
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00003339/00001
Material Information
 Title:
 Fluid oscillations and enhanced heat transfer in a parallel plate channel bounded by end reservoirs maintained at different temperature
 Creator:
 Zhao, Alex X., 1949
 Publication Date:
 1990
 Language:
 English
 Physical Description:
 xviii, 120 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Analytics ( jstor )
Cross flow ( jstor ) Geometric lines ( jstor ) Heat transfer ( jstor ) Oscillating flow ( jstor ) Oscillation ( jstor ) Pistons ( jstor ) Temperature distribution ( jstor ) Temperature gradients ( jstor ) Velocity ( jstor )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1990.
 Bibliography:
 Includes bibliographical references (leaves 116118).
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Alex X. Zhao.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 024005704 ( ALEPH )
AHZ5391 ( NOTIS ) 25034727 ( OCLC )

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FLUID OSCILLATIONS AND ENHANCED HEAT TRANSFER
IN A PARALLEL PLATE CHANNEL BOUNDED BY END RESERVOIRS
MAINTAINED AT DIFFERENT TEMPERATURE
BY
ALEX X. ZHAO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1990
To my parents and my beloved motherland
ACKNOWLEDGEMENTS
The successful completion of this dissertation would have been impossible if
it had not been for the assistance and help I received from numerous people. First
mention must be made for the academic guidance and financial support made
possible by my graduate adviser Dr. Ulrich H. Kurzweg. He, with his strong
background and extensive knowledge in fluid mechanics, heat transfer and applied
mathematics, gave me clear guidance in maintaining the correct research direction.
What I learned from him are not only the relevant knowledge in those fields but also
the skills needed to conduct scientific research. Two other persons who helped me
significantly in the dissertation were Dr. Wei Shyy and Dr. Ranga Narayanan.
Several fruitful discussions with Dr. Shyy helped me to construct the numerical
procedure used successfully and more efficiently for the time periodic problems in
this dissertation. The experience of working with Dr. Narayanan put me in contact
with many interesting topics in fluid mechanics, heat transfer and numerical analysis,
which made my understanding of the enhancement of heat transfer problem
considered here much more clear. I am thankful to the other and former members
of my supervisory committee, Dr. E Rune. Lindgren, Dr. Joseph L. Hammack, Dr.
ChiaShun Yih, Dr. Edward K. Walsh and Dr. Tom IP. Shih, for their time and
effort.
Thanks are also expressed here to the Department of Aerospace Engineering,
Mechanics and Engineering Science and the Department of Chemical Engineering
of the University of Florida for their financial assistance in allowing me to pursue my
educational goals within this lovely country. I am also grateful to the early support
of this work by National Science Foundation through the grant CBT8611254 and the
allocation of 95 Service Units for this study on the CRAY/YMP supercomputer in
the Pittsburgh Supercomputing Center.
I would also like to thank my wife Wei Zhu for being here with me in a
"foreign" country and working hard to help support the family. Without her help, I
would not have been able to successfully complete my education. I also wish to
express gratitude to my wonderful office mate Bill Harter. Through many discussions
with him, I learned a lot about PC computers, about the English language, about
American customs and etc. Finally I wish to thank myself for putting in four hard
working years on this dissertation effort.
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS
LIST OF FIGURES
LIST OF TABLES
NOMENCLATURE
ABSTRACT
CHAPTERS
I INTRODUCTION ...................................... 1
1.1 Background ...................................... 1
1.2 Literature Review .................................... 5
1.3 Analytical Solutions Within the Connecting Channel ........ 9
II MATHEMATICAL MODEL .............................. 24
2.1 Geometric Configurations ........... .................. 24
2.2 Governing Equations and Boundary Conditions ............. 28
III NUMERICAL TECHNIQUE EMPLOYED ..................... 32
3.1 Grid and Transformation of Governing Equations ......... 33
3.2 A Brief Derivation of the Associated Finite Difference
Equations and Boundary Conditions ..................... 38
3.3 Calculation Procedure .............................. 43
3.4 A Vectorized Line Group Method for Solving
the System of Algebraic Equations ...................... 47
IV RESULTS AND DISCUSSION ............................. 55
4.1 Validation of the Results ............................. 56
4.2 Velocity Field ...................................... 70
4.3 Temperature and Heat Transfer Coefficient ............... 88
...................................... iii
..................... ................ vii
................... ................... xii
..................................... xvii
...................................... xvii
V CONCLUDING REMARKS
.............................. 106
APPENDIX THE VELOCITYPRESSURE CORRECTION .............. 110
REFERENCES ................................................ 116
LIST OF FIGURES
Figure M
Fig. 11 Heat transfer in a parallel plate channel: (a) pure conduction;
(b) enhancement by oscillating flows. .... 3
Fig. 12 A schematic of the Thermal Pump (U.S. Patent No. 4590993).
5
Fig. 13 Geometric configuration for the analytic solution of flow
oscillations and the associated heat transfer in parallel plate
channels. .... 10
Fig. 14 Analytic solution of the dimensionless velocity of oscillating
flow in a parallel plate channel as a function of dimensionless
transverse position at different oscillation phases ot: (a) a=1;
(b) a=10. .... 12
Fig. 15 Analytic solution of the dimensionless temperature distribution
in oscillating flow in a parallel plate channel and its walls as a
function of dimensionless transverse position at different
oscillation phases wt: (a) a=1; (b) a=10. .... 14
Fig. 16 Analytic solution of the dimensionless time averaged effective
thermal diffusivity enhanced by oscillating flow in a parallel
plate channel as a function of Womersley number a at
different Prandlt numbers: (a) e=2 (conducting walls with
a = =1); (b) e=l (insulating walls). .... 16
Fig. 17 Analytic solution of dimensionless time averaged effective
thermal diffusivity enhanced by oscillating flow in a parallel
plate channel at e=2 (conducting walls) as a function of
Womersley number a for several different liquidsolid
combinations. .... 19
Fig. 18 Analytic solution of dimensionless time averaged effective
thermal diffusivity enhanced by oscillating flow as a function of
cladding thickness for liquid lithium in a parallel plate channel
with stainless steel walls. .... 19
Fig. 19 Analytic solution of heat transferred by an oscillating flow of
water in a parallel plate channel with glass walls for different
a: (a) time averaged value as a function of dimensionless
transverse position; (b) cross section integrated value as a
function of oscillation phase. .... 20
Fig. 21 Configurations of the thermal pump used in the numerical
investigation: (a) conduction model; (b) cross flow model. .... 23
Fig. 31 The correspondence of blocks between the physical domain
and the computational domain: (a) grid in the physical domain
at ot=0; (b) time independent and uniform grid in the
computational domain; (c) grid in the physical domain at
wt=180o. .... 32
Fig. 32 The movable and nonuniform grid in the physical domain: (a)
at=0; (b) at=90; (c) wt=180. .... 33
Fig. 33 The staggered grid in the computational domain. .... 36
Fig. 34 Typical cells for the velocity component U: (a) an interior cell;
(b) a boundary cell with its south wall at the domain boundary. .... 38
Fig. 35 Flowchart for determining the time periodic velocity
components U, V and pressure P. .... 42
Fig. 36 Flowchart for solving the time periodic temperature
distribution based on the calculated velocity. .... 45
Fig. 37 Flowchart for the LineByLine iteration method for solving the
associated system of algebraic equations. Dashed lines
represent the innermost DO loops. ... 48
Fig. 38 Flowchart for the Vectorized Line Group iteration method for
solving the associated system of algebraic equations. Dashed
lines represent the innermost DO loops. .... 50
Fig. 41 Stream line patterns for Case No.6 calculated with different
number of grid points: (a) grid 91x71; (b) grid 121x101.
Increment between neighboring lines is A r=20 in non
dimensional units. .... 54
Fig. 42 Nondimensional velocity component U for Case No.5 at the
middle of the connecting channel as a function of oscillation
phase wt: solid line numerical result; dashed line analytic
solution. .... 57
Fig. 43 Nondimensional temperature T for Case No.5 at the middle of
the connecting channel as a function of oscillation phase ft:
solid line numerical result; dashed line analytic solution. .... 57
Fig. 44 Nondimensional velocity component U for Case No.5 at the
middle of the connecting channel as a function of non
dimensional position y at several oscillation phases: (a)
analytic solution; (b) numerical result. .... 60
Fig. 45 Nondimensional temperature T for Case No.5 at the middle of
the connecting channel as a function of nondimensional
position y at several oscillation phases: (a) analytic solution;
(b) numerical result. .... 61
Fig. 46 Nondimensional velocity component U for Case No.6 at the
middle of the connecting channel as a function of non
dimensional position y at several oscillation phases: (a)
analytic solution; (b) numerical result. .... 62
Fig. 47 Nondimensional x direction pressure gradient for Case No.5 at
the middle of connecting channel as a function of oscillation
phase ft: solid line numerical result; dashed line analytic
solution. .... 64
Fig. 48 Stream line pattern for Case No.3 at oscillation phases of
interval A(ot) =150 in the lower half cycle 0tot<1800. .... 66
Fig. 49 Temporal approach to the final time periodic state of the non
dimensional velocity component U(x,O, ot) for Case No.3 at
three different locations: (a) x=152; (b) x=175; (c) x=190. .... 69
Fig. 410 Stream line pattern for Case No.4 at oscillation intervals of
A(ot) =15 in the lower half cycle Oscot <180. .... 70
Fig. 411 Stream line pattern for Case No.6 at oscillation intervals of
A(ct) =150 in the lower half cycle 00 t<1800. .... 71
Fig. 412 Nondimensional pressure distribution along the x axis in Case
No.6 at oscillation intervals of A(wt) =450 over the whole cycle
0" cot<3600. .... 73
Fig. 413 Stream line pattern for Case No.6 at the oscillation intervals of
A(at) =450 over the whole period 00Oat<3600. .... 74
Fig. 414 Stream line pattern for Case No.1 at the oscillation intervals of
A(ct) =150 in the lower half cycle 00t<1800. .... 76
Fig. 415 Stream line pattern for Case No.8 in the right reservoir at
oscillation intervals of A(at)=900 over the whole period
0at <360. .... 78
Fig. 416 Stream line pattern for Case No.9 at oscillation intervals of
A (ot) =15 over the lower half period 00
Fig. 417 Stream line pattern for Case No.10 at oscillation intervals of
A (t) = 150 in the lower half cycle 00s t <1800. .... 80
Fig. 418 Temperature contours for Case No.3 at oscillation intervals of
A (at) =150 in the lower half cycle 00 t <180. .... 83
Fig. 419 Nondimensional temperature profile for Case No.3 along the
vertical line x=153 in the left reservoir: (a) at several
oscillation phases; (b) time averaged. .... 85
Fig. 420 Temperature contours for Case No.4 at oscillation intervals of
A(act) =150 in the lower half cycle 00ot<1800. .... 86
Fig. 421 Nondimensional temperature profile for Case No.4 along the
vertical line x=153 in the left reservoir: (a) at several
oscillation phases; (b) time averaged. .... 87
Fig. 422 Temperature contours for Case No.6 at oscillation intervals of
A(ot) =150 in the lower half cycle 0o
Fig. 423 Nondimensional temperature profile for Case No.6 along the
vertical line x=153 in the left reservoir: (a) at several
oscillation phases; (b) time averaged. .... 89
Fig. 424 Temperature contours for Case No.9 at oscillation intervals of
A(at) =150 in the lower half cycle 0t <1800. .... 91
Fig. 425 Nondimensional temperature profile for Case No.9 along the
vertical line x=153 in the left reservoir: (a) at several
oscillation phases; (b) time averaged. .... 92
Fig. 426 Temperature contours for Case No.10 at oscillation intervals of
A (t) = 150 in the lower half cycle 00ot <1800. .... 93
Fig. 427 Nondimensional temperature profile for Case No.10 along the
vertical line x=153 in the left reservoir: (a) at several
oscillation phases; (b) time averaged. .... 94
LIST OF TABLES
The list of ten cases studied in the numerical investigation.
Heat transfer coefficients for several cases in the study.
... 51
... 95
Table 41
Table 42
NOMENCLATURE
a the half width of the connecting channel
A coefficients in the Finite Difference Equations
Ao the cross section area of the connecting channel
b the distance from the channel center line to the wall center line in the
analytic solution or the half width of the end reservoir in the numerical study
c time averaged length of the end reservoir
Cp specific heat of working fluid
D molecular mass diffusivity of working fluid
h, heat transfer coefficient from the analytic solution
h, heat transfer coefficient from the numerical results
L channel length
P pressure
Pr Prandtl number
q local instantaneous axial heat flux in the channel
Q total time averaged heat rate in the channel
r, the ratio of tidal displacement to channel length
Re Reynolds number based on the maximum of piston velocity and reservoir half
width
RHS
S
Sc
T
T,
Tc
T,
u
U
Up
v
V
Vc
x
6x
y
6y
terms on the Right Hand Side of the associated system of algebraic equations
source term in the Finite Difference Equations
Schmidt number
temperature
high temperature specified at hot end for supplying heat
low temperature specified at cold end for removing heat
the approximate time averaged temperature in right reservoir from the
numerical results
velocity component in axial (x) direction at the walls of U cells
velocity component in axial (x) direction
maximum piston velocity
velocity component in transverse (y) direction at the walls of U cells
velocity component in transverse (y) direction
cross flow velocity
axial coordinate in the physical domain
x direction size of a grid cell in the physical domain
transverse coordinate in the physical domain
y direction size of a grid cell in the physical domain
a Womersley number
p stretching parameter for generating nonuniform grid
y time averaged axial temperature gradient (constant)
6i variable time step size in calculation
6 B basic time step size in calculation
Ax tidal displacement
e the ratio of total channel half width (including the thickness of solid wall) to
the channel half width filled by fluid in the analytic solutions or general
convergence criterion in the numerical calculations
qe convergence criterion for the VelocityPressure correction
ep convergence criterion for the final time periodic states
r1 transverse coordinate in the computational domain
0 magnitude of the sinusoidal pressure gradient applied in the analytic solutions
K molecular thermal diffusivity of working fluid
Keff effective thermal diffusivity by oscillating flow
p the ratio of fluidtosolid (wall) thermal conductivities
v kinematic viscosity of working fluid
t axial coordinate in the computational domain
p density of working fluid
a the ratio of fluidtosolid (wall) thermal diffusivities in analytic solutions or
general convergence indicator in numerical calculations
Ca convergence indicator for VelocityPressure correction
a, convergence indicator for the final time periodic state
r time in the computational domain
ir stream function of the oscillating flow field
o angular frequency of the oscillation
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
FLUID OSCILLATIONS AND HEAT TRANSFER
IN A PARALLEL PLATE CHANNEL BOUNDED BY END RESERVOIRS
MAINTAINED AT DIFFERENT TEMPERATURE
by
Alex X. Zhao
December 1990
Chairman: Ulrich H. Kurzweg
Major Department: Aerospace Engineering, Mechanics,
and Engineering Science
Existing analytic and experimental studies have shown that sinusoidal
oscillations of a viscous fluid within open ended conduits connected to reservoirs can
enhance the thermal diffusivity between the hot fluid and cold fluid in the opposite
end reservoirs by some four orders of magnitude in excess of that possible in the
absence of oscillation. The heat transfer coefficients achieved in this process can be
very high and can readily exceed those possible via heat pipes, yet involve no net
convective mass exchange.
A numerical investigation of laminar oscillating flows of incompressible
viscous fluid and the associated enhanced heat transfer in a 2D parallel plate channel
bounded by rectangular end reservoirs which have sinusoidally oscillating piston
boundaries and are maintained at different temperatures forms the topic of the
present dissertation.
xvii
A modification of the widely used SIMPLE algorithm, termed SIMPLETP
(for Time Periodic), has been developed to take time periodicity and boundary
movements into account. The method has been successfully applied to the present
time periodic flow and heat transfer problem.
Two different models of heat supply and removal, namely, conduction through
the reservoir walls and convection by cross flow, were considered in this study.
Several case studies involving different combinations of Womersley number, tidal
displacement, length of reservoirs and cross flow velocities were carried out. The
numerical results obtained were compared with analytic solutions where such
comparisons were possible.
It was found that the velocity field in each reservoir is characterized by a high
velocity jet emanating from the channel end during a portion of the oscillation cycle
and that one or more counterrotating vortex(es) or vortex pair(s) exist during the
whole oscillation cycle as long as the Reynolds number, based on the maximum
piston velocity and reservoir width, is large enough. The tidal displacement and cross
flow velocity have a strong effect on the flow patterns in the reservoir and the
reservoir temperature field is mainly determined by the temperature boundary
conditions and less so by the velocity field. High oscillation frequencies coupled with
a large tidal displacement and a relatively large cross flow velocity are found to yield
large enhancements of heat transfer. Heat transfer coefficient as high as 1.8x106
W/m2K can readily be achieved when using water as the working fluid.
xviii
CHAPTER I
INTRODUCTION
1.1 Background
For a sinusoidal oscillatory fluid flow over a flat solid surface, the influence
of the noslip condition on the solid surface penetrates an approximate distance of
2v_/&c into the fluid, where v is the kinematic viscosity of the fluid and a is the
angular frequency of the flow oscillation. For an oscillatory flow inside a flat plate
channel, this is still true as long as the channel half width, a, is greater than the
penetration distance. Under this condition, the fluid flow can be considered to
consist of two parts, the boundary layers and the core region, which are shown in Fig.
11(b).
If a property of the fluid, namely, concentration or temperature, is maintained
at different values at the two ends of the channel, a very strong transport from the
high side to low side will occur. Some theoretical and experimental studies have
shown that the effective coefficient for the transport can be much larger than the
values which characterize the transport without oscillation. The reason for the
enhancement of the transport is that both the area over which the exchange takes
place and the property gradient across the area are increased significantly over the
values existing in the absence of oscillations.
1
2
Consider the case where the temperature of the liquid is different at the two
ends of the channel. Here the heat transfer from the hot end to the cool end can
be enhanced by a flow oscillation in the axial direction. The time averaged heat
transferred axially, without a accompanying mass exchange, can be expressed as
= AopCpKg (1,1)
where Kef, p and Cp stand for the effective thermal diffusivity, fluid density and
specific heat, respectively, Ao is the channel cross section, and y is the time averaged
axial temperature gradient which is maintained as a constant.
For pure molecular conduction, Kfe simply is the molecular thermal diffusivity
, Ao is equal to 2a when the channel depth is equal to one, as is shown in Fig. 1
l(a), and the amount of heat transferred per unit depth will be
Qc = 2apCpKY (1,2)
With oscillations, it is necessary to examine the heat transfer process in some
detail since it becomes a time dependent problem. During the axial fluid oscillation
towards the colder side, hot liquid is brought into the core region and moved in
contact with the cooled boundary layers, which in the previous half stroke are in
contact with cooler liquid in the core region. The colder boundary layers will absorb
heat from the hot liquid core and heat up. During the next half stroke the oscillation
is towards the hot side, and cold liquid is brought into contact with the heated
boundary layers. The boundary layers then release heat to the cool core region and
thus become cool again. These two steps of heat transfer in oscillating flows at
(a)
(b)
Heat transfer in a parallel plate channel: (a) pure conduction; (b)
enhancement by oscillating flows.
I
2a Th Tc
2v/w
T
/ c
boundary layers
2a core region Tc
Th
Fig. 11
4
Prandtl number 1, for which the thicknesses of momentum and thermal boundary
layers are equal, are shown in Fig. 11(b) with the dotted areas representing the cold
fluid. The exchange process is repeated during subsequent oscillation cycles. This
leads to a time periodic problem. Because the heat exchange happens between the
boundary layers and core region, the effective heat transfer surface become much
larger than Ao and can be expressed in terms of a quantity Ax named the tidal
displacement (equal to twice the cross section averaged axial amplitude of the
sinusoidal oscillations). The temperature gradient in the normal of interface between
the boundary layers and core region is also much larger than y and is a function of
time and position. By averaging over the whole oscillation period and channel cross
section, the heat transferred per unit depth from the hot end to the cool end now can
be written as
S= 2apCp, fy (1,3)
where now the effective thermal diffusivity Krg is a function of a, v, <, Ax and some
other factors. Krf can be much larger than K provided a good choice for those
factors is made. Note that there will be no net convection between the two ends
when Ax is kept at a fraction of the channel length, and the process thus can be
considered to be free of mass exchange between the channel ends under proper
conditions (explained in Section 1.3).
If the concentration is maintained at different values at the ends of the
channel, an enhanced axial mass diffusion will occur for appropriate conditions, thus
producing an effective diffusivity much larger than the molecular diffusivity. In
5
recent years, these phenomena have received considerable research attention from
engineers and scientists because of their potential practical applications such as for
nuclear reactor cooling and for gas separation, etc.
A typical device for enhancing heat transfer by oscillating flows, termed the
Thermal Pump, is shown in Fig. 12. This represents essentially US Patent No.
4590993 held by U. Kurzweg1 and the University of Florida. This dissertation will
be concerned with a numerical investigation of the laminar flows of incompressible
fluids and the associated heat transfer in a 2D version of the thermal pump when
sinusoidally moving piston boundaries are located in the end reservoirs. The
connecting channel will consist of a single flat plate channel formed by two parallel
plates separated by a fixed distance of 2a.
1.2 Literature Review
The phenomenon of transport enhancement of contaminants was initially
studied by Taylor2(1953) and Aris3(1956) in their investigations of axial dispersion
in steady flows within capillary tubes. Their results indicated that a significant
increase in axial contaminant dispersion will occur in steady laminar flows. Similar
effects were found to exist for oscillating flows by Bowden4 in 1965. Harris and
Goren5(1967) studied, both analytically and experimentally, the mass transfer
through a long tube connecting two reservoirs maintained at different but constant
concentrations by oscillating the flow in the tube and found that the increase of mass
transfer rates is a function of the Womersley number a =a o/v the tidal
Cooler Fluid Out
Heat Transport
Channels
Hot Fluid In Hotter Fluid Out
A schematic of the Thermal Pump (U.S. Patent No. 4590993)1.
Left End
Chamber
Right End
Chamber
Oscillator
Cold Fluid In
Fig. 12
7
displacement Ax and the Schmidt number v/D. Rice and Eagleton6(1970) conducted
a similar study to that of Harris5 and found that the mass transfer increase is pro
portional to the square of tidal displacement Ax. In 1975, Chatwin7 showed theo
retically that the effective diffusion coefficient in oscillating flows is a harmonic
function of time with a period equal to one half that of the imposed time dependent
pressure gradient. Further theoretical studies by Watsons(1983) and experiments by
Joshi et al.9(1983) gave similar results. Kurzweg and Jaeger"0(1986) first showed
the existence of a tuning effect in enhanced gas dispersion in oscillating flows. All
these studies prove that a contaminant will spread axially in oscillating laminar pipe
flow at rates as much as five orders of magnitude higher than those achievable by
pure molecular diffusion in the absence of oscillations.
The same enhancement in oscillating flow for heat transfer was discovered by
Kurzweg in 1983 in view of the mathematical similarity between the governing
equations for heat conduction and mass diffusion. The equations for contaminant
diffusion with superimposed oscillating flow are mathematically nearly identical to
the equation of heat conduction, with the only major differences being the boundary
conditions and the use of the Prandtl number Pr= v/I instead of the Schmidt number
Sc=D/c. He solved the governing equations for enhanced axial heat transfer in
oscillating flows in a tube of infinite length for low value of CftPr and found the
effective conductivity can readily reach values three orders of magnitude greater than
normal heat conductivity. This information was published in 198511. Kurzweg and
Zhao'2(1984) conducted some experiments to measure the effective axial heat con
8
duction rates through a capillary bundle connecting two fluid reservoirs maintained
at different temperatures and found very good agreement with theoretical
predictions. Two other papers by Kurzweg(1985)13(1986)14 explained the tuning
effect in the enhancement of heat transfer in detail. In May of 1986, a U.S. patent
was granted to the University of Florida and Professor U. Kurzweg1 on this process.
Zhang15 performed some numerical studies on tubes of finite length with some
portions of their walls or ends maintained at different temperatures and concluded
that the highest rates of heat transfer predicted earlier can only be reached when the
amount of heat supplied and removed from the tube ends is large enough.
Otherwise a heat "bottle neck" will occur and the temperature gradient in the axial
direction of the tube will be reduced considerably. Kaviany's16 recent study on heat
transfer by oscillating flows in tubes bounded by two end reservoirs maintained at
different temperatures verified, both theoretically and experimentally, the results
obtained by Kurzweg and Zhao12.
It should be noted that most of theoretical studies to this point, both analytic
and numerical, have considered only the tube or connecting channel without any
attention to the flow behavior and heat transfer process occurring in the two end
reservoirs which generally have much larger cross section than the connecting
conduit. Kaviany16 has made some calculations on this flow using a very simplified
model of the end chambers. Durst et al.17(1989) did some numerical studies on the
transient laminar flow over a sharp pipe expansion driven by a piston in the wide end
moving as a rectangular function of time starting from an initial rest condition and
9
compared the numerical result with their experimental data18. Neither of them
considered the temperature distribution and their results for velocity can not be
directly used in the analysis of the thermal pump process because of the
simplification made in former and the different velocity conditions used in the latter.
The motivation for this dissertation research was to obtain a clearer under
standing of the flow and temperature patterns inside of the end reservoirs of a
thermal pump in order to lend further guidance for the practical construction and the
usage of thermal pumps. Also, associated with this study, a proper numerical
procedure based on the SIMPLE algorithm was developed for the simulation of heat
transfer in time periodic flows of viscous incompressible fluids with moving
boundaries.
