cylinder exerts a great influence. Compared
with the results at the eccentricity ratio of 0.5, the faster rotated outer cylinder (0.1m/s)
49
associated with the further narrow throat decided the flow pattern. The inner cylinder is
set to rotate in the clockwise direction. If the outer cylinder rotates in the clockwise, the
right kidney cell will occupy the whole computed space except the small area close the
narrow throat. On the other hand, the anti-clockwise rotation of the outer cylinder
produces the opposite activities. Corresponding to the flow pattern, the low temperature
stagnant zone follows the rotation direction of the outer cylinder, which is shown in
Fig. 4.5.3 (a) and (d).
In the case of rotating the inner cylinder down at the eccentric ratio of 0.5,
moving the inner cylinder vertically down without rotation produces an increase in the
size of the rotating cells accompanied by a reduction in the size of stagnation zone. The
thermal boundary layer on the outer cylinder is increased in length from that of the
concentric case. When the rotation is applied to both cylinders, the out layer of the
thermal plume is moved with the outer cylinder and the root of the thermal plume is
shifted by the inner cylinder. The narrow throat is moved downwards, which reduces
the effect of the outer cylinder’s rotation and increases the influence of the inner
cylinder’s rotation because it is difficult for the particles of the cooled fluid driven by
the outer cylinder to pass the smaller stagnant part. The temperature and fluid flow
fields of both cylinders rotating in the same and opposite direction are respectively
presented in Figure 4.5.4 (a-c) and (e-f). Corresponding to Fig. 4.5.2, Fig. 4.5.5
numerically shows the temperature distribution along the shortest line and longest line
shooting from the inner cylinder to the outer cylinder.
50
CHAPTER 5
ANALYSIS OF RESULTS FOR THE FINITE DIFFERENCE METHOD
The chapter solves the numerical model through using finite difference method.
The numerical model from the finite difference method yields details of the flow and
the temperature fields through the streamline and isotherm plots and then computes the
local heat transfer values to predict the overall heat transfer coefficients. These plots
help to show the physics of the flow and the heat transfer behavior. All the isotherm
and streamline plots in this chapter are drawn by the software Tecplot 8.0, so that one
grid line can not be deleted because of the mesh divisions.
The overall heat transfer coefficients yielded by the numerical model at the
radius ratio of 2.6 are compared with similar results of other investigators. The good
agreement is a confirmation of the validity of the numerical model (see Fig. 5.0.1).
The effects of Rayleigh number, radius ratio and Reynolds number on the flow
and heat transfer between concentric cylinders are discussed in Sections 5.1, 5.2 and
5.3 respectively. Section 5.4 focuses on the heat transfer and fluid flow between
eccentric cylinders without inner cylinder rotating and Section 5.5 for eccentric
cylinders with inner cylinder rotating. In these sections, the discussions of the heat
transfer results are preceded by an analysis of the physics of flow as revealed by the
streamline and the isotherm plots. The working fluid with a Prandtl number of 0.701
(air) is assumed for the numerical studies of the present investigation.
51
5.1 The effects of Rayleigh numbers
At low Rayleigh numbers, the rate of convection of heat energy from the inner
cylinder to the outer cylinder is low because the buoyancy force proportional to the
temperature difference is very small and the induced flow velocity is low. At
2102×=lRa (Fig. 5.1.1 (a)), the isotherm plots are almost concentric circles like those
of pure conduction since the convective flow velocity is small. However, the counter-
rotating pair of kidney-shaped vortices (Fig. 5.1.2 (a)) is already clear. The overall heat
transfer coefficient for Rayleigh numbers up to just a few thousands is close to 1.0,
indicating that the heat transfer is predominantly conductive. Though the convective
flow exits at low Rayleigh numbers, the overall heat transfer is dominated by the
conduction. The temperature in the annuli falls monotonically, shooting outwards.
With the Rayleigh number increasing, the increased flow velocity on the
surfaces of both cylinders caused by the increased buoyancy results in the distortion of
the hitherto circle-like isotherms. At 410=lRa (Fig. 5.1.1 (c)), the isotherm plots have
lost its circular forms. As the Rayleigh number is increased to be strong enough, a
situation of temperature inversion occurs in which cooler fluid is separated from the
cooling surface by a layer of warmer fluid. The ‘extent’ of inversion is greater at higher
Rayleigh numbers. True boundary layer flow occurs for Rayleigh numbers above
4103× . As temperature inversion develops with increasing Rayleigh number, the shape
of a thermal plume begins to form at the top of the inner cylinder. By 4105× , a well-
defined thermal plume can be seen clearly (Fig. 5.1.1 (d)). The thermal plume becomes
narrower as the Rayleigh number is increased. The streamline plots of Fig. 5.1.2 reveals
that the kidney-shaped cells become progressively elongated and the center of rotation
move higher up though further increase in the Rayleigh number merely further
distorting the isotherms. At the Rayleigh number of 510 , the centers are about 030±
52
from the upward vertical. The thermal plume becomes narrower as the Rayleigh
number is increased. A pair of very weak counter-rotating cells is formed at the
stagnation zone of the bottom of the annulus. The effect is manifested by two additional
zero-valued streamlines. These cells are formed as more fluid is being drawn from the
sides of the annulus at high Rayleigh numbers.