1.3 Analytic Solutions Within the Connecting Channel
The numerical investigations presented in the chapters II through V can be
considered as an extension of the analytic work done by Kurzweg13, which gave
solutions of the momentum and energy equations in a parallel plate channel of
infinite length. The numerical results in the connecting channel for the more
complicated geometry with end reservoirs should approximate those analytic solutions
when the connecting channel length becomes much longer than the tidal displace
ment. Thus, it is essential to first briefly evaluate the existing analytic solutions in
10
order to later make a comparison with the numerical results to be obtained wherever
such a comparison is possible.
The configuration dealt with in Ref. 13 is shown in Fig. 13. It consists of a
set of channels filled with a viscous fluid and connected to chambers at x = The
individual channels of width 2a each are confined between thermally conducting walls
aP
of width 2(ba) each and a time periodic pressure gradient a = Oe'"' and a
aT
constant axial temperature gradient = Y are superimposed. The velocity of the
oscillating flow in the central channel generated by the pressure gradient is given by
1 cosh(Ta y/a)
cosh(ri a)
U(y,t) = REAL Ii Ax cosh(ITa) e (1,4)
2 1 tanh(va)
where y is the coordinate perpendicular to the channel axis as shown in Fig. 13, W
is the angular frequency of the time periodic pressure gradient, a = a / v the
Womersley number, with v representing fluid kinematic viscosity, and Ax the tidal
displacement for the maximum cross stream averaged axial excursion of fluid
elements during one oscillation cycle. The vertical bars indicate the absolute value
of the quantity shown.
The relation between the magnitude of the axial pressure gradient and the
tidal displacement can be expressed as
K>
I I
f
Periodic
velocity profile
b
Thermally conducting
fluid
T.
x
Geometric configuration for the analytic solution of flow oscillations
and the associated heat transfer in parallel plate channels.
2(ba)
LA
\10\1~x
ic
ie
t
Fig. 13
r
K~
a~a
solid thermal
conductor
Period
pressure
gradier

i
S: P 02,&x
2 1 tanh(a) (1,5)
In practice, the channel half width a, the tidal displacement Ax, and the
oscillation frequency o as well as the properties of the working fluid are easy to
control. Hence, either they or a combination of some of them, in particular the
Womersley number, are used as the control parameter for this heat transfer problem.
A numerical evaluation of the nondimensional velocity, normalized by CoAx,
as a function of nondimensional transverse position in the y direction at different
oscillation phases for a=1 and a=10 is shown in Fig. 14(a) and (b), respectively.
It is clear that the velocity profile varies considerably with change in Womersley
number a, from an essentially parabolic shape at very low a to one having a nearly
constant velocity core connected to thin boundary layers of thickness 6 = ~/2 v /
at the walls for large a.
By assuming
T(x,y,t) = y(x + REAL[ag(y)ei"t] (1,6)
,the crosssection dependent portion of the temperature perturbation gly) in the
fluid and g,(y) in solid walls have the forms13
(a)
0.5
0.0
0.5
1.0 1 1 I 1 '
0.8 0.6 0.4 0.2 0.0
U/cAx
(b)
0.5
N 0.0 
0.5
1.0 L
0.8
Fig. 14 An
par
pos
0.2 0.4 0.6 0.8
0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8
U/WAx
alytic solution of the dimensionless velocity of oscillating flow in a
rallel plate channel as a function of dimensionless transverse
sition at different oscillation phases ct: (a) a=1; (b) a=10.
1 a2pr 1 cosh(a y/a) + H cosh(ViPa y/a)
gf (y) _[ cosh(V a) cosh(ia) (1,7)
2aa2(Pr1) 1 tanh( iJa)
hx 4ia
g Y) 1 H cosh[V7 Pr a(ey/a)]
2a2(Pr1) 1 tanh(Ta) cosh[VTaPr a (e1)] (1,8)
2a a2Pr 1) tanh(fa) v '0/
Ax Fa
with
H = 11"Pr tanh(F a) + Ytanh[VaPr a(e 1)] (1,9)
p tanh(Ti Pr a) + fV tanh[V/i a Pr a(e 1)]
where i =kf/k, is the ratio of fluidtosolid (wall) thermal conductivities, o= r/Kr
is the ratio of thermal diffusivities, and e=b/a.
The nondimensional temperature profiles T/yAx versusy/a for p = a=1 and
e=2 but different Womersley number a are shown in Fig. 15(a) and (b). One
observes an appreciable penetration of the temperature variation into the solid wall
at a=l, but almost no penetration occurs for a=10. This is to be expected in view
of the fact that the typical penetration distance into a solid of an oscillating
temperature field is proportional to the inverse square root of the oscillation
frequency (i.e. skin effect). Note that the frequency ratio at a=1 to that at a=10 is
one hundred for the same channel width and the same fluid.
(a) 2.0
a= 1
wall = 1 
a=1
Pr=1
1.0 channel 10 
150 60
a I / 300 / t=0 120 90
S0.0 
> 210 180 30
270
1.0 240 330
wall
2.0
0.4 0.2 0.0 0.2 0.4
T/yAx
(b) 2.0 i i I '
wall I= 1 
a=1
Pr= I 
channel
30 60 90 120
0 ut=O 150
S 330A 180
30 270 240 210
1.0
wall
2.0
0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8
T/yAx
Fig. 15 Analytic solution of the dimensionless temperature distribution in
oscillating flow in a parallel plate channel and its walls as a function
of dimensionless transverse position at different oscillation phases (t:
(a) a=I; (b) a=10.
16
Neglecting axial conduction in both the wall and the fluid, the local and
instantaneous axial heat flux can be expressed as
4(y,t) = p C U(y,t) T(0,y,t) (1,10)
The time averaged total heat flow rate, then, will be obtained by integrating
this quantity over the channel width and the oscillation period, i.e.
C 2 n +a
_= p f U(y,t) T(0,y,t)d dd(t) (1,11)
2 o a
By means of Equations of (1,3), (1,4) and (1,7), and after some rather lengthy
but straight forward manipulations, the effective thermal diffusivity of the oscillation
flow is found to be
Pr[(1 H)I + (1H)1] + (Im H) + (1mH)
eff2 (1,12)
16 a2 (Pr2 1) 1 tanh(tfa)
d0x2 VXa
where
I = F/ca tanh(vTa) (1,13)
m = VPr a tanh(Vi7Pr a) (1,14)
with the bar indicating the complex conjugate of the function shown and H being
defined by equation (1,9).
17
Two graphs of the nondimensional rff/a x2, as a function of Womersley
number for different Prandtl numbers, are presented in Fig. 16(a) and (b). The first
one, (a), for e=2 representing the case of wall thickness being the same as channel
width and the second one corresponding to insulating walls which is mathematically
equivalent to the case of conducting walls with zero thickness. The curves shown are
seen to have a single maximum point which shifts to lower values of a as Pr is
increased. This is termed the tuning effect because the peak indicates the value of
channel width at which the heat transfer is optimized. These tuning points corre
spond to the condition where the thermal diffusion time from channel axis to the
walls is approximately equal to half of the oscillation period. The observed drop off
in the peak value of r!C/alx2 for very small Prandtl number in the insulating case
is attributed to the inability to store much heat in the boundary layers which become
quite thin for fluids with small kinematic viscosity v such as in liquid metals.
Note that the second set of curves, Fig. 16(b), also present information for
the enhanced axial mass diffusion by oscillating flows since insulating walls give the
same boundary conditions as those for diffusional mass transfer and both of them
come from the same type of equations with the only difference being the use of
Prandtl number for heat transfer but the Schmidt number for mass transfer. One
finds that the peak value of rgff/Ax2 for Pr=l is about 102 in the vicinity of a =3
while the corresponding value for the effective mass diffusivity for Schmidt number
1000 at the same Womersley number falls down to 5x10s by considering the curve
for Pr=103 as a curve for mass transfer for a Sc=10 fluid. Since most liquids have
(a)
e=b/a=2 (conducting walls)
10 
10 a Pr=10 /
< 10 4
10
10 IT 3
10
103 10 a 10 ' 1 10 10 z 10 S
a
(b) 1
(b ) i n1  i iiiiil i ,i,,. ii 1'" 1 ii i i l "
Se=b/a=l (insulating walls)
10 1
10 
< 10 Pr= 103
3 10
100
10 4
10
10 10 10 1 10 10 2 10 3
O(
Fig. 16 Analytic solution of the dimensionless time averaged effective thermal
diffusivity enhanced by oscillating flow in a parallel plate channel as
a function of Womersley number a at different Prandit numbers: (a)
e=2 (conducting walls with a=p ==1); (b) e=1 (insulating walls).
19
a large value for the ratio of Schmidt number to Prandtl number (about 1000 to 1
for water, for example), a really large increase in heat transfer with a negligible
enhancement of diffusional mass transfer can be achieved, and thus, the process of
heat transfer by thermal pumping in liquids can be considered as essentially free of
mass transfer provided the connecting channel length is longer than the tidal
displacement. Note that this same observations does not follow for gases where
Schmidt number and Prandtl number are approximately the same.
The tuning curves of effective thermal diffusivity for several different liquid
solid combinations in configurations of equal wall and channel thickness (e=2) are
shown in Fig. 17. There is no major difference in the maximum Ki, among these
different material combinations and all curves give a maximum ratio of Kff to oAx2
of about 0.05. It is clear that the effective maximum thermal diffusivity at the peak
of the tuning curves is directly proportional to the frequency of the fluid oscillation
and to the square of the tidal displacement. This means that in the optimization pro
cedure for a thermal pump, o and Ax should be made as large as possible. The
channel width a, then, should be determined from the valve of a corresponding to
the maximum effective thermal diffusivity and the kinematic viscosity v of the
working fluid. For example, at wo=2xr rad/s in the waterglass combination, the
optimum charmel half width will be a =2mm for optimum heat transfer.
The effect of wall thickness on the heat transfer process in the thermal pump
is shown in Fig. 18, where a plot of ./foAx2e versus e for several values of a2Pr
for the liquidsolid combination of liquid lithium and stainless steel are presented.
10 a
101 C
102
Lx
3
10 '
10
10 6
10 3
Fig. 17
0.025
0.020
0.015
3 0.010
0.005
0.000
Fig. 18
10 a 10 1 1 10 10 2
10
a
Analytic solution of dimensionless time averaged effective thermal
diffusivity enhanced by oscillating flow in a parallel plate channel at
e=2 (conducting walls) as a function of Womersley number a for
several different liquidsolid combinations.
2 3 4 5
Analytic solution of dimensionless time averaged effective thermal
diffusivity enhanced by oscillating flow as a function of cladding
thickness for liquid lithium in a parallel plate channel with stainless
steel walls.
21
These curves show that an e=2 geometry, corresponding to equal wall thickness and
channel width, is about optimum when an appreciable fraction of the heat can be
stored in the bounding walls. The reason for normalizing Kef by the extra factor e
is to take into account the fact that the walls take away a fraction 1/e of the cross
sectional area of the thermal pump available for the fluid oscillations.
Integration of the local insantaneous heat flux over the oscillation period
yields the time averaged axial heat flux distribution over the cross section, while an
integration over the channel width yields the instantaneous axial heat flow through
the whole channel as a function of time. Fig. 19(a) and (b) show the behavior of
the two partially integrated, nondimensional products of U(y,t) and T(0,y,t) for the
case of a waterglass combination with e=2. From (a) one observes that the largest
value of UT/yoAx2 occurs for a=1 which is near the tuning point for this particular
condition. Note the much smaller values at a=0.1 and a=10 which occur at de
tuned conditions. The flux transport becomes confined to the vicinity of the walls as
the Womersley number becomes large. Fig. 19(b) shows that the thermal flux is
time periodic at a frequency twice that of the fluid oscillations and that only near
tuning conditions is the positive flux not essentially canceled by the negative heat
flux.
In the following chapter, Chapter II, the geometry for which this numerical
study was performed will be explained in detail. The assumptions and the mathemat
ical models used to present two different physical situations will also be shown there.
Chapter II will focus on the numerical method employed including a brief derivation
(a) 1.0 
0.8
0.6
0.4
0.2
"' 0.0
0.2
0.4
0.6
0.8
1.0
0.02
(b)
0.5
0.5
1.0
Fig. 19
0.00 0.02 0.04 0.06
UT/7yAx2
60 120 180 240 300
0.08
360
Ct
Analytic solution of heat transferred by an oscillating flow of water in
a parallel plate channel with glass walls for different a: (a) time
averaged value as a function of dimensionless transverse position; (b)
cross section integrated value as a function of oscillation phase.
23
of the associated Finite Difference Equations. The results obtained from the numeri
cal investigation will be presented in Chapter IV, with some comparisons of the
results with analytic solutions discussed in the present section. Concluding remarks
will be found in Chapter V. Some materials useful for a clearer understanding of the
VelocityPressure correction in the SIMPLE algorithm are summarized in Appendix.
CHAPTER II
MATHEMATICAL MODEL
The mathematical model for the numerical study presented in this dissertation
is explained in this chapter. First, the geometric configurations for which the
numerical investigation were performed are described. Then, the governing
equations to be solved and boundary conditions assigned are given.
2.1 Geometric Configurations
Two different configurations based upon the thermal pump geometry shown
in Fig. 12 were chosen for the present numerical study. These are shown in Fig. 21.
Some simplifications were introduced for the purpose of computational convenience.
Note also that, to more clearly show details, the axialtransverse ratio of the
configurations shown is different from their true value used in the calculations as
discussed in Chapter IV. Both configurations essentially consist of a 2D parallel
plate channel of width 2a and length L connected to two 2D rectangular end reser
voirs of width 2b and time averaged length c. There is a piston moving back and
forth sinusoidally in the axial direction in each of the reservoirs. Choosing a rectan
gular shape for the reservoirs allows calculations in Cartesian coordinates, although
a coordinate transformation is still necessary because of the moving boundaries.
24
(a) x=(c+L). fcosW1
Th insulating Tc
I 1 I I
X= (c+L+ ACOS(A
2\r 2bZ
b
 tttttttt" tttttttt
Th Tc
Fig. 21 Configurations of the thermal pump used in the numerical
investigation: (a) conduction model; (b) cross flow model.
(b) x=(C+L) 1,coswtt
26
The coordinate system used in the physical domain is also shown in Fig. 21.
The origin is set at the center point in both axial and transverse directions. The axis
of the connecting channel and reservoirs was chosen as the x axis and then they axis
is in transverse direction. Based on this coordinate system, the positions of the
moving boundaries (piston surfaces) can be expressed as
XL, = (c + ) Acos(Ot) (2,1)
2 2b
x, = (c + ) cos(ot) (2,2)
2 2b
for the left and right pistons, respectively. Note that the piston movements are in
phase in order to satisfy the law of mass conservation for the assumed incompressible
working fluid entirely filling the channel and reservoirs.
The first configuration, shown in Fig. 21(a) and named the conduction model
because of the way that heat is supplied and removed from the end reservoirs, is a
closed system for mass transfer. All walls except the piston surfaces are stationary.
The channel side walls are thermally insulating as indicated by the oblique shading
lines, while the reservoir side walls including the piston surfaces are conducting and
maintained at different temperature at the opposite sides displayed by the blank (for
cold at T,) and black (for hot at T,) regions. The configuration is symmetrical about
the horizontal center line corresponding to the x axis. This symmetry allows the
calculation to be undertaken in only the upper half of the domain as long as suitable
boundary conditions are applied at the center line.
27
The second configuration is shown in Fig. 21(b). It uses fluid cross flow to
add and remove heat from the end reservoirs and is termed the cross flow model.
It is assumed that the cross flow enters and exits the reservoirs at a constant but
small velocity V, perpendicular to the reservoir side walls. All walls including the
piston surfaces are insulating in this model. The cross flow is maintained at different
temperatures on the opposite sides as they come into the system. The thermal
conduction at the cross flow exits (i.e. transpiration) is excluded in the numerical
study since it is negligible compared with the heat carried by the flow. This configu
ration no longer produces symmetric flows about the center horizontal line and the
computation thus has to be done over the entire flow domain.
A very useful characteristic for both configurations is that all information
needed to describe the flow and heat transfer in one oscillation phase step can be
readily obtained from the solution at another phase step one half period earlier as
soon as the flow pattern and temperature distribution in whole physical domain have
reached a final time periodic state. Time periodic conditions existing in the right end
reservoir at a fixed phase point in the oscillation cycle will correspond to those found
in the left reservoir half an oscillation cycle later. Furthermore, if temperature is
normalized so that the given temperatures Th and T, in opposite reservoirs are the
negative of each other, the temperature distribution at wt=1800 are the same as that
in the reflected image of the geometry at wt= 0 except for a sign change. The same
thing happens for every pair of solutions at wt and ot+1800. It is because of this time
symmetry that the computations need be taken over only the range of O"stsl800,
28
which significantly reduced the CPU time and storage required in the numerical
investigation.
2.2 Governing Equations and Boundary Conditions
The general assumptions in this study are following.
a) The working fluid is a homogeneous, Newtonian fluid;
b) The working fluid is incompressible
constant density;
c) The working fluid has constant material properties
constant viscosity,
constant thermal diffusivity,
constant thermal conductivity;
d) All body forces are negligible;
e) The viscous heating term in the energy equation is negligible;
f) The flow is laminar.
As shown in Section 1.3, liquid is a suitable working fluid in the thermal pump
for heat transfer without mass exchange. The incompressible fluid assumption is
valid for most liquids. The main advantage of the constant material properties
assumption is the ability to separate velocity and pressure solution from the tempera
ture solution. It was found that in order to reach the time periodic state, the number
of oscillation periods needed for temperature is much more than that for velocity,
while the CPU time used for solving velocity and pressure distribution in each period
29
is much longer than that for temperature. Calculating velocity and temperature
separately did save considerable CPU time.
The dependent variables to be solved for are the x component of velocity, U,
the y component of velocity, V, the pressure P and the temperature T. They and the
independent variables x, y and time t are nondimensionalized as follows.
x* x y t* =t (2,3)
= .. y =_ = (2,3)
a a
U* U V* VP' (2,4)
wa wa p o2a2
ST (Th + T)/2
T* (2,5)
(T T,)/2
where the quantities with are dimensionless and
a half width of the channel;
W angular frequency of the piston movement;
p fluid density;
Th temperature at the hot end;
Tc temperature at the cold end.
After nondimensionalization, the governing equations read
V + V = 0 (2,6)
U, + (UU)x + (VU)y = Px + (1/a2) (U + U )
(2,7)
V, + (UV) + (WV) = P + (1/a2) (V, + V) (2,8)
T, + (UT)~ + (VT), = (1/Pra2) (T,, + T) (2,9)
where the subscripts indicate partial derivatives and
Pr= v/K Prandlt number;
a= a / v Womersley number;
Note that a conservation form of the governing equations is used here. All
quantities in the equations are dimensionless and the nondimensional mark has
been omitted for simplicity. This notation, in which dimensionless quantities are
termed without superscript *, will be used from this point through the end of the
dissertation except where the quantity mentioned needs to be expressed in dimen
sional form.
In view of the geometry explained in Section 2.1 and the nondimensiona
lization shown above, the boundary conditions can be mathematically expressed as
follows:
Velocity:
1) U 0 at the stationary walls; (2,10)
= V 0
2) {U = sin t
2b at the piston surfaces; (2,11)
V= 0
3) u=0
V
a
at the cross flow entrance;
au
ay
4)
along the symmetry line y = 0.
(2,12)
(2,13)
V= 0
Temperature:
1) aT
an
2) T =+1
3) T =1
4) 0
ere in Eq. (2,12) is
where VF in Eq. (2,12) is
at the insulating walls and symmetry line y =0; (2,14)
at the hot walls and hot cross flow entrance; (2,15)
at the cold walls and cold cross flow entrance; (2,16)
at the cross flow exits. (2,17)
dimensional and in Eq. (2,14) indicates that the partial
iT
derivative normal to the walls.
CHAPTER III
NUMERICAL TECHNIQUE EMPLOYED
The governing equations (2,6)(2,9) with the boundary conditions (2,10)(2,17)
specified in the physical domain, which have been presented in Chapter II, have to
be solved numerically and thus require a suitable numerical method. The existing
SemiImplicit Method for Pressure Linked Equations (SIMPLE) of Patankar19 and
of Patankar and Spalding20 is one such method available for dealing with heat
transfer problems in incompressible viscous flows, but it needs to be modified here
by taking into account time periodicity and boundary movement, which are present
in the fluid oscillation and associated heat transfer problems investigated in this
dissertation. Such a consideration leads to the extended SIMPLE algorithm, termed
SIMPLETP (for Time Periodic), developed in this investigation and to be described
in some detail in this chapter. First, the coordinate systems in both the physical
domain and computational domain and the transformation of the governing equations
from the former to latter are described. Second, a brief derivation of the Finite
Difference Equations and the treatment of boundary conditions are presented. In
Section 3.3, an overview of the numerical iteration procedure SIMPLETP is given.
Finally, a Vectorized Line Group algorithm for solving the associated system of
algebraic equations also developed in this investigation is presented.
33
3.1 Grid and Transformation of Governing Equations
It is most convenient to transfer the time periodic physical domain to a time
independent computational domain. In other words, the grid system in the physical
domain which is movable and nonuniform needs to be mapped to a grid system in
the computational domain which is fixed and has even spacing in order to allow the
numerical calculations to be carry out properly. For the physical domain geometry
shown in Fig. 21, it is found convenient to break both of the domains into three
subregions when making the mapping. The two grid systems and their block corre
spondence are shown in Fig. 31.
The grid in the physical domain at three different phases of piston movement
are shown in Fig. 32. One can observe from this figure that the grid is fixed for the
middle blocks corresponding to the connecting channel, but movable in the other two
blocks representing the end reservoirs of the thermal pump. The grid spacing in the
area of each reservoir near the connecting channel, where a cross flow can also enter
or leave the reservoir in they direction, is also independent of time. This subdivision
is of advantage for specifying the boundary conditions correctly and easily when using
cross flow. Note that the same grid structure was used for all cases in both con
duction and cross flow models in order to allow a good comparison, even if it is not
necessary for the former. The grid points in the rest of the area of the reservoirs
move like an accordion in the x direction because of the piston oscillation. The y
components of the grids were established at the beginning of each run, while the x
components of the grids had to be generated at each time step.
(a)
(b)
(c)
Fig. 31 The correspondence of blocks between the physical domain and the
computational domain: (a) grid in the physical domain at ot=0; (b)
time independent and uniform grid in the computational domain; (c)
grid in the physical domain at ot=180.
(a)
(b)
(c)
Fig. 32 The movable and nonuniform grid in the physical domain:
(a) =t=00; (b) at=900; (c) ot=1800
36
In all the three subregions, the grid points are nonuniformly spaced such that
those regions near the walls and at the edge of the jet region at y=1 where large
velocity and/or temperature gradients are expected to exist have higher grid density.
In order to generate the nonuniform grid spacing, the stretching function found in
the book by Anderson21 was used. It has the form
(<+1) (p1)
(p +12 .1 (3,1)
,where X and x are the position of grid points before and after stretching; 0 is the
stretching parameter which controls the stretching strength. P is different in diffe
rent subregions and in different directions. The values of P's in the same direction
for different subregions were chosen so that the grid size nearest the dividing line
between subregions were equal.
The relationship between the coordinates (x,y,t) in physical domain and
(,, rl, r) in computational domain can be expressed as
= ((x,t)
n =1 (y) (3,2)
r=t
Under the condition that the product of the Jacobian and its inverse are equal to
unity, as is required for one to one mapping, the following expressions hold.
=1 1 x x (3,3)
x y7 xt
By means of these, the governing equations (2,6)(2,9) in the physical domain
can, after some chain rule manipulations, be rewritten in terms of 4, r7 and r as
1 + V, = 0 (3,4)
x Y7
x 1 1 t P 1 1 U (3,5)
U7JU + (UU),+(VU), = Pg+4! U]+iUj
x 1 + (3,6)
x x y
TXTT+(U), () 1 1 1+ (3,7)
x X y j. Pra2 x t x Ex P "y
Note that there is a new source term appearing in each of the momentum and
energy equations which represents the effect of the grid movement in the physical
domain.
38
3.2 A Brief Derivation of the Associated
Finite Difference Equations and Boundary Conditions
As in the SIMPLE algorithm, the same staggered grid, as shown in Fig. 33,
was also used in the SIMPLETP. The pressure and temperature are calculated at
the grid cell center designated by o's while the velocity components U and V are
determined at the center of the cell walls indicated by arrows. By means of this
staggered grid configuration, the checkerboard pressure effect introduced for non
staggered grids can be eliminated and thus allows the VelocityPressure correction
to converge easily. In SIMPLE, a control volume approach was used to derive the
Finite Difference Equations (FDE's) in this grid configuration, and a derivation,
based on a stationary grid in the physical domain can, be found in the book by
Patankar19. The derivation of the FDE's in the present SIMPLETP followed the
same procedure, but some modifications had to be made since it was carried out in
the computational domain. A brief derivation of the x momentum equation is
presented below as an example of the derivation applied in the SIMPLETP.
The notation used here is the same as that of Patankar19 and is shown in Fig.
34. The location of U where the differential equation is going to be discretized is
designated by "P" while the location of its four neighboring Us are indicated by
"E"(East), "W'(West), "N"(North) and "S"(South), respectively. The four centers of
cell walls are marked by "e", "w", "n", and "s" also, but in lower case, according to
their directions away from the cell's center. Note that the position indices are placed
on the position of superscripts in order to differ from the partial derivatives which
are expressed by subscripts in the governing equations.
j+l 1
0
I I 
 0  0 0 0 
 t t t t 
o 0  o 0
o 0 0
L 
I b
The staggered grid in the computational domain.
I I I I I
Fig. 33
9
O
 O0
L
* O
n
P
1 i i+1
(b)
i1 i i+1
Typical cells for the velocity component U: (a) an interior cell; (b) a
boundary cell with its south wall at the domain boundary.
0 t
j+1
j
j
jl
a
r0
" O 
Fig. 34
I I 1
I ) I 1 1 1 1
t
41
The equation (3,5) can be rewritten as
xhy Ur yrU + y PE + Y7 UU + x, V o (3,8)
An integration of this equation over the grid cell S f6r and the time step Sr yields
6X6Y(UP'UO) L6Y(xPXO)(Ueu) + 6y(PePW)
6r ST
\ r (3,9)
U r U U U,
+ Sy uU E uU L] + Sx6vU vU = 0
a 2X: C 2X a } 2y
,where all U's have their values at the current time step except Uo which takes the
value of UP at the previous time step. The u and v here represent the velocity
components in the current time step also, but have their values at the previous itera
tion and thus will go into the coefficients A's or source term S. It is in this way that
the nonlinear NavierStokes equations can be solved as linear equations in each step
of the iteration.
Note that the IUs are never located at the centers of the cell walls, thus the
U's with lowercase superscripts have to be replaced by those with uppercase super
scripts (see Fig. 34(a)). The, socalled, hybrid scheme, which is a combination of
central differencing and upwind schemes described in detail by Patankar19, was used
for the convective and diffusion terms. Linear interpolation was used for the
replacement in other terms. Because of the staggered grid, PC and P" are available
without any further action.
The finally obtained FDE for U looks like
APUP =AEUE +AWUW +ANUN +ASUS +AOUO +Sc (3,10)
Where
AE = by 0, 1 uel + MAX(u e,0) (3,11)
S a 2(XE X P) 2
A by 0, 1 lul + MAX( u',) (3,12)
AW= y[ (, a2(XPx W) 2
AN = 6x M 1 lvn + MAX(v n,0) (3,13)
A a2( N y ) 2 )
Sa2y P y S) 2
Ao Sxby (3,15)
8'
AP =AE +Aw N A N +As +A (3,16)
Sc = Y x o)(ue UW) (3,17)
Linear interpolation was also needed here to obtain the values of u and v at the
centers of the cell walls.
At the cells, which have one wall and sometimes two coinciding with the
domain boundary, the neighboring velocity component will locate at the center of the
43
cell wall. In Fig. 34(b), one of the boundary cells having its south wall coinciding
with the domain boundary is shown as an example. For all of the boundary cells,
proper adjustments to the right hand side in some of the equations (3,11)(3,17) were
taken according to the geometry and boundary conditions specified. Particularly for
the boundary cell shown in Fig. 34(b), one has
As = 0, 1 vI + MAX( v,0) (3,18)
x 2( P s) 2 1
The changes made in the superscript of yS and v' come from the change of the
location of Us. Equation (3,10) for this boundary cell should be changed also since
US here is known from the boundary conditions.
The derivation of all other FDE's and the treatment of other boundary cells
are straight forward and follow the same procedure explained above. More detail
and some discussion about the discretization scheme can be found in Ref. 19.
3.3 Calculation Procedure
Because of the assumptions of constant density and temperature independent
viscosity, the energy equation does not couple with the continuity and momentum
equations. This simplifies the problem. In this investigation, the velocity and
pressure field was first obtained without involving any temperature consideration.
The energy equation was then solved and the temperature distribution determined
44
based on the calculated velocity field. It was found that the temperature converges
much slower than the velocity does, which showed that a big saving in CPU time was
accomplished by being able to solve for velocity and temperature separately.
A flowchart for solving for U, V and P is shown in Fig. 35. The calculation
procedure was begun from an initial condition which was for all cases in this investi
gation as that of the fluid at rest and the pressure uniform everywhere in the entire
physical domain. In the meantime, the dimensionless time rk=1 was assigned as zero
and a basic time step size rB was defined. Starting from this point, U, V and P at
tk+1= rk+ r were determined based on the grid generated in the physical domain
for this time step by means of the standard SIMPLE algorithm. In the iteration
process, the values for the dependent variables either at the previous time step or
those existing at the same rk but in the previous oscillation period, depending on
which was closer, were chosen as the initial guesses. Typically after stepping off
several oscillation cycles, the values at the same t in the previous cycle gave the most
rapid convergence.
The maximum ratio of the residue at each grid cell which came from substitu
ting the nonconverged velocity into the continuity equation (3,4) to the product of
the local velocity and the cell boundary size was used as a convergence indicator cc.
Whenever the value of a, was less than a prescribed small value e, the solution was
considered to have converged and one could proceed to the next time step. It was
found that sometimes the VelocityPressure correction was unable to achieve conver
gence and the value of a, oscillated around a constant value larger than e, after
Fig. 35 Flowchart for determining the time periodic velocity components U, V
and pressure P.
46
several time steps, from the initial conditions or from a converged solution, were
taken. Under this situation, convergence could be reestablished by a continual
halving of the time step increments bt until they were sufficiently small. Once
convergence was again obtained, an attempt of doubling tb was taken in order to
use as large a time step size as possible.
After the time had been stepped off to r = t, a check for the convergence to
the final expected time periodic state was undertaken. This consisted of essentially
determining the quantities
1 imax ]max
a = imamax '=1 i=l (3,19)
(U j
i mamax i=1 j=1
and
mx max max
SYmaax i=1 j=1 (v(2=o)
ima max i=1 j=1
which represent the difference between the values of the velocity components at fixed
spatial position between period n and n+1. Note that the symmetric conditions
about the vertical center line have to be used in these formulas too. It was
considered that the periodic solution had been reached when both of them were less
47
than a given small value ep. If the solution was not periodic under the criteria
mentioned above, another cycle of calculation would be started by setting z = 0 and
6r= 6rB.
An almost identical procedure was followed for determining the temperature
distribution based on the velocity obtained (see Fig. 36). The major difference here
from that for calculating U, V and P was that there were no iterations corresponding
to the VelocityPressure correction in each time step, thus the variable time step size
was not needed for T. Since the calculation for temperature was associated with the
final time periodic velocity even at the beginning of each run, there were no initial
conditions existing in this situation. A linear temperature distribution in the t
direction and uniform in rn direction was used as the initial guess at r = 0 of the first
period.
3.4 A Vectorized Line Group Method
for Solving the System of Algebraic Equations
The method for solving the associated system of algebraic equations always
play an important role in any overall numerical method for multidimensional flow
and heat transfer problems. A LineByLine iteration method (also called Successive
Line Relaxation method) is commonly used in the SIMPLE algorithm for 2D or 3D
problems. In this method, one picks a grid line, say in the t direction, and assumes
that the dependent variable at its neighboring lines (i.e. the '1 direction neighbors
Fig. 36 Flowchart for solving the time periodic temperature distribution based
on the calculated velocity.
49
of the points on the chosen line) are known from the last iteration. By doing this,
a system of simultaneous equations with a tridiagnal coefficient matrix is formed,
which reads
A A 0
A A A
3j 3j 3j
0 ... ...
O  .. ...
o
0
* *" 0
0
W P E
A A1M AJ
AIM2j AIM2j AIM2
w P
.... AIM_ij AM1j.
where A's are coefficients,
right hand side which can be expressed as
RHS2,
RHS3,
=W + N +A S +AO 0 OC
= A2 (P+A2 2jP1+l 2j*P2j1+A2 92j+S2
N +AS ,0+ 0 ,+SC
3j *(P3jtl 3j* 3j1 + 3j *3j +3j
(3,22)
RHSM2j = AIM2j* IM2j++AIM2j* qIM2j +AIM2j*(IM2j+SIM 2j
RHSIM J (PIMj l*' M2 N
RHSIMlj = AAIM,. pQMj+A Itlj* (IM1j.+
AS j ,0 0 o C
IMlj ffIMljI'llM1j +A lMl +jM1j
This kind of system of algebraic equations can be solved by means of the Thomas
algorithm, a direct elimination method (see Anderson21). The forming and solving
(p2j
(p3j
(PIM2j
(PIMljj
RHS2j
RHSzj
RHSIMj
RHSIM
(3,21)
> =
50
procedure, then, was used for all lines in the same direction (the E direction in our
example) and repeated until there was no significant changes in the dependent vari
ables. The LineByLine method is widely preferred because it takes the advantages
of direct and iteration methods together to gain a fast convergence and good accu
racy for the VelocityPressure correction iterations. More discussion concerning the
details of this method can be found in Ref. 19.
The above described method, however, is not suitable for vectorized calcula
tions possible with supercomputers such as the CRAY and other newer machines.
Roughly speaking, vectorized calculation requires that there is no dependency
between the calculation operations in any pair of DO loop indices within the inner
most DO loops. Once this requirement is satisfied, the calculations are performed
almost in parallel and the CPU time used is reduced accordingly (see Grenzsch22).
A flowchart for the LineByLine method with lines in Z direction is shown in Fig.
37. There are two innermost DO loops which are marked by the dashed lines in the
flowchart and both have "i" as their indices according to the line direction. It is easy
to see that Aj depends on A j, RHSi, is calculated from RHSi., in the first inner
most DO loop and that (ij is obtained from (pi,j in the second innermost DO loop,
and that this procedure does not meet the requirement for a vectorized calculation.
A Vectorized Line Group method was developed in this study to overcome
the difficulties mentioned above especially since it was clear that the computations
for time periodic problems involve a great number of time steps, and thus some
attention have to be paid to the reduction of the CPU time used. Also a super
Fig. 37 Flowchart for the LineByLine iteration method for solving the
associated system of algebraic equations. Dashed lines represent the
innermost DO loops.
52
computer resource (CRAY/YMP), employing vectorized processor, was available for
this study through the Pittsburgh Supercomputing Center.
By use of this new vectorized method, all lines in one direction, the f
direction for example, were divided into two groups (one for the even and the other
for the odd numbered lines) which lead to two groups of simultaneous algebraic
equations and a corresponding two sets of tridiagnal matrixes in the same way as in
the LineByLine method. The Thomas algorithm could then be applied to each line
in one group at the same time and the calculation could be vectorized. By altering
to update the dependent variable for the two groups, a converged solution could be
obtained some three times faster than that obtained by the LineByLine method.
The flowchart of the Vectorized Line Group method is shown in Fig. 38.
There are three innermost DO loops marked by dashed lines, in which the first and
third are corresponding to the innermost DO loops in the LineByLine method
shown in Fig.37. The dependencies between the pairs ofA's, RHS's and (p's are still
on the subscripts "i", the same as that in the LineByLine method, because the same
line direction is chosen. However, the indices of the innermost DO loops are "j",
which is not the same as that in the LineByLine method. It is because of this
difference that the requirement of vectorized calculation is satisfied. The only
drawback of the Vectorized Line Group method is that it requires a rectangular
computational domain. Fortunately this is achievable for most anticipated problems,
when using domain subdivisions.
Fig. 38 Flowchart for the Vectorized Line Group iteration method for solving
the associated system of algebraic equations. Dashed lines represent
the innermost DO loops.
54
When the Vectorized Line Group method was applied to one of the three
subdomains shown in Fig. 31, the values of the dependent variables from previous
iteration in the neighboring subdomain were used as the boundary condition along
the interface boundary between the two subdomains. The same treatment was
undertaken for the all three subdomains. Since the solving procedure is iterative, the
continuity condition at such boundaries is insured by updating the values of depen
dent variables there alteratively.
CHAPTER IV
RESULTS AND DISCUSSION
Ten cases with different values on the control parameters were investigated
in this numerical study. Those parameter values are listed in Table 41.
Table 41. The list of 10 cases studied in the numerical investigation.
Case b c Ax rs a Up Re V(
No. (cm) (cm) (cm) (cm/s) (cm/s)
1 2 5 0.4 0.013 0.5 0.003 1 0
2 2 5 5 0.167 1 0.125 50 0
3 2 5 6 0.2 3 1.35 540 0
4 2 5 20 0.667 3 4.5 1,800 0
5 2 5 6 0.2 10 15. 6,000 0
6 2 5 20 0.667 10 50. 20,000 0
7 2 5 40 1.333 10 100. 40,000 0
8 0.6 10 6 0.2 3 4.5 540 0
9 2 5 20 0.667 10 50. 20,000 1
10 2 5 20 0.667 10 50. 20,000 10
Note that the connecting channel has half width a = 0.1cm and length L = 30cm
for all cases. The working fluid in this study is assumed to be water with a kinematic
56
viscosity of v=0.01cm2/s, density p=lg/cm3, specific heat C,= 4.18J/gK, and Prandtl
number Pr=1 (corresponding to water at high temperature). The zero cross flow
(Vc=0) represents the conduction model, and Re= a2Ax/a is the corresponding
Reynolds number based on the maximum piston velocity Up= oaAx/2b and the
reservoir half width b, but expressed in terms of Womersley number a, tidal
displacement Ax and the channel half width a through the requirement of mass
conservation and the definition of Womersley number.
The numerical solutions of the governing equations described in Chapter II
obtained by use of the SIMPLETP algorithm explained in Chapter III for the ten
cases and some discussion about several interesting phenomena found from those
results will be presented in this chapter with some work done to validate the
numerical results at the beginning. Note that all dependent variables, the velocity
components U, V, the stream function 0i, which is calculated from U, V, the pressure
P, the Temperature T, as well as the independent variables x, y shown in the figures
of this chapter are dimensionless quantities which are nondimensionalized by the
method shown in equations (2,1)(2,3), although the marker is omitted as
mentioned in section 2.2.
4.1 Validation of Results
In any numerical process involving calculation by computers, errors always
arise. It is because of this that some estimate of the magnitude of these quantities
is included in all numerical studies.
57
Since the purpose of this dissertation is to obtain a qualitative guidance to the
usage of the thermal pump technique and no detailed experimental data for the time
periodic velocity patterns and temperature profiles in such configuration exist with
which to compare the numerical data, only a qualitative error analysis is carried out.
That is, to demonstrate that the numerical results obtained are qualitatively close to
the true solution of the governing equations for the specified boundary conditions.
The number of grid points chosen is very important for numerical studies of
heat transfer in fluid flows. If a few grid points are taken, the results will be unable
to display the correct structure of the velocity pattern and temperature profile. A
large number of grid points, on the other hand, will need too much computing time,
which may either waste the computer resources or require such long calculations that
they can not be handled by present computers. For time periodic problems, it is
more desirable to use a small number grid points or, in other words, a large grid cell
size because generally a smaller time step size is required for a smaller grid cell size
in order to achieve convergence. If the time step size is very small, the number of
necessary time steps per oscillating cycle will become very large, and thus an
extremely large number of the total time steps will be needed since several
oscillation cycles are required in order to reach the final time periodic state.
A relatively low number of 91x71 nonuniform grid points were used in this
numerical investigation. Fig. 41 compares the stream line patterns for Case No.6 at
several different oscillation phases for 91x71 grid points (left part of the figure) and
121 x01 grid points (right part). The procedure used to obtain the stream lines from
grid: 91x71 grid: 121x101
Fig. 41 Stream line patterns for Case No.6 calculated with different number of
grid points: (a) grid 91x71; (b) grid 121x101. Increment between
neighboring lines is A or=20 in nondimensional units.
59
the numerically calculated velocity components will be explained in the next section.
There is some difference between the two flow patterns but it is not significant. The
changes from one to the other are only in detail and the basic structure of the flows
does not change. The dimensionless basic time step size 6 r is ir/360 for the grid
system 91x71 with several halvings occurring from the basic size to Cr/1440 only in
the first oscillation cycle (see Section 3.3), while the smaller basic time increment
6 B= ir/1440 was needed for the 121x101 grid in order to get a converged solution.
The CPU time used for the latter is more than 4 times of that for the former for the
same convergence criterion. The slight changes in flow structure and the big saving
in calculation time lead to the decision to choose the sparser grid structure of 91x71
in the calculations for other cases.
The value of e, which is the criterion on the convergence indicator oc of the
VelocityPressure correction, is set at 3%. A typical number of VelocityPressure
corrections required at each time step in the SIMPLETP technique for Case No.6
is 11 for this setting in most oscillation cycles except the first one. In the first cycle,
especially in the early part of the cycle, it needs more VelocityPressure iterations to
obtain a converged solution for e =3%. The criteria for reaching the final time
periodic state was taken as 1% for the velocity and as 0.01% for the temperature.
Twelve oscillation cycles were typically needed for velocity convergence, while 200
cycles were required for temperature. There were three other criteria needed for
solving the system of algebraic equations associated with velocity, pressure and
temperature. The domain averaged ratio of the difference between calculated
60
dependent variables from two successive iterations to the values of the dependent
variables obtained from the current iteration was chosen as a convergence indicator.
The criteria for velocity, pressure and temperature were 1xlO4, 1x10l, and 1x10"
respectively. For velocity and temperature, less than 10 iterations were needed to
meet the criteria but usually more than 300 iterations were needed to have the
pressure converged. The numbers of iterations needed to satisfy these criteria
settings varied slightly from case to case for the different values of parameters listed
in Table 41. 3.5 hours of CPU time on a CRAY/YMP at the Pittsburgh Super
computing Center was typically required for obtaining the velocity and pressure
values for Case No.6 using the 91x71 grid under the above converge criteria. An
extra 0.5 hour was needed to find the corresponding temperature field.
As shown in Section 1.3, there are closed form solutions for the velocity and
temperature profiles in oscillating flow in a parallel plate channel connected to end
reservoirs at x= + The flow is 1D in the axial direction and the variations of both
velocity and temperature along the transverse direction are independent of x. Note
that the ratio of the tidal displacement to the channel length, r,, in this situation is
always zero for whatever value of Ax except infinity. When the end reservoirs are at
a finite distance x= L/2, and thus r, is nonzero, a distortion of the velocity and
temperature profiles from what they are for infinite channel length will occur and the
above x independence will disappear. The distortion and the x dependence increase
when rL becomes large and under these conditions the solutions have to be obtained
numerically. On the other hand, when r, is small, the distortion should be small and
61
the numerical results for velocity profile and temperature distribution at the middle
of channel should be close to the analytic solution for the channel of infinite length
as long as other conditions are the same and, of course, the numerical results have
converged. Based on this point, a comparison of numerical results obtained in the
middle of connecting channel for the case of small r, with the analytic solution
under the same a and same insulating wall conditions for temperature will show
whether the numerical solutions are correct or not.
The first comparison involves the curves, shown in Fig. 42 and Fig. 43, for
the velocity component U and temperature T in Case No.5 versus the oscillation
phase ot for an entire oscillation cycle after the time periodic state has been reached.
The numerical results were taken at the position x= y=0.819 in the dimensionless
coordinate system shown in Fig. 21. A 5 phase shift was introduced in comparing
the analytic solutions with the numerical results in order to allow the best com
parison. This phase shift comes from the difference of phase accounting methods in
the two solutions: the analytic solutions are based on the oscillation phase of the
pressure gradient, while the numerical results are based on the phase of the piston
oscillation.
The analytic solution is valid only for a channel of infinite length. A uniform
in space but time sinusoidal pressure gradient exists for this very simple geometry.
The analytic solving procedure starts from introducing this pressure gradient into the
NavierStokes equation (see Section 1.3), from which all time periodic solutions are
obtained. The phases of all other quantities, velocity, temperature etc., are delimited
40
30
20
i 10
CO 0
o
T 10
S 20
30
40 '" 1
0 60 120 180 240 300 360
ut
Fig. 42 Nondimensional velocity component U for Case No.5 at the middle of
the connecting channel as a function of oscillation phase
solid line numerical result; dashed line analytic solution.
63
0.15
0.10
S0.05
'0 0.00
Ci
S0.05
0.10
0.15 l""" i l tii l lll ll n r
0 60 120 180 240 300 360
wt
Fig. 43 Nondimensional temperature T for Case No.5 at the middle of the
connecting channel as a function of oscillation phase
solid line numerical result; dashed line analytic solution.
64
from this pressure gradient template. For a geometry of finite channel length, the
pressure gradient is no longer uniform in the entire domain, but uniform in the
region far from the ends when rL is small enough, and thus can not be introduced
a prior at the beginning of the calculation. Instead, its behavior comes part of the
numerical results. The numerical solving procedure begins by giving the movement
of pistons, their position and velocity, in end reservoirs and thus the oscillation phase
of the piston oscillation becomes the basic template. The two templates do not
coincide with each other. In other words, the nearly uniform and time periodic
pressure gradient in the middle of connecting channel generated by oscillation of the
pistons in end reservoirs does not have the same phase as that of piston movement.
Introducing a phase shift is necessary for the best comparison.
One observes from the two figures (42 and 43) that the numerical results
(solid lines) match the analytic solutions (dashed lines) very well over most parts of
oscillation cycle. By comparing the corresponding zero points of the curves in these
two figures, one finds a 87, not a 90, phase delay of the temperature curves from
those of velocity. It is this difference of 30 in phase delay that makes the net heat
transfer in axial direction possible without an accompanying net mass transfer along
the axis. In the region near the x axis (center line), the phase delay is 90 so that
there is no net heat flux along the axis there (see the curve for a =10 in Fig. 19(a)).
However, for a=3, which is near the maximum point in Kff curve (see Fig. 16), the
phase delay between temperature and velocity is 750 at the x axis where the
difference from 900 is as much as 15.
65
The second comparison was made for the curves of velocity component U and
temperature T versus y. Fig. 44 and Fig. 45 show these curves at several oscillation
phases for Case No.5 with rL=0.2. The same phase shift was introduced when
comparing the analytic curves with numerical results. The curves in (a) come from
the analytic solutions (Eq. (1,4) and Eq. (1,6)), while those in (b) are from the
numerical results at x=0. One sees that the shape of the curves are essentially
identical except for the kinks produced by the finite grid point structure used in the
numerical calculations. The velocity curves for Case No.6 are presented in Fig. 46.
The curves are still almost the same in the region near the channel walls, but there
are larger differences between the numerical and analytic solutions near the center
line. Since rL=0.67 in this case, it makes sense that large differences should exist
in this case as explained above.
A comparison of curves of pressure gradient for Case No.5 versus oscillation
phase between numerical results atx=0 and that used for analytic solutions is shown
in Fig. 47. The dashed line again represents the analytic pressure gradient and the
same phase shift as above was introduced. The behavior of the numerical pressure
gradient does match the analytic one but there is an error larger than that for
velocity and temperature shown in Fig. 42 and 43, especially near the points at
which the pistons change their movement direction. It was found that this error
could not be reduced by reducing the grid size and/or time step size. Some
simplification is made in the SIMPLE algorithm when the pressure correction
equation is established from the continuity equation (explained in the Appendix, see
(a)
0.5 
S0.0 
0.5
1.0
50 40 30 20 10 0
U(y,At)
(b)
10 20 30 40 50
0.5
0.0
0.5
1.0
Fig. 44
50 40 30 20 10 0 10
U(O,y,wt)
20 30 40 50
Nondimensional velocity component U for Case No.5 at the middle of
the connecting channel as a function of nondimensional position y at
several oscillation phases: (a) analytic solution; (b) numerical result.
(a) 1.0
analytic
0.5
ctt=0 30 60 90 120 150 180
> 0.0
0.5
1.0
0.2 0.1 0.0 0.1 0.2
T(y,wt)
(b) 1.o
numerical
0.5 tt=0 30 60 90 120 150 180
> 0.0 
0.5
1.0
0.2 0.1 0.0 0.1 0.2
T(O,y,wt)
Fig. 45 Nondimensional temperature T for Case No.5 at the middle of the
connecting channel as a function of nondimensional position y at
several oscillation phases: (a) analytic solution; (b) numerical result.
(a)
0.5
> 0.0
0.5
 1.0 ... .. ..
50 40 30 20 10 0 10 20 30 40 50
U(y,wt)
(b)
0.5
> 0.0
0.5
1.0 
50
Fig. 46 Nc
the
sev
40 30 20 10 0 10
U(O,y,wt)
20 30 40 50
ndimensional velocity component U for Case No.6 at the middle of
connecting channel as a function of nondimensional position y at
reral oscillation phases: (a) analytic solution; (b) numerical result.
69
5 0 I i i i i I i i i i l
25 _
x 0
25
5 0 I I I I I I I I I I I I I I I I I I
0 90 180 270 360
ot
Fig. 47 Nondimensional x direction pressure gradient for Case No.5 at the
middle of connecting channel as a function of oscillation phase
solid line numerical result; dashed line analytic solution.
70
also reference 19). This simplification does not affect obtaining a good pressure
field for steady flows but may introduce more errors for unsteady and especially for
oscillatory flows.
4.2 Velocity Field
It was found that the velocity field under the conditions considered in this
study can not be well presented by the normal ways, i.e. by drawing arrows at several
points in the physical domain with length proportional to the speed and in the
velocity direction at that particular location, because of the complexity of the
oscillation flows and the large differences in speed at different locations within the
end reservoirs. Therefore, plotting the isovalue lines of the time varying stream
function, the stream lines, was chosen as the mode of velocity field presentation.
From the definition of stream function r(xy,t)
U(x,y,t) = a(x,y,t) V(x,y,t) = (x,y,t) (4,1)
ay x
,the value of the stream function (xy,t) at a particular point (xy,t) can be calculated
from r(xOyOt), the value of stream function at its neighboring point (xyet). The
following formula will do this.