The experimental investigation of Powe et al. (1969) manifested that for radius
ratios greater than about 1.8, instability at high Rayleigh number initially sets in as a
form of two-dimensional oscillatory flow about the longitudinal axis. The flow in a
plane perpendicular to the longitudinal axis exhibits the same kidney-shaped pattern
and the patterns in adjacent planes oscillates out of phase with each other resulting in
an apparent wave motion along the axis of the annulus. At the Rayleigh number of
5103× , these oscillations were still only intermittent and presumably of small
amplitude. Above the Rayleigh number of about 6105.1 × there was no definite
frequency in the oscillation and the plume became turbulent at the Rayleigh number of
about 7102 × . It may be deduced that for Rayleigh numbers between 510 and 610 , the
time-averaged flow behavior and heat transfer of the actual physical flow between the
two cylinders are not very different from those of a hypothetical two-dimensional flow
without the oscillatory motion, and this lends credence to the possibility that the
numerical method may yet yield a reasonable solution. From Fig. (5.1.1(f)) and Fig.
(5.1.2(f)), the highly distorted form of the isotherms and streamlines manifests the
unstable nature of the flow at so high Rayleigh numbers.
5.2 The effects of Radius ratio
This section focuses on the effects of radius ratio on the flow and the heat
transfer in concentric annulus for radius ratios up to 5.0. The higher radius ratios make
53
the effects of the outer cylinder so small that the flow and the heat transfer around the
inner cylinder would be almost close to those of a free convecting cylinder. The three
radius ratios of 1.25, 2.6 and 5.0 were investigated for the flow and temperature
distribution.
The radius ratio is one of the important parameters for measuring the heat
transfer and the flow between the horizontal annuli. When the radius is low, free
movement of the fluid is more difficult and the conductive mode of heat transfer
becomes dominant, whereas at the large radius ratios, the heat transfer from the inner
cylinder approaches that of natural convection from a heated horizontal cylinder.
The isotherm and streamlines for radius ratio of 1.25, 2.6 and 5.0 at various
Rayleigh numbers are described in Figs. 5.2.1-5.2.3 and in Figs. 5.1.1-5.1.2. For the
larger radius ratios and the smaller radius ratios, the convection and conduction
dominate the flow differently. Convective heat transfer which is dependent on the
velocity of the flow over the surface is suppressed at the small radius ratios (1.25)
because the proximity of the walls forces the opposite-direction flows on the two
cylinders to interact strongly with each other (through viscosity) and slow each other
down. Therefore, the isotherms remain more nearly circular up to a higher Rayleigh
number. At high radius ratios, the flows on the cylinders are separated by greater
distances and do not affect each other that the convection have a great effect. The
boundary layer flow at high radius ratios thus has a higher momentum which enables it
easily to distort the isotherms to create the effect of temperature inversion. It is thus
easy for the higher radius ratios to yield temperature inversion.
5.3 The effect of the inner cylinder rotating
54
The effects of the rotation of the inner cylinder on the flow and the heat transfer
in concentric annular spaces are the subjects of this section. The computed results are
for the radius ratios of 2.6, and as in the previous section, a Prandtl number Pr of about
0.7 is assumed. The actual numerical results and the physics of the flow as revealed by
the numerically obtained streamline and isotherm plots are discussed.
Figs. 5.3.1-5.3.2 show the streamline plots and the isotherm plots when the
inner cylinder is rotated with increasing speed at a fixed Rayleigh number of 4105.5 × .
The fluid adjacent to the surface of the inner cylinder moves with the inner cylinder for
the virtue of viscous drag yielded by the rotating inner cylinder. Rotating the inner
cylinder has the effect on rising the hot fluid on the ascending side of the cylinder while
on the descending side, the rising hot fluid is slowed down. Near the surface of the
inner cylinder on the descending side is a small region of very low flow velocity. The
effect of rotation on the flow field at the Reynolds number (Reyl=140) is depicted in
Fig. 5.3.1(a). The right-hand cell is elongated while the left-hand cell is reduced. The
circulation, defined as the integral ∫ −−⋅ dsU , of the left-hand is reduced while that in the
right-hand cell is increased. Since the zero-value streamline has lifted off the surface of
the inner cylinder, the thermal plume is shifted in the direction of the rotation which is
depicted in the isotherm plot Fig. 5.3.2 (a). The point of impingement on the outer
cylinder is correspondingly moved. The right-hand cell grows further in size while the
left-hand cell becomes smaller and weaker with increasing the angular speed to the
Reynolds number of 280. The left-hand cell is now separated from the inner cylinder.