(xo,y,)Y (xyt)
*(x,y,t) = *(xo,yo,t) + f U(x, Y y,t)dy f V(XO,y,t)dx (4,2)
(xoyor) (x,,yt)
71
Because of the staggered grid, the values of stream function are easily obtained at
the covers of grid cells without any interpolation (see Fig. 33).
The spacing between the streamline isovalues is equal to a constant which may
vary from case to case. This method of representing a time dependent flow field
allows one to take advantage of the fact that the absolute value of the instantaneous
streamfunction gradient is equal to flow speed and hence that regions of high stream
line concentration (i.e. darkened regions) corresponding to parts of the flow field
having high velocity. The velocity is always in the tangential direction of the stream
lines, and the solid curves represent counterclockwise rotation, the dotted curves zero
rotation and the dashed curves clockwise rotation.
The stream lines in the end reservoirs at the oscillation phases of 150 intervals
for Case No.3 are presented in Fig. 48. Only the flow patterns at the phases in
lower half period Oast < 1800 are shown here. The flow profile at any point in upper
half period 180awt <3600 can be found by reflecting the picture at the corresponding
1800 earlier phase about the vertical center line (i.e. y axis). The constant interval
between two neighboring lines is 5 nondimensional units.
It is observed that as the piston in the left reservoir moves toward the
connecting channel starting from its extreme position of x = c at at=0, there
2b
is essentially one large vortex pair present with the flow along the horizontal center
line (i.e. x axis) toward the piston surface. A weaker and smaller second vortex pair
of opposite rotation sense sits in the covers nearest the channel end. Flow is seen
leaving the left reservoir starting at about wt=300 and continuing on to about
Fig. 48 Stream line pattern for Case No.3 at oscillation phases of interval
A(wt) =15" in the lower half cycle *00ot<180*.
73
ot=1500 mainly via a side flow into the connecting channel. In the right reservoir,
one can see a distinct high velocity jet emanating from the connecting channel and
moving across the entire reservoir along the x axis. It begins at about at =30, the
same time as flow starts leaving the left reservoir, continues flowing along the
channel exit until at least ot=165. The jet collides with the right piston surface at
around ot =210 and disappears at ct =285. The images of these pictures are present
in the left reservoir at ot=300 and wt=1050, respectively. The origin for this high
velocity pulse is clearly the inability for the liquid entering the reservoir during the
suction stage to accommodate itself to the wider cross section of the reservoir as it
would if the Reynolds number were low enough so that flow separation at the
entrance would not occur (see Case No.1 in Fig. 414). The appearance of a velocity
jet during the intake portion of the piston stroke has also been noted in the related
experimental work by Durst23 as well as by Kurzweg, Lindgren and Lothrop2. Note
that the large counter rotating vortex pair filling most of the reservoirs remains
during the entire oscillation cycle and its rotating sense is always such that fluid
motion along the x axis is always from the end of connecting channel and toward the
moving pistons, no matter what oscillation phase one is in. The smaller second
vortex pair also exists during the entire oscillation cycle, although it is much weaker
than the primary one. There is third vortex pair showing up in Fig. 48. It is
generated near the flow separation point near the channel end in the right reservoir
at about ot=450, shortly after the velocity jet starts, pushed toward the receding
piston by the velocity jet and finally joins the large vortex pair at about ot=135C.
74
The numerical development of the nondimensional velocity component U at
three different locations along the x axis within the left reservoir under the conditions
corresponding to Case No. 3 are plotted in Fig. 49 with (a) at a position of 2a from
the channel end, (b) at 25a, and (c) at 40a. It shows that except at the position
nearest the connecting channel end, the velocity remains negative throughout the
oscillation cycle and is thus always moving the liquid toward the piston surface. The
sharp velocity spikes seen correspond to the periodic velocity jets crossing the
reservoir which are shown in the right reservoirs of Fig. 48 since they exist in the left
reservoir during the upper half period 1800 Mt <3600. The time delay in the velocity
peak between the nondimensional position x =2 and x =25 is 0.0875 seconds and
thus indicates a velocity pulse traveling at 26.3cm/s which is only slightly less than
the rms axial velocity of 27cm/s in the channel but considerably larger than the
piston face velocity at the same instant.
The stream lines for Case No.4 and Case No.6 are shown in Fig. 410 and 4
11, respectively, using the same format as that in Fig. 48 but employing 20 non
dimensional units of interval in the isovalues of r. Similar to the results shown in
Fig. 48, there are three vortex pairs, two remaining during the entire oscillation cycle
and one existing in a portion of the cycle, and also a velocity jet in both cases.
For Case No.4(Fig. 410), the jet within the right reservoir begins at wt=30,
the same as in Case No.3, but collides with the piston surface at about ot=900, much
earlier than the wt =210 value of Case No.3, and disappears at about at=180, also
earlier than the wt=2850 result in Case No.3. The third vortex pair starts at about
a) 10
0
.10 
20
30
40
50 I
0 720 1440 2160
b) ,o
3 10
0 
n 20
S30
40
50
0 720 1440 2160
wt
C) 10
0
3 10
0 20 
'
30
40
50
0 720 1440 2160
wt
Fig. 49 Temporal approach to the final time periodic state of the non
dimensional velocity component U(x,Ot) for Case No.3 at three
different locations: (a) x=152; (b) x=175; (c) x=190.
.ot=30
I!,I i ? i
1 aOt=5600
oJ13<75}
^^^r'L ^ t
Fig. 410 Stream line pattern for Case No.4 at oscillation intervals of A (et) =15"
in the lower half cycle 00a
Fig. 411 Stream line pattern of Case No.6 at oscillation intervals of A(wt) =15
in the lower half cycle 0*ot<180.
78
tot=30, almost at the same time as when the jet pulse begins and a little earlier than
at ot= 45 in Case No.3, and joins the big vortex pair at tf= 750, which is also much
earlier than wt=1500 phase point in Case No.3. The only difference between the
conditions in Case No.3 and Case No.4 is the different values of the tidal displace
ment Ax. Case No.4 has Ax=20cm, or 10/3 times more than that in Case No.3, which
means there is a larger rms velocity in the channel for Case No.4. It is the faster
fluid movement in channel which produces the faster jet and associated shorter
appearance of the third vortex pair.
Comparing the stream lines of Case No.6 with those in Case No.4, one can not
find as large a difference as that between the stream lines of Case No.3 and Case
No.4. Note that Case No.6 has a=10 and Case No.4 has a=3, which is the only
difference between the two cases and means that the oscillation in Case No.6 is 10/3
times faster than that in Case No.4 since the fluid viscosity and channel width do not
change. The above observation reveals the fact that Womersley number no longer
has a large effect on the flow profile in the end reservoirs for large b/a ratio like its
influence on the axial velocity distributions within the connecting channel (see
Section 1.3). Instead, the tidal displacement Ax has more influence on the flow
profile in reservoirs than Womersley number does. Actually, significant changes in
the reservoir flow patterns are characterized by the Reynolds number, as will be
shown below.
Fig. 412 presents the pressure distribution for Case No.6 along the x axis at
several oscillation phases over the whole period O0sot<3600, and Fig. 413 shows the
79
corresponding stream lines at the same phase points. The pressure in most parts of
the channel is seen to vary linearly with x for most of the oscillation phases except
those near ot = 900 and wt = 270 when the pistons' movement changes from accelera
tion into deceleration. The linear pressure distribution in the channel corresponds
to the expected constant pressure gradient there, and as used in the analytic solutions
(see Section 1.3). Examining these last two figures, one finds that the largest
pressure drop occurs at the channel end during that time when the fluid enters into
the channel through that end and when the velocity jet exists from the other end.
It is apparent that the pressure drop existing at one end of the connecting channel
provides the necessary power to form the jet flow at another end. The two ends alter
their roles as the pistons change their directions of motion. The magnitude of the
pressure gradient found in the channel agrees well with the analytic solution given
by Eq. (1,5) for the same values of parameters.
Stream lines with a spacing interval of 0.05 units for Case No.1 are presented
in Fig. 414. The corresponding Reynolds number in this case has the low value of
1, which is small enough to avoid flow separations in the case of steady flow over a
step. For a part of the oscillation cycle (about half), when the pistons move essen
tially in one direction, no separations exists also in this oscillating flow situation. The
fact that flow separation does occur over rest of the oscillation cycle even for such
low Reynolds number is easily understood by noticing that this part of oscillation
period is around the points at which the pistons' motion changes its direction. The
flow pattern in Case No.l, in which both the existing vortex pairs and flow jet have
0
X
1
0
a0
X
0
250 150 0 50 150 250
250 150 50 50 1,50 250
0
x
1
250 150 60 50 150 250
0
1 0
a I
z
2 L I 1 1 i I
250 150 50 50 160 250
X
2
1
0
x
1
250 150 50 50 150 250
X
2
1
0
x 0
a 1
I I
250 10 60 50 10
250 150 60 50 150 250
cat=135"
1
2
250 150 60 50 150 250
50 0 0 50 150 50
250 150 60 50 150 250
t=315
I
0
x
14
2 
250 150 60 50 150 250
Fig. 412 Nondimensional pressure distribution along the x axis in Case No.6 at
oscillation intervals of A(ot) =45* over the whole cycle 0 art<3600.
r
ot=45' I
I
1\ I
I
I \ I
I \ I
I \ I
I \I
I
i I
I ot=225'
I
I
I
I / I
I / I
I I
I
I/ (
I ot=9O"
I I
I I
II I
nI I
I co270" I
I
I i
I I
^I I
[
e I I I
i
ot=O" I
I
i
i
I \ I
I
f
f I
I \I
ot=180'
I /,
I /(
I / (
I / I
I / 1
I / I
I / I
r/ I
I/ I
J
I
J
I
ii i:r x=2250
NI;,
Fig. 413 Stream line pattern for Case No.6 at the oscillation intervals of
d4(t) =45* over the whole period 0*at<360*
Ca=t=105 IAIII 9W, 
1.9
ot= 120(
1.8<11<0
fl450
31.411<
()t=600
(ol=750
81 9
(ft=1500
1 .3
Fig. 414 Stream line pattern for Case No.1 at the oscillation intervals of
A (t) = 150 in the lower half cycle Ostot <1800.
, rr c= 15 0
' c2>
0.6<%VO
01=900
r;
r?~y
I
wt=1 350
1H "as0.
 I
 
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81,9(56,7< 2) )/25,'$
THEl>
THEREOl:
FLUID OSCILLATIONS AND ENHANCED HEAT TRANSFER
IN A PARALLEL PLATE CHANNEL BOUNDED BY END RESERVOIRS
MAINTAINED AT DIFFERENT TEMPERATURE
BY
ALEX X. ZHAO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1990
To my parents and my beloved motherland
ACKNOWLEDGEMENTS
The successful completion of this dissertation would have been impossible if
it had not been for the assistance and help I received from numerous people. First
mention must be made for the academic guidance and financial support made
possible by my graduate adviser Dr. Ulrich H. Kurzweg. He, with his strong
background and extensive knowledge in fluid mechanics, heat transfer and applied
mathematics, gave me clear guidance in maintaining the correct research direction.
What I learned from him are not only the relevant knowledge in those fields but also
the skills needed to conduct scientific research. Two other persons who helped me
significantly in the dissertation were Dr. Wei Shyy and Dr. Ranga Narayanan.
Several fruitful discussions with Dr. Shyy helped me to construct the numerical
procedure used successfully and more efficiently for the time periodic problems in
this dissertation. The experience of working with Dr. Narayanan put me in contact
with many interesting topics in fluid mechanics, heat transfer and numerical analysis,
which made my understanding of the enhancement of heat transfer problem
considered here much more clear. I am thankful to the other and former members
of my supervisory committee, Dr. E Rune. Lindgren, Dr. Joseph L. Hammack, Dr.
ChiaShun Yih, Dr. Edward K. Walsh and Dr. Tom IP. Shih, for their time and
effort.
m
Thanks are also expressed here to the Department of Aerospace Engineering,
Mechanics and Engineering Science and the Department of Chemical Engineering
of the University of Florida for their financial assistance in allowing me to pursue my
educational goals within this lovely country. I am also grateful to the early support
of this work by National Science Foundation through the grant CBT8611254 and the
allocation of 95 Service Units for this study on the CRAY/YMP supercomputer in
the Pittsburgh Supercomputing Center.
I would also like to thank my wife Wei Zhu for being here with me in a
"foreign" country and working hard to help support the family. Without her help, 1
would not have been able to successfully complete my education. I also wish to
express gratitude to my wonderful office mate Bill Harter. Through many discussions
with him, I learned a lot about PC computers, about the English language, about
American customs and etc. Finally I wish to thank myself for putting in four hard
working years on this dissertation effort.
IV
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS iii
LIST OF FIGURES vii
LIST OF TABLES xn
NOMENCLATURE xiii
ABSTRACT xvii
CHAPTERS
I INTRODUCTION 1
1.1 Background 1
1.2 Literature Review 5
1.3 Analytical Solutions Within the Connecting Channel 9
II MATHEMATICAL MODEL 24
2.1 Geometric Configurations 24
2.2 Governing Equations and Boundary Conditions 28
III NUMERICAL TECHNIQUE EMPLOYED 32
3.1 Grid and Transformation of Governing Equations 33
3.2 A Brief Derivation of the Associated Finite Difference
Equations and Boundary Conditions 38
3.3 Calculation Procedure 43
3.4 A Vectorized Line Group Method for Solving
the System of Algebraic Equations 47
IV RESULTS AND DISCUSSION 55
4.1 Validation of the Results 56
4.2 Velocity Field 70
4.3 Temperature and Heat Transfer Coefficient 88
v
V CONCLUDING REMARKS
106
APPENDIX THE VELOCITYPRESSURE CORRECTION 110
REFERENCES 116
vi
LIST OF FIGURES
Figure
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Heat transfer in a parallel plate channel: (a) pure conduction;
(b) enhancement by oscillating flows.
A schematic of the Thermal Pump (U.S. Patent No. 4590993).
Geometric configuration for the analytic solution of flow
oscillations and the associated heat transfer in parallel plate
channels.
Analytic solution of the dimensionless velocity of oscillating
flow in a parallel plate channel as a function of dimensionless
transverse position at different oscillation phases cot: (a) a = l\
(b) a = 10.
Analytic solution of the dimensionless temperature distribution
in oscillating flow in a parallel plate channel and its walls as a
function of dimensionless transverse position at different
oscillation phases cot: (a) a = 1; (b) a = 10.
Analytic solution of the dimensionless time averaged effective
thermal diffusivity enhanced by oscillating flow in a parallel
plate channel as a function of Womersley number a at
different Prandlt numbers: (a) e=2 (conducting walls with
o=p = l)\ (b) e=l (insulating walls).
Analytic solution of dimensionless time averaged effective
thermal diffusivity enhanced by oscillating flow in a parallel
plate channel at e=2 (conducting walls) as a function of
Womersley number a for several different liquidsolid
combinations.
vii
Bigs
. . . 3
. . . 5
. . . 10
. . . 12
. . . 14
. . . 16
. . . 19
Fig. 18 Analytic solution of dimensionless time averaged effective
thermal diffusivity enhanced by oscillating flow as a function of
cladding thickness for liquid lithium in a parallel plate channel
with stainless steel walls. ....
Fig. 19 Analytic solution of heat transferred by an oscillating flow of
water in a parallel plate channel with glass walls for different
a: (a) time averaged value as a function of dimensionless
transverse position; (b) cross section integrated value as a
function of oscillation phase. ....
Fig. 21 Configurations of the thermal pump used in the numerical
investigation: (a) conduction model; (b) cross flow model.
Fig. 31 The correspondence of blocks between the physical domain
and the computational domain: (a) grid in the physical domain
at at = 0Â°\ (b) time independent and uniform grid in the
computational domain; (c) grid in the physical domain at
at = 180Â°.
Fig. 32 The movable and nonuniform grid in the physical domain: (a)
at=0Â°; (b) at = 90Â°; (c) at=180Â°.
Fig. 33 The staggered grid in the computational domain.
Fig. 34 Typical cells for the velocity component U: (a) an interior cell;
(b) a boundary cell with its south wall at the domain boundary
Fig. 35 Flowchart for determining the time periodic velocity
components U, V and pressure P.
Fig. 36 Flowchart for solving the time periodic temperature
distribution based on the calculated velocity.
Fig. 37 Flowchart for the LineByLine iteration method for solving the
associated system of algebraic equations. Dashed lines
represent the innermost DO loops.
Fig. 38 Flowchart for the Vectorized Line Group iteration method for
solving the associated system of algebraic equations. Dashed
lines represent the innermost DO loops.
19
20
23
32
33
36
38
42
45
48
50
vm
Fig. 41 Stream line patterns for Case No.6 calculated with different
number of grid points: (a) grid 91x71\ (b) grid 121x101.
Increment between neighboring lines is Ai/r=20 in non
dimensional units.
Fig. 42 Nondimensional velocity component U for Case No.5 at the
middle of the connecting channel as a function of oscillation
phase ot: solid line  numerical result; dashed line  analytic
solution.
Fig. 43 Nondimensional temperature T for Case No.5 at the middle of
the connecting channel as a function of oscillation phase ot:
solid line  numerical result; dashed line  analytic solution.
Fig. 44 Nondimensional velocity component U for Case No.5 at the
middle of the connecting channel as a function of non
dimensional position y at several oscillation phases: (a)
analytic solution; (b) numerical result.
Fig. 45 Nondimensional temperature T for Case No.5 at the middle of
the connecting channel as a function of nondimensional
position y at several oscillation phases: (a) analytic solution;
(b) numerical result.
Fig. 46 Nondimensional velocity component U for Case No.6 at the
middle of the connecting channel as a function of non
dimensional position y at several oscillation phases: (a)
analytic solution; (b) numerical result.
Fig. 47 Nondimensional x direction pressure gradient for Case No.5 at
the middle of connecting channel as a function of oscillation
phase ot: solid line  numerical result; dashed line  analytic
solution.
Fig. 48 Stream line pattern for Case No.3 at oscillation phases of
interval A(ot)=15Â° in the lower half cycle 0Â°^ot<180Â°.
Fig. 49 Temporal approach to the final time periodic state of the non
dimensional velocity component U(x,0,ot) for Case No.3 at
three different locations: (a) x=152\ (b) x = 175; (c)x=190.
Fig. 410 Stream line pattern for Case No.4 at oscillation intervals of
A(ot)=15Â° in the lower half cycle 0Â°^ot<180Â°.
54
57
57
60
61
62
64
66
69
70
IX
Fig. 411
Stream line pattern for Case No.6 at oscillation intervals of
A(Ãº)t)=15Â° in the lower half cycle 0Â°^o)t<180Â°.
.. 71
Fig. 412
Nondimensional pressure distribution along the x axis in Case
No.6 at oscillation intervals of A(o>t) =45Â° over the whole cycle
0Â°i(ot<360Â°.
.. 73
Fig. 413
Stream line pattern for Case No.6 at the oscillation intervals of
A(cot)=45Â° over the whole period 0Â°^(ot<360Â°.
. . 74
Fig. 414
Stream line pattern for Case No.l at the oscillation intervals of
A(ot)=15Â° in the lower half cycle 0Â°^(ot<180Â°.
. . 76
Fig. 415
Stream line pattern for Case No.8 in the right reservoir at
oscillation intervals of A((ot)=90Â° over the whole period
0Â°io>t<360Â°.
. . 78
Fig. 416
Stream line pattern for Case No.9 at oscillation intervals of
A(o)t) =15Â° over the lower half period 0Â°
.. 79
Fig. 417
Stream line pattern for Case No. 10 at oscillation intervals of
A((ot) = 15Â° in the lower half cycle 0Â°zo)t<180Â°.
. . 80
Fig. 418
Temperature contours for Case No.3 at oscillation intervals of
A(u>t)=15Â° in the lower half cycle 0Â°zo)t<180Â°.
. . 83
Fig. 419
Nondimensional temperature profile for Case No.3 along the
vertical line x = 153 in the left reservoir: (a) at several
oscillation phases; (b) time averaged.
. . 85
Fig. 420
Temperature contours for Case No.4 at oscillation intervals of
A(o)t) =15Â° in the lower half cycle 0Â°^ot<180Â°.
.. 86
Fig. 421
Nondimensional temperature profile for Case No.4 along the
vertical line x = 153 in the left reservoir: (a) at several
oscillation phases; (b) time averaged.
. . 87
Fig. 422
Temperature contours for Case No.6 at oscillation intervals of
A((ot)=15Â° in the lower half cycle 0Â°^o>t<180Â°.
. . 88
Fig. 423
Nondimensional temperature profile for Case No.6 along the
vertical line x = 153 in the left reservoir: (a) at several
oscillation phases; (b) time averaged.
. . 89
X
Fig. 424 Temperature contours for Case No.9 at oscillation intervals of
A(o)t)=15Â° in the lower half cycle 0Â°^o>t<180Â°. .... 91
Fig. 425 Nondimensional temperature profile for Case No.9 along the
vertical line x = 153 in the left reservoir: (a) at several
oscillation phases; (b) time averaged. .... 92
Fig. 426 Temperature contours for Case No. 10 at oscillation intervals of
A((ot)15Â° in the lower half cycle 0Â°^(ot<180Â°. .... 93
Fig. 427 Nondimensional temperature profile for Case No.10 along the
vertical line x = 153 in the left reservoir: (a) at several
oscillation phases; (b) time averaged. .... 94
xi
LIST OF TABLES
Table 41 The list of ten cases studied in the numerical investigation. ... 51
Table 42 Heat transfer coefficients for several cases in the study. ... 95
xii
NOMENCLATURE
a the half width of the connecting channel
A coefficients in the Finite Difference Equations
A0 the cross section area of the connecting channel
b the distance from the channel center line to the wall center line in the
analytic solution or the half width of the end reservoir in the numerical study
c time averaged length of the end reservoir
Cp specific heat of working fluid
D molecular mass diffusivity of working fluid
ha heat transfer coefficient from the analytic solution
hn heat transfer coefficient from the numerical results
L channel length
P pressure
Pr Prandtl number
q local instantaneous axial heat flux in the channel
Q total time averaged heat rate in the channel
rxL the ratio of tidal displacement to channel length
Re Reynolds number based on the maximum of piston velocity and reservoir half
width
xiii
RHS terms on the /?ight Hand Side of the associated system of algebraic equations
S source term in the Finite Difference Equations
Sc Schmidt number
T temperature
Th high temperature specified at hot end for supplying heat
Tc low temperature specified at cold end for removing heat
T, the approximate time averaged temperature in right reservoir from the
numerical results
u velocity component in axial (x) direction at the walls of U cells
U velocity component in axial (x) direction
Up maximum piston velocity
v velocity component in transverse (y) direction at the walls of U cells
V velocity component in transverse (y) direction
Vc cross flow velocity
x axial coordinate in the physical domain
6x x direction size of a grid cell in the physical domain
y transverse coordinate in the physical domain
6y y direction size of a grid cell in the physical domain
xiv
a Womersley number
P stretching parameter for generating nonuniform grid
Y time averaged axial temperature gradient (constant)
6t variable time step size in calculation
6tb basic time step size in calculation
Ax tidal displacement
e the ratio of total channel half width (including the thickness of solid wall) to
the channel half width filled by fluid in the analytic solutions or general
convergence criterion in the numerical calculations
ec convergence criterion for the VelocityPressure correction
ep convergence criterion for the final time periodic states
p transverse coordinate in the computational domain
Â© magnitude of the sinusoidal pressure gradient applied in the analytic solutions
k molecular thermal diffusivity of working fluid
Keff effective thermal diffusivity by oscillating flow
P the ratio of fluidtosolid (wall) thermal conductivities
v kinematic viscosity of working fluid
i axial coordinate in the computational domain
p density of working fluid
o the ratio of fluidtosolid (wall) thermal diffusivities in analytic solutions or
general convergence indicator in numerical calculations
ac convergence indicator for VelocityPressure correction
xv
op convergence indicator for the final time periodic state
t time in the computational domain
ii stream function of the oscillating flow field
6) angular frequency of the oscillation
xvi
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
FLUID OSCILLATIONS AND HEAT TRANSFER
IN A PARALLEL PLATE CHANNEL BOUNDED BY END RESERVOIRS
MAINTAINED AT DIFFERENT TEMPERATURE
by
Alex X. Zhao
December 1990
Chairman: Ulrich H. Kurzweg
Major Department: Aerospace Engineering, Mechanics,
and Engineering Science
Existing analytic and experimental studies have shown that sinusoidal
oscillations of a viscous fluid within open ended conduits connected to reservoirs can
enhance the thermal diffusivity between the hot fluid and cold fluid in the opposite
end reservoirs by some four orders of magnitude in excess of that possible in the
absence of oscillation. The heat transfer coefficients achieved in this process can be
very high and can readily exceed those possible via heat pipes, yet involve no net
convective mass exchange.
A numerical investigation of laminar oscillating flows of incompressible
viscous fluid and the associated enhanced heat transfer in a 2D parallel plate channel
bounded by rectangular end reservoirs which have sinusoidally oscillating piston
boundaries and are maintained at different temperatures forms the topic of the
present dissertation.
XVII
A modification of the widely used SIMPLE algorithm, termed SIMPLETP
(for Time Periodic), has been developed to take time periodicity and boundary
movements into account. The method has been successfully applied to the present
time periodic flow and heat transfer problem.
Two different models of heat supply and removal, namely, conduction through
the reservoir walls and convection by cross flow, were considered in this study.