Fig. 5.3.1 (b) showed that there is a zone of Couette-like flow around the inner
cylinder. Fig. 5.3.2 (b) depicted that the thermal plume has shifted even further in the
direction of rotation and its origin or ‘root’, the region near to the surface of the inner
cylinder where the thermal plume rises from, is lightly below the horizontal level of the
55
center of the inner cylinder. On the outer cylinder, the point of plume impinge has
shifted further down and temperature gradient has dropped.
With increasing Reynolds number, the right-hand cell elongates further as the
left-hand cell continues to shrink in size. When the Reynolds number rises to 416, the
right-hand cell is nearly close to its maximum length. Therefore, the growing ‘Couette-
flow’ region around inner cylinder encroached the cells. At the Reynolds number of
800 and 1120, the kidney-shaped eddy has been changed into a layer of very weak flow
pressed against the left-hand surface of the outer cylinder. At the Reynolds number of
1120, the ‘Couette-flow’ zone has stretched to a third of the clearance. This can been
seen in Fig. 5.3.1 (f).
The gradual disappearance of the thermal plume can be seen from the Fig. 5.3.2
(c) to 5.3.2 (f). The plume becomes less distinct and the ‘root’ of the plume, a region of
relatively low temperature gradient is extended when the Reynolds number is
increased. Therefore, this results in a rapid reduction of the temperature gradient around
the inner cylinder. From the Reynolds number of 560 to 800, increased numbers of
nearly circular rings of isotherms are formed around the inner cylinder. At the Reynolds
number of 1120, the isotherms away from the inner cylinder almost become circular,
and the temperature they represent is approaching that of pure conduction. However,
this situation is unlike in real flow because it is known that the two-dimensional
circular Couette flows at high Reynolds numbers are unstable.
5.4 The effects of eccentricity
Eccentricity has a great effect on the flow and heat transfer behavior in the
horizontal annulus. The flow field is in fact strongly influenced by the geometry of the
annulus as well as its orientation. The effects of eccentricity on the heat transfer were
56
studied at the radius ratios of 2.6 using the numerical model of the finite difference
method. The present study focuses on the eccentricities in the angular ±450 against the
horizontal direction.
The streamline and isotherm lines at the eccentricity ratios re of 1/3 and 2/3
and the radius ratio of 2.6 are depicted in Figs. 5.4.1-5.4.3. The Streamline plots
depicted that the immediate effect of moving the inner cylinder upwards is to yield a
reduction in the size of the two counter-rotating flow cells and to increase the size of
the stagnation region at the bottom of the annulus. The thermal boundary layer on the
outer cylinder is reduced in length from that of the concentric case. The increase of
conduction can be predicted. Moving the inner cylinder downwards has quite the
opposite effect. However, For eccentricity ratio 31≤re , the inner cylinder seem to be
unaffected by its own position. At the upward eccentricity )32,0(=
−re , the isotherm plots
depict that the isotherms around the inner cylinder yield a pair of symmetrical humps.
These indicate the inner thermal boundary layer has begun to separate from the cylinder
before reaching the top of the annulus. There is no distinct thermal plume as in the
other cases. At the downward eccentricity )32,0( −=
−re the outer cylinder has a thermal
boundary layer almost except near the bottom of the annulus. When the Rayleigh
number is increased to 510 , there are two smaller humps instead of the usual one big
hump, and the rising warm fluid in the two plumes yields a pair of relatively weak
counter-rotating vortices at the top of the annulus. A pair of very weak cells is formed
by viewing the zero-streamline in the stagnation zone. Moving the inner cylinder seems
only to cause one of the kidney-shape cells to be squashed in size and the other to grow
in size. The size of the stagnation zone is relatively unaffected. The thermal plume is
inclined towards the wider side of the annulus, which can be found in the isotherm
57
plots. However, as the increase of the Rayleigh number, the thermal plume tilts towards
the side of the narrow throat.
The effect of moving the inner cylinder to angular 450 against the horizontal
line (x axis see Fig. 1.1.1) at the eccentricity )32( =re has the characteristics like moving
the inner cylinder both upward and right. The size of the stagnation region at the
bottom of the annulus is increased when the size of the top region is reduced. However,
there is still a distinct thermal plume inclined towards the wider side of the annulus.
From the streamline plots, one cell appears in the kidney-shaped cells near the narrower
throat of the annulus. This is one weak cell. The effect of moving the inner cylinder to
the angular -450 at the eccentricity )32( =re has the characteristics like moving the inner
cylinder both downward and right. The size of the stagnation region at the top region is
increased when the size of the stagnation region at the bottom of the annulus is reduced.
The outer cylinder has a thermal boundary layer that occupied great space except the
narrower throat of the annulus. (Fig. 5.4.3)
5.5 The effects of rotating inner cylinder in the eccentric annulus
Fig. 5.5.1 and Fig. 5.5.2 displayed the streamline and isotherm plots
respectively at various eccentric positions with the eccentricity ratio re of 1/3. The
adopted Reyleigh number and Reynolds number are 5×104 and 140 respectively.
Compared with the effects of the rotation for the concentric case, it is found that the
heat transfer between the two cylinders is strongly determined by what happens in a
small region around the inner cylinder, the ‘Couette-flow’ region. From the streamline
and isotherm plots at the eccentricity ratio re of 1/3, the flow and the temperature
distribution adjacent to the inner cylinder surface are close to those at the concentric
58
position. The amount of circulating flow through the annulus, as measured by the
Stream Function value on the inner cylinder, has fallen from its corresponding
concentric value at the same Reynolds number. This is maybe due to the narrower
throat in the annular region that present an obstruction to a free flow. Compared with
the flow in the concentric annulus, the thermal plumes are shifted to a slightly lesser
extent. The ‘root’ of the thermal plumes at all the eight eccentric positions appears at
about the same location on the inner cylinder.
For the eccentricity ratio of 2/3 (Fig. 5.5.3-5.5.4), the effect of rotating the inner
cylinder is more significant than the eccentricity ratio of 1/3. The narrow throat
between the inner and outer cylinders exerts a great influence on both the flow and
temperature distribution around the inner cylinder. The amounts by which the ‘root’ of
the thermal plume has moved are different at all the eight eccentric positions, whereas
they were about the same at the eccentricity ratio of 1/3. At the eccentricity ratios
)0,32(−=
−re , the right cell is enlarged significantly so that it reaches most of the annulus
and the left cell is reduced to only a small region near the narrow throat. The
narrowness of the throat region makes it difficult for the thermal plume to go through
the narrow throat. At the eccentricity ratios )32,0( −=
−re , compared with rotating inner
cylinder in the concentric case, the thermal plume that is shifted by the rotating is close
to that of the concentric case. The reason for the phenomena is perhaps that the narrow
throat delays the flow velocity to rise. Compared with the stationary eccentric case, the
thermal plume is shifted further. At the eccentricity ratios )32,0(=
−re , the symmetry is
destroyed by the rotation. There is one larger thermal plume on the left side and one
small thermal plume on the right. The Figure depicts the left plume and its ‘root’ are
moved a great distance while the right plume has difficulty passing through the throat.
59
At the eccentricity ratios )0,32(−=
−re the right cell has been squeezed to the left. The
thermal plume and its ‘root’ are shifted more than at )0,32(−=
−re , possibly the joint
effect of the rotation and the eccentricity.
At the eccentricity ratio 3/2=ee and the inner positions has an angle ±450
against the horizontal line, both the narrowness of the throat region and the rotation of
the inner cylinder have the effect on the flow and the heat transfer on the annulus. Fig.
5.5.5-5.5.6 separately show the streamline and temperature fields.
60
CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS
The flow and the heat transfer in horizontal concentric and eccentric annuli have
been investigated. The two-dimensional model is used to obtain the numerical results
from two methods. The results from the finite element method are compared with those
from the finite difference method. The finite difference method is easily implemented
in computer codes. However, the finite element can provide more accurate results than
the finite difference method though the sparse matrix is created, which need more
memory and executed time. For the 2-D steady state cases, the author prefers the finite
element method. The numerical results are in good agreement with the experimental
and numerical results of other investigators such as Kuehn & Goldstein (1976) and
Projahn et al. (1981). This is a confirmation of the validity of the numerical model
which can be applied with reasonable confidence to obtain useful results in cases where
practical experiment could be difficult or time consuming. The effects of parameters
such as radius ratio, eccentricity, Rayleigh number, Reynolds number and Prandtl
number on the flow and the heat transfer were investigated numerically. The effects of
rotating inner cylinder on the flow and the heat transfer in eccentric annuli of the
moderate radius ratios were investigated numerically as the important part of the study.