Several case studies involving different combinations of Womersley number, tidal
displacement, length of reservoirs and cross flow velocities were carried out. The
numerical results obtained were compared with analytic solutions where such
comparisons were possible.
It was found that the velocity field in each reservoir is characterized by a high
velocity jet emanating from the channel end during a portion of the oscillation cycle
and that one or more counterrotating vortex(es) or vortex pair(s) exist during the
whole oscillation cycle as long as the Reynolds number, based on the maximum
piston velocity and reservoir width, is large enough. The tidal displacement and cross
flow velocity have a strong effect on the flow patterns in the reservoir and the
reservoir temperature field is mainly determined by the temperature boundary
conditions and less so by the velocity field. High oscillation frequencies coupled with
a large tidal displacement and a relatively large cross flow velocity are found to yield
large enhancements of heat transfer. Heat transfer coefficient as high as 1.8x1o6
W/nrK can readily be achieved when using water as the working fluid.
xvi 11
CHAPTER I
INTRODUCTION
1.1 Background
For a sinusoidal oscillatory fluid flow over a flat solid surface, the influence
of the noslip condition on the solid surface penetrates an approximate distance of
\J 2v/o) into the fluid, where v is the kinematic viscosity of the fluid and o> is the
angular frequency of the flow oscillation. For an oscillatory flow inside a flat plate
channel, this is still true as long as the channel half width, a, is greater than the
penetration distance. Under this condition, the fluid flow can be considered to
consist of two parts, the boundary layers and the core region, which are shown in Fig.
ll(b).
If a property of the fluid, namely, concentration or temperature, is maintained
at different values at the two ends of the channel, a very strong transport from the
high side to low side will occur. Some theoretical and experimental studies have
shown that the effective coefficient for the transport can be much larger than the
values which characterize the transport without oscillation. The reason for the
enhancement of the transport is that both the area over which the exchange takes
place and the property gradient across the area are increased significantly over the
values existing in the absence of oscillations.
1
2
Consider the case where the temperature of the liquid is different at the two
ends of the channel. Here the heat transfer from the hot end to the cool end can
be enhanced by a flow oscillation in the axial direction. The time averaged heat
transferred axially, without a accompanying mass exchange, can be expressed as
Q = ~AoP CpKeffy (1,1)
where Keff, p and Cp stand for the effective thermal diffusivity, fluid density and
specific heat, respectively, A0 is the channel cross section, and y is the time averaged
axial temperature gradient which is maintained as a constant.
For pure molecular conduction, Keff simply is the molecular thermal diffusivity
k, A0 is equal to 2a when the channel depth is equal to one, as is shown in Fig. 1
1(a), and the amount of heat transferred per unit depth will be
Qc = ~ 2a pCpKy O*2)
With oscillations, it is necessary to examine the heat transfer process in some
detail since it becomes a time dependent problem. During the axial fluid oscillation
towards the colder side, hot liquid is brought into the core region and moved in
contact with the cooled boundary layers, which in the previous half stroke are in
contact with cooler liquid in the core region. The colder boundary layers will absorb
heat from the hot liquid core and heat up. During the next half stroke the oscillation
is towards the hot side, and cold liquid is brought into contact with the heated
boundary layers. The boundary layers then release heat to the cool core region and
thus become cool again. These two steps of heat transfer in oscillating flows at
3
(a)
;'
2a
Th
//
'/S*
'///////////z
2a core region Tc
Th
/////////////// / '///////////
Fig. 11
Heat transfer in a parallel plate channel: (a) pure conduction; (b)
enhancement by oscillating flows.
4
Prandtl number 1, for which the thicknesses of momentum and thermal boundary
layers are equal, are shown in Fig. ll(b) with the dotted areas representing the cold
fluid. The exchange process is repeated during subsequent oscillation cycles. This
leads to a time periodic problem. Because the heat exchange happens between the
boundary layers and core region, the effective heat transfer surface become much
larger than A0 and can be expressed in terms of a quantity Ax named the tidal
displacement (equal to twice the cross section averaged axial amplitude of the
sinusoidal oscillations). The temperature gradient in the normal of interface between
the boundary layers and core region is also much larger than y and is a function of
time and position. By averaging over the whole oscillation period and channel cross
section, the heat transferred per unit depth from the hot end to the cool end now can
be written as
Q = 2apCpKeffy 03)
where now the effective thermal diffusivity Keff is a function of a, v, o, Ax and some
other factors. tceff can be much larger than k provided a good choice for those
factors is made. Note that there will be no net convection between the two ends
when Ax is kept at a fraction of the channel length, and the process thus can be
considered to be free of mass exchange between the channel ends under proper
conditions (explained in Section 1.3).
If the concentration is maintained at different values at the ends of the
channel, an enhanced axial mass diffusion will occur for appropriate conditions, thus
producing an effective diffusivity much larger than the molecular diffusivity. In
5
recent years, these phenomena have received considerable research attention from
engineers and scientists because of their potential practical applications such as for
nuclear reactor coolings and for gas separation, etc.
A typical device for enhancing heat transfer by oscillating flows, termed the
Thermal Pump, is shown in Fig. 12. This represents essentially US Patent No.
4590993 held by U. Kurzweg1 and the University of Florida. This dissertation will
be concerned with a numerical investigation of the laminar flows of incompressible
fluids and the associated heat transfer in a 2D version of the thermal pump when
sinusoidally moving piston boundaries are located in the end reservoirs. The
connecting channel will consist of a single flat plate channel formed by two parallel
plates separated by a fixed distance of 2a.
1.2 Literature Review
The phenomenon of transport enhancement of contaminants was initially
studied by Taylor(1953) and Aris3(1956) in their investigations of axial dispersion
in steady flows within capillary tubes. Their results indicated that a significant
increase in axial contaminant dispersion will occur in steady laminar flows. Similar
effects were found to exist for oscillating flows by Bowden4 in 1965. Harris and
Goren5(1967) studied, both analytically and experimentally, the mass transfer
through a long tube connecting two reservoirs maintained at different but constant
concentrations by oscillating the flow in the tube and found that the increase of mass
transfer rates is a function of the Womersley number a =a\J a>/v , the tidal
6
Fig. 12
A schematic of the Thermal Pump (U.S. Patent No. 4590993)1.
7
displacement Ax and the Schmidt number v/D. Rice and Eagleton6(1970) conducted
a similar study to that of Harris5 and found that the mass transfer increase is proÂ¬
portional to the square of tidal displacement Ax. In 1975, Chatwin7 showed theoÂ¬
retically that the effective diffusion coefficient in oscillating flows is a harmonic
function of time with a period equal to one half that of the imposed time dependent
pressure gradient. Further theoretical studies by Watson8(1983) and experiments by
Joshi et al.9(1983) gave similar results. Kurzweg and Jaeger10(1986) first showed
the existence of a tuning effect in enhanced gas dispersion in oscillating flows. All
these studies prove that a contaminant will spread axially in oscillating laminar pipe
flow at rates as much as five orders of magnitude higher than those achievable by
pure molecular diffusion in the absence of oscillations.
The same enhancement in oscillating flow for heat transfer was discovered by
Kurzweg in 1983 in view of the mathematical similarity between the governing
equations for heat conduction and mass diffusion. The equations for contaminant
diffusion with superimposed oscillating flow are mathematically nearly identical to
the equation of heat conduction, with the only major differences being the boundary
conditions and the use of the Prandtl number Pr= v/k instead of the Schmidt number
Sc=D/k. He solved the governing equations for enhanced axial heat transfer in
oscillating flows in a tube of infinite length for low value of a2Pr and found the
effective conductivity can readily reach values three orders of magnitude greater than
normal heat conductivity. This information was published in 198511. Kurzweg and
Zhao12(1984) conducted some experiments to measure the effective axial heat con
8
duction rates through a capillary bundle connecting two fluid reservoirs maintained
at different temperatures and found very good agreement with theoretical
predictions. Two other papers by Kurzweg(1985)13(1986)14 explained the tuning
effect in the enhancement of heat transfer in detail. In May of 1986, a U.S. patent
was granted to the University of Florida and Professor U. Kurzweg1 on this process.
Zhang15 performed some numerical studies on tubes of finite length with some
portions of their walls or ends maintained at different temperatures and concluded
that the highest rates of heat transfer predicted earlier can only be reached when the
amount of heat supplied and removed from the tube ends is large enough.
Otherwise a heat "bottle neck" will occur and the temperature gradient in the axial
direction of the tube will be reduced considerably. Kavianyâ€™s16 recent study on heat
transfer by oscillating flows in tubes bounded by two end reservoirs maintained at
different temperatures verified, both theoretically and experimentally, the results
obtained by Kurzweg and Zhao12.
It should be noted that most of theoretical studies to this point, both analytic
and numerical, have considered only the tube or connecting channel without any
attention to the flow behavior and heat transfer process occurring in the two end
reservoirs which generally have much larger cross section than the connecting
conduit. Kaviany16 has made some calculations on this flow using a very simplified
model of the end chambers. Durst et al.17(1989) did some numerical studies on the
transient laminar flow over a sharp pipe expansion driven by a piston in the wide end
moving as a rectangular function of time starting from an initial rest condition and
9
compared the numerical result with their experimental data18. Neither of them
considered the temperature distribution and their results for velocity can not be
directly used in the analysis of the thermal pump process because of the
simplification made in former and the different velocity conditions used in the latter.
The motivation for this dissertation research was to obtain a clearer underÂ¬
standing of the flow and temperature patterns inside of the end reservoirs of a
thermal pump in order to lend further guidance for the practical construction and the
usage of thermal pumps. Also, associated with this study, a proper numerical
procedure based on the SIMPLE algorithm was developed for the simulation of heat
transfer in time periodic flows of viscous incompressible fluids with moving
boundaries.
1.3 Analytic Solutions Within the Connecting Channel
The numerical investigations presented in the chapters II through V can be
considered as an extension of the analytic work done by Kurzweg13, which gave
solutions of the momentum and energy equations in a parallel plate channel of
infinite length. The numerical results in the connecting channel for the more
complicated geometry with end reservoirs should approximate those analytic solutions
when the connecting channel length becomes much longer than the tidal displaceÂ¬
ment. Thus, it is essential to first briefly evaluate the existing analytic solutions in
10
order to later make a comparison with the numerical results to be obtained wherever
such a comparison is possible.
The configuration dealt with in Ref. 13 is shown in Fig. 13. It consists of a
set of channels filled with a viscous fluid and connected to chambers at x= Â± Â«>. The
individual channels of width 2a each are confined between thermally conducting walls
dP
of width 2(ba) each and a time periodic pressure gradient â€” = and a
dT
constant axial temperature gradient â€” = Y are superimposed. The velocity of the
oscillating flow in the central channel generated by the pressure gradient is given by
U(y,t) = REAL â€¢
Ã Cl) AjC
1 _ cosh(\/Ta y/a)
cosh (\fT a)
1 _ tanh(77~a)
\[T a
(1,4)
where y is the coordinate perpendicular to the channel axis as shown in Fig. 13, o>
is the angular frequency of the time periodic pressure gradient, â€œ = a \J o> / v the
Womersley number, with v representing fluid kinematic viscosity, and Ax the tidal
displacement for the maximum cross stream averaged axial excursion of fluid
elements during one oscillation cycle. The vertical bars indicate the absolute value
of the quantity shown.
The relation between the magnitude of the axial pressure gradient and the
tidal displacement can be expressed as
11
Fig. 13
Geometric configuration for the analytic solution of flow oscillations
and the associated heat transfer in parallel plate channels.
(1.5)
2 \  tanh(/Ta)
>fi a
In practice, the channel half width a, the tidal displacement Ax, and the
oscillation frequency a) as well as the properties of the working fluid are easy to
control. Hence, either they or a combination of some of them, in particular the
Womersley number, are used as the control parameter for this heat transfer problem.
A numerical evaluation of the nondimensional velocity, normalized by oAx,
as a function of nondimensional transverse position in the y direction at different
oscillation phases for a1 and a=10 is shown in Fig. l4(a) and (b), respectively.
It is clear that the velocity profile varies considerably with change in Womersley
number a, from an essentially parabolic shape at very low a to one having a nearly
constant velocity core connected to thin boundary layers of thickness 6 = \Â¡ 2 v / g>
at the walls for large a.
By assuming
T(x,y,t) = y I* + REAL[flg(y)eiw'j (1>6)
, the crosssection dependent portion of the temperature perturbation g/y) in the
fluid and gs(y) in solid walls have the forms13
13
U/cjAx
U/cjAx
Fig. 14 Analytic solution of the dimensionless velocity of oscillating flow in a
parallel plate channel as a function of dimensionless transverse
position at different oscillation phases cot: (a) a = l\ (b) a = 10.
14
gf(y) =
1  a2Pr
j cosh (\Â¡Tay/a)
cosh(y/T a)
j_j cosh(\//Pr a y/a)
cosh(v//Pr a)
2 a a2 ( Pr  1 )
Ax
1 
tanh(/T a)
\[i a
(1,7)
8s(y) =
1
 H
2 a a2 ( Pr  1)
1 tanh(\ZT a)
Ax
\[T a
cosh[\J i o Pr a(ey/a)]
cosh[/7aPr a(e 1)]
(1,8)
with
_ p \J Pr tanh(\Z/~ a) + \J a tanh[^i a Pr a(el)] ^ ^
p tanh(\/ i Pr a) + \J a tanh[\/i a Pr a(el)]
where fi=kj/ks is the ratio of fluidtosolid (wall) thermal conductivities, o = k^/ks
is the ratio of thermal diffusivities, and e=b/a.
The nondimensional temperature profiles T/yAx versusy/a for n = o=l and
e=2 but different Womersley number a are shown in Fig. l5(a) and (b). One
observes an appreciable penetration of the temperature variation into the solid wall
at a=1, but almost no penetration occurs for a=10. This is to be expected in view
of the fact that the typical penetration distance into a solid of an oscillating
temperature field is proportional to the inverse square root of the oscillation
frequency (i.e. skin effect). Note that the frequency ratio at a=l to that at a = 10 is
one hundred for the same channel width and the same fluid.
15
0.4 0.2 0.0 0.2 0.4
T/yAx
T/yAx
Fig. 15 Analytic solution of the dimensionless temperature distribution in
oscillating flow in a parallel plate channel and its walls as a function
of dimensionless transverse position at different oscillation phases cot:
(a) a = l\ (b) a = 10.
16
Neglecting axial conduction in both the wall and the fluid, the local and
instantaneous axial heat flux can be expressed as
q(y,t) = pCpU(y,t)T(0,y,t) (1,10)
The time averaged total heat flow rate, then, will be obtained by integrating
this quantity over the channel width and the oscillation period, i.e.
pq,
2 n
5
J U(y,t) T(0,y,t)dy
(1.11)
By means of Equations of (1,3), (1,4) and (1,7), and after some rather lengthy
but straight forward manipulations, the effective thermal diffusivity of the oscillation
flow is found to be
eff
Pr [ (1  H ) / + (1H)/] + (/ m H ) + (1mH)
16 a2 ( Pr2  1)
03 Ax2
where
/ = \JTa tanh(\/Ta)
m = s/ i Pr a tanh(\/ / Pr a)
(1,12)
(U3)
(1.14)
with the bar indicating the complex conjugate of the function shown and H being
defined by equation (1,9).
17
Two graphs of the nondimensional k^/oAx2, as a function of Womersley
number for different Prandtl numbers, are presented in Fig. l6(a) and (b). The first
one, (a), for e=2 representing the case of wall thickness being the same as channel
width and the second one corresponding to insulating walls which is mathematically
equivalent to the case of conducting walls with zero thickness. The curves shown are
seen to have a single maximum point which shifts to lower values of a as Pr is
increased. This is termed the tuning effect because the peak indicates the value of
channel width at which the heat transfer is optimized. These tuning points correÂ¬
spond to the condition where the thermal diffusion time from channel axis to the
walls is approximately equal to half of the oscillation period. The observed drop off
in the peak value of k^/cjAx2 for very small Prandtl number in the insulating case
is attributed to the inability to store much heat in the boundary layers which become
quite thin for fluids with small kinematic viscosity v such as in liquid metals.
Note that the second set of curves, Fig. l6(b), also present information for
the enhanced axial mass diffusion by oscillating flows since insulating walls give the
same boundary conditions as those for diffusional mass transfer and both of them
come from the same type of equations with the only difference being the use of
Prandtl number for heat transfer but the Schmidt number for mass transfer. One
finds that the peak value of k^/cjAx2 for Pr=l is about IP2 in the vicinity of a =3
while the corresponding value for the effective mass diffusivity for Schmidt number
1000 at the same Womersley number falls down to 5xlP5 by considering the curve
for Pr=103 as a curve for mass transfer for a Sc = 103 fluid. Since most liquids have
18
a
Fig. 16 Analytic solution of the dimensionless time averaged effective thermal
diffusivity enhanced by oscillating flow in a parallel plate channel as
a function of Womersley number a at different Prandlt numbers: (a)
e=2 (conducting walls with o = Â¡i = l); (b) el (insulating walls).
19
a large value for the ratio of Schmidt number to Prandtl number (about 1000 to 1
for water, for example), a really large increase in heat transfer with a negligible
enhancement of diffusional mass transfer can be achieved, and thus, the process of
heat transfer by thermal pumping in liquids can be considered as essentially free of
mass transfer provided the connecting channel length is longer than the tidal
displacement. Note that this same observations does not follow for gases where
Schmidt number and Prandtl number are approximately the same.
The tuning curves of effective thermal diffusivity for several different liquid
solid combinations in configurations of equal wall and channel thickness (e=2) are
shown in Fig. 17. There is no major difference in the maximum Keff among these
different material combinations and all curves give a maximum ratio of Keffto o>Axr
of about 0.05. It is clear that the effective maximum thermal diffusivity at the peak
of the tuning curves is directly proportional to the frequency of the fluid oscillation
and to the square of the tidal displacement. This means that in the optimization proÂ¬
cedure for a thermal pump, and Ax should be made as large as possible. The
channel width a, then, should be determined from the valve of a corresponding to
the maximum effective thermal diffusivity and the kinematic viscosity v of the
working fluid. For example, at o>=2n rad/s in the waterglass combination, the
optimum channel half width will be a=2mm for optimum heat transfer.
The effect of wall thickness on the heat transfer process in the thermal pump
is shown in Fig. 18, where a plot of Keff/a)Ax2e versus e for several values of crPr
for the liquidsolid combination of liquid lithium and stainless steel are presented.
20
a
Fig. 17 Analytic solution of dimensionless time averaged effective thermal
diffusivity enhanced by oscillating flow in a parallel plate channel at
e=2 (conducting walls) as a function of Womersley number a for
several different liquidsolid combinations.
e
Fig. 18 Analytic solution of dimensionless time averaged effective thermal
diffusivity enhanced by oscillating flow as a function of cladding
thickness for liquid lithium in a parallel plate channel with stainless
steel walls.
21
These curves show that an e=2 geometry, corresponding to equal wall thickness and
channel width, is about optimum when an appreciable fraction of the heat can be
stored in the bounding walls. The reason for normalizing Keff by the extra factor e
is to take into account the fact that the walls take away a fraction 1/e of the cross
sectional area of the thermal pump available for the fluid oscillations.
Integration of the local instantaneous heat flux over the oscillation period
yields the time averaged axial heat flux distribution over the cross section, while an
integration over the channel width yields the instantaneous axial heat flow through
the whole channel as a function of time. Fig. l9(a) and (b) show the behavior of
the two partially integrated, nondimensional products of U(y,t) and T(0,y,t) for the
case of a waterglass combination with e=2. From (a) one observes that the largest
value of UT/yoAx2 occurs for a = l which is near the tuning point for this particular
condition. Note the much smaller values at a=0.1 and a = 10 which occur at deÂ¬
tuned conditions. The flux transport becomes confined to the vicinity of the walls as
the Womersley number becomes large. Fig. l9(b) shows that the thermal flux is
time periodic at a frequency twice that of the fluid oscillations and that only near
tuning conditions is the positive flux not essentially canceled by the negative heat
flux.
In the following chapter, Chapter II, the geometry for which this numerical
study was performed will be explained in detail. The assumptions and the mathematÂ¬
ical models used to present two different physical situations will also be shown there.
Chapter III will focus on the numerical method employed including a brief derivation
22
UT/ywAx2
cut
Fig. 19 Analytic solution of heat transferred by an oscillating flow of water in
a parallel plate channel with glass walls for different a: (a) time
averaged value as a function of dimensionless transverse position; (b)
cross section integrated value as a function of oscillation phase.
23
of the associated Finite Difference Equations. The results obtained from the numeriÂ¬
cal investigation will be presented in Chapter IV, with some comparisons of the
results with analytic solutions discussed in the present section. Concluding remarks
will be found in Chapter V. Some materials useful for a clearer understanding of the
VelocityPressure correction in the SIMPLE algorithm are summarized in Appendix.
CHAPTER II
MATHEMATICAL MODEL
The mathematical model for the numerical study presented in this dissertation
is explained in this chapter. First, the geometric configurations for which the
numerical investigation were performed are described. Then, the governing
equations to be solved and boundary conditions assigned are given.
2.1 Geometric Configurations
Two different configurations based upon the thermal pump geometry shown
in Fig. 12 were chosen for the present numerical study. These are shown in Fig. 21.
Some simplifications were introduced for the purpose of computational convenience.
Note also that, to more clearly show details, the axialtransverse ratio of the
configurations shown is different from their true value used in the calculations as
discussed in Chapter IV. Both configurations essentially consist of a 2D parallel
plate channel of width 2a and length L connected to two 2D rectangular end reserÂ¬
voirs of width 2b and time averaged length c. There is a piston moving back and
forth sinusoidally in the axial direction in each of the reservoirs. Choosing a rectanÂ¬
gular shape for the reservoirs allows calculations in Cartesian coordinates, although
a coordinate transformation is still necessary because of the moving boundaries.
24
25
Fig. 21
Configurations of the thermal pump used in the numerical
investigation: (a) conduction model; (b) cross flow model.
26
The coordinate system used in the physical domain is also shown in Fig. 21.
The origin is set at the center point in both axial and transverse directions. The axis
of the connecting channel and reservoirs was chosen as the* axis and then they axis
is in transverse direction. Based on this coordinate system, the positions of the
moving boundaries (piston surfaces) can be expressed as
(2,1)
(c + â€” )  â€” cos(oit)
2 2b
(2,2)
for the left and right pistons, respectively. Note that the piston movements are in
phase in order to satisfy the law of mass conservation for the assumed incompressible
working fluid entirely filling the channel and reservoirs.
The first configuration, shown in Fig. 2l(a) and named the conduction model
because of the way that heat is supplied and removed from the end reservoirs, is a
closed system for mass transfer. All walls except the piston surfaces are stationary.
The channel side walls are thermally insulating as indicated by the oblique shading
lines, while the reservoir side walls including the piston surfaces are conducting and
maintained at different temperature at the opposite sides displayed by the blank (for
cold at Tc) and black (for hot at Th) regions. The configuration is symmetrical about
the horizontal center line corresponding to the x axis. This symmetry allows the
calculation to be undertaken in only the upper half of the domain as long as suitable
boundary conditions are applied at the center line.
27
The second configuration is shown in Fig. 21 (b). It uses fluid cross flow to
add and remove heat from the end reservoirs and is termed the cross flow model.
It is assumed that the cross flow enters and exits the reservoirs at a constant but
small velocity Vc perpendicular to the reservoir side walls. All walls including the
piston surfaces are insulating in this model. The cross flow is maintained at different
temperatures on the opposite sides as they come into the system. The thermal
conduction at the cross flow exits (i.e. transpiration) is excluded in the numerical
study since it is negligible compared with the heat carried by the flow. This configuÂ¬
ration no longer produces symmetric flows about the center horizontal line and the
computation thus has to be done over the entire flow domain.
A very useful characteristic for both configurations is that all information
needed to describe the flow and heat transfer in one oscillation phase step can be
readily obtained from the solution at another phase step one half period earlier as
soon as the flow pattern and temperature distribution in whole physical domain have
reached a final time periodic state. Time periodic conditions existing in the right end
reservoir at a fixed phase point in the oscillation cycle will correspond to those found
in the left reservoir half an oscillation cycle later. Furthermore, if temperature is
normalized so that the given temperatures Th and Tc in opposite reservoirs are the
negative of each other, the temperature distribution at (ot = 180Â°aTe the same as that
in the reflected image of the geometry at a)t=0Â° except for a sign change. The same
thing happens for every pair of solutions at ot and o)t + 180Â°. It is because of this time
symmetry that the computations need be taken over only the range of 0Â°<(ot^l80Â°,
28
which significantly reduced the CPU time and storage required in the numerical
investigation.
2.2 Governing Equations and Boundary Conditions
The general assumptions in this study are following.
a) The working fluid is a homogeneous, Newtonian fluid;
b) The working fluid is incompressible
constant density;
c) The working fluid has constant material properties
constant viscosity,
constant thermal diffusivity,
constant thermal conductivity;
d) All body forces are negligible;
e) The viscous heating term in the energy equation is negligible;
f) The flow is laminar.
As shown in Section 1.3, liquid is a suitable working fluid in the thermal pump
for heat transfer without mass exchange. The incompressible fluid assumption is
valid for most liquids. The main advantage of the constant material properties
assumption is the ability to separate velocity and pressure solution from the temperaÂ¬
ture solution. It was found that in order to reach the time periodic state, the number
of oscillation periods needed for temperature is much more than that for velocity,
while the CPU time used for solving velocity and pressure distribution in each period
29
is much longer than that for temperature. Calculating velocity and temperature
separately did save considerable CPU time.