Finally, the effects of the rotating the inner cylinder and outer cylinder and both were
studied. It is found that the moving boundary condition can increase the internal
convection. The rotation can also change the fluid flow distribution such as
elongating/reducing the stagnant zone. Particularly, for the concentric and eccentric
cases with both cylinders rotating, the outer cylinder rotating can cause the increase of
one side of the cooled fluid flow and restrict another side of the cooled fluid flow. It
has a great influence on the low temperature stagnant zone.
61
In the future, the turbulence and instability situation is the objective which
researchers need to search for. When the radius of the inner and outer cylinders is
increased, or the temperature difference between the two cylinders is higher, or the
Reynolds number for the inner or outer cylinder rotation is larger, the instability
appears and further the turbulence appears to occur. In the situation, the numerical
model is not good to calculate the flow and heat transfer between the two horizontal
cylinders. Therefore, new turbulence model needs to be developed to solve the
problem.
62
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WAKITANI S. 1997, “Development of Multicellular solutions in heat convection in air-filled vertical cavity”, ASME Journal of Heat Transfer Vol. 119, pp 97-101. YEO K. S. 1984, “A study of thermally-influenced fluid flows between concentric and eccentric horizontal cylinders”, Master Thesis. National University of Singapore. YOO J. S. 1998, “Natural convection in a narrow horizontal cylindrical annulus: 3.0Pr ≤ ”, International J. Heat Transfer, 41, pp 3055-3073.
68
Figure 1.1.1 Geometry of annular region and the gravity direction
Mesh for finite element method for 2-D
computations
Mesh for finite difference method for 2-D
computations
Figure 2.1.1 Mesh used for the numerical computation
Co
Ci
g
ϕ0
Ri
R0
Vi er
V0
xΦ
Ω
y
69
(a) (b) (c)
Figure 3.1.1 2-D view of three kinds of shape function: (a) four node element; (b) eight node element; (c) nine node element
70
Table 3.1 Shape Functions of Quadrilateral Elements (4-9)
Shape Function i = 5 i = 6 i = 7 i = 8 i = 9
))(( ηξφ −−−−−−−−==== 1141
1 521 φ−−−− 82
1 φ−−−− 941 φ
))(( ηξφ −−−−++++==== 1141
2 521 φ−−−− 62
1 φ−−−− 941 φ
))(( ηξφ ++++++++==== 1141
3 621 φ−−−− 72
1 φ−−−− 941 φ
))(( ηξφ ++++−−−−==== 1141
4 721 φ−−−− 82
1 φ−−−− 941 φ
))(( ηξφ −−−−−−−−==== 1121 2
5 921 φ−−−−
))(( 26 11
21 ηξφ −−−−++++==== 92
1 φ−−−−
))(( ηξφ ++++−−−−==== 1121 2
7 921 φ−−−−
))(( 28 11
21 ηξφ −−−−−−−−==== 92
1 φ−−−−
))(( 221 11
21 ηξφ −−−−−−−−====
Table 4.1 Parameters used in calculations
Parameters Air ρ (kg/m3) 1.1774 µ (kg/m-s) 1.829×10-5
Κ (W/m-K) 2.624×10-2
CP (J/kg-K) 1005.7 β (K-1) 3.3×10-3
Pr 0.701
71
(a) Temperature
(b) Streamline
(c) Velocity
Fig 4.1.1 Thermal and fluid flow distribution in concentric annuli without rotation The working fluid is air with a Prandtl number of 0.701 Radius ratio: 2.6; inner radius: 0.01 m; outer radius: 0.026 m The inner temperature: 304 K The outer temperature: 301 K Rayleigh number: 1124
(a) Temperature
(b) Streamline
(c) Velocity
Fig 4.1.2 Thermal and fluid flow distribution in concentric annuli without rotation The working fluid is air with a Prandtl number of 0.701 Radius ratio: 2.6; inner radius: 0.0431 m; outer radius: 0.112 m The inner temperature: 304 K The outer temperature: 301 K Rayleigh number: 9×104
72
(a) Temperature
(b) Streamline
(c) Velocity
Fig 4.1.3 Thermal and fluid flow distribution in concentric annuli without rotation The working fluid is air with a Prandtl number of 0.701 Radius ratio: 2.6; inner radius: 0.01 m; outer radius: 0.026 m The inner temperature: 541 K The outer temperature: 301 K Rayleigh number: 9×104
(a) Temperature
(b) Streamline
(c) Velocity
Fig 4.1.4 Thermal and fluid flow distribution in concentric annuli without rotation The working fluid is air with a Prandtl number of 0.701 Radius ratio: 2.6; inner radius: 0.065 m; outer radius: 0.169 m The inner temperature: 304 K The outer temperature: 301 K Rayleigh number: 3.085×105
73
(a) Temperature
(b) Streamline
(c) Velocity