The dependent variables to be solved for are the x component of velocity, U,
they component of velocity, V, the pressure P and the temperature T. They and the
independent variables x, y and time t are nondimensionalized as follows.
=
x
a
t* = wt
(2,3)
(Â¿a
P*
P
pora2
(2,4)
T. T(Tt + T')l 2
(TtTc)/2
where the quantities with * are dimensionless and
a half width of the channel;
co angular frequency of the piston movement;
p fluid density;
Th temperature at the hot end;
Tc temperature at the cold end.
After nondimensionalization, the governing equations read
(2,5)
Ux + Vy = 0
(2,6)
U,  (UU)X + (VU)y = Px + (1/a2)+ Uyy)
(2,7)
30
K + (UV)X + (W), Â« ~Py + (1/a2) (Vxx + Vyy)
(2,8)
T, + (UT)x + (VT)y = (1/Pra2) (7^ + )
(2,9)
where the subscripts indicate partial derivatives and
Prv/K  Prandlt number;
a = a \Â¡ o) / v  Womersley number;
Note that a conservation form of the governing equations is used here. All
quantities in the equations are dimensionless and the nondimensional mark * has
been omitted for simplicity. This notation, in which dimensionless quantities are
termed without superscript *, will be used from this point through the end of the
dissertation except where the quantity mentioned needs to be expressed in dimenÂ¬
sional form.
In view of the geometry explained in Section 2.1 and the nondimensiona
lization shown above, the boundary conditions can be mathematically expressed as
follows:
Velocity:
U = 0
at the stationary walls;
(2,10)
V = 0
2)
at the piston surfaces;
(2,11)
2b
V = 0
31
3)
4)
U = O
coa
ay
V = o
at the cross flow entrance; (2,12)
along the symmetry line y=0. (2,13)
Temperature:
1) Â»o
dn
2) T = + 1
3) T =l
4) â€” = 0
dy
at the insulating walls and symmetry liney = 0; (2,14)
at the hot walls and hot cross flow entrance; (2,15)
at the cold walls and cold cross flow entrance; (2,16)
at the cross flow exits. (2,17)
where Vc in Eq. (2,12) is dimensional and â€” in Eq. (2,14) indicates that the partial
derivative normal to the walls.
CHAPTER III
NUMERICAL TECHNIQUE EMPLOYED
The governing equations (2,6)(2,9) with the boundary conditions (2,10)(2,17)
specified in the physical domain, which have been presented in Chapter II, have to
be solved numerically and thus require a suitable numerical method. The existing
SemiImplicit Method for Pressure Linked Equations (SIMPLE) of Patankar19 and
of Patankar and Spalding20 is one such method available for dealing with heat
transfer problems in incompressible viscous flows, but it needs to be modified here
by taking into account time periodicity and boundary movement, which are present
in the fluid oscillation and associated heat transfer problems investigated in this
dissertation. Such a consideration leads to the extended SIMPLE algorithm, termed
SIMPLETP (for Time Periodic), developed in this investigation and to be described
in some detail in this chapter. First, the coordinate systems in both the physical
domain and computational domain and the transformation of the governing equations
from the former to latter are described. Second, a brief derivation of the Finite
Difference Equations and the treatment of boundary conditions are presented. In
Section 3.3, an overview of the numerical iteration procedure SIMPLETP is given.
Finally, a Vectorized Line Group algorithm for solving the associated system of
algebraic equations also developed in this investigation is presented.
32
33
3.1 Grid and Transformation of Governing Equations
It is most convenient to transfer the time periodic physical domain to a time
independent computational domain. In other words, the grid system in the physical
domain which is movable and nonuniform needs to be mapped to a grid system in
the computational domain which is fixed and has even spacing in order to allow the
numerical calculations to be carry out properly. For the physical domain geometry
shown in Fig. 21, it is found convenient to break both of the domains into three
subregions when making the mapping. The two grid systems and their block correÂ¬
spondence are shown in Fig. 31.
The grid in the physical domain at three different phases of piston movement
are shown in Fig. 32. One can observe from this figure that the grid is fixed for the
middle blocks corresponding to the connecting channel, but movable in the other two
blocks representing the end reservoirs of the thermal pump. The grid spacing in the
area of each reservoir near the connecting channel, where a cross flow can also enter
or leave the reservoir in they direction, is also independent of time. This subdivision
is of advantage for specifying the boundary conditions correctly and easily when using
cross flow. Note that the same grid structure was used for all cases in both conÂ¬
duction and cross flow models in order to allow a good comparison, even if it is not
necessary for the former. The grid points in the rest of the area of the reservoirs
move like an accordion in the x direction because of the piston oscillation. The y
components of the grids were established at the beginning of each run, while the x
components of the grids had to be generated at each time step.
34
Fig. 31 The correspondence of blocks between the physical domain and the
computational domain: (a) grid in the physical domain at ot=0Â°\ (b)
time independent and uniform grid in the computational domain; (c)
grid in the physical domain at
35
Fig. 32 The movable and nonuniform grid in the physical domain:
(a) ot=0Â°; (b) ot = 90Â°\ (c) at=180Â°.
36
In all the three subregions, the grid points are nonuniformly spaced such that
those regions near the walls and at the edge of the jet region at y=l where large
velocity and/or temperature gradients are expected to exist have higher grid density.
In order to generate the nonuniform grid spacing, the stretching function found in
the book by Anderson21 was used. It has the form
x
(3,1)
fp+n2*'1 .
â€”â€” +i
, where x and x are the position of grid points before and after stretching; p is the
stretching parameter which controls the stretching strength, p is different in diffeÂ¬
rent subregions and in different directions. The values of Pâ€™s in the same direction
for different subregions were chosen so that the grid size nearest the dividing line
between subregions were equal.
The relationship between the coordinates (x,y,t) in physical domain and
(Â£ i), r) in computational domain can be expressed as
n=n(y)
(3,2)
Under the condition that the product of the Jacobian and its inverse are equal to
unity, as is required for one to one mapping, the following expressions hold.
37
lx = 
xl
7V = â€”
1
y~v
(3,3)
By means of these, the governing equations (2,6)(2,9) in the physical domain
can, after some chain rule manipulations, be rewritten in terms of Â£, 77 and r as
â€”Uf + â€” Vn = 0
x, 1 V v
l J V
Ur^.U(+Â±(UU)(+Â±(VU),  byL\
Xl xi yV XE
1
1
2
or
Uc
y*
In
V
V
KfV(WVW,
xt y.
1
1
a
V'
/ \
yy,
V_
{yv
X 1
â€”7>Â¿1
x?
.1
y*
1
1
Pra2
i y*
_Lt
yâ€ž 71
(3,4)
(3,5)
(3,6)
(3,7)
Note that there is a new source term appearing in each of the momentum and
energy equations which represents the effect of the grid movement in the physical
domain.
38
3.2 A Brief Derivation of the Associated
Finite Difference Equations and Boundary Conditions
As in the SIMPLE algorithm, the same staggered grid, as shown in Fig. 33,
was also used in the SIMPLETP. The pressure and temperature are calculated at
the grid cell center designated by oâ€™s while the velocity components U and V are
determined at the center of the cell walls indicated by arrows. By means of this
staggered grid configuration, the checkerboard pressure effect introduced for non
staggered grids can be eliminated and thus allows the VelocityPressure correction
to converge easily. In SIMPLE, a control volume approach was used to derive the
Finite Difference Equations (FDEâ€™s) in this grid configuration, and a derivation,
based on a stationary grid in the physical domain can, be found in the book by
Patankar19. The derivation of the FDEâ€™s in the present SIMPLETP followed the
same procedure, but some modifications had to be made since it was carried out in
the computational domain. A brief derivation of the x momentum equation is
presented below as an example of the derivation applied in the SIMPLETP.
The notation used here is the same as that of Patankar19 and is shown in Fig.
34. The location of U where the differential equation is going to be discretized is
designated by "P" while the location of its four neighboring ITs are indicated by
"Â£"(East), "IF"(West), 'W'(North) and "5"(South), respectively. The four centers of
cell walls are marked by "e", "w", V, and "s" also, but in lower case, according to
their directions away from the cellâ€™s center. Note that the position indices are placed
on the position of superscripts in order to differ from the partial derivatives which
are expressed by subscripts in the governing equations.
39
Fig. 33
The staggered grid in the computational domain.
40
(a)
(b)
Fig. 34 Typical cells for the velocity component U: (a) an interior cell; (b) a
boundary cell with its south wall at the domain boundary.
41
The equation (3,5) can be rewritten as
x&flr  vrui+ y/i+ y*
uu 
uc)
2_
a~xt
+ x,
VU 
= o
(3,8)
An integration of this equation over the grid cell SÂ£Sr) and the time step St yields
8x8y{UpUÂ°)  8l(xpxÂ°)(ueuw) + Sy(PePw)
St St
Sy
Ur
uU *â–
a2x
e (
Ur
uU 1
\w
a2Xr
+ Sx
vU^l
\n
\Uâ€”!L
a2yv
\S
(3,9)
, where all Us have their values at the current time step except UÂ° which takes the
value of Up at the previous time step. The u and v here represent the velocity
components in the current time step also, but have their values at the previous iteraÂ¬
tion and thus will go into the coefficients >lâ€™s or source term S. It is in this way that
the nonlinear NavierStokes equations can be solved as linear equations in each step
of the iteration.
Note that the Us are never located at the centers of the cell walls, thus the
Us with lowercase superscripts have to be replaced by those with uppercase superÂ¬
scripts (see Fig. 34(a)). The, socalled, hybrid scheme, which is a combination of
central differencing and upwind schemes described in detail by Patankar19, was used
for the convective and diffusion terms. Linear interpolation was used for the
replacement in other terms. Because of the staggered grid, F* and Pw are available
without any further action.
The finally obtained FDE for U looks like
42
Ap Up = Ae Ue + Aw Uw + AN UN + As Us + AÂ° UÂ° + Sc (3â€™10)
Where
A E = by
MAX
0,
A w = by
MAX 0,
AN = bx
MAX
0,
As = bx
MAX 0,
1
Kl
a\xExp)
2
/
1
\uw\
a2(xpx w)
2
1
1 vn f
a2(yNyp)
2
/
1
Vâ€™l
a2(ypys)
2
/
MAX(m e,0)
MAX( u H',0)
MAX(v n,0)
MAX( vs,0)
(3,11)
(3,12)
(3,13)
(3,14)
Ao _ Sxfy
bx
(3,15)
A p = A e + A ^ + A n + A s + A o
(3,16)
x\, _uW)
bx
(3,17)
Linear interpolation was also needed here to obtain the values of u and v at the
centers of the cell walls.
At the cells, which have one wall and sometimes two coinciding with the
domain boundary, the neighboring velocity component will locate at the center of the
43
cell wall. In Fig. 34(b), one of the boundary cells having its south wall coinciding
with the domain boundary is shown as an example. For all of the boundary cells,
proper adjustments to the right hand side in some of the equations (3,11)(3,17) were
taken according to the geometry and boundary conditions specified. Particularly for
the boundary cell shown in Fig. 34(b), one has
Co
II
O*
MAX
\
Â©
i iâ€”*
i
1 X
âœ“
+ MAX( v5,0)
l *(yFys) 2 j
(3,18)
The changes made in the superscript of y5 and Vs come from the change of the
location of IIs. Equation (3,10) for this boundary cell should be changed also since
Us here is known from the boundary conditions.
The derivation of all other FDEâ€™s and the treatment of other boundary cells
are straight forward and follow the same procedure explained above. More detail
and some discussion about the discretization scheme can be found in Ref. 19.
3.3 Calculation Procedure
Because of the assumptions of constant density and temperature independent
viscosity, the energy equation does not couple with the continuity and momentum
equations. This simplifies the problem. In this investigation, the velocity and
pressure field was first obtained without involving any temperature consideration.
The energy equation was then solved and the temperature distribution determined
44
based on the calculated velocity field. It was found that the temperature converges
much slower than the velocity does, which showed that a big saving in CPU time was
accomplished by being able to solve for velocity and temperature separately.
A flowchart for solving for U, V and P is shown in Fig. 35. The calculation
procedure was begun from an initial condition which was for all cases in this investiÂ¬
gation as that of the fluid at rest and the pressure uniform everywhere in the entire
physical domain. In the meantime, the dimensionless time vk=1 was assigned as zero
and a basic time step size <5 tb was defined. Starting from this point, U, V and P at
rk+i= vk+ St were determined based on the grid generated in the physical domain
for this time step by means of the standard SIMPLE algorithm. In the iteration
process, the values for the dependent variables either at the previous time step or
those existing at the same rk but in the previous oscillation period, depending on
which was closer, were chosen as the initial guesses. Typically after stepping off
several oscillation cycles, the values at the same t in the previous cycle gave the most
rapid convergence.
The maximum ratio of the residue at each grid cell which came from substituÂ¬
ting the nonconverged velocity into the continuity equation (3,4) to the product of
the local velocity and the cell boundary size was used as a convergence indicator oc.
Whenever the value of oc was less than a prescribed small value ec, the solution was
considered to have converged and one could proceed to the next time step. It was
found that sometimes the VelocityPressure correction was unable to achieve converÂ¬
gence and the value of oc oscillated around a constant value larger than ec after
45
Fig. 35
Flowchart for determining the time periodic velocity components U, V
and pressure P.
46
several time steps, from the initial conditions or from a converged solution, were
taken. Under this situation, convergence could be reestablished by a continual
halving of the time step increments 6c until they were sufficiently small. Once
convergence was again obtained, an attempt of doubling 6t was taken in order to
use as large a time step size as possible.
After the time had been stepped off to t = n, a check for the convergence to
the final expected time periodic state was undertaken. This consisted of essentially
determining the quantities
pu
N
* max J max
r7 E E (%Â»*
^mad max ,=1 j~ 1
N
.. max 'max
~ E E M
l mad max ,=1 j1
(3,19)
and
N
*max J
i j ^ 5 " ^maxÂ«+V.â€”)
pV
N
(3,20)
fmax J:
max j max
r7â€” E E (Kvâ€ž=â€ž)2
l mad max j~^
which represent the difference between the values of the velocity components at fixed
spatial position between period n and n + 1. Note that the symmetric conditions
about the vertical center line have to be used in these formulas too. It was
considered that the periodic solution had been reached when both of them were less
47
than a given small value ep. If the solution was not periodic under the criteria
mentioned above, another cycle of calculation would be started by setting rÂ¡=0 and
<5 r = <5 tb.
An almost identical procedure was followed for determining the temperature
distribution based on the velocity obtained (see Fig. 36). The major difference here
from that for calculating U, V and P was that there were no iterations corresponding
to the VelocityPressure correction in each time step, thus the variable time step size
was not needed for T. Since the calculation for temperature was associated with the
final time periodic velocity even at the beginning of each run, there were no initial
conditions existing in this situation. A linear temperature distribution in the \
direction and uniform in r\ direction was used as the initial guess at rÂ¡=0 of the first
period.
3.4 A Vectorized Line Group Method
for Solving the System of Algebraic Equations
The method for solving the associated system of algebraic equations always
play an important role in any overall numerical method for multidimensional flow
and heat transfer problems. A LineByLine iteration method (also called Successive
Line Relaxation method) is commonly used in the SIMPLE algorithm for 2D or 3D
problems. In this method, one picks a grid line, say in the Â£ direction, and assumes
that the dependent variable at its neighboring lines (i.e. the rj direction neighbors
48
Fig. 36
Flowchart for solving the time periodic temperature distribution based
on the calculated velocity.
49
of the points on the chosen line) are known from the last iteration. By doing this,
a system of simultaneous equations with a tridiagnal coefficient matrix is formed,
which reads
<
0
0
rt
9
RHS2j '
<
<
ae
^3 j
0
9
RHSy
â–º â€” *
0
AW AP
AIM2j AIM2j
ae
AIM2j
V/M2J
MS,u2j
0
A W
AP
WSiMy
where y4â€™s are coefficients, (pâ€™s are dependent variables and RHSâ€™s are terms on the
right hand side which can be expressed as
RHS2j
RHSy
,W . N A S A O O c C
= A2J *(Vlj+A2j*2j.l+A2j*Vyl+A2j*
A N A S A O O cC
= A3j * +A3.j*
(3,22)
RHSÂ¡m_2j  AIM2j* WÂ¡M2j+1 +AIM2j* VIM2jl+AIM2j* ^IM2.j+^IM2j
RHSIMy  AIMlj*VlMj +AIMlj* VlMlj+l
+AIM1 j * VlMlj1 +AIM1j * V/Mlj +$IMlj
This kind of system of algebraic equations can be solved by means of the Thomas
algorithm, a direct elimination method (see Anderson21). The forming and solving
50
procedure, then, was used for all lines in the same direction (the i direction in our
example) and repeated until there was no significant changes in the dependent variÂ¬
ables. The LineByLine method is widely preferred because it takes the advantages
of direct and iteration methods together to gain a fast convergence and good accuÂ¬
racy for the VelocityPressure correction iterations. More discussion concerning the
details of this method can be found in Ref. 19.
The above described method, however, is not suitable for vectorized calculaÂ¬
tions possible with supercomputers such as the CRAY and other newer machines.
Roughly speaking, vectorized calculation requires that there is no dependency
between the calculation operations in any pair of DO loop indices within the innerÂ¬
most DO loops. Once this requirement is satisfied, the calculations are performed
almost in parallel and the CPU time used is reduced accordingly (see Grenzsch22).
A flowchart for the LineByLine method with lines in Â£ direction is shown in Fig.
37. There are two innermost DO loops which are marked by the dashed lines in the
flowchart and both have "i" as their indices according to the line direction. It is easy
to see that Api} depends on^.jj, RHSU] is calculated from RHSlA j in the first innerÂ¬
most DO loop and that
and that this procedure does not meet the requirement for a vectorized calculation.
A Vectorized Line Group method was developed in this study to overcome
the difficulties mentioned above especially since it was clear that the computations
for time periodic problems involve a great number of time steps, and thus some
attention have to be paid to the reduction of the CPU time used. Also a super
51
(start)
T=2^
i = 3
Agâ€” /^mj
RHS g = RHS g + RHS MJ * A * / A?,u
p
ZÂ£Z
â–º i = i  1
fgâ€”(RHS g + Ag*
Fig. 37 Flowchart for the LineByLine iteration method for solving the
associated system of algebraic equations. Dashed lines represent the
innermost DO loops.
52
computer resource (CRAY/YMP), employing vectorized processor, was available for
this study through the Pittsburgh Supercomputing Center.
By use of this new vectorized method, all lines in one direction, the \
direction for example, were divided into two groups (one for the even and the other
for the odd numbered lines) which lead to two groups of simultaneous algebraic
equations and a corresponding two sets of tridiagnal matrixes in the same way as in
the LineByLine method. The Thomas algorithm could then be applied to each line
in one group at the same time and the calculation could be vectorized. By altering
to update the dependent variable for the two groups, a converged solution could be
obtained some three times faster than that obtained by the LineByLine method.
The flowchart of the Vectorized Line Group method is shown in Fig. 38.
There are three innermost DO loops marked by dashed lines, in which the first and
third are corresponding to the innermost DO loops in the LineByLine method
shown in Fig.37. The dependencies between the pairs ofAâ€™s, RHSâ€™s and (pâ€™s are still
on the subscripts "i", the same as that in the LineByLine method, because the same
line direction is chosen. However, the indices of the innermost DO loops are "j",
which is not the same as that in the LineByLine method. It is because of this
difference that the requirement of vectorized calculation is satisfied. The only
drawback of the Vectorized Line Group method is that it requires a rectangular
computational domain. Fortunately this is achievable for most anticipated problems,
when using domain subdivisions.
53
group No.l
JS = 2, JE = JM1
I
i = 3
l
HE
RHSu = RHSg + RHSnj *
F
v,f anj /A M1J
T
; M ky
group No.2
JS = 3, JE = JM2
i = i 1
I
r
j=JS
9^ RHS g+ <**,.â€ž)/<
T
j=j + 2
!
(f> JE^>
Fig. 38 Flowchart for the Vectorized Line Group iteration method for solving
the associated system of algebraic equations. Dashed lines represent
the innermost DO loops.
54
When the Vectorized Line Group method was applied to one of the three
subdomains shown in Fig. 31, the values of the dependent variables from previous
iteration in the neighboring subdomain were used as the boundary condition along
the interface boundary between the two subdomains. The same treatment was
undertaken for the all three subdomains. Since the solving procedure is iterative, the
continuity condition at such boundaries is insured by updating the values of depenÂ¬
dent variables there alteratively.
CHAPTER IV
RESULTS AND DISCUSSION
Ten cases with different values on the control parameters were investigated
in this numerical study. Those parameter values are listed in Table 41.
Table 41. The list of 10 cases studied in the numerical investigation.
Case
No.
b
(cm)
c
(cm)
Ax
(cm)
rxL
a
Up
(cm/s)
Re
K
(cm/s)
1
2
5
0.4
0.013
0.5
0.003
1
0
2
2
5
5
0.167
1
0.125
50
0
3
2
5
6
0.2
3
1.35
540
0
4
2
5
20
0.667
3
4.5
1,800
0
5
2
5
6
0.2
10
15.
6,000
0
6
2
5
20
0.667
10
50.
20,000
0
7
2
5
40
1.333
10
100.
40,000
0
8
0.6
10
6
0.2
3
4.5
540
0
9
2
5
20
0.667
10
50.
20,000
1
10
2
5
20
0.667
10
50.
20,000
10
Note that the connecting channel has half width a = 0. lcm and length L =30cm
for all cases. The working fluid in this study is assumed to be water with a kinematic
55
56
viscosity of v = 0.01cm2/s, density p = lg/cm3, specific heat Cp=4.18J/gK, and Prandtl
number Pr=l (corresponding to water at high temperature). The zero cross flow
(Vc=0) represents the conduction model, and Re = a2Ax/a is the corresponding
Reynolds number based on the maximum piston velocity Up = (oaAx/2b and the
reservoir half width b, but expressed in terms of Womersley number a, tidal
displacement Ax and the channel half width a through the requirement of mass
conservation and the definition of Womersley number.
The numerical solutions of the governing equations described in Chapter II
obtained by use of the SIMPLETP algorithm explained in Chapter III for the ten
cases and some discussion about several interesting phenomena found from those
results will be presented in this chapter with some work done to validate the
numerical results at the beginning. Note that all dependent variables, the velocity
components U, V, the stream function ifr, which is calculated from U, V, the pressure
P, the Temperature T, as well as the independent variables x, y shown in the figures
of this chapter are dimensionless quantities which are nondimensionalized by the
method shown in equations (2,l)(2,3), although the marker * is omitted as
mentioned in section 2.2.
4.1 Validation of Results
In any numerical process involving calculation by computers, errors always
arise. It is because of this that some estimate of the magnitude of these quantities
is included in all numerical studies.
57
Since the purpose of this dissertation is to obtain a qualitative guidance to the
usage of the thermal pump technique and no detailed experimental data for the time
periodic velocity patterns and temperature profiles in such configuration exist with
which to compare the numerical data, only a qualitative error analysis is carried out.
That is, to demonstrate that the numerical results obtained are qualitatively close to
the true solution of the governing equations for the specified boundary conditions.
The number of grid points chosen is very important for numerical studies of
heat transfer in fluid flows. If a few grid points are taken, the results will be unable
to display the correct structure of the velocity pattern and temperature profile. A
large number of grid points, on the other hand, will need too much computing time,
which may either waste the computer resources or require such long calculations that
they can not be handled by present computers. For time periodic problems, it is
more desirable to use a small number grid points or, in other words, a large grid cell
size because generally a smaller time step size is required for a smaller grid cell size
in order to achieve convergence. If the time step size is very small, the number of
necessary time steps per oscillating cycle will become very large, and thus an
extremely large number of the total time steps will be needed since several
oscillation cycles are required in order to reach the final time periodic state.
A relatively low number of 91 x71 nonuniform grid points were used in this
numerical investigation. Fig. 41 compares the stream line patterns for Case No.6 at
several different oscillation phases for 91 x71 grid points (left part of the figure) and
121 xlOl grid points (right part). The procedure used to obtain the stream lines from
58
cÃ¼t=0Â°
iSiSIlJ
U0 >)'Vvâ€¢
cÃºt=0Â°
..ÃsÃ¡vOÂ¿'ÃÃ
<Â¿'5Ã$3Â£Â£Â¡2Â£Ã/
B
Â¿izrAa/j

o)t=60Â°
Â¡0Â¡
(ot=60Â°
HÃ‰ÃÃP^
grid: 121x101
Fig. 41 Stream line patterns for Case No.6 calculated with different number of
grid points: (a) grid 91x71\ (b) grid 121x101. Increment between
neighboring lines is AiÂ¡r=20 in nondimensional units.
59
the numerically calculated velocity components will be explained in the next section.
There is some difference between the two flow patterns but it is not significant. The
changes from one to the other are only in detail and the basic structure of the flows
does not change. The dimensionless basic time step size 6tb is jc/360 for the grid
system 91x71 with several halvings occurring from the basic size to n/1440 only in
the first oscillation cycle (see Section 3.3), while the smaller basic time increment
6tb= k/1440 was needed for the 121x101 grid in order to get a converged solution.
The CPU time used for the latter is more than 4 times of that for the former for the
same convergence criterion. The slight changes in flow structure and the big saving
in calculation time lead to the decision to choose the sparser grid structure of 91 x71
in the calculations for other cases.