Fig 4.1.5 Thermal and fluid flow distribution in concentric annuli without rotation The working fluid is air with a Prandtl number of 0.701 Radius ratio: 2.6; inner radius: 0.07 m; outer radius: 0.182 m The inner temperature: 304 K The outer temperature: 301 K Rayleigh number: 3.85×105
74
(a) Temperature
(b) Streamline
(c) Velocity
(d) Temperature
(e) Streamline
(f) Velocity
Fig 4.2.1 Thermal and fluid flow distribution in concentric annuli without rotation The working fluid is air with a Prandtl number of 0.701 Rayleigh number: 1.5×105
(a-c) Radius ratio: 5.0; inner radius: 0.02044 m; outer radius: 0.1022 m The inner temperature: 304 K The outer temperature: 301 K (d-f) Radius ratio: 2.6; inner radius: 0.02044 m; outer radius: 0.053144 m The inner temperature: 347.9 K The outer temperature: 301 K
75
(a) Temperature
(b) Streamline
(c) Velocity
Fig 4.2.2 Thermal and fluid flow distribution in concentric annuli without rotation The working fluid is air with a Prandtl number of 0.701 Radius ratio: 5.0, inner radius: 0.03054 m; outer radius: 0.1527 m The inner temperature: 304 K The outer temperature: 301 K Rayleigh number: 5×105
76
(a) Temperature
(b) Streamline
(c) Velocity
Fig 4.3.1 Thermal and fluid flow distribution in eccentric annuli without rotation The working fluid is air with a Prandtl number of 0.701 Radius ratio: 2.6, inner radius: 0.05 m; outer radius: 0.13 m The inner temperature: 305 K The outer temperature: 301 K Rayleigh number: 1.87×105
Eccentric ratio: - 0.75
(a) Temperature
(b) Streamline
(c) Velocity
Fig 4.3.2 Thermal and fluid flow distribution in eccentric annuli without rotation The working fluid is air with a Prandtl number of 0.701 Radius ratio: 2.6, inner radius: 0.05 m; outer radius: 0.13 m The inner temperature: 305 K The outer temperature: 301 K Rayleigh number: 1.87×105 Eccentric ratio: 0.75
77
(a) Temperature (b) Streamline (c) Velocity
(d) Temperature (e) Streamline (f) Velocity
Fig 4.3.3 Thermal and fluid flow distribution in eccentric annuli without rotation The working fluid is air with a Prandtl number of 0.701 Radius ratio: 2.6, inner radius: 0.028m; outer radius: 0.084m The inner temperature: 307 K; The outer temperature: 301 K Rayleigh number: 4.95×104; Eccentric ratio: 0.5 (a-c) for moving the inner cylinder to angular 450 against the horizontal line; (d-e) for moving the inner cylinder to angular -450 against the horizontal line;
78
00.10.20.30.40.50.60.70.80.9
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Fig 4.3.4
Temperature distribution along the shortest line and longest line shooting from the inner cylinder to outer cylinder corresponding to picture in Figure 4.4.3
1. The shortest distance between Ro and Ri of Fig. 4.3.3 (a) 2. The shortest distance between Ro and Ri of Fig. 4.3.3 (d) 3. The longest distance between Ro and Ri of Fig. 4.3.3 (a) 4. The longest distance between Ro and Ri of Fig. 4.3.3 (d)
79
(a) Temperature (b) Streamline
(c) Temperature (d) Streamline
Figure 4.4.1
Temperature and Streamline velocity plot in the concentric annulus with the inner or outer cylinder rotating with Pr = 0.7, Ro=0.0728m, Ri=0.028m, To=307K, Ti=301K, Ral=4.95×104, (a-b) for inner cylinder rotating with tangential velocity 0.1 m/s, (c-d) for outer cylinder rotating with tangential velocity 0.1 m/s
80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.01 0.02 0.03 0.04 0.05
Figure 4.4.2
Temperature distribution along different angular line shooting from the inner cylinder to outer cylinder corresponding to picture in Figure 4.4.1.
1
2
3
4
56
7
8
1. 00 (Vo= 0.1) 2. 900 (Vo= 0.1) 3. 1800 (Vo= 0.1) 4. 2700 (Vo= 0.1) 5. 00 (Vi = 0.1) 6. 900 (Vi= 0.1) 7. 1800 (Vi= 0.1) 8. 2700 (Vi= 0.1)
Distance from Ro to Ri (m)
(T-T
o)/(T
i-To)
81
(a) Temperature
(b) Streamline
(c) Velocity
(d) Temperature (e) Streamline (f) Velocity
Figure 4.4.3
Temperature and Streamline velocity plot in the concentric annulus with both inner and outer cylinder rotating with Pr = 0.7, Ro=0.0728m, Ri=0.028m, To=307K, Ti=301K, Ral=4.95×104, (a-c) for inner and outer cylinder rotating with tangential velocity Vi = 0.05 m/s and Vo = 0.05 m/s, (c-d) for inner and outer cylinder rotating with tangential velocity Vi = 0.05 m/s and Vo = -0.05 m/s.
82
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.01 0.02 0.03 0.04 0.05
Figure 4.4.4
Temperature distribution along different angular line shooting from the inner cylinder to outer cylinder corresponding to picture in Figure 4.4.3
1
2
34
5
6
7
8
Distance from Ro to Ri (m)
(T-T
o)/(T
i-To)
1. 00 (Vo=Vi) 2. 900 (Vo=Vi) 3. 1800 (Vo=Vi) 4. 2700 (Vo=Vi) 5. 00 (Vo = -Vi) 6. 900 (Vo = -Vi) 7. 1800 (Vo = -Vi) 8. 2700 (Vo = -Vi)
83
(a) Temperature
(b) Streamline
(c) Velocity
(d) Temperature
(e) Streamline
(f) Velocity
Figure 4.5.1
Temperature and Streamline velocity plot in the eccentric annulus of inner cylinder moving up with both inner and outer cylinder rotating with Pr = 0.7, Ro=0.0728m, Ri=0.028m, To=307K, Ti=301K, Ral=4.95×104 and er=0.5, (a-c) for inner and outer cylinder rotating with tangential velocity Vi = 0.05 m/s and Vo = 0.05 m/s, (c-d) for inner and outer cylinder rotating with tangential velocity Vi = 0.05 m/s and Vo = -0.05 m/s.
84
00.1
0.20.30.40.5
0.60.70.8
0.91
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Figure 4.5.2
Temperature distribution along the shortest line and longest line shooting from the inner cylinder to outer cylinder corresponding to picture in Figure 4.5.1.
1
2
3
4
Distance from Ro to Ri (m)
(T-T
o)/(T
i-To)
1. 900 (Vo=Vi) 2. 2700 (Vo=Vi) 3. 900 (Vo = -Vi) 4. 2700 (Vo = -Vi)
85
(a) Temperature
(b) Streamline
(c) Velocity
(d) Temperature
(e) Streamline
(f) Velocity
Fig 4.5.3
Both of inner and outer cylinder rotating at 0.1m/s The working fluid is air with a Prandtl number of 0.701 Radius ratio: 2.6, inner radius: 0.03m; The inner temperature: 304 K; the outer temperature: 301 K; (a-c) Inner cylinder rotating vel: 0.1m/s; Outer cylinder rotating vel: 0.1m/s; (d-f) Inner cylinder rotating vel: 0.1m/s; Outer cylinder rotating vel: -0.1m/s; Eccentric ratios: 0.75
86
(a) Temperature (b) Streamline
Velocity
(a) Temperature
(b) Streamline Velocity
Figure 4.5.4
Temperature and Streamline velocity plot in the eccentric annulus of inner cylinder moving down with both inner and outer cylinder rotating with Pr = 0.7, Ro=0.0728m, Ri=0.028, To=307K, Ti=301K, Ral=4.95×104 and er=0.5, (a-c) for inner and outer cylinder rotating with tangential velocity Vi = 0.05 m/s and Vo = 0.05 m/s, (c-d) for inner and outer cylinder rotating with tangential velocity Vi = 0.05 m/s and Vo = -0.05 m/s.
87
00.1
0.20.3
0.40.50.6
0.70.8
0.91
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Figure 4.5.5
Temperature distribution along the shortest line and longest line shooting from the inner cylinder to outer cylinder corresponding to picture in Figure 5.5.4.
1
2
3
4
1. 900 (Vo=Vi) 2. 2700 (Vo=Vi) 3. 900 (Vo = -Vi) 4. 2700 (Vo = -Vi)
Distance from Ro to Ri (m)
(T-T
o)/(T
i-To)
88
1
2
3
4
5
6
7
100 1000 10000 100000 1000000
Rayleigh number
Ove
rall
equi
vale
nt h
eat t
rans
fer c
oeffi
cien
t
Figure 5.0.1 Overall heat transfer coefficient versus Rayleigh number at the radius ratio
of 2.6, 0=re .