The value of ec, which is the criterion on the convergence indicator oc of the
VelocityPressure correction, is set at 3%. A typical number of VelocityPressure
corrections required at each time step in the SIMPLETP technique for Case No.6
is 11 for this setting in most oscillation cycles except the first one. In the first cycle,
especially in the early part of the cycle, it needs more VelocityPressure iterations to
obtain a converged solution for ec=3%. The criteria for reaching the final time
periodic state was taken as 1% for the velocity and as 0.01% for the temperature.
Twelve oscillation cycles were typically needed for velocity convergence, while 200
cycles were required for temperature. There were three other criteria needed for
solving the system of algebraic equations associated with velocity, pressure and
temperature. The domain averaged ratio of the difference between calculated
60
dependent variables from two successive iterations to the values of the dependent
variables obtained from the current iteration was chosen as a convergence indicator.
The criteria for velocity, pressure and temperature were lxlfr4, lxl O'3, and lxlO'4
respectively. For velocity and temperature, less than 10 iterations were needed to
meet the criteria but usually more than 300 iterations were needed to have the
pressure converged. The numbers of iterations needed to satisfy these criteria
settings varied slightly from case to case for the different values of parameters listed
in Table 41. 3.5 hours of CPU time on a CRAY/YMP at the Pittsburgh SuperÂ¬
computing Center was typically required for obtaining the velocity and pressure
values for Case No.6 using the 91x71 grid under the above converge criteria. An
extra 0.5 hour was needed to find the corresponding temperature field.
As shown in Section 1.3, there are closed form solutions for the velocity and
temperature profiles in oscillating flow in a parallel plate channel connected to end
reservoirs at x= Â± Â«>. The flow is ID in the axial direction and the variations of both
velocity and temperature along the transverse direction are independent of x. Note
that the ratio of the tidal displacement to the channel length, rxL, in this situation is
always zero for whatever value of Ax except infinity. When the end reservoirs are at
a finite distance x=Â±L/2, and thus rxL is nonzero, a distortion of the velocity and
temperature profiles from what they are for infinite channel length will occur and the
above x independence will disappear. The distortion and the x dependence increase
when rxL becomes large and under these conditions the solutions have to be obtained
numerically. On the other hand, when rxL is small, the distortion should be small and
61
the numerical results for velocity profile and temperature distribution at the middle
of channel should be close to the analytic solution for the channel of infinite length
as long as other conditions are the same and, of course, the numerical results have
converged. Based on this point, a comparison of numerical results obtained in the
middle of connecting channel for the case of small rxL with the analytic solution
under the same a and same insulating wall conditions for temperature will show
whether the numerical solutions are correct or not.
The first comparison involves the curves, shown in Fig. 42 and Fig. 43, for
the velocity component U and temperature T in Case No.5 versus the oscillation
phase (ot for an entire oscillation cycle after the time periodic state has been reached.
The numerical results were taken at the position x = 0, y=0.819 in the dimensionless
coordinate system shown in Fig. 21. A 5Â° phase shift was introduced in comparing
the analytic solutions with the numerical results in order to allow the best comÂ¬
parison. This phase shift comes from the difference of phase accounting methods in
the two solutions: the analytic solutions are based on the oscillation phase of the
pressure gradient, while the numerical results are based on the phase of the piston
oscillation.
The analytic solution is valid only for a channel of infinite length. A uniform
in space but time sinusoidal pressure gradient exists for this very simple geometry.
The analytic solving procedure starts from introducing this pressure gradient into the
NavierStokes equation (see Section 1.3), from which all time periodic solutions are
obtained. The phases of all other quantities, velocity, temperature etc., are delimited
62
Cut
Fig. 42 Nondimensional velocity component U for Case No.5 at the middle of
the connecting channel as a function of oscillation phase (Jt:
solid line  numerical result; dashed line  analytic solution.
T(0,0.819,wt)
63
Fig. 43
Nondimensional temperature T for Case No.5 at the middle of the
connecting channel as a function of oscillation phase cot:
solid line  numerical result; dashed line  analytic solution.
64
from this pressure gradient template. For a geometry of finite channel length, the
pressure gradient is no longer uniform in the entire domain, but uniform in the
region far from the ends when rxL is small enough, and thus can not be introduced
a priori at the beginning of the calculation. Instead, its behavior comes part of the
numerical results. The numerical solving procedure begins by giving the movement
of pistons, their position and velocity, in end reservoirs and thus the oscillation phase
of the piston oscillation becomes the basic template. The two templates do not
coincide with each other. In other words, the nearly uniform and time periodic
pressure gradient in the middle of connecting channel generated by oscillation of the
pistons in end reservoirs does not have the same phase as that of piston movement.
Introducing a phase shift is necessary for the best comparison.
One observes from the two figures (42 and 43) that the numerical results
(solid lines) match the analytic solutions (dashed lines) very well over most parts of
oscillation cycle. By comparing the corresponding zero points of the curves in these
two figures, one finds a 87Â°, not a 90Â°, phase delay of the temperature curves from
those of velocity. It is this difference of 3Â° in phase delay that makes the net heat
transfer in axial direction possible without an accompanying net mass transfer along
the axis. In the region near the x axis (center line), the phase delay is 90Â° so that
there is no net heat flux along the axis there (see the curve for a = 10 in Fig. l9(a)).
However, for a =3, which is near the maximum point in Keff curve (see Fig. 16), the
phase delay between temperature and velocity is 75Â° at the x axis where the
difference from 90Â° is as much as 15Â°.
65
The second comparison was made for the curves of velocity component U and
temperature T versus y. Fig. 44 and Fig. 45 show these curves at several oscillation
phases for Case No.5 with rxL=0.2. The same phase shift was introduced when
comparing the analytic curves with numerical results. The curves in (a) come from
the analytic solutions (Eq. (1,4) and Eq. (1,6)), while those in (b) are from the
numerical results at x = 0. One sees that the shape of the curves are essentially
identical except for the kinks produced by the finite grid point structure used in the
numerical calculations. The velocity curves for Case No.6 are presented in Fig. 46.
The curves are still almost the same in the region near the channel walls, but there
are larger differences between the numerical and analytic solutions near the center
line. Since rxL = 0.67 in this case, it makes sense that large differences should exist
in this case as explained above.
A comparison of curves of pressure gradient for Case No.5 versus oscillation
phase between numerical results at x = 0 and that used for analytic solutions is shown
in Fig. 47. The dashed line again represents the analytic pressure gradient and the
same phase shift as above was introduced. The behavior of the numerical pressure
gradient does match the analytic one but there is an error larger than that for
velocity and temperature shown in Fig. 42 and 43, especially near the points at
which the pistons change their movement direction. It was found that this error
could not be reduced by reducing the grid size and/or time step size. Some
simplification is made in the SIMPLE algorithm when the pressure correction
equation is established from the continuity equation (explained in the Appendix, see
66
U(y,cjt)
U(0,y,cjt)
Fig. 44 Nondimensional velocity component U for Case No.5 at the middle of
the connecting channel as a function of nondimensional position y at
several oscillation phases: (a) analytic solution; (b) numerical result.
67
T(y,cjt)
T(O.y.cjt)
Fig. 45 Nondimensional temperature T for Case No.5 at the middle of the
connecting channel as a function of nondimensional position y at
several oscillation phases: (a) analytic solution; (b) numerical result.
68
U(y,cjt)
U(0,y,cjt)
Fig. 46 Nondimensional velocity component U for Case No.6 at the middle of
the connecting channel as a function of nondimensional position y at
several oscillation phases: (a) analytic solution; (b) numerical result.
69
O 90 180 270 360
cot
Fig. 47
Non dimensional x direction pressure gradient for Case No.5 at the
middle of connecting channel as a function of oscillation phase vt:
solid line  numerical result; dashed line  analytic solution.
70
also reference 19). This simplification does not affect obtaining a good pressure
field for steady flows but may introduce more errors for unsteady and especially for
oscillatory flows.
4.2 Velocity Field
It was found that the velocity field under the conditions considered in this
study can not be well presented by the normal ways, i.e. by drawing arrows at several
points in the physical domain with length proportional to the speed and in the
velocity direction at that particular location, because of the complexity of the
oscillation flows and the large differences in speed at different locations within the
end reservoirs. Therefore, plotting the isovalue lines of the time varying stream
function, the stream lines, was chosen as the mode of velocity field presentation.
From the definition of stream function iÂ¡r(x,y,t)
U(x,y,t)
dty(x,y,t)
dy
V(x,y,0  
ax
(4,1)
,the value of the stream function i/r(x,y,t) at a particular point (x,y,t) can be calculated
from i/rix^y^t), the value of stream function at its neighboring point (x^y^t). The
following formula will do this.
(W0 (x,y, t)
4Â»(x,y,0 = if(x0,y0,t) + J U(p0,?2l ,t) dy  J V(?^,y,t)dx (4<2)
(Wo*) (w0
2
71
Because of the staggered grid, the values of stream function are easily obtained at
the corners of grid cells without any interpolation (see Fig. 33).
The spacing between the streamline isovalues is equal to a constant which may
vary from case to case. This method of representing a time dependent flow field
allows one to take advantage of the fact that the absolute value of the instantaneous
streamfunction gradient is equal to flow speed and hence that regions of high streamÂ¬
line concentration (i.e. darkened regions) corresponding to parts of the flow field
having high velocity. The velocity is always in the tangential direction of the stream
lines, and the solid curves represent counterclockwise rotation, the dotted curves zero
rotation and the dashed curves clockwise rotation.
The stream lines in the end reservoirs at the oscillation phases of 15Â° intervals
for Case No.3 are presented in Fig. 48. Only the flow patterns at the phases in
lower half period 0Â°Â¿a>t<180Â° are shown here. The flow profile at any point in upper
half period 180Â°^ot <360Â° can be found by reflecting the picture at the corresponding
180Â° earlier phase about the vertical center line (i.e. y axis). The constant interval
between two neighboring lines is 5 nondimensional units.
It is observed that as the piston in the left reservoir moves toward the
â€¢ â€¢ A y
connecting channel starting from its extreme position of x = câ€” at ot = 0Â°, there
is essentially one large vortex pair present with the flow along the horizontal center
line (i.e. x axis) toward the piston surface. A weaker and smaller second vortex pair
of opposite rotation sense sits in the corners nearest the channel end. Flow is seen
leaving the left reservoir starting at about cot =30Â° and continuing on to about
72
cot=0Â°
sum
^ :
r
C0t=90Â°
^pgjgl
55
69
â€”'r.'.'V^'o^vSvf
* ^Si>' c
at=15Â°
Â®fÂ¡53
59
Ã“Ã8S
Fig. 48
Stream line pattern for Case No.3 at oscillation phases of interval
A(cot)=15Â° in the lower half cycle 0Â°Â¿cot<180Â°.
73
o)t=150Â° mainly via a side flow into the connecting channel. In the right reservoir,
one can see a distinct high velocity jet emanating from the connecting channel and
moving across the entire reservoir along the x axis. It begins at about ot=30Â°, the
same time as flow starts leaving the left reservoir, continues flowing along the
channel exit until at least (ot = 165Â°. The jet collides with the right piston surface at
around (ot =210Â° and disappears at
in the left reservoir at (ot=30Â° and o)t = 105Â°, respectively. The origin for this high
velocity pulse is clearly the inability for the liquid entering the reservoir during the
suction stage to accommodate itself to the wider cross section of the reservoir as it
would if the Reynolds number were low enough so that flow separation at the
entrance would not occur (see Case No.l in Fig. 414). The appearance of a velocity
jet during the intake portion of the piston stroke has also been noted in the related
experimental work by Durst23 as well as by Kurzweg, Lindgren and Lothrop24. Note
that the large counter rotating vortex pair filling most of the reservoirs remains
during the entire oscillation cycle and its rotating sense is always such that fluid
motion along the x axis is always from the end of connecting channel and toward the
moving pistons, no matter what oscillation phase one is in. The smaller second
vortex pair also exists during the entire oscillation cycle, although it is much weaker
than the primary one. There is third vortex pair showing up in Fig. 48. It is
generated near the flow separation point near the channel end in the right reservoir
at about (ot=45Â°, shortly after the velocity jet starts, pushed toward the receding
piston by the velocity jet and finally joins the large vortex pair at about a>t = 135c.
74
The numerical development of the nondimensional velocity component U at
three different locations along the x axis within the left reservoir under the conditions
corresponding to Case No. 3 are plotted in Fig. 49 with (a) at a position of 2a from
the channel end, (b) at 25a, and (c) at 40a. It shows that except at the position
nearest the connecting channel end, the velocity remains negative throughout the
oscillation cycle and is thus always moving the liquid toward the piston surface. The
sharp velocity spikes seen correspond to the periodic velocity jets crossing the
reservoir which are shown in the right reservoirs of Fig. 48 since they exist in the left
reservoir during the upper half period 180Â°z(ot<360Â°. The time delay in the velocity
peak between the nondimensional position x = 2 and x = 25 is 0.0875 seconds and
thus indicates a velocity pulse traveling at 26.3cm/s which is only slightly less than
the nns axial velocity of 27cm/s in the channel but considerably larger than the
piston face velocity at the same instant.
The stream lines for Case No.4 and Case No.6 are shown in Fig. 410 and 4
11, respectively, using the same format as that in Fig. 48 but employing 20 non
dimensional units of interval in the isovalues of i/r. Similar to the results shown in
Fig. 48, there are three vortex pairs, two remaining during the entire oscillation cycle
and one existing in a portion of the cycle, and also a velocity jet in both cases.
For Case No.4(Fig. 410), the jet within the right reservoir begins at o)t=30Â°,
the same as in Case No.3, but collides with the piston surface at about ot = 90Â°, much
earlier than the (ot=210Â° value of Case No.3, and disappears at about ot180Â°, also
earlier than the o)t=285Â° result in Case No.3. The third vortex pair starts at about
75
wt
Fig. 49 Temporal approach to the final time periodic state of the non
dimensional velocity component U(x,0,cot) for Case No.3 at three
different locations: (a)x=152\ (b)* = 775; (c) x=190.
76
cot=0Â°
(WBM
E9.
â€¢189
\ â€˜Ã³AC o')'ill
â€”OrV^â€”'V i '
â€œt=15Â°
J \y 1
\
. rr'Vi^ â€¢?
Ã‰SÃ‰PM
cot=45Â°
h~&5Mjyy.
192
: f^^asysisc. â–
;Sp
o> j J
â€¢
cot=60Â°
193
C Co
1
j
pf5fli>vx
SÃ1A Or=^/j sÂ£Â¿^
(Ot=75Â°
193<\y<95
~MSI
cSosi
ISlk
ox=105Â°
wÃÃw.
182
Ã'^czSf)^\
I j tVs\
tot=120Â°
172<\k171
ox=135Â°
1S7
cot=150Â°
lES^
â€¢141
Ã(iksÂ¿S)V
cot=165Â°
VjvQ*' ~â€”â€¢.â€”)
123
â– â€¢' â€”â– >?ggi=gfa. ;
Fig. 410
Stream line pattern for Case No.4 at oscillation intervals of A(
in the lower half cycle 0Â°i(ot<180Â°.
77
ox=0Â°
:
â– KtÃ©r
211
_:vN^>^ror.r^ <
CT5 â€¢
tea
ox=15Â°
211
O' ^s?:=rÂ¿Ã f
cot=30Â°
Jg
208
tÂ®S2Â§% a*si2Â°B
I 185
MU'
Ã¼*=135Â°
1CT/
172
^ '^/ÃÃÃXâ€™iÃrÃSÃ¡
'â€”â€¢> 'C^aSÂ»v?Ã±KÃ‰sÂ¡wk
: i,
; â€”
ox=45Â°
a a
Â¿SCI
209Â«y<73
â€”j}\\
O,'^:
Fig. 411
Stream line pattern of Case No.6 at oscillation intervals of A((ot)=15
in the lower half cycle 0Â°i
78
cot30Â°, almost at the same time as when the jet pulse begins and a little earlier than
at cot=45Â° in Case No.3, and joins the big vortex pair at cot = 75Â°, which is also much
earlier than cot = 150Â° phase point in Case No.3. The only difference between the
conditions in Case No.3 and Case No.4 is the different values of the tidal displaceÂ¬
ment Ax. Case No.4 has Ax=20cm, or 10/3 times more than that in Case No.3, which
means there is a larger rms velocity in the channel for Case No.4. It is the faster
fluid movement in channel which produces the faster jet and associated shorter
appearance of the third vortex pair.
Comparing the stream lines of Case No.6 with those in Case No.4, one can not
find as large a difference as that between the stream lines of Case No.3 and Case
No.4. Note that Case No.6 has a = 10 and Case No.4 has a=3, which is the only
difference between the two cases and means that the oscillation in Case No.6 is 10/3
times faster than that in Case No.4 since the fluid viscosity and channel width do not
change. The above observation reveals the fact that Womersley number no longer
has a large effect on the flow profile in the end reservoirs for large b/a ratio like its
influence on the axial velocity distributions within the connecting channel (see
Section 1.3). Instead, the tidal displacement Ax has more influence on the flow
profile in reservoirs than Womersley number does. Actually, significant changes in
the reservoir flow patterns are characterized by the Reynolds number, as will be
shown below.
Fig. 412 presents the pressure distribution for Case No.6 along the x axis at
several oscillation phases over the whole period 0Â°^cot<360Â°, and Fig. 413 shows the
79
corresponding stream lines at the same phase points. The pressure in most parts of
the channel is seen to vary linearly with x for most of the oscillation phases except
those near ot = 90Â° and o>t=270Â° when the pistonsâ€™ movement changes from acceleraÂ¬
tion into deceleration. The linear pressure distribution in the channel corresponds
to the expected constant pressure gradient there, and as used in the analytic solutions
(see Section 1.3). Examining these last two figures, one finds that the largest
pressure drop occurs at the channel end during that time when the fluid enters into
the channel through that end and when the velocity jet exists from the other end.
It is apparent that the pressure drop existing at one end of the connecting channel
provides the necessary power to form the jet flow at another end. The two ends alter
their roles as the pistons change their directions of motion. The magnitude of the
pressure gradient found in the channel agrees well with the analytic solution given
by Eq. (1,5) for the same values of parameters.
Stream lines with a spacing interval of 0.05 units for Case No.l are presented
in Fig. 414. The corresponding Reynolds number in this case has the low value of
2, which is small enough to avoid flow separations in the case of steady flow over a
step. For a part of the oscillation cycle (about half), when the pistons move essenÂ¬
tially in one direction, no separations exists also in this oscillating flow situation. The
fact that flow separation does occur over rest of the oscillation cycle even for such
low Reynolds number is easily understood by noticing that this part of oscillation
period is around the points at which the pistonsâ€™ motion changes its direction. The
flow pattern in Case No.l, in which both the existing vortex pairs and flow jet have
x10 p xlO4 p xlO4 p x 10
80
Fig. 412
Nondimensional pressure distribution along the x axis in Case No. 6 at
oscillation intervals of A(ot)=45Â° over the whole cycle 0Â°Â¿o)t<360Â°.
81
Fig. 413 Stream line pattern for Case No.6 at the oscillation intervals of
A((ot)=45Â° over the whole period 0Â°z(ot<360Â°.
82
Fig. 414 Stream line pattern for Case No.l at the oscillation intervals of
A(a>t) = 15Â° in the lower half cycle 0Â°*cot <180Â°.
83
disappeared, is totally different from Case No.3, No.4 and No.6 discussed earlier.
This clearly shows Reynolds number dependent flow patterns. To determine the
critical point at which the flow pattern changes from the type of Case No.l to that
of Case No.3 is not included in the dissertation topics, but it can be found from the
numerical results that the point is below Re = or Ax/a50 because the stream lines for
Case No.2 (not shown) have the same shape as those of Case No.3.
The stream lines in the right reservoir at four oscillation phases separated by
an interval of A(o)t)=90Â° for Case No.8, with longer and narrower reservoirs, are
shown in Fig. 415. The interval for the equally spaced isovalues of ijr is 3 non
dimensional units. It is certain that vortex pair remain during the most parts of
oscillation cycle, since the corresponding Reynolds number here is 540, the same as
that for Case No.3. Two almost equally large, but not equally strong, vortex pairs are
observed and it is seen that the fluid jet no longer impinges on the piston surface.
A qualitative explanation for the multiple vortex pair pattern observed is that the
fluid jet formed at the channel end can, from continuity considerations, penetrate the
reservoir during the oscillation cycle by no more than the tidal displacement of the
fluid elements within the connecting channel. In the present case, this distance is
6cm which is less than the reservoir length of 10cm but about equal to the first vortex
pair axial length. Case. No.8 was the only situation in this study in which one finds
more than one equally large vortex pairs existing during most parts of the oscillation
cycle. All other cases do not have this since either the reservoir length is short
compared with the tidal displacement or the Reynolds number is too low.
84
Fig. 415 Stream line pattern for Case No.S in the right reservoir at oscillation
intervals of A(ot)=90Â° over the whole period 0Â°za)t<360Â°.
85
Both Case No.l and Case No.8 are not good for the enhancement of heat
transfer by oscillating flow, but they help make a clear understanding of the flow
patterns.
Case No.9 and Case No.10 present the cross flow model. Stream lines
representing these cases are shown in Fig. 416 and 417 using the same uniform
spaced isovalues of ifr as those in Fig. 410. The weak cross flow case, Case No.9,
exhibits a flow profile which looks like a small distortion superposed on the flows of
Case No.6. This is expected since the cross flow velocity here is relatively small.
Despite of the weak cross flow velocity, the symmetry of the flows about the
horizontal center line (x axis) disappears. There still is a large vortex pair, although
the two sides of the vortex pair no longer always have same size, but they remain
during the entire oscillation cycle. The weaker and smaller vortex pair no longer
always exists over the entire oscillation cycle. The one near the cross flow entrance
disappears for some values of
vortex pair found. Instead, a single vortex is generated near the channel end on that
side of the fluid jet nearest the cross flow exit. It begins at (ot=30Â° and joins the
large vortex at about o)t = 75Â°, the same as that in Case No.6. The flow jet starts at
(ot =30Â° and disappears around a>t = 180Â°, also the same as that in Case No.6, but never
touches the piston surface because of more resistance from the cross flow. It is no
longer strictly along the horizontal center line but is bent toward the cross flow exit
at the beginning and bowed back before vanishing.
86
CÃ¼t=45Â°
218
m m
cot=60Â°
Â¿s^ySl
259
* VA'\
0%Â£$ÃN
â€”â€”â€”* m m
cot=75Â°
318
jgggj
, , !
Fig. 416 Stream line pattern for Case No.9 at oscillation intervals of A(o)t) = 15Â°
over the lower half period 0Â°zo>t<180Â°.
87
m
0)t=15Â°
281
â™¦ Â« 4
iâ€”; ! 1 i i â€”â€”
; 1â€”2â€”zâ€”
Â¡Â¡Â¡Â¡IB
cot=30Â°
299
/ / fifi//' \
*#({ Ã wÃ¡
Â» .
Fig. 417
Stream line pattern for Case No. 10 at oscillation intervals of A (cot)=15
in the lower half cycle 0Â°^(ot
88
The reservoir stream line pattern corresponding to Case No. 10, however, is
significantly different from that in Case No.6. There no longer exists a pair of
vortices in the end reservoirs. Instead, the large counterrotating vortex pair of Case
No.6 is replaced by a single large vortex which fills almost the entire reservoir over
the entire oscillation cycle. A very weak vortex occurs at the corner near the cross
flow entrance but does not remain for the entire oscillation cycle. Only one vortex
is generated by the flow jet, the same as that in Case No.9, but it does not join the
large vortex as happens in Case No.6 and also in Case No.9. It does not existing
during the whole cycle. The velocity jet is always bent toward the cross flow exit and
never penetrates half of the reservoir length because of the presence of the strong
cross flow. The portion of the oscillation cycle during which the jet pulse shows up
remains from cot =30Â° to cot=180Â°, the same as in Case No.6 and Case No.9. The
behavior of the velocity jet in Case No.9 and Case No.10 has some similarities with
that encountered in fluidic devices where the direction of the main flow jet is
controlled by the magnitude and direction of a cross flow.25
4.3 Temperature and Heat Transfer Coefficients
The associated temperature distributions and heat transfer coefficients for the
thermal pump configurations presented by Cases No.3, No.4, No.6, No.9 and No. 10
were determined numerically.
89
Fig. 418 shows the equally spaced temperature contours for Case No.3 at 12
phase intervals A(o)t)=15Â° over the lower half period 0Â°Â¿a>t<180Â°. Similar to the
process used for velocity, the temperature contours at varying oscillation phases over
the upper half period 180Â°
picture about the y axis. The isotherms shown are at nondimensional interval of
0.02 temperature units with the maximum dimensionless temperature difference given
as 2. Thus one can realize that the temperatures in the end reservoirs are fairly
uniform. Larger temperature gradients occur only near the reservoir walls including
the piston surface and along the reservoir center line. A thermal jet is observed to
be associated with the fluid jet exiting from the connecting channel. The thermal jet
lasts as long as the velocity jet does, but there is a phase delay between them as
already discussed in Section 4.1. Note the impingement of the thermal jet onto the
piston surface can lead to a very high heat exchange at the piston surface. The fluid
motion produces a good mixing of the temperature in the reservoirs with the preÂ¬
dominant conductive heat interchange between the reservoir fluid and the oscillating
flow in the connecting channel occurring across the jet interface. The non
dimensional temperature profiles along the line in they direction atx = 153, which
is located in the left reservoir close to the channel end, are presented in Fig. 419,
where (a) shows the temperature at A(o>t) = 60Â° intervals starting from o)t=0Â°, and (b)
gives the time averaged value of T. It is seen that there is no big temperature
change with time along most parts of the line except near the center line where the
pulse jet occurs. There are two large temperature drops along the line, one at the
90
cot=0Â°
O^i
lSo
wt=30Â°
;  Tt'f1 njrm i; â– ' 1
' ***''â€”* /
^ 'x
r' ^ â€”i
cot=120Â°
f i'
1
p>^
dx=135Â°
i" <=> i
ox=150Â°
â€™  J $
r ^ cr j
Jiiâ€”w mm \ Ti av> ^ 1
L ^ 
Â»^Saa.v/./$
Fig. 418
Temperature contours for Case No.3 at oscillation intervals of
A(ot)=15Â° in the lower half cycle 0Â°iot<180Â°.