: Present number 6.2/ =io rr : Kuehn & Goldstein, expt., 6.2/ =io rr : Grigull & Hauf, expt., 45.2/ =io rr : Projahn et al., num., 6.2/ =io rr
89
(a) 2102×=Ral (b) 310=Ral
(c) 410=Ral (d) 4105×=Ral
(e) 510=Ral (f) 610=Ral
Fig. 5.1.1 Isotherm plots at various Rayleigh number, ,6.2/ =io rr .0=re
90
(a) 2102×=Ral (b) 310=Ral
(c) 410=Ral (d) 4105×=Ral
(e) 510=Ral (f) 610=Ral
Fig. 5.1.2 Streamline plots at various Rayleigh numbers, ,6.2/ =io rr .0=re
91
(a) 2102×=Ral (b) 310=Ral
(c) 410=Ral (d) 4105×=Ral
(e) 510=Ral (f) 610=Ral
Fig. 5.2.1 Isotherm plots at various Rayleigh numbers, ,0.5/ =io rr .0=re
92
(a) 2102×=Ral (b) 310=Ral
(c) 410=Ral (d) 4105×=Ral
(e) 510=Ral (f) 610=Ral
Fig. 5.2.2 Isotherm plots at various Rayleigh numbers, ,0.5/ =io rr .0=re
93
(c) 410=Ral (d) 410=Ral
(e) 510=Ral (f) 510=Ral
Isotherms Streamlines
Fig. 5.2.3 Isotherm and Streamline plots at various Rayleigh numbers, ,25.1/ =io rr
.0=re
94
(a) Reyl=140 (b) Reyl=280
(c) Reyl=416 (d) Reyl=560
(e) Reyl=800 (f) Reyl=1120
Fig. 5.3.1 Streamline plots at various Reynolds numbers, 6.2/ =io rr , ,0=re
4105.5 ×=Ral .
95
(a) Reyl=140 (b) Reyl=280
(c) Reyl=416 (d) Reyl=560
(e) Reyl=800 (f) Reyl=1120
Fig. 5.3.2 Isotherm plots at various Reynolds numbers, 6.2/ =io rr , ,0=re
4105.5 ×=Ral .
96
(a) )31,0(=
− re (b) )
31,0(=
− re
(c) )31,0( −=
− re (d) )
31,0( −=
− re
(e) )0,31(−=
− re (f) )0,
31(−=
− re
Fig. 5.4.1 Isotherm and streamline plots at various eccentric positions, ,6.2/0 =irr
,31=e 4105×=Ral .
97
(a) )
32,0(=
− re (b) )
32,0(=
− re
(c) )32,0( −=
− re (d) )
32,0( −=
− re
(e) )0,
32(−=
− re (f) )0,
32(−=
− re
Streamlines Isotherms
Fig. 5.4.2 Isotherm and streamlines plots at various eccentric positions, ,6.2/ =io rr
,32=re 4105×=Ral .
98
(a) )
21
32,
21
32( ⋅⋅=
− re (b) )
21
32,
21
32( ⋅⋅=
− re
(c) )
21
32,
21
32( ⋅−⋅=
− re (d) )
21
32,
21
32( ⋅−⋅=
− re
Streamline Isotherms
Fig. 5.4.3 Isotherm and Streamline plots at various eccentric positions ±450, ,6.2/ =io rr
,32=re 4105×=Ral .
99
(a) )31,0(=
− re (b) )
31,0( −=
− re
(c) )0,31(−=
− re (d) )0,
31(=
− re
Fig. 5.5.1 Streamline plots at eccentric positions, ,6.2/ =io rr ,31=re ,105.5 4×=Ral
Rey1=280.
100
(a) )31,0(=
− re (b) )
31,0( −=
− re
(c) )0,31(−=
− re (d) )0,
31(=
− re
Fig. 5.5.2 Isotherm plots at eccentric positions, ,6.2/ =io rr ,31=re ,105.5 4×=Ral
Rey1=280.
101
(a) )3/2,0(=
− re (b) )
32,0( −=
− re
(c) )0,32(−=
− re (d) )0,
32(=
− re
Fig. 5.5.3. Streamline plots at eccentric positions, ,6.2/ =io rr ,32=re ,105.5 4×=Ral
Rey1=280,
102
(a) )
32,0(=
− re (b) )
32,0( −=
− re
(c) )0,
32(−=
− re (d) )0,
32(=
− re
Fig. 5.5.4 Isotherm plots at eccentric positions, ,6.2/ =io rr ,32=
− re ,105.5 4×=Ral
Rey1=280.
103
(a) )
21
32,
21
32( ⋅⋅−=
− re (b) )
21
32,
21
32( ⋅−⋅−=
− re
(c) )2
132,
21
32( ⋅⋅=
− re (d) )
21
32,
21
32( ⋅−⋅=
− re
Fig. 5.5.5 Streamline plots at eccentric positions ,135,45 00 ±± , ,6.2/ =io rr ,32=
− re
,105.5 4×=Ral Rey1=280.
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