91
T
Fig. 419 Nondimensional temperature profile for Case No.3 along the vertical
linex = 75J in the left reservoir: (a) at several oscillation phases; (b)
time averaged.
92
reservoir walls and one near the center line. From Fig. 418, it is clear that
temperature curves along they direction lines at most positions of x=constant within
both left and right reservoirs have the same behavior as those shown in Fig. 419.
Figs. 420, 421 and Figs. 422, 423 are quite similar to Figs. 418, 419 for
Case No.4 and Case No.6, respectively. From the temperature contours, one finds
that they look almost identical to Case No.3, i.e. fairly uniform over most of the
reservoirs, with high gradients at walls and center line, and a jet pulse from the
channel to piston surface. The thermal jets in Case No.4 and Case No.6 start earlier,
travel faster and last longer than the thermal jet does in Case No.3. In these two
cases, the thermal jets last from cot = 75Â° to cot =330Â° for Case No.4 and from cot=60Â°
to cot =360Â° for Case No.6, while the corresponding fluid jets last from cot=30Â° to
cot = 180Â° in both cases. This phenomenon further shows that the heat exchange
between the reservoir fluid and the entering flow from connecting channel is
dominated by conduction which introduces the phase lags and also needs a longer
time to complete. From Fig. 421(a) and 423(a), one observes that the variation of
temperature with time is still small in the most areas of the reservoirs except near
the center line and that the two large temperature drops along the vertical line occur
at the reservoir walls and near the center line.
Temperature fields in the two cases for the cross flow model are presented
in the same way as those for the conduction model. Fig. 424 and 425 correspond
to Case No.9, and Fig. 426 and 427 to Case No. 10.
93
Fig. 420 Temperature contours for Case No.4 at oscillation intervals of
A(
94
T
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T
Fig. 421 Nondimensional temperature profile for Case No.4 along the vertical
line x = 153 in the left reservoir: (a) at several oscillation phases;
(b) time averaged.
95
y X â€˜
ox=90Â°
, 'â€”^72 \f
â€” SsEa
Fig. 422 Temperature contours for Case No.6 at oscillation intervals of
A(o)t) = 150 in the lower half cycle 0Â°*cot <180Â°.
96
T
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T
Fig. 423 Nondimensional temperature profile for Case No.6 along the vertical
line x = 153 in the left reservoir: (a) at several oscillation phases;
(b) time averaged.
97
(Ãºt=90Â°
*1
ck
H
,i â€¢? â€¢
â– til
ox=1058
â€”> J ,
'â€”Â£
BJ
l'rrl*^ â€¢ â€˜
(Ãºt=120Â°
''1
,, ftmrTr=*  _ p
rrr
U â€¢ ' â€™
cot=i358
Â» * *
\ !
J
ml
n Â«=2^
&â€”=^J.
(ot=150Â°
t â€¢ 1 u
31
^ f : :
( ^\\ / .\
cot=165Â°
L3
JJBtmrn 3Ãœ
tj J
Fig. 424
Temperature contours for Case No.9 at oscillation intervals of
A((ot) = 15Â° in the lower half cycle 0Â°*(ot<180Â°.
98
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T
Fig. 425 Nondimensional temperature profile for Case No.9 along the vertical
line x = 153 in the left reservoir: (a) at several oscillation phases;
(b) time averaged.
99
Fig. 426 Temperature contours for Case No. 10 at oscillation intervals of
A(o)t) = 15Â° in the lower half cycle 0Â°i
100
T
T
Fig. 427 Nondimensional temperature profile for Case No.10 along the vertical
line x = 153 in the left reservoir: (a) at several oscillation phases;
(b) time averaged.
101
The temperature distribution in Case No.9 is very uniform in most of the end
reservoirs except at the cross flow entrance and near the thermal pulse. The average
temperature over the entire reservoir has the approximate value at the outflow
section of the cross flow and is also nearly equal to the value at the channel end. It
is shown by Fig. 425 that the cross flow has a big temperature drop when first
entering the reservoir, then has almost no change until running into the thermal jet,
gets another temperature drop from the jet and maintains the new temperature value
until exiting through the opposite wall. For the cross flow model, the temperature
along other vertical lines is also almost constant as can be figured out by examining
Fig. 424.
The temperature field shown in Fig. 426 for Case No. 10 has almost the same
characteristics as that for Case No.9, with a few differences. One difference is the
lower uniformness of the temperature in the reservoirs, and another is that a high
temperature gradient exists at the cross flow entrance into the reservoir. Both of
these changes come from the high velocity of the cross flow in Case No. 10. It is
because of this high velocity that the cross flow in Case No. 10 keeps its original
temperature for a little distance after it comes into the reservoir. It then starts the
same procedure for temperature variation as that of the cross flow in Case No.9.
These are also presented in Fig. 427. Note that the cross flow velocity Vc = 10cm/s
in Case No. 10. This is still quite small compared to the value 1000cm/s mis axial
velocity value existing in the channel.
102
One sees a totally different temperature field for Case No.9 in Fig. 424 from
that of Case No.6 in Fig. 422, although the velocity fields for the two cases are close
to each other (see Fig. 416 and 411). On the other hand, very close temperature
fields are found for Case No.9 and Case No. 10 which have a totally different velocity
field (see Fig. 416 and 417). This phenomenon shows that the temperature disÂ¬
tribution in the end reservoirs is mainly affected by the temperature boundary
conditions and less by the velocity field.
With the velocity and temperature fields on hand, one can calculate the heat
transferr rate through the channel by the same way as used for analytic solution. In
particular, one has
Q =
coa2p Cp(ThTc)
2 TC
2ic
+1
J J* U(0,y,t)T(0,y,t)dy
d((jst)
(4,3)
where U, T and y are still nondimensional, but Q is dimensional and expressed in
Watts.
The heat transfer coefficient between the connecting channel ends will then
be obtained by
h
Q
2a(ThTc)
(4,4)
The numerical heat transfer coefficients, defined by hn, obtained from the
numerical results of Cases No.3, 4, 6, 9 and 10 are listed in Table 42. ha there is the
heat transfer coefficient calculated according to the values of effective thermal
103
diffusivity Kg* from the analytic solution discussed in Section 1.3 and the
corresponding values of parameters used in the numerical investigation. The
corresponding angular frequency o, the tidal displacement Ax and approximate non
dimensional mean temperature T, in the left reservoir are also listed in the table for
convenience of discussion.
Table 42 Heat transfer coefficients for the several cases in the study.
Case
No.
hn (W/cm2 K)
ha (W/cm2 K)
o) (rad/s)
Ax (cm)
T,
3
0.67
0.45
9
6
0.94
4
12.3
5.1
9
20
0.81
6
49.3
13.9
100
20
0.77
9
37.7
13.9
100
20
0.18
10
180.6
13.9
100
20
0.64
The data in Table 42 show that the heat transfer coefficients achievable by
oscillating flows in thermal pump for most of these cases are in the order of 10s
Watts per square meters per degree which is a value only possible in convective heat
transfer when using phase changes26 and is the same order as that possible via heat
pipes. The Case No. 10 produces an even larger value of hn by a factor of ten, thus
making it possible in this case to transfer more heat than via heat pipes.
It is expected that the higher the tidal displacement Ax, the larger will be the
heat transfer coefficient. However, hn is no longer exactly proportional to (Ax)2 for
104
fixed (o as is ha (see data for Case No.3 and Case No.4) because of the nonzero
values of rxL there. It is also because of the not very small value of rxL that the y
direction velocity component V does not totally vanish. This causes more heat
exchange between the core region and the boundary layers and therefore produces
a higher value of heat transfer coefficient hn than for ha in each of the cases
considered.
Having examined the hn and TÂ¡ in three cases of the conduction model, one
finds that lower heat transfer coefficient corresponds to higher mean temperature in
the right reservoir and vice versa. Because the heat transferred through the conÂ¬
necting channel is supplied into the system by conduction across the reservoir walls,
which is dependent on the temperature drop from the walls to the fluid, a lower
mean temperature in the right reservoir fluid is necessary in order to absorb enough
heat from the walls at fixed temperature and then to transfer it to the other
reservoir. Note that the largest temperature drop in the three cases is 27% of the
half total drop for the entire system. This is not so big that an obvious "bottle neck"
for heat transfer exists at the reservoir walls. The reason for this is that the size of
the reservoir walls is large enough to provide the required area for supplying enough
heat.
The requirement for large size reservoirs does not exist in the cross flow
model because of the different way of heat supply, namely convection by cross flows.
Instead, a large enough cross flow velocity Vc is necessary in order to provide enough
heat. For too small a value of Vc, it is still possible that there is a "bottle neck" in
105
this model as happens in Case No.9 where the temperature drop at the cross flow
entrance is 82% of the half temperature difference for the whole system and
therefore the amount of heat transferred is even smaller than that in Case No.6. On
the other hand, if the purpose for using the thermal pump is to cool the liquid
coming into the right reservoir, the conditions in Case No.9 give a good example.
Case No. 10 has the highest heat transfer coefficient among all the cases in this study,
which means that a combination of high 6>, large Ax and large Vc in a cross flow
model is the preferred model of heat transfer enhancement by oscillating flows in a
thermal pump.
CHAPTER V
CONCLUDING REMARKS
The enhancement of heat transfer is an old topic for engineers and scientists,
specially in the thermal sciences. In recent years, numerical simulation has become
more and more popular as a research tool for seeking new methods to enhance heat
transfer. In this dissertation, a numerical simulation of a simplified Thermal Pump
model for enhancing heat transfer was carried out.
Some interesting phenomena were discovered from the numerical results. The
velocity field in each reservoir was found to be characterized by a high velocity jet
pulse emanating from the channel end during a portion of the oscillation cycle
starting when the piston recedes from the channel end and one or more, depending
on the ratio of tidal displacement and the reservoir length, counterrotating vortex
pair(s) existing during the whole oscillation cycle as long as the Reynolds number,
based on the maximum of piston velocity and reservoir half width, is large enough
and the cross flow velocity is small enough. For very small Reynolds number, the jet
pulse disappears totally and only one vortex pair shows up in the reservoirs near the
channel exits during that portion of the oscillating cycle when the pistons change
their movement direction. Under the conditions of large cross flow velocity, the fluid
jet still exists but is bent in the direction of the cross flow, a single large vortex exists
106
107
during the whole cycle, and a small one can be found only during part of the cycle.
It is also shown, from the presented computer graphics of the stream lines, that the
change of Womersley number does not affect the flow pattern in reservoir as much
as it does in the channel, but that the change in tidal displacement, established by
changing the piston displacement, and also a change in cross flow velocity have a
strong affect on reservoir flow structure.
Temperature profiles were also obtained for several cases. One finds that the
temperature in the end reservoirs is fairly uniform and that a thermal jet associated
with the velocity jet exits in the reservoir during part of the oscillation cycle. In the
case of constant temperature along the reservoir walls and piston surface (conduction
model), large temperature gradients exist near the wall and center line, while in the
case of cross flow, a high temperature gradient exists in the region near the cross
flow entrances. The temperature distribution in the end reservoirs mainly depends
on the temperature boundary conditions instead of the velocity field associated with
it.
Heat transfer coefficients between the channel ends were determined using
water as the working fluid. It was found that in most cases investigated the heat
transfer coefficients reach the order of 105 Watts per square meter per degree and
the highest values was as much as 1.8x1o6 W/m2 K in the cross flow model, which
is superior to the heat transfer coefficient achievable with heat pipes. The high
oscillation frequency combined with large tidal displacement will produce more heat
transferred through the connecting channel but requires a large reservoir size,
108
compared to the channel width, in the conduction model or a large cross flow
velocity to supply enough heat in the cross flow model.
A good understanding of the flow structure and temperature distribution in
the end reservoirs obtained from the numerical results provides more information on
the use of sinusoidal oscillating flows for the enhancement of heat transfer. The
information may also find applications for flow studies on internal combustion
engines and displacement pumps or other related configurations.
Meanwhile, a modification of the widely used SemiImplicit Method for
Pressure Linked Equations (SIMPLE), which has been termed SIMPLETP (for
Time Periodic), was made to take the time periodicity and boundary movements into
account. Variable time step sizes are used in the SIMPLETP for identical conÂ¬
vergence in each time step. Two types of initial guesses for iteratively solving the
NavierStokes equations are used depending on which is closer to the final periodic
velocity and pressure field. A Vectorized Line Group method for solving the
associated system of algebraic equations was developed and used in conjunction with
a multiblock algorithm to speed up the calculations on a CRAY/YMP superÂ¬
computer. It was found that the SIMPLETP is successful for the 2D, time periodic
flow and heat transfer problems considered in this dissertation and there should be
no major difficulty in extending the present method to 3D problems and/or to other
methods related to the SIMPLE algorithm as well as the other discretizing schemes.
Further research topics in the simulation of thermal pump processes and
related physical phenomena should include turbulence modeling, temperature
109
dependent viscosity, nonrectangular reservoirs and multichannel connections. The
extension to the development of numerical methods for other more complicated time
periodic recirculating flows of incompressible viscous fluids would first require an
increase in the accuracy of calculations for the instantaneous pressure field. Using
SIMPLER, instead of SIMPLE, may improve the calculation, otherwise it will be
necessary to make a direct connection between the pressure fields in two successive
time steps. The calculations involving pressure corrections converge much more
slowly than either velocity or temperature. An application of multigrid techniques
may help speed up such calculations.
APPENDIX
THE VELOCITYPRESSURE CORRECTION
The incompressible fluid assumption that the fluid density is independent of
pressure is commonly used in the theoretical study of flow and heat transfer
problems involving liquids. It both well represents the real property of liquids and
also much simplifies the governing equations. The dimensional governing equations
under the all assumptions mentioned in Section 2.2 read
Ux + Vy = 0 (A,l)
U, + (UU)X + (VU\ = px + V (Uâ€ž + Uâ€ž) (A,2)
Vt + (UF), + (FK)^ = Py + v (Kâ€ž + F^) (A,3)
Tt  (UT)X + (FT)^ = k (7^ + ) (A,4)
There are four equations corresponding to four unknowns so that the system
of equations is complete. Note that here pressure P replaces density p to become
the dependent variable governed by the continuity equation but it does not appear
in the continuity equation. The relationship between pressure and velocity in
110
Ill
incompressible flows can be expressed so that only the pressure field, which satisfies
the given boundary conditions, both for itself and for velocity, will produce a correct
velocity field which satisfies not only the boundary conditions but also the continuity
equation. Based on this concept, a VelocityPressure correction procedure was
constructed in the SIMPLE algorithm as follows.
1) A guess for the pressure distribution is substituted into the momentum
equations (A,2) and (A,3) and the corresponding velocity components
are solved out under the velocity boundary conditions given.
2) In general, the resultant velocity field does not satisfy the continuity
equation (A,l) since the guessed pressure is usually incorrect. There
will be some information about where and how the incorrect scaler
pressure field should be changed, which is included in the residual
obtained by substituting the incorrect velocity into the continuity
equation. By use of the pressure correction equation coming from the
residual, the amount of required pressure correction is calculated.
3) The obtained pressure correction is used to correct both guessed
pressure and incorrect velocity.
4) The corrected pressure is considered as a new guess for the correct
pressure field and is put into the momentum equations.
112
5) These four steps are repeated many times until a velocity field satisfying
the continuity equation is found. The corresponding pressure field is
then considered the correct pressure.
In deriving the pressure correction equation, one introduces
(A,5)
P = P* + P
(A,6)
U = U* + U
(A,7)
V = V* + V
where P, U and V are the correct pressure and velocity components satisfying the
momentum and continuity equations; the quantities with * are either the guessed or
the imperfect pressure and velocity; and the quantities with ~ are the amount of
corrections.
The discretized momentum equations for the correct pressure and velocity
components are
ApUp '52/AiUi + BU{PW Pe) + Sv
(A,8)
(A.9)
apvp = + bap.p.) * sv
113
where the summation is taken for all neighbors and Sv, Sv are source terms which
do not contain the pressure.
If the guessed pressure is employed, the discretized equations will be
afv;  Y,A,u; * bu(pwp;) * su
(A, 10)
apv; = Y.Ay; * By(p; />;> * sâ€ž
(A.H)
Subtracting equation (A,10) from (A,8), (A,ll)
equations (A,5), (A,6) and (A,7), one has
from (A,9), and using
ArÂ°r  Y.APi +Bv(PÂ»P,)
(A, 12)
ApVp  Â£A,V, *BV(PSPn)
(A, 13)
A further assumption that the correction at the point is independent of the
correction at its neighboring points is made here to simplify equations (A, 12) and
(A, 13). This shows that
0 = = C(PWP,) (A, 14)
Ap
v=^Psrn) =D(PSPn)
Ap
(A, 15)
The discretized continuity equation reads
114
(Ue  UJ Ay H (Vn  Vs) Ax = 0 (A,16)
or (from (A,5), (A,6) and (A,7))
([/;  K) Ay * (v;  VÂ¡) Ax * (0,  ÃœJAy * (Kâ€ž  Vs)Ax = 0 (A17)
Using equation (A, 14) and (A, 15) and making the necessary adjustments on
the subscripts according to the location of pressure and velocity components shown
in Fig. 34, the equation for pressure correction at a typical interior pressure cell will
be found as
ApPp ApPE + A^P w + AjyPN + AfPs + RES (A,18)
where
Ae = Ce\y
(A, 19)
Aw = QAv
(A,20)
AN = Dn*X
(A,21)
As = DsAx
(A,22)
115
Ap AÂ£ + Aw + An + As
(A,23)
RES = l(u;  Uw) Ay + (Vâ€ž  VÂ¡) Ax ]
More information about the VelocityPressure correction procedure can be
found in the book by Patankar19.
REFERENCE
1. Kurzweg, U. H., "Heat Transfer Device for the Transport of Large
Conduction Flux Without Net Mass Transfer," US Patent 4590993 1986
2. Taylor, G. I., "Dispersion of Soluble Matter in Solvents Flowing Slowly
Through a Tube," Proc. Roy. Soc., A219, pp.186203, 1953
3. Aris, R., "On the Dispersion of a Solute in a Fluid Flowing through a Tube,"
Proc. Roy. Soc., A235, pp.6777, 1956
4. Bowden, K. F., "Horizontal Mixing in the Sea due to a Shearing Current," J.
Fluid Mech., 21, pp.8496, 1965
5. Harris, H. G. and Goren, S. L. "Axial Diffusion in a Cylinder with Pulsed
Flow," Chem. Eng. Science, 22, pp.15711576, 1967
6. Rice, R. G. and Eagleton, L. C, "Mass Transfer Produced by Laminar Flow
Oscillations," Canadian J. Chem. Eng., 48, pp.4651, 1970
7. Chatwin, P. C, "On the Longitudinal Dispersion of Passive Contaminant in
Oscillatory Flows in Tubes,"/. Fluid Mech., 71, pp.513527, 1975
8. Waston, E. J., "Diffusion in Oscillatory Pipe Flow," J. Fluid Mech., 133,
pp.233244, 1983
9. Joshi, C. H., Kamm, R. D., Drazen, J. M. and Slutsky, A. S., "An
Experimental Study of Gas Exchange in Laminar Oscillatory Flows," J. Fluid
Mech., 133, pp.245254, 1983
10. Kurzweg, U. H. and Jaeger, M. J., "Turning Effect in Enhanced Gas
Dispersion Under Oscillatory Conditions," Phys. Fluids, 29, pp. 13241326,
1986
11. Kurzweg, U. H., "Enhanced Heat Conduction in Fluids Subjected to
Sinusoidal Oscillations," J. Heat Transfer, 107, pp.459467, 1985
116
117
12. Kurzweg, U. H. and Zhao, L. D., "Heat Transfer by High Frequency OscillaÂ¬
tions; a New Hydrodynamics Technique for Achieving Large Effective Axial
Conductivities," Phys. Fluids, 27, pp.26242627, 1984
13. Kurzweg, U. H. "Enhanced Heat Conduction in Oscillating Viscose Flows
Within Parallel Plate Channels,"/. Fluid Mech. 156, pp.291300, 1985
14. Kurzweg, U. H., "Temporal and Spatial Distribution of Heat Flux in
Oscillating Flow Subjected to an Axial Temperature Gradient," Int. J. Heat
Mass Trans. 29, pp. 19691977, 1986
15. Zhang, G., "TimeDependent Enhanced Heat Transfer in Oscillating Pipe
Flow," PhD Dissertation, University of Florida, 1988
16. Kaviany, M., "Enhanced Thermal Diffusion in Oscillating Flows in a Capillary
Heat Exchanger," /. Heat Transfer, to appear 1990
17. Durst, F., Peric, M., Scheuerer, G. and Stroll, H., "Numerical Study of
Laminar, Unsteady PistonCylinder Flows," Finite Approximations in Fluid
Mechanics II. Friedr. Vieweg & Sohn, Braunschweig, pp.8092 1989
18. Durst, F., Periera, J. C. F. and Scheuerer, G., "Calculations and Experimental
Investigations of the Laminar Unsteady Flow in a Pipe Expansion," Finite
Approximations in Fluid Mechanics. Friedr. Vieweg & Sohn, Braunschweig,
pp.4355 1986
19. Patankar, S. V., Numerical Heat Transfer and Fluid Flow. Hemisphere,
Washington DC., 1980
20. Patankar, S. V. and Spalding, D. B., "A Calculation Procedure for Heat, Mass
and Momentum Transfer in Three Dimensional Parabolic Flows," Int. J. Heat
and Mass Trans., 15, pp. 17871806, 1978
21. Anderson, D. A., Tannehill, T. C. and Pletcher, R. H., Computational Fluid
Mechanics and Heat Transfer. Hemisphere, Washington DC., 1984
22. Gentzsch, W., Vectorization of Computer Programs with Applications to
Computational Fluid Dynamics. Friedr. Vieweg & Sohn, Braunschweig, 1984
23. Durst, F., Maxworthy, T. and Pereira, J. c. f., "PistonDriven, Unsteady
Separation at a Sudden Expansion in a Tube: Flow Visualization and LDA
Measurements," Phy. Fluids A, 1, pp.12491260, 1989
118
24. Kurzweg, U. H., Lindgren, E. R. and Lothrop, B., "Onset of Turbulence in
Oscillating Flow at Low Womersley Number," Phy. Fluids A, 1, pp.19721975,
1989
25. Belsterling, C. A., Fluidic Systems Design. WileyInterscience, New York,
1971
26. Holman, J. P., Heat Transfer. Sixth Edition, McGrawHill, New York, 1986
BIOGRAPHICAL SKETCH
Alex X. Zhao was born on September 30, 1949, in Beijing, The Peopleâ€™s
Republic of China. In 1956, he moved to Tianjin, the third biggest city in China, and
there spent most of years before he came to the United States. Because of the
"cultural revolution movement" in China from 1966 to 1976, he missed college
education for ten years, but fortunately in 1978, when he was 29 years old, was
admitted to Tianjin University, the oldest university in China, to their applied physics
program. With a GPA of 91.3/100, he graduated from the Department of Basic
Sciences with a Bachelor of Science degree in February 1982 and obtained a faculty
position in the Department of Thermophysics at the same university. Since then, he
has taught courses in engineering thermodynamics, heat transfer and some other
topics and also did some research on solar energy and optical measurements of the
temperature field at solid surfaces. In June of 1985, he was selected by Tianjin
University to study for an advanced degree abroad and came to the United States.
He entered the masterâ€™s degree program in the Department of Mechanical EngineeÂ¬
ring at the University of Florida in January 1986 and transferred to the Department
of Aerospace Engineering, Mechanics and Engineering Science in August of that
year. It was at that time that he started the research on enhancement of heat
transfer by oscillating flows under the guidance of Dr. Ulrich H. Kurzweg. Mr. Zhao
119
120
received his Master of Science in engineering sciences in August of 1987, became a
Ph.D. candidate in May of 1988 and is expected to receive the degree of Doctor of
Philosophy in the late summer of 1990. He is married to Wei Zhu and has one son
Licheng Zhao.
I certify that 1 have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
XA l *1*cX AA VWLWf Y
Ulrich H. Kurzweg, Chairman
Professor of Aerospace Engineering,
Mechanics and Engineering Science
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
E. Rune Lindgren
Professor of Aerospace Engineering,
Mechanics and Engineering Science
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Professor of Aerospace Engineering,
Mechanics and Engineering Science
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
\ ( ' '
Am *
Wei Shyy ^ {
Associate Professor of Aerospace
Engineering, Mechanics and Engineering
Science
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
c~
/V
Ranganathan Narayanan
Associate Professor of Chemical
Engineering
This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of
the requirements for the degree of Doctor of Philosophy.
December 1990
Winfred M. Phillips
Dean, College of Engineering
Madelyn M. Lockhart
Dean, Graduate School
UNIVERSITY OF FLORIDA
